TPTP Problem File: COM091_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : COM091_5 : TPTP v9.0.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 217
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Nip08] Nipkow (2008), Linear Quantifier Elimination
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : qe_217 [Bla13]
% Status : Theorem
% Rating : 0.67 v7.4.0, 1.00 v6.4.0
% Syntax : Number of formulae : 186 ( 53 unt; 52 typ; 0 def)
% Number of atoms : 252 ( 82 equ)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 175 ( 57 ~; 16 |; 17 &)
% ( 26 <=>; 59 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 19 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 36 ( 22 >; 14 *; 0 +; 0 <<)
% Number of predicates : 13 ( 12 usr; 0 prp; 1-3 aty)
% Number of functors : 37 ( 37 usr; 8 con; 0-6 aty)
% Number of variables : 511 ( 449 !; 9 ?; 511 :)
% ( 53 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:22:49
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_List_Olist,type,
list: $tType > $tType ).
tff(ty_tc_PresArith_Oatom,type,
atom: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tc_prod,type,
product_prod: ( $tType * $tType ) > $tType ).
%----Explicit typings (46)
tff(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ominus,type,
cl_Groups_Ominus:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(B,fun(A,C))) ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_Finite__Set_Ofinite,type,
finite_finite1:
!>[A: $tType] : ( fun(A,bool) > $o ) ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_ListVector_Oiprod,type,
iprod:
!>[A: $tType] : fun(list(A),fun(list(A),A)) ).
tff(sy_c_ListVector_Ozipwith0,type,
zipwith0:
!>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(list(A),fun(list(B),list(C))) ) ).
tff(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord467138063of_set:
!>[A: $tType] : ( fun(A,bool) > list(A) ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : fun(A,fun(list(A),list(A))) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olist_Olist__case,type,
list_case:
!>[T: $tType,A: $tType] : ( ( T * fun(A,fun(list(A),T)) * list(A) ) > T ) ).
tff(sy_c_List_Olist__all,type,
list_all:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Olistset,type,
listset:
!>[A: $tType] : ( list(fun(A,bool)) > fun(list(A),bool) ) ).
tff(sy_c_List_Oset,type,
set:
!>[A: $tType] : ( list(A) > fun(A,bool) ) ).
tff(sy_c_List_Oset__Cons,type,
set_Cons:
!>[A: $tType] : ( ( fun(A,bool) * fun(list(A),bool) ) > fun(list(A),bool) ) ).
tff(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
tff(sy_c_PresArith_OI_092_060_094isub_062Z,type,
i_Z: ( atom * list(int) ) > $o ).
tff(sy_c_PresArith_Olbounds,type,
lbounds: list(atom) > list(product_prod(int,list(int))) ).
tff(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : fun(A,fun(B,product_prod(A,B))) ).
tff(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(product_prod(A,B),C) * A * B ) > C ) ).
tff(sy_c_Product__Type_Ointernal__split,type,
produc1605651328_split:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * product_prod(A,B) ) > C ) ).
tff(sy_c_Product__Type_Oprod_Oprod__rec,type,
product_prod_rec:
!>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fEx,type,
fEx:
!>[A: $tType] : fun(fun(A,bool),bool) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fNot,type,
fNot: fun(bool,bool) ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fdisj,type,
fdisj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,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_as,type,
as: list(atom) ).
tff(sy_v_x____,type,
x: int ).
tff(sy_v_xs,type,
xs: list(int) ).
%----Relevant facts (100)
tff(fact_0__096lbounds_Aas_A_126_061_A_091_093_096,axiom,
lbounds(as) != nil(product_prod(int,list(int))) ).
tff(fact_1_calculation,axiom,
finite_finite1(int,collect(int,aa(fun(int,fun(list(int),bool)),fun(int,bool),aa(fun(fun(list(int),bool),bool),fun(fun(int,fun(list(int),bool)),fun(int,bool)),combb(fun(list(int),bool),bool,int),fEx(list(int))),aa(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),bool)),aa(fun(fun(list(int),fun(int,bool)),fun(list(int),bool)),fun(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),bool))),combb(fun(list(int),fun(int,bool)),fun(list(int),bool),int),aa(fun(fun(int,bool),bool),fun(fun(list(int),fun(int,bool)),fun(list(int),bool)),combb(fun(int,bool),bool,list(int)),fEx(int))),aa(fun(list(int),fun(int,bool)),fun(int,fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),fun(fun(list(int),fun(int,bool)),fun(int,fun(list(int),fun(int,bool)))),combc(int,fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))),aa(fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),aa(fun(fun(list(int),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),fun(fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))))),combb(fun(list(int),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))),int),combs(list(int),fun(int,bool),fun(int,bool))),aa(fun(int,fun(list(int),fun(int,fun(bool,bool)))),fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),aa(fun(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(fun(int,fun(list(int),fun(int,fun(bool,bool)))),fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool))))),combb(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool))),int),aa(fun(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool)))),combb(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),list(int)),combs(int,bool,bool))),aa(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),fun(int,fun(bool,bool)))),aa(fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool)))),fun(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),fun(int,fun(bool,bool))))),combb(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool))),int),aa(fun(fun(int,bool),fun(int,fun(bool,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool)))),combb(fun(int,bool),fun(int,fun(bool,bool)),list(int)),aa(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj))),aa(fun(list(int),fun(int,int)),fun(int,fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