TPTP Problem File: COM042_5.p
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%------------------------------------------------------------------------------
% File : COM042_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 76
% 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_76 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.25 v7.1.0, 0.33 v6.4.0
% Syntax : Number of formulae : 173 ( 46 unt; 54 typ; 0 def)
% Number of atoms : 214 ( 121 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 105 ( 10 ~; 1 |; 9 &)
% ( 9 <=>; 76 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 5 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 45 ( 22 >; 23 *; 0 +; 0 <<)
% Number of predicates : 20 ( 19 usr; 0 prp; 1-2 aty)
% Number of functors : 32 ( 32 usr; 10 con; 0-5 aty)
% Number of variables : 406 ( 369 !; 0 ?; 406 :)
% ( 37 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:16:54
%------------------------------------------------------------------------------
%----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 (49)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oplus,type,
cl_Groups_Oplus:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Divides_Osemiring__div,type,
semiring_div:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Oab__semigroup__idem__mult,type,
ab_sem1668676832m_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : $o ).
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_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > fun(A,A) ) ).
tff(sy_c_Int_Osucc,type,
succ: int > int ).
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,C2: $tType] : ( fun(A,fun(B,C2)) > fun(list(A),fun(list(B),list(C2))) ) ).
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:
!>[T: $tType,A: $tType] : ( ( T * fun(A,fun(list(A),T)) * list(A) ) > T ) ).
tff(sy_c_List_Olist_Olist__rec,type,
list_rec:
!>[T: $tType,A: $tType] : ( ( T * fun(A,fun(list(A),fun(T,T))) * list(A) ) > T ) ).
tff(sy_c_List_Omap,type,
map:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) ) > list(B) ) ).
tff(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
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_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))) * atom ) > T ) ).
tff(sy_c_PresArith_Oatom_Oatom__rec,type,
atom_rec:
!>[T: $tType] : ( ( fun(int,fun(list(int),T)) * fun(int,fun(int,fun(list(int),T))) * fun(int,fun(int,fun(list(int),T))) * atom ) > T ) ).
tff(sy_c_PresArith_Odivisor,type,
divisor: atom > int ).
tff(sy_c_PresArith_Ohd__coeff,type,
hd_coeff: atom > int ).
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_pp,type,
pp: bool > $o ).
tff(sy_v_a,type,
a: atom ).
tff(sy_v_d____,type,
d: int ).
tff(sy_v_e,type,
e: list(int) ).
tff(sy_v_i,type,
i: int ).
tff(sy_v_j,type,
j: int ).
tff(sy_v_ks_H____,type,
ks: list(int) ).
tff(sy_v_ks____,type,
ks1: list(int) ).
tff(sy_v_l____,type,
l: int ).
%----Relevant facts (99)
tff(fact_0_calculation,axiom,
div_mod(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),l),aa(int,int,aa(int,fun(int,int),plus_plus(int),i),iprod(int,ks,e))),d) = div_mod(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),div_mod(int,l,d)),div_mod(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),i),iprod(int,ks,e)),d)),d) ).
tff(fact_1_mod__add__self2,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [B1: A,A1: A] : div_mod(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1),B1) = div_mod(A,A1,B1) ) ).
tff(fact_2_mod__add__self1,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [A1: A,B1: A] : div_mod(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),A1),B1) = div_mod(A,A1,B1) ) ).
tff(fact_3_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_4_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,B3: A,Aa: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa),C1) )
<=> ( B3 = C1 ) ) ) ).
tff(fact_5_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,Aa: A,B3: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),Aa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C1),Aa) )
<=> ( B3 = C1 ) ) ) ).
tff(fact_6_iprod__left__add__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [Zs: list(A),Ys1: list(A),Xs1: list(A)] : iprod(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),Xs1),Ys1),Zs) = aa(A,A,aa(A,fun(A,A),plus_plus(A),iprod(A,Xs1,Zs)),iprod(A,Ys1,Zs)) ) ).
tff(fact_7_mod__add__right__eq,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [C: A,B1: A,A1: A] : div_mod(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1),C) = div_mod(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),div_mod(A,B1,C)),C) ) ).
