TPTP Problem File: COM084_5.p
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%------------------------------------------------------------------------------
% File : COM084_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 199
% 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_199 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 176 ( 67 unt; 55 typ; 0 def)
% Number of atoms : 217 ( 137 equ)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 134 ( 38 ~; 7 |; 13 &)
% ( 22 <=>; 54 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 13 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 55 ( 31 >; 24 *; 0 +; 0 <<)
% Number of predicates : 7 ( 6 usr; 0 prp; 1-2 aty)
% Number of functors : 45 ( 45 usr; 9 con; 0-5 aty)
% Number of variables : 412 ( 364 !; 11 ?; 412 :)
% ( 37 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:21:54
%------------------------------------------------------------------------------
%----Should-be-implicit typings (7)
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 ).
tff(ty_tc_prod,type,
product_prod: ( $tType * $tType ) > $tType ).
%----Explicit typings (48)
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_cl_Divides_Osemiring__div,type,
semiring_div:
!>[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_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_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_Divides_Odiv__class_Omod,type,
div_mod:
!>[A: $tType] : ( ( A * A ) > 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)) * list(A) ) > T ) ).
tff(sy_c_List_Omap,type,
map:
!>[A: $tType,B: $tType] : ( ( fun(A,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_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__case,type,
atom_case:
!>[T: $tType] : ( ( fun(int,fun(list(int),T)) * fun(int,fun(int,fun(list(int),T))) * fun(int,fun(int,fun(list(int),T))) ) > fun(atom,T) ) ).
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: fun(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_Olbounds,type,
lbounds: list(atom) > list(product_prod(int,list(int))) ).
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_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_fconj,type,
fconj: 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_a____,type,
a: atom ).
tff(sy_v_as,type,
as: list(atom) ).
tff(sy_v_x____,type,
x: int ).
tff(sy_v_xs,type,
xs: list(int) ).
%----Relevant facts (98)
tff(fact_0__096EX_Ax_O_AALL_Aa_058set_Aas_O_AI_092_060_094isub_062Z_Aa_A_Ix_A_D_Axs_J_096,axiom,
? [X1: int] :
! [Xa2: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),Xa2),set(atom,as)))
=> i_Z(Xa2,aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X1),xs)) ) ).
tff(fact_1_x,axiom,
! [X3: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X3),set(atom,as)))
=> i_Z(X3,aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),x),xs)) ) ).
tff(fact_2_norm,axiom,
! [X3: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X3),set(atom,as)))
=> ( aa(atom,int,divisor,X3) != zero_zero(int) ) ) ).
tff(fact_3__096_B_Bthesis_O_A_I_B_Bx_O_AALL_Aa_058set_Aas_O_AI_092_060_094isub_062Z_Aa_A_Ix_A_D_Axs_J_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [X1: int] :
~ ! [Xa2: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),Xa2),set(atom,as)))
=> i_Z(Xa2,aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X1),xs)) ) ).
tff(fact_4_filter_Osimps_I2_J,axiom,
! [A: $tType,Xsa: list(A),Xa: A,P1: fun(A,bool)] :
( ( pp(aa(A,bool,P1,Xa))
=> ( filter(A,P1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),filter(A,P1,Xsa)) ) )
& ( ~ pp(aa(A,bool,P1,Xa))
=> ( filter(A,P1,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa)) = filter(A,P1,Xsa) ) ) ) ).
tff(fact_5_map_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Xsa: list(B),Xa: B,F: fun(B,A)] : map(B,A,F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xa),Xsa)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F,Xa)),map(B,A,F,Xsa)) ).
tff(fact_6_I__cyclic,axiom,
! [E: list(int),J: int,I3: int,Aa: atom] :
( pp(aa(atom,bool,atom_case(bool,aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),aa(bool,fun(list(int),bool),combk(bool,list(int)),fFalse)),aa(fun(int,fun(list(int),bool)),fun(int,fun(int,fun(list(int),bool))),combk(fun(int,fun(list(int),bool)),int),aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),aa(bool,fun(list(int),bool),combk(bool,list(int)),fTrue))),aa(fun(int,fun(list(int),bool)),fun(int,fun(int,fun(list(int),bool))),combk(fun(int,fun(list(int),bool)),int),aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),aa(bool,fun(list(int),bool),combk(bool,list(int)),fTrue)))),Aa))
=> ( ( aa(atom,int,hd_coeff,Aa) = one_one(int) )
=> ( ( div_mod(int,I3,aa(atom,int,divisor,Aa)) = div_mod(int,J,aa(atom,int,divisor,Aa)) )
=> ( i_Z(Aa,aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I3),E))
<=> i_Z(Aa,aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),E)) ) ) ) ) ).
