TPTP Problem File: SCT249_5.p
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%------------------------------------------------------------------------------
% File : SCT249_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 268
% 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_268 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 164 ( 44 unt; 39 typ; 0 def)
% Number of atoms : 281 ( 55 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 191 ( 35 ~; 8 |; 8 &)
% ( 22 <=>; 118 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 5 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 26 ( 16 >; 10 *; 0 +; 0 <<)
% Number of predicates : 11 ( 10 usr; 0 prp; 1-4 aty)
% Number of functors : 25 ( 25 usr; 8 con; 0-5 aty)
% Number of variables : 349 ( 315 !; 3 ?; 349 :)
% ( 31 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:26:05
%------------------------------------------------------------------------------
%----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 (33)
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_Ounanimity,type,
arrow_2069624013nimity: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)) > $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * B ) > fun(A,C) ) ).
tff(sy_c_COMBI,type,
combi:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_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_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_SetInterval_Oord__class_OatLeastLessThan,type,
ord_atLeastLessThan:
!>[A: $tType] : ( ( A * A ) > fun(A,bool) ) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,bool) ) > fun(B,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] : fun(A,fun(fun(A,bool),bool)) ).
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_h____,type,
h: fun(arrow_159774573e_indi,nat) ).
%----Relevant facts (100)
tff(fact_0__096Lba_A_058_ALin_096,axiom,
pp(aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),bool,aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),bool),member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),lba),arrow_1985332922le_Lin)) ).
tff(fact_1__096Lab_A_058_ALin_096,axiom,
pp(aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),bool,aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),bool),member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),lab),arrow_1985332922le_Lin)) ).
tff(fact_2__096range_Ah_A_060_061_A_1230_O_O_060N_125_096,axiom,
ord_less_eq(fun(nat,bool),image(arrow_159774573e_indi,nat,h,top_top(fun(arrow_159774573e_indi,bool))),ord_atLeastLessThan(nat,zero_zero(nat),finite_card(arrow_159774573e_indi,top_top(fun(arrow_159774573e_indi,bool))))) ).
tff(fact_3_surjh,axiom,
image(arrow_159774573e_indi,nat,h,top_top(fun(arrow_159774573e_indi,bool))) = ord_atLeastLessThan(nat,zero_zero(nat),finite_card(arrow_159774573e_indi,top_top(fun(arrow_159774573e_indi,bool)))) ).
tff(fact_4_injh,axiom,
inj_on(arrow_159774573e_indi,nat,h,top_top(fun(arrow_159774573e_indi,bool))) ).
tff(fact_5_PiProf,axiom,
! [N1: nat] : pp(aa(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool,aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool),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_6_top1I,axiom,
! [A: $tType,X2: A] : pp(aa(A,bool,top_top(fun(A,bool)),X2)) ).
tff(fact_7_UNIV__I,axiom,
! [A: $tType,X2: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),top_top(fun(A,bool)))) ).
tff(fact_8_iso__tuple__UNIV__I,axiom,
! [A: $tType,X2: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),top_top(fun(A,bool)))) ).
tff(fact_9_top__apply,axiom,
! [B: $tType,A: $tType] :
( top(A)
=> ! [X2: B] : aa(B,A,top_top(fun(B,A)),X2) = top_top(A) ) ).
tff(fact_10_not__top__less,axiom,
! [A: $tType] :
( top(A)
=> ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),A2)) ) ).
tff(fact_11_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_12__C0_C,axiom,
! [N1: nat] : pp(aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),bool,aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),bool),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_UNIV__def,axiom,
! [A: $tType] : top_top(fun(A,bool)) = collect(A,combk(bool,A,fTrue)) ).
tff(fact_14_less__not__refl,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).
tff(fact_15_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_16_linorder__neqE__nat,axiom,
! [Y: nat,X1: nat] :
( ( X1 != Y )
=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X1),Y))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X1)) ) ) ).
tff(fact_17_u,axiom,
arrow_2069624013nimity(f) ).
