TPTP Problem File: COM122_5.p
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%------------------------------------------------------------------------------
% File : COM122_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 294
% 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_294 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.00 v6.4.0
% Syntax : Number of formulae : 166 ( 47 unt; 50 typ; 0 def)
% Number of atoms : 234 ( 107 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 159 ( 41 ~; 4 |; 14 &)
% ( 31 <=>; 69 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 4 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 38 ( 24 >; 14 *; 0 +; 0 <<)
% Number of predicates : 16 ( 15 usr; 0 prp; 1-3 aty)
% Number of functors : 31 ( 31 usr; 6 con; 0-4 aty)
% Number of variables : 281 ( 239 !; 11 ?; 281 :)
% ( 31 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:25:46
%------------------------------------------------------------------------------
%----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 (44)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Osgn__if,type,
sgn_if:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_c_COMBI,type,
combi:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Int_Oiszero,type,
iszero:
!>[A: $tType] : ( A > $o ) ).
tff(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
tff(sy_c_List_OListMem,type,
listMem:
!>[A: $tType] : ( ( A * list(A) ) > $o ) ).
tff(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olist__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) * list(A) ) > list(B) ) ).
tff(sy_c_List_Omember,type,
member1:
!>[A: $tType] : fun(list(A),fun(A,bool)) ).
tff(sy_c_List_Oset,type,
set:
!>[A: $tType] : fun(list(A),fun(A,bool)) ).
tff(sy_c_List_Osublist,type,
sublist:
!>[A: $tType] : ( ( list(A) * fun(nat,bool) ) > list(A) ) ).
tff(sy_c_Nat_Osemiring__1__class_ONats,type,
semiring_1_Nats:
!>[A: $tType] : fun(A,bool) ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $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_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_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_a,type,
a: atom ).
tff(sy_v_as,type,
as: list(atom) ).
tff(sy_v_b____,type,
b: atom ).
%----Relevant facts (98)
tff(fact_0_div0,axiom,
! [X: atom] :
( member(atom,X,aa(list(atom),fun(atom,bool),set(atom),as))
=> ( divisor(X) != zero_zero(int) ) ) ).
tff(fact_1_a,axiom,
member(atom,a,aa(list(atom),fun(atom,bool),set(atom),qEpres1907640072oeffs1(as))) ).
tff(fact_2_hd0,axiom,
! [X: atom] :
( member(atom,X,aa(list(atom),fun(atom,bool),set(atom),as))
=> ( aa(atom,int,hd_coeff,X) != zero_zero(int) ) ) ).
tff(fact_3__096b_A_058_Aset_Aas_096,axiom,
member(atom,b,aa(list(atom),fun(atom,bool),set(atom),as)) ).
tff(fact_4__096a_A_061_Ahd__coeff1_A_Izlcms_A_Imap_Ahd__coeff_Aas_J_J_Ab_096,axiom,
a = aa(atom,atom,hd_coeff1(zlcms(map(atom,int,hd_coeff,as))),b) ).
tff(fact_5__096EX_Ab_058set_Aas_O_Aa_A_061_Ahd__coeff1_A_Izlcms_A_Imap_Ahd__coeff_Aas_J_J_Ab_096,axiom,
? [X3: atom] :
( member(atom,X3,aa(list(atom),fun(atom,bool),set(atom),as))
& ( a = aa(atom,atom,hd_coeff1(zlcms(map(atom,int,hd_coeff,as))),X3) ) ) ).
tff(fact_6_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X1: A] :
( ( zero_zero(A) = X1 )
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_7_of__int__0,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A,zero_zero(int)) = zero_zero(A) ) ) ).
tff(fact_8_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z1: int] :
( ( zero_zero(A) = ring_1_of_int(A,Z1) )
<=> ( Z1 = zero_zero(int) ) ) ) ).
tff(fact_9_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z1: int] :
( ( ring_1_of_int(A,Z1) = zero_zero(A) )
<=> ( Z1 = zero_zero(int) ) ) ) ).
