TPTP Problem File: COM089_5.p
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%------------------------------------------------------------------------------
% File : COM089_5 : TPTP v9.0.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 213
% 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_213 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 175 ( 46 unt; 47 typ; 0 def)
% Number of atoms : 252 ( 80 equ)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 176 ( 52 ~; 15 |; 17 &)
% ( 15 <=>; 77 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 19 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 27 ( 17 >; 10 *; 0 +; 0 <<)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-3 aty)
% Number of functors : 35 ( 35 usr; 8 con; 0-5 aty)
% Number of variables : 607 ( 547 !; 10 ?; 607 :)
% ( 50 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:22:35
%------------------------------------------------------------------------------
%----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 (41)
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_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[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_COMBI,type,
combi:
!>[A: $tType] : fun(A,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_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__ex1,type,
list_ex1:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
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_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) > fun(A,fun(B,C)) ) ).
tff(sy_c_Product__Type_Ointernal__split,type,
produc1605651328_split:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).
tff(sy_c_Product__Type_Oprod_Oprod__case,type,
product_prod_case:
!>[A: $tType,B: $tType,T: $tType] : fun(fun(A,fun(B,T)),fun(product_prod(A,B),T)) ).
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_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,bool) ) > fun(B,bool) ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_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,image(product_prod(int,list(int)),int,aa(fun(int,fun(list(int),int)),fun(product_prod(int,list(int)),int),product_prod_case(int,list(int),int),aa(fun(list(int),int),fun(int,fun(list(int),int)),aa(fun(int,fun(fun(list(int),int),fun(list(int),int))),fun(fun(list(int),int),fun(int,fun(list(int),int))),combc(int,fun(list(int),int),fun(list(int),int)),aa(fun(int,fun(int,int)),fun(int,fun(fun(list(int),int),fun(list(int),int))),aa(fun(fun(int,int),fun(fun(list(int),int),fun(list(int),int))),fun(fun(int,fun(int,int)),fun(int,fun(fun(list(int),int),fun(list(int),int)))),combb(fun(int,int),fun(fun(list(int),int),fun(list(int),int)),int),combb(int,int,list(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))),set(product_prod(int,list(int)),lbounds(as)))) ).
tff(fact_2_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C)),A1: fun(product_prod(A,B),bool),B1: B,A2: A] :
( 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),A2),B1)),A1))
=> pp(aa(fun(C,bool),bool,aa(C,fun(fun(C,bool),bool),member(C),aa(B,C,aa(A,fun(B,C),F,A2),B1)),image(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),F),A1))) ) ).
tff(fact_3_split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,B1: C,A2: B,F: fun(B,fun(C,A))] : ( aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B1)) = aa(C,A,aa(B,fun(C,A),F,A2),B1) ) ).
tff(fact_4_image__eqI,axiom,
! [A: $tType,B: $tType,A1: fun(B,bool),Xa: B,F: fun(B,A),B1: A] :
( ( B1 = aa(B,A,F,Xa) )
=> ( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),Xa),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B1),image(B,A,F,A1))) ) ) ).
tff(fact_5_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_6_Pair__eq,axiom,
! [A: $tType,B: $tType,B7: B,A4: A,B1: B,A2: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B7) )
<=> ( ( A2 = A4 )
& ( B1 = B7 ) ) ) ).
tff(fact_7_split__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),aa(fun(A,fun(B,product_prod(A,B))),fun(A,fun(B,C)),aa(fun(fun(B,product_prod(A,B)),fun(B,C)),fun(fun(A,fun(B,product_prod(A,B))),fun(A,fun(B,C))),combb(fun(B,product_prod(A,B)),fun(B,C),A),aa(fun(product_prod(A,B),C),fun(fun(B,product_prod(A,B)),fun(B,C)),combb(product_prod(A,B),C,B),F)),product_Pair(A,B))) = F ) ).
tff(fact_8_iprod__left__diff__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [Zs: list(A),Ys1: 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),Ys1)),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),Ys1),Zs)) ) ) ).