tff(fact_18_assms_I3_J,axiom,
arrow_1958449194le_IIA(f) ).
tff(fact_19_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X1: A] : ord_less_eq(A,X1,X1) ) ).
tff(fact_20_subsetD,axiom,
! [A: $tType,C3: A,B2: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B2)
=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C3),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C3),B2)) ) ) ).
tff(fact_21_equalityI,axiom,
! [A: $tType,B2: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B2)
=> ( ord_less_eq(fun(A,bool),B2,A1)
=> ( A1 = B2 ) ) ) ).
tff(fact_22_image__eqI,axiom,
! [A: $tType,B: $tType,A1: fun(B,bool),X2: B,F: fun(B,A),Ba: A] :
( ( Ba = aa(B,A,F,X2) )
=> ( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Ba),image(B,A,F,A1))) ) ) ).
tff(fact_23__096F_A_I_Fi_O_ALba_J_A_058_ALin_096,axiom,
pp(aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),bool,aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),bool),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,combk(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_159774573e_indi,lba))),arrow_1985332922le_Lin)) ).
tff(fact_24_less__zeroE,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).
tff(fact_25_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_26_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_27_assms_I1_J,axiom,
pp(aa(fun(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)),bool),bool,aa(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)),fun(fun(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)),bool),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))),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_28__096_B_Bthesis_O_A_I_B_Bh_O_A_091_124_Ainj_Ah_059_Arange_Ah_A_061_A_1230_O_O_060N_125_A_124_093_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [H: fun(arrow_159774573e_indi,nat)] :
( inj_on(arrow_159774573e_indi,nat,H,top_top(fun(arrow_159774573e_indi,bool)))
=> ( image(arrow_159774573e_indi,nat,H,top_top(fun(arrow_159774573e_indi,bool))) != ord_atLeastLessThan(nat,zero_zero(nat),finite_card(arrow_159774573e_indi,top_top(fun(arrow_159774573e_indi,bool)))) ) ) ).
tff(fact_29_subset__refl,axiom,
! [A: $tType,A1: fun(A,bool)] : ord_less_eq(fun(A,bool),A1,A1) ).
tff(fact_30_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
<=> ! [X3: A] : ord_less_eq(B,aa(A,B,F,X3),aa(A,B,G,X3)) ) ) ).
tff(fact_31_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X1: A] :
( ord_less_eq(A,X1,Y)
| ord_less_eq(A,Y,X1) ) ) ).
tff(fact_32_psubset__eq,axiom,
! [A: $tType,B2: fun(A,bool),A1: fun(A,bool)] :
( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less(fun(A,bool)),A1),B2))
<=> ( ord_less_eq(fun(A,bool),A1,B2)
& ( A1 != B2 ) ) ) ).
tff(fact_33_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F: fun(A,B)] :
( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less(fun(A,B)),F),G))
<=> ( ord_less_eq(fun(A,B),F,G)
& ~ ord_less_eq(fun(A,B),G,F) ) ) ) ).
tff(fact_34_set__eq__subset,axiom,
! [A: $tType,B2: fun(A,bool),A1: fun(A,bool)] :
( ( A1 = B2 )
<=> ( ord_less_eq(fun(A,bool),A1,B2)
& ord_less_eq(fun(A,bool),B2,A1) ) ) ).
tff(fact_35_order__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X2: A] :
( ( X2 = Y1 )
<=> ( ord_less_eq(A,X2,Y1)
& ord_less_eq(A,Y1,X2) ) ) ) ).
tff(fact_36_subset__iff__psubset__eq,axiom,
! [A: $tType,B2: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B2)
<=> ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less(fun(A,bool)),A1),B2))
| ( A1 = B2 ) ) ) ).
tff(fact_37_subset__image__iff,axiom,
! [A: $tType,B: $tType,A1: fun(B,bool),F: fun(B,A),B2: fun(A,bool)] :
( ord_less_eq(fun(A,bool),B2,image(B,A,F,A1))
<=> ? [AA: fun(B,bool)] :
( ord_less_eq(fun(B,bool),AA,A1)
& ( B2 = image(B,A,F,AA) ) ) ) ).
