TPTP Problem File: COM027_5.p
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%------------------------------------------------------------------------------
% File : COM027_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 23
% 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_23 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 167 ( 27 unt; 45 typ; 0 def)
% Number of atoms : 353 ( 83 equ)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 254 ( 23 ~; 12 |; 13 &)
% ( 37 <=>; 169 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 31 ( 18 >; 13 *; 0 +; 0 <<)
% Number of predicates : 24 ( 23 usr; 0 prp; 1-3 aty)
% Number of functors : 19 ( 19 usr; 5 con; 0-5 aty)
% Number of variables : 419 ( 359 !; 27 ?; 419 :)
% ( 33 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:16:02
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
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_PresArith_Oatom,type,
atom: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (40)
tff(sy_cl_Rings_Odvd,type,
dvd:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oidom,type,
idom:
!>[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_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ono__zero__divisors,type,
no_zero_divisors:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__no__zero__divisors,type,
ring_n68954251visors:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__ring__strict,type,
linord581940658strict:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Oab__semigroup__idem__mult,type,
ab_sem1668676832m_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semiring__strict,type,
linord20386208strict:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord893533164strict:
!>[A: $tType] : $o ).
tff(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_Olist__case,type,
list_case:
!>[T2: $tType,A: $tType] : ( ( T2 * fun(A,fun(list(A),T2)) * list(A) ) > T2 ) ).
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_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_PresArith_OI_092_060_094isub_062Z,type,
i_Z: ( atom * list(int) ) > $o ).
tff(sy_c_PresArith_Odecr_092_060_094isub_062Z,type,
decr_Z: atom > atom ).
tff(sy_c_PresArith_Ohd__coeff,type,
hd_coeff: fun(atom,int) ).
tff(sy_c_PresArith_Ohd__coeff1,type,
hd_coeff1: ( int * atom ) > atom ).
tff(sy_c_PresArith_Ozlcms,type,
zlcms: list(int) > int ).
tff(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( ( A * 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 (99)
tff(fact_0__C0_C,axiom,
! [X1: atom] :
( member(atom,X1,set(atom,as))
=> ( aa(atom,int,hd_coeff,X1) != zero_zero(int) ) ) ).
tff(fact_1_map__eq__conv,axiom,
! [A: $tType,B: $tType,G: fun(B,A),Xsa: list(B),F3: fun(B,A)] :
( ( map(B,A,F3,Xsa) = map(B,A,G,Xsa) )
<=> ! [X: B] :
( member(B,X,set(B,Xsa))
=> ( aa(B,A,F3,X) = aa(B,A,G,X) ) ) ) ).
tff(fact_2_map_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Xsa: list(B),X4: B,F3: fun(B,A)] : map(B,A,F3,cons(B,X4,Xsa)) = cons(A,aa(B,A,F3,X4),map(B,A,F3,Xsa)) ).
tff(fact_3_list_Oinject,axiom,
! [A: $tType,List1: list(A),A4: A,List: list(A),A1: A] :
( ( cons(A,A1,List) = cons(A,A4,List1) )
<=> ( ( A1 = A4 )
& ( List = List1 ) ) ) ).
tff(fact_4_mult__left__idem,axiom,
! [A: $tType] :
( ab_sem1668676832m_mult(A)
=> ! [B1: A,A2: A] : times_times(A,A2,times_times(A,A2,B1)) = times_times(A,A2,B1) ) ).
tff(fact_5_set__ConsD,axiom,
! [A: $tType,Xsa: list(A),X4: A,Y1: A] :
( member(A,Y1,set(A,cons(A,X4,Xsa)))
=> ( ( Y1 = X4 )
| member(A,Y1,set(A,Xsa)) ) ) ).
tff(fact_6__0960_A_060_Azlcms_A_Imap_Ahd__coeff_Aas_J_096,axiom,
ord_less(int,zero_zero(int),zlcms(map(atom,int,hd_coeff,as))) ).
tff(fact_7_ex__map__conv,axiom,
! [B: $tType,A: $tType,F3: fun(A,B),Ys1: list(B)] :
( ? [Xs1: list(A)] : Ys1 = map(A,B,F3,Xs1)
<=> ! [X: B] :
( member(B,X,set(B,Ys1))
=> ? [Xa: A] : X = aa(A,B,F3,Xa) ) ) ).