tff(fact_9_prod_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType,B1: C,A2: B,F1: fun(B,fun(C,A))] : ( aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),F1),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B1)) = aa(C,A,aa(B,fun(C,A),F1,A2),B1) ) ).
tff(fact_10_split__twice,axiom,
! [B: $tType,C: $tType,A: $tType,E: $tType,D: $tType,P3: product_prod(D,E),G: fun(D,fun(E,product_prod(B,C))),F: fun(B,fun(C,A))] : ( aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),F),aa(product_prod(D,E),product_prod(B,C),aa(fun(D,fun(E,product_prod(B,C))),fun(product_prod(D,E),product_prod(B,C)),product_prod_case(D,E,product_prod(B,C)),G),P3)) = aa(product_prod(D,E),A,aa(fun(D,fun(E,A)),fun(product_prod(D,E),A),product_prod_case(D,E,A),aa(fun(D,fun(E,product_prod(B,C))),fun(D,fun(E,A)),aa(fun(fun(E,product_prod(B,C)),fun(E,A)),fun(fun(D,fun(E,product_prod(B,C))),fun(D,fun(E,A))),combb(fun(E,product_prod(B,C)),fun(E,A),D),aa(fun(product_prod(B,C),A),fun(fun(E,product_prod(B,C)),fun(E,A)),combb(product_prod(B,C),A,E),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),F))),G)),P3) ) ).
tff(fact_11_image__image,axiom,
! [B: $tType,A: $tType,C: $tType,A1: fun(C,bool),G: fun(C,B),F: fun(B,A)] : ( image(B,A,F,image(C,B,G,A1)) = image(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),combb(B,A,C),F),G),A1) ) ).
tff(fact_12_image__ident,axiom,
! [A: $tType,Y4: fun(A,bool)] : ( image(A,A,combi(A),Y4) = Y4 ) ).
tff(fact_13_prod__caseI,axiom,
! [A: $tType,B: $tType,B1: B,A2: A,F1: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),F1,A2),B1))
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),F1),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1))) ) ).
tff(fact_14_splitI,axiom,
! [A: $tType,B: $tType,B1: B,A2: A,F: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),F,A2),B1))
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1))) ) ).
tff(fact_15_mem__splitI,axiom,
! [A: $tType,B: $tType,C: $tType,B1: C,A2: B,C2: fun(B,fun(C,fun(A,bool))),Z: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Z),aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),C2,A2),B1)))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Z),aa(product_prod(B,C),fun(A,bool),aa(fun(B,fun(C,fun(A,bool))),fun(product_prod(B,C),fun(A,bool)),product_prod_case(B,C,fun(A,bool)),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B1)))) ) ).
tff(fact_16_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_17_splitD,axiom,
! [A: $tType,B: $tType,B1: B,A2: A,F: fun(A,fun(B,bool))] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1)))
=> pp(aa(B,bool,aa(A,fun(B,bool),F,A2),B1)) ) ).
tff(fact_18_splitD_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: C,B1: B,A2: A,R1: fun(A,fun(B,fun(C,bool)))] :
( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_prod_case(A,B,fun(C,bool)),R1),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1)),C2))
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),R1,A2),B1),C2)) ) ).
tff(fact_19_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_20_rev__image__eqI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),B1: B,A1: fun(A,bool),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),A1))
=> ( ( B1 = aa(A,B,F,Xa) )
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),B1),image(A,B,F,A1))) ) ) ).
tff(fact_21_imageI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A1: fun(A,bool),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),A1))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),aa(A,B,F,Xa)),image(A,B,F,A1))) ) ).
tff(fact_22_image__iff,axiom,
! [A: $tType,B: $tType,A1: fun(B,bool),F: fun(B,A),Z: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Z),image(B,A,F,A1)))
<=> ? [X3: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X3),A1))
& ( Z = aa(B,A,F,X3) ) ) ) ).