tff(fact_7_filter__filter,axiom,
! [A: $tType,Xsa: list(A),Q1: fun(A,bool),P1: fun(A,bool)] : filter(A,P1,filter(A,Q1,Xsa)) = filter(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fconj,Q1),P1),Xsa) ).
tff(fact_8_map__ident,axiom,
! [A: $tType,X3: list(A)] : map(A,A,combi(A),X3) = X3 ).
tff(fact_9_calculation,axiom,
lbounds(as) = nil(product_prod(int,list(int))) ).
tff(fact_10_mod__mod__trivial,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [B1: A,A1: A] : div_mod(A,div_mod(A,A1,B1),B1) = div_mod(A,A1,B1) ) ).
tff(fact_11_list_Oinject,axiom,
! [A: $tType,List3: list(A),A5: A,List: list(A),Aa: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa),List) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),List3) )
<=> ( ( Aa = A5 )
& ( List = List3 ) ) ) ).
tff(fact_12_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys1: list(B),F: fun(B,A),Xsa: list(A),Xa: A] :
( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) = map(B,A,F,Ys1) )
<=> ? [Z: B,Zs: list(B)] :
( ( Ys1 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z),Zs) )
& ( Xa = aa(B,A,F,Z) )
& ( Xsa = map(B,A,F,Zs) ) ) ) ).
tff(fact_13_map__eq__Cons__conv,axiom,
! [B: $tType,A: $tType,Ys1: list(A),Y2: A,Xsa: list(B),F: fun(B,A)] :
( ( map(B,A,F,Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys1) )
<=> ? [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) = Y2 )
& ( map(B,A,F,Zs) = Ys1 ) ) ) ).
tff(fact_14_mod__self,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [A1: A] : div_mod(A,A1,A1) = zero_zero(A) ) ).
tff(fact_15_mod__0,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [A1: A] : div_mod(A,zero_zero(A),A1) = zero_zero(A) ) ).
tff(fact_16_map__eq__conv,axiom,
! [A: $tType,B: $tType,G: fun(B,A),Xsa: list(B),F: fun(B,A)] :
( ( map(B,A,F,Xsa) = map(B,A,G,Xsa) )
<=> ! [X2: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),set(B,Xsa)))
=> ( aa(B,A,F,X2) = aa(B,A,G,X2) ) ) ) ).
tff(fact_17_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,Xsa: list(B),F: fun(B,A)] :
( ( map(B,A,F,Xsa) = nil(A) )
<=> ( Xsa = nil(B) ) ) ).
tff(fact_18_map_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] : map(B,A,F,nil(B)) = nil(A) ).
tff(fact_19_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,Xsa: list(B),F: fun(B,A)] :
( ( nil(A) = map(B,A,F,Xsa) )
<=> ( Xsa = nil(B) ) ) ).
tff(fact_20_zmod__self,axiom,
! [A1: int] : div_mod(int,A1,A1) = zero_zero(int) ).
tff(fact_21_zmod__zero,axiom,
! [B1: int] : div_mod(int,zero_zero(int),B1) = zero_zero(int) ).
tff(fact_22_set__filter,axiom,
! [A: $tType,Xsa: list(A),P1: fun(A,bool)] : set(A,filter(A,P1,Xsa)) = collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fconj,combc(A,fun(A,bool),bool,member(A),set(A,Xsa))),P1)) ).
tff(fact_23_filter_Osimps_I1_J,axiom,
! [A: $tType,P1: fun(A,bool)] : filter(A,P1,nil(A)) = nil(A) ).
tff(fact_24_mod__by__1,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [A1: A] : div_mod(A,A1,one_one(A)) = zero_zero(A) ) ).
tff(fact_25_filter__empty__conv,axiom,
! [A: $tType,Xsa: list(A),P1: fun(A,bool)] :
( ( filter(A,P1,Xsa) = nil(A) )
<=> ! [X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),set(A,Xsa)))
=> ~ pp(aa(A,bool,P1,X2)) ) ) ).
