TPTP Problem File: SCT251_5.p
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%------------------------------------------------------------------------------
% File : SCT251_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 272
% 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_272 [Bla13]
% Status : Theorem
% Rating : 1.00 v8.2.0, 0.67 v7.5.0, 1.00 v6.4.0
% Syntax : Number of formulae : 169 ( 51 unt; 44 typ; 0 def)
% Number of atoms : 329 ( 97 equ)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 279 ( 75 ~; 9 |; 8 &)
% ( 33 <=>; 154 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 33 ( 18 >; 15 *; 0 +; 0 <<)
% Number of predicates : 13 ( 12 usr; 0 prp; 1-4 aty)
% Number of functors : 28 ( 28 usr; 10 con; 0-5 aty)
% Number of variables : 360 ( 322 !; 5 ?; 360 :)
% ( 33 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:26:18
%------------------------------------------------------------------------------
%----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_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tc_prod,type,
product_prod: ( $tType * $tType ) > $tType ).
%----Explicit typings (38)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[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_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] : fun(A,fun(B,A)) ).
tff(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[A: $tType] : ( fun(A,bool) > nat ) ).
tff(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,bool) ) > $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_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_If,type,
if:
!>[A: $tType] : fun(bool,fun(A,fun(A,A))) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
tff(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( ( A * B ) > product_prod(A,B) ) ).
tff(sy_c_Product__Type_Oprod_Oprod__rec,type,
product_prod_rec:
!>[A: $tType,B: $tType,T1: $tType] : ( ( fun(A,fun(B,T1)) * product_prod(A,B) ) > T1 ) ).
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_Lab____,type,
lab: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) ).
tff(sy_v_Lba____,type,
lba: 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_h____,type,
h: fun(arrow_159774573e_indi,nat) ).
%----Relevant facts (99)
tff(fact_0_u,axiom,
arrow_2069624013nimity(f) ).
tff(fact_1__096Lab_A_058_ALin_096,axiom,
member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),lab,arrow_1985332922le_Lin) ).
tff(fact_2_assms_I3_J,axiom,
arrow_1958449194le_IIA(f) ).
tff(fact_3__096Lba_A_058_ALin_096,axiom,
member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),lba,arrow_1985332922le_Lin) ).
tff(fact_4__096F_A_I_Fi_O_ALab_J_A_058_ALin_096,axiom,
member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),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,aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),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),lab)),arrow_1985332922le_Lin) ).
tff(fact_5__096a_A_126_061_Ab_096,axiom,
a != b ).
tff(fact_6__096a_A_060_092_060_094bsub_062Lab_092_060_094esub_062_Ab_096,axiom,
member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,a,b),lab) ).
tff(fact_7__096b_A_060_092_060_094bsub_062Lba_092_060_094esub_062_Aa_096,axiom,
member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),lba) ).
tff(fact_8__096_I_Fi_O_Aif_Ah_Ai_A_060_AN_Athen_ALab_Aelse_ALba_J_A_061_A_I_Fp_O_ALab_J_096,axiom,
( ? [X4: arrow_159774573e_indi] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(arrow_159774573e_indi,nat,h,X4)),finite_card(arrow_159774573e_indi,top_top(fun(arrow_159774573e_indi,bool)))))
=> ( lba = lab ) ) ).
tff(fact_9__096F_A_I_Fi_O_ALba_J_A_058_ALin_096,axiom,
member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),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,aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),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),lba)),arrow_1985332922le_Lin) ).
tff(fact_10__096_Ib_M_Aa_J_A_126_058_ALab_096,axiom,
~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),lab) ).
tff(fact_11__096_Ia_M_Ab_J_A_126_058_ALba_096,axiom,
~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,a,b),lba) ).
tff(fact_12__C0_C,axiom,
! [N1: nat] : member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),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,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),N1)),lab),lba)),arrow_1985332922le_Lin) ).
tff(fact_13__096_B_Bthesis_O_A_I_B_BLab_O_A_091_124_Aa_A_060_092_060_094bsub_062Lab_092_060_094esub_062_Ab_059_ALab_A_058_ALin_A_124_093_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [Lab: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,a,b),Lab)
=> ~ member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Lab,arrow_1985332922le_Lin) ) ).