tff(fact_23_split__weak__cong,axiom,
! [C: $tType,B: $tType,A: $tType,C2: fun(A,fun(B,C)),Q2: product_prod(A,B),P3: product_prod(A,B)] :
( ( P3 = Q2 )
=> ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),C2),P3) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),C2),Q2) ) ) ).
tff(fact_24_finite__imageI,axiom,
! [B: $tType,A: $tType,H: fun(A,B),F5: fun(A,bool)] :
( finite_finite1(A,F5)
=> finite_finite1(B,image(A,B,H,F5)) ) ).
tff(fact_25_finite__set,axiom,
! [A: $tType,Xsa: list(A)] : finite_finite1(A,set(A,Xsa)) ).
tff(fact_26_imageE,axiom,
! [A: $tType,B: $tType,A1: fun(B,bool),F: fun(B,A),B1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B1),image(B,A,F,A1)))
=> ~ ! [X1: B] :
( ( B1 = aa(B,A,F,X1) )
=> ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X1),A1)) ) ) ).
tff(fact_27_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_28_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_29_splitE,axiom,
! [A: $tType,B: $tType,P3: product_prod(A,B),C2: fun(A,fun(B,bool))] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),C2),P3))
=> ~ ! [X1: A,Y3: B] :
( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),Y3) )
=> ~ pp(aa(B,bool,aa(A,fun(B,bool),C2,X1),Y3)) ) ) ).
tff(fact_30_splitI2,axiom,
! [B: $tType,A: $tType,C2: fun(A,fun(B,bool)),P3: product_prod(A,B)] :
( ! [A5: A,B2: B] :
( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2) )
=> pp(aa(B,bool,aa(A,fun(B,bool),C2,A5),B2)) )
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),C2),P3)) ) ).
tff(fact_31_splitE_H,axiom,
! [B: $tType,A: $tType,C: $tType,Z: C,P3: product_prod(A,B),C2: fun(A,fun(B,fun(C,bool)))] :
( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_prod_case(A,B,fun(C,bool)),C2),P3),Z))
=> ~ ! [X1: A,Y3: B] :
( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),Y3) )
=> ~ pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C2,X1),Y3),Z)) ) ) ).
tff(fact_32_mem__splitE,axiom,
! [B: $tType,A: $tType,C: $tType,P3: product_prod(B,C),C2: fun(B,fun(C,fun(A,bool))),Z: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Z),aa(product_prod(B,C),fun(A,bool),aa(fun(B,fun(C,fun(A,bool))),fun(product_prod(B,C),fun(A,bool)),product_prod_case(B,C,fun(A,bool)),C2),P3)))
=> ~ ! [X1: B,Y3: C] :
( ( P3 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X1),Y3) )
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Z),aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),C2,X1),Y3))) ) ) ).
tff(fact_33_finite__Diff,axiom,
! [A: $tType,B3: fun(A,bool),A1: fun(A,bool)] :
( finite_finite1(A,A1)
=> 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)),A1),B3)) ) ).
tff(fact_34_finite__code,axiom,
! [A: $tType] :
( finite_finite(A)
=> ! [A1: fun(A,bool)] : finite_finite1(A,A1) ) ).
tff(fact_35_split__part,axiom,
! [B: $tType,A: $tType,Q1: fun(A,fun(B,bool)),P1: bool,X2: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(fun(B,bool),fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),combb(fun(B,bool),fun(B,bool),A),aa(fun(bool,bool),fun(fun(B,bool),fun(B,bool)),combb(bool,bool,B),aa(bool,fun(bool,bool),fconj,P1))),Q1)),X2))
<=> ( pp(P1)
& pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),Q1),X2)) ) ) ).
tff(fact_36_finite__Diff2,axiom,
! [A: $tType,A1: 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)),A1),B3))
<=> finite_finite1(A,A1) ) ) ).
tff(fact_37_finite,axiom,
! [A: $tType] :
( finite_finite(A)
=> ! [A1: fun(A,bool)] : finite_finite1(A,A1) ) ).
