TPTP Problem File: COM119_5.p
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%------------------------------------------------------------------------------
% File : COM119_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 287
% 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_287 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 170 ( 69 unt; 55 typ; 0 def)
% Number of atoms : 196 ( 141 equ)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 113 ( 32 ~; 6 |; 14 &)
% ( 20 <=>; 41 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 13 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 52 ( 32 >; 20 *; 0 +; 0 <<)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-3 aty)
% Number of functors : 44 ( 44 usr; 6 con; 0-5 aty)
% Number of variables : 394 ( 344 !; 11 ?; 394 :)
% ( 39 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:25:39
%------------------------------------------------------------------------------
%----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_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_PresArith_Oatom,type,
atom: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (49)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(B,fun(A,C)) ) ).
tff(sy_c_COMBI,type,
combi:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : fun(A,fun(B,A)) ).
tff(sy_c_Enum_On__lists,type,
n_lists:
!>[A: $tType] : ( ( nat * list(A) ) > list(list(A)) ) ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( list(list(A)) > list(A) ) ).
tff(sy_c_List_Ofilter,type,
filter:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).
tff(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : fun(A,fun(list(A),list(A))) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olist_Olist__case,type,
list_case:
!>[T: $tType,A: $tType] : ( ( T * fun(A,fun(list(A),T)) ) > fun(list(A),T) ) ).
tff(sy_c_List_Omap,type,
map:
!>[A: $tType,B: $tType] : fun(fun(A,B),fun(list(A),list(B))) ).
tff(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( fun(A,list(B)) * list(A) ) > list(B) ) ).
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_List_Otl,type,
tl:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_List_Otranspose,type,
transpose:
!>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).
tff(sy_c_List_Otranspose__rel,type,
transpose_rel:
!>[A: $tType] : fun(list(list(A)),fun(list(list(A)),bool)) ).
tff(sy_c_Nat_OSuc,type,
suc: nat > nat ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_PresArith_OI_092_060_094isub_062Z,type,
i_Z: ( atom * list(int) ) > $o ).
tff(sy_c_PresArith_Oasubst,type,
asubst: ( int * list(int) * atom ) > atom ).
tff(sy_c_PresArith_Oatom_ODvd,type,
dvd: ( int * int * list(int) ) > atom ).
tff(sy_c_PresArith_Oatom_OLe,type,
c_PresArith_Oatom_OLe: ( int * list(int) ) > atom ).
tff(sy_c_PresArith_Oatom_ONDvd,type,
nDvd: ( int * int * list(int) ) > atom ).
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_Ohd__coeff,type,
hd_coeff: fun(atom,int) ).
tff(sy_c_PresArith_Ohd__coeff1,type,
hd_coeff1: int > fun(atom,atom) ).
tff(sy_c_PresArith_Ozlcms,type,
zlcms: list(int) > int ).
tff(sy_c_QEpres__Mirabelle__iocckttzyp_Ohd__coeffs1,type,
qEpres1907640072oeffs1: list(atom) > list(atom) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * A ) > $o ) ).
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_fNot,type,
fNot: fun(bool,bool) ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_member,type,
member:
!>[A: $tType] : ( ( A * fun(A,bool) ) > $o ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_a,type,
a: atom ).
tff(sy_v_as,type,
as: list(atom) ).
%----Relevant facts (98)
tff(fact_0_a,axiom,
member(atom,a,set(atom,qEpres1907640072oeffs1(as))) ).
tff(fact_1_hd0,axiom,
! [X: atom] :
( member(atom,X,set(atom,as))
=> ( aa(atom,int,hd_coeff,X) != zero_zero(int) ) ) ).
tff(fact_2_div0,axiom,
! [X: atom] :
( member(atom,X,set(atom,as))
=> ( divisor(X) != zero_zero(int) ) ) ).
tff(fact_3_hd__coeff1_Osimps_I5_J,axiom,
! [Va: int,V: int,Uu: int] : aa(atom,atom,hd_coeff1(Uu),dvd(V,Va,nil(int))) = dvd(V,Va,nil(int)) ).
tff(fact_4_zlcms_Osimps_I1_J,axiom,
zlcms(nil(int)) = one_one(int) ).
