TPTP Problem File: COM057_5.p
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%------------------------------------------------------------------------------
% File : COM057_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 121
% 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_121 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 172 ( 52 unt; 54 typ; 0 def)
% Number of atoms : 249 ( 136 equ)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 173 ( 42 ~; 15 |; 24 &)
% ( 24 <=>; 68 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 62 ( 36 >; 26 *; 0 +; 0 <<)
% Number of predicates : 6 ( 5 usr; 0 prp; 1-5 aty)
% Number of functors : 45 ( 45 usr; 8 con; 0-6 aty)
% Number of variables : 444 ( 382 !; 13 ?; 444 :)
% ( 49 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:18:38
%------------------------------------------------------------------------------
%----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_Ozero,type,
zero:
!>[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)) * B ) > fun(A,C) ) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).
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_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)) * list(A) * list(B) ) > list(C) ) ).
tff(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( list(list(A)) > list(A) ) ).
tff(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).
tff(sy_c_List_Oinsert,type,
insert1:
!>[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__rec,type,
list_rec:
!>[T: $tType,A: $tType] : ( ( T * fun(A,fun(list(A),fun(T,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_Omap,type,
map:
!>[A: $tType,B: $tType] : ( 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_Oremdups,type,
remdups:
!>[A: $tType] : ( list(A) > list(A) ) ).
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_Otranspose,type,
transpose:
!>[A: $tType] : ( list(list(A)) > list(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_PresArith_OI_092_060_094isub_062Z,type,
i_Z: fun(atom,fun(list(int),bool)) ).
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_Set_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * fun(A,bool) ) > fun(A,bool) ) ).
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_fdisj,type,
fdisj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_fimplies,type,
fimplies: fun(bool,fun(bool,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__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_1__096lbounds_Aas_A_061_A_091_093_096,axiom,
lbounds(as) = nil(product_prod(int,list(int))) ).
tff(fact_2_list_Oinject,axiom,
! [A: $tType,List3: list(A),A5: A,List: list(A),A2: A] :
( ( cons(A,A2,List) = cons(A,A5,List3) )
<=> ( ( A2 = A5 )
& ( List = List3 ) ) ) ).
tff(fact_3_set__ConsD,axiom,
! [A: $tType,Xsa: list(A),X2: A,Y1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y1),set(A,cons(A,X2,Xsa))))
=> ( ( Y1 = X2 )
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y1),set(A,Xsa))) ) ) ).
tff(fact_4_norm,axiom,
! [X: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X),set(atom,as)))
=> ( divisor(X) != zero_zero(int) ) ) ).
tff(fact_5_not__Cons__self,axiom,
! [A: $tType,X1: A,Xs1: list(A)] : Xs1 != cons(A,X1,Xs1) ).
tff(fact_6_not__Cons__self2,axiom,
! [A: $tType,Xs1: list(A),X1: A] : cons(A,X1,Xs1) != Xs1 ).
tff(fact_7_List_Oinsert__def,axiom,
! [A: $tType,Xsa: list(A),X2: A] :
( ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),set(A,Xsa)))
=> ( insert1(A,X2,Xsa) = Xsa ) )
& ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),set(A,Xsa)))
=> ( insert1(A,X2,Xsa) = cons(A,X2,Xsa) ) ) ) ).
tff(fact_8_not__in__set__insert,axiom,
! [A: $tType,Xsa: list(A),X2: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),set(A,Xsa)))
=> ( insert1(A,X2,Xsa) = cons(A,X2,Xsa) ) ) ).
tff(fact_9_remdups_Osimps_I2_J,axiom,
! [A: $tType,Xsa: list(A),X2: A] :
( ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),set(A,Xsa)))
=> ( remdups(A,cons(A,X2,Xsa)) = remdups(A,Xsa) ) )
& ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),set(A,Xsa)))
=> ( remdups(A,cons(A,X2,Xsa)) = cons(A,X2,remdups(A,Xsa)) ) ) ) ).