),fun(fun(list(int),fun(int,int)),fun(int,fun(list(int),fun(int,bool)))),combc(int,fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(int,int),fun(int,bool))),fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),aa(fun(fun(fun(int,int),fun(int,bool)),fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),fun(fun(int,fun(fun(int,int),fun(int,bool))),fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))))),combb(fun(fun(int,int),fun(int,bool)),fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))),int),combb(fun(int,int),fun(int,bool),list(int))),aa(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool))),aa(fun(fun(int,bool),fun(fun(int,int),fun(int,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool)))),combb(fun(int,bool),fun(fun(int,int),fun(int,bool)),int),combb(int,bool,int)),fequal(int)))),aa(fun(list(int),int),fun(list(int),fun(int,int)),aa(fun(int,fun(int,int)),fun(fun(list(int),int),fun(list(int),fun(int,int))),combb(int,fun(int,int),list(int)),aa(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),minus_minus(int))),aa(list(int),fun(list(int),int),aa(fun(list(int),fun(list(int),int)),fun(list(int),fun(list(int),int)),combc(list(int),list(int),int),iprod(int)),xs))))))),aa(fun(product_prod(int,list(int)),bool),fun(list(int),fun(int,bool)),aa(fun(list(int),fun(fun(product_prod(int,list(int)),bool),fun(int,bool))),fun(fun(product_prod(int,list(int)),bool),fun(list(int),fun(int,bool))),combc(list(int),fun(product_prod(int,list(int)),bool),fun(int,bool)),aa(fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),fun(list(int),fun(fun(product_prod(int,list(int)),bool),fun(int,bool))),aa(fun(fun(int,fun(fun(product_prod(int,list(int)),bool),bool)),fun(fun(product_prod(int,list(int)),bool),fun(int,bool))),fun(fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),fun(list(int),fun(fun(product_prod(int,list(int)),bool),fun(int,bool)))),combb(fun(int,fun(fun(product_prod(int,list(int)),bool),bool)),fun(fun(product_prod(int,list(int)),bool),fun(int,bool)),list(int)),combc(int,fun(product_prod(int,list(int)),bool),bool)),aa(fun(list(int),fun(int,product_prod(int,list(int)))),fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),aa(fun(fun(int,product_prod(int,list(int))),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),fun(fun(list(int),fun(int,product_prod(int,list(int)))),fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool)))),combb(fun(int,product_prod(int,list(int))),fun(int,fun(fun(product_prod(int,list(int)),bool),bool)),list(int)),aa(fun(product_prod(int,list(int)),fun(fun(product_prod(int,list(int)),bool),bool)),fun(fun(int,product_prod(int,list(int))),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),combb(product_prod(int,list(int)),fun(fun(product_prod(int,list(int)),bool),bool),int),member(product_prod(int,list(int))))),aa(fun(int,fun(list(int),product_prod(int,list(int)))),fun(list(int),fun(int,product_prod(int,list(int)))),combc(int,list(int),product_prod(int,list(int))),product_Pair(int,list(int)))))),set(product_prod(int,list(int)),lbounds(as)))))))) ).
tff(fact_2_all__not__in__conv,axiom,
! [A: $tType,A2: fun(A,bool)] :
( ! [X2: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A2))
<=> ( A2 = bot_bot(fun(A,bool)) ) ) ).
tff(fact_3_empty__Collect__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( bot_bot(fun(A,bool)) = collect(A,P1) )
<=> ! [X2: A] : ~ pp(aa(A,bool,P1,X2)) ) ).
tff(fact_4_empty__iff,axiom,
! [A: $tType,C1: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),bot_bot(fun(A,bool)))) ).
tff(fact_5_Collect__empty__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( collect(A,P1) = bot_bot(fun(A,bool)) )
<=> ! [X2: A] : ~ pp(aa(A,bool,P1,X2)) ) ).
tff(fact_6_emptyE,axiom,
! [A: $tType,A4: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A4),bot_bot(fun(A,bool)))) ).
tff(fact_7_split__paired__All,axiom,
! [A: $tType,B: $tType,P1: fun(product_prod(A,B),bool)] :
( ! [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P1,X11))
<=> ! [A6: A,B4: B] : pp(aa(product_prod(A,B),bool,P1,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B4))) ) ).
tff(fact_8_Pair__eq,axiom,
! [A: $tType,B: $tType,B7: B,A5: A,B1: B,A4: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B1) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B7) )
<=> ( ( A4 = A5 )
& ( B1 = B7 ) ) ) ).
tff(fact_9_iprod__left__diff__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [Zs: list(A),Ys2: list(A),Xs: list(A)] : ( aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),minus_minus(list(A)),Xs),Ys2)),Zs) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Xs),Zs)),aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Ys2),Zs)) ) ) ).
tff(fact_10_bot__fun__def,axiom,
! [A: $tType,B: $tType] :
( bot(B)
=> ! [X3: A] : ( aa(A,B,bot_bot(fun(A,B)),X3) = bot_bot(B) ) ) ).
tff(fact_11_bot__apply,axiom,
! [B: $tType,A: $tType] :
( bot(A)
=> ! [Xa: B] : ( aa(B,A,bot_bot(fun(B,A)),Xa) = bot_bot(A) ) ) ).
tff(fact_12_List_Oset_Osimps_I1_J,axiom,
! [A: $tType] : ( set(A,nil(A)) = bot_bot(fun(A,bool)) ) ).
tff(fact_13_Diff__cancel,axiom,
! [A: $tType,A2: fun(A,bool)] : ( aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),A2) = bot_bot(fun(A,bool)) ) ).
tff(fact_14_empty__Diff,axiom,
! [A: $tType,A2: fun(A,bool)] : ( aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),bot_bot(fun(A,bool))),A2) = bot_bot(fun(A,bool)) ) ).
tff(fact_15_finite__set,axiom,
! [A: $tType,Xsa: list(A)] : finite_finite1(A,set(A,Xsa)) ).
tff(fact_16_set__empty,axiom,
! [A: $tType,Xsa: list(A)] :
( ( set(A,Xsa) = bot_bot(fun(A,bool)) )
<=> ( Xsa = nil(A) ) ) ).
tff(fact_17_set__empty2,axiom,
! [A: $tType,Xsa: list(A)] :
( ( bot_bot(fun(A,bool)) = set(A,Xsa) )
<=> ( Xsa = nil(A) ) ) ).
tff(fact_18_list__diff__Nil2,axiom,
! [A: $tType] :
( group_add(A)
=> ! [Xs: list(A)] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),minus_minus(list(A)),Xs),nil(A)) = Xs ) ) ).
tff(fact_19_Diff__empty,axiom,
! [A: $tType,A2: fun(A,bool)] : ( aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),bot_bot(fun(A,bool))) = A2 ) ).