tff(fact_10__096_B_Bthesis_O_A_I_B_Bb_O_A_091_124_Ab_A_058_Aset_Aas_059_Aa_A_061_Ahd__coeff1_A_Izlcms_A_Imap_Ahd__coeff_Aas_J_J_Ab_A_124_093_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [B1: atom] :
( member(atom,B1,aa(list(atom),fun(atom,bool),set(atom),as))
=> ( a != aa(atom,atom,hd_coeff1(zlcms(map(atom,int,hd_coeff,as))),B1) ) ) ).
tff(fact_11_Z_Oanormal__decr,axiom,
! [A3: atom] :
( ( aa(atom,int,hd_coeff,A3) = zero_zero(int) )
=> ( ( divisor(A3) != zero_zero(int) )
=> ( divisor(decr_Z(A3)) != zero_zero(int) ) ) ) ).
tff(fact_12_sgn0,axiom,
! [A: $tType] :
( sgn_if(A)
=> ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).
tff(fact_13_Nats__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> member(A,zero_zero(A),semiring_1_Nats(A)) ) ).
tff(fact_14_iszero__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> iszero(A,zero_zero(A)) ) ).
tff(fact_15_iszero__def,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Z1: A] :
( iszero(A,Z1)
<=> ( Z1 = zero_zero(A) ) ) ) ).
tff(fact_16_of__int__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z1: int,W: int] :
( ( ring_1_of_int(A,W) = ring_1_of_int(A,Z1) )
<=> ( W = Z1 ) ) ) ).
tff(fact_17_map__eq__conv,axiom,
! [A: $tType,B: $tType,G: fun(B,A),Xs: list(B),F: fun(B,A)] :
( ( map(B,A,F,Xs) = map(B,A,G,Xs) )
<=> ! [X4: B] :
( member(B,X4,aa(list(B),fun(B,bool),set(B),Xs))
=> ( aa(B,A,F,X4) = aa(B,A,G,X4) ) ) ) ).
tff(fact_18_sgn__sgn,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] : sgn_sgn(A,sgn_sgn(A,A3)) = sgn_sgn(A,A3) ) ).
tff(fact_19_sgn__0__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Aa: A] :
( ( sgn_sgn(A,Aa) = zero_zero(A) )
<=> ( Aa = zero_zero(A) ) ) ) ).
tff(fact_20_divisor__asubst,axiom,
! [A3: atom,Ks: list(int),I: int] : divisor(asubst(I,Ks,A3)) = divisor(A3) ).
tff(fact_21_ex__map__conv,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Ys1: list(B)] :
( ? [Xs2: list(A)] : Ys1 = map(A,B,F,Xs2)
<=> ! [X4: B] :
( member(B,X4,aa(list(B),fun(B,bool),set(B),Ys1))
=> ? [Xa: A] : X4 = aa(A,B,F,Xa) ) ) ).
tff(fact_22_map__ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),Xs: list(A)] :
( ! [X3: A] :
( member(A,X3,aa(list(A),fun(A,bool),set(A),Xs))
=> ( aa(A,B,F,X3) = aa(A,B,G,X3) ) )
=> ( map(A,B,F,Xs) = map(A,B,G,Xs) ) ) ).
tff(fact_23_map__idI,axiom,
! [A: $tType,F: fun(A,A),Xs: list(A)] :
( ! [X3: A] :
( member(A,X3,aa(list(A),fun(A,bool),set(A),Xs))
=> ( aa(A,A,F,X3) = X3 ) )
=> ( map(A,A,F,Xs) = Xs ) ) ).
tff(fact_24_map__cong,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),Ys1: list(A),Xs: list(A)] :
( ( Xs = Ys1 )
=> ( ! [X3: A] :
( member(A,X3,aa(list(A),fun(A,bool),set(A),Ys1))
=> ( aa(A,B,F,X3) = aa(A,B,G,X3) ) )
=> ( map(A,B,F,Xs) = map(A,B,G,Ys1) ) ) ) ).