tff(fact_10_list_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,List: list(B),A2: B,F2: fun(B,fun(list(B),A)),F1: A] : list_case(A,B,F1,F2,cons(B,A2,List)) = aa(list(B),A,aa(B,fun(list(B),A),F2,A2),List) ).
tff(fact_11_List_Oset_Osimps_I2_J,axiom,
! [A: $tType,Xsa: list(A),X2: A] : set(A,cons(A,X2,Xsa)) = insert(A,X2,set(A,Xsa)) ).
tff(fact_12_set__remdups,axiom,
! [A: $tType,Xsa: list(A)] : set(A,remdups(A,Xsa)) = set(A,Xsa) ).
tff(fact_13_remdups_Osimps_I1_J,axiom,
! [A: $tType] : remdups(A,nil(A)) = nil(A) ).
tff(fact_14_remdups__eq__nil__right__iff,axiom,
! [A: $tType,X2: list(A)] :
( ( nil(A) = remdups(A,X2) )
<=> ( X2 = nil(A) ) ) ).
tff(fact_15_remdups__eq__nil__iff,axiom,
! [A: $tType,X2: list(A)] :
( ( remdups(A,X2) = nil(A) )
<=> ( X2 = nil(A) ) ) ).
tff(fact_16_List_Oset__insert,axiom,
! [A: $tType,Xsa: list(A),X2: A] : set(A,insert1(A,X2,Xsa)) = insert(A,X2,set(A,Xsa)) ).
tff(fact_17_insert__Nil,axiom,
! [A: $tType,X1: A] : insert1(A,X1,nil(A)) = cons(A,X1,nil(A)) ).
tff(fact_18_remdups__remdups,axiom,
! [A: $tType,Xs1: list(A)] : remdups(A,remdups(A,Xs1)) = remdups(A,Xs1) ).
tff(fact_19_insert__remdups,axiom,
! [A: $tType,Xs1: list(A),X1: A] : insert1(A,X1,remdups(A,Xs1)) = remdups(A,insert1(A,X1,Xs1)) ).
tff(fact_20_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_21_list_Osimps_I3_J,axiom,
! [A: $tType,List2: list(A),A4: A] : cons(A,A4,List2) != nil(A) ).
tff(fact_22_list_Osimps_I2_J,axiom,
! [A: $tType,List2: list(A),A4: A] : nil(A) != cons(A,A4,List2) ).
tff(fact_23_in__set__insert,axiom,
! [A: $tType,Xsa: list(A),X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),set(A,Xsa)))
=> ( insert1(A,X2,Xsa) = Xsa ) ) ).
tff(fact_24_insert__absorb2,axiom,
! [A: $tType,A1: fun(A,bool),X2: A] : insert(A,X2,insert(A,X2,A1)) = insert(A,X2,A1) ).
tff(fact_25_insert__iff,axiom,
! [A: $tType,A1: fun(A,bool),B3: A,A2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),insert(A,B3,A1)))
<=> ( ( A2 = B3 )
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),A1)) ) ) ).
tff(fact_26_insertE,axiom,
! [A: $tType,A1: fun(A,bool),B3: A,A2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),insert(A,B3,A1)))
=> ( ( A2 != B3 )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),A1)) ) ) ).
tff(fact_27_insertCI,axiom,
! [A: $tType,B3: A,B2: fun(A,bool),A2: A] :
( ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),B2))
=> ( A2 = B3 ) )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),insert(A,B3,B2))) ) ).
tff(fact_28_list_Oexhaust,axiom,
! [A: $tType,Y: list(A)] :
( ( Y != nil(A) )
=> ~ ! [A3: A,List1: list(A)] : Y != cons(A,A3,List1) ) ).
tff(fact_29_neq__Nil__conv,axiom,
! [A: $tType,Xsa: list(A)] :
( ( Xsa != nil(A) )
<=> ? [Y2: A,Ys2: list(A)] : Xsa = cons(A,Y2,Ys2) ) ).
tff(fact_30_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_31_list__ex1__simps_I1_J,axiom,
! [A: $tType,P1: fun(A,bool)] : ~ list_ex1(A,P1,nil(A)) ).