tff(fact_8_map__ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F3: fun(A,B),Xsa: list(A)] :
( ! [X2: A] :
( member(A,X2,set(A,Xsa))
=> ( aa(A,B,F3,X2) = aa(A,B,G,X2) ) )
=> ( map(A,B,F3,Xsa) = map(A,B,G,Xsa) ) ) ).
tff(fact_9_map__idI,axiom,
! [A: $tType,F3: fun(A,A),Xsa: list(A)] :
( ! [X2: A] :
( member(A,X2,set(A,Xsa))
=> ( aa(A,A,F3,X2) = X2 ) )
=> ( map(A,A,F3,Xsa) = Xsa ) ) ).
tff(fact_10_map__cong,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F3: fun(A,B),Ys1: list(A),Xsa: list(A)] :
( ( Xsa = Ys1 )
=> ( ! [X2: A] :
( member(A,X2,set(A,Ys1))
=> ( aa(A,B,F3,X2) = aa(A,B,G,X2) ) )
=> ( map(A,B,F3,Xsa) = map(A,B,G,Ys1) ) ) ) ).
tff(fact_11_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys1: list(B),F3: fun(B,A),Xsa: list(A),X4: A] :
( ( cons(A,X4,Xsa) = map(B,A,F3,Ys1) )
<=> ? [Z4: B,Zs: list(B)] :
( ( Ys1 = cons(B,Z4,Zs) )
& ( X4 = aa(B,A,F3,Z4) )
& ( Xsa = map(B,A,F3,Zs) ) ) ) ).
tff(fact_12_map__eq__Cons__conv,axiom,
! [B: $tType,A: $tType,Ys1: list(A),Y1: A,Xsa: list(B),F3: fun(B,A)] :
( ( map(B,A,F3,Xsa) = cons(A,Y1,Ys1) )
<=> ? [Z4: B,Zs: list(B)] :
( ( Xsa = cons(B,Z4,Zs) )
& ( aa(B,A,F3,Z4) = Y1 )
& ( map(B,A,F3,Zs) = Ys1 ) ) ) ).
tff(fact_13_mult__idem,axiom,
! [A: $tType] :
( ab_sem1668676832m_mult(A)
=> ! [X3: A] : times_times(A,X3,X3) = X3 ) ).
tff(fact_14_times_Oidem,axiom,
! [A: $tType] :
( ab_sem1668676832m_mult(A)
=> ! [A2: A] : times_times(A,A2,A2) = A2 ) ).
tff(fact_15_not__Cons__self2,axiom,
! [A: $tType,Xs: list(A),X3: A] : cons(A,X3,Xs) != Xs ).
tff(fact_16_not__Cons__self,axiom,
! [A: $tType,X3: A,Xs: list(A)] : Xs != cons(A,X3,Xs) ).
tff(fact_17_zlcms0__iff,axiom,
! [Is: list(int)] :
( ( zlcms(Is) = zero_zero(int) )
<=> member(int,zero_zero(int),set(int,Is)) ) ).
tff(fact_18_mult__eq__0__iff,axiom,
! [A: $tType] :
( ring_n68954251visors(A)
=> ! [B2: A,A1: A] :
( ( times_times(A,A1,B2) = zero_zero(A) )
<=> ( ( A1 = zero_zero(A) )
| ( B2 = zero_zero(A) ) ) ) ) ).
tff(fact_19_mult__zero__right,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A2: A] : times_times(A,A2,zero_zero(A)) = zero_zero(A) ) ).
tff(fact_20_mult__zero__left,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A2: A] : times_times(A,zero_zero(A),A2) = zero_zero(A) ) ).
tff(fact_21_zmult__zless__mono2,axiom,
! [K2: int,J: int,I1: int] :
( ord_less(int,I1,J)
=> ( ord_less(int,zero_zero(int),K2)
=> ord_less(int,times_times(int,K2,I1),times_times(int,K2,J)) ) ) ).
tff(fact_22_not__square__less__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [A2: A] : ~ ord_less(A,times_times(A,A2,A2),zero_zero(A)) ) ).