tff(fact_26_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_27_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_28_mod__by__0,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [A1: A] : div_mod(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_29_set__ConsD,axiom,
! [A: $tType,Xsa: list(A),Xa: A,Y2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y2),set(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa))))
=> ( ( Y2 = Xa )
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y2),set(A,Xsa))) ) ) ).
tff(fact_30_filter__id__conv,axiom,
! [A: $tType,Xsa: list(A),P1: fun(A,bool)] :
( ( filter(A,P1,Xsa) = Xsa )
<=> ! [X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),set(A,Xsa)))
=> pp(aa(A,bool,P1,X2)) ) ) ).
tff(fact_31_not__Cons__self2,axiom,
! [A: $tType,Xs1: list(A),X: A] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1) != Xs1 ).
tff(fact_32_not__Cons__self,axiom,
! [A: $tType,X: A,Xs1: list(A)] : Xs1 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1) ).
tff(fact_33_filter__False,axiom,
! [A: $tType,P1: fun(A,bool),Xsa: list(A)] :
( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),set(A,Xsa)))
=> ~ pp(aa(A,bool,P1,X1)) )
=> ( filter(A,P1,Xsa) = nil(A) ) ) ).
tff(fact_34_I__hd__coeffs1,axiom,
! [Xsa: list(int),Asa: list(atom)] :
( ! [X1: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X1),set(atom,Asa)))
=> ( aa(atom,int,hd_coeff,X1) != zero_zero(int) ) )
=> ( ? [X2: int] :
! [Xa1: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),Xa1),set(atom,qEpres1907640072oeffs1(Asa))))
=> i_Z(Xa1,aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X2),Xsa)) )
<=> ? [X2: int] :
! [Xa1: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),Xa1),set(atom,Asa)))
=> i_Z(Xa1,aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X2),Xsa)) ) ) ) ).
tff(fact_35_filter__True,axiom,
! [A: $tType,P1: fun(A,bool),Xsa: list(A)] :
( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),set(A,Xsa)))
=> pp(aa(A,bool,P1,X1)) )
=> ( filter(A,P1,Xsa) = Xsa ) ) ).
tff(fact_36_filter__cong,axiom,
! [A: $tType,Q1: fun(A,bool),P1: fun(A,bool),Ys1: list(A),Xsa: list(A)] :
( ( Xsa = Ys1 )
=> ( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),set(A,Ys1)))
=> ( pp(aa(A,bool,P1,X1))
<=> pp(aa(A,bool,Q1,X1)) ) )
=> ( filter(A,P1,Xsa) = filter(A,Q1,Ys1) ) ) ) ).
tff(fact_37_map__cong,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),Ys1: list(A),Xsa: list(A)] :
( ( Xsa = Ys1 )
=> ( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),set(A,Ys1)))
=> ( aa(A,B,F,X1) = aa(A,B,G,X1) ) )
=> ( map(A,B,F,Xsa) = map(A,B,G,Ys1) ) ) ) ).
tff(fact_38_map__idI,axiom,
! [A: $tType,F: fun(A,A),Xsa: list(A)] :
( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),set(A,Xsa)))
=> ( aa(A,A,F,X1) = X1 ) )
=> ( map(A,A,F,Xsa) = Xsa ) ) ).
tff(fact_39_map__ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),Xsa: list(A)] :
( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),set(A,Xsa)))
=> ( aa(A,B,F,X1) = aa(A,B,G,X1) ) )
=> ( map(A,B,F,Xsa) = map(A,B,G,Xsa) ) ) ).
tff(fact_40_ex__map__conv,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Ys1: list(B)] :
( ? [Xs: list(A)] : Ys1 = map(A,B,F,Xs)
<=> ! [X2: B] :
( pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),X2),set(B,Ys1)))
=> ? [Xa1: A] : X2 = aa(A,B,F,Xa1) ) ) ).
tff(fact_41_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [Xa: A] :
( ( zero_zero(A) = Xa )
<=> ( Xa = zero_zero(A) ) ) ) ).
tff(fact_42_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [Xa: A] :
( ( one_one(A) = Xa )
<=> ( Xa = one_one(A) ) ) ) ).
tff(fact_43_zlcms_Osimps_I1_J,axiom,
zlcms(nil(int)) = one_one(int) ).
tff(fact_44_zlcms0__iff,axiom,
! [Is: list(int)] :
( ( zlcms(Is) = zero_zero(int) )
<=> pp(aa(fun(int,bool),bool,aa(int,fun(fun(int,bool),bool),member(int),zero_zero(int)),set(int,Is))) ) ).