tff(fact_32_splice_Osimps_I3_J,axiom,
! [A: $tType,Ys1: list(A),Y: A,Xs1: list(A),X1: A] : splice(A,cons(A,X1,Xs1),cons(A,Y,Ys1)) = cons(A,X1,cons(A,Y,splice(A,Xs1,Ys1))) ).
tff(fact_33_list__ex1__iff,axiom,
! [A: $tType,Xsa: list(A),P1: fun(A,bool)] :
( list_ex1(A,P1,Xsa)
<=> ? [X4: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),set(A,Xsa)))
& pp(aa(A,bool,P1,X4))
& ! [Y2: A] :
( ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y2),set(A,Xsa)))
& pp(aa(A,bool,P1,Y2)) )
=> ( Y2 = X4 ) ) ) ) ).
tff(fact_34_splice__Nil2,axiom,
! [A: $tType,Xs1: list(A)] : splice(A,Xs1,nil(A)) = Xs1 ).
tff(fact_35_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys1: list(A)] : splice(A,nil(A),Ys1) = Ys1 ).
tff(fact_36_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X2: A] :
( ( zero_zero(A) = X2 )
<=> ( X2 = zero_zero(A) ) ) ) ).
tff(fact_37_insert__absorb,axiom,
! [A: $tType,A1: fun(A,bool),A2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),A1))
=> ( insert(A,A2,A1) = A1 ) ) ).
tff(fact_38_insertI2,axiom,
! [A: $tType,B3: A,B2: fun(A,bool),A2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),B2))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),insert(A,B3,B2))) ) ).
tff(fact_39_insert__eq__iff,axiom,
! [A: $tType,B2: fun(A,bool),B3: A,A1: fun(A,bool),A2: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),A1))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B3),B2))
=> ( ( insert(A,A2,A1) = insert(A,B3,B2) )
<=> ( ( ( A2 = B3 )
=> ( A1 = B2 ) )
& ( ( A2 != B3 )
=> ? [C1: fun(A,bool)] :
( ( A1 = insert(A,B3,C1) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B3),C1))
& ( B2 = insert(A,A2,C1) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),C1)) ) ) ) ) ) ) ).
tff(fact_40_insert__ident,axiom,
! [A: $tType,B2: fun(A,bool),A1: fun(A,bool),X2: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A1))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),B2))
=> ( ( insert(A,X2,A1) = insert(A,X2,B2) )
<=> ( A1 = B2 ) ) ) ) ).
tff(fact_41_insert__code,axiom,
! [A: $tType,X2: A,A1: fun(A,bool),Y1: A] :
( pp(aa(A,bool,insert(A,Y1,A1),X2))
<=> ( ( Y1 = X2 )
| pp(aa(A,bool,A1,X2)) ) ) ).
tff(fact_42_insert__commute,axiom,
! [A: $tType,A1: fun(A,bool),Y1: A,X2: A] : insert(A,X2,insert(A,Y1,A1)) = insert(A,Y1,insert(A,X2,A1)) ).
tff(fact_43_insert__Collect,axiom,
! [A: $tType,P1: fun(A,bool),A2: A] : insert(A,A2,collect(A,P1)) = collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fimplies,combb(bool,bool,A,fNot,combc(A,A,bool,fequal(A),A2))),P1)) ).
tff(fact_44_insert__compr,axiom,
! [A: $tType,B2: fun(A,bool),A2: A] : insert(A,A2,B2) = collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fdisj,combc(A,A,bool,fequal(A),A2)),combc(A,fun(A,bool),bool,member(A),B2))) ).
tff(fact_45_insertI1,axiom,
! [A: $tType,B2: fun(A,bool),A2: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),insert(A,A2,B2))) ).
tff(fact_46_Set_Oset__insert,axiom,
! [A: $tType,A1: fun(A,bool),X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A1))
=> ~ ! [B1: fun(A,bool)] :
( ( A1 = insert(A,X2,B1) )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),B1)) ) ) ).