tff(fact_8_mod__add__left__eq,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [C: A,B1: A,A1: A] : div_mod(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1),C) = div_mod(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),div_mod(A,A1,C)),B1),C) ) ).
tff(fact_9_mod__add__eq,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [C: A,B1: A,A1: A] : div_mod(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1),C) = div_mod(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),div_mod(A,A1,C)),div_mod(A,B1,C)),C) ) ).
tff(fact_10_zmod__simps_I2_J,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [C: A,B1: A,A1: A] : div_mod(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),div_mod(A,B1,C)),C) = div_mod(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1),C) ) ).
tff(fact_11_zmod__simps_I1_J,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [B1: A,C: A,A1: A] : div_mod(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),div_mod(A,A1,C)),B1),C) = div_mod(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1),C) ) ).
tff(fact_12_mod__add__cong,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [B2: A,B1: A,A3: A,C: A,A1: A] :
( ( div_mod(A,A1,C) = div_mod(A,A3,C) )
=> ( ( div_mod(A,B1,C) = div_mod(A,B2,C) )
=> ( div_mod(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1),C) = div_mod(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2),C) ) ) ) ) ).
tff(fact_13_list__add__assoc,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [Zs: list(A),Ys1: list(A),Xs1: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),Xs1),Ys1)),Zs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),Xs1),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),Ys1),Zs)) ) ).
tff(fact_14_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,A1: A,B1: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),A1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C),A1) )
=> ( B1 = C ) ) ) ).
tff(fact_15_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C: A,B1: A,A1: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),C) )
=> ( B1 = C ) ) ) ).
tff(fact_16_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,B1: A,A1: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),C) )
=> ( B1 = C ) ) ) ).
tff(fact_17_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C: A,B1: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),C) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),C)) ) ).
tff(fact_18_NDvd,axiom,
a = nDvd(d,l,ks1) ).
tff(fact_19_assms_I3_J,axiom,
div_mod(int,i,divisor(a)) = div_mod(int,j,divisor(a)) ).
tff(fact_20_list__add__def,axiom,
! [A: $tType] :
( ( cl_Groups_Oplus(A)
& zero(A) )
=> ( plus_plus(list(A)) = zipwith0(A,A,A,plus_plus(A)) ) ) ).
tff(fact_21_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C: A,B1: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C),D)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),C)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),D)) ) ).
tff(fact_22_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,B1: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),C) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),C)),B1) ) ).
tff(fact_23_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,B1: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),C) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),C)) ) ).
tff(fact_24_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),aa(A,A,aa(A,fun(A,A),plus_plus(A),C),D)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),C)),D) ) ).
tff(fact_25_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),aa(A,A,aa(A,fun(A,A),plus_plus(A),C),D)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),D)) ) ).
tff(fact_26_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),C) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C),A1) ) ).
tff(fact_27_atom_Osimps_I3_J,axiom,
! [List3: list(int),Int22: int,Int13: int,List: list(int),Int21: int,Int12: int] :
( ( nDvd(Int12,Int21,List) = nDvd(Int13,Int22,List3) )
<=> ( ( Int12 = Int13 )
& ( Int21 = Int22 )
& ( List = List3 ) ) ) ).
tff(fact_28_divisor_Osimps_I3_J,axiom,
! [Ks: list(int),I1: int,D: int] : divisor(nDvd(D,I1,Ks)) = D ).
tff(fact_29__096ks_A_061_A1_A_D_Aks_H_096,axiom,
ks1 = cons(int,one_one(int),ks) ).
tff(fact_30_assms_I2_J,axiom,
hd_coeff(a) = one_one(int) ).
tff(fact_31_list__add__Cons,axiom,
! [A: $tType] :
( ( cl_Groups_Oplus(A)
& zero(A) )
=> ! [Ys1: list(A),Y: A,Xs1: list(A),X1: A] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),cons(A,X1,Xs1)),cons(A,Y,Ys1)) = cons(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X1),Y),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),plus_plus(list(A)),Xs1),Ys1)) ) ).
tff(fact_32_mod__mult__self2,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [C: A,B1: A,A1: A] : div_mod(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),aa(A,A,times_times(A,B1),C)),B1) = div_mod(A,A1,B1) ) ).