tff(fact_38_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_39_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_40_mem__splitI2,axiom,
! [C: $tType,B: $tType,A: $tType,C2: fun(A,fun(B,fun(C,bool))),Z: C,P3: product_prod(A,B)] :
( ! [A5: A,B2: B] :
( ( P3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2) )
=> pp(aa(fun(C,bool),bool,aa(C,fun(fun(C,bool),bool),member(C),Z),aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C2,A5),B2))) )
=> pp(aa(fun(C,bool),bool,aa(C,fun(fun(C,bool),bool),member(C),Z),aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_prod_case(A,B,fun(C,bool)),C2),P3))) ) ).
tff(fact_41_splitI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,Xa: C,C2: fun(A,fun(B,fun(C,bool))),P3: product_prod(A,B)] :
( ! [A5: A,B2: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2) = P3 )
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C2,A5),B2),Xa)) )
=> pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_prod_case(A,B,fun(C,bool)),C2),P3),Xa)) ) ).
tff(fact_42_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_43_cond__split__eta,axiom,
! [C: $tType,B: $tType,A: $tType,G: fun(product_prod(A,B),C),F: fun(A,fun(B,C))] :
( ! [X1: A,Y3: B] : ( aa(B,C,aa(A,fun(B,C),F,X1),Y3) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),Y3)) )
=> ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),F) = G ) ) ).
tff(fact_44_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A1: fun(A,bool)] :
( ~ finite_finite1(A,A1)
=> ( finite_finite1(B,image(A,B,F,A1))
=> ? [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),A1))
& ~ 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),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)),A1))),aa(B,fun(A,bool),aa(fun(A,fun(B,bool)),fun(B,fun(A,bool)),combc(A,B,bool),aa(fun(A,B),fun(A,fun(B,bool)),aa(fun(B,fun(B,bool)),fun(fun(A,B),fun(A,fun(B,bool))),combb(B,fun(B,bool),A),fequal(B)),F)),aa(A,B,F,X1))))) ) ) ) ).
tff(fact_45_splitE2,axiom,
! [B: $tType,A: $tType,C: $tType,Z: product_prod(B,C),P1: fun(B,fun(C,A)),Q1: fun(A,bool)] :
( pp(aa(A,bool,Q1,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),P1),Z)))
=> ~ ! [X1: B,Y3: C] :
( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X1),Y3) )
=> ~ pp(aa(A,bool,Q1,aa(C,A,aa(B,fun(C,A),P1,X1),Y3))) ) ) ).
tff(fact_46_DiffE,axiom,
! [A: $tType,B3: fun(A,bool),A1: fun(A,bool),C2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A1),B3)))
=> ~ ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B3)) ) ) ).
tff(fact_47_DiffI,axiom,
! [A: $tType,B3: fun(A,bool),A1: fun(A,bool),C2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A1))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B3))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A1),B3))) ) ) ).
tff(fact_48_Diff__idemp,axiom,
! [A: $tType,B3: fun(A,bool),A1: 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)),A1),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)),A1),B3) ) ).
tff(fact_49_Diff__iff,axiom,
! [A: $tType,B3: fun(A,bool),A1: fun(A,bool),C2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A1),B3)))
<=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A1))
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B3)) ) ) ).
tff(fact_50_DiffD2,axiom,
! [A: $tType,B3: fun(A,bool),A1: fun(A,bool),C2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A1),B3)))
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),B3)) ) ).
tff(fact_51_DiffD1,axiom,
! [A: $tType,B3: fun(A,bool),A1: fun(A,bool),C2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A1),B3)))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C2),A1)) ) ).
tff(fact_52_set__diff__eq,axiom,
! [A: $tType,B3: fun(A,bool),A1: 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)),A1),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)),A1))),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_53_finite__list,axiom,
! [A: $tType,A1: fun(A,bool)] :
( finite_finite1(A,A1)
=> ? [Xs1: list(A)] : ( set(A,Xs1) = A1 ) ) ).