tff(fact_20_Pair__inject,axiom,
! [A: $tType,B: $tType,B6: B,A3: A,B5: B,A7: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B5) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B6) )
=> ~ ( ( A7 = A3 )
=> ( B5 != B6 ) ) ) ).
tff(fact_21_equals0D,axiom,
! [A: $tType,A4: A,A2: fun(A,bool)] :
( ( A2 = bot_bot(fun(A,bool)) )
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A4),A2)) ) ).
tff(fact_22_ex__in__conv,axiom,
! [A: $tType,A2: fun(A,bool)] :
( ? [X2: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A2))
<=> ( A2 != bot_bot(fun(A,bool)) ) ) ).
tff(fact_23_empty__def,axiom,
! [A: $tType] : ( bot_bot(fun(A,bool)) = collect(A,combk(bool,A,fFalse)) ) ).
tff(fact_24_finite_OemptyI,axiom,
! [A: $tType] : finite_finite1(A,bot_bot(fun(A,bool))) ).
tff(fact_25_finite__Collect__disjI,axiom,
! [A: $tType,Q1: fun(A,bool),P1: fun(A,bool)] :
( finite_finite1(A,collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fdisj),P1)),Q1)))
<=> ( finite_finite1(A,collect(A,P1))
& finite_finite1(A,collect(A,Q1)) ) ) ).
tff(fact_26_finite__Collect__conjI,axiom,
! [A: $tType,Q1: fun(A,bool),P1: fun(A,bool)] :
( ( finite_finite1(A,collect(A,P1))
| finite_finite1(A,collect(A,Q1)) )
=> finite_finite1(A,collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),P1)),Q1))) ) ).
tff(fact_27_split__paired__Ex,axiom,
! [A: $tType,B: $tType,P1: fun(product_prod(A,B),bool)] :
( ? [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P1,X11))
<=> ? [A6: A,B4: B] : pp(aa(product_prod(A,B),bool,P1,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B4))) ) ).
tff(fact_28_finite__code,axiom,
! [A: $tType] :
( finite_finite(A)
=> ! [A2: fun(A,bool)] : finite_finite1(A,A2) ) ).
tff(fact_29_finite__Diff,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool)] :
( finite_finite1(A,A2)
=> finite_finite1(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3)) ) ).
tff(fact_30_finite__list,axiom,
! [A: $tType,A2: fun(A,bool)] :
( finite_finite1(A,A2)
=> ? [Xs1: list(A)] : ( set(A,Xs1) = A2 ) ) ).
tff(fact_31_finite__Collect__bounded__ex,axiom,
! [B: $tType,A: $tType,Q1: fun(B,fun(A,bool)),P1: fun(A,bool)] :
( finite_finite1(A,collect(A,P1))
=> ( finite_finite1(B,collect(B,aa(fun(B,fun(A,bool)),fun(B,bool),aa(fun(fun(A,bool),bool),fun(fun(B,fun(A,bool)),fun(B,bool)),combb(fun(A,bool),bool,B),fEx(A)),aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(fun(A,bool),fun(A,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),combb(fun(A,bool),fun(A,bool),B),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),P1))),Q1))))
<=> ! [Y1: A] :
( pp(aa(A,bool,P1,Y1))
=> finite_finite1(B,collect(B,aa(A,fun(B,bool),aa(fun(B,fun(A,bool)),fun(A,fun(B,bool)),combc(B,A,bool),Q1),Y1))) ) ) ) ).
tff(fact_32_finite__image__set,axiom,
! [B: $tType,A: $tType,F: fun(A,B),P1: fun(A,bool)] :
( finite_finite1(A,collect(A,P1))
=> finite_finite1(B,collect(B,aa(fun(B,fun(A,bool)),fun(B,bool),aa(fun(fun(A,bool),bool),fun(fun(B,fun(A,bool)),fun(B,bool)),combb(fun(A,bool),bool,B),fEx(A)),aa(fun(A,bool),fun(B,fun(A,bool)),aa(fun(B,fun(fun(A,bool),fun(A,bool))),fun(fun(A,bool),fun(B,fun(A,bool))),combc(B,fun(A,bool),fun(A,bool)),aa(fun(B,fun(A,fun(bool,bool))),fun(B,fun(fun(A,bool),fun(A,bool))),aa(fun(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool))),fun(fun(B,fun(A,fun(bool,bool))),fun(B,fun(fun(A,bool),fun(A,bool)))),combb(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),B),combs(A,bool,bool)),aa(fun(B,fun(A,bool)),fun(B,fun(A,fun(bool,bool))),aa(fun(fun(A,bool),fun(A,fun(bool,bool))),fun(fun(B,fun(A,bool)),fun(B,fun(A,fun(bool,bool)))),combb(fun(A,bool),fun(A,fun(bool,bool)),B),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj)),aa(fun(A,B),fun(B,fun(A,bool)),aa(fun(B,fun(fun(A,B),fun(A,bool))),fun(fun(A,B),fun(B,fun(A,bool))),combc(B,fun(A,B),fun(A,bool)),aa(fun(B,fun(B,bool)),fun(B,fun(fun(A,B),fun(A,bool))),aa(fun(fun(B,bool),fun(fun(A,B),fun(A,bool))),fun(fun(B,fun(B,bool)),fun(B,fun(fun(A,B),fun(A,bool)))),combb(fun(B,bool),fun(fun(A,B),fun(A,bool)),B),combb(B,bool,A)),fequal(B))),F)))),P1)))) ) ).
tff(fact_33_Diff__iff,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3)))
<=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A2))
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B3)) ) ) ).
tff(fact_34_Diff__idemp,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool)] : ( aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3)),B3) = aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3) ) ).
tff(fact_35_DiffI,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A2))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B3))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3))) ) ) ).
tff(fact_36_DiffE,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3)))
=> ~ ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A2))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B3)) ) ) ).