tff(fact_23_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B2: A,C2: A,A1: A] :
( ord_less(A,times_times(A,A1,C2),times_times(A,B2,C2))
<=> ( ( ord_less(A,zero_zero(A),C2)
& ord_less(A,A1,B2) )
| ( ord_less(A,C2,zero_zero(A))
& ord_less(A,B2,A1) ) ) ) ) ).
tff(fact_24_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B2: A,A1: A,C2: A] :
( ord_less(A,times_times(A,C2,A1),times_times(A,C2,B2))
<=> ( ( ord_less(A,zero_zero(A),C2)
& ord_less(A,A1,B2) )
| ( ord_less(A,C2,zero_zero(A))
& ord_less(A,B2,A1) ) ) ) ) ).
tff(fact_25_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B2: A,A1: A,C2: A] :
( ord_less(A,zero_zero(A),C2)
=> ( ord_less(A,times_times(A,C2,A1),times_times(A,C2,B2))
<=> ord_less(A,A1,B2) ) ) ) ).
tff(fact_26_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y: A,X3: A] :
( ( X3 != Y )
=> ( ~ ord_less(A,X3,Y)
=> ord_less(A,Y,X3) ) ) ) ).
tff(fact_27_no__zero__divisors,axiom,
! [A: $tType] :
( no_zero_divisors(A)
=> ! [B1: A,A2: A] :
( ( A2 != zero_zero(A) )
=> ( ( B1 != zero_zero(A) )
=> ( times_times(A,A2,B1) != zero_zero(A) ) ) ) ) ).
tff(fact_28_divisors__zero,axiom,
! [A: $tType] :
( no_zero_divisors(A)
=> ! [B1: A,A2: A] :
( ( times_times(A,A2,B1) = zero_zero(A) )
=> ( ( A2 = zero_zero(A) )
| ( B1 = zero_zero(A) ) ) ) ) ).
tff(fact_29_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [C1: A,A2: A,B1: A] :
( ord_less(A,B1,A2)
=> ( ord_less(A,C1,zero_zero(A))
=> ord_less(A,times_times(A,C1,A2),times_times(A,C1,B1)) ) ) ) ).
tff(fact_30_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [C1: A,A2: A,B1: A] :
( ord_less(A,B1,A2)
=> ( ord_less(A,C1,zero_zero(A))
=> ord_less(A,times_times(A,A2,C1),times_times(A,B1,C1)) ) ) ) ).
tff(fact_31_comm__mult__strict__left__mono,axiom,
! [A: $tType] :
( linord893533164strict(A)
=> ! [C1: A,B1: A,A2: A] :
( ord_less(A,A2,B1)
=> ( ord_less(A,zero_zero(A),C1)
=> ord_less(A,times_times(A,C1,A2),times_times(A,C1,B1)) ) ) ) ).
tff(fact_32_mult__strict__left__mono,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [C1: A,B1: A,A2: A] :
( ord_less(A,A2,B1)
=> ( ord_less(A,zero_zero(A),C1)
=> ord_less(A,times_times(A,C1,A2),times_times(A,C1,B1)) ) ) ) ).
tff(fact_33_mult__strict__right__mono,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [C1: A,B1: A,A2: A] :
( ord_less(A,A2,B1)
=> ( ord_less(A,zero_zero(A),C1)
=> ord_less(A,times_times(A,A2,C1),times_times(A,B1,C1)) ) ) ) ).
tff(fact_34_mult__neg__neg,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B1: A,A2: A] :
( ord_less(A,A2,zero_zero(A))
=> ( ord_less(A,B1,zero_zero(A))
=> ord_less(A,zero_zero(A),times_times(A,A2,B1)) ) ) ) ).
tff(fact_35_mult__neg__pos,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [B1: A,A2: A] :
( ord_less(A,A2,zero_zero(A))
=> ( ord_less(A,zero_zero(A),B1)
=> ord_less(A,times_times(A,A2,B1),zero_zero(A)) ) ) ) ).
tff(fact_36_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B2: A,A1: A,C2: A] :
( ord_less(A,C2,zero_zero(A))
=> ( ord_less(A,times_times(A,C2,A1),times_times(A,C2,B2))
<=> ord_less(A,B2,A1) ) ) ) ).
tff(fact_37_zero__less__mult__pos2,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [A2: A,B1: A] :
( ord_less(A,zero_zero(A),times_times(A,B1,A2))
=> ( ord_less(A,zero_zero(A),A2)
=> ord_less(A,zero_zero(A),B1) ) ) ) ).