tff(fact_5_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,Xs: list(B),F: fun(B,A)] :
( ( nil(A) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) )
<=> ( Xs = nil(B) ) ) ).
tff(fact_6_map_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),nil(B)) = nil(A) ).
tff(fact_7_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,Xs: list(B),F: fun(B,A)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) = nil(A) )
<=> ( Xs = nil(B) ) ) ).
tff(fact_8_map__eq__conv,axiom,
! [A: $tType,B: $tType,G: fun(B,A),Xs: list(B),F: fun(B,A)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xs) )
<=> ! [X3: B] :
( member(B,X3,set(B,Xs))
=> ( aa(B,A,F,X3) = aa(B,A,G,X3) ) ) ) ).
tff(fact_9_map_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Xs: list(B),X2: B,F: fun(B,A)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F,X2)),aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs)) ).
tff(fact_10_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_11_list_Oinject,axiom,
! [A: $tType,List3: list(A),A5: A,List2: list(A),Aa: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa),List2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),List3) )
<=> ( ( Aa = A5 )
& ( List2 = List3 ) ) ) ).
tff(fact_12_list_Osimps_I2_J,axiom,
! [A: $tType,List1: list(A),A4: A] : nil(A) != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),List1) ).
tff(fact_13_zlcms0__iff,axiom,
! [Is: list(int)] :
( ( zlcms(Is) = zero_zero(int) )
<=> member(int,zero_zero(int),set(int,Is)) ) ).
tff(fact_14_divisor_Osimps_I2_J,axiom,
! [Ks: list(int),I: int,D: int] : divisor(dvd(D,I,Ks)) = D ).
tff(fact_15_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X2: A] :
( ( zero_zero(A) = X2 )
<=> ( X2 = zero_zero(A) ) ) ) ).
tff(fact_16_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X2: A] :
( ( one_one(A) = X2 )
<=> ( X2 = one_one(A) ) ) ) ).
tff(fact_17_not__Cons__self2,axiom,
! [A: $tType,Xs1: list(A),X1: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs1) != Xs1 ).
tff(fact_18_not__Cons__self,axiom,
! [A: $tType,X1: A,Xs1: list(A)] : Xs1 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs1) ).
tff(fact_19_set__ConsD,axiom,
! [A: $tType,Xs: list(A),X2: A,Y2: A] :
( member(A,Y2,set(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)))
=> ( ( Y2 = X2 )
| member(A,Y2,set(A,Xs)) ) ) ).
tff(fact_20_list_Osimps_I3_J,axiom,
! [A: $tType,List1: list(A),A4: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),List1) != nil(A) ).
tff(fact_21_atom_Osize_I2_J,axiom,
! [List: list(int),Int21: int,Int11: int] : atom_size(dvd(Int11,Int21,List)) = zero_zero(nat) ).
tff(fact_22_map__cong,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),Ys2: list(A),Xs: list(A)] :
( ( Xs = Ys2 )
=> ( ! [X4: A] :
( member(A,X4,set(A,Ys2))
=> ( aa(A,B,F,X4) = aa(A,B,G,X4) ) )
=> ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Ys2) ) ) ) ).
tff(fact_23_map__idI,axiom,
! [A: $tType,F: fun(A,A),Xs: list(A)] :
( ! [X4: A] :
( member(A,X4,set(A,Xs))
=> ( aa(A,A,F,X4) = X4 ) )
=> ( aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),F),Xs) = Xs ) ) ).
tff(fact_24_map__ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),Xs: list(A)] :
( ! [X4: A] :
( member(A,X4,set(A,Xs))
=> ( aa(A,B,F,X4) = aa(A,B,G,X4) ) )
=> ( aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Xs) ) ) ).
tff(fact_25_ex__map__conv,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Ys2: list(B)] :
( ? [Xs2: list(A)] : Ys2 = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs2)
<=> ! [X3: B] :
( member(B,X3,set(B,Ys2))
=> ? [Xa: A] : X3 = aa(A,B,F,Xa) ) ) ).
tff(fact_26_map__eq__Cons__conv,axiom,
! [B: $tType,A: $tType,Ys2: list(A),Y2: A,Xs: list(B),F: fun(B,A)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) )
<=> ? [Z: B,Zs: list(B)] :
( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z),Zs) )
& ( aa(B,A,F,Z) = Y2 )
& ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Zs) = Ys2 ) ) ) ).