tff(fact_14__096_B_Bthesis_O_A_I_B_BLba_O_A_091_124_Ab_A_060_092_060_094bsub_062Lba_092_060_094esub_062_Aa_059_ALba_A_058_ALin_A_124_093_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [Lba: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),Lba)
=> ~ member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Lba,arrow_1985332922le_Lin) ) ).
tff(fact_15_injh,axiom,
inj_on(arrow_159774573e_indi,nat,h,top_top(fun(arrow_159774573e_indi,bool))) ).
tff(fact_16_PiProf,axiom,
! [N1: nat] : member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),N1)),lab),lba),arrow_610318064e_Prof) ).
tff(fact_17__C1_C,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,combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),zero_zero(nat))),lab),lba))) ).
tff(fact_18_top1I,axiom,
! [A: $tType,X1: A] : pp(aa(A,bool,top_top(fun(A,bool)),X1)) ).
tff(fact_19_UNIV__I,axiom,
! [A: $tType,X1: A] : member(A,X1,top_top(fun(A,bool))) ).
tff(fact_20_iso__tuple__UNIV__I,axiom,
! [A: $tType,X1: A] : member(A,X1,top_top(fun(A,bool))) ).
tff(fact_21_split__paired__All,axiom,
! [A: $tType,B: $tType,P1: fun(product_prod(A,B),bool)] :
( ! [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P1,X11))
<=> ! [A4: A,B4: B] : pp(aa(product_prod(A,B),bool,P1,product_Pair(A,B,A4,B4))) ) ).
tff(fact_22_Pair__eq,axiom,
! [A: $tType,B: $tType,B6: B,A6: A,Ba: B,Aa: A] :
( ( product_Pair(A,B,Aa,Ba) = product_Pair(A,B,A6,B6) )
<=> ( ( Aa = A6 )
& ( Ba = B6 ) ) ) ).
tff(fact_23__C2_C,axiom,
! [P2: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),B6: arrow_411405190le_alt,A6: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( A6 != B6 )
=> ( ( Aa != B6 )
=> ( ( Ba != A6 )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P2,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),P1,I2))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B6),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P2,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,P1))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,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,P2)) ) ) ) ) ) ) ) ) ).
tff(fact_24__096_B_Bb_H_Ab_Aa_H_Aa_AP_H_AP_O_A_091_124_Aa_A_126_061_Ab_059_Aa_H_A_126_061_Ab_H_059_Aa_A_126_061_Ab_H_059_Ab_A_126_061_Aa_H_059_AP_A_058_AProf_059_AP_H_A_058_AProf_059_AALL_Ai_O_A_Ia_A_060_092_060_094bsub_062P_Ai_092_060_094esub_062_Ab_J_A_061_A_Ia_H_A_060_092_060_094bsub_062P_H_Ai_092_060_094esub_062_Ab_H_J_059_Aa_A_060_092_060_094bsub_062F_AP_092_060_094esub_062_Ab_A_124_093_A_061_061_062_Aa_H_A_060_092_060_094bsub_062F_AP_H_092_060_094esub_062_Ab_H_096,axiom,
! [P2: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),B6: arrow_411405190le_alt,A6: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( A6 != B6 )
=> ( ( Aa != B6 )
=> ( ( Ba != A6 )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P2,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),P1,I2))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B6),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P2,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,P1))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,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,P2)) ) ) ) ) ) ) ) ) ).
tff(fact_25__C4_C,axiom,
! [P2: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),C2: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( Ba != C2 )
=> ( ( Aa != C2 )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P2,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),P1,I2))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,C2),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P2,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,P1))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,C2),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,P2)) ) ) ) ) ) ) ) ).
tff(fact_26_pairwise__neutrality,axiom,
! [P2: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),B6: arrow_411405190le_alt,A6: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( A6 != B6 )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P2,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),P1,I2))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B6),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P2,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,P1))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,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,P2)) ) ) ) ) ) ) ).
tff(fact_27__C3_C,axiom,
! [P2: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P1,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P2,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),P1,I2))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,Aa),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P2,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,P1))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,Aa),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,P2)) ) ) ) ) ) ).
tff(fact_28_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,aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),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_29_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)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),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_30_Pair__inject,axiom,
! [A: $tType,B: $tType,B5: B,A5: A,B1: B,A1: A] :
( ( product_Pair(A,B,A1,B1) = product_Pair(A,B,A5,B5) )
=> ~ ( ( A1 = A5 )
=> ( B1 != B5 ) ) ) ).