tff(fact_37_set__diff__eq,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool)] : ( aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3) = collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(fun(A,bool),bool)),fun(fun(A,bool),fun(A,bool)),combc(A,fun(A,bool),bool),member(A)),A2))),aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),combb(bool,bool,A),fNot),aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(fun(A,bool),bool)),fun(fun(A,bool),fun(A,bool)),combc(A,fun(A,bool),bool),member(A)),B3)))) ) ).
tff(fact_38_DiffD1,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3)))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A2)) ) ).
tff(fact_39_DiffD2,axiom,
! [A: $tType,B3: fun(A,bool),A2: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3)))
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B3)) ) ).
tff(fact_40_finite__Diff2,axiom,
! [A: $tType,A2: fun(A,bool),B3: fun(A,bool)] :
( finite_finite1(A,B3)
=> ( finite_finite1(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A2),B3))
<=> finite_finite1(A,A2) ) ) ).
tff(fact_41_finite,axiom,
! [A: $tType] :
( finite_finite(A)
=> ! [A2: fun(A,bool)] : finite_finite1(A,A2) ) ).
tff(fact_42_prod_Orecs,axiom,
! [B: $tType,A: $tType,C: $tType,B1: C,A4: B,F1: fun(B,fun(C,A))] : ( product_prod_rec(B,C,A,F1,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A4),B1)) = aa(C,A,aa(B,fun(C,A),F1,A4),B1) ) ).
tff(fact_43_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( cl_Groups_Ominus(B)
=> ! [B3: fun(A,B),A2: fun(A,B),X3: A] : ( aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),minus_minus(fun(A,B)),A2),B3),X3) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A2,X3)),aa(A,B,B3,X3)) ) ) ).
tff(fact_44_minus__apply,axiom,
! [A: $tType,B: $tType] :
( cl_Groups_Ominus(A)
=> ! [Xa: B,B3: fun(B,A),A2: fun(B,A)] : ( aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),minus_minus(fun(B,A)),A2),B3),Xa) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,A2,Xa)),aa(B,A,B3,Xa)) ) ) ).
tff(fact_45_sorted__list__of__set__empty,axiom,
! [A: $tType] :
( linorder(A)
=> ( linord467138063of_set(A,bot_bot(fun(A,bool))) = nil(A) ) ) ).
tff(fact_46_list_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,fun(list(B),A)),F1: A] : ( list_case(A,B,F1,F2,nil(B)) = F1 ) ).
tff(fact_47_equals0I,axiom,
! [A: $tType,A2: fun(A,bool)] :
( ! [Y3: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y3),A2))
=> ( A2 = bot_bot(fun(A,bool)) ) ) ).
tff(fact_48_bot__empty__eq2,axiom,
! [B: $tType,A: $tType,X3: A,Xa1: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X3),Xa1))
<=> pp(aa(fun(product_prod(A,B),bool),bool,aa(product_prod(A,B),fun(fun(product_prod(A,B),bool),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa1)),bot_bot(fun(product_prod(A,B),bool)))) ) ).
tff(fact_49_prod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod(A,B)] :
~ ! [A1: A,B2: B] : ( Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B2) ) ).
tff(fact_50_PairE,axiom,
! [A: $tType,B: $tType,P2: product_prod(A,B)] :
~ ! [X1: A,Y3: B] : ( P2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),Y3) ) ).
tff(fact_51_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
~ ! [A1: A,B2: B,C2: C] : ( Y != aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A1),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C2)) ) ).
tff(fact_52_bot2E,axiom,
! [A: $tType,B: $tType,Y2: B,Xa: A] : ~ pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),Xa),Y2)) ).
tff(fact_53_bot__empty__eq,axiom,
! [A: $tType,X3: A] :
( pp(aa(A,bool,bot_bot(fun(A,bool)),X3))
<=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),bot_bot(fun(A,bool)))) ) ).
tff(fact_54_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,S: fun(product_prod(A,B),bool),R1: fun(product_prod(A,B),bool)] :
( ! [X2: A,Xa2: B] :
( pp(aa(fun(product_prod(A,B),bool),bool,aa(product_prod(A,B),fun(fun(product_prod(A,B),bool),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa2)),R1))
<=> pp(aa(fun(product_prod(A,B),bool),bool,aa(product_prod(A,B),fun(fun(product_prod(A,B),bool),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Xa2)),S)) )
<=> ( R1 = S ) ) ).
tff(fact_55_prod__induct6,axiom,
! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),P1: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),bool)] :
( ! [A1: A,B2: B,C2: C,D1: D,E1: E,F4: F3] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),bool,P1,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),A1),aa(product_prod(C,product_prod(D,product_prod(E,F3))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F3))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F3)))),B2),aa(product_prod(D,product_prod(E,F3)),product_prod(C,product_prod(D,product_prod(E,F3))),aa(C,fun(product_prod(D,product_prod(E,F3)),product_prod(C,product_prod(D,product_prod(E,F3)))),product_Pair(C,product_prod(D,product_prod(E,F3))),C2),aa(product_prod(E,F3),product_prod(D,product_prod(E,F3)),aa(D,fun(product_prod(E,F3),product_prod(D,product_prod(E,F3))),product_Pair(D,product_prod(E,F3)),D1),aa(F3,product_prod(E,F3),aa(E,fun(F3,product_prod(E,F3)),product_Pair(E,F3),E1),F4)))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),bool,P1,Xa)) ) ).
tff(fact_56_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))))] :
~ ! [A1: A,B2: B,C2: C,D1: D,E1: E,F4: F3] : ( Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),A1),aa(product_prod(C,product_prod(D,product_prod(E,F3))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F3))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F3)))),B2),aa(product_prod(D,product_prod(E,F3)),product_prod(C,product_prod(D,product_prod(E,F3))),aa(C,fun(product_prod(D,product_prod(E,F3)),product_prod(C,product_prod(D,product_prod(E,F3)))),product_Pair(C,product_prod(D,product_prod(E,F3))),C2),aa(product_prod(E,F3),product_prod(D,product_prod(E,F3)),aa(D,fun(product_prod(E,F3),product_prod(D,product_prod(E,F3))),product_Pair(D,product_prod(E,F3)),D1),aa(F3,product_prod(E,F3),aa(E,fun(F3,product_prod(E,F3)),product_Pair(E,F3),E1),F4))))) ) ).
