TPTP Problem File: COM085_5.p
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%------------------------------------------------------------------------------
% File : COM085_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 201
% 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_201 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 168 ( 35 unt; 41 typ; 0 def)
% Number of atoms : 310 ( 96 equ)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 229 ( 46 ~; 14 |; 13 &)
% ( 26 <=>; 130 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 6 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 36 ( 21 >; 15 *; 0 +; 0 <<)
% Number of predicates : 11 ( 10 usr; 0 prp; 1-3 aty)
% Number of functors : 28 ( 28 usr; 7 con; 0-5 aty)
% Number of variables : 392 ( 350 !; 9 ?; 392 :)
% ( 33 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:22:07
%------------------------------------------------------------------------------
%----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_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 (35)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[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] : ( 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_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_List_Ofilter,type,
filter:
!>[A: $tType] : ( ( fun(A,bool) * 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_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_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[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_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_Odivisor,type,
divisor: fun(atom,int) ).
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_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_as,type,
as: list(atom) ).
tff(sy_v_x____,type,
x: int ).
tff(sy_v_xs,type,
xs: list(int) ).
%----Relevant facts (99)
tff(fact_0_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_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,cons(int,x,xs)) ) ).
tff(fact_2__096EX_Ax_O_AALL_Aa_058set_Aas_O_AI_092_060_094isub_062Z_Aa_A_Ix_A_D_Axs_J_096,axiom,
? [X: 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,cons(int,X,xs)) ) ).
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,
~ ! [X: 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,cons(int,X,xs)) ) ).
tff(fact_4_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_5_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,cons(A,Xa,Xsa)) = cons(A,Xa,filter(A,P1,Xsa)) ) )
& ( ~ pp(aa(A,bool,P1,Xa))
=> ( filter(A,P1,cons(A,Xa,Xsa)) = filter(A,P1,Xsa) ) ) ) ).
tff(fact_6_zmod__zero,axiom,
! [B1: int] : div_mod(int,zero_zero(int),B1) = zero_zero(int) ).
tff(fact_7_zmod__self,axiom,
! [A1: int] : div_mod(int,A1,A1) = zero_zero(int) ).
tff(fact_8_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_9_map_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Xsa: list(B),Xa: B,F: fun(B,A)] : map(B,A,F,cons(B,Xa,Xsa)) = cons(A,aa(B,A,F,Xa),map(B,A,F,Xsa)) ).
tff(fact_10_mod__0,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [A1: A] : div_mod(A,zero_zero(A),A1) = zero_zero(A) ) ).
tff(fact_11_mod__self,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [A1: A] : div_mod(A,A1,A1) = zero_zero(A) ) ).
tff(fact_12_pos__mod__sign,axiom,
! [A1: int,B1: int] :
( ord_less(int,zero_zero(int),B1)
=> ord_less_eq(int,zero_zero(int),div_mod(int,A1,B1)) ) ).
tff(fact_13_pos__mod__conj,axiom,
! [A1: int,B1: int] :
( ord_less(int,zero_zero(int),B1)
=> ( ord_less_eq(int,zero_zero(int),div_mod(int,A1,B1))
& ord_less(int,div_mod(int,A1,B1),B1) ) ) ).
tff(fact_14_mod__pos__pos__trivial,axiom,
! [B1: int,A1: int] :
( ord_less_eq(int,zero_zero(int),A1)
=> ( ord_less(int,A1,B1)
=> ( div_mod(int,A1,B1) = A1 ) ) ) ).
tff(fact_15_neg__mod__sign,axiom,
! [A1: int,B1: int] :
( ord_less(int,B1,zero_zero(int))
=> ord_less_eq(int,div_mod(int,A1,B1),zero_zero(int)) ) ).
tff(fact_16_list_Oinject,axiom,
! [A: $tType,List3: list(A),A6: A,List2: list(A),A5: A] :
( ( cons(A,A5,List2) = cons(A,A6,List3) )
<=> ( ( A5 = A6 )
& ( List2 = List3 ) ) ) ).
tff(fact_17_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_18_map__ident,axiom,
! [A: $tType,X3: list(A)] : map(A,A,combi(A),X3) = X3 ).
tff(fact_19_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_20_not__Cons__self2,axiom,
! [A: $tType,Xs1: list(A),X1: A] : cons(A,X1,Xs1) != Xs1 ).