tff(fact_33__096_B_Bthesis_O_A_I_B_Bks_H_O_Aks_A_061_A1_A_D_Aks_H_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [Ks1: list(int)] : ks1 != cons(int,one_one(int),Ks1) ).
tff(fact_34_zipwith0_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C2: $tType] :
( ( zero(C2)
& zero(B) )
=> ! [Ys: list(C2),Y1: C2,Xs: list(B),X: B,F: fun(B,fun(C2,A))] : aa(list(C2),list(A),aa(list(B),fun(list(C2),list(A)),zipwith0(B,C2,A,F),cons(B,X,Xs)),cons(C2,Y1,Ys)) = cons(A,aa(C2,A,aa(B,fun(C2,A),F,X),Y1),aa(list(C2),list(A),aa(list(B),fun(list(C2),list(A)),zipwith0(B,C2,A,F),Xs),Ys)) ) ).
tff(fact_35_mod__mult__self1,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [B1: A,C: A,A1: A] : div_mod(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),aa(A,A,times_times(A,C),B1)),B1) = div_mod(A,A1,B1) ) ).
tff(fact_36_iprod__Cons,axiom,
! [A: $tType] :
( ring(A)
=> ! [Ys1: list(A),Y: A,Xs1: list(A),X1: A] : iprod(A,cons(A,X1,Xs1),cons(A,Y,Ys1)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,X1),Y)),iprod(A,Xs1,Ys1)) ) ).
tff(fact_37_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [M: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),M),M) = aa(A,A,times_times(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))),M) ) ).
tff(fact_38_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A,M: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),M),aa(A,A,times_times(A,A1),M)) = aa(A,A,times_times(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),one_one(A))),M) ) ).
tff(fact_39_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [M: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,A1),M)),M) = aa(A,A,times_times(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),one_one(A))),M) ) ).
tff(fact_40_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Ly)),aa(A,A,times_times(A,Rx),Ry)) = aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Rx)),aa(A,A,times_times(A,Ly),Ry)) ) ).
tff(fact_41_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Ly)),aa(A,A,times_times(A,Rx),Ry)) = aa(A,A,times_times(A,Rx),aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Ly)),Ry)) ) ).
tff(fact_42_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Ly)),aa(A,A,times_times(A,Rx),Ry)) = aa(A,A,times_times(A,Lx),aa(A,A,times_times(A,Ly),aa(A,A,times_times(A,Rx),Ry))) ) ).
tff(fact_43_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Rx: A,Ly: A,Lx: A] : aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Ly)),Rx) = aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Rx)),Ly) ) ).
tff(fact_44_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Rx: A,Ly: A,Lx: A] : aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Ly)),Rx) = aa(A,A,times_times(A,Lx),aa(A,A,times_times(A,Ly),Rx)) ) ).
tff(fact_45_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Lx: A] : aa(A,A,times_times(A,Lx),aa(A,A,times_times(A,Rx),Ry)) = aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Rx)),Ry) ) ).
tff(fact_46_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Lx: A] : aa(A,A,times_times(A,Lx),aa(A,A,times_times(A,Rx),Ry)) = aa(A,A,times_times(A,Rx),aa(A,A,times_times(A,Lx),Ry)) ) ).
tff(fact_47_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [B1: A,A1: A] : aa(A,A,times_times(A,A1),B1) = aa(A,A,times_times(A,B1),A1) ) ).
tff(fact_48_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : aa(A,A,times_times(A,A1),one_one(A)) = A1 ) ).
tff(fact_49_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : aa(A,A,times_times(A,one_one(A)),A1) = A1 ) ).
tff(fact_50_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [C: A,B1: A,A1: A] : aa(A,A,times_times(A,aa(A,A,times_times(A,A1),B1)),C) = aa(A,A,times_times(A,A1),aa(A,A,times_times(A,B1),C)) ) ).
tff(fact_51_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A1: A] : aa(A,A,times_times(A,A1),one_one(A)) = A1 ) ).
tff(fact_52_mult__1__right,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : aa(A,A,times_times(A,A1),one_one(A)) = A1 ) ).
tff(fact_53_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X: A] :
( ( one_one(A) = X )
<=> ( X = one_one(A) ) ) ) ).
