TPTP Problem File: SCT257_5.p
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%------------------------------------------------------------------------------
% File : SCT257_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 288
% 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_288 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 180 ( 50 unt; 52 typ; 0 def)
% Number of atoms : 309 ( 113 equ)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 258 ( 77 ~; 12 |; 18 &)
% ( 25 <=>; 126 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 37 ( 21 >; 16 *; 0 +; 0 <<)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-4 aty)
% Number of functors : 39 ( 39 usr; 15 con; 0-5 aty)
% Number of variables : 462 ( 413 !; 7 ?; 462 :)
% ( 42 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:26:37
%------------------------------------------------------------------------------
%----Should-be-implicit typings (7)
tff(ty_tc_Arrow__Order__Mirabelle__qkbtqzkjxu_Oalt,type,
arrow_411405190le_alt: $tType ).
tff(ty_tc_Arrow__Order__Mirabelle__qkbtqzkjxu_Oindi,type,
arrow_159774573e_indi: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_List_Olist,type,
list: $tType > $tType ).
tff(ty_tc_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 (45)
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_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(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)))) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_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_Omkbot,type,
arrow_276188178_mkbot: fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_Omktop,type,
arrow_424895264_mktop: fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))) ).
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_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).
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_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(A,B) ) > fun(B,A) ) ).
tff(sy_c_If,type,
if:
!>[A: $tType] : fun(bool,fun(A,fun(A,A))) ).
tff(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( list(A) > $o ) ).
tff(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_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_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( fun(product_prod(A,A),bool) * fun(product_prod(B,B),bool) ) > fun(product_prod(product_prod(A,B),product_prod(A,B)),bool) ) ).
tff(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( fun(A,nat) > fun(product_prod(A,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_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_member,type,
member:
!>[A: $tType] : ( ( A * fun(A,bool) ) > $o ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_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_P____,type,
p: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)) ).
tff(sy_v_c____,type,
c: arrow_411405190le_alt ).
tff(sy_v_d____,type,
d: arrow_411405190le_alt ).
tff(sy_v_e____,type,
e: arrow_411405190le_alt ).
tff(sy_v_h____,type,
h: fun(arrow_159774573e_indi,nat) ).
tff(sy_v_n____,type,
n: nat ).
%----Relevant facts (100)
tff(fact_0__096P_A_058_AProf_096,axiom,
member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),p,arrow_610318064e_Prof) ).
tff(fact_1__096c_A_126_061_Ad_096,axiom,
c != d ).
tff(fact_2_dist,axiom,
distinct(arrow_411405190le_alt,cons(arrow_411405190le_alt,c,cons(arrow_411405190le_alt,d,cons(arrow_411405190le_alt,e,nil(arrow_411405190le_alt))))) ).
tff(fact_3_in__mkbot,axiom,
! [Z1: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_276188178_mkbot,L),Z1))
<=> ( ( Y1 != Z1 )
& ( ( X1 = Z1 )
=> ( X1 != Y1 ) )
& ( ( X1 != Z1 )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),L) ) ) ) ).
tff(fact_4_in__mktop,axiom,
! [Z1: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_424895264_mktop,L),Z1))
<=> ( ( X1 != Z1 )
& ( ( Y1 = Z1 )
=> ( X1 != Y1 ) )
& ( ( Y1 != Z1 )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),L) ) ) ) ).
tff(fact_5__096c_A_060_092_060_094bsub_062P_A_Iinv_Ah_An_J_092_060_094esub_062_Ad_096,axiom,
member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,d),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,aa(nat,arrow_159774573e_indi,hilbert_inv_into(arrow_159774573e_indi,nat,top_top(fun(arrow_159774573e_indi,bool)),h),n))) ).
tff(fact_6__096_B_Bthesis_O_A_I_B_Be_O_Adistinct_A_091c_M_Ad_M_Ae_093_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [E1: arrow_411405190le_alt] : ~ distinct(arrow_411405190le_alt,cons(arrow_411405190le_alt,c,cons(arrow_411405190le_alt,d,cons(arrow_411405190le_alt,E1,nil(arrow_411405190le_alt))))) ).
tff(fact_7__096_I_Fi_O_Aif_Ah_Ai_A_060_An_Athen_Amktop_A_IP_Ai_J_Ae_Aelse_Aif_Ah_Ai_A_061_An_Athen_Aabove_A_IP_Ai_J_Ac_Ae_Aelse_Amkbot_A_IP_Ai_J_Ae_J_058_AProf_096,axiom,
member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combs(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),n)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_159774573e_indi,arrow_424895264_mktop,p),e)),combs(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combs(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,fequal(nat),h),n)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_1158827142_above,p),c),e)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_159774573e_indi,arrow_276188178_mkbot,p),e))),arrow_610318064e_Prof) ).