tff(fact_38_image__iff,axiom,
! [A: $tType,B: $tType,A1: fun(B,bool),F: fun(B,A),Z1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Z1),image(B,A,F,A1)))
<=> ? [X3: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X3),A1))
& ( Z1 = aa(B,A,F,X3) ) ) ) ).
tff(fact_39_rev__predicate1D,axiom,
! [A: $tType,Q1: fun(A,bool),X2: A,P1: fun(A,bool)] :
( pp(aa(A,bool,P1,X2))
=> ( ord_less_eq(fun(A,bool),P1,Q1)
=> pp(aa(A,bool,Q1,X2)) ) ) ).
tff(fact_40_const__Lin__Prof,axiom,
! [L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( pp(aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),bool,aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),bool),member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),L),arrow_1985332922le_Lin))
=> pp(aa(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool,aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool),member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),combk(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_159774573e_indi,L)),arrow_610318064e_Prof)) ) ).
tff(fact_41_equalityD1,axiom,
! [A: $tType,B2: fun(A,bool),A1: fun(A,bool)] :
( ( A1 = B2 )
=> ord_less_eq(fun(A,bool),A1,B2) ) ).
tff(fact_42_equalityD2,axiom,
! [A: $tType,B2: fun(A,bool),A1: fun(A,bool)] :
( ( A1 = B2 )
=> ord_less_eq(fun(A,bool),B2,A1) ) ).
tff(fact_43_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X1: A] :
( ( X1 = Y )
=> ord_less_eq(A,X1,Y) ) ) ).
tff(fact_44_psubset__imp__subset,axiom,
! [A: $tType,B2: fun(A,bool),A1: fun(A,bool)] :
( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less(fun(A,bool)),A1),B2))
=> ord_less_eq(fun(A,bool),A1,B2) ) ).
tff(fact_45_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [X2: A,G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
=> ord_less_eq(B,aa(A,B,F,X2),aa(A,B,G,X2)) ) ) ).
tff(fact_46_imageI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A1: fun(A,bool),X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),aa(A,B,F,X2)),image(A,B,F,A1))) ) ).
tff(fact_47_in__mono,axiom,
! [A: $tType,X2: A,B2: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B2)
=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),B2)) ) ) ).
tff(fact_48_image__mono,axiom,
! [B: $tType,A: $tType,F: fun(A,B),B2: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B2)
=> ord_less_eq(fun(B,bool),image(A,B,F,A1),image(A,B,F,B2)) ) ).
tff(fact_49_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [X2: A,Y1: A] :
( ord_less_eq(A,Y1,X2)
=> ( ord_less_eq(A,X2,Y1)
<=> ( X2 = Y1 ) ) ) ) ).
tff(fact_50_predicate1D,axiom,
! [A: $tType,X2: A,Q1: fun(A,bool),P1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),P1,Q1)
=> ( pp(aa(A,bool,P1,X2))
=> pp(aa(A,bool,Q1,X2)) ) ) ).
tff(fact_51_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B1: A,A2: A] :
( ( A2 = B1 )
=> ( ord_less_eq(A,B1,C1)
=> ord_less_eq(A,A2,C1) ) ) ) ).
tff(fact_52_set__rev__mp,axiom,
! [A: $tType,B2: fun(A,bool),A1: fun(A,bool),X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A1))
=> ( ord_less_eq(fun(A,bool),A1,B2)
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),B2)) ) ) ).
tff(fact_53_psubset__subset__trans,axiom,
! [A: $tType,C2: fun(A,bool),B2: fun(A,bool),A1: fun(A,bool)] :
( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less(fun(A,bool)),A1),B2))
=> ( ord_less_eq(fun(A,bool),B2,C2)
=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less(fun(A,bool)),A1),C2)) ) ) ).
tff(fact_54_set__mp,axiom,
! [A: $tType,X2: A,B2: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B2)
=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),B2)) ) ) ).
