TPTP Problem File: SCT253_5.p
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%------------------------------------------------------------------------------
% File : SCT253_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 278
% 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_278 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.4.0, 0.25 v7.1.0, 0.33 v6.4.0
% Syntax : Number of formulae : 162 ( 41 unt; 45 typ; 0 def)
% Number of atoms : 289 ( 61 equ)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 213 ( 41 ~; 5 |; 4 &)
% ( 30 <=>; 133 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 28 ( 17 >; 11 *; 0 +; 0 <<)
% Number of predicates : 17 ( 16 usr; 1 prp; 0-4 aty)
% Number of functors : 25 ( 25 usr; 10 con; 0-5 aty)
% Number of variables : 341 ( 304 !; 3 ?; 341 :)
% ( 34 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:26:25
%------------------------------------------------------------------------------
%----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 (39)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[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_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_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( 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_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( ( A * A ) > 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_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( ( A * A ) > $o ) ).
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_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) ).
tff(sy_v_thesis____,type,
thesis: $o ).
%----Relevant facts (99)
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__096Lab_A_058_ALin_096,axiom,
member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),lab,arrow_1985332922le_Lin) ).
tff(fact_4__096Lba_A_058_ALin_096,axiom,
member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),lba,arrow_1985332922le_Lin) ).
tff(fact_5__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_6__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_7__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_8__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_9__096EX_An_060N_O_A_IALL_Am_060_061n_O_Ab_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_Am_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Aa_J_A_G_Aa_A_060_092_060_094bsub_062F_A_I_Fi_O_Aif_Ah_Ai_A_060_An_A_L_A1_Athen_ALab_Aelse_ALba_J_092_060_094esub_062_Ab_096,axiom,
? [N1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N1),finite_card(arrow_159774573e_indi,top_top(fun(arrow_159774573e_indi,bool)))))
& ! [M3: nat] :
( ord_less_eq(nat,M3,N1)
=> 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,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),M3)),lab),lba))) )
& 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),plus_plus(nat,N1,one_one(nat)))),lab),lba))) ) ).
tff(fact_10_injh,axiom,
inj_on(arrow_159774573e_indi,nat,h,top_top(fun(arrow_159774573e_indi,bool))) ).
tff(fact_11_PiProf,axiom,
! [N3: 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),N3)),lab),lba),arrow_610318064e_Prof) ).
tff(fact_12_nat__add__left__cancel__le,axiom,
! [N3: nat,M1: nat,K: nat] :
( ord_less_eq(nat,plus_plus(nat,K,M1),plus_plus(nat,K,N3))
<=> ord_less_eq(nat,M1,N3) ) ).
tff(fact_13_nat__add__left__cancel__less,axiom,
! [N3: nat,M1: nat,K: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,K,M1)),plus_plus(nat,K,N3)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),N3)) ) ).
tff(fact_14_add__less__cancel__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [Ba: A,C2: A,Aa: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),plus_plus(A,Aa,C2)),plus_plus(A,Ba,C2)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Aa),Ba)) ) ) ).
tff(fact_15_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [Ba: A,Aa: A,C2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),plus_plus(A,C2,Aa)),plus_plus(A,C2,Ba)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Aa),Ba)) ) ) ).
tff(fact_16_add__le__cancel__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [Ba: A,C2: A,Aa: A] :
( ord_less_eq(A,plus_plus(A,Aa,C2),plus_plus(A,Ba,C2))
<=> ord_less_eq(A,Aa,Ba) ) ) ).
tff(fact_17_add__le__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [Ba: A,Aa: A,C2: A] :
( ord_less_eq(A,plus_plus(A,C2,Aa),plus_plus(A,C2,Ba))
<=> ord_less_eq(A,Aa,Ba) ) ) ).
tff(fact_18__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_19__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_20_less__add__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),plus_plus(A,A3,one_one(A)))) ) ).
tff(fact_21_add__less__le__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [D: A,C1: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( ord_less_eq(A,C1,D)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),plus_plus(A,A3,C1)),plus_plus(A,B2,D))) ) ) ) ).
tff(fact_22_add__le__less__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [D: A,C1: A,B2: A,A3: A] :
( ord_less_eq(A,A3,B2)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C1),D))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),plus_plus(A,A3,C1)),plus_plus(A,B2,D))) ) ) ) ).
tff(fact_23_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C2: A,Aa: A,Ba: A] :
( ( plus_plus(A,Ba,Aa) = plus_plus(A,C2,Aa) )
<=> ( Ba = C2 ) ) ) ).
tff(fact_24_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C2: A,Ba: A,Aa: A] :
( ( plus_plus(A,Aa,Ba) = plus_plus(A,Aa,C2) )
<=> ( Ba = C2 ) ) ) ).
tff(fact_25_nat__add__right__cancel,axiom,
! [N3: nat,K: nat,M1: nat] :
( ( plus_plus(nat,M1,K) = plus_plus(nat,N3,K) )
<=> ( M1 = N3 ) ) ).
tff(fact_26_nat__add__left__cancel,axiom,
! [N3: nat,M1: nat,K: nat] :
( ( plus_plus(nat,K,M1) = plus_plus(nat,K,N3) )
<=> ( M1 = N3 ) ) ).