tff(fact_21_not__Cons__self,axiom,
! [A: $tType,X1: A,Xs1: list(A)] : Xs1 != cons(A,X1,Xs1) ).
tff(fact_22_mod__by__0,axiom,
! [A: $tType] :
( semiring_div(A)
=> ! [A1: A] : div_mod(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_23_set__ConsD,axiom,
! [A: $tType,Xsa: list(A),Xa: A,Y: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y),set(A,cons(A,Xa,Xsa))))
=> ( ( Y = Xa )
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y),set(A,Xsa))) ) ) ).
tff(fact_24_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_25_Divides_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y1: int,X1: int] :
( ord_less_eq(int,zero_zero(int),X1)
=> ( ord_less_eq(int,zero_zero(int),Y1)
=> ord_less_eq(int,zero_zero(int),div_mod(int,X1,Y1)) ) ) ).
tff(fact_26_zmod__le__nonneg__dividend,axiom,
! [K: int,M: int] :
( ord_less_eq(int,zero_zero(int),M)
=> ord_less_eq(int,div_mod(int,M,K),M) ) ).
tff(fact_27_neg__mod__bound,axiom,
! [A1: int,B1: int] :
( ord_less(int,B1,zero_zero(int))
=> ord_less(int,B1,div_mod(int,A1,B1)) ) ).
tff(fact_28_pos__mod__bound,axiom,
! [A1: int,B1: int] :
( ord_less(int,zero_zero(int),B1)
=> ord_less(int,div_mod(int,A1,B1),B1) ) ).
tff(fact_29_mod__neg__neg__trivial,axiom,
! [B1: int,A1: int] :
( ord_less_eq(int,A1,zero_zero(int))
=> ( ord_less(int,B1,A1)
=> ( div_mod(int,A1,B1) = A1 ) ) ) ).
tff(fact_30_neg__mod__conj,axiom,
! [A1: int,B1: int] :
( ord_less(int,B1,zero_zero(int))
=> ( ord_less_eq(int,div_mod(int,A1,B1),zero_zero(int))
& ord_less(int,B1,div_mod(int,A1,B1)) ) ) ).
tff(fact_31_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_32_calculation,axiom,
lbounds(as) = nil(product_prod(int,list(int))) ).
tff(fact_33_filter__True,axiom,
! [A: $tType,P1: fun(A,bool),Xsa: list(A)] :
( ! [X: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),set(A,Xsa)))
=> pp(aa(A,bool,P1,X)) )
=> ( filter(A,P1,Xsa) = Xsa ) ) ).
tff(fact_34_filter__cong,axiom,
! [A: $tType,Q1: fun(A,bool),P1: fun(A,bool),Ys1: list(A),Xsa: list(A)] :
( ( Xsa = Ys1 )
=> ( ! [X: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),set(A,Ys1)))
=> ( pp(aa(A,bool,P1,X))
<=> pp(aa(A,bool,Q1,X)) ) )
=> ( filter(A,P1,Xsa) = filter(A,Q1,Ys1) ) ) ) ).
tff(fact_35_map__cong,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),Ys1: list(A),Xsa: list(A)] :
( ( Xsa = Ys1 )
=> ( ! [X: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),set(A,Ys1)))
=> ( aa(A,B,F,X) = aa(A,B,G,X) ) )
=> ( map(A,B,F,Xsa) = map(A,B,G,Ys1) ) ) ) ).
tff(fact_36_map__idI,axiom,
! [A: $tType,F: fun(A,A),Xsa: list(A)] :
( ! [X: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),set(A,Xsa)))
=> ( aa(A,A,F,X) = X ) )
=> ( map(A,A,F,Xsa) = Xsa ) ) ).
tff(fact_37_map__ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),Xsa: list(A)] :
( ! [X: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),set(A,Xsa)))
=> ( aa(A,B,F,X) = aa(A,B,G,X) ) )
=> ( map(A,B,F,Xsa) = map(A,B,G,Xsa) ) ) ).
tff(fact_38_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_39_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_40_map_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] : map(B,A,F,nil(B)) = nil(A) ).
tff(fact_41_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_42_filter_Osimps_I1_J,axiom,
! [A: $tType,P1: fun(A,bool)] : filter(A,P1,nil(A)) = nil(A) ).
