TPTP Problem File: COM079_5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : COM079_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 184
% 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_184 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 178 ( 55 unt; 54 typ; 0 def)
% Number of atoms : 241 ( 121 equ)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 142 ( 25 ~; 3 |; 20 &)
% ( 22 <=>; 72 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 49 ( 30 >; 19 *; 0 +; 0 <<)
% Number of predicates : 18 ( 17 usr; 1 prp; 0-5 aty)
% Number of functors : 33 ( 33 usr; 5 con; 0-5 aty)
% Number of variables : 341 ( 292 !; 8 ?; 341 :)
% ( 41 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:21:30
%------------------------------------------------------------------------------
%----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 (46)
tff(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oplus,type,
cl_Groups_Oplus:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : $o ).
tff(sy_c_Enum_On__lists,type,
n_lists:
!>[A: $tType] : ( ( nat * list(A) ) > list(list(A)) ) ).
tff(sy_c_Enum_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_ListVector_Oiprod,type,
iprod:
!>[A: $tType] : ( ( list(A) * list(A) ) > A ) ).
tff(sy_c_ListVector_Ozipwith0,type,
zipwith0:
!>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(list(A),fun(list(B),list(C))) ) ).
tff(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_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_Olist__size,type,
list_size:
!>[A: $tType] : ( ( fun(A,nat) * list(A) ) > nat ) ).
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_Osplice,type,
splice:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
tff(sy_c_List_Osublist,type,
sublist:
!>[A: $tType] : ( ( list(A) * fun(nat,bool) ) > 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_Logic_Omap__fm,type,
map_fm:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * fm(A) ) > fm(B) ) ).
tff(sy_c_Logic_Oneg,type,
neg:
!>[A: $tType] : ( fm(A) > fm(A) ) ).
tff(sy_c_Logic_Oqfree,type,
qfree:
!>[A: $tType] : ( fm(A) > $o ) ).
tff(sy_c_Nat_OSuc,type,
suc: nat > nat ).
tff(sy_c_PresArith_OI_092_060_094isub_062Z,type,
i_Z: fun(atom,fun(list(int),bool)) ).
tff(sy_c_PresArith_Oasubst,type,
asubst: ( int * list(int) ) > fun(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_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_member,type,
member:
!>[A: $tType] : ( ( A * fun(A,bool) ) > $o ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_as,type,
as: list(atom) ).
tff(sy_v_thesis____,type,
thesis: $o ).
tff(sy_v_xs,type,
xs: list(int) ).
%----Relevant facts (99)
tff(fact_0__096lbounds_Aas_A_126_061_A_091_093_096,axiom,
lbounds(as) != nil(product_prod(int,list(int))) ).
tff(fact_1__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_2_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C2: A,B2: A,A1: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),C2) )
<=> ( B2 = C2 ) ) ) ).
tff(fact_3_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C2: A,A1: A,B2: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A1) )
<=> ( B2 = C2 ) ) ) ).
tff(fact_4_calculation,axiom,
( ( lbounds(as) = nil(product_prod(int,list(int))) )
=> ? [X: int] :
! [Xa: atom] :
( member(atom,Xa,set(atom,as))
=> pp(aa(list(int),bool,aa(atom,fun(list(int),bool),i_Z,Xa),cons(int,X,xs))) ) ) ).
tff(fact_5_norm,axiom,
! [X4: atom] :
( member(atom,X4,set(atom,as))
=> ( divisor(X4) != zero_zero(int) ) ) ).
tff(fact_6_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C1: A,A6: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),C1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C1),A6) ) ).
tff(fact_7_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C1: A,A6: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),aa(A,A,aa(A,fun(A,A),plus_plus(A),C1),D)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C1),aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),D)) ) ).
tff(fact_8_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C1: A,A6: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),aa(A,A,aa(A,fun(A,A),plus_plus(A),C1),D)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),C1)),D) ) ).
tff(fact_9_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C1: A,B1: A,A6: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),B1)),C1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),C1)) ) ).
tff(fact_10_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C1: A,B1: A,A6: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),B1)),C1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),C1)) ) ).