tff(fact_38_zero__less__mult__pos,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [B1: A,A2: A] :
( ord_less(A,zero_zero(A),times_times(A,A2,B1))
=> ( ord_less(A,zero_zero(A),A2)
=> ord_less(A,zero_zero(A),B1) ) ) ) ).
tff(fact_39_mult__pos__neg2,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [B1: A,A2: A] :
( ord_less(A,zero_zero(A),A2)
=> ( ord_less(A,B1,zero_zero(A))
=> ord_less(A,times_times(A,B1,A2),zero_zero(A)) ) ) ) ).
tff(fact_40_mult__pos__neg,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [B1: A,A2: A] :
( ord_less(A,zero_zero(A),A2)
=> ( ord_less(A,B1,zero_zero(A))
=> ord_less(A,times_times(A,A2,B1),zero_zero(A)) ) ) ) ).
tff(fact_41_mult__pos__pos,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [B1: A,A2: A] :
( ord_less(A,zero_zero(A),A2)
=> ( ord_less(A,zero_zero(A),B1)
=> ord_less(A,zero_zero(A),times_times(A,A2,B1)) ) ) ) ).
tff(fact_42_zlcms__pos,axiom,
! [Is: list(int)] :
( ! [X2: int] :
( member(int,X2,set(int,Is))
=> ( X2 != zero_zero(int) ) )
=> ord_less(int,zero_zero(int),zlcms(Is)) ) ).
tff(fact_43_I__hd__coeff1__mult__a,axiom,
! [Xsa: list(int),X4: int,A1: atom,M: int] :
( ord_less(int,zero_zero(int),M)
=> ( ( dvd_dvd(int,aa(atom,int,hd_coeff,A1),M)
| ( aa(atom,int,hd_coeff,A1) = zero_zero(int) ) )
=> ( i_Z(hd_coeff1(M,A1),cons(int,times_times(int,M,X4),Xsa))
<=> i_Z(A1,cons(int,X4,Xsa)) ) ) ) ).
tff(fact_44_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A2: A] : times_times(A,zero_zero(A),A2) = zero_zero(A) ) ).
tff(fact_45_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A2: A] : times_times(A,A2,zero_zero(A)) = zero_zero(A) ) ).
tff(fact_46_Z_Onot__dep__decr,axiom,
! [Xsa: list(int),X4: int,A1: atom] :
( ( aa(atom,int,hd_coeff,A1) = zero_zero(int) )
=> ( i_Z(A1,cons(int,X4,Xsa))
<=> i_Z(decr_Z(A1),Xsa) ) ) ).
tff(fact_47_of__int__less__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( ord_less(A,ring_1_of_int(A,Z2),zero_zero(A))
<=> ord_less(int,Z2,zero_zero(int)) ) ) ).
tff(fact_48_of__int__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z2: int,W1: int] :
( ( ring_1_of_int(A,W1) = ring_1_of_int(A,Z2) )
<=> ( W1 = Z2 ) ) ) ).
tff(fact_49_dvd__0__right,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A2: A] : dvd_dvd(A,A2,zero_zero(A)) ) ).
tff(fact_50_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( idom(A)
=> ! [B2: A,C2: A,A1: A] :
( dvd_dvd(A,times_times(A,A1,C2),times_times(A,B2,C2))
<=> ( ( C2 = zero_zero(A) )
| dvd_dvd(A,A1,B2) ) ) ) ).
tff(fact_51_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( idom(A)
=> ! [B2: A,A1: A,C2: A] :
( dvd_dvd(A,times_times(A,C2,A1),times_times(A,C2,B2))
<=> ( ( C2 = zero_zero(A) )
| dvd_dvd(A,A1,B2) ) ) ) ).
tff(fact_52_of__int__0,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A,zero_zero(int)) = zero_zero(A) ) ) ).
tff(fact_53_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z2: int] :
( ( zero_zero(A) = ring_1_of_int(A,Z2) )
<=> ( Z2 = zero_zero(int) ) ) ) ).
tff(fact_54_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z2: int] :
( ( ring_1_of_int(A,Z2) = zero_zero(A) )
<=> ( Z2 = zero_zero(int) ) ) ) ).