tff(fact_43_list_Osimps_I2_J,axiom,
! [A: $tType,List1: list(A),A4: A] : nil(A) != cons(A,A4,List1) ).
tff(fact_44_list_Osimps_I3_J,axiom,
! [A: $tType,List1: list(A),A4: A] : cons(A,A4,List1) != nil(A) ).
tff(fact_45_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_46_zlcms__pos,axiom,
! [Is: list(int)] :
( ! [X: int] :
( pp(aa(fun(int,bool),bool,aa(int,fun(fun(int,bool),bool),member(int),X),set(int,Is)))
=> ( X != zero_zero(int) ) )
=> ord_less(int,zero_zero(int),zlcms(Is)) ) ).
tff(fact_47_elem__le__zlcms,axiom,
! [I: int,Is: list(int)] :
( ! [X: int] :
( pp(aa(fun(int,bool),bool,aa(int,fun(fun(int,bool),bool),member(int),X),set(int,Is)))
=> ( X != zero_zero(int) ) )
=> ( pp(aa(fun(int,bool),bool,aa(int,fun(fun(int,bool),bool),member(int),I),set(int,Is)))
=> ord_less_eq(int,I,zlcms(Is)) ) ) ).
tff(fact_48_filter__False,axiom,
! [A: $tType,P1: fun(A,bool),Xsa: list(A)] :
( ! [X: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),set(A,Xsa)))
=> ~ pp(aa(A,bool,P1,X)) )
=> ( filter(A,P1,Xsa) = nil(A) ) ) ).
tff(fact_49_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys1: list(B),F: fun(B,A),Xsa: list(A),Xa: A] :
( ( cons(A,Xa,Xsa) = map(B,A,F,Ys1) )
<=> ? [Z1: B,Zs: list(B)] :
( ( Ys1 = cons(B,Z1,Zs) )
& ( Xa = aa(B,A,F,Z1) )
& ( Xsa = map(B,A,F,Zs) ) ) ) ).
tff(fact_50_map__eq__Cons__conv,axiom,
! [B: $tType,A: $tType,Ys1: list(A),Y: A,Xsa: list(B),F: fun(B,A)] :
( ( map(B,A,F,Xsa) = cons(A,Y,Ys1) )
<=> ? [Z1: B,Zs: list(B)] :
( ( Xsa = cons(B,Z1,Zs) )
& ( aa(B,A,F,Z1) = Y )
& ( map(B,A,F,Zs) = Ys1 ) ) ) ).
tff(fact_51_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X1: A] : ord_less_eq(A,X1,X1) ) ).
tff(fact_52_list_Oexhaust,axiom,
! [A: $tType,Y1: list(A)] :
( ( Y1 != nil(A) )
=> ~ ! [A3: A,List: list(A)] : Y1 != cons(A,A3,List) ) ).
tff(fact_53_neq__Nil__conv,axiom,
! [A: $tType,Xsa: list(A)] :
( ( Xsa != nil(A) )
<=> ? [Y2: A,Ys: list(A)] : Xsa = cons(A,Y2,Ys) ) ).
tff(fact_54_set__subset__Cons,axiom,
! [A: $tType,Xa: A,Xsa: list(A)] : ord_less_eq(fun(A,bool),set(A,Xsa),set(A,cons(A,Xa,Xsa))) ).
tff(fact_55_filter__is__subset,axiom,
! [A: $tType,Xsa: list(A),P1: fun(A,bool)] : ord_less_eq(fun(A,bool),set(A,filter(A,P1,Xsa)),set(A,Xsa)) ).
tff(fact_56_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [Xa: A] :
( ( zero_zero(A) = Xa )
<=> ( Xa = zero_zero(A) ) ) ) ).
tff(fact_57_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
<=> ! [X2: A] : ord_less_eq(B,aa(A,B,F,X2),aa(A,B,G,X2)) ) ) ).
tff(fact_58_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
| ord_less_eq(A,Y1,X1) ) ) ).
tff(fact_59_order__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,Xa: A] :
( ( Xa = Y )
<=> ( ord_less_eq(A,Xa,Y)
& ord_less_eq(A,Y,Xa) ) ) ) ).
tff(fact_60_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X1: A] :
( ( X1 = Y1 )
=> ord_less_eq(A,X1,Y1) ) ) ).