tff(fact_11_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C1: A,B1: A,A6: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),B1)),C1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),C1)),B1) ) ).
tff(fact_12_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C1: A,B1: A,A6: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),B1)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C1),D)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),C1)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),D)) ) ).
tff(fact_13_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,B1: A,A6: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),B1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),C1) )
=> ( B1 = C1 ) ) ) ).
tff(fact_14_double__zero__sym,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),A1) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_15_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X2: A] :
( ( zero_zero(A) = X2 )
<=> ( X2 = zero_zero(A) ) ) ) ).
tff(fact_16_add__0__iff,axiom,
! [A: $tType] :
( semiri456707255roduct(A)
=> ! [A1: A,B2: A] :
( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A1) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_17_add_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A6: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),zero_zero(A)) = A6 ) ).
tff(fact_18_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A6: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),zero_zero(A)) = A6 ) ).
tff(fact_19_add__0__right,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A6: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),zero_zero(A)) = A6 ) ).
tff(fact_20_add__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A6: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A6) = A6 ) ).
tff(fact_21_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A6: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A6) = A6 ) ).
tff(fact_22_add__0__left,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A6: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A6) = A6 ) ).
tff(fact_23_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,A6: A,B1: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),A6) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C1),A6) )
=> ( B1 = C1 ) ) ) ).
tff(fact_24_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C1: A,B1: A,A6: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),B1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),C1) )
=> ( B1 = C1 ) ) ) ).
tff(fact_25_divisor__asubst,axiom,
! [A6: atom,Ks1: list(int),I2: int] : divisor(aa(atom,atom,asubst(I2,Ks1),A6)) = divisor(A6) ).
tff(fact_26_list__add__Cons,axiom,
! [A: $tType] :
( ( cl_Groups_Oplus(A)
& zero(A) )
=> ! [Ys: list(A),Y: A,Xs1: list(A),X1: A] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),cons(A,X1,Xs1)),cons(A,Y,Ys)) = cons(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X1),Y),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),Xs1),Ys)) ) ).
tff(fact_27_double__eq__0__iff,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),A1) = zero_zero(A) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_28_list_Oinject,axiom,
! [A: $tType,List3: list(A),A5: A,List: list(A),A1: A] :
( ( cons(A,A1,List) = cons(A,A5,List3) )
<=> ( ( A1 = A5 )
& ( List = List3 ) ) ) ).
tff(fact_29_list_Osimps_I2_J,axiom,
! [A: $tType,List2: list(A),A4: A] : nil(A) != cons(A,A4,List2) ).
tff(fact_30_list_Osimps_I3_J,axiom,
! [A: $tType,List2: list(A),A4: A] : cons(A,A4,List2) != nil(A) ).
tff(fact_31_set__ConsD,axiom,
! [A: $tType,Xsa: list(A),X2: A,Y2: A] :
( member(A,Y2,set(A,cons(A,X2,Xsa)))
=> ( ( Y2 = X2 )
| member(A,Y2,set(A,Xsa)) ) ) ).
tff(fact_32_IZ__asubst,axiom,
! [Xsa: list(int),A1: atom,Ks: list(int),I1: int] :
( pp(aa(list(int),bool,aa(atom,fun(list(int),bool),i_Z,aa(atom,atom,asubst(I1,Ks),A1)),Xsa))
<=> pp(aa(list(int),bool,aa(atom,fun(list(int),bool),i_Z,A1),cons(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I1),iprod(int,Ks,Xsa)),Xsa))) ) ).
tff(fact_33_product_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Uu: list(B)] : product(A,B,nil(A),Uu) = nil(product_prod(A,B)) ).
tff(fact_34_iprod__Nil2,axiom,
! [A: $tType] :
( ring(A)
=> ! [Xs1: list(A)] : iprod(A,Xs1,nil(A)) = zero_zero(A) ) ).
tff(fact_35_iprod__Nil,axiom,
! [A: $tType] :
( ring(A)
=> ! [Ys: list(A)] : iprod(A,nil(A),Ys) = zero_zero(A) ) ).