tff(fact_45_neq__Nil__conv,axiom,
! [A: $tType,Xsa: list(A)] :
( ( Xsa != nil(A) )
<=> ? [Y1: A,Ys: list(A)] : Xsa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y1),Ys) ) ).
tff(fact_46_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_47_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(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)))),map(atom,atom,hd_coeff1(zlcms(map(atom,int,hd_coeff,Asa))),Asa)) ).
tff(fact_48_atom_Osimps_I2_J,axiom,
! [List3: list(int),Int23: int,Int13: int,List: list(int),Int2: int,Int1: int] :
( ( dvd(Int1,Int2,List) = dvd(Int13,Int23,List3) )
<=> ( ( Int1 = Int13 )
& ( Int2 = Int23 )
& ( List = List3 ) ) ) ).
tff(fact_49_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_50_divisor_Osimps_I2_J,axiom,
! [Ks: list(int),I1: int,D: int] : aa(atom,int,divisor,dvd(D,I1,Ks)) = D ).
tff(fact_51_atom_Osimps_I11_J,axiom,
! [A: $tType,List: list(int),Int2: int,Int1: int,F3: fun(int,fun(int,fun(list(int),A))),F2: fun(int,fun(int,fun(list(int),A))),F1: fun(int,fun(list(int),A))] : aa(atom,A,atom_case(A,F1,F2,F3),dvd(Int1,Int2,List)) = aa(list(int),A,aa(int,fun(list(int),A),aa(int,fun(int,fun(list(int),A)),F2,Int1),Int2),List) ).
tff(fact_52_hd__coeff_Osimps_I2_J,axiom,
! [Ks2: list(int),I3: int,D1: int] : aa(atom,int,hd_coeff,dvd(D1,I3,Ks2)) = list_case(int,int,zero_zero(int),combk(int,list(int)),Ks2) ).
tff(fact_53_atom_Osize_I2_J,axiom,
! [List2: list(int),Int22: int,Int12: int] : atom_size(dvd(Int12,Int22,List2)) = zero_zero(nat) ).
tff(fact_54_asubst_Osimps_I5_J,axiom,
! [Va: int,V: int,Ks1: list(int),I2: int] : asubst(I2,Ks1,dvd(V,Va,nil(int))) = dvd(V,Va,nil(int)) ).
tff(fact_55_atom_Osize_I5_J,axiom,
! [List2: list(int),Int22: int,Int12: int] : size_size(atom,dvd(Int12,Int22,List2)) = zero_zero(nat) ).
tff(fact_56_divisor__asubst,axiom,
! [A1: atom,Ks: list(int),I1: int] : aa(atom,int,divisor,asubst(I1,Ks,A1)) = aa(atom,int,divisor,A1) ).
tff(fact_57_list_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,List: list(B),Aa: B,F2: fun(B,fun(list(B),A)),F1: A] : list_case(A,B,F1,F2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Aa),List)) = aa(list(B),A,aa(B,fun(list(B),A),F2,Aa),List) ).
tff(fact_58_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_59_hd__coeff_Osimps_I3_J,axiom,
! [Ks2: list(int),I3: int,D1: int] : aa(atom,int,hd_coeff,nDvd(D1,I3,Ks2)) = list_case(int,int,zero_zero(int),combk(int,list(int)),Ks2) ).
tff(fact_60_hd__coeff_Osimps_I1_J,axiom,
! [Ks2: list(int),I3: int] : aa(atom,int,hd_coeff,c_PresArith_Oatom_OLe(I3,Ks2)) = list_case(int,int,zero_zero(int),combk(int,list(int)),Ks2) ).
tff(fact_61_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
tff(fact_62_atom_Osimps_I3_J,axiom,
! [List3: list(int),Int23: int,Int13: int,List: list(int),Int2: int,Int1: int] :
( ( nDvd(Int1,Int2,List) = nDvd(Int13,Int23,List3) )
<=> ( ( Int1 = Int13 )
& ( Int2 = Int23 )
& ( List = List3 ) ) ) ).
tff(fact_63_atom_Osimps_I1_J,axiom,
! [List3: list(int),Int4: int,List: list(int),Int: int] :
( ( c_PresArith_Oatom_OLe(Int,List) = c_PresArith_Oatom_OLe(Int4,List3) )
<=> ( ( Int = Int4 )
& ( List = List3 ) ) ) ).