tff(fact_47_mk__disjoint__insert,axiom,
! [A: $tType,A1: fun(A,bool),A2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),A1))
=> ? [B1: fun(A,bool)] :
( ( A1 = insert(A,A2,B1) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A2),B1)) ) ) ).
tff(fact_48_sublist__singleton,axiom,
! [A: $tType,X2: A,A1: fun(nat,bool)] :
( ( pp(aa(fun(nat,bool),bool,aa(nat,fun(fun(nat,bool),bool),member(nat),zero_zero(nat)),A1))
=> ( sublist(A,cons(A,X2,nil(A)),A1) = cons(A,X2,nil(A)) ) )
& ( ~ pp(aa(fun(nat,bool),bool,aa(nat,fun(fun(nat,bool),bool),member(nat),zero_zero(nat)),A1))
=> ( sublist(A,cons(A,X2,nil(A)),A1) = nil(A) ) ) ) ).
tff(fact_49_list__nonempty__induct,axiom,
! [A: $tType,P1: fun(list(A),bool),Xsa: list(A)] :
( ( Xsa != nil(A) )
=> ( ! [X3: A] : pp(aa(list(A),bool,P1,cons(A,X3,nil(A))))
=> ( ! [X3: A,Xs2: list(A)] :
( ( Xs2 != nil(A) )
=> ( pp(aa(list(A),bool,P1,Xs2))
=> pp(aa(list(A),bool,P1,cons(A,X3,Xs2))) ) )
=> pp(aa(list(A),bool,P1,Xsa)) ) ) ) ).
tff(fact_50_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_51_Z_OI__list__conj,axiom,
! [Xsa: list(int),Fs: list(fm(atom))] :
( interpret(atom,int,i_Z,list_conj(atom,Fs),Xsa)
<=> ! [X4: fm(atom)] :
( pp(aa(fun(fm(atom),bool),bool,aa(fm(atom),fun(fun(fm(atom),bool),bool),member(fm(atom)),X4),set(fm(atom),Fs)))
=> interpret(atom,int,i_Z,X4,Xsa) ) ) ).
tff(fact_52_sublist__nil,axiom,
! [A: $tType,A1: fun(nat,bool)] : sublist(A,nil(A),A1) = nil(A) ).
tff(fact_53_in__set__sublistD,axiom,
! [A: $tType,I: fun(nat,bool),Xsa: list(A),X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),set(A,sublist(A,Xsa,I))))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),set(A,Xsa))) ) ).
tff(fact_54_notin__set__sublistI,axiom,
! [A: $tType,I: fun(nat,bool),Xsa: list(A),X2: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),set(A,Xsa)))
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),set(A,sublist(A,Xsa,I)))) ) ).
tff(fact_55_Z_OI__list__disj,axiom,
! [Xsa: list(int),Fs: list(fm(atom))] :
( interpret(atom,int,i_Z,list_disj(atom,Fs),Xsa)
<=> ? [X4: fm(atom)] :
( pp(aa(fun(fm(atom),bool),bool,aa(fm(atom),fun(fun(fm(atom),bool),bool),member(fm(atom)),X4),set(fm(atom),Fs)))
& interpret(atom,int,i_Z,X4,Xsa) ) ) ).
tff(fact_56_n__lists__Nil,axiom,
! [A: $tType,N: nat] :
( ( ( N = zero_zero(nat) )
=> ( n_lists(A,N,nil(A)) = cons(list(A),nil(A),nil(list(A))) ) )
& ( ( N != zero_zero(nat) )
=> ( n_lists(A,N,nil(A)) = nil(list(A)) ) ) ) ).
tff(fact_57_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_58_list_Osize_I1_J,axiom,
! [A: $tType,Fa: fun(A,nat)] : list_size(A,Fa,nil(A)) = zero_zero(nat) ).
tff(fact_59_transpose_Osimps_I2_J,axiom,
! [A: $tType,Xss1: list(list(A))] : transpose(A,cons(list(A),nil(A),Xss1)) = transpose(A,Xss1) ).
tff(fact_60_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))] : 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)),zipwith0(B,C,A,F,Xsa,nil(C))) ) ).
