TPTP Problem File: COM060_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : COM060_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 130
% 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_130 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 194 ( 54 unt; 59 typ; 0 def)
% Number of atoms : 267 ( 87 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 167 ( 35 ~; 16 |; 11 &)
% ( 31 <=>; 74 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Maximal term depth : 19 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 37 ( 24 >; 13 *; 0 +; 0 <<)
% Number of predicates : 14 ( 13 usr; 0 prp; 1-5 aty)
% Number of functors : 42 ( 42 usr; 9 con; 0-4 aty)
% Number of variables : 373 ( 326 !; 3 ?; 373 :)
% ( 44 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:18:54
%------------------------------------------------------------------------------
%----Should-be-implicit typings (8)
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_Logic_Ofm,type,
fm: $tType > $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $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 (51)
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_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Finite__Set_Ofinite,type,
finite_finite1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[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] : fun(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_finite:
!>[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_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
tff(sy_c_ListVector_Oiprod,type,
iprod:
!>[A: $tType] : fun(list(A),fun(list(A),A)) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olist_Olist__case,type,
list_case:
!>[T1: $tType,A: $tType] : ( ( T1 * fun(A,fun(list(A),T1)) ) > fun(list(A),T1) ) ).
tff(sy_c_List_Oset,type,
set:
!>[A: $tType] : ( list(A) > fun(A,bool) ) ).
tff(sy_c_List_Otl,type,
tl:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_Logic_Ointerpret,type,
interpret:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(list(B),bool)) * fm(A) * list(B) ) > $o ) ).
tff(sy_c_Logic_Olist__conj,type,
list_conj:
!>[A: $tType] : ( list(fm(A)) > fm(A) ) ).
tff(sy_c_Logic_Olist__disj,type,
list_disj:
!>[A: $tType] : ( list(fm(A)) > fm(A) ) ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_PresArith_OI_092_060_094isub_062Z,type,
i_Z: fun(atom,fun(list(int),bool)) ).
tff(sy_c_PresArith_Oatom_ODvd,type,
dvd: ( int * int * list(int) ) > atom ).
tff(sy_c_PresArith_Oatom_OLe,type,
c_PresArith_Oatom_OLe: fun(int,fun(list(int),atom)) ).
tff(sy_c_PresArith_Oatom_ONDvd,type,
nDvd: ( int * int * list(int) ) > atom ).
tff(sy_c_PresArith_Oatom_Oatom__case,type,
atom_case:
!>[T1: $tType] : ( ( fun(int,fun(list(int),T1)) * fun(int,fun(int,fun(list(int),T1))) * fun(int,fun(int,fun(list(int),T1))) ) > fun(atom,T1) ) ).
tff(sy_c_PresArith_Oatom_Oatom__size,type,
atom_size: atom > nat ).
tff(sy_c_PresArith_Odecr_092_060_094isub_062Z,type,
decr_Z: atom > atom ).
tff(sy_c_PresArith_Odivisor,type,
divisor: atom > int ).
tff(sy_c_PresArith_Olbounds,type,
lbounds: list(atom) > list(product_prod(int,list(int))) ).
tff(sy_c_QEpres__Mirabelle__iocckttzyp_Oqe__pres_092_060_094isub_0621,type,
qEpres896714165pres_1: list(atom) > fm(atom) ).
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_xs,type,
xs: list(int) ).
%----Relevant facts (99)
tff(fact_0_norm,axiom,
! [X4: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X4),set(atom,as)))
=> ( divisor(X4) != zero_zero(int) ) ) ).
tff(fact_1_finite__set,axiom,
! [A: $tType,Xsa: list(A)] : finite_finite(A,set(A,Xsa)) ).
tff(fact_2_diff__self,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),A3) = zero_zero(A) ) ).
tff(fact_3__096lbounds_Aas_A_061_A_091_093_096,axiom,
lbounds(as) = nil(product_prod(int,list(int))) ).