tff(fact_27_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys2: list(B),F: fun(B,A),Xs: list(A),X2: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ys2) )
<=> ? [Z: B,Zs: list(B)] :
( ( Ys2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z),Zs) )
& ( X2 = aa(B,A,F,Z) )
& ( Xs = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Zs) ) ) ) ).
tff(fact_28_list_Oexhaust,axiom,
! [A: $tType,Y: list(A)] :
( ( Y != nil(A) )
=> ~ ! [A3: A,List4: list(A)] : Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),List4) ) ).
tff(fact_29_atom_Osize_I5_J,axiom,
! [List: list(int),Int21: int,Int11: int] : size_size(atom,dvd(Int11,Int21,List)) = zero_zero(nat) ).
tff(fact_30_neq__Nil__conv,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
<=> ? [Y1: A,Ys1: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y1),Ys1) ) ).
tff(fact_31_asubst_Osimps_I5_J,axiom,
! [Va: int,V: int,Ks1: list(int),I1: int] : asubst(I1,Ks1,dvd(V,Va,nil(int))) = dvd(V,Va,nil(int)) ).
tff(fact_32_one__neq__zero,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( one_one(A) != zero_zero(A) ) ) ).
tff(fact_33_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
tff(fact_34_hd__coeffs1__def,axiom,
! [Asa: list(atom)] : qEpres1907640072oeffs1(Asa) = aa(list(atom),list(atom),aa(atom,fun(list(atom),list(atom)),cons(atom),dvd(zlcms(aa(list(atom),list(int),aa(fun(atom,int),fun(list(atom),list(int)),map(atom,int),hd_coeff),Asa)),zero_zero(int),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),one_one(int)),nil(int)))),aa(list(atom),list(atom),aa(fun(atom,atom),fun(list(atom),list(atom)),map(atom,atom),hd_coeff1(zlcms(aa(list(atom),list(int),aa(fun(atom,int),fun(list(atom),list(int)),map(atom,int),hd_coeff),Asa)))),Asa)) ).
tff(fact_35_map__ident,axiom,
! [A: $tType,X: list(A)] : aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),combi(A)),X) = X ).
tff(fact_36_divisor__asubst,axiom,
! [A1: atom,Ks: list(int),I: int] : divisor(asubst(I,Ks,A1)) = divisor(A1) ).
tff(fact_37_hd__coeff_Osimps_I2_J,axiom,
! [Ks2: list(int),I2: int,D1: int] : aa(atom,int,hd_coeff,dvd(D1,I2,Ks2)) = aa(list(int),int,list_case(int,int,zero_zero(int),combk(int,list(int))),Ks2) ).
tff(fact_38_insert__Nil,axiom,
! [A: $tType,X1: A] : insert(A,X1,nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),nil(A)) ).
tff(fact_39_sublist__singleton,axiom,
! [A: $tType,X2: A,A2: fun(nat,bool)] :
( ( member(nat,zero_zero(nat),A2)
=> ( sublist(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)),A2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) ) )
& ( ~ member(nat,zero_zero(nat),A2)
=> ( sublist(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)),A2) = nil(A) ) ) ) ).
tff(fact_40_concat__map__singleton,axiom,
! [A: $tType,B: $tType,Xs: list(B),F: fun(B,A)] : concat(A,aa(list(B),list(list(A)),aa(fun(B,list(A)),fun(list(B),list(list(A))),map(B,list(A)),aa(list(A),fun(B,list(A)),combc(B,list(A),list(A),combb(A,fun(list(A),list(A)),B,cons(A),F)),nil(A))),Xs)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xs) ).
tff(fact_41_I__hd__coeffs1,axiom,
! [Xs: list(int),Asa: list(atom)] :
( ! [X4: atom] :
( member(atom,X4,set(atom,Asa))
=> ( aa(atom,int,hd_coeff,X4) != zero_zero(int) ) )
=> ( ? [X3: int] :
! [Xa: atom] :
( member(atom,Xa,set(atom,qEpres1907640072oeffs1(Asa)))
=> i_Z(Xa,aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X3),Xs)) )
<=> ? [X3: int] :
! [Xa: atom] :
( member(atom,Xa,set(atom,Asa))
=> i_Z(Xa,aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X3),Xs)) ) ) ) ).