tff(fact_8_split__paired__All,axiom,
! [A: $tType,B: $tType,Pa: fun(product_prod(A,B),bool)] :
( ! [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,Pa,X11))
<=> ! [A6: A,B4: B] : pp(aa(product_prod(A,B),bool,Pa,product_Pair(A,B,A6,B4))) ) ).
tff(fact_9_Pair__eq,axiom,
! [A: $tType,B: $tType,B3: B,A4: A,Ba: B,Aa: A] :
( ( product_Pair(A,B,Aa,Ba) = product_Pair(A,B,A4,B3) )
<=> ( ( Aa = A4 )
& ( Ba = B3 ) ) ) ).
tff(fact_10_in__measure,axiom,
! [A: $tType,F: fun(A,nat),Y1: A,X1: A] :
( member(product_prod(A,A),product_Pair(A,A,X1,Y1),measure(A,F))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X1)),aa(A,nat,F,Y1))) ) ).
tff(fact_11_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),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_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_12_n_I1_J,axiom,
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))))) ).
tff(fact_13_less__not__refl,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).
tff(fact_14_nat__neq__iff,axiom,
! [Na: nat,M1: nat] :
( ( M1 != Na )
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),Na))
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Na),M1)) ) ) ).
tff(fact_15_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_16_less__irrefl__nat,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).
tff(fact_17__096Lba_A_058_ALin_096,axiom,
member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),lba,arrow_1985332922le_Lin) ).
tff(fact_18__096Lab_A_058_ALin_096,axiom,
member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),lab,arrow_1985332922le_Lin) ).
tff(fact_19_injh,axiom,
inj_on(arrow_159774573e_indi,nat,h,top_top(fun(arrow_159774573e_indi,bool))) ).
tff(fact_20_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_21_Lin__irrefl,axiom,
! [Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),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_22_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_23_mktop__Lin,axiom,
! [X1: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_424895264_mktop,L),X1),arrow_1985332922le_Lin) ) ).
tff(fact_24_mkbot__Lin,axiom,
! [X1: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_276188178_mkbot,L),X1),arrow_1985332922le_Lin) ) ).
tff(fact_25_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),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_1158827142_above,L),X1),Y1),arrow_1985332922le_Lin) ) ) ).
tff(fact_26_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_27_nat__less__cases,axiom,
! [Pa: fun(nat,fun(nat,bool)),Na: nat,M1: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),Na))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),Pa,Na),M1)) )
=> ( ( ( M1 = Na )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),Pa,Na),M1)) )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Na),M1))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),Pa,Na),M1)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),Pa,Na),M1)) ) ) ) ).
tff(fact_28_less__not__refl3,axiom,
! [T: nat,S2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S2),T))
=> ( S2 != T ) ) ).
tff(fact_29_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_30_distinct_Osimps_I1_J,axiom,
! [A: $tType] : distinct(A,nil(A)) ).
tff(fact_31_third__alt,axiom,
! [B1: arrow_411405190le_alt,A1: arrow_411405190le_alt] :
( ( A1 != B1 )
=> ? [C2: arrow_411405190le_alt] : distinct(arrow_411405190le_alt,cons(arrow_411405190le_alt,A1,cons(arrow_411405190le_alt,B1,cons(arrow_411405190le_alt,C2,nil(arrow_411405190le_alt))))) ) ).
tff(fact_32_split__paired__Ex,axiom,
! [A: $tType,B: $tType,Pa: fun(product_prod(A,B),bool)] :
( ? [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,Pa,X11))
<=> ? [A6: A,B4: B] : pp(aa(product_prod(A,B),bool,Pa,product_Pair(A,B,A6,B4))) ) ).
tff(fact_33_distinct__singleton,axiom,
! [A: $tType,X: A] : distinct(A,cons(A,X,nil(A))) ).
tff(fact_34_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_35_top1I,axiom,
! [A: $tType,X1: A] : pp(aa(A,bool,top_top(fun(A,bool)),X1)) ).
tff(fact_36_UNIV__I,axiom,
! [A: $tType,X1: A] : member(A,X1,top_top(fun(A,bool))) ).
tff(fact_37_iso__tuple__UNIV__I,axiom,
! [A: $tType,X1: A] : member(A,X1,top_top(fun(A,bool))) ).
tff(fact_38_list_Oinject,axiom,
! [A: $tType,List3: list(A),A4: A,List2: list(A),Aa: A] :
( ( cons(A,Aa,List2) = cons(A,A4,List3) )
<=> ( ( Aa = A4 )
& ( List2 = List3 ) ) ) ).
tff(fact_39_PiProf,axiom,
! [Na: 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),Na)),lab),lba),arrow_610318064e_Prof) ).