tff(fact_31_UNIV__def,axiom,
! [A: $tType] : top_top(fun(A,bool)) = collect(A,aa(bool,fun(A,bool),combk(bool,A),fTrue)) ).
tff(fact_32_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,S1: fun(product_prod(A,B),bool),R1: fun(product_prod(A,B),bool)] :
( ! [X3: A,Xa: B] :
( member(product_prod(A,B),product_Pair(A,B,X3,Xa),R1)
<=> member(product_prod(A,B),product_Pair(A,B,X3,Xa),S1) )
<=> ( R1 = S1 ) ) ).
tff(fact_33_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_34_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),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),L)
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Y1,X1),L) ) ) ) ).
tff(fact_35_less__zeroE,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).
tff(fact_36_less__nat__zero__code,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).
tff(fact_37_neq0__conv,axiom,
! [N1: nat] :
( ( N1 != zero_zero(nat) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N1)) ) ).
tff(fact_38_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)
<=> ! [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)
=> ! [Xa: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Xa,arrow_610318064e_Prof)
=> ! [A4: 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,A4,B4),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X3,I1))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A4,B4),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Xa,I1)) )
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A4,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,X3))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A4,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,Xa)) ) ) ) ) ) ).
tff(fact_39_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)
<=> ! [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)
=> ! [A4: 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,A4,B4),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X3,I1))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A4,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,X3)) ) ) ) ).
tff(fact_40_split__paired__Ex,axiom,
! [A: $tType,B: $tType,P1: fun(product_prod(A,B),bool)] :
( ? [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P1,X11))
<=> ? [A4: A,B4: B] : pp(aa(product_prod(A,B),bool,P1,product_Pair(A,B,A4,B4))) ) ).
tff(fact_41_less__not__refl,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).
tff(fact_42_nat__neq__iff,axiom,
! [N1: nat,M1: nat] :
( ( M1 != N1 )
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),N1))
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N1),M1)) ) ) ).
tff(fact_43_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X)) ) ) ).
tff(fact_44_less__irrefl__nat,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).
tff(fact_45_less__not__refl2,axiom,
! [M: nat,N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
=> ( M != N ) ) ).
tff(fact_46_less__not__refl3,axiom,
! [T: nat,S: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S),T))
=> ( S != T ) ) ).
tff(fact_47_nat__less__cases,axiom,
! [P1: fun(nat,fun(nat,bool)),N1: nat,M1: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),N1))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M1)) )
=> ( ( ( M1 = N1 )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M1)) )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N1),M1))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M1)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M1)) ) ) ) ).
tff(fact_48_not__less0,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).
tff(fact_49_gr__implies__not0,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> ( N != zero_zero(nat) ) ) ).
tff(fact_50_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).
tff(fact_51_Pi__UNIV,axiom,
! [B: $tType,A: $tType,A2: fun(A,bool)] : pi(A,B,A2,aa(fun(B,bool),fun(A,fun(B,bool)),combk(fun(B,bool),A),top_top(fun(B,bool)))) = top_top(fun(fun(A,B),bool)) ).
tff(fact_52_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,aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),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)
=> ! [A3: arrow_411405190le_alt,B3: arrow_411405190le_alt] :
( ( A3 != B3 )
=> ( member(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),X2,I))
=> member(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,X2)) ) ) )
=> arrow_987702531ctator(Fa,I) ) ) ).
tff(fact_53_PiE,axiom,
! [B: $tType,A: $tType,X1: A,B2: fun(A,fun(B,bool)),A2: fun(A,bool),F: fun(A,B)] :
( member(fun(A,B),F,pi(A,B,A2,B2))
=> ( ~ member(B,aa(A,B,F,X1),aa(A,fun(B,bool),B2,X1))
=> ~ member(A,X1,A2) ) ) ).
tff(fact_54_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),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),X2) ) ) ).
tff(fact_55_inj__fun,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(fun(A,bool)))
=> inj_on(A,fun(C,B),combb(B,fun(C,B),A,combk(B,C),F),top_top(fun(A,bool))) ) ).