tff(fact_25_zlcms0__iff,axiom,
! [Is: list(int)] :
( ( zlcms(Is) = zero_zero(int) )
<=> member(int,zero_zero(int),aa(list(int),fun(int,bool),set(int),Is)) ) ).
tff(fact_26__096a_A_061_ADvd_A_Izlcms_A_Imap_Ahd__coeff_Aas_J_J_A0_A_0911_093_A_124_IEX_Ab_058set_Aas_O_Aa_A_061_Ahd__coeff1_A_Izlcms_A_Imap_Ahd__coeff_Aas_J_J_Ab_J_096,axiom,
( ( a = dvd(zlcms(map(atom,int,hd_coeff,as)),zero_zero(int),cons(int,one_one(int),nil(int))) )
| ? [X3: atom] :
( member(atom,X3,aa(list(atom),fun(atom,bool),set(atom),as))
& ( a = aa(atom,atom,hd_coeff1(zlcms(map(atom,int,hd_coeff,as))),X3) ) ) ) ).
tff(fact_27_sgn__less,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Aa: A] :
( ord_less(A,sgn_sgn(A,Aa),zero_zero(A))
<=> ord_less(A,Aa,zero_zero(A)) ) ) ).
tff(fact_28_sgn__greater,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Aa: A] :
( ord_less(A,zero_zero(A),sgn_sgn(A,Aa))
<=> ord_less(A,zero_zero(A),Aa) ) ) ).
tff(fact_29_in__set__member,axiom,
! [A: $tType,Xs: list(A),X1: A] :
( member(A,X1,aa(list(A),fun(A,bool),set(A),Xs))
<=> pp(aa(A,bool,aa(list(A),fun(A,bool),member1(A),Xs),X1)) ) ).
tff(fact_30_ListMem__iff,axiom,
! [A: $tType,Xs: list(A),X1: A] :
( listMem(A,X1,Xs)
<=> member(A,X1,aa(list(A),fun(A,bool),set(A),Xs)) ) ).
tff(fact_31_List_Omember__def,axiom,
! [A: $tType,X1: A,Xs: list(A)] :
( pp(aa(A,bool,aa(list(A),fun(A,bool),member1(A),Xs),X1))
<=> member(A,X1,aa(list(A),fun(A,bool),set(A),Xs)) ) ).
tff(fact_32_list__ex1__iff,axiom,
! [A: $tType,Xs: list(A),P1: fun(A,bool)] :
( list_ex1(A,P1,Xs)
<=> ? [X4: A] :
( member(A,X4,aa(list(A),fun(A,bool),set(A),Xs))
& pp(aa(A,bool,P1,X4))
& ! [Y1: A] :
( ( member(A,Y1,aa(list(A),fun(A,bool),set(A),Xs))
& pp(aa(A,bool,P1,Y1)) )
=> ( Y1 = X4 ) ) ) ) ).
tff(fact_33_list_Oinject,axiom,
! [A: $tType,List4: list(A),A5: A,List3: list(A),Aa: A] :
( ( cons(A,Aa,List3) = cons(A,A5,List4) )
<=> ( ( Aa = A5 )
& ( List3 = List4 ) ) ) ).
tff(fact_34_atom_Osimps_I2_J,axiom,
! [List4: list(int),Int22: int,Int12: int,List3: list(int),Int21: int,Int11: int] :
( ( dvd(Int11,Int21,List3) = dvd(Int12,Int22,List4) )
<=> ( ( Int11 = Int12 )
& ( Int21 = Int22 )
& ( List3 = List4 ) ) ) ).
tff(fact_35_map_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Xs: list(B),X1: B,F: fun(B,A)] : map(B,A,F,cons(B,X1,Xs)) = cons(A,aa(B,A,F,X1),map(B,A,F,Xs)) ).
tff(fact_36_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,Xs: list(B),F: fun(B,A)] :
( ( nil(A) = map(B,A,F,Xs) )
<=> ( Xs = nil(B) ) ) ).