tff(fact_42_concat_Osimps_I1_J,axiom,
! [A: $tType] : concat(A,nil(list(A))) = nil(A) ).
tff(fact_43_concat__eq__Nil__conv,axiom,
! [A: $tType,Xss: list(list(A))] :
( ( concat(A,Xss) = nil(A) )
<=> ! [X3: list(A)] :
( member(list(A),X3,set(list(A),Xss))
=> ( X3 = nil(A) ) ) ) ).
tff(fact_44_Nil__eq__concat__conv,axiom,
! [A: $tType,Xss: list(list(A))] :
( ( nil(A) = concat(A,Xss) )
<=> ! [X3: list(A)] :
( member(list(A),X3,set(list(A),Xss))
=> ( X3 = nil(A) ) ) ) ).
tff(fact_45_sublist__nil,axiom,
! [A: $tType,A2: fun(nat,bool)] : sublist(A,nil(A),A2) = nil(A) ).
tff(fact_46_map__concat,axiom,
! [A: $tType,B: $tType,Xs: list(list(B)),F: fun(B,A)] : aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),concat(B,Xs)) = concat(A,aa(list(list(B)),list(list(A)),aa(fun(list(B),list(A)),fun(list(list(B)),list(list(A))),map(list(B),list(A)),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F)),Xs)) ).
tff(fact_47_notin__set__sublistI,axiom,
! [A: $tType,I3: fun(nat,bool),Xs: list(A),X2: A] :
( ~ member(A,X2,set(A,Xs))
=> ~ member(A,X2,set(A,sublist(A,Xs,I3))) ) ).
tff(fact_48_in__set__sublistD,axiom,
! [A: $tType,I3: fun(nat,bool),Xs: list(A),X2: A] :
( member(A,X2,set(A,sublist(A,Xs,I3)))
=> member(A,X2,set(A,Xs)) ) ).
tff(fact_49_in__set__insert,axiom,
! [A: $tType,Xs: list(A),X2: A] :
( member(A,X2,set(A,Xs))
=> ( insert(A,X2,Xs) = Xs ) ) ).
tff(fact_50_List_Oinsert__def,axiom,
! [A: $tType,Xs: list(A),X2: A] :
( ( member(A,X2,set(A,Xs))
=> ( insert(A,X2,Xs) = Xs ) )
& ( ~ member(A,X2,set(A,Xs))
=> ( insert(A,X2,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs) ) ) ) ).
tff(fact_51_not__in__set__insert,axiom,
! [A: $tType,Xs: list(A),X2: A] :
( ~ member(A,X2,set(A,Xs))
=> ( insert(A,X2,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs) ) ) ).
tff(fact_52_list_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,List2: list(B),Aa: B,F2: fun(B,fun(list(B),A)),F1: A] : aa(list(B),A,list_case(A,B,F1,F2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Aa),List2)) = aa(list(B),A,aa(B,fun(list(B),A),F2,Aa),List2) ).
tff(fact_53_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_54_hd__coeff_Osimps_I1_J,axiom,
! [Ks2: list(int),I2: int] : aa(atom,int,hd_coeff,c_PresArith_Oatom_OLe(I2,Ks2)) = aa(list(int),int,list_case(int,int,zero_zero(int),combk(int,list(int))),Ks2) ).
tff(fact_55_hd__coeff_Osimps_I3_J,axiom,
! [Ks2: list(int),I2: int,D1: int] : aa(atom,int,hd_coeff,nDvd(D1,I2,Ks2)) = aa(list(int),int,list_case(int,int,zero_zero(int),combk(int,list(int))),Ks2) ).
tff(fact_56_maps__def,axiom,
! [A: $tType,B: $tType,Xs: list(B),F: fun(B,list(A))] : maps(B,A,F,Xs) = concat(A,aa(list(B),list(list(A)),aa(fun(B,list(A)),fun(list(B),list(list(A))),map(B,list(A)),F),Xs)) ).