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_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)),B3: arrow_411405190le_alt,A4: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( A4 != B3 )
=> ( 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,A4,B3),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,A4,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),f,P2)) ) ) ) ) ) ) ).
tff(fact_29__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_30__C1_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)),B3: arrow_411405190le_alt,A4: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( A4 != B3 )
=> ( ( Aa != B3 )
=> ( ( Ba != A4 )
=> ( 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,A4,B3),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,A4,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),f,P2)) ) ) ) ) ) ) ) ) ).
tff(fact_31__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)),B3: arrow_411405190le_alt,A4: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( A4 != B3 )
=> ( ( Aa != B3 )
=> ( ( Ba != A4 )
=> ( 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,A4,B3),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,A4,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),f,P2)) ) ) ) ) ) ) ) ) ).
tff(fact_32_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_33_const__Lin__Prof,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)
=> 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,L1),arrow_610318064e_Prof) ) ).
tff(fact_34_Lin__irrefl,axiom,
! [Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt,L1: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L1,arrow_1985332922le_Lin)
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),L1)
=> ~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,Aa),L1) ) ) ).
tff(fact_35_notin__Lin__iff,axiom,
! [Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt,L1: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L1,arrow_1985332922le_Lin)
=> ( ( X1 != Y1 )
=> ( ~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),L1)
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Y1,X1),L1) ) ) ) ).
tff(fact_36_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(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_37_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,A3: A,B2: A] :
( ( plus_plus(A,B2,A3) = plus_plus(A,C1,A3) )
=> ( B2 = C1 ) ) ) ).
tff(fact_38_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C1: A,B2: A,A3: A] :
( ( plus_plus(A,A3,B2) = plus_plus(A,A3,C1) )
=> ( B2 = C1 ) ) ) ).
tff(fact_39_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,B2: A,A3: A] :
( ( plus_plus(A,A3,B2) = plus_plus(A,A3,C1) )
=> ( B2 = C1 ) ) ) ).
tff(fact_40_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C1: A,B2: A,A3: A] : plus_plus(A,plus_plus(A,A3,B2),C1) = plus_plus(A,A3,plus_plus(A,B2,C1)) ) ).
tff(fact_41_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X1: A] :
( ( one_one(A) = X1 )
<=> ( X1 = one_one(A) ) ) ) ).
tff(fact_42_nat__less__cases,axiom,
! [P1: fun(nat,fun(nat,bool)),N3: nat,M1: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),N3))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N3),M1)) )
=> ( ( ( M1 = N3 )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N3),M1)) )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),M1))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N3),M1)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N3),M1)) ) ) ) ).
tff(fact_43_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_44_less__not__refl2,axiom,
! [M2: nat,N2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),M2))
=> ( M2 != N2 ) ) ).
tff(fact_45_less__irrefl__nat,axiom,
! [N2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),N2)) ).
tff(fact_46_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_47_nat__neq__iff,axiom,
! [N3: nat,M1: nat] :
( ( M1 != N3 )
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),N3))
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N3),M1)) ) ) ).
tff(fact_48_less__not__refl,axiom,
! [N2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),N2)) ).
tff(fact_49_nat__add__assoc,axiom,
! [K2: nat,N2: nat,M2: nat] : plus_plus(nat,plus_plus(nat,M2,N2),K2) = plus_plus(nat,M2,plus_plus(nat,N2,K2)) ).
tff(fact_50_nat__add__left__commute,axiom,
! [Z: nat,Y: nat,X: nat] : plus_plus(nat,X,plus_plus(nat,Y,Z)) = plus_plus(nat,Y,plus_plus(nat,X,Z)) ).
tff(fact_51_nat__add__commute,axiom,
! [N2: nat,M2: nat] : plus_plus(nat,M2,N2) = plus_plus(nat,N2,M2) ).
tff(fact_52_le__antisym,axiom,
! [N2: nat,M2: nat] :
( ord_less_eq(nat,M2,N2)
=> ( ord_less_eq(nat,N2,M2)
=> ( M2 = N2 ) ) ) ).
tff(fact_53_le__trans,axiom,
! [K2: nat,J: nat,I1: nat] :
( ord_less_eq(nat,I1,J)
=> ( ord_less_eq(nat,J,K2)
=> ord_less_eq(nat,I1,K2) ) ) ).