tff(fact_57_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),P1: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool)] :
( ! [A1: 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,P1,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A1),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B2),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D1),E1))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P1,Xa)) ) ).
tff(fact_58_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
~ ! [A1: A,B2: B,C2: C,D1: D,E1: E] : ( Y != aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A1),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B2),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D1),E1)))) ) ).
tff(fact_59_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A1: A,B2: B,C2: C,D1: D] : ( Y != aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A1),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B2),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D1))) ) ).
tff(fact_60_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,product_prod(C,D))),P1: fun(product_prod(A,product_prod(B,product_prod(C,D))),bool)] :
( ! [A1: A,B2: B,C2: C,D1: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P1,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A1),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B2),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D1)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P1,Xa)) ) ).
tff(fact_61_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,C)),P1: fun(product_prod(A,product_prod(B,C)),bool)] :
( ! [A1: A,B2: B,C2: C] : pp(aa(product_prod(A,product_prod(B,C)),bool,P1,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A1),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C2))))
=> pp(aa(product_prod(A,product_prod(B,C)),bool,P1,Xa)) ) ).
tff(fact_62_internal__split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,B1: C,A4: B,C1: fun(B,fun(C,A))] : ( produc1605651328_split(B,C,A,C1,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A4),B1)) = aa(C,A,aa(B,fun(C,A),C1,A4),B1) ) ).
tff(fact_63_curry__conv,axiom,
! [A: $tType,B: $tType,C: $tType,B1: C,A4: B,F: fun(product_prod(B,C),A)] : ( product_curry(B,C,A,F,A4,B1) = aa(product_prod(B,C),A,F,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A4),B1)) ) ).
tff(fact_64_list__diff__def,axiom,
! [A: $tType] :
( ( cl_Groups_Ominus(A)
& zero(A) )
=> ( minus_minus(list(A)) = zipwith0(A,A,A,minus_minus(A)) ) ) ).
tff(fact_65_curryI,axiom,
! [A: $tType,B: $tType,B1: B,A4: A,F: fun(product_prod(A,B),bool)] :
( pp(aa(product_prod(A,B),bool,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B1)))
=> pp(product_curry(A,B,bool,F,A4,B1)) ) ).
tff(fact_66_zipwith0_Osimps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( zero(B)
& zero(C) )
=> ! [F: fun(B,fun(C,A))] : ( aa(list(C),list(A),aa(list(B),fun(list(C),list(A)),zipwith0(B,C,A,F),nil(B)),nil(C)) = nil(A) ) ) ).
tff(fact_67_curryE,axiom,
! [A: $tType,B: $tType,B1: B,A4: A,F: fun(product_prod(A,B),bool)] :
( pp(product_curry(A,B,bool,F,A4,B1))
=> pp(aa(product_prod(A,B),bool,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B1))) ) ).
tff(fact_68_curryD,axiom,
! [A: $tType,B: $tType,B1: B,A4: A,F: fun(product_prod(A,B),bool)] :
( pp(product_curry(A,B,bool,F,A4,B1))
=> pp(aa(product_prod(A,B),bool,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B1))) ) ).
tff(fact_69_curry__def,axiom,
! [C: $tType,A: $tType,B: $tType,X3: fun(product_prod(A,B),C),Xa1: A,Xb: B] : ( product_curry(A,B,C,X3,Xa1,Xb) = aa(product_prod(A,B),C,X3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa1),Xb)) ) ).
tff(fact_70_list__ex1__simps_I1_J,axiom,
! [A: $tType,P1: fun(A,bool)] : ~ list_ex1(A,P1,nil(A)) ).
tff(fact_71_x,axiom,
! [X3: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X3),set(atom,as)))
=> i_Z(X3,aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),x),xs)) ) ).
tff(fact_72__096EX_Ax_O_AALL_Aa_058set_Aas_O_AI_092_060_094isub_062Z_Aa_A_Ix_A_D_Axs_J_096,axiom,
? [X1: int] :
! [Xa1: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),Xa1),set(atom,as)))
=> i_Z(Xa1,aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X1),xs)) ) ).
tff(fact_73_list_Oinject,axiom,
! [A: $tType,List3: list(A),A5: A,List2: list(A),A4: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),List2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),List3) )
<=> ( ( A4 = A5 )
& ( List2 = List3 ) ) ) ).
tff(fact_74_zipwith0_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( zero(C)
& zero(B) )
=> ! [Ys: list(C),Y2: C,Xsa: list(B),Xa: B,F: fun(B,fun(C,A))] : ( aa(list(C),list(A),aa(list(B),fun(list(C),list(A)),zipwith0(B,C,A,F),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xa),Xsa)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y2),Ys)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(C,A,aa(B,fun(C,A),F,Xa),Y2)),aa(list(C),list(A),aa(list(B),fun(list(C),list(A)),zipwith0(B,C,A,F),Xsa),Ys)) ) ) ).
tff(fact_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X1: A] : ( aa(A,B,F,X1) = aa(A,B,G,X1) )
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A2: fun(A,bool),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),A2))
<=> pp(aa(A,bool,A2,Xa)) ) ).
tff(fact_77_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : ( collect(A,P1) = P1 ) ).
tff(fact_78_list__diff__Cons__Cons,axiom,
! [A: $tType] :
( ( cl_Groups_Ominus(A)
& zero(A) )
=> ! [Ys2: list(A),Y: A,Xs: list(A),X: A] : ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),minus_minus(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys2)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),X),Y)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),minus_minus(list(A)),Xs),Ys2)) ) ) ).
tff(fact_79__096_B_Bthesis_O_A_I_B_Bx_O_AALL_Aa_058set_Aas_O_AI_092_060_094isub_062Z_Aa_A_Ix_A_D_Axs_J_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [X1: int] :
~ ! [Xa1: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),Xa1),set(atom,as)))
=> i_Z(Xa1,aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X1),xs)) ) ).
tff(fact_80_list_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,List2: list(B),A4: B,F2: fun(B,fun(list(B),A)),F1: A] : ( list_case(A,B,F1,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A4),List2)) = aa(list(B),A,aa(B,fun(list(B),A),F2,A4),List2) ) ).