tff(fact_64_divisor_Osimps_I1_J,axiom,
! [Ks: list(int),I1: int] : aa(atom,int,divisor,c_PresArith_Oatom_OLe(I1,Ks)) = one_one(int) ).
tff(fact_65_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_66_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_67_asubst_Osimps_I6_J,axiom,
! [Va: int,V: int,Ks1: list(int),I2: int] : asubst(I2,Ks1,nDvd(V,Va,nil(int))) = nDvd(V,Va,nil(int)) ).
tff(fact_68_asubst_Osimps_I4_J,axiom,
! [V: int,Ks1: list(int),I2: int] : asubst(I2,Ks1,c_PresArith_Oatom_OLe(V,nil(int))) = c_PresArith_Oatom_OLe(V,nil(int)) ).
tff(fact_69_atom_Osize_I3_J,axiom,
! [List2: list(int),Int22: int,Int12: int] : atom_size(nDvd(Int12,Int22,List2)) = zero_zero(nat) ).
tff(fact_70_atom_Osize_I6_J,axiom,
! [List2: list(int),Int22: int,Int12: int] : size_size(atom,nDvd(Int12,Int22,List2)) = zero_zero(nat) ).
tff(fact_71_atom_Osize_I1_J,axiom,
! [List2: list(int),Int3: int] : atom_size(c_PresArith_Oatom_OLe(Int3,List2)) = zero_zero(nat) ).
tff(fact_72_atom_Osize_I4_J,axiom,
! [List2: list(int),Int3: int] : size_size(atom,c_PresArith_Oatom_OLe(Int3,List2)) = zero_zero(nat) ).
tff(fact_73_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X1: A] : aa(A,B,F,X1) = aa(A,B,G,X1)
=> ( F = G ) ) ).
tff(fact_74_mem__def,axiom,
! [A: $tType,A2: fun(A,bool),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),A2))
<=> pp(aa(A,bool,A2,Xa)) ) ).
tff(fact_75_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_76_divisor_Osimps_I3_J,axiom,
! [Ks: list(int),I1: int,D: int] : aa(atom,int,divisor,nDvd(D,I1,Ks)) = D ).
tff(fact_77_atom_Osimps_I7_J,axiom,
! [List2: list(int),Int3: int,List1: list(int),Int21: int,Int11: int] : nDvd(Int11,Int21,List1) != c_PresArith_Oatom_OLe(Int3,List2) ).
tff(fact_78_atom_Osimps_I6_J,axiom,
! [List1: list(int),Int21: int,Int11: int,List2: list(int),Int3: int] : c_PresArith_Oatom_OLe(Int3,List2) != nDvd(Int11,Int21,List1) ).
tff(fact_79_atom_Osimps_I4_J,axiom,
! [List1: list(int),Int21: int,Int11: int,List2: list(int),Int3: int] : c_PresArith_Oatom_OLe(Int3,List2) != dvd(Int11,Int21,List1) ).
tff(fact_80_atom_Osimps_I5_J,axiom,
! [List2: list(int),Int3: int,List1: list(int),Int21: int,Int11: int] : dvd(Int11,Int21,List1) != c_PresArith_Oatom_OLe(Int3,List2) ).
tff(fact_81_atom_Osimps_I8_J,axiom,
! [List1: list(int),Int21: int,Int11: int,List2: list(int),Int22: int,Int12: int] : dvd(Int12,Int22,List2) != nDvd(Int11,Int21,List1) ).
tff(fact_82_atom_Osimps_I9_J,axiom,
! [List2: list(int),Int22: int,Int12: int,List1: list(int),Int21: int,Int11: int] : nDvd(Int11,Int21,List1) != dvd(Int12,Int22,List2) ).
tff(fact_83_atom_Osimps_I12_J,axiom,
! [A: $tType,List: list(int),Int2: int,Int1: int,F3: fun(int,fun(int,fun(list(int),A))),F2: fun(int,fun(int,fun(list(int),A))),F1: fun(int,fun(list(int),A))] : aa(atom,A,atom_case(A,F1,F2,F3),nDvd(Int1,Int2,List)) = aa(list(int),A,aa(int,fun(list(int),A),aa(int,fun(int,fun(list(int),A)),F3,Int1),Int2),List) ).