tff(fact_61_zipwith0_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( zero(C)
& zero(B) )
=> ! [Ys: list(C),Y1: C,F: fun(B,fun(C,A))] : zipwith0(B,C,A,F,nil(B),cons(C,Y1,Ys)) = cons(A,aa(C,A,aa(B,fun(C,A),F,zero_zero(B)),Y1),zipwith0(B,C,A,F,nil(B),Ys)) ) ).
tff(fact_62_zipwith0_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( zero(C)
& zero(B) )
=> ! [Ys: list(C),Y1: C,Xsa: list(B),X2: B,F: fun(B,fun(C,A))] : zipwith0(B,C,A,F,cons(B,X2,Xsa),cons(C,Y1,Ys)) = cons(A,aa(C,A,aa(B,fun(C,A),F,X2),Y1),zipwith0(B,C,A,F,Xsa,Ys)) ) ).
tff(fact_63_zipwith0_Osimps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( zero(B)
& zero(C) )
=> ! [F: fun(B,fun(C,A))] : zipwith0(B,C,A,F,nil(B),nil(C)) = nil(A) ) ).
tff(fact_64_transpose_Osimps_I1_J,axiom,
! [A: $tType] : transpose(A,nil(list(A))) = nil(list(A)) ).
tff(fact_65_transpose__empty,axiom,
! [A: $tType,Xsa: list(list(A))] :
( ( transpose(A,Xsa) = nil(list(A)) )
<=> ! [X4: list(A)] :
( pp(aa(fun(list(A),bool),bool,aa(list(A),fun(fun(list(A),bool),bool),member(list(A)),X4),set(list(A),Xsa)))
=> ( X4 = nil(A) ) ) ) ).
tff(fact_66_iprod__Nil,axiom,
! [A: $tType] :
( ring(A)
=> ! [Ys1: list(A)] : iprod(A,nil(A),Ys1) = zero_zero(A) ) ).
tff(fact_67_iprod__Nil2,axiom,
! [A: $tType] :
( ring(A)
=> ! [Xs1: list(A)] : iprod(A,Xs1,nil(A)) = zero_zero(A) ) ).
tff(fact_68_list_Orecs_I1_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,fun(list(B),fun(A,A))),F1: A] : list_rec(A,B,F1,F2,nil(B)) = F1 ).
tff(fact_69_list_Orecs_I2_J,axiom,
! [A: $tType,B: $tType,List: list(B),A2: B,F2: fun(B,fun(list(B),fun(A,A))),F1: A] : list_rec(A,B,F1,F2,cons(B,A2,List)) = aa(A,A,aa(list(B),fun(A,A),aa(B,fun(list(B),fun(A,A)),F2,A2),List),list_rec(A,B,F1,F2,List)) ).
tff(fact_70_iprod0__if__coeffs0,axiom,
! [A: $tType] :
( ring(A)
=> ! [Xsa: list(A),Cs: list(A)] :
( ! [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),set(A,Cs)))
=> ( X3 = zero_zero(A) ) )
=> ( iprod(A,Cs,Xsa) = zero_zero(A) ) ) ) ).
tff(fact_71_dropWhile__eq__Nil__conv,axiom,
! [A: $tType,Xsa: list(A),P1: fun(A,bool)] :
( ( dropWhile(A,P1,Xsa) = nil(A) )
<=> ! [X4: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),set(A,Xsa)))
=> pp(aa(A,bool,P1,X4)) ) ) ).
tff(fact_72_Nil__eq__concat__conv,axiom,
! [A: $tType,Xss: list(list(A))] :
( ( nil(A) = concat(A,Xss) )
<=> ! [X4: list(A)] :
( pp(aa(fun(list(A),bool),bool,aa(list(A),fun(fun(list(A),bool),bool),member(list(A)),X4),set(list(A),Xss)))
=> ( X4 = nil(A) ) ) ) ).