tff(fact_37_map_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] : map(B,A,F,nil(B)) = nil(A) ).
tff(fact_38_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,Xs: list(B),F: fun(B,A)] :
( ( map(B,A,F,Xs) = nil(A) )
<=> ( Xs = nil(B) ) ) ).
tff(fact_39_of__int__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z1: int,W: int] :
( ord_less(A,ring_1_of_int(A,W),ring_1_of_int(A,Z1))
<=> ord_less(int,W,Z1) ) ) ).
tff(fact_40_list__ex1__simps_I1_J,axiom,
! [A: $tType,P1: fun(A,bool)] : ~ list_ex1(A,P1,nil(A)) ).
tff(fact_41_of__int__1,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A,one_one(int)) = one_one(A) ) ) ).
tff(fact_42_zlcms_Osimps_I1_J,axiom,
zlcms(nil(int)) = one_one(int) ).
tff(fact_43_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_44_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_45_of__int__less__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z1: int] :
( ord_less(A,ring_1_of_int(A,Z1),zero_zero(A))
<=> ord_less(int,Z1,zero_zero(int)) ) ) ).
tff(fact_46_of__int__0__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z1: int] :
( ord_less(A,zero_zero(A),ring_1_of_int(A,Z1))
<=> ord_less(int,zero_zero(int),Z1) ) ) ).
tff(fact_47_not__one__less__zero,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ~ ord_less(A,one_one(A),zero_zero(A)) ) ).
tff(fact_48_zero__less__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ord_less(A,zero_zero(A),one_one(A)) ) ).
tff(fact_49_insert,axiom,
! [A: $tType,Y: A,Xs1: list(A),X2: A] :
( listMem(A,X2,Xs1)
=> listMem(A,X2,cons(A,Y,Xs1)) ) ).
tff(fact_50_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y: A,X2: A] :
( ( X2 != Y )
=> ( ~ ord_less(A,X2,Y)
=> ord_less(A,Y,X2) ) ) ) ).
tff(fact_51_member__rec_I1_J,axiom,
! [A: $tType,Y2: A,Xs: list(A),X1: A] :
( pp(aa(A,bool,aa(list(A),fun(A,bool),member1(A),cons(A,X1,Xs)),Y2))
<=> ( ( X1 = Y2 )
| pp(aa(A,bool,aa(list(A),fun(A,bool),member1(A),Xs),Y2)) ) ) ).
tff(fact_52_member__rec_I2_J,axiom,
! [A: $tType,Y: A] : ~ pp(aa(A,bool,aa(list(A),fun(A,bool),member1(A),nil(A)),Y)) ).
tff(fact_53_elem,axiom,
! [A: $tType,Xs1: list(A),X2: A] : listMem(A,X2,cons(A,X2,Xs1)) ).
tff(fact_54_not__Cons__self2,axiom,
! [A: $tType,Xs1: list(A),X2: A] : cons(A,X2,Xs1) != Xs1 ).
tff(fact_55_list_Osimps_I3_J,axiom,
! [A: $tType,List2: list(A),A4: A] : cons(A,A4,List2) != nil(A) ).
tff(fact_56_not__Cons__self,axiom,
! [A: $tType,X2: A,Xs1: list(A)] : Xs1 != cons(A,X2,Xs1) ).
tff(fact_57_list_Osimps_I2_J,axiom,
! [A: $tType,List2: list(A),A4: A] : nil(A) != cons(A,A4,List2) ).
tff(fact_58_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X1: A] :
( ( one_one(A) = X1 )
<=> ( X1 = one_one(A) ) ) ) ).
tff(fact_59_sgn__1__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Aa: A] :
( ( sgn_sgn(A,Aa) = one_one(A) )
<=> ord_less(A,zero_zero(A),Aa) ) ) ).
tff(fact_60_sgn__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A3: A] :
( ord_less(A,zero_zero(A),A3)
=> ( sgn_sgn(A,A3) = one_one(A) ) ) ) ).