tff(fact_55_subset__psubset__trans,axiom,
! [A: $tType,C2: fun(A,bool),B2: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B2)
=> ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less(fun(A,bool)),B2),C2))
=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less(fun(A,bool)),A1),C2)) ) ) ).
tff(fact_56_subset__trans,axiom,
! [A: $tType,C2: fun(A,bool),B2: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B2)
=> ( ord_less_eq(fun(A,bool),B2,C2)
=> ord_less_eq(fun(A,bool),A1,C2) ) ) ).
tff(fact_57_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B1: A,A2: A] :
( ord_less_eq(A,A2,B1)
=> ( ( B1 = C1 )
=> ord_less_eq(A,A2,C1) ) ) ) ).
tff(fact_58_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X1: A] :
( ord_less_eq(A,X1,Y)
=> ( ord_less_eq(A,Y,X1)
=> ( X1 = Y ) ) ) ) ).
tff(fact_59_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X1: A] :
( ord_less_eq(A,X1,Y)
=> ( ord_less_eq(A,Y,Z)
=> ord_less_eq(A,X1,Z) ) ) ) ).
tff(fact_60_rev__image__eqI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Ba: B,A1: fun(A,bool),X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A1))
=> ( ( Ba = aa(A,B,F,X2) )
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Ba),image(A,B,F,A1))) ) ) ).
tff(fact_61_equalityE,axiom,
! [A: $tType,B2: fun(A,bool),A1: fun(A,bool)] :
( ( A1 = B2 )
=> ~ ( ord_less_eq(fun(A,bool),A1,B2)
=> ~ ord_less_eq(fun(A,bool),B2,A1) ) ) ).
tff(fact_62_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [X2: A,G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
=> ord_less_eq(B,aa(A,B,F,X2),aa(A,B,G,X2)) ) ) ).
tff(fact_63_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X1: A] :
( ~ ord_less_eq(A,X1,Y)
=> ord_less_eq(A,Y,X1) ) ) ).
tff(fact_64_image__ident,axiom,
! [A: $tType,Y2: fun(A,bool)] : image(A,A,combi(A),Y2) = Y2 ).
tff(fact_65_image__image,axiom,
! [B: $tType,A: $tType,C: $tType,A1: fun(C,bool),G: fun(C,B),F: fun(B,A)] : image(B,A,F,image(C,B,G,A1)) = image(C,A,combb(B,A,C,F,G),A1) ).
tff(fact_66_pred__subset__eq,axiom,
! [A: $tType,S: fun(A,bool),R1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),combc(A,fun(A,bool),bool,member(A),R1),combc(A,fun(A,bool),bool,member(A),S))
<=> ord_less_eq(fun(A,bool),R1,S) ) ).
tff(fact_67_range__eqI,axiom,
! [A: $tType,B: $tType,X2: B,F: fun(B,A),Ba: A] :
( ( Ba = aa(B,A,F,X2) )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Ba),image(B,A,F,top_top(fun(B,bool))))) ) ).
tff(fact_68_rangeI,axiom,
! [A: $tType,B: $tType,X2: B,F: fun(B,A)] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),aa(B,A,F,X2)),image(B,A,F,top_top(fun(B,bool))))) ).
tff(fact_69_order__le__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X1: A] :
( ord_less_eq(A,X1,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),X1),Z)) ) ) ) ).
tff(fact_70_order__less__le__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X1: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),Y))
=> ( ord_less_eq(A,Y,Z)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),Z)) ) ) ) ).
tff(fact_71_order__le__neq__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B1: A,A2: A] :
( ord_less_eq(A,A2,B1)
=> ( ( A2 != B1 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B1)) ) ) ) ).
tff(fact_72_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X1: A] :
( ord_less_eq(A,X1,Y)
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),Y))
| ( X1 = Y ) ) ) ) ).
tff(fact_73_linorder__antisym__conv2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X2: A] :
( ord_less_eq(A,X2,Y1)
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y1))
<=> ( X2 = Y1 ) ) ) ) ).