tff(fact_54_eq__imp__le,axiom,
! [N2: nat,M2: nat] :
( ( M2 = N2 )
=> ord_less_eq(nat,M2,N2) ) ).
tff(fact_55_nat__le__linear,axiom,
! [N2: nat,M2: nat] :
( ord_less_eq(nat,M2,N2)
| ord_less_eq(nat,N2,M2) ) ).
tff(fact_56_le__refl,axiom,
! [N2: nat] : ord_less_eq(nat,N2,N2) ).
tff(fact_57_add__le__imp__le__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B2: A,A3: A,C1: A] :
( ord_less_eq(A,plus_plus(A,C1,A3),plus_plus(A,C1,B2))
=> ord_less_eq(A,A3,B2) ) ) ).
tff(fact_58_add__le__imp__le__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B2: A,C1: A,A3: A] :
( ord_less_eq(A,plus_plus(A,A3,C1),plus_plus(A,B2,C1))
=> ord_less_eq(A,A3,B2) ) ) ).
tff(fact_59_add__mono,axiom,
! [A: $tType] :
( ordere779506340up_add(A)
=> ! [D: A,C1: A,B2: A,A3: A] :
( ord_less_eq(A,A3,B2)
=> ( ord_less_eq(A,C1,D)
=> ord_less_eq(A,plus_plus(A,A3,C1),plus_plus(A,B2,D)) ) ) ) ).
tff(fact_60_add__left__mono,axiom,
! [A: $tType] :
( ordere779506340up_add(A)
=> ! [C1: A,B2: A,A3: A] :
( ord_less_eq(A,A3,B2)
=> ord_less_eq(A,plus_plus(A,C1,A3),plus_plus(A,C1,B2)) ) ) ).
tff(fact_61_add__right__mono,axiom,
! [A: $tType] :
( ordere779506340up_add(A)
=> ! [C1: A,B2: A,A3: A] :
( ord_less_eq(A,A3,B2)
=> ord_less_eq(A,plus_plus(A,A3,C1),plus_plus(A,B2,C1)) ) ) ).
tff(fact_62_add__less__imp__less__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B2: A,A3: A,C1: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),plus_plus(A,C1,A3)),plus_plus(A,C1,B2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
tff(fact_63_add__less__imp__less__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B2: A,C1: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),plus_plus(A,A3,C1)),plus_plus(A,B2,C1)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2)) ) ) ).
tff(fact_64_add__strict__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [D: A,C1: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C1),D))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),plus_plus(A,A3,C1)),plus_plus(A,B2,D))) ) ) ) ).
tff(fact_65_add__strict__left__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [C1: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),plus_plus(A,C1,A3)),plus_plus(A,C1,B2))) ) ) ).
tff(fact_66_add__strict__right__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [C1: A,B2: A,A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),plus_plus(A,A3,C1)),plus_plus(A,B2,C1))) ) ) ).
tff(fact_67_add__lessD1,axiom,
! [K2: nat,J: nat,I1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I1,J)),K2))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I1),K2)) ) ).
tff(fact_68_less__add__eq__less,axiom,
! [N2: nat,M2: nat,L: nat,K2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),L))
=> ( ( plus_plus(nat,M2,L) = plus_plus(nat,K2,N2) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N2)) ) ) ).
tff(fact_69_add__less__mono,axiom,
! [L: nat,K2: nat,J: nat,I1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I1),J))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),L))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I1,K2)),plus_plus(nat,J,L))) ) ) ).
tff(fact_70_add__less__mono1,axiom,
! [K2: nat,J: nat,I1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I1),J))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I1,K2)),plus_plus(nat,J,K2))) ) ).
tff(fact_71_trans__less__add2,axiom,
! [M2: nat,J: nat,I1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I1),J))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I1),plus_plus(nat,M2,J))) ) ).
tff(fact_72_trans__less__add1,axiom,
! [M2: nat,J: nat,I1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I1),J))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I1),plus_plus(nat,J,M2))) ) ).
tff(fact_73_not__add__less2,axiom,
! [I1: nat,J: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,J,I1)),I1)) ).
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_not__add__less1,axiom,
! [J: nat,I1: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),plus_plus(nat,I1,J)),I1)) ).
tff(fact_78_less__or__eq__imp__le,axiom,
! [N2: nat,M2: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N2))
| ( M2 = N2 ) )
=> ord_less_eq(nat,M2,N2) ) ).
tff(fact_79_le__neq__implies__less,axiom,
! [N2: nat,M2: nat] :
( ord_less_eq(nat,M2,N2)
=> ( ( M2 != N2 )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N2)) ) ) ).
tff(fact_80_less__imp__le__nat,axiom,
! [N2: nat,M2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N2))
=> ord_less_eq(nat,M2,N2) ) ).