tff(fact_55_of__int__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int,W1: int] :
( ord_less(A,ring_1_of_int(A,W1),ring_1_of_int(A,Z2))
<=> ord_less(int,W1,Z2) ) ) ).
tff(fact_56_of__int__mult,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Z3: int,W: int] : ring_1_of_int(A,times_times(int,W,Z3)) = times_times(A,ring_1_of_int(A,W),ring_1_of_int(A,Z3)) ) ).
tff(fact_57_of__int__0__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( ord_less(A,zero_zero(A),ring_1_of_int(A,Z2))
<=> ord_less(int,zero_zero(int),Z2) ) ) ).
tff(fact_58_dvd__trans,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C1: A,B1: A,A2: A] :
( dvd_dvd(A,A2,B1)
=> ( dvd_dvd(A,B1,C1)
=> dvd_dvd(A,A2,C1) ) ) ) ).
tff(fact_59_dvd__refl,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A2: A] : dvd_dvd(A,A2,A2) ) ).
tff(fact_60_dvd__0__left,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A2: A] :
( dvd_dvd(A,zero_zero(A),A2)
=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_61_dvd__triv__left,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [B1: A,A2: A] : dvd_dvd(A,A2,times_times(A,A2,B1)) ) ).
tff(fact_62_dvd__triv__right,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [B1: A,A2: A] : dvd_dvd(A,A2,times_times(A,B1,A2)) ) ).
tff(fact_63_dvd__mult2,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C1: A,B1: A,A2: A] :
( dvd_dvd(A,A2,B1)
=> dvd_dvd(A,A2,times_times(A,B1,C1)) ) ) ).
tff(fact_64_dvd__mult,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [B1: A,C1: A,A2: A] :
( dvd_dvd(A,A2,C1)
=> dvd_dvd(A,A2,times_times(A,B1,C1)) ) ) ).
tff(fact_65_mult__dvd__mono,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D1: A,C1: A,B1: A,A2: A] :
( dvd_dvd(A,A2,B1)
=> ( dvd_dvd(A,C1,D1)
=> dvd_dvd(A,times_times(A,A2,C1),times_times(A,B1,D1)) ) ) ) ).
tff(fact_66_dvdI,axiom,
! [A: $tType] :
( dvd(A)
=> ! [K2: A,B1: A,A2: A] :
( ( A2 = times_times(A,B1,K2) )
=> dvd_dvd(A,B1,A2) ) ) ).
tff(fact_67_dvd__mult__left,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C1: A,B1: A,A2: A] :
( dvd_dvd(A,times_times(A,A2,B1),C1)
=> dvd_dvd(A,A2,C1) ) ) ).
tff(fact_68_dvd__mult__right,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C1: A,B1: A,A2: A] :
( dvd_dvd(A,times_times(A,A2,B1),C1)
=> dvd_dvd(A,B1,C1) ) ) ).
tff(fact_69_zdvd__not__zless,axiom,
! [N: int,M1: int] :
( ord_less(int,zero_zero(int),M1)
=> ( ord_less(int,M1,N)
=> ~ dvd_dvd(int,N,M1) ) ) ).
tff(fact_70_zdvd__mult__cancel,axiom,
! [N: int,M1: int,K2: int] :
( dvd_dvd(int,times_times(int,K2,M1),times_times(int,K2,N))
=> ( ( K2 != zero_zero(int) )
=> dvd_dvd(int,M1,N) ) ) ).
tff(fact_71_dvd__zlcms,axiom,
! [Is: list(int),I: int] :
( member(int,I,set(int,Is))
=> dvd_dvd(int,I,zlcms(Is)) ) ).
tff(fact_72_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X4: A] :
( ( zero_zero(A) = X4 )
<=> ( X4 = zero_zero(A) ) ) ) ).
tff(fact_73_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),times_times(A,Rx,Ry)) = times_times(A,times_times(A,Lx,Rx),times_times(A,Ly,Ry)) ) ).
tff(fact_74_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F3: fun(A,B)] :
( ! [X2: A] : aa(A,B,F3,X2) = aa(A,B,G,X2)
=> ( F3 = G ) ) ).
tff(fact_75_mem__def,axiom,
! [A: $tType,A3: fun(A,bool),X4: A] :
( member(A,X4,A3)
<=> pp(aa(A,bool,A3,X4)) ) ).