tff(fact_56_inj__on__def,axiom,
! [B: $tType,A: $tType,A2: fun(A,bool),F: fun(A,B)] :
( inj_on(A,B,F,A2)
<=> ! [X3: A] :
( member(A,X3,A2)
=> ! [Xa: A] :
( member(A,Xa,A2)
=> ( ( aa(A,B,F,X3) = aa(A,B,F,Xa) )
=> ( X3 = Xa ) ) ) ) ) ).
tff(fact_57_inj__on__contraD,axiom,
! [B: $tType,A: $tType,Y1: A,X1: A,A2: fun(A,bool),F: fun(A,B)] :
( inj_on(A,B,F,A2)
=> ( ( X1 != Y1 )
=> ( member(A,X1,A2)
=> ( member(A,Y1,A2)
=> ( aa(A,B,F,X1) != aa(A,B,F,Y1) ) ) ) ) ) ).
tff(fact_58_inj__on__iff,axiom,
! [B: $tType,A: $tType,Y1: A,X1: A,A2: fun(A,bool),F: fun(A,B)] :
( inj_on(A,B,F,A2)
=> ( member(A,X1,A2)
=> ( member(A,Y1,A2)
=> ( ( aa(A,B,F,X1) = aa(A,B,F,Y1) )
<=> ( X1 = Y1 ) ) ) ) ) ).
tff(fact_59_inj__onD,axiom,
! [B: $tType,A: $tType,Y1: A,X1: A,A2: fun(A,bool),F: fun(A,B)] :
( inj_on(A,B,F,A2)
=> ( ( aa(A,B,F,X1) = aa(A,B,F,Y1) )
=> ( member(A,X1,A2)
=> ( member(A,Y1,A2)
=> ( X1 = Y1 ) ) ) ) ) ).
tff(fact_60_Pi__mem,axiom,
! [B: $tType,A: $tType,X1: A,B2: fun(A,fun(B,bool)),A2: fun(A,bool),F: fun(A,B)] :
( member(fun(A,B),F,pi(A,B,A2,B2))
=> ( member(A,X1,A2)
=> member(B,aa(A,B,F,X1),aa(A,fun(B,bool),B2,X1)) ) ) ).
tff(fact_61_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)),aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),fun(arrow_159774573e_indi,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),combk(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),arrow_159774573e_indi),arrow_1985332922le_Lin)) ).
tff(fact_62_inj__on__id2,axiom,
! [A: $tType,A2: fun(A,bool)] : inj_on(A,A,combi(A),A2) ).
tff(fact_63_funcset__id,axiom,
! [A: $tType,A2: fun(A,bool)] : member(fun(A,A),combi(A),pi(A,A,A2,aa(fun(A,bool),fun(A,fun(A,bool)),combk(fun(A,bool),A),A2))) ).
tff(fact_64_funcset__mem,axiom,
! [A: $tType,B: $tType,X1: A,B2: fun(B,bool),A2: fun(A,bool),F: fun(A,B)] :
( member(fun(A,B),F,pi(A,B,A2,aa(fun(B,bool),fun(A,fun(B,bool)),combk(fun(B,bool),A),B2)))
=> ( member(A,X1,A2)
=> member(B,aa(A,B,F,X1),B2) ) ) ).
tff(fact_65_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)
<=> ! [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)
=> ( 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) = aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X3,I) ) ) ) ).
tff(fact_66_injD,axiom,
! [B: $tType,A: $tType,Y1: A,X1: A,F: fun(A,B)] :
( inj_on(A,B,F,top_top(fun(A,bool)))
=> ( ( aa(A,B,F,X1) = aa(A,B,F,Y1) )
=> ( X1 = Y1 ) ) ) ).
tff(fact_67_inj__eq,axiom,
! [B: $tType,A: $tType,Y1: A,X1: A,F: fun(A,B)] :
( inj_on(A,B,F,top_top(fun(A,bool)))
=> ( ( aa(A,B,F,X1) = aa(A,B,F,Y1) )
<=> ( X1 = Y1 ) ) ) ).
tff(fact_68_Pi__I,axiom,
! [B: $tType,A: $tType,B2: fun(A,fun(B,bool)),F: fun(A,B),A2: fun(A,bool)] :
( ! [X2: A] :
( member(A,X2,A2)
=> member(B,aa(A,B,F,X2),aa(A,fun(B,bool),B2,X2)) )
=> member(fun(A,B),F,pi(A,B,A2,B2)) ) ).
