TPTP Problem File: COM081_5.p
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%------------------------------------------------------------------------------
% File : COM081_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 187
% 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_187 [Bla13]
% Status : Theorem
% Rating : 0.33 v7.5.0, 0.00 v7.4.0, 0.25 v7.1.0, 0.33 v6.4.0
% Syntax : Number of formulae : 161 ( 53 unt; 53 typ; 0 def)
% Number of atoms : 216 ( 138 equ)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 134 ( 26 ~; 1 |; 23 &)
% ( 21 <=>; 63 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 13 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 60 ( 35 >; 25 *; 0 +; 0 <<)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-5 aty)
% Number of functors : 42 ( 42 usr; 5 con; 0-6 aty)
% Number of variables : 409 ( 344 !; 17 ?; 409 :)
% ( 48 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:21:33
%------------------------------------------------------------------------------
%----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 (45)
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)) > 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_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,
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_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(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_List_Otranspose__rel,type,
transpose_rel:
!>[A: $tType] : fun(list(list(A)),fun(list(list(A)),bool)) ).
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_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) * 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_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_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_xs,type,
xs: list(int) ).
%----Relevant facts (98)
tff(fact_0__096lbounds_Aas_A_126_061_A_091_093_A_061_061_062_AEX_Ax_O_AALL_Aa_058set_Aas_O_AI_092_060_094isub_062Z_Aa_A_Ix_A_D_Axs_J_096,axiom,
( ( lbounds(as) != nil(product_prod(int,list(int))) )
=> ? [X4: int] :
! [Xa2: atom] :
( member(atom,Xa2,set(atom,as))
=> pp(aa(list(int),bool,aa(atom,fun(list(int),bool),i_Z,Xa2),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X4),xs))) ) ) ).
tff(fact_1__096lbounds_Aas_A_061_A_091_093_A_061_061_062_AEX_Ax_O_AALL_Aa_058set_Aas_O_AI_092_060_094isub_062Z_Aa_A_Ix_A_D_Axs_J_096,axiom,
( ( lbounds(as) = nil(product_prod(int,list(int))) )
=> ? [X4: int] :
! [Xa2: atom] :
( member(atom,Xa2,set(atom,as))
=> pp(aa(list(int),bool,aa(atom,fun(list(int),bool),i_Z,Xa2),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X4),xs))) ) ) ).
tff(fact_2__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_3_list_Oinject,axiom,
! [A: $tType,List3: list(A),A6: A,List: list(A),A3: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),List) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),List3) )
<=> ( ( A3 = A6 )
& ( List = List3 ) ) ) ).
tff(fact_4_set__ConsD,axiom,
! [A: $tType,Xsa: list(A),X2: A,Y1: A] :
( member(A,Y1,set(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xsa)))
=> ( ( Y1 = X2 )
| member(A,Y1,set(A,Xsa)) ) ) ).
tff(fact_5_norm,axiom,
! [X: atom] :
( member(atom,X,set(atom,as))
=> ( divisor(X) != zero_zero(int) ) ) ).
tff(fact_6_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_7_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_8_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) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xsa) ) ) ) ).
tff(fact_9_not__in__set__insert,axiom,
! [A: $tType,Xsa: list(A),X2: A] :
( ~ member(A,X2,set(A,Xsa))
=> ( insert(A,X2,Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xsa) ) ) ).
tff(fact_10_remdups_Osimps_I2_J,axiom,
! [A: $tType,Xsa: list(A),X2: A] :
( ( member(A,X2,set(A,Xsa))
=> ( remdups(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xsa)) = remdups(A,Xsa) ) )
& ( ~ member(A,X2,set(A,Xsa))
=> ( remdups(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xsa)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),remdups(A,Xsa)) ) ) ) ).
tff(fact_11_list_Osimps_I2_J,axiom,
! [A: $tType,List2: list(A),A5: A] : nil(A) != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),List2) ).
tff(fact_12_list_Osimps_I3_J,axiom,
! [A: $tType,List2: list(A),A5: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),List2) != nil(A) ).
tff(fact_13_list_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,List: list(B),A3: 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),A3),List)) = aa(list(B),A,aa(B,fun(list(B),A),F2,A3),List) ).
tff(fact_14_set__remdups,axiom,
! [A: $tType,Xsa: list(A)] : set(A,remdups(A,Xsa)) = set(A,Xsa) ).
tff(fact_15_remdups_Osimps_I1_J,axiom,
! [A: $tType] : remdups(A,nil(A)) = nil(A) ).
tff(fact_16_remdups__eq__nil__right__iff,axiom,
! [A: $tType,X2: list(A)] :
( ( nil(A) = remdups(A,X2) )
<=> ( X2 = nil(A) ) ) ).