tff(fact_54_prod_Orecs,axiom,
! [B: $tType,A: $tType,C: $tType,B1: C,A2: 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),A2),B1)) = aa(C,A,aa(B,fun(C,A),F1,A2),B1) ) ).
tff(fact_55_minus__apply,axiom,
! [A: $tType,B: $tType] :
( cl_Groups_Ominus(A)
=> ! [Xa: B,B3: fun(B,A),A1: 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)),A1),B3),Xa) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,A1,Xa)),aa(B,A,B3,Xa)) ) ) ).
tff(fact_56_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( cl_Groups_Ominus(B)
=> ! [B3: fun(A,B),A1: fun(A,B),X2: 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)),A1),B3),X2) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A1,X2)),aa(A,B,B3,X2)) ) ) ).
tff(fact_57_internal__split__def,axiom,
! [C: $tType,B: $tType,A: $tType] : ( produc1605651328_split(A,B,C) = product_prod_case(A,B,C) ) ).
tff(fact_58_image__cong,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),N: fun(A,bool),M: fun(A,bool)] :
( ( M = N )
=> ( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),N))
=> ( aa(A,B,F,X1) = aa(A,B,G,X1) ) )
=> ( image(A,B,F,M) = image(A,B,G,N) ) ) ) ).
tff(fact_59_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_60_internal__split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,B1: C,A2: B,C2: fun(B,fun(C,A))] : ( aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),produc1605651328_split(B,C,A),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B1)) = aa(C,A,aa(B,fun(C,A),C2,A2),B1) ) ).
tff(fact_61_prod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod(A,B)] :
~ ! [A5: A,B2: B] : ( Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B2) ) ).
tff(fact_62_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_63_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
~ ! [A5: A,B2: B,C1: 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)),A5),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C1)) ) ).
tff(fact_64_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)] :
( ! [A5: A,B2: B,C1: 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))))),A5),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))),C1),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_65_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)))))] :
~ ! [A5: A,B2: B,C1: 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))))),A5),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))),C1),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_66_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)] :
( ! [A5: A,B2: B,C1: 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)))),A5),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)),C1),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_67_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))))] :
~ ! [A5: A,B2: B,C1: 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)))),A5),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)),C1),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D1),E1)))) ) ).
tff(fact_68_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A5: A,B2: B,C1: 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))),A5),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),C1),D1))) ) ).
tff(fact_69_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)] :
( ! [A5: A,B2: B,C1: 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))),A5),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),C1),D1)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P1,Xa)) ) ).
tff(fact_70_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)] :
( ! [A5: A,B2: B,C1: 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)),A5),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C1))))
=> pp(aa(product_prod(A,product_prod(B,C)),bool,P1,Xa)) ) ).
tff(fact_71_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,S: fun(product_prod(A,B),bool),R1: fun(product_prod(A,B),bool)] :
( ! [X3: 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),X3),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),X3),Xa2)),S)) )
<=> ( R1 = S ) ) ).
tff(fact_72_curry__conv,axiom,
! [A: $tType,B: $tType,C: $tType,B1: C,A2: B,F: fun(product_prod(B,C),A)] : ( aa(C,A,aa(B,fun(C,A),product_curry(B,C,A,F),A2),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),A2),B1)) ) ).
tff(fact_73_curryI,axiom,
! [A: $tType,B: $tType,B1: B,A2: 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),A2),B1)))
=> pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),A2),B1)) ) ).
tff(fact_74_curryE,axiom,
! [A: $tType,B: $tType,B1: B,A2: A,F: fun(product_prod(A,B),bool)] :
( pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),A2),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),A2),B1))) ) ).