tff(fact_36_list__add__assoc,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [Zs: list(A),Ys: list(A),Xs1: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),Xs1),Ys)),Zs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),Xs1),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),Ys),Zs)) ) ).
tff(fact_37_iprod__left__add__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [Zs: list(A),Ys: list(A),Xs1: list(A)] : iprod(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),Xs1),Ys),Zs) = aa(A,A,aa(A,fun(A,A),plus_plus(A),iprod(A,Xs1,Zs)),iprod(A,Ys,Zs)) ) ).
tff(fact_38_not__Cons__self2,axiom,
! [A: $tType,Xs1: list(A),X1: A] : cons(A,X1,Xs1) != Xs1 ).
tff(fact_39_not__Cons__self,axiom,
! [A: $tType,X1: A,Xs1: list(A)] : Xs1 != cons(A,X1,Xs1) ).
tff(fact_40_list__add__Nil2,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [Xs1: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),Xs1),nil(A)) = Xs1 ) ).
tff(fact_41_list__add__Nil,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [Xs1: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),nil(A)),Xs1) = Xs1 ) ).
tff(fact_42_iprod0__if__coeffs0,axiom,
! [A: $tType] :
( ring(A)
=> ! [Xsa: list(A),Cs: list(A)] :
( ! [X: A] :
( member(A,X,set(A,Cs))
=> ( X = zero_zero(A) ) )
=> ( iprod(A,Cs,Xsa) = zero_zero(A) ) ) ) ).
tff(fact_43_Z_OI__list__conj,axiom,
! [Xsa: list(int),Fs: list(fm(atom))] :
( interpret(atom,int,i_Z,list_conj(atom,Fs),Xsa)
<=> ! [X3: fm(atom)] :
( member(fm(atom),X3,set(fm(atom),Fs))
=> interpret(atom,int,i_Z,X3,Xsa) ) ) ).
tff(fact_44_list_Oexhaust,axiom,
! [A: $tType,Y: list(A)] :
( ( Y != nil(A) )
=> ~ ! [A3: A,List1: list(A)] : Y != cons(A,A3,List1) ) ).
tff(fact_45_neq__Nil__conv,axiom,
! [A: $tType,Xsa: list(A)] :
( ( Xsa != nil(A) )
<=> ? [Y1: A,Ys2: list(A)] : Xsa = cons(A,Y1,Ys2) ) ).
tff(fact_46_zipwith0_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( zero(C)
& zero(B) )
=> ! [Xsa: list(B),X2: 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),cons(B,X2,Xsa)),nil(C)) = cons(A,aa(C,A,aa(B,fun(C,A),F,X2),zero_zero(C)),aa(list(C),list(A),aa(list(B),fun(list(C),list(A)),zipwith0(B,C,A,F),Xsa),nil(C))) ) ).
tff(fact_47_zipwith0_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( zero(C)
& zero(B) )
=> ! [Ys1: list(C),Y2: C,Xsa: list(B),X2: 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),cons(B,X2,Xsa)),cons(C,Y2,Ys1)) = cons(A,aa(C,A,aa(B,fun(C,A),F,X2),Y2),aa(list(C),list(A),aa(list(B),fun(list(C),list(A)),zipwith0(B,C,A,F),Xsa),Ys1)) ) ).
tff(fact_48_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_49_zipwith0_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( zero(C)
& zero(B) )
=> ! [Ys1: list(C),Y2: 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)),cons(C,Y2,Ys1)) = cons(A,aa(C,A,aa(B,fun(C,A),F,zero_zero(B)),Y2),aa(list(C),list(A),aa(list(B),fun(list(C),list(A)),zipwith0(B,C,A,F),nil(B)),Ys1)) ) ).
tff(fact_50_list__add__def,axiom,
! [A: $tType] :
( ( cl_Groups_Oplus(A)
& zero(A) )
=> ( plus_plus(list(A)) = zipwith0(A,A,A,plus_plus(A)) ) ) ).