tff(fact_54_mult__1,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A1: A] : aa(A,A,times_times(A,one_one(A)),A1) = A1 ) ).
tff(fact_55_mult__1__left,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : aa(A,A,times_times(A,one_one(A)),A1) = A1 ) ).
tff(fact_56_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Z3: A,Y: A,X1: A] : aa(A,A,times_times(A,X1),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,X1),Y)),aa(A,A,times_times(A,X1),Z3)) ) ).
tff(fact_57_crossproduct__noteq,axiom,
! [A: $tType] :
( semiri456707255roduct(A)
=> ! [Da: A,C1: A,B3: A,Aa: A] :
( ( ( Aa != B3 )
& ( C1 != Da ) )
<=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,Aa),C1)),aa(A,A,times_times(A,B3),Da)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,Aa),Da)),aa(A,A,times_times(A,B3),C1)) ) ) ) ).
tff(fact_58_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,B1: A,A1: A] : aa(A,A,times_times(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),C) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,A1),C)),aa(A,A,times_times(A,B1),C)) ) ).
tff(fact_59_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [B1: A,M: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,A1),M)),aa(A,A,times_times(A,B1),M)) = aa(A,A,times_times(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),M) ) ).
tff(fact_60_crossproduct__eq,axiom,
! [A: $tType] :
( semiri456707255roduct(A)
=> ! [Z: A,X: A,Y1: A,W1: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,W1),Y1)),aa(A,A,times_times(A,X),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,W1),Z)),aa(A,A,times_times(A,X),Y1)) )
<=> ( ( W1 = X )
| ( Y1 = Z ) ) ) ) ).
tff(fact_61_mod__mult__cong,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [B2: A,B1: A,A3: A,C: A,A1: A] :
( ( div_mod(A,A1,C) = div_mod(A,A3,C) )
=> ( ( div_mod(A,B1,C) = div_mod(A,B2,C) )
=> ( div_mod(A,aa(A,A,times_times(A,A1),B1),C) = div_mod(A,aa(A,A,times_times(A,A3),B2),C) ) ) ) ) ).
tff(fact_62_zmod__simps_I4_J,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [B1: A,C: A,A1: A] : div_mod(A,aa(A,A,times_times(A,div_mod(A,A1,C)),B1),C) = div_mod(A,aa(A,A,times_times(A,A1),B1),C) ) ).
tff(fact_63_mod__mult__mult2,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [B1: A,C: A,A1: A] : div_mod(A,aa(A,A,times_times(A,A1),C),aa(A,A,times_times(A,B1),C)) = aa(A,A,times_times(A,div_mod(A,A1,B1)),C) ) ).
tff(fact_64_mod__mult__mult1,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [B1: A,A1: A,C: A] : div_mod(A,aa(A,A,times_times(A,C),A1),aa(A,A,times_times(A,C),B1)) = aa(A,A,times_times(A,C),div_mod(A,A1,B1)) ) ).
tff(fact_65_mod__mult__eq,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [C: A,B1: A,A1: A] : div_mod(A,aa(A,A,times_times(A,A1),B1),C) = div_mod(A,aa(A,A,times_times(A,div_mod(A,A1,C)),div_mod(A,B1,C)),C) ) ).
tff(fact_66_mod__mult__left__eq,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [C: A,B1: A,A1: A] : div_mod(A,aa(A,A,times_times(A,A1),B1),C) = div_mod(A,aa(A,A,times_times(A,div_mod(A,A1,C)),B1),C) ) ).
tff(fact_67_mod__mult__right__eq,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [C: A,B1: A,A1: A] : div_mod(A,aa(A,A,times_times(A,A1),B1),C) = div_mod(A,aa(A,A,times_times(A,A1),div_mod(A,B1,C)),C) ) ).
tff(fact_68_zmod__simps_I3_J,axiom,
! [C: int,B1: int,A1: int] : div_mod(int,aa(int,int,times_times(int,A1),div_mod(int,B1,C)),C) = div_mod(int,aa(int,int,times_times(int,A1),B1),C) ).