tff(fact_73_dropWhile_Osimps_I2_J,axiom,
! [A: $tType,Xsa: list(A),X2: A,P1: fun(A,bool)] :
( ( pp(aa(A,bool,P1,X2))
=> ( dropWhile(A,P1,cons(A,X2,Xsa)) = dropWhile(A,P1,Xsa) ) )
& ( ~ pp(aa(A,bool,P1,X2))
=> ( dropWhile(A,P1,cons(A,X2,Xsa)) = cons(A,X2,Xsa) ) ) ) ).
tff(fact_74_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X3: A] : aa(A,B,F,X3) = aa(A,B,G,X3)
=> ( F = G ) ) ).
tff(fact_75_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A1))
<=> pp(aa(A,bool,A1,X2)) ) ).
tff(fact_76_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_77_dropWhile_Osimps_I1_J,axiom,
! [A: $tType,P1: fun(A,bool)] : dropWhile(A,P1,nil(A)) = nil(A) ).
tff(fact_78_concat_Osimps_I1_J,axiom,
! [A: $tType] : concat(A,nil(list(A))) = nil(A) ).
tff(fact_79_concat__eq__Nil__conv,axiom,
! [A: $tType,Xss: list(list(A))] :
( ( concat(A,Xss) = nil(A) )
<=> ! [X4: list(A)] :
( pp(aa(fun(list(A),bool),bool,aa(list(A),fun(fun(list(A),bool),bool),member(list(A)),X4),set(list(A),Xss)))
=> ( X4 = nil(A) ) ) ) ).
tff(fact_80_dropWhile__cong,axiom,
! [A: $tType,Q1: fun(A,bool),P1: fun(A,bool),K: list(A),L: list(A)] :
( ( L = K )
=> ( ! [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),set(A,L)))
=> ( pp(aa(A,bool,P1,X3))
<=> pp(aa(A,bool,Q1,X3)) ) )
=> ( dropWhile(A,P1,L) = dropWhile(A,Q1,K) ) ) ) ).
tff(fact_81_zipwith0__Nil,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( zero(C)
& zero(B) )
=> ! [Ys: list(C),F: fun(B,fun(C,A))] : zipwith0(B,C,A,F,nil(B),Ys) = aa(list(C),list(A),map(C,A,aa(B,fun(C,A),F,zero_zero(B))),Ys) ) ).
tff(fact_82_map_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Xsa: list(B),X2: B,F: fun(B,A)] : aa(list(B),list(A),map(B,A,F),cons(B,X2,Xsa)) = cons(A,aa(B,A,F,X2),aa(list(B),list(A),map(B,A,F),Xsa)) ).
tff(fact_83_map__eq__conv,axiom,
! [A: $tType,B: $tType,G: fun(B,A),Xsa: list(B),F: fun(B,A)] :
( ( aa(list(B),list(A),map(B,A,F),Xsa) = aa(list(B),list(A),map(B,A,G),Xsa) )
<=> ! [X4: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X4),set(B,Xsa)))
=> ( aa(B,A,F,X4) = aa(B,A,G,X4) ) ) ) ).
tff(fact_84_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,Xsa: list(B),F: fun(B,A)] :
( ( aa(list(B),list(A),map(B,A,F),Xsa) = nil(A) )
<=> ( Xsa = nil(B) ) ) ).
tff(fact_85_map_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] : aa(list(B),list(A),map(B,A,F),nil(B)) = nil(A) ).
tff(fact_86_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,Xsa: list(B),F: fun(B,A)] :
( ( nil(A) = aa(list(B),list(A),map(B,A,F),Xsa) )
<=> ( Xsa = nil(B) ) ) ).
tff(fact_87_map__concat,axiom,
! [A: $tType,B: $tType,Xsa: list(list(B)),F: fun(B,A)] : aa(list(B),list(A),map(B,A,F),concat(B,Xsa)) = concat(A,aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F)),Xsa)) ).
tff(fact_88_transpose__map__map,axiom,
! [A: $tType,B: $tType,Xsa: list(list(B)),F: fun(B,A)] : transpose(A,aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F)),Xsa)) = aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F)),transpose(B,Xsa)) ).