tff(fact_69_injI,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( ! [X2: A,Y2: A] :
( ( aa(A,B,F,X2) = aa(A,B,F,Y2) )
=> ( X2 = Y2 ) )
=> inj_on(A,B,F,top_top(fun(A,bool))) ) ).
tff(fact_70_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X1: A] :
( ( zero_zero(A) = X1 )
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_71_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),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),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),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Aa),L) ) )
& ( ( Y1 != Ba )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),L) ) ) ) ) ) ) ) ).
tff(fact_72_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),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),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),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,Y1),L) ) )
& ( ( X1 != Aa )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),L) ) ) ) ) ) ) ) ).
tff(fact_73_prod_Orecs,axiom,
! [B: $tType,A: $tType,C: $tType,Ba: C,Aa: B,F1: fun(B,fun(C,A))] : product_prod_rec(B,C,A,F1,product_Pair(B,C,Aa,Ba)) = aa(C,A,aa(B,fun(C,A),F1,Aa),Ba) ).
tff(fact_74_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_75_mem__def,axiom,
! [A: $tType,A2: fun(A,bool),X1: A] :
( member(A,X1,A2)
<=> pp(aa(A,bool,A2,X1)) ) ).
tff(fact_76_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_77_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_78_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_79_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_80_less__top,axiom,
! [A: $tType] :
( top(A)
=> ! [Aa: A] :
( ( Aa != top_top(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Aa),top_top(A))) ) ) ).
tff(fact_81_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X)) ) ).
tff(fact_82_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ( X1 != Y1 )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),Y1))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y1),X1)) ) ) ) ).
tff(fact_83_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),Y1))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y1),X1))
| ( X1 = Y1 ) ) ) ) ).
tff(fact_84_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
| ( X = Y )
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
tff(fact_85_linorder__antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X1: A,Y1: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y1),X1))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),Y1))
<=> ( X1 = Y1 ) ) ) ) ).
tff(fact_86_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ( X != Y )
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).
tff(fact_87_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( X != Y ) ) ) ).
tff(fact_88_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
tff(fact_89_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
tff(fact_90_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( X != Y ) ) ) ).
tff(fact_91_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( Y != X ) ) ) ).
tff(fact_92_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [P1: bool,Y1: A,X1: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),Y1))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y1),X1))
=> pp(P1) ) ) ) ).
tff(fact_93_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B1: A,A1: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A1),B1))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B1),A1)) ) ) ).
tff(fact_94_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B1: A,A1: A] :
( ( A1 = B1 )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B1),C1))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A1),C1)) ) ) ) ).
tff(fact_95_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B1: A,A1: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A1),B1))
=> ( ( B1 = C1 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A1),C1)) ) ) ) ).
tff(fact_96_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z)) ) ) ) ).
tff(fact_97_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
tff(fact_98_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( ( X != Y )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).
%----Arities (14)
tff(arity_fun___Orderings_Opreorder,axiom,
! [T_1: $tType,T_2: $tType] :
( preorder(T_2)
=> preorder(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oorder,axiom,
! [T_1: $tType,T_2: $tType] :
( order(T_2)
=> order(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Otop,axiom,
! [T_1: $tType,T_2: $tType] :
( top(T_2)
=> top(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oord,axiom,
! [T_1: $tType,T_2: $tType] :
( ord(T_2)
=> ord(fun(T_1,T_2)) ) ).
tff(arity_Nat_Onat___Orderings_Opreorder,axiom,
preorder(nat) ).
tff(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
tff(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
tff(arity_Nat_Onat___Orderings_Oord,axiom,
ord(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
tff(arity_HOL_Obool___Orderings_Otop,axiom,
top(bool) ).
tff(arity_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
%----Helper facts (11)
tff(help_If_1_1_T,axiom,
! [A: $tType,Y: A,X: A] : aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),fTrue),X),Y) = X ).
tff(help_If_2_1_T,axiom,
! [A: $tType,Y: A,X: A] : aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,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,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,aa(A,fun(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,combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(bool,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,nat,bool,combb(nat,fun(nat,bool),arrow_159774573e_indi,ord_less(nat),h),finite_card(arrow_159774573e_indi,top_top(fun(arrow_159774573e_indi,bool))))),lab),lba))) ).
%------------------------------------------------------------------------------