tff(fact_69_zmod__zmult1__eq,axiom,
! [C: int,B1: int,A1: int] : div_mod(int,aa(int,int,times_times(int,A1),B1),C) = div_mod(int,aa(int,int,times_times(int,A1),div_mod(int,B1,C)),C) ).
tff(fact_70_list_Oinject,axiom,
! [A: $tType,List3: list(A),A2: A,List: list(A),Aa: A] :
( ( cons(A,Aa,List) = cons(A,A2,List3) )
<=> ( ( Aa = A2 )
& ( List = List3 ) ) ) ).
tff(fact_71_mult__left__idem,axiom,
! [A: $tType] :
( ab_sem1668676832m_mult(A)
=> ! [B1: A,A1: A] : aa(A,A,times_times(A,A1),aa(A,A,times_times(A,A1),B1)) = aa(A,A,times_times(A,A1),B1) ) ).
tff(fact_72_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( comm_semiring(A)
=> ! [C: A,B1: A,A1: A] : aa(A,A,times_times(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),C) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,A1),C)),aa(A,A,times_times(A,B1),C)) ) ).
tff(fact_73_combine__common__factor,axiom,
! [A: $tType] :
( semiring(A)
=> ! [C: A,B1: A,E: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,A1),E)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,B1),E)),C)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),E)),C) ) ).
tff(fact_74_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X2: A] : aa(A,B,F,X2) = aa(A,B,G,X2)
=> ( F = G ) ) ).
tff(fact_75_atom_Orecs_I3_J,axiom,
! [A: $tType,List: list(int),Int21: int,Int12: 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))] : atom_rec(A,F1,F2,F3,nDvd(Int12,Int21,List)) = aa(list(int),A,aa(int,fun(list(int),A),aa(int,fun(int,fun(list(int),A)),F3,Int12),Int21),List) ).
tff(fact_76_atom_Osimps_I12_J,axiom,
! [A: $tType,List: list(int),Int21: int,Int12: 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))] : atom_case(A,F1,F2,F3,nDvd(Int12,Int21,List)) = aa(list(int),A,aa(int,fun(list(int),A),aa(int,fun(int,fun(list(int),A)),F3,Int12),Int21),List) ).
tff(fact_77_mult__idem,axiom,
! [A: $tType] :
( ab_sem1668676832m_mult(A)
=> ! [X1: A] : aa(A,A,times_times(A,X1),X1) = X1 ) ).
tff(fact_78_times_Oidem,axiom,
! [A: $tType] :
( ab_sem1668676832m_mult(A)
=> ! [A1: A] : aa(A,A,times_times(A,A1),A1) = A1 ) ).
tff(fact_79_not__Cons__self2,axiom,
! [A: $tType,Xs1: list(A),X1: A] : cons(A,X1,Xs1) != Xs1 ).
tff(fact_80_not__Cons__self,axiom,
! [A: $tType,X1: A,Xs1: list(A)] : Xs1 != cons(A,X1,Xs1) ).
tff(fact_81_int__distrib_I1_J,axiom,
! [W: int,Z2: int,Z1: int] : aa(int,int,times_times(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z2)),W) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,times_times(int,Z1),W)),aa(int,int,times_times(int,Z2),W)) ).
tff(fact_82_int__distrib_I2_J,axiom,
! [Z2: int,Z1: int,W: int] : aa(int,int,times_times(int,W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,times_times(int,W),Z1)),aa(int,int,times_times(int,W),Z2)) ).
tff(fact_83_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,cons(B,Aa,List)) = aa(list(B),A,aa(B,fun(list(B),A),F2,Aa),List) ).
tff(fact_84_splice_Osimps_I3_J,axiom,
! [A: $tType,Ys1: list(A),Y: A,Xs1: list(A),X1: A] : splice(A,cons(A,X1,Xs1),cons(A,Y,Ys1)) = cons(A,X1,cons(A,Y,splice(A,Xs1,Ys1))) ).
tff(fact_85_divisor_Osimps_I1_J,axiom,
! [Ks: list(int),I1: int] : divisor(c_PresArith_Oatom_OLe(I1,Ks)) = one_one(int) ).