tff(fact_51_Z_OI__list__disj,axiom,
! [Xsa: list(int),Fs: list(fm(atom))] :
( interpret(atom,int,i_Z,list_disj(atom,Fs),Xsa)
<=> ? [X3: fm(atom)] :
( member(fm(atom),X3,set(fm(atom),Fs))
& interpret(atom,int,i_Z,X3,Xsa) ) ) ).
tff(fact_52_insert__Nil,axiom,
! [A: $tType,X1: A] : insert(A,X1,nil(A)) = cons(A,X1,nil(A)) ).
tff(fact_53_splice_Osimps_I2_J,axiom,
! [A: $tType,Va: list(A),V: A] : splice(A,cons(A,V,Va),nil(A)) = cons(A,V,Va) ).
tff(fact_54_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_55_splice_Osimps_I3_J,axiom,
! [A: $tType,Ys: list(A),Y: A,Xs1: list(A),X1: A] : splice(A,cons(A,X1,Xs1),cons(A,Y,Ys)) = cons(A,X1,cons(A,Y,splice(A,Xs1,Ys))) ).
tff(fact_56_in__set__insert,axiom,
! [A: $tType,Xsa: list(A),X2: A] :
( member(A,X2,set(A,Xsa))
=> ( insert(A,X2,Xsa) = Xsa ) ) ).
tff(fact_57_splice__Nil2,axiom,
! [A: $tType,Xs1: list(A)] : splice(A,Xs1,nil(A)) = Xs1 ).
tff(fact_58_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] : splice(A,nil(A),Ys) = Ys ).
tff(fact_59_not__in__set__insert,axiom,
! [A: $tType,Xsa: list(A),X2: A] :
( ~ member(A,X2,set(A,Xsa))
=> ( insert(A,X2,Xsa) = cons(A,X2,Xsa) ) ) ).
tff(fact_60_List_Oinsert__def,axiom,
! [A: $tType,Xsa: list(A),X2: A] :
( ( member(A,X2,set(A,Xsa))
=> ( insert(A,X2,Xsa) = Xsa ) )
& ( ~ member(A,X2,set(A,Xsa))
=> ( insert(A,X2,Xsa) = cons(A,X2,Xsa) ) ) ) ).
tff(fact_61_list_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,List: list(B),A1: B,F2: fun(B,fun(list(B),A)),F1: A] : list_case(A,B,F1,F2,cons(B,A1,List)) = aa(list(B),A,aa(B,fun(list(B),A),F2,A1),List) ).
tff(fact_62_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: fun(A,bool)] : ~ list_ex1(A,P,nil(A)) ).
tff(fact_63_I__subst,axiom,
! [Xsa: list(int),Ks: list(int),I1: int,Phi: fm(atom)] :
( qfree(atom,Phi)
=> ( interpret(atom,int,i_Z,Phi,cons(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I1),iprod(int,Ks,Xsa)),Xsa))
<=> interpret(atom,int,i_Z,map_fm(atom,atom,asubst(I1,Ks),Phi),Xsa) ) ) ).
tff(fact_64_sublist__singleton,axiom,
! [A: $tType,X2: A,A2: fun(nat,bool)] :
( ( member(nat,zero_zero(nat),A2)
=> ( sublist(A,cons(A,X2,nil(A)),A2) = cons(A,X2,nil(A)) ) )
& ( ~ member(nat,zero_zero(nat),A2)
=> ( sublist(A,cons(A,X2,nil(A)),A2) = nil(A) ) ) ) ).
tff(fact_65_sublist__nil,axiom,
! [A: $tType,A2: fun(nat,bool)] : sublist(A,nil(A),A2) = nil(A) ).
tff(fact_66_in__set__sublistD,axiom,
! [A: $tType,I: fun(nat,bool),Xsa: list(A),X2: A] :
( member(A,X2,set(A,sublist(A,Xsa,I)))
=> member(A,X2,set(A,Xsa)) ) ).
tff(fact_67_notin__set__sublistI,axiom,
! [A: $tType,I: fun(nat,bool),Xsa: list(A),X2: A] :
( ~ member(A,X2,set(A,Xsa))
=> ~ member(A,X2,set(A,sublist(A,Xsa,I))) ) ).