tff(fact_17_remdups__eq__nil__iff,axiom,
! [A: $tType,X2: list(A)] :
( ( remdups(A,X2) = nil(A) )
<=> ( X2 = nil(A) ) ) ).
tff(fact_18_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_19_remdups__remdups,axiom,
! [A: $tType,Xs1: list(A)] : remdups(A,remdups(A,Xs1)) = remdups(A,Xs1) ).
tff(fact_20_insert__remdups,axiom,
! [A: $tType,Xs1: list(A),X1: A] : insert(A,X1,remdups(A,Xs1)) = remdups(A,insert(A,X1,Xs1)) ).
tff(fact_21_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_22_in__set__insert,axiom,
! [A: $tType,Xsa: list(A),X2: A] :
( member(A,X2,set(A,Xsa))
=> ( insert(A,X2,Xsa) = Xsa ) ) ).
tff(fact_23_list_Oexhaust,axiom,
! [A: $tType,Y: list(A)] :
( ( Y != nil(A) )
=> ~ ! [A4: A,List1: list(A)] : Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),List1) ) ).
tff(fact_24_neq__Nil__conv,axiom,
! [A: $tType,Xsa: list(A)] :
( ( Xsa != nil(A) )
<=> ? [Y2: A,Ys2: list(A)] : Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys2) ) ).
tff(fact_25_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_26_list__ex1__simps_I1_J,axiom,
! [A: $tType,P1: fun(A,bool)] : ~ list_ex1(A,P1,nil(A)) ).
tff(fact_27_splice_Osimps_I3_J,axiom,
! [A: $tType,Ys1: 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),Ys1)) = 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,Ys1))) ).
tff(fact_28_list__nonempty__induct,axiom,
! [A: $tType,P1: fun(list(A),bool),Xsa: list(A)] :
( ( Xsa != nil(A) )
=> ( ! [X4: A] : pp(aa(list(A),bool,P1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A))))
=> ( ! [X4: A,Xs2: list(A)] :
( ( Xs2 != nil(A) )
=> ( pp(aa(list(A),bool,P1,Xs2))
=> pp(aa(list(A),bool,P1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs2))) ) )
=> pp(aa(list(A),bool,P1,Xsa)) ) ) ) ).
tff(fact_29_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_30_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_31_list__ex1__iff,axiom,
! [A: $tType,Xsa: list(A),P1: fun(A,bool)] :
( list_ex1(A,P1,Xsa)
<=> ? [X3: A] :
( member(A,X3,set(A,Xsa))
& pp(aa(A,bool,P1,X3))
& ! [Y2: A] :
( ( member(A,Y2,set(A,Xsa))
& pp(aa(A,bool,P1,Y2)) )
=> ( Y2 = X3 ) ) ) ) ).
tff(fact_32_splice__Nil2,axiom,
! [A: $tType,Xs1: list(A)] : splice(A,Xs1,nil(A)) = Xs1 ).
tff(fact_33_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys1: list(A)] : splice(A,nil(A),Ys1) = Ys1 ).
tff(fact_34_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X2: A] :
( ( zero_zero(A) = X2 )
<=> ( X2 = zero_zero(A) ) ) ) ).
tff(fact_35_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_36_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_37_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,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xsa),nil(C)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),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_38_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),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y1),Ys)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),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_39_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_40_list_Orecs_I2_J,axiom,
! [A: $tType,B: $tType,List: list(B),A3: B,F2: fun(B,fun(list(B),fun(A,A))),F1: A] : list_rec(A,B,F1,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A3),List)) = aa(A,A,aa(list(B),fun(A,A),aa(B,fun(list(B),fun(A,A)),F2,A3),List),list_rec(A,B,F1,F2,List)) ).
tff(fact_41_sublist__nil,axiom,
! [A: $tType,A2: fun(nat,bool)] : sublist(A,nil(A),A2) = nil(A) ).
tff(fact_42_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,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xsa),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Y1),Ys)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(C,A,aa(B,fun(C,A),F,X2),Y1)),zipwith0(B,C,A,F,Xsa,Ys)) ) ).
tff(fact_43_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_44_in__set__sublistD,axiom,
! [A: $tType,I1: fun(nat,bool),Xsa: list(A),X2: A] :
( member(A,X2,set(A,sublist(A,Xsa,I1)))
=> member(A,X2,set(A,Xsa)) ) ).
tff(fact_45_notin__set__sublistI,axiom,
! [A: $tType,I1: fun(nat,bool),Xsa: list(A),X2: A] :
( ~ member(A,X2,set(A,Xsa))
=> ~ member(A,X2,set(A,sublist(A,Xsa,I1))) ) ).
tff(fact_46_n__lists__Nil,axiom,
! [A: $tType,N: nat] :
( ( ( N = zero_zero(nat) )
=> ( n_lists(A,N,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))) ) )
& ( ( N != zero_zero(nat) )
=> ( n_lists(A,N,nil(A)) = nil(list(A)) ) ) ) ).