tff(fact_86_list_Orecs_I2_J,axiom,
! [A: $tType,B: $tType,List: list(B),Aa: B,F2: fun(B,fun(list(B),fun(A,A))),F1: A] : list_rec(A,B,F1,F2,cons(B,Aa,List)) = aa(A,A,aa(list(B),fun(A,A),aa(B,fun(list(B),fun(A,A)),F2,Aa),List),list_rec(A,B,F1,F2,List)) ).
tff(fact_87_atom_Osimps_I1_J,axiom,
! [List3: list(int),Int3: int,List: list(int),Int: int] :
( ( c_PresArith_Oatom_OLe(Int,List) = c_PresArith_Oatom_OLe(Int3,List3) )
<=> ( ( Int = Int3 )
& ( List = List3 ) ) ) ).
tff(fact_88_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))] : 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_89_atom_Osimps_I7_J,axiom,
! [List1: list(int),Int1: int,List2: list(int),Int2: int,Int11: int] : nDvd(Int11,Int2,List2) != c_PresArith_Oatom_OLe(Int1,List1) ).
tff(fact_90_atom_Osimps_I6_J,axiom,
! [List2: list(int),Int2: int,Int11: int,List1: list(int),Int1: int] : c_PresArith_Oatom_OLe(Int1,List1) != nDvd(Int11,Int2,List2) ).
tff(fact_91_atom_Orecs_I1_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))] : atom_rec(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_92_divisor__asubst,axiom,
! [A1: atom,Ks: list(int),I1: int] : divisor(asubst(I1,Ks,A1)) = divisor(A1) ).
tff(fact_93_succ__def,axiom,
! [K1: int] : succ(K1) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K1),one_one(int)) ).
tff(fact_94_asubst_Osimps_I3_J,axiom,
! [Ksa: list(int),K: int,Ia: int,Da: int,Ks_a: list(int),I: int] : asubst(I,Ks_a,nDvd(Da,Ia,cons(int,K,Ksa))) = nDvd(Da,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),aa(int,int,times_times(int,K),I)),aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),plus_plus(list(int)),map(int,int,times_times(int,K),Ks_a)),Ksa)) ).
tff(fact_95_IZ__asubst,axiom,
! [Xs: list(int),Aa: atom,Ksa: list(int),Ia: int] :
( i_Z(asubst(Ia,Ksa,Aa),Xs)
<=> i_Z(Aa,cons(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ia),iprod(int,Ksa,Xs)),Xs)) ) ).
tff(fact_96_map_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Xs: list(B),X: B,F: fun(B,A)] : map(B,A,F,cons(B,X,Xs)) = cons(A,aa(B,A,F,X),map(B,A,F,Xs)) ).
tff(fact_97_iprod__assoc,axiom,
! [A: $tType] :
( ring(A)
=> ! [Ys: list(A),Xs: list(A),X: A] : iprod(A,map(A,A,times_times(A,X),Xs),Ys) = aa(A,A,times_times(A,X),iprod(A,Xs,Ys)) ) ).
tff(fact_98_scale1,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [Xs: list(A)] : map(A,A,times_times(A,one_one(A)),Xs) = Xs ) ).
%----Arities (17)
tff(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct(int) ).
tff(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(int) ).
tff(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(int) ).
tff(arity_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(int) ).
tff(arity_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(int) ).
tff(arity_Int_Oint___Divides_Osemiring__div,axiom,
semiring_div(int) ).
tff(arity_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring(int) ).
tff(arity_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult(int) ).
tff(arity_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add(int) ).
tff(arity_Int_Oint___Rings_Osemiring,axiom,
semiring(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Groups_Oplus,axiom,
cl_Groups_Oplus(int) ).
tff(arity_Int_Oint___Rings_Oring,axiom,
ring(int) ).
tff(arity_Int_Oint___Groups_Oone,axiom,
one(int) ).
tff(arity_List_Olist___Groups_Oplus,axiom,
! [T_1: $tType] :
( ( zero(T_1)
& cl_Groups_Oplus(T_1) )
=> cl_Groups_Oplus(list(T_1)) ) ).
%----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,
div_mod(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),i),iprod(int,ks,e)),d) = div_mod(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),div_mod(int,i,d)),div_mod(int,iprod(int,ks,e),d)),d) ).
%------------------------------------------------------------------------------