tff(fact_4_finite__Collect__disjI,axiom,
! [A: $tType,Q1: fun(A,bool),P1: fun(A,bool)] :
( finite_finite(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_finite(A,collect(A,P1))
& finite_finite(A,collect(A,Q1)) ) ) ).
tff(fact_5_finite__Collect__conjI,axiom,
! [A: $tType,Q1: fun(A,bool),P1: fun(A,bool)] :
( ( finite_finite(A,collect(A,P1))
| finite_finite(A,collect(A,Q1)) )
=> finite_finite(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_6_less__bin__lemma,axiom,
! [L1: int,K1: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K1),L1))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K1),L1)),zero_zero(int))) ) ).
tff(fact_7_atom_Osimps_I1_J,axiom,
! [List3: list(int),Int4: int,List2: list(int),Int3: int] :
( ( aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int3),List2) = aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int4),List3) )
<=> ( ( Int3 = Int4 )
& ( List2 = List3 ) ) ) ).
tff(fact_8_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2)),zero_zero(A))) ) ) ).
tff(fact_9_atom_Osimps_I10_J,axiom,
! [A: $tType,List2: list(int),Int3: int,F3: fun(int,fun(int,fun(list(int),A))),F2: fun(int,fun(int,fun(list(int),A))),F1: fun(int,fun(list(int),A))] : aa(atom,A,atom_case(A,F1,F2,F3),aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int3),List2)) = aa(list(int),A,aa(int,fun(list(int),A),F1,Int3),List2) ).
tff(fact_10_finite__code,axiom,
! [A: $tType] :
( finite_finite1(A)
=> ! [A1: fun(A,bool)] : finite_finite(A,A1) ) ).
tff(fact_11_finite__image__set,axiom,
! [B: $tType,A: $tType,F: fun(A,B),P1: fun(A,bool)] :
( finite_finite(A,collect(A,P1))
=> finite_finite(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_12_finite__Diff,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( finite_finite(A,A1)
=> finite_finite(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),B1)) ) ).
tff(fact_13_tl_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(A),list(A),tl(A),nil(A)) = nil(A) ).
tff(fact_14_finite__Diff2,axiom,
! [A: $tType,A1: fun(A,bool),B1: fun(A,bool)] :
( finite_finite(A,B1)
=> ( finite_finite(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),B1))
<=> finite_finite(A,A1) ) ) ).
tff(fact_15_list_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,fun(list(B),A)),F1: A] : aa(list(B),A,list_case(A,B,F1,F2),nil(B)) = F1 ).
tff(fact_16_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X3: A] :
( ( zero_zero(A) = X3 )
<=> ( X3 = zero_zero(A) ) ) ) ).
tff(fact_17_finite,axiom,
! [A: $tType] :
( finite_finite1(A)
=> ! [A1: fun(A,bool)] : finite_finite(A,A1) ) ).
tff(fact_18_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [D1: A,C1: A,B2: A,A2: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C1),D1) )
=> ( ( A2 = B2 )
<=> ( C1 = D1 ) ) ) ) ).
tff(fact_19_right__minus__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B2: A,A2: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = zero_zero(A) )
<=> ( A2 = B2 ) ) ) ).
tff(fact_20_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B2: A,A2: A] :
( ( A2 = B2 )
<=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = zero_zero(A) ) ) ) ).
tff(fact_21_diff__0__right,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A3: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),zero_zero(A)) = A3 ) ).
tff(fact_22_diff__eq__diff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [D1: A,C1: A,B2: A,A2: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C1),D1) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C1),D1)) ) ) ) ).
tff(fact_23_finite__Collect__bounded__ex,axiom,
! [B: $tType,A: $tType,Q1: fun(B,fun(A,bool)),P1: fun(A,bool)] :
( finite_finite(A,collect(A,P1))
=> ( finite_finite(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_finite(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_24_iprod__Nil2,axiom,
! [A: $tType] :
( ring(A)
=> ! [Xs: list(A)] : aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Xs),nil(A)) = zero_zero(A) ) ).
