TPTP Problem File: COM083_5.p
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%------------------------------------------------------------------------------
% File : COM083_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 196
% 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_196 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 165 ( 33 unt; 39 typ; 0 def)
% Number of atoms : 324 ( 87 equ)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 247 ( 49 ~; 16 |; 11 &)
% ( 29 <=>; 142 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 6 avg)
% Maximal term depth : 9 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 34 ( 20 >; 14 *; 0 +; 0 <<)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-3 aty)
% Number of functors : 27 ( 27 usr; 7 con; 0-5 aty)
% Number of variables : 399 ( 364 !; 4 ?; 399 :)
% ( 31 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:21:47
%------------------------------------------------------------------------------
%----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 (33)
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_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_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,
! [X: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X),set(atom,as)))
=> ( aa(atom,int,divisor,X) != zero_zero(int) ) ) ).
tff(fact_1_x,axiom,
! [X: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X),set(atom,as)))
=> i_Z(X,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,
? [X1: int] :
! [Xa3: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),Xa3),set(atom,as)))
=> i_Z(Xa3,cons(int,X1,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,
~ ! [X1: int] :
~ ! [Xa3: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),Xa3),set(atom,as)))
=> i_Z(Xa3,cons(int,X1,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),Xa1: A,P1: fun(A,bool)] :
( ( pp(aa(A,bool,P1,Xa1))
=> ( filter(A,P1,cons(A,Xa1,Xsa)) = cons(A,Xa1,filter(A,P1,Xsa)) ) )
& ( ~ pp(aa(A,bool,P1,Xa1))
=> ( filter(A,P1,cons(A,Xa1,Xsa)) = filter(A,P1,Xsa) ) ) ) ).
tff(fact_6_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_7_map_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Xsa: list(B),Xa1: B,F: fun(B,A)] : map(B,A,F,cons(B,Xa1,Xsa)) = cons(A,aa(B,A,F,Xa1),map(B,A,F,Xsa)) ).
tff(fact_8_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_9_map__ident,axiom,
! [A: $tType,X: list(A)] : map(A,A,combi(A),X) = X ).
tff(fact_10_calculation,axiom,
lbounds(as) = nil(product_prod(int,list(int))) ).
tff(fact_11_list_Oinject,axiom,
! [A: $tType,List2: list(A),A5: A,List1: list(A),A4: A] :
( ( cons(A,A4,List1) = cons(A,A5,List2) )
<=> ( ( A4 = A5 )
& ( List1 = List2 ) ) ) ).
tff(fact_12_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X3: A] : ord_less_eq(A,X3,X3) ) ).
tff(fact_13_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_14_less__int__def,axiom,
! [W: int,Z1: int] :
( ord_less(int,Z1,W)
<=> ( ord_less_eq(int,Z1,W)
& ( Z1 != W ) ) ) ).
tff(fact_15_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_16_map_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] : map(B,A,F,nil(B)) = nil(A) ).
tff(fact_17_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_18_filter_Osimps