tff(fact_81_set__ConsD,axiom,
! [A: $tType,Xsa: list(A),Xa: A,Y2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y2),set(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))))
=> ( ( Y2 = Xa )
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y2),set(A,Xsa))) ) ) ).
tff(fact_82_list_Osimps_I2_J,axiom,
! [A: $tType,List1: list(A),A3: A] : ( nil(A) != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),List1) ) ).
tff(fact_83_list_Osimps_I3_J,axiom,
! [A: $tType,List1: list(A),A3: A] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),List1) != nil(A) ) ).
tff(fact_84_not__Cons__self,axiom,
! [A: $tType,X: A,Xs: list(A)] : ( Xs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) ).
tff(fact_85_not__Cons__self2,axiom,
! [A: $tType,Xs: list(A),X: A] : ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) != Xs ) ).
tff(fact_86_list__ex1__iff,axiom,
! [A: $tType,Xsa: list(A),P1: fun(A,bool)] :
( list_ex1(A,P1,Xsa)
<=> ? [X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),set(A,Xsa)))
& pp(aa(A,bool,P1,X2))
& ! [Y1: A] :
( ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y1),set(A,Xsa)))
& pp(aa(A,bool,P1,Y1)) )
=> ( Y1 = X2 ) ) ) ) ).
tff(fact_87_set__Cons__def,axiom,
! [A: $tType,XS: fun(list(A),bool),A2: fun(A,bool)] : ( set_Cons(A,A2,XS) = collect(list(A),aa(fun(list(A),fun(A,bool)),fun(list(A),bool),aa(fun(fun(A,bool),bool),fun(fun(list(A),fun(A,bool)),fun(list(A),bool)),combb(fun(A,bool),bool,list(A)),fEx(A)),aa(fun(list(A),fun(A,fun(list(A),bool))),fun(list(A),fun(A,bool)),aa(fun(fun(A,fun(list(A),bool)),fun(A,bool)),fun(fun(list(A),fun(A,fun(list(A),bool))),fun(list(A),fun(A,bool))),combb(fun(A,fun(list(A),bool)),fun(A,bool),list(A)),aa(fun(fun(list(A),bool),bool),fun(fun(A,fun(list(A),bool)),fun(A,bool)),combb(fun(list(A),bool),bool,A),fEx(list(A)))),aa(fun(A,fun(list(A),bool)),fun(list(A),fun(A,fun(list(A),bool))),aa(fun(list(A),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool)))),fun(fun(A,fun(list(A),bool)),fun(list(A),fun(A,fun(list(A),bool)))),combc(list(A),fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool))),aa(fun(list(A),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),fun(list(A),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool)))),aa(fun(fun(A,fun(fun(list(A),bool),fun(list(A),bool))),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool)))),fun(fun(list(A),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),fun(list(A),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool))))),combb(fun(A,fun(fun(list(A),bool),fun(list(A),bool))),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),bool))),list(A)),combs(A,fun(list(A),bool),fun(list(A),bool))),aa(fun(list(A),fun(A,fun(list(A),fun(bool,bool)))),fun(list(A),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),aa(fun(fun(A,fun(list(A),fun(bool,bool))),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),fun(fun(list(A),fun(A,fun(list(A),fun(bool,bool)))),fun(list(A),fun(A,fun(fun(list(A),bool),fun(list(A),bool))))),combb(fun(A,fun(list(A),fun(bool,bool))),fun(A,fun(fun(list(A),bool),fun(list(A),bool))),list(A)),aa(fun(fun(list(A),fun(bool,bool)),fun(fun(list(A),bool),fun(list(A),bool))),fun(fun(A,fun(list(A),fun(bool,bool))),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),combb(fun(list(A),fun(bool,bool)),fun(fun(list(A),bool),fun(list(A),bool)),A),combs(list(A),bool,bool))),aa(fun(list(A),fun(A,fun(list(A),bool))),fun(list(A),fun(A,fun(list(A),fun(bool,bool)))),aa(fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),fun(bool,bool)))),fun(fun(list(A),fun(A,fun(list(A),bool))),fun(list(A),fun(A,fun(list(A),fun(bool,bool))))),combb(fun(A,fun(list(A),bool)),fun(A,fun(list(A),fun(bool,bool))),list(A)),aa(fun(fun(list(A),bool),fun(list(A),fun(bool,bool))),fun(fun(A,fun(list(A),bool)),fun(A,fun(list(A),fun(bool,bool)))),combb(fun(list(A),bool),fun(list(A),fun(bool,bool)),A),aa(fun(bool,fun(bool,bool)),fun(fun(list(A),bool),fun(list(A),fun(bool,bool))),combb(bool,fun(bool,bool),list(A)),fconj))),aa(fun(A,fun(list(A),list(A))),fun(list(A),fun(A,fun(list(A),bool))),aa(fun(list(A),fun(fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool)))),fun(fun(A,fun(list(A),list(A))),fun(list(A),fun(A,fun(list(A),bool)))),combc(list(A),fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool))),aa(fun(list(A),fun(fun(list(A),list(A)),fun(list(A),bool))),fun(list(A),fun(fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool)))),aa(fun(fun(fun(list(A),list(A)),fun(list(A),bool)),fun(fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool)))),fun(fun(list(A),fun(fun(list(A),list(A)),fun(list(A),bool))),fun(list(A),fun(fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool))))),combb(fun(fun(list(A),list(A)),fun(list(A),bool)),fun(fun(A,fun(list(A),list(A))),fun(A,fun(list(A),bool))),list(A)),combb(fun(list(A),list(A)),fun(list(A),bool),A)),aa(fun(list(A),fun(list(A),bool)),fun(list(A),fun(fun(list(A),list(A)),fun(list(A),bool))),aa(fun(fun(list(A),bool),fun(fun(list(A),list(A)),fun(list(A),bool))),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(fun(list(A),list(A)),fun(list(A),bool)))),combb(fun(list(A),bool),fun(fun(list(A),list(A)),fun(list(A),bool)),list(A)),combb(list(A),bool,list(A))),fequal(list(A))))),cons(A)))))),aa(fun(list(A),bool),fun(A,fun(list(A),bool)),aa(fun(A,fun(fun(list(A),bool),fun(list(A),bool))),fun(fun(list(A),bool),fun(A,fun(list(A),bool))),combc(A,fun(list(A),bool),fun(list(A),bool)),aa(fun(A,fun(bool,bool)),fun(A,fun(fun(list(A),bool),fun(list(A),bool))),aa(fun(fun(bool,bool),fun(fun(list(A),bool),fun(list(A),bool))),fun(fun(A,fun(bool,bool)),fun(A,fun(fun(list(A),bool),fun(list(A),bool)))),combb(fun(bool,bool),fun(fun(list(A),bool),fun(list(A),bool)),A),combb(bool,bool,list(A))),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(fun(A,bool),bool)),fun(fun(A,bool),fun(A,bool)),combc(A,fun(A,bool),bool),member(A)),A2)))),aa(fun(list(A),bool),fun(list(A),bool),aa(fun(list(A),fun(fun(list(A),bool),bool)),fun(fun(list(A),bool),fun(list(A),bool)),combc(list(A),fun(list(A),bool),bool),member(list(A))),XS)))))) ) ).