tff(fact_68_list__ex1__iff,axiom,
! [A: $tType,Xsa: list(A),P: fun(A,bool)] :
( list_ex1(A,P,Xsa)
<=> ? [X3: A] :
( member(A,X3,set(A,Xsa))
& pp(aa(A,bool,P,X3))
& ! [Y1: A] :
( ( member(A,Y1,set(A,Xsa))
& pp(aa(A,bool,P,Y1)) )
=> ( Y1 = X3 ) ) ) ) ).
tff(fact_69_add__is__0,axiom,
! [N: nat,M2: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) = zero_zero(nat) )
<=> ( ( M2 = zero_zero(nat) )
& ( N = zero_zero(nat) ) ) ) ).
tff(fact_70_qfree__map__fm,axiom,
! [A: $tType,B: $tType,Phi: fm(B),F: fun(B,A)] :
( qfree(A,map_fm(B,A,F,Phi))
<=> qfree(B,Phi) ) ).
tff(fact_71_qfree__list__conj,axiom,
! [A: $tType,Fs: list(fm(A))] :
( ! [X: fm(A)] :
( member(fm(A),X,set(fm(A),Fs))
=> qfree(A,X) )
=> qfree(A,list_conj(A,Fs)) ) ).
tff(fact_72_nat__add__right__cancel,axiom,
! [N: nat,K1: nat,M2: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),K1) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K1) )
<=> ( M2 = N ) ) ).
tff(fact_73_nat__add__left__cancel,axiom,
! [N: nat,M2: nat,K1: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K1),M2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K1),N) )
<=> ( M2 = N ) ) ).
tff(fact_74_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X: A] : aa(A,B,F,X) = aa(A,B,G,X)
=> ( F = G ) ) ).
tff(fact_75_mem__def,axiom,
! [A: $tType,A2: fun(A,bool),X2: A] :
( member(A,X2,A2)
<=> pp(aa(A,bool,A2,X2)) ) ).
tff(fact_76_nat__add__assoc,axiom,
! [K: nat,N1: nat,M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),N1)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N1),K)) ).
tff(fact_77_nat__add__left__commute,axiom,
! [Z: nat,Y: nat,X1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X1),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),Z)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X1),Z)) ).
tff(fact_78_nat__add__commute,axiom,
! [N1: nat,M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),N1) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N1),M1) ).
tff(fact_79_add__eq__self__zero,axiom,
! [N1: nat,M1: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),N1) = M1 )
=> ( N1 = zero_zero(nat) ) ) ).
tff(fact_80_Nat_Oadd__0__right,axiom,
! [M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),zero_zero(nat)) = M1 ).
tff(fact_81_plus__nat_Oadd__0,axiom,
! [N1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N1) = N1 ).
tff(fact_82_qfree__list__disj,axiom,
! [A: $tType,Fs: list(fm(A))] :
( ! [X: fm(A)] :
( member(fm(A),X,set(fm(A),Fs))
=> qfree(A,X) )
=> qfree(A,list_disj(A,Fs)) ) ).
tff(fact_83_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs1: list(A)] : n_lists(A,zero_zero(nat),Xs1) = cons(list(A),nil(A),nil(list(A))) ).
tff(fact_84_n__lists__Nil,axiom,
! [A: $tType,N1: nat] :
( ( ( N1 = zero_zero(nat) )
=> ( n_lists(A,N1,nil(A)) = cons(list(A),nil(A),nil(list(A))) ) )
& ( ( N1 != zero_zero(nat) )
=> ( n_lists(A,N1,nil(A)) = nil(list(A)) ) ) ) ).
tff(fact_85_qfree__neg,axiom,
! [A: $tType,Phi: fm(A)] :
( qfree(A,neg(A,Phi))
<=> qfree(A,Phi) ) ).
tff(fact_86_list_Osize_I1_J,axiom,
! [A: $tType,Fa: fun(A,nat)] : list_size(A,Fa,nil(A)) = zero_zero(nat) ).