tff(fact_61_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [Xa: A,G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
=> ord_less_eq(B,aa(A,B,F,Xa),aa(A,B,G,Xa)) ) ) ).
tff(fact_62_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [Xa: A,Y: A] :
( ord_less_eq(A,Y,Xa)
=> ( ord_less_eq(A,Xa,Y)
<=> ( Xa = Y ) ) ) ) ).
tff(fact_63_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B1: A,A1: A] :
( ( A1 = B1 )
=> ( ord_less_eq(A,B1,C1)
=> ord_less_eq(A,A1,C1) ) ) ) ).
tff(fact_64_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B1: A,A1: A] :
( ord_less_eq(A,A1,B1)
=> ( ( B1 = C1 )
=> ord_less_eq(A,A1,C1) ) ) ) ).
tff(fact_65_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
=> ( ord_less_eq(A,Y1,X1)
=> ( X1 = Y1 ) ) ) ) ).
tff(fact_66_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
=> ( ord_less_eq(A,Y1,Z)
=> ord_less_eq(A,X1,Z) ) ) ) ).
tff(fact_67_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [Xa: A,G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
=> ord_less_eq(B,aa(A,B,F,Xa),aa(A,B,G,Xa)) ) ) ).
tff(fact_68_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ~ ord_less_eq(A,X1,Y1)
=> ord_less_eq(A,Y1,X1) ) ) ).
tff(fact_69_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X1: A] : ~ ord_less(A,X1,X1) ) ).
tff(fact_70_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,Xa: A] :
( ( Xa != Y )
<=> ( ord_less(A,Xa,Y)
| ord_less(A,Y,Xa) ) ) ) ).
tff(fact_71_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,Xa: A] :
( ~ ord_less(A,Xa,Y)
<=> ( ord_less(A,Y,Xa)
| ( Xa = Y ) ) ) ) ).
tff(fact_72_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
| ( X1 = Y1 )
| ord_less(A,Y1,X1) ) ) ).
tff(fact_73_linorder__antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xa: A,Y: A] :
( ~ ord_less(A,Y,Xa)
=> ( ~ ord_less(A,Xa,Y)
<=> ( Xa = Y ) ) ) ) ).
tff(fact_74_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X: A] : aa(A,B,F,X) = aa(A,B,G,X)
=> ( F = G ) ) ).
tff(fact_75_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_76_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_77_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ( X1 != Y1 )
=> ( ~ ord_less(A,X1,Y1)
=> ord_less(A,Y1,X1) ) ) ) ).
tff(fact_78_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ( X1 != Y1 ) ) ) ).
tff(fact_79_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ~ ord_less(A,Y1,X1) ) ) ).
tff(fact_80_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ~ ord_less(A,Y1,X1) ) ) ).
tff(fact_81_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ( X1 != Y1 ) ) ) ).
tff(fact_82_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ( Y1 != X1 ) ) ) ).
tff(fact_83_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [P1: bool,Y: A,Xa: A] :
( ord_less(A,Xa,Y)
=> ( ord_less(A,Y,Xa)
=> pp(P1) ) ) ) ).
tff(fact_84_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ~ ord_less(A,B1,A1) ) ) ).
tff(fact_85_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B1: A,A1: A] :
( ( A1 = B1 )
=> ( ord_less(A,B1,C1)
=> ord_less(A,A1,C1) ) ) ) ).
tff(fact_86_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ( ( B1 = C1 )
=> ord_less(A,A1,C1) ) ) ) ).
tff(fact_87_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ( ord_less(A,Y1,Z)
=> ord_less(A,X1,Z) ) ) ) ).
tff(fact_88_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ~ ord_less(A,Y1,X1) ) ) ).
tff(fact_89_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ~ ord_less(A,X1,Y1)
=> ( ( X1 != Y1 )
=> ord_less(A,Y1,X1) ) ) ) ).
tff(fact_90_linorder__not__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,Xa: A] :
( ~ ord_less(A,Xa,Y)
<=> ord_less_eq(A,Y,Xa) ) ) ).
tff(fact_91_linorder__not__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,Xa: A] :
( ~ ord_less_eq(A,Xa,Y)
<=> ord_less(A,Y,Xa) ) ) ).
tff(fact_92_linorder__le__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
| ord_less(A,Y1,X1) ) ) ).