tff(fact_57_concat__map__maps,axiom,
! [A: $tType,B: $tType,Xs: list(B),F: fun(B,list(A))] : concat(A,aa(list(B),list(list(A)),aa(fun(B,list(A)),fun(list(B),list(list(A))),map(B,list(A)),F),Xs)) = maps(B,A,F,Xs) ).
tff(fact_58_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_59_atom_Osimps_I1_J,axiom,
! [List3: list(int),Int4: int,List2: list(int),Int3: int] :
( ( c_PresArith_Oatom_OLe(Int3,List2) = c_PresArith_Oatom_OLe(Int4,List3) )
<=> ( ( Int3 = Int4 )
& ( List2 = List3 ) ) ) ).
tff(fact_60_divisor_Osimps_I1_J,axiom,
! [Ks: list(int),I: int] : divisor(c_PresArith_Oatom_OLe(I,Ks)) = one_one(int) ).
tff(fact_61_asubst_Osimps_I6_J,axiom,
! [Va: int,V: int,Ks1: list(int),I1: int] : asubst(I1,Ks1,nDvd(V,Va,nil(int))) = nDvd(V,Va,nil(int)) ).
tff(fact_62_asubst_Osimps_I4_J,axiom,
! [V: int,Ks1: list(int),I1: int] : asubst(I1,Ks1,c_PresArith_Oatom_OLe(V,nil(int))) = c_PresArith_Oatom_OLe(V,nil(int)) ).
tff(fact_63_hd__coeff1_Osimps_I6_J,axiom,
! [Va: int,V: int,Uu: int] : aa(atom,atom,hd_coeff1(Uu),nDvd(V,Va,nil(int))) = nDvd(V,Va,nil(int)) ).
tff(fact_64_hd__coeff1_Osimps_I4_J,axiom,
! [V: int,Uu: int] : aa(atom,atom,hd_coeff1(Uu),c_PresArith_Oatom_OLe(V,nil(int))) = c_PresArith_Oatom_OLe(V,nil(int)) ).
tff(fact_65_atom_Osize_I6_J,axiom,
! [List: list(int),Int21: int,Int11: int] : size_size(atom,nDvd(Int11,Int21,List)) = zero_zero(nat) ).
tff(fact_66_atom_Osize_I4_J,axiom,
! [List: list(int),Int: int] : size_size(atom,c_PresArith_Oatom_OLe(Int,List)) = zero_zero(nat) ).
tff(fact_67_atom_Osize_I3_J,axiom,
! [List: list(int),Int21: int,Int11: int] : atom_size(nDvd(Int11,Int21,List)) = zero_zero(nat) ).
tff(fact_68_atom_Osize_I1_J,axiom,
! [List: list(int),Int: int] : atom_size(c_PresArith_Oatom_OLe(Int,List)) = zero_zero(nat) ).
tff(fact_69_atom_Osimps_I7_J,axiom,
! [List: list(int),Int: int,List1: list(int),Int2: int,Int1: int] : nDvd(Int1,Int2,List1) != c_PresArith_Oatom_OLe(Int,List) ).
tff(fact_70_atom_Osimps_I6_J,axiom,
! [List1: list(int),Int2: int,Int1: int,List: list(int),Int: int] : c_PresArith_Oatom_OLe(Int,List) != nDvd(Int1,Int2,List1) ).
tff(fact_71_divisor_Osimps_I3_J,axiom,
! [Ks: list(int),I: int,D: int] : divisor(nDvd(D,I,Ks)) = D ).
tff(fact_72_atom_Osimps_I9_J,axiom,
! [List: list(int),Int21: int,Int11: int,List1: list(int),Int2: int,Int1: int] : nDvd(Int1,Int2,List1) != dvd(Int11,Int21,List) ).
tff(fact_73_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X4: A] : aa(A,B,F,X4) = aa(A,B,G,X4)
=> ( F = G ) ) ).
tff(fact_74_mem__def,axiom,
! [A: $tType,A2: fun(A,bool),X2: A] :
( member(A,X2,A2)
<=> pp(aa(A,bool,A2,X2)) ) ).
tff(fact_75_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_76_atom_Osimps_I8_J,axiom,
! [List1: list(int),Int2: int,Int1: int,List: list(int),Int21: int,Int11: int] : dvd(Int11,Int21,List) != nDvd(Int1,Int2,List1) ).