tff(fact_40_not__Cons__self2,axiom,
! [A: $tType,Xs1: list(A),X: A] : cons(A,X,Xs1) != Xs1 ).
tff(fact_41_not__Cons__self,axiom,
! [A: $tType,X: A,Xs1: list(A)] : Xs1 != cons(A,X,Xs1) ).
tff(fact_42_UNIV__def,axiom,
! [A: $tType] : top_top(fun(A,bool)) = collect(A,aa(bool,fun(A,bool),combk(bool,A),fTrue)) ).
tff(fact_43_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,S1: fun(product_prod(A,B),bool),R2: fun(product_prod(A,B),bool)] :
( ! [X3: A,Xa: B] :
( member(product_prod(A,B),product_Pair(A,B,X3,Xa),R2)
<=> member(product_prod(A,B),product_Pair(A,B,X3,Xa),S1) )
<=> ( R2 = S1 ) ) ).
tff(fact_44_list_Osimps_I3_J,axiom,
! [A: $tType,List1: list(A),A5: A] : cons(A,A5,List1) != nil(A) ).
tff(fact_45_list_Osimps_I2_J,axiom,
! [A: $tType,List1: list(A),A5: A] : nil(A) != cons(A,A5,List1) ).
tff(fact_46_distinct__length__2__or__more,axiom,
! [A: $tType,Xs: list(A),Ba: A,Aa: A] :
( distinct(A,cons(A,Aa,cons(A,Ba,Xs)))
<=> ( ( Aa != Ba )
& distinct(A,cons(A,Aa,Xs))
& distinct(A,cons(A,Ba,Xs)) ) ) ).
tff(fact_47_inv__f__eq,axiom,
! [B: $tType,A: $tType,Y1: B,X1: A,F: fun(A,B)] :
( inj_on(A,B,F,top_top(fun(A,bool)))
=> ( ( aa(A,B,F,X1) = Y1 )
=> ( aa(B,A,hilbert_inv_into(A,B,top_top(fun(A,bool)),F),Y1) = X1 ) ) ) ).
tff(fact_48_inv__f__f,axiom,
! [B: $tType,A: $tType,X1: A,F: fun(A,B)] :
( inj_on(A,B,F,top_top(fun(A,bool)))
=> ( aa(B,A,hilbert_inv_into(A,B,top_top(fun(A,bool)),F),aa(A,B,F,X1)) = X1 ) ) ).
tff(fact_49_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_50_list_Oexhaust,axiom,
! [A: $tType,Y: list(A)] :
( ( Y != nil(A) )
=> ~ ! [A3: A,List: list(A)] : Y != cons(A,A3,List) ) ).
tff(fact_51_neq__Nil__conv,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
<=> ? [Y3: A,Ys: list(A)] : Xs = cons(A,Y3,Ys) ) ).
tff(fact_52_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_53_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_54_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_55_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_56_inj__on__id2,axiom,
! [A: $tType,A2: fun(A,bool)] : inj_on(A,A,combi(A),A2) ).
tff(fact_57_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_58_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_59_inv__into__f__eq,axiom,
! [B: $tType,A: $tType,Y1: B,X1: A,A2: fun(A,bool),F: fun(A,B)] :
( inj_on(A,B,F,A2)
=> ( member(A,X1,A2)
=> ( ( aa(A,B,F,X1) = Y1 )
=> ( aa(B,A,hilbert_inv_into(A,B,A2,F),Y1) = X1 ) ) ) ) ).
tff(fact_60_inv__into__f__f,axiom,
! [B: $tType,A: $tType,X1: A,A2: fun(A,bool),F: fun(A,B)] :
( inj_on(A,B,F,A2)
=> ( member(A,X1,A2)
=> ( aa(B,A,hilbert_inv_into(A,B,A2,F),aa(A,B,F,X1)) = X1 ) ) ) ).
tff(fact_61_inj__imp__inv__eq,axiom,
! [A: $tType,B: $tType,G: fun(B,A),F: fun(A,B)] :
( inj_on(A,B,F,top_top(fun(A,bool)))
=> ( ! [X2: B] : aa(A,B,F,aa(B,A,G,X2)) = X2
=> ( hilbert_inv_into(A,B,top_top(fun(A,bool)),F) = G ) ) ) ).
tff(fact_62_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_63_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_64_in__lex__prod,axiom,
! [A: $tType,B: $tType,S: fun(product_prod(B,B),bool),R1: fun(product_prod(A,A),bool),B3: B,A4: A,Ba: B,Aa: A] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),product_Pair(product_prod(A,B),product_prod(A,B),product_Pair(A,B,Aa,Ba),product_Pair(A,B,A4,B3)),lex_prod(A,B,R1,S))
<=> ( member(product_prod(A,A),product_Pair(A,A,Aa,A4),R1)
| ( ( Aa = A4 )
& member(product_prod(B,B),product_Pair(B,B,Ba,B3),S) ) ) ) ).