tff(fact_47_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_48_list_Osize_I1_J,axiom,
! [A: $tType,Fa: fun(A,nat)] : list_size(A,Fa,nil(A)) = zero_zero(nat) ).
tff(fact_49_iprod__Nil,axiom,
! [A: $tType] :
( ring(A)
=> ! [Ys1: list(A)] : iprod(A,nil(A),Ys1) = zero_zero(A) ) ).
tff(fact_50_iprod__Nil2,axiom,
! [A: $tType] :
( ring(A)
=> ! [Xs1: list(A)] : iprod(A,Xs1,nil(A)) = zero_zero(A) ) ).
tff(fact_51_iprod0__if__coeffs0,axiom,
! [A: $tType] :
( ring(A)
=> ! [Xsa: list(A),Cs: list(A)] :
( ! [X4: A] :
( member(A,X4,set(A,Cs))
=> ( X4 = zero_zero(A) ) )
=> ( iprod(A,Cs,Xsa) = zero_zero(A) ) ) ) ).
tff(fact_52_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_53_dropWhile__eq__Nil__conv,axiom,
! [A: $tType,Xsa: list(A),P1: fun(A,bool)] :
( ( dropWhile(A,P1,Xsa) = nil(A) )
<=> ! [X3: A] :
( member(A,X3,set(A,Xsa))
=> pp(aa(A,bool,P1,X3)) ) ) ).
tff(fact_54_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),aa(fun(C,A),fun(list(C),list(A)),map(C,A),aa(B,fun(C,A),F,zero_zero(B))),Ys) ) ).
tff(fact_55_map_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Xsa: 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),Xsa)) = 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),Xsa)) ).
tff(fact_56_map__eq__conv,axiom,
! [A: $tType,B: $tType,G: fun(B,A),Xsa: list(B),F: fun(B,A)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xsa) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),G),Xsa) )
<=> ! [X3: B] :
( member(B,X3,set(B,Xsa))
=> ( aa(B,A,F,X3) = aa(B,A,G,X3) ) ) ) ).
tff(fact_57_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,Xsa: 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),Xsa) )
<=> ( Xsa = nil(B) ) ) ).
tff(fact_58_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_59_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,Xsa: list(B),F: fun(B,A)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xsa) = nil(A) )
<=> ( Xsa = nil(B) ) ) ).
tff(fact_60_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,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xsa)) = dropWhile(A,P1,Xsa) ) )
& ( ~ pp(aa(A,bool,P1,X2))
=> ( dropWhile(A,P1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xsa)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xsa) ) ) ) ).
tff(fact_61_dropWhile_Osimps_I1_J,axiom,
! [A: $tType,P1: fun(A,bool)] : dropWhile(A,P1,nil(A)) = nil(A) ).
tff(fact_62_transpose_Osimps_I1_J,axiom,
! [A: $tType] : transpose(A,nil(list(A))) = nil(list(A)) ).
tff(fact_63_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)),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)),Xsa)) = 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,Xsa)) ).
tff(fact_64_remdups__map__remdups,axiom,
! [A: $tType,B: $tType,Xsa: list(B),F: fun(B,A)] : remdups(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),remdups(B,Xsa))) = remdups(A,aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xsa)) ).
tff(fact_65_transpose__empty,axiom,
! [A: $tType,Xsa: list(list(A))] :
( ( transpose(A,Xsa) = nil(list(A)) )
<=> ! [X3: list(A)] :
( member(list(A),X3,set(list(A),Xsa))
=> ( X3 = nil(A) ) ) ) ).
tff(fact_66_dropWhile__cong,axiom,
! [A: $tType,Q1: fun(A,bool),P1: fun(A,bool),K: list(A),L: list(A)] :
( ( L = K )
=> ( ! [X4: A] :
( member(A,X4,set(A,L))
=> ( pp(aa(A,bool,P1,X4))
<=> pp(aa(A,bool,Q1,X4)) ) )
=> ( dropWhile(A,P1,L) = dropWhile(A,Q1,K) ) ) ) ).
tff(fact_67_ex__map__conv,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Ys: list(B)] :
( ? [Xs: list(A)] : Ys = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),F),Xs)
<=> ! [X3: B] :
( member(B,X3,set(B,Ys))
=> ? [Xa1: A] : X3 = aa(A,B,F,Xa1) ) ) ).
tff(fact_68_map__ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),Xsa: list(A)] :
( ! [X4: A] :
( member(A,X4,set(A,Xsa))
=> ( 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),Xsa) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Xsa) ) ) ).
tff(fact_69_map__cong,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),Ys: list(A),Xsa: list(A)] :
( ( Xsa = Ys )
=> ( ! [X4: A] :
( member(A,X4,set(A,Ys))
=> ( 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),Xsa) = aa(list(A),list(B),aa(fun(A,B),fun(list(A),list(B)),map(A,B),G),Ys) ) ) ) ).