tff(fact_77_atom_Osimps_I5_J,axiom,
! [List: list(int),Int: int,List1: list(int),Int2: int,Int1: int] : dvd(Int1,Int2,List1) != c_PresArith_Oatom_OLe(Int,List) ).
tff(fact_78_atom_Osimps_I4_J,axiom,
! [List1: list(int),Int2: int,Int1: int,List: list(int),Int: int] : c_PresArith_Oatom_OLe(Int,List) != dvd(Int1,Int2,List1) ).
tff(fact_79_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: fun(B,list(A))] : maps(B,A,F,nil(B)) = nil(A) ).
tff(fact_80_transpose_Osimps_I3_J,axiom,
! [A: $tType,Xss: list(list(A)),Xs: list(A),X2: A] : transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),Xss)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),concat(A,aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),list_case(list(A),A,nil(A),combb(list(A),fun(list(A),list(A)),A,combk(list(A),list(A)),aa(list(A),fun(A,list(A)),combc(A,list(A),list(A),cons(A)),nil(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),concat(list(A),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),list_case(list(list(A)),A,nil(list(A)),aa(fun(list(A),list(list(A))),fun(A,fun(list(A),list(list(A)))),combk(fun(list(A),list(list(A))),A),aa(list(list(A)),fun(list(A),list(list(A))),combc(list(A),list(list(A)),list(list(A)),cons(list(A))),nil(list(A)))))),Xss))))) ).
tff(fact_81_Z_Onot__dep__decr,axiom,
! [Xs: list(int),X2: int,Aa: atom] :
( ( aa(atom,int,hd_coeff,Aa) = zero_zero(int) )
=> ( i_Z(Aa,aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X2),Xs))
<=> i_Z(decr_Z(Aa),Xs) ) ) ).
tff(fact_82_splice_Osimps_I2_J,axiom,
! [A: $tType,Va: list(A),V: A] : splice(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va),nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Va) ).
tff(fact_83_transpose_Osimps_I1_J,axiom,
! [A: $tType] : transpose(A,nil(list(A))) = nil(list(A)) ).
tff(fact_84_splice_Osimps_I3_J,axiom,
! [A: $tType,Ys: list(A),Y: A,Xs1: list(A),X1: A] : splice(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),splice(A,Xs1,Ys))) ).
tff(fact_85_transpose_Osimps_I2_J,axiom,
! [A: $tType,Xss1: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss1)) = transpose(A,Xss1) ).
tff(fact_86_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys: list(A)] : splice(A,nil(A),Ys) = Ys ).
tff(fact_87_splice__Nil2,axiom,
! [A: $tType,Xs1: list(A)] : splice(A,Xs1,nil(A)) = Xs1 ).
tff(fact_88_transpose__map__map,axiom,
! [A: $tType,B: $tType,Xs: list(list(B)),F: fun(B,A)] : transpose(A,aa(list(list(B)),list(list(A)),aa(fun(list(B),list(A)),fun(list(list(B)),list(list(A))),map(list(B),list(A)),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F)),Xs)) = aa(list(list(B)),list(list(A)),aa(fun(list(B),list(A)),fun(list(list(B)),list(list(A))),map(list(B),list(A)),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F)),transpose(B,Xs)) ).
tff(fact_89_Z_Oanormal__decr,axiom,
! [A1: atom] :
( ( aa(atom,int,hd_coeff,A1) = zero_zero(int) )
=> ( ( divisor(A1) != zero_zero(int) )
=> ( divisor(decr_Z(A1)) != zero_zero(int) ) ) ) ).
tff(fact_90_transpose__empty,axiom,
! [A: $tType,Xs: list(list(A))] :
( ( transpose(A,Xs) = nil(list(A)) )
<=> ! [X3: list(A)] :
( member(list(A),X3,set(list(A),Xs))
=> ( X3 = nil(A) ) ) ) ).