tff(fact_61_one__neq__zero,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( one_one(A) != zero_zero(A) ) ) ).
tff(fact_62_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
tff(fact_63_set__ConsD,axiom,
! [A: $tType,Xs: list(A),X1: A,Y2: A] :
( member(A,Y2,aa(list(A),fun(A,bool),set(A),cons(A,X1,Xs)))
=> ( ( Y2 = X1 )
| member(A,Y2,aa(list(A),fun(A,bool),set(A),Xs)) ) ) ).
tff(fact_64_not__iszero__1,axiom,
! [A: $tType] :
( semiring_1(A)
=> ~ iszero(A,one_one(A)) ) ).
tff(fact_65_Nats__1,axiom,
! [A: $tType] :
( semiring_1(A)
=> member(A,one_one(A),semiring_1_Nats(A)) ) ).
tff(fact_66_divisor_Osimps_I2_J,axiom,
! [Ks: list(int),I: int,D: int] : divisor(dvd(D,I,Ks)) = D ).
tff(fact_67_member__set,axiom,
! [A: $tType] : member1(A) = set(A) ).
tff(fact_68_zlcms__pos,axiom,
! [Is: list(int)] :
( ! [X3: int] :
( member(int,X3,aa(list(int),fun(int,bool),set(int),Is))
=> ( X3 != zero_zero(int) ) )
=> ord_less(int,zero_zero(int),zlcms(Is)) ) ).
tff(fact_69_atom_Osize_I2_J,axiom,
! [List: list(int),Int2: int,Int1: int] : atom_size(dvd(Int1,Int2,List)) = zero_zero(nat) ).
tff(fact_70_map__eq__Cons__conv,axiom,
! [B: $tType,A: $tType,Ys1: list(A),Y2: A,Xs: list(B),F: fun(B,A)] :
( ( map(B,A,F,Xs) = cons(A,Y2,Ys1) )
<=> ? [Z: B,Zs: list(B)] :
( ( Xs = cons(B,Z,Zs) )
& ( aa(B,A,F,Z) = Y2 )
& ( map(B,A,F,Zs) = Ys1 ) ) ) ).
tff(fact_71_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys1: list(B),F: fun(B,A),Xs: list(A),X1: A] :
( ( cons(A,X1,Xs) = map(B,A,F,Ys1) )
<=> ? [Z: B,Zs: list(B)] :
( ( Ys1 = cons(B,Z,Zs) )
& ( X1 = aa(B,A,F,Z) )
& ( Xs = map(B,A,F,Zs) ) ) ) ).
tff(fact_72_list_Oexhaust,axiom,
! [A: $tType,Y: list(A)] :
( ( Y != nil(A) )
=> ~ ! [A2: A,List1: list(A)] : Y != cons(A,A2,List1) ) ).
tff(fact_73_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_74_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),X1: A] :
( member(A,X1,A1)
<=> pp(aa(A,bool,A1,X1)) ) ).
tff(fact_75_neq__Nil__conv,axiom,
! [A: $tType,Xs: list(A)] :
( ( Xs != nil(A) )
<=> ? [Y1: A,Ys: list(A)] : Xs = cons(A,Y1,Ys) ) ).
tff(fact_76_less__zeroE,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_77_less__nat__zero__code,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_78_neq0__conv,axiom,
! [N1: nat] :
( ( N1 != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),N1) ) ).
tff(fact_79_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_80_gr__implies__not0,axiom,
! [N: nat,M1: nat] :
( ord_less(nat,M1,N)
=> ( N != zero_zero(nat) ) ) ).
tff(fact_81_not__less0,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_82_atom_Osize_I5_J,axiom,
! [List: list(int),Int2: int,Int1: int] : size_size(atom,dvd(Int1,Int2,List)) = zero_zero(nat) ).
tff(fact_83_insert__Nil,axiom,
! [A: $tType,X2: A] : insert(A,X2,nil(A)) = cons(A,X2,nil(A)) ).