tff(fact_76_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),times_times(A,Rx,Ry)) = times_times(A,Rx,times_times(A,times_times(A,Lx,Ly),Ry)) ) ).
tff(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),times_times(A,Rx,Ry)) = times_times(A,Lx,times_times(A,Ly,times_times(A,Rx,Ry))) ) ).
tff(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),Rx) = times_times(A,times_times(A,Lx,Rx),Ly) ) ).
tff(fact_79_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),Rx) = times_times(A,Lx,times_times(A,Ly,Rx)) ) ).
tff(fact_80_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [C1: A,B1: A,A2: A] : times_times(A,times_times(A,A2,B1),C1) = times_times(A,A2,times_times(A,B1,C1)) ) ).
tff(fact_81_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Lx: A] : times_times(A,Lx,times_times(A,Rx,Ry)) = times_times(A,times_times(A,Lx,Rx),Ry) ) ).
tff(fact_82_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Lx: A] : times_times(A,Lx,times_times(A,Rx,Ry)) = times_times(A,Rx,times_times(A,Lx,Ry)) ) ).
tff(fact_83_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [B1: A,A2: A] : times_times(A,A2,B1) = times_times(A,B1,A2) ) ).
tff(fact_84_zdvd__mono,axiom,
! [T1: int,M: int,K1: int] :
( ( K1 != zero_zero(int) )
=> ( dvd_dvd(int,M,T1)
<=> dvd_dvd(int,times_times(int,K1,M),times_times(int,K1,T1)) ) ) ).
tff(fact_85_dvdE,axiom,
! [A: $tType] :
( dvd(A)
=> ! [A2: A,B1: A] :
( dvd_dvd(A,B1,A2)
=> ~ ! [K: A] : A2 != times_times(A,B1,K) ) ) ).
tff(fact_86_list_Osimps_I5_J,axiom,
! [A: $tType,B: $tType,List: list(B),A1: B,F2: fun(B,fun(list(B),A)),F1: A] : list_case(A,B,F1,F2,cons(B,A1,List)) = aa(list(B),A,aa(B,fun(list(B),A),F2,A1),List) ).
tff(fact_87_splice_Osimps_I3_J,axiom,
! [A: $tType,Ys: list(A),Y: A,Xs: list(A),X3: A] : splice(A,cons(A,X3,Xs),cons(A,Y,Ys)) = cons(A,X3,cons(A,Y,splice(A,Xs,Ys))) ).
tff(fact_88_pinf_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Q1: fun(A,bool),Q: fun(A,bool),P1: fun(A,bool),P: fun(A,bool)] :
( ? [Z1: A] :
! [X2: A] :
( ord_less(A,Z1,X2)
=> ( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,P1,X2)) ) )
=> ( ? [Z1: A] :
! [X2: A] :
( ord_less(A,Z1,X2)
=> ( pp(aa(A,bool,Q,X2))
<=> pp(aa(A,bool,Q1,X2)) ) )
=> ? [Z: A] :
! [X1: A] :
( ord_less(A,Z,X1)
=> ( ( pp(aa(A,bool,P,X1))
| pp(aa(A,bool,Q,X1)) )
<=> ( pp(aa(A,bool,P1,X1))
| pp(aa(A,bool,Q1,X1)) ) ) ) ) ) ) ).
tff(fact_89_pinf_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Q1: fun(A,bool),Q: fun(A,bool),P1: fun(A,bool),P: fun(A,bool)] :
( ? [Z1: A] :
! [X2: A] :
( ord_less(A,Z1,X2)
=> ( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,P1,X2)) ) )
=> ( ? [Z1: A] :
! [X2: A] :
( ord_less(A,Z1,X2)
=> ( pp(aa(A,bool,Q,X2))
<=> pp(aa(A,bool,Q1,X2)) ) )
=> ? [Z: A] :
! [X1: A] :
( ord_less(A,Z,X1)
=> ( ( pp(aa(A,bool,P,X1))
& pp(aa(A,bool,Q,X1)) )
<=> ( pp(aa(A,bool,P1,X1))
& pp(aa(A,bool,Q1,X1)) ) ) ) ) ) ) ).
tff(fact_90_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ord(C)
=> ! [F: D] :
? [Z: C] :
! [X1: C] :
( ord_less(C,X1,Z)
=> ( F = F ) ) ) ).