tff(fact_84_atom_Osimps_I10_J,axiom,
! [A: $tType,List: list(int),Int: int,F3: fun(int,fun(int,fun(list(int),A))),F2: fun(int,fun(int,fun(list(int),A))),F1: fun(int,fun(list(int),A))] : aa(atom,A,atom_case(A,F1,F2,F3),c_PresArith_Oatom_OLe(Int,List)) = aa(list(int),A,aa(int,fun(list(int),A),F1,Int),List) ).
tff(fact_85_one__neq__zero,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( one_one(A) != zero_zero(A) ) ) ).
tff(fact_86_insert__Nil,axiom,
! [A: $tType,X: A] : insert(A,X,nil(A)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ).
tff(fact_87_sublist__singleton,axiom,
! [A: $tType,Xa: A,A2: fun(nat,bool)] :
( ( pp(aa(fun(nat,bool),bool,aa(nat,fun(fun(nat,bool),bool),member(nat),zero_zero(nat)),A2))
=> ( sublist(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)),A2) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)) ) )
& ( ~ pp(aa(fun(nat,bool),bool,aa(nat,fun(fun(nat,bool),bool),member(nat),zero_zero(nat)),A2))
=> ( sublist(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),nil(A)),A2) = nil(A) ) ) ) ).
tff(fact_88_Z_Onot__dep__decr,axiom,
! [Xsa: list(int),Xa: 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),Xa),Xsa))
<=> i_Z(decr_Z(Aa),Xsa) ) ) ).
tff(fact_89_sublist__nil,axiom,
! [A: $tType,A2: fun(nat,bool)] : sublist(A,nil(A),A2) = nil(A) ).
tff(fact_90_notin__set__sublistI,axiom,
! [A: $tType,I: fun(nat,bool),Xsa: list(A),Xa: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,Xsa)))
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,sublist(A,Xsa,I)))) ) ).
tff(fact_91_in__set__sublistD,axiom,
! [A: $tType,I: fun(nat,bool),Xsa: list(A),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,sublist(A,Xsa,I))))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,Xsa))) ) ).
tff(fact_92_in__set__insert,axiom,
! [A: $tType,Xsa: list(A),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,Xsa)))
=> ( insert(A,Xa,Xsa) = Xsa ) ) ).
tff(fact_93_List_Oinsert__def,axiom,
! [A: $tType,Xsa: list(A),Xa: A] :
( ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,Xsa)))
=> ( insert(A,Xa,Xsa) = Xsa ) )
& ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,Xsa)))
=> ( insert(A,Xa,Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) ) ) ) ).
tff(fact_94_not__in__set__insert,axiom,
! [A: $tType,Xsa: list(A),Xa: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),set(A,Xsa)))
=> ( insert(A,Xa,Xsa) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xa),Xsa) ) ) ).
tff(fact_95_Z_Oanormal__decr,axiom,
! [A1: atom] :
( ( aa(atom,int,hd_coeff,A1) = zero_zero(int) )
=> ( ( aa(atom,int,divisor,A1) != zero_zero(int) )
=> ( aa(atom,int,divisor,decr_Z(A1)) != zero_zero(int) ) ) ) ).
tff(fact_96_concat__map__singleton,axiom,
! [A: $tType,B: $tType,Xsa: list(B),F: fun(B,A)] : concat(A,map(B,list(A),combc(B,list(A),list(A),combb(A,fun(list(A),list(A)),B,cons(A),F),nil(A)),Xsa)) = map(B,A,F,Xsa) ).
tff(fact_97_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) ).
%----Arities (8)
tff(arity_Int_Oint___Divides_Osemiring__div,axiom,
semiring_div(int) ).
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___Divides_Osemiring__div,axiom,
semiring_div(nat) ).
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 (14)
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,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_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_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(Q) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
~ ( i_Z(a,aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),div_mod(int,x,zlcms(map(atom,int,divisor,filter(atom,atom_case(bool,aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),aa(bool,fun(list(int),bool),combk(bool,list(int)),fFalse)),aa(fun(int,fun(list(int),bool)),fun(int,fun(int,fun(list(int),bool))),combk(fun(int,fun(list(int),bool)),int),aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),aa(bool,fun(list(int),bool),combk(bool,list(int)),fTrue))),aa(fun(int,fun(list(int),bool)),fun(int,fun(int,fun(list(int),bool))),combk(fun(int,fun(list(int),bool)),int),aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),aa(bool,fun(list(int),bool),combk(bool,list(int)),fTrue)))),as))))),xs))
<=> ~ i_Z(a,aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),x),xs)) ) ).
%------------------------------------------------------------------------------