tff(fact_91_transpose_Opsimps_I3_J,axiom,
! [A: $tType,Xss: list(list(A)),Xs: list(A),X2: A] :
( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),Xss))
=> ( transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),Xss)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),concat(A,aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),list_case(list(A),A,nil(A),combb(list(A),fun(list(A),list(A)),A,combk(list(A),list(A)),aa(list(A),fun(A,list(A)),combc(A,list(A),list(A),cons(A)),nil(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),concat(list(A),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),list_case(list(list(A)),A,nil(list(A)),aa(fun(list(A),list(list(A))),fun(A,fun(list(A),list(list(A)))),combk(fun(list(A),list(list(A))),A),aa(list(list(A)),fun(list(A),list(list(A))),combc(list(A),list(list(A)),list(list(A)),cons(list(A))),nil(list(A)))))),Xss))))) ) ) ).
tff(fact_92_n__lists__Nil,axiom,
! [A: $tType,N1: nat] :
( ( ( N1 = zero_zero(nat) )
=> ( n_lists(A,N1,nil(A)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ) )
& ( ( N1 != zero_zero(nat) )
=> ( n_lists(A,N1,nil(A)) = nil(list(A)) ) ) ) ).
tff(fact_93_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs1: list(A)] : n_lists(A,zero_zero(nat),Xs1) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ).
tff(fact_94_transpose_Opsimps_I1_J,axiom,
! [A: $tType] :
( accp(list(list(A)),transpose_rel(A),nil(list(A)))
=> ( transpose(A,nil(list(A))) = nil(list(A)) ) ) ).
tff(fact_95_transpose_Opsimps_I2_J,axiom,
! [A: $tType,Xss: list(list(A))] :
( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss))
=> ( transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss)) = transpose(A,Xss) ) ) ).
tff(fact_96_n__lists_Osimps_I2_J,axiom,
! [A: $tType,Xs: list(A),N: nat] : n_lists(A,suc(N),Xs) = concat(list(A),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),aa(list(A),fun(list(A),list(list(A))),combc(list(A),list(A),list(list(A)),combb(fun(A,list(A)),fun(list(A),list(list(A))),list(A),map(A,list(A)),combc(A,list(A),list(A),cons(A)))),Xs)),n_lists(A,N,Xs))) ).
tff(fact_97_transpose__aux__filter__tail,axiom,
! [A: $tType,Xss: list(list(A))] : concat(list(A),aa(list(list(A)),list(list(list(A))),aa(fun(list(A),list(list(A))),fun(list(list(A)),list(list(list(A)))),map(list(A),list(list(A))),list_case(list(list(A)),A,nil(list(A)),aa(fun(list(A),list(list(A))),fun(A,fun(list(A),list(list(A)))),combk(fun(list(A),list(list(A))),A),aa(list(list(A)),fun(list(A),list(list(A))),combc(list(A),list(list(A)),list(list(A)),cons(list(A))),nil(list(A)))))),Xss)) = aa(list(list(A)),list(list(A)),aa(fun(list(A),list(A)),fun(list(list(A)),list(list(A))),map(list(A),list(A)),tl(A)),filter(list(A),combb(bool,bool,list(A),fNot,aa(list(A),fun(list(A),bool),combc(list(A),list(A),bool,fequal(list(A))),nil(A))),Xss)) ).
%----Arities (6)
tff(arity_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Groups_Oone,axiom,
one(int) ).
tff(arity_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_Nat_Onat___Groups_Oone,axiom,
one(nat) ).
%----Helper facts (10)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
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,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),combc(A,B,C,P),Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
tff(help_COMBI_1_1_U,axiom,
! [A: $tType,P: A] : aa(A,A,combi(A),P) = P ).
tff(help_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,aa(A,fun(B,A),combk(A,B),P),Q) = P ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,Y: A,X1: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X1),Y))
| ( X1 = Y ) ) ).
tff(help_fequal_2_1_T,axiom,
! [A: $tType,Y: A,X1: A] :
( ( X1 != Y )
| pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X1),Y)) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
( ( a = dvd(zlcms(aa(list(atom),list(int),aa(fun(atom,int),fun(list(atom),list(int)),map(atom,int),hd_coeff),as)),zero_zero(int),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),one_one(int)),nil(int))) )
| ? [X: atom] :
( member(atom,X,set(atom,as))
& ( a = aa(atom,atom,hd_coeff1(zlcms(aa(list(atom),list(int),aa(fun(atom,int),fun(list(atom),list(int)),map(atom,int),hd_coeff),as))),X) ) ) ) ).
%------------------------------------------------------------------------------