tff(fact_25_iprod__Nil,axiom,
! [A: $tType] :
( ring(A)
=> ! [Ys: list(A)] : aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),nil(A)),Ys) = zero_zero(A) ) ).
tff(fact_26_atom_Osize_I1_J,axiom,
! [List: list(int),Int: int] : atom_size(aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int),List)) = zero_zero(nat) ).
tff(fact_27_finite__list,axiom,
! [A: $tType,A1: fun(A,bool)] :
( finite_finite(A,A1)
=> ? [Xs1: list(A)] : set(A,Xs1) = A1 ) ).
tff(fact_28_iprod__left__diff__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [Zs: list(A),Ys: 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),Ys)),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),Ys),Zs)) ) ).
tff(fact_29_atom_Osize_I4_J,axiom,
! [List: list(int),Int: int] : size_size(atom,aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int),List)) = zero_zero(nat) ).
tff(fact_30__096Z_OI_A_Iqe__pres_092_060_094isub_0621_Aas_J_Axs_096,axiom,
interpret(atom,int,i_Z,qEpres896714165pres_1(as),xs) ).
tff(fact_31_iprod0__if__coeffs0,axiom,
! [A: $tType] :
( ring(A)
=> ! [Xsa: list(A),Cs: list(A)] :
( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),set(A,Cs)))
=> ( X1 = zero_zero(A) ) )
=> ( aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Cs),Xsa) = zero_zero(A) ) ) ) ).
tff(fact_32_of__int__less__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),ring_1_of_int(A,Z)),zero_zero(A)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),zero_zero(int))) ) ) ).
tff(fact_33_of__int__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z: int,W: int] :
( ( ring_1_of_int(A,W) = ring_1_of_int(A,Z) )
<=> ( W = Z ) ) ) ).
tff(fact_34_of__int__0,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A,zero_zero(int)) = zero_zero(A) ) ) ).
tff(fact_35_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z: int] :
( ( zero_zero(A) = ring_1_of_int(A,Z) )
<=> ( Z = zero_zero(int) ) ) ) ).
tff(fact_36_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z: int] :
( ( ring_1_of_int(A,Z) = zero_zero(A) )
<=> ( Z = zero_zero(int) ) ) ) ).
tff(fact_37_of__int__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int,W: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),ring_1_of_int(A,W)),ring_1_of_int(A,Z)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ) ).
tff(fact_38_of__int__diff,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Z1: int,W1: int] : ring_1_of_int(A,aa(int,int,aa(int,fun(int,int),minus_minus(int),W1),Z1)) = aa(A,A,aa(A,fun(A,A),minus_minus(A),ring_1_of_int(A,W1)),ring_1_of_int(A,Z1)) ) ).
tff(fact_39_of__int__0__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),ring_1_of_int(A,Z)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) ) ) ).
tff(fact_40_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_41_Z_OI__list__conj,axiom,
! [Xsa: list(int),Fs: list(fm(atom))] :
( interpret(atom,int,i_Z,list_conj(atom,Fs),Xsa)
<=> ! [X2: fm(atom)] :
( pp(aa(fun(fm(atom),bool),bool,aa(fm(atom),fun(fun(fm(atom),bool),bool),member(fm(atom)),X2),set(fm(atom),Fs)))
=> interpret(atom,int,i_Z,X2,Xsa) ) ) ).
tff(fact_42_Diff__iff,axiom,
! [A: $tType,B1: fun(A,bool),A1: 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)),A1),B1)))
<=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A1))
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B1)) ) ) ).
tff(fact_43_Diff__idemp,axiom,
! [A: $tType,B1: 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),B1)),B1) = aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),minus_minus(fun(A,bool)),A1),B1) ).
tff(fact_44_DiffI,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A1))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B1))
=> 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)),A1),B1))) ) ) ).
tff(fact_45_DiffE,axiom,
! [A: $tType,B1: fun(A,bool),A1: 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)),A1),B1)))
=> ~ ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B1)) ) ) ).