tff(fact_65_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_66_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,X1: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),Pa: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool)] :
( ! [A3: A,B2: B,C2: C,D1: D,E1: E,F3: F2] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool,Pa,product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),A3,product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F2))),B2,product_Pair(C,product_prod(D,product_prod(E,F2)),C2,product_Pair(D,product_prod(E,F2),D1,product_Pair(E,F2,E1,F3)))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool,Pa,X1)) ) ).
tff(fact_67_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))] :
~ ! [A3: A,B2: B,C2: C,D1: D,E1: E,F3: F2] : Y != product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),A3,product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F2))),B2,product_Pair(C,product_prod(D,product_prod(E,F2)),C2,product_Pair(D,product_prod(E,F2),D1,product_Pair(E,F2,E1,F3))))) ).
tff(fact_68_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,X1: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),Pa: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool)] :
( ! [A3: A,B2: B,C2: C,D1: D,E1: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,Pa,product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E))),A3,product_Pair(B,product_prod(C,product_prod(D,E)),B2,product_Pair(C,product_prod(D,E),C2,product_Pair(D,E,D1,E1))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,Pa,X1)) ) ).
tff(fact_69_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
~ ! [A3: A,B2: B,C2: C,D1: D,E1: E] : Y != product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E))),A3,product_Pair(B,product_prod(C,product_prod(D,E)),B2,product_Pair(C,product_prod(D,E),C2,product_Pair(D,E,D1,E1)))) ).
tff(fact_70_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A3: A,B2: B,C2: C,D1: D] : Y != product_Pair(A,product_prod(B,product_prod(C,D)),A3,product_Pair(B,product_prod(C,D),B2,product_Pair(C,D,C2,D1))) ).
tff(fact_71_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,X1: product_prod(A,product_prod(B,product_prod(C,D))),Pa: fun(product_prod(A,product_prod(B,product_prod(C,D))),bool)] :
( ! [A3: A,B2: B,C2: C,D1: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,Pa,product_Pair(A,product_prod(B,product_prod(C,D)),A3,product_Pair(B,product_prod(C,D),B2,product_Pair(C,D,C2,D1)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,Pa,X1)) ) ).
tff(fact_72_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,X1: product_prod(A,product_prod(B,C)),Pa: fun(product_prod(A,product_prod(B,C)),bool)] :
( ! [A3: A,B2: B,C2: C] : pp(aa(product_prod(A,product_prod(B,C)),bool,Pa,product_Pair(A,product_prod(B,C),A3,product_Pair(B,C,B2,C2))))
=> pp(aa(product_prod(A,product_prod(B,C)),bool,Pa,X1)) ) ).
tff(fact_73_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
~ ! [A3: A,B2: B,C2: C] : Y != product_Pair(A,product_prod(B,C),A3,product_Pair(B,C,B2,C2)) ).
tff(fact_74_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_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X2: A] : aa(A,B,F,X2) = aa(A,B,G,X2)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A2: fun(A,bool),X1: A] :
( member(A,X1,A2)
<=> pp(aa(A,bool,A2,X1)) ) ).
tff(fact_77_Collect__def,axiom,
! [A: $tType,Pa: fun(A,bool)] : collect(A,Pa) = Pa ).
tff(fact_78_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_79_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_80_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_81_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_82_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_83_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_84_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_85_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_86_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_87_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_88_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_89_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_90_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Pa: 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(Pa) ) ) ) ).
tff(fact_91_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_92_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_93_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_94_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_95_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_96_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)) ) ) ) ).
tff(fact_97_not__top__less,axiom,
! [A: $tType] :
( top(A)
=> ! [A1: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),A1)) ) ).
tff(fact_98_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_99_insert__Nil,axiom,
! [A: $tType,X: A] : insert(A,X,nil(A)) = cons(A,X,nil(A)) ).
%----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 (14)
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_COMBS_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(A,fun(B,C))] : aa(A,C,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).
tff(help_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ( X != Y )
| pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y)) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
! [I: arrow_159774573e_indi] :
~ ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,d),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I))
<=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(arrow_159774573e_indi,nat,h,I)),n))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,d),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_424895264_mktop,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I)),e)) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(arrow_159774573e_indi,nat,h,I)),n))
=> ( ( ( aa(arrow_159774573e_indi,nat,h,I) = n )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,d),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_1158827142_above,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I)),c),e)) )
& ( ( aa(arrow_159774573e_indi,nat,h,I) != n )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,c,d),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_276188178_mkbot,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I)),e)) ) ) ) ) ) ).
%------------------------------------------------------------------------------