tff(fact_81_le__eq__less__or__eq,axiom,
! [N3: nat,M1: nat] :
( ord_less_eq(nat,M1,N3)
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),N3))
| ( M1 = N3 ) ) ) ).
tff(fact_82_nat__less__le,axiom,
! [N3: nat,M1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),N3))
<=> ( ord_less_eq(nat,M1,N3)
& ( M1 != N3 ) ) ) ).
tff(fact_83_add__leE,axiom,
! [N2: nat,K2: nat,M2: nat] :
( ord_less_eq(nat,plus_plus(nat,M2,K2),N2)
=> ~ ( ord_less_eq(nat,M2,N2)
=> ~ ord_less_eq(nat,K2,N2) ) ) ).
tff(fact_84_add__leD1,axiom,
! [N2: nat,K2: nat,M2: nat] :
( ord_less_eq(nat,plus_plus(nat,M2,K2),N2)
=> ord_less_eq(nat,M2,N2) ) ).
tff(fact_85_add__leD2,axiom,
! [N2: nat,K2: nat,M2: nat] :
( ord_less_eq(nat,plus_plus(nat,M2,K2),N2)
=> ord_less_eq(nat,K2,N2) ) ).
tff(fact_86_add__le__mono,axiom,
! [L: nat,K2: nat,J: nat,I1: nat] :
( ord_less_eq(nat,I1,J)
=> ( ord_less_eq(nat,K2,L)
=> ord_less_eq(nat,plus_plus(nat,I1,K2),plus_plus(nat,J,L)) ) ) ).
tff(fact_87_add__le__mono1,axiom,
! [K2: nat,J: nat,I1: nat] :
( ord_less_eq(nat,I1,J)
=> ord_less_eq(nat,plus_plus(nat,I1,K2),plus_plus(nat,J,K2)) ) ).
tff(fact_88_trans__le__add2,axiom,
! [M2: nat,J: nat,I1: nat] :
( ord_less_eq(nat,I1,J)
=> ord_less_eq(nat,I1,plus_plus(nat,M2,J)) ) ).
tff(fact_89_trans__le__add1,axiom,
! [M2: nat,J: nat,I1: nat] :
( ord_less_eq(nat,I1,J)
=> ord_less_eq(nat,I1,plus_plus(nat,J,M2)) ) ).
tff(fact_90_le__iff__add,axiom,
! [N3: nat,M1: nat] :
( ord_less_eq(nat,M1,N3)
<=> ? [K1: nat] : N3 = plus_plus(nat,M1,K1) ) ).
tff(fact_91_le__add1,axiom,
! [M2: nat,N2: nat] : ord_less_eq(nat,N2,plus_plus(nat,N2,M2)) ).
tff(fact_92_le__add2,axiom,
! [M2: nat,N2: nat] : ord_less_eq(nat,N2,plus_plus(nat,M2,N2)) ).
tff(fact_93_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)
=> ! [A1: arrow_411405190le_alt,B1: arrow_411405190le_alt] :
( ! [I: arrow_159774573e_indi] :
( 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(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Xa,I)) )
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,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))
<=> 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,Xa)) ) ) ) ) ) ).
tff(fact_94_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)
=> ! [A1: arrow_411405190le_alt,B1: arrow_411405190le_alt] :
( ! [I: arrow_159774573e_indi] : 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)) ) ) ) ).
tff(fact_95_mono__nat__linear__lb,axiom,
! [K: nat,M1: nat,F: fun(nat,nat)] :
( ! [M: nat,N1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N1))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F,M)),aa(nat,nat,F,N1))) )
=> ord_less_eq(nat,plus_plus(nat,aa(nat,nat,F,M1),K),aa(nat,nat,F,plus_plus(nat,M1,K))) ) ).
tff(fact_96_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_97_iso__tuple__UNIV__I,axiom,
! [A: $tType,X1: A] : member(A,X1,top_top(fun(A,bool))) ).
tff(fact_98_UNIV__I,axiom,
! [A: $tType,X1: A] : member(A,X1,top_top(fun(A,bool))) ).
%----Arities (8)
tff(arity_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(nat) ).
tff(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(nat) ).
tff(arity_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
tff(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
tff(arity_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(nat) ).
tff(arity_Nat_Onat___Groups_Oone,axiom,
one(nat) ).
%----Helper facts (8)
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_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,combk(A,B,P),Q) = P ).
%----Conjectures (2)
tff(conj_0,hypothesis,
! [N: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),finite_card(arrow_159774573e_indi,top_top(fun(arrow_159774573e_indi,bool)))))
=> ( ! [M: nat] :
( ord_less_eq(nat,M,N)
=> 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,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),M)),lab),lba))) )
=> ( 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),plus_plus(nat,N,one_one(nat)))),lab),lba)))
=> thesis ) ) ) ).
tff(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------