tff(fact_74_order__less__imp__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X1: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),Y))
=> ord_less_eq(A,X1,Y) ) ) ).
tff(fact_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X: A] : aa(A,B,F,X) = aa(A,B,G,X)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A1))
<=> pp(aa(A,bool,A1,X2)) ) ).
tff(fact_77_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_78_leD,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X1: A,Y: A] :
( ord_less_eq(A,Y,X1)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),Y)) ) ) ).
tff(fact_79_order__neq__le__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B1: A,A2: A] :
( ( A2 != B1 )
=> ( ord_less_eq(A,A2,B1)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B1)) ) ) ) ).
tff(fact_80_linorder__antisym__conv1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X2: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y1))
=> ( ord_less_eq(A,X2,Y1)
<=> ( X2 = Y1 ) ) ) ) ).
tff(fact_81_not__leE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X1: A,Y: A] :
( ~ ord_less_eq(A,Y,X1)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),Y)) ) ) ).
tff(fact_82_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X1: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),Y))
=> ord_less_eq(A,Y,X1) ) ) ).
tff(fact_83_order__le__less,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X2: A] :
( ord_less_eq(A,X2,Y1)
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y1))
| ( X2 = Y1 ) ) ) ) ).
tff(fact_84_less__le__not__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y1))
<=> ( ord_less_eq(A,X2,Y1)
& ~ ord_less_eq(A,Y1,X2) ) ) ) ).
tff(fact_85_order__less__le,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y1))
<=> ( ord_less_eq(A,X2,Y1)
& ( X2 != Y1 ) ) ) ) ).
tff(fact_86_linorder__le__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X1: A] :
( ord_less_eq(A,X1,Y)
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X1)) ) ) ).
tff(fact_87_linorder__not__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X2: A] :
( ~ ord_less_eq(A,X2,Y1)
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y1),X2)) ) ) ).
tff(fact_88_linorder__not__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X2: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y1))
<=> ord_less_eq(A,Y1,X2) ) ) ).
tff(fact_89_top__le,axiom,
! [A: $tType] :
( top(A)
=> ! [A2: A] :
( ord_less_eq(A,top_top(A),A2)
=> ( A2 = top_top(A) ) ) ) ).
tff(fact_90_top__unique,axiom,
! [A: $tType] :
( top(A)
=> ! [Aa: A] :
( ord_less_eq(A,top_top(A),Aa)
<=> ( Aa = top_top(A) ) ) ) ).
tff(fact_91_top__greatest,axiom,
! [A: $tType] :
( top(A)
=> ! [A2: A] : ord_less_eq(A,A2,top_top(A)) ) ).
tff(fact_92_subset__UNIV,axiom,
! [A: $tType,A1: fun(A,bool)] : ord_less_eq(fun(A,bool),A1,top_top(fun(A,bool))) ).
tff(fact_93_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_94_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_95_not__less0,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).
tff(fact_96_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,G: fun(B,C),F: fun(C,A)] : image(B,A,combb(C,A,B,F,G),top_top(fun(B,bool))) = image(C,A,F,image(B,C,G,top_top(fun(B,bool)))) ).
tff(fact_97_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X1: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),Y))
=> ( ( X1 != Y )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X1)) ) ) ) ).
tff(fact_98_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X1: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X1)) ) ) ).
tff(fact_99_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X1: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),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),X1),Z)) ) ) ) ).
%----Arities (13)
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_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,X1: A] : aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),fTrue),X1),Y) = X1 ).
tff(help_If_2_1_T,axiom,
! [A: $tType,Y: A,X1: A] : aa(A,A,aa(A,fun(A,A),aa(bool,fun(A,fun(A,A)),if(A),fFalse),X1),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,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,
( ? [X: arrow_159774573e_indi] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(arrow_159774573e_indi,nat,h,X)),finite_card(arrow_159774573e_indi,top_top(fun(arrow_159774573e_indi,bool)))))
=> ( lba = lab ) ) ).
%------------------------------------------------------------------------------