tff(fact_88_neq__Nil__conv,axiom,
! [A: $tType,Xsa: list(A)] :
( ( Xsa != nil(A) )
<=> ? [Y1: A,Ys1: list(A)] : ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y1),Ys1) ) ) ).
tff(fact_89_listset_Osimps_I2_J,axiom,
! [A: $tType,As: list(fun(A,bool)),A2: fun(A,bool)] : ( listset(A,aa(list(fun(A,bool)),list(fun(A,bool)),aa(fun(A,bool),fun(list(fun(A,bool)),list(fun(A,bool))),cons(fun(A,bool)),A2),As)) = set_Cons(A,A2,listset(A,As)) ) ).
tff(fact_90_list_Oexhaust,axiom,
! [A: $tType,Y: list(A)] :
( ( Y != nil(A) )
=> ~ ! [A1: A,List: list(A)] : ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A1),List) ) ) ).
tff(fact_91_insert__Nil,axiom,
! [A: $tType,X: A] : ( insert(A,X,nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) ).
tff(fact_92_list__ex1__simps_I2_J,axiom,
! [A: $tType,Xsa: list(A),Xa: A,P1: fun(A,bool)] :
( list_ex1(A,P1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))
<=> ( ( pp(aa(A,bool,P1,Xa))
=> list_all(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fdisj),aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),combb(bool,bool,A),fNot),P1))),aa(A,fun(A,bool),fequal(A),Xa)),Xsa) )
& ( ~ pp(aa(A,bool,P1,Xa))
=> list_ex1(A,P1,Xsa) ) ) ) ).
tff(fact_93_list__all__simps_I1_J,axiom,
! [A: $tType,Xsa: list(A),Xa: A,P1: fun(A,bool)] :
( list_all(A,P1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))
<=> ( pp(aa(A,bool,P1,Xa))
& list_all(A,P1,Xsa) ) ) ).
tff(fact_94_list__all__simps_I2_J,axiom,
! [A: $tType,P1: fun(A,bool)] : list_all(A,P1,nil(A)) ).
tff(fact_95_list__all__iff,axiom,
! [A: $tType,Xsa: list(A),P1: fun(A,bool)] :
( list_all(A,P1,Xsa)
<=> ! [X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),set(A,Xsa)))
=> pp(aa(A,bool,P1,X2)) ) ) ).
tff(fact_96_in__set__insert,axiom,
! [A: $tType,Xsa: list(A),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,Xsa)))
=> ( insert(A,Xa,Xsa) = Xsa ) ) ).
tff(fact_97_not__in__set__insert,axiom,
! [A: $tType,Xsa: list(A),Xa: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,Xsa)))
=> ( insert(A,Xa,Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) ) ) ).
tff(fact_98_List_Oinsert__def,axiom,
! [A: $tType,Xsa: list(A),Xa: A] :
( ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,Xsa)))
=> ( insert(A,Xa,Xsa) = Xsa ) )
& ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,Xsa)))
=> ( insert(A,Xa,Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) ) ) ) ).
tff(fact_99_list__all__cong,axiom,
! [A: $tType,G: fun(A,bool),F: fun(A,bool),Ys: list(A),Xsa: list(A)] :
( ( Xsa = Ys )
=> ( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),set(A,Ys)))
=> ( pp(aa(A,bool,F,X1))
<=> pp(aa(A,bool,G,X1)) ) )
=> ( list_all(A,F,Xsa)
<=> list_all(A,G,Ys) ) ) ) ).
%----Arities (14)
tff(arity_fun___Finite__Set_Ofinite,axiom,
! [T_1: $tType,T_2: $tType] :
( ( finite_finite(T_2)
& finite_finite(T_1) )
=> finite_finite(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Obot,axiom,
! [T_1: $tType,T_2: $tType] :
( bot(T_2)
=> bot(fun(T_1,T_2)) ) ).
tff(arity_fun___Groups_Ominus,axiom,
! [T_1: $tType,T_2: $tType] :
( cl_Groups_Ominus(T_2)
=> cl_Groups_Ominus(fun(T_1,T_2)) ) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Groups_Ogroup__add,axiom,
group_add(int) ).
tff(arity_Int_Oint___Groups_Ominus,axiom,
cl_Groups_Ominus(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Rings_Oring,axiom,
ring(int) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Finite__Set_Ofinite,axiom,
finite_finite(bool) ).
tff(arity_HOL_Obool___Orderings_Obot,axiom,
bot(bool) ).
tff(arity_HOL_Obool___Groups_Ominus,axiom,
cl_Groups_Ominus(bool) ).
tff(arity_List_Olist___Groups_Ominus,axiom,
! [T_1: $tType] :
( ( zero(T_1)
& cl_Groups_Ominus(T_1) )
=> cl_Groups_Ominus(list(T_1)) ) ).
tff(arity_prod___Finite__Set_Ofinite,axiom,
! [T_1: $tType,T_2: $tType] :
( ( finite_finite(T_2)
& finite_finite(T_1) )
=> finite_finite(product_prod(T_1,T_2)) ) ).