tff(fact_46_neq0__conv,axiom,
! [Na: nat] :
( ( Na != zero_zero(nat) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ).
tff(fact_47_diff__self__eq__0,axiom,
! [M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M1),M1) = zero_zero(nat) ).
tff(fact_48_diff__0__eq__0,axiom,
! [N1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),N1) = zero_zero(nat) ).
tff(fact_49_finite__Collect__less__nat,axiom,
! [K1: nat] : finite_finite(nat,collect(nat,aa(nat,fun(nat,bool),aa(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),K1))) ).
tff(fact_50_less__zeroE,axiom,
! [N1: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N1),zero_zero(nat))) ).
tff(fact_51_less__nat__zero__code,axiom,
! [N1: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N1),zero_zero(nat))) ).
tff(fact_52_zero__less__diff,axiom,
! [M2: nat,Na: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M2)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),Na)) ) ).
tff(fact_53_less__imp__diff__less,axiom,
! [N1: nat,K: nat,J: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),K))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),J),N1)),K)) ) ).
tff(fact_54_diff__less__mono2,axiom,
! [L: nat,N1: nat,M1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),N1))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),L))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),N1)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),L),M1))) ) ) ).
tff(fact_55_nat__less__cases,axiom,
! [P1: fun(nat,fun(nat,bool)),Na: nat,M2: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),Na))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M2)) )
=> ( ( ( M2 = Na )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M2)) )
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Na),M2))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M2)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M2)) ) ) ) ).
tff(fact_56_less__not__refl3,axiom,
! [T: nat,S: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S),T))
=> ( S != T ) ) ).
tff(fact_57_less__not__refl2,axiom,
! [M1: nat,N1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N1),M1))
=> ( M1 != N1 ) ) ).
tff(fact_58_less__irrefl__nat,axiom,
! [N1: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N1),N1)) ).
tff(fact_59_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X)) ) ) ).
tff(fact_60_nat__neq__iff,axiom,
! [Na: nat,M2: nat] :
( ( M2 != Na )
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),Na))
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Na),M2)) ) ) ).
tff(fact_61_less__not__refl,axiom,
! [N1: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N1),N1)) ).
tff(fact_62_gr0I,axiom,
! [N1: nat] :
( ( N1 != zero_zero(nat) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N1)) ) ).
tff(fact_63_diffs0__imp__equal,axiom,
! [N1: nat,M1: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M1),N1) = zero_zero(nat) )
=> ( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N1),M1) = zero_zero(nat) )
=> ( M1 = N1 ) ) ) ).
tff(fact_64_gr__implies__not0,axiom,
! [N1: nat,M1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),N1))
=> ( N1 != zero_zero(nat) ) ) ).
tff(fact_65_diff__less,axiom,
! [M1: nat,N1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N1))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M1))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M1),N1)),M1)) ) ) ).
tff(fact_66_minus__nat_Odiff__0,axiom,
! [M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M1),zero_zero(nat)) = M1 ).
tff(fact_67_not__less0,axiom,
! [N1: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N1),zero_zero(nat))) ).
tff(fact_68_DiffD2,axiom,
! [A: $tType,B1: fun(A,bool),A1: 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)),A1),B1)))
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B1)) ) ).
tff(fact_69_DiffD1,axiom,
! [A: $tType,B1: fun(A,bool),A1: 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)),A1),B1)))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A1)) ) ).
tff(fact_70_set__diff__eq,axiom,
! [A: $tType,B1: 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),B1) = 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)),B1)))) ).
tff(fact_71_Z_OI__list__disj,axiom,
! [Xsa: list(int),Fs: list(fm(atom))] :
( interpret(atom,int,i_Z,list_disj(atom,Fs),Xsa)
<=> ? [X2: fm(atom)] :
( pp(aa(fun(fm(atom),bool),bool,aa(fm(atom),fun(fun(fm(atom),bool),bool),member(fm(atom)),X2),set(fm(atom),Fs)))
& interpret(atom,int,i_Z,X2,Xsa) ) ) ).
tff(fact_72_minus__apply,axiom,
! [A: $tType,B: $tType] :
( cl_Groups_Ominus(A)
=> ! [X3: B,B1: 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),B1),X3) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(B,A,A1,X3)),aa(B,A,B1,X3)) ) ).
tff(fact_73_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( cl_Groups_Ominus(B)
=> ! [B1: fun(A,B),A1: fun(A,B),X4: 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),B1),X4) = aa(B,B,aa(B,fun(B,B),minus_minus(B),aa(A,B,A1,X4)),aa(A,B,B1,X4)) ) ).
tff(fact_74_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_75_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A1))
<=> pp(aa(A,bool,A1,X3)) ) ).
tff(fact_76_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_77_decr_092_060_094isub_062Z_Osimps_I1_J,axiom,
! [Ks1: list(int),I1: int] : decr_Z(aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,I1),Ks1)) = aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,I1),aa(list(int),list(int),tl(int),Ks1)) ).
tff(fact_78_diff__commute,axiom,
! [K: nat,J: nat,I1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I1),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),I1),K)),J) ).
tff(fact_79_finite__M__bounded__by__nat,axiom,
! [I: nat,P1: fun(nat,bool)] : finite_finite(nat,collect(nat,aa(fun(nat,bool),fun(nat,bool),aa(fun(nat,fun(bool,bool)),fun(fun(nat,bool),fun(nat,bool)),combs(nat,bool,bool),aa(fun(nat,bool),fun(nat,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(nat,bool),fun(nat,fun(bool,bool))),combb(bool,fun(bool,bool),nat),fconj),P1)),aa(nat,fun(nat,bool),aa(fun(nat,fun(nat,bool)),fun(nat,fun(nat,bool)),combc(nat,nat,bool),ord_less(nat)),I)))) ).
tff(fact_80_finite__nat__set__iff__bounded,axiom,
! [N: fun(nat,bool)] :
( finite_finite(nat,N)
<=> ? [M: nat] :
! [X2: nat] :
( pp(aa(fun(nat,bool),bool,aa(nat,fun(fun(nat,bool),bool),member(nat),X2),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),M)) ) ) ).
tff(fact_81_bounded__nat__set__is__finite,axiom,
! [Na: nat,N: fun(nat,bool)] :
( ! [X1: nat] :
( pp(aa(fun(nat,bool),bool,aa(nat,fun(fun(nat,bool),bool),member(nat),X1),N))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X1),Na)) )
=> finite_finite(nat,N) ) ).
tff(fact_82_decr_092_060_094isub_062Z_Osimps_I3_J,axiom,
! [Ks1: list(int),I1: int,D: int] : decr_Z(nDvd(D,I1,Ks1)) = nDvd(D,I1,aa(list(int),list(int),tl(int),Ks1)) ).
tff(fact_83_atom_Osimps_I3_J,axiom,
! [List3: list(int),Int23: int,Int13: int,List2: list(int),Int22: int,Int12: int] :
( ( nDvd(Int12,Int22,List2) = nDvd(Int13,Int23,List3) )
<=> ( ( Int12 = Int13 )
& ( Int22 = Int23 )
& ( List2 = List3 ) ) ) ).
tff(fact_84_atom_Osize_I6_J,axiom,
! [List: list(int),Int2: int,Int1: int] : size_size(atom,nDvd(Int1,Int2,List)) = zero_zero(nat) ).
tff(fact_85_atom_Osize_I3_J,axiom,
! [List: list(int),Int2: int,Int1: int] : atom_size(nDvd(Int1,Int2,List)) = zero_zero(nat) ).
tff(fact_86_atom_Osimps_I12_J,axiom,
! [A: $tType,List2: list(int),Int22: int,Int12: int,F3: fun(int,fun(int,fun(list(int),A))),F2: fun(int,fun(int,fun(list(int),A))),F1: fun(int,fun(list(int),A))] : aa(atom,A,atom_case(A,F1,F2,F3),nDvd(Int12,Int22,List2)) = aa(list(int),A,aa(int,fun(list(int),A),aa(int,fun(int,fun(list(int),A)),F3,Int12),Int22),List2) ).
tff(fact_87_atom_Osimps_I7_J,axiom,
! [List: list(int),Int: int,List1: list(int),Int21: int,Int11: int] : nDvd(Int11,Int21,List1) != aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int),List) ).
tff(fact_88_atom_Osimps_I6_J,axiom,
! [List1: list(int),Int21: int,Int11: int,List: list(int),Int: int] : aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int),List) != nDvd(Int11,Int21,List1) ).
tff(fact_89_divisor_Osimps_I3_J,axiom,
! [Ks1: list(int),I1: int,D: int] : divisor(nDvd(D,I1,Ks1)) = D ).
tff(fact_90_decr_092_060_094isub_062Z_Osimps_I2_J,axiom,
! [Ks1: list(int),I1: int,D: int] : decr_Z(dvd(D,I1,Ks1)) = dvd(D,I1,aa(list(int),list(int),tl(int),Ks1)) ).
tff(fact_91_I_092_060_094isub_062Z_Osimps_I1_J,axiom,
! [Xsa: list(int),Ks: list(int),I: int] :
( pp(aa(list(int),bool,aa(atom,fun(list(int),bool),i_Z,aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,I),Ks)),Xsa))
<=> ord_less_eq(int,I,aa(list(int),int,aa(list(int),fun(list(int),int),iprod(int),Ks),Xsa)) ) ).
tff(fact_92_atom_Osimps_I2_J,axiom,
! [List3: list(int),Int23: int,Int13: int,List2: list(int),Int22: int,Int12: int] :
( ( dvd(Int12,Int22,List2) = dvd(Int13,Int23,List3) )
<=> ( ( Int12 = Int13 )
& ( Int22 = Int23 )
& ( List2 = List3 ) ) ) ).
tff(fact_93_of__int__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int,W: int] :
( ord_less_eq(A,ring_1_of_int(A,W),ring_1_of_int(A,Z))
<=> ord_less_eq(int,W,Z) ) ) ).
tff(fact_94_atom_Osize_I2_J,axiom,
! [List: list(int),Int2: int,Int1: int] : atom_size(dvd(Int1,Int2,List)) = zero_zero(nat) ).
tff(fact_95_atom_Osize_I5_J,axiom,
! [List: list(int),Int2: int,Int1: int] : size_size(atom,dvd(Int1,Int2,List)) = zero_zero(nat) ).
tff(fact_96_of__int__0__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int] :
( ord_less_eq(A,zero_zero(A),ring_1_of_int(A,Z))
<=> ord_less_eq(int,zero_zero(int),Z) ) ) ).
tff(fact_97_of__int__le__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int] :
( ord_less_eq(A,ring_1_of_int(A,Z),zero_zero(A))
<=> ord_less_eq(int,Z,zero_zero(int)) ) ) ).
tff(fact_98_atom_Osimps_I8_J,axiom,
! [List1: list(int),Int21: int,Int11: int,List: list(int),Int2: int,Int1: int] : dvd(Int1,Int2,List) != nDvd(Int11,Int21,List1) ).
%----Arities (16)
tff(arity_fun___Finite__Set_Ofinite,axiom,
! [T_1: $tType,T_2: $tType] :
( ( finite_finite1(T_2)
& finite_finite1(T_1) )
=> finite_finite1(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_Oordered__ab__group__add,axiom,
ordered_ab_group_add(int) ).
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Groups_Ogroup__add,axiom,
group_add(int) ).
tff(arity_Int_Oint___Int_Oring__char__0,axiom,
ring_char_0(int) ).
tff(arity_Int_Oint___Rings_Oring__1,axiom,
ring_1(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_Nat_Onat___Groups_Ominus,axiom,
cl_Groups_Ominus(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_HOL_Obool___Finite__Set_Ofinite,axiom,
finite_finite1(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_finite1(T_2)
& finite_finite1(T_1) )
=> finite_finite1(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,aa(A,fun(B,A),combk(A,B),P),Q) = P ).
tff(help_COMBS_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(A,fun(B,C))] : aa(A,C,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,
finite_finite(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)),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),aa(fun(list(int),list(int)),fun(list(int),fun(list(int),int)),aa(fun(list(int),fun(list(int),int)),fun(fun(list(int),list(int)),fun(list(int),fun(list(int),int))),combb(list(int),fun(list(int),int),list(int)),iprod(int)),tl(int))),xs))))))),aa(fun(atom,bool),fun(list(int),fun(int,bool)),aa(fun(list(int),fun(fun(atom,bool),fun(int,bool))),fun(fun(atom,bool),fun(list(int),fun(int,bool))),combc(list(int),fun(atom,bool),fun(int,bool)),aa(fun(list(int),fun(int,fun(fun(atom,bool),bool))),fun(list(int),fun(fun(atom,bool),fun(int,bool))),aa(fun(fun(int,fun(fun(atom,bool),bool)),fun(fun(atom,bool),fun(int,bool))),fun(fun(list(int),fun(int,fun(fun(atom,bool),bool))),fun(list(int),fun(fun(atom,bool),fun(int,bool)))),combb(fun(int,fun(fun(atom,bool),bool)),fun(fun(atom,bool),fun(int,bool)),list(int)),combc(int,fun(atom,bool),bool)),aa(fun(list(int),fun(int,atom)),fun(list(int),fun(int,fun(fun(atom,bool),bool))),aa(fun(fun(int,atom),fun(int,fun(fun(atom,bool),bool))),fun(fun(list(int),fun(int,atom)),fun(list(int),fun(int,fun(fun(atom,bool),bool)))),combb(fun(int,atom),fun(int,fun(fun(atom,bool),bool)),list(int)),aa(fun(atom,fun(fun(atom,bool),bool)),fun(fun(int,atom),fun(int,fun(fun(atom,bool),bool))),combb(atom,fun(fun(atom,bool),bool),int),member(atom))),aa(fun(int,fun(list(int),atom)),fun(list(int),fun(int,atom)),combc(int,list(int),atom),c_PresArith_Oatom_OLe)))),collect(atom,aa(fun(atom,bool),fun(atom,bool),aa(fun(atom,fun(bool,bool)),fun(fun(atom,bool),fun(atom,bool)),combs(atom,bool,bool),aa(fun(atom,bool),fun(atom,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(atom,bool),fun(atom,fun(bool,bool))),combb(bool,fun(bool,bool),atom),fconj),aa(fun(atom,bool),fun(atom,bool),aa(fun(atom,fun(fun(atom,bool),bool)),fun(fun(atom,bool),fun(atom,bool)),combc(atom,fun(atom,bool),bool),member(atom)),set(atom,as)))),atom_case(bool,aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),list_case(bool,int,fFalse,aa(fun(int,bool),fun(int,fun(list(int),bool)),aa(fun(bool,fun(list(int),bool)),fun(fun(int,bool),fun(int,fun(list(int),bool))),combb(bool,fun(list(int),bool),int),combk(bool,list(int))),aa(int,fun(int,bool),aa(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),zero_zero(int))))),aa(fun(int,fun(list(int),bool)),fun(int,fun(int,fun(list(int),bool))),combk(fun(int,fun(list(int),bool)),int),aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),aa(bool,fun(list(int),bool),combk(bool,list(int)),fFalse))),aa(fun(int,fun(list(int),bool)),fun(int,fun(int,fun(list(int),bool))),combk(fun(int,fun(list(int),bool)),int),aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),aa(bool,fun(list(int),bool),combk(bool,list(int)),fFalse)))))))))))) ).
%------------------------------------------------------------------------------