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,A1: fun(A,bool),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),A1))
<=> pp(aa(A,bool,A1,Xa)) ) ).
tff(fact_77_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : ( collect(A,P1) = P1 ) ).
tff(fact_78_curryD,axiom,
! [A: $tType,B: $tType,B1: B,A2: A,F: fun(product_prod(A,B),bool)] :
( pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),A2),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),A2),B1))) ) ).
tff(fact_79_split__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),product_curry(A,B,C,F)) = F ) ).
tff(fact_80_curry__split,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C))] : ( product_curry(A,B,C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),F)) = F ) ).
tff(fact_81_curry__def,axiom,
! [C: $tType,A: $tType,B: $tType,X2: fun(product_prod(A,B),C),Xa1: A,Xb: B] : ( aa(B,C,aa(A,fun(B,C),product_curry(A,B,C,X2),Xa1),Xb) = aa(product_prod(A,B),C,X2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa1),Xb)) ) ).
tff(fact_82_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_83_list__ex1__simps_I1_J,axiom,
! [A: $tType,P1: fun(A,bool)] : ~ list_ex1(A,P1,nil(A)) ).
tff(fact_84_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_85_list__ex1__iff,axiom,
! [A: $tType,Xsa: list(A),P1: fun(A,bool)] :
( list_ex1(A,P1,Xsa)
<=> ? [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),set(A,Xsa)))
& pp(aa(A,bool,P1,X3))
& ! [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 = X3 ) ) ) ) ).
tff(fact_86_x,axiom,
! [X2: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X2),set(atom,as)))
=> i_Z(X2,aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),x),xs)) ) ).
tff(fact_87__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_88_list_Oinject,axiom,
! [A: $tType,List2: list(A),A4: A,List: list(A),A2: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),List) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),List2) )
<=> ( ( A2 = A4 )
& ( List = List2 ) ) ) ).
tff(fact_89_zipwith0_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( zero(C)
& zero(B) )
=> ! [Ys2: 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),Ys2)) = 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),Ys2)) ) ) ).
tff(fact_90_list__diff__Cons__Cons,axiom,
! [A: $tType] :
( ( cl_Groups_Ominus(A)
& zero(A) )
=> ! [Ys1: 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),Ys1)) = 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),Ys1)) ) ) ).
tff(fact_91__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_92_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_93_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_94_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_95_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_96_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_97_list_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,List: list(B),A2: 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),A2),List)) = aa(list(B),A,aa(B,fun(list(B),A),F2,A2),List) ) ).
tff(fact_98_set__Cons__def,axiom,
! [A: $tType,XS: fun(list(A),bool),A1: fun(A,bool)] : ( set_Cons(A,A1,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)),A1)))),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_99_neq__Nil__conv,axiom,
! [A: $tType,Xsa: list(A)] :
( ( Xsa != nil(A) )
<=> ? [Y1: A,Ys: list(A)] : ( Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y1),Ys) ) ) ).
%----Arities (10)
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___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___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___Finite__Set_Ofinite,axiom,
finite_finite(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 (17)
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_COMBI_1_1_U,axiom,
! [A: $tType,P: A] : ( aa(A,A,combi(A),P) = 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_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,
image(product_prod(int,list(int)),int,aa(fun(int,fun(list(int),int)),fun(product_prod(int,list(int)),int),product_prod_case(int,list(int),int),aa(fun(list(int),int),fun(int,fun(list(int),int)),aa(fun(int,fun(fun(list(int),int),fun(list(int),int))),fun(fun(list(int),int),fun(int,fun(list(int),int))),combc(int,fun(list(int),int),fun(list(int),int)),aa(fun(int,fun(int,int)),fun(int,fun(fun(list(int),int),fun(list(int),int))),aa(fun(fun(int,int),fun(fun(list(int),int),fun(list(int),int))),fun(fun(int,fun(int,int)),fun(int,fun(fun(list(int),int),fun(list(int),int)))),combb(fun(int,int),fun(fun(list(int),int),fun(list(int),int)),int),combb(int,int,list(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))),set(product_prod(int,list(int)),lbounds(as))) = 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))))))) ).
%------------------------------------------------------------------------------