tff(fact_89_remdups__map__remdups,axiom,
! [A: $tType,B: $tType,Xsa: list(B),F: fun(B,A)] : remdups(A,aa(list(B),list(A),map(B,A,F),remdups(B,Xsa))) = remdups(A,aa(list(B),list(A),map(B,A,F),Xsa)) ).
tff(fact_90_map__cong,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),Ys: list(A),Xsa: list(A)] :
( ( Xsa = Ys )
=> ( ! [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),set(A,Ys)))
=> ( aa(A,B,F,X3) = aa(A,B,G,X3) ) )
=> ( aa(list(A),list(B),map(A,B,F),Xsa) = aa(list(A),list(B),map(A,B,G),Ys) ) ) ) ).
tff(fact_91_map__idI,axiom,
! [A: $tType,F: fun(A,A),Xsa: list(A)] :
( ! [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),set(A,Xsa)))
=> ( aa(A,A,F,X3) = X3 ) )
=> ( aa(list(A),list(A),map(A,A,F),Xsa) = Xsa ) ) ).
tff(fact_92_ex__map__conv,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Ys: list(B)] :
( ? [Xs: list(A)] : Ys = aa(list(A),list(B),map(A,B,F),Xs)
<=> ! [X4: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X4),set(B,Ys)))
=> ? [Xa1: A] : X4 = aa(A,B,F,Xa1) ) ) ).
tff(fact_93_map__ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),Xsa: list(A)] :
( ! [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),set(A,Xsa)))
=> ( aa(A,B,F,X3) = aa(A,B,G,X3) ) )
=> ( aa(list(A),list(B),map(A,B,F),Xsa) = aa(list(A),list(B),map(A,B,G),Xsa) ) ) ).
tff(fact_94_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list(B),F: fun(B,A),Xsa: list(A),X2: A] :
( ( cons(A,X2,Xsa) = aa(list(B),list(A),map(B,A,F),Ys) )
<=> ? [Z: B,Zs: list(B)] :
( ( Ys = cons(B,Z,Zs) )
& ( X2 = aa(B,A,F,Z) )
& ( Xsa = aa(list(B),list(A),map(B,A,F),Zs) ) ) ) ).
tff(fact_95_map__eq__Cons__conv,axiom,
! [B: $tType,A: $tType,Ys: list(A),Y1: A,Xsa: list(B),F: fun(B,A)] :
( ( aa(list(B),list(A),map(B,A,F),Xsa) = cons(A,Y1,Ys) )
<=> ? [Z: B,Zs: list(B)] :
( ( Xsa = cons(B,Z,Zs) )
& ( aa(B,A,F,Z) = Y1 )
& ( aa(list(B),list(A),map(B,A,F),Zs) = Ys ) ) ) ).
tff(fact_96_concat__map__maps,axiom,
! [A: $tType,B: $tType,Xsa: list(B),F: fun(B,list(A))] : concat(A,aa(list(B),list(list(A)),map(B,list(A),F),Xsa)) = maps(B,A,F,Xsa) ).
tff(fact_97_maps__def,axiom,
! [A: $tType,B: $tType,Xsa: list(B),F: fun(B,list(A))] : maps(B,A,F,Xsa) = concat(A,aa(list(B),list(list(A)),map(B,list(A),F),Xsa)) ).
tff(fact_98_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: fun(B,list(A))] : maps(B,A,F,nil(B)) = nil(A) ).
%----Arities (3)
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Rings_Oring,axiom,
ring(int) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
%----Helper facts (15)
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,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
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,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).
tff(help_fdisj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q))
| pp(P)
| pp(Q) ) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,Y: A,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)) ) ).
tff(help_fimplies_1_1_U,axiom,
! [Q: bool,P: bool] :
( pp(P)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).
tff(help_fimplies_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).
tff(help_fimplies_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q))
| ~ pp(P)
| pp(Q) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
? [X: int] :
! [Xa: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),Xa),set(atom,as)))
=> pp(aa(list(int),bool,aa(atom,fun(list(int),bool),i_Z,Xa),cons(int,X,xs))) ) ).
%------------------------------------------------------------------------------