tff(fact_91_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ord(C)
=> ! [F: D] :
? [Z: C] :
! [X1: C] :
( ord_less(C,Z,X1)
=> ( F = F ) ) ) ).
tff(fact_92_minf_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Q1: fun(A,bool),Q: fun(A,bool),P1: fun(A,bool),P: fun(A,bool)] :
( ? [Z1: A] :
! [X2: A] :
( ord_less(A,X2,Z1)
=> ( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,P1,X2)) ) )
=> ( ? [Z1: A] :
! [X2: A] :
( ord_less(A,X2,Z1)
=> ( pp(aa(A,bool,Q,X2))
<=> pp(aa(A,bool,Q1,X2)) ) )
=> ? [Z: A] :
! [X1: A] :
( ord_less(A,X1,Z)
=> ( ( pp(aa(A,bool,P,X1))
| pp(aa(A,bool,Q,X1)) )
<=> ( pp(aa(A,bool,P1,X1))
| pp(aa(A,bool,Q1,X1)) ) ) ) ) ) ) ).
tff(fact_93_minf_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Q1: fun(A,bool),Q: fun(A,bool),P1: fun(A,bool),P: fun(A,bool)] :
( ? [Z1: A] :
! [X2: A] :
( ord_less(A,X2,Z1)
=> ( pp(aa(A,bool,P,X2))
<=> pp(aa(A,bool,P1,X2)) ) )
=> ( ? [Z1: A] :
! [X2: A] :
( ord_less(A,X2,Z1)
=> ( pp(aa(A,bool,Q,X2))
<=> pp(aa(A,bool,Q1,X2)) ) )
=> ? [Z: A] :
! [X1: A] :
( ord_less(A,X1,Z)
=> ( ( pp(aa(A,bool,P,X1))
& pp(aa(A,bool,Q,X1)) )
<=> ( pp(aa(A,bool,P1,X1))
& pp(aa(A,bool,Q1,X1)) ) ) ) ) ) ) ).
tff(fact_94_minf_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z: A] :
! [X1: A] :
( ord_less(A,X1,Z)
=> ( X1 != T ) ) ) ).
tff(fact_95_pinf_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z: A] :
! [X1: A] :
( ord_less(A,Z,X1)
=> ( X1 != T ) ) ) ).
tff(fact_96_pinf_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z: A] :
! [X1: A] :
( ord_less(A,Z,X1)
=> ~ ord_less(A,X1,T) ) ) ).
tff(fact_97_pinf_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z: A] :
! [X1: A] :
( ord_less(A,Z,X1)
=> ( X1 != T ) ) ) ).
tff(fact_98_minf_I7_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z: A] :
! [X1: A] :
( ord_less(A,X1,Z)
=> ~ ord_less(A,T,X1) ) ) ).
%----Arities (20)
tff(arity_fun___Orderings_Oord,axiom,
! [T_1: $tType,T_2: $tType] :
( ord(T_2)
=> ord(fun(T_1,T_2)) ) ).
tff(arity_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(int) ).
tff(arity_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(int) ).
tff(arity_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord581940658strict(int) ).
tff(arity_Int_Oint___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(int) ).
tff(arity_Int_Oint___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(int) ).
tff(arity_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring(int) ).
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(int) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Rings_Omult__zero,axiom,
mult_zero(int) ).
tff(arity_Int_Oint___Int_Oring__char__0,axiom,
ring_char_0(int) ).
tff(arity_Int_Oint___Orderings_Oord,axiom,
ord(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___Rings_Oidom,axiom,
idom(int) ).
tff(arity_Int_Oint___Rings_Odvd,axiom,
dvd(int) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (1)
tff(conj_0,conjecture,
~ ( ? [X: int] :
! [Xa: atom] :
( member(atom,Xa,set(atom,as))
=> i_Z(Xa,cons(int,X,xs)) )
<=> ~ ? [X: int] :
! [Xa: atom] :
( member(atom,Xa,set(atom,as))
=> i_Z(hd_coeff1(zlcms(map(atom,int,hd_coeff,as)),Xa),cons(int,times_times(int,zlcms(map(atom,int,hd_coeff,as)),X),xs)) ) ) ).
%------------------------------------------------------------------------------