tff(fact_93_order__less__le,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,Xa: A] :
( ord_less(A,Xa,Y)
<=> ( ord_less_eq(A,Xa,Y)
& ( Xa != Y ) ) ) ) ).
tff(fact_94_less__le__not__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,Xa: A] :
( ord_less(A,Xa,Y)
<=> ( ord_less_eq(A,Xa,Y)
& ~ ord_less_eq(A,Y,Xa) ) ) ) ).
tff(fact_95_order__le__less,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,Xa: A] :
( ord_less_eq(A,Xa,Y)
<=> ( ord_less(A,Xa,Y)
| ( Xa = Y ) ) ) ) ).
tff(fact_96_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ~ ord_less(A,X1,Y1)
=> ord_less_eq(A,Y1,X1) ) ) ).
tff(fact_97_not__leE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X1: A,Y1: A] :
( ~ ord_less_eq(A,Y1,X1)
=> ord_less(A,X1,Y1) ) ) ).
tff(fact_98_linorder__antisym__conv1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,Xa: A] :
( ~ ord_less(A,Xa,Y)
=> ( ord_less_eq(A,Xa,Y)
<=> ( Xa = Y ) ) ) ) ).
%----Arities (13)
tff(arity_fun___Orderings_Opreorder,axiom,
! [T_1: $tType,T_2: $tType] :
( preorder(T_2)
=> preorder(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oorder,axiom,
! [T_1: $tType,T_2: $tType] :
( order(T_2)
=> order(fun(T_1,T_2)) ) ).
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___Divides_Osemiring__div,axiom,
semiring_div(int) ).
tff(arity_Int_Oint___Orderings_Opreorder,axiom,
preorder(int) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Orderings_Oorder,axiom,
order(int) ).
tff(arity_Int_Oint___Orderings_Oord,axiom,
ord(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
tff(arity_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
%----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,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,
( ord_less_eq(int,zero_zero(int),div_mod(int,x,zlcms(map(atom,int,divisor,filter(atom,atom_case(bool,combk(fun(list(int),bool),int,combk(bool,list(int),fFalse)),combk(fun(int,fun(list(int),bool)),int,combk(fun(list(int),bool),int,combk(bool,list(int),fTrue))),combk(fun(int,fun(list(int),bool)),int,combk(fun(list(int),bool),int,combk(bool,list(int),fTrue)))),as)))))
& ord_less(int,div_mod(int,x,zlcms(map(atom,int,divisor,filter(atom,atom_case(bool,combk(fun(list(int),bool),int,combk(bool,list(int),fFalse)),combk(fun(int,fun(list(int),bool)),int,combk(fun(list(int),bool),int,combk(bool,list(int),fTrue))),combk(fun(int,fun(list(int),bool)),int,combk(fun(list(int),bool),int,combk(bool,list(int),fTrue)))),as)))),zlcms(map(atom,int,divisor,filter(atom,atom_case(bool,combk(fun(list(int),bool),int,combk(bool,list(int),fFalse)),combk(fun(int,fun(list(int),bool)),int,combk(fun(list(int),bool),int,combk(bool,list(int),fTrue))),combk(fun(int,fun(list(int),bool)),int,combk(fun(list(int),bool),int,combk(bool,list(int),fTrue)))),as))))
& ! [X: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X),collect(atom,combs(atom,bool,bool,combb(bool,fun(bool,bool),atom,fconj,combc(atom,fun(atom,bool),bool,member(atom),set(atom,as))),atom_case(bool,combk(fun(list(int),bool),int,combk(bool,list(int),fFalse)),combk(fun(int,fun(list(int),bool)),int,combk(fun(list(int),bool),int,combk(bool,list(int),fTrue))),combk(fun(int,fun(list(int),bool)),int,combk(fun(list(int),bool),int,combk(bool,list(int),fTrue))))))))
=> i_Z(X,cons(int,div_mod(int,x,zlcms(map(atom,int,divisor,filter(atom,atom_case(bool,combk(fun(list(int),bool),int,combk(bool,list(int),fFalse)),combk(fun(int,fun(list(int),bool)),int,combk(fun(list(int),bool),int,combk(bool,list(int),fTrue))),combk(fun(int,fun(list(int),bool)),int,combk(fun(list(int),bool),int,combk(bool,list(int),fTrue)))),as)))),xs)) ) ) ).
%------------------------------------------------------------------------------