tff(fact_70_map__idI,axiom,
! [A: $tType,F: fun(A,A),Xsa: list(A)] :
( ! [X4: A] :
( member(A,X4,set(A,Xsa))
=> ( aa(A,A,F,X4) = X4 ) )
=> ( aa(list(A),list(A),aa(fun(A,A),fun(list(A),list(A)),map(A,A),F),Xsa) = Xsa ) ) ).
tff(fact_71_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list(B),F: fun(B,A),Xsa: list(A),X2: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xsa) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ys) )
<=> ? [Z1: B,Zs1: list(B)] :
( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z1),Zs1) )
& ( X2 = aa(B,A,F,Z1) )
& ( Xsa = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Zs1) ) ) ) ).
tff(fact_72_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),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y1),Ys) )
<=> ? [Z1: B,Zs1: list(B)] :
( ( Xsa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z1),Zs1) )
& ( aa(B,A,F,Z1) = Y1 )
& ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Zs1) = Ys ) ) ) ).
tff(fact_73_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,Ys: list(B),F: fun(B,A),Xsa: list(A),X2: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xsa) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Ys) )
=> ? [Z: B,Zs: list(B)] :
( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z),Zs) )
& ( X2 = aa(B,A,F,Z) )
& ( Xsa = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Zs) ) ) ) ).
tff(fact_74_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_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_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,Ys: list(A),Y1: A,Xsa: list(B),F: fun(B,A)] :
( ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y1),Ys) )
=> ? [Z: B,Zs: list(B)] :
( ( Xsa = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z),Zs) )
& ( aa(B,A,F,Z) = Y1 )
& ( aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Zs) = Ys ) ) ) ).
tff(fact_77_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_78_concat_Osimps_I1_J,axiom,
! [A: $tType] : concat(A,nil(list(A))) = nil(A) ).
tff(fact_79_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_80_map__concat,axiom,
! [A: $tType,B: $tType,Xsa: 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,Xsa)) = 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)),Xsa)) ).
tff(fact_81_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)),aa(fun(B,list(A)),fun(list(B),list(list(A))),map(B,list(A)),F),Xsa)) ).
tff(fact_82_concat__map__maps,axiom,
! [A: $tType,B: $tType,Xsa: 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),Xsa)) = maps(B,A,F,Xsa) ).
tff(fact_83_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: fun(B,list(A))] : maps(B,A,F,nil(B)) = nil(A) ).
tff(fact_84_transpose_Osimps_I3_J,axiom,
! [A: $tType,Xss: list(list(A)),Xsa: 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),Xsa)),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)),Xsa),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_85_divisor__asubst,axiom,
! [A1: atom,Ks: list(int),I: int] : divisor(asubst(I,Ks,A1)) = divisor(A1) ).
tff(fact_86_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_87_concat__map__singleton,axiom,
! [A: $tType,B: $tType,Xsa: 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))),Xsa)) = aa(list(B),list(A),aa(fun(B,A),fun(list(B),list(A)),map(B,A),F),Xsa) ).
tff(fact_88_transpose_Opsimps_I3_J,axiom,
! [A: $tType,Xss: list(list(A)),Xsa: 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),Xsa)),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),Xsa)),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)),Xsa),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_89_n__lists_Osimps_I2_J,axiom,
! [A: $tType,Xsa: list(A),N1: nat] : n_lists(A,suc(N1),Xsa) = 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)))),Xsa)),n_lists(A,N1,Xsa))) ).
tff(fact_90_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_91_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_92_nat_Oinject,axiom,
! [Nat1: nat,Nat: nat] :
( ( suc(Nat) = suc(Nat1) )
<=> ( Nat = Nat1 ) ) ).
tff(fact_93_n__not__Suc__n,axiom,
! [N: nat] : N != suc(N) ).
tff(fact_94_Suc__n__not__n,axiom,
! [N: nat] : suc(N) != N ).
tff(fact_95_Suc__inject,axiom,
! [Y: nat,X1: nat] :
( ( suc(X1) = suc(Y) )
=> ( X1 = Y ) ) ).
tff(fact_96_Suc__neq__Zero,axiom,
! [M: nat] : suc(M) != zero_zero(nat) ).
tff(fact_97_Zero__neq__Suc,axiom,
! [M: nat] : zero_zero(nat) != suc(M) ).
%----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 (6)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
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 ).
%----Conjectures (1)
tff(conj_0,conjecture,
? [X: int] :
! [Xa: atom] :
( member(atom,Xa,set(atom,as))
=> pp(aa(list(int),bool,aa(atom,fun(list(int),bool),i_Z,Xa),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),xs))) ) ).
%------------------------------------------------------------------------------