tff(fact_84_nat__less__cases,axiom,
! [P1: fun(nat,fun(nat,bool)),N1: nat,M: nat] :
( ( ord_less(nat,M,N1)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M)) )
=> ( ( ( M = N1 )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M)) )
=> ( ( ord_less(nat,N1,M)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M)) ) ) ) ).
tff(fact_85_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_86_less__not__refl2,axiom,
! [M1: nat,N: nat] :
( ord_less(nat,N,M1)
=> ( M1 != N ) ) ).
tff(fact_87_less__irrefl__nat,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_88_linorder__neqE__nat,axiom,
! [Y: nat,X2: nat] :
( ( X2 != Y )
=> ( ~ ord_less(nat,X2,Y)
=> ord_less(nat,Y,X2) ) ) ).
tff(fact_89_nat__neq__iff,axiom,
! [N1: nat,M: nat] :
( ( M != N1 )
<=> ( ord_less(nat,M,N1)
| ord_less(nat,N1,M) ) ) ).
tff(fact_90_less__not__refl,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_91_in__set__insert,axiom,
! [A: $tType,Xs: list(A),X1: A] :
( member(A,X1,aa(list(A),fun(A,bool),set(A),Xs))
=> ( insert(A,X1,Xs) = Xs ) ) ).
tff(fact_92_List_Oinsert__def,axiom,
! [A: $tType,Xs: list(A),X1: A] :
( ( member(A,X1,aa(list(A),fun(A,bool),set(A),Xs))
=> ( insert(A,X1,Xs) = Xs ) )
& ( ~ member(A,X1,aa(list(A),fun(A,bool),set(A),Xs))
=> ( insert(A,X1,Xs) = cons(A,X1,Xs) ) ) ) ).
tff(fact_93_not__in__set__insert,axiom,
! [A: $tType,Xs: list(A),X1: A] :
( ~ member(A,X1,aa(list(A),fun(A,bool),set(A),Xs))
=> ( insert(A,X1,Xs) = cons(A,X1,Xs) ) ) ).
tff(fact_94_sublist__singleton,axiom,
! [A: $tType,X1: A,A1: fun(nat,bool)] :
( ( member(nat,zero_zero(nat),A1)
=> ( sublist(A,cons(A,X1,nil(A)),A1) = cons(A,X1,nil(A)) ) )
& ( ~ member(nat,zero_zero(nat),A1)
=> ( sublist(A,cons(A,X1,nil(A)),A1) = nil(A) ) ) ) ).
tff(fact_95_hd__coeffs1__def,axiom,
! [Asa: list(atom)] : qEpres1907640072oeffs1(Asa) = cons(atom,dvd(zlcms(map(atom,int,hd_coeff,Asa)),zero_zero(int),cons(int,one_one(int),nil(int))),map(atom,atom,hd_coeff1(zlcms(map(atom,int,hd_coeff,Asa))),Asa)) ).
tff(fact_96_map__ident,axiom,
! [A: $tType,X: list(A)] : map(A,A,combi(A),X) = X ).
tff(fact_97_sublist__nil,axiom,
! [A: $tType,A1: fun(nat,bool)] : sublist(A,nil(A),A1) = nil(A) ).
%----Arities (14)
tff(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one(int) ).
tff(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(arity_Int_Oint___Int_Oring__char__0,axiom,
ring_char_0(int) ).
tff(arity_Int_Oint___Groups_Osgn__if,axiom,
sgn_if(int) ).
tff(arity_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Groups_Oone,axiom,
one(int) ).
tff(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
tff(arity_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_Nat_Onat___Groups_Oone,axiom,
one(nat) ).
%----Helper facts (3)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_COMBI_1_1_U,axiom,
! [A: $tType,P: A] : aa(A,A,combi(A),P) = P ).
%----Conjectures (1)
tff(conj_0,conjecture,
aa(atom,int,hd_coeff,b) != zero_zero(int) ).
%------------------------------------------------------------------------------