%----Helper facts (19)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_fEx_1_1_U,axiom,
! [A: $tType,X: A,P: fun(A,bool)] :
( ~ pp(aa(A,bool,P,X))
| pp(aa(fun(A,bool),bool,fEx(A),P)) ) ).
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ pp(aa(bool,bool,fNot,P))
| ~ pp(P) ) ).
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( pp(P)
| pp(aa(bool,bool,fNot,P)) ) ).
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,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(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,aa(B,fun(A,C),aa(fun(A,fun(B,C)),fun(B,fun(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 ) ).
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,aa(fun(A,B),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,B),fun(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_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(Q) ) ).
tff(help_fdisj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q))
| pp(P)
| pp(Q) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_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,
collect(int,aa(fun(int,fun(list(int),bool)),fun(int,bool),aa(fun(fun(list(int),bool),bool),fun(fun(int,fun(list(int),bool)),fun(int,bool)),combb(fun(list(int),bool),bool,int),fEx(list(int))),aa(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),bool)),aa(fun(fun(list(int),fun(int,bool)),fun(list(int),bool)),fun(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),bool))),combb(fun(list(int),fun(int,bool)),fun(list(int),bool),int),aa(fun(fun(int,bool),bool),fun(fun(list(int),fun(int,bool)),fun(list(int),bool)),combb(fun(int,bool),bool,list(int)),fEx(int))),aa(fun(list(int),fun(int,bool)),fun(int,fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),fun(fun(list(int),fun(int,bool)),fun(int,fun(list(int),fun(int,bool)))),combc(int,fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))),aa(fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),aa(fun(fun(list(int),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),fun(fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))))),combb(fun(list(int),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))),int),combs(list(int),fun(int,bool),fun(int,bool))),aa(fun(int,fun(list(int),fun(int,fun(bool,bool)))),fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),aa(fun(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(fun(int,fun(list(int),fun(int,fun(bool,bool)))),fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool))))),combb(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool))),int),aa(fun(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool)))),combb(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),list(int)),combs(int,bool,bool))),aa(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),fun(int,fun(bool,bool)))),aa(fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool)))),fun(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),fun(int,fun(bool,bool))))),combb(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool))),int),aa(fun(fun(int,bool),fun(int,fun(bool,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool)))),combb(fun(int,bool),fun(int,fun(bool,bool)),list(int)),aa(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj))),aa(fun(list(int),fun(int,int)),fun(int,fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),fun(fun(list(int),fun(int,int)),fun(int,fun(list(int),fun(int,bool)))),combc(int,fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(int,int),fun(int,bool))),fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),aa(fun(fun(fun(int,int),fun(int,bool)),fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),fun(fun(int,fun(fun(int,int),fun(int,bool))),fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))))),combb(fun(fun(int,int),fun(int,bool)),fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))),int),combb(fun(int,int),fun(int,bool),list(int))),aa(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool))),aa(fun(fun(int,bool),fun(fun(int,int),fun(int,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool)))),combb(fun(int,bool),fun(fun(int,int),fun(int,bool)),int),combb(int,bool,int)),fequal(int)))),aa(fun(list(int),int),fun(list(int),fun(int,int)),aa(fun(int,fun(int,int)),fun(fun(list(int),int),fun(list(int),fun(int,int))),combb(int,fun(int,int),list(int)),aa(fun(int,fun(int,int)),fun(int,fun(int,int)),combc(int,int,int),minus_minus(int))),aa(list(int),fun(list(int),int),aa(fun(list(int),fun(list(int),int)),fun(list(int),fun(list(int),int)),combc(list(int),list(int),int),iprod(int)),xs))))))),aa(fun(product_prod(int,list(int)),bool),fun(list(int),fun(int,bool)),aa(fun(list(int),fun(fun(product_prod(int,list(int)),bool),fun(int,bool))),fun(fun(product_prod(int,list(int)),bool),fun(list(int),fun(int,bool))),combc(list(int),fun(product_prod(int,list(int)),bool),fun(int,bool)),aa(fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),fun(list(int),fun(fun(product_prod(int,list(int)),bool),fun(int,bool))),aa(fun(fun(int,fun(fun(product_prod(int,list(int)),bool),bool)),fun(fun(product_prod(int,list(int)),bool),fun(int,bool))),fun(fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),fun(list(int),fun(fun(product_prod(int,list(int)),bool),fun(int,bool)))),combb(fun(int,fun(fun(product_prod(int,list(int)),bool),bool)),fun(fun(product_prod(int,list(int)),bool),fun(int,bool)),list(int)),combc(int,fun(product_prod(int,list(int)),bool),bool)),aa(fun(list(int),fun(int,product_prod(int,list(int)))),fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),aa(fun(fun(int,product_prod(int,list(int))),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),fun(fun(list(int),fun(int,product_prod(int,list(int)))),fun(list(int),fun(int,fun(fun(product_prod(int,list(int)),bool),bool)))),combb(fun(int,product_prod(int,list(int))),fun(int,fun(fun(product_prod(int,list(int)),bool),bool)),list(int)),aa(fun(product_prod(int,list(int)),fun(fun(product_prod(int,list(int)),bool),bool)),fun(fun(int,product_prod(int,list(int))),fun(int,fun(fun(product_prod(int,list(int)),bool),bool))),combb(product_prod(int,list(int)),fun(fun(product_prod(int,list(int)),bool),bool),int),member(product_prod(int,list(int))))),aa(fun(int,fun(list(int),product_prod(int,list(int)))),fun(list(int),fun(int,product_prod(int,list(int)))),combc(int,list(int),product_prod(int,list(int))),product_Pair(int,list(int)))))),set(product_prod(int,list(int)),lbounds(as))))))) != bot_bot(fun(int,bool)) ).
%------------------------------------------------------------------------------