tff(fact_87_list_Osize_I2_J,axiom,
! [A: $tType,List: list(A),A1: A,Fa: fun(A,nat)] : list_size(A,Fa,cons(A,A1,List)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Fa,A1)),list_size(A,Fa,List))),suc(zero_zero(nat))) ).
tff(fact_88_list__nonempty__induct,axiom,
! [A: $tType,P: fun(list(A),bool),Xsa: list(A)] :
( ( Xsa != nil(A) )
=> ( ! [X: A] : pp(aa(list(A),bool,P,cons(A,X,nil(A))))
=> ( ! [X: A,Xs: list(A)] :
( ( Xs != nil(A) )
=> ( pp(aa(list(A),bool,P,Xs))
=> pp(aa(list(A),bool,P,cons(A,X,Xs))) ) )
=> pp(aa(list(A),bool,P,Xsa)) ) ) ) ).
tff(fact_89_nat_Oinject,axiom,
! [Nat3: nat,Nat2: nat] :
( ( suc(Nat2) = suc(Nat3) )
<=> ( Nat2 = Nat3 ) ) ).
tff(fact_90_add__Suc,axiom,
! [N1: nat,M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),suc(M1)),N1) = suc(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),N1)) ).
tff(fact_91_add__Suc__right,axiom,
! [N1: nat,M1: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),suc(N1)) = suc(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M1),N1)) ).
tff(fact_92_Zero__not__Suc,axiom,
! [M1: nat] : zero_zero(nat) != suc(M1) ).
tff(fact_93_nat_Osimps_I2_J,axiom,
! [Nat1: nat] : zero_zero(nat) != suc(Nat1) ).
tff(fact_94_Suc__not__Zero,axiom,
! [M1: nat] : suc(M1) != zero_zero(nat) ).
tff(fact_95_nat_Osimps_I3_J,axiom,
! [Nat: nat] : suc(Nat) != zero_zero(nat) ).
tff(fact_96_one__is__add,axiom,
! [N: nat,M2: nat] :
( ( suc(zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) )
<=> ( ( ( M2 = suc(zero_zero(nat)) )
& ( N = zero_zero(nat) ) )
| ( ( M2 = zero_zero(nat) )
& ( N = suc(zero_zero(nat)) ) ) ) ) ).
tff(fact_97_add__is__1,axiom,
! [N: nat,M2: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),N) = suc(zero_zero(nat)) )
<=> ( ( ( M2 = suc(zero_zero(nat)) )
& ( N = zero_zero(nat) ) )
| ( ( M2 = zero_zero(nat) )
& ( N = suc(zero_zero(nat)) ) ) ) ) ).
tff(fact_98_Zero__neq__Suc,axiom,
! [M1: nat] : zero_zero(nat) != suc(M1) ).
%----Arities (21)
tff(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct(int) ).
tff(arity_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(int) ).
tff(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(int) ).
tff(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(int) ).
tff(arity_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(int) ).
tff(arity_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(int) ).
tff(arity_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Groups_Oplus,axiom,
cl_Groups_Oplus(int) ).
tff(arity_Int_Oint___Rings_Oring,axiom,
ring(int) ).
tff(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
tff(arity_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(nat) ).
tff(arity_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(nat) ).
tff(arity_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_Nat_Onat___Groups_Oplus,axiom,
cl_Groups_Oplus(nat) ).
tff(arity_List_Olist___Groups_Oplus,axiom,
! [T_1: $tType] :
( ( zero(T_1)
& cl_Groups_Oplus(T_1) )
=> cl_Groups_Oplus(list(T_1)) ) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (2)
tff(conj_0,hypothesis,
( ? [Il: int,M: int,Ksl: list(int)] :
! [X: atom] :
( member(atom,X,set(atom,as))
=> pp(aa(list(int),bool,aa(atom,fun(list(int),bool),i_Z,aa(atom,atom,asubst(aa(int,int,aa(int,fun(int,int),plus_plus(int),Il),M),Ksl),X)),xs)) )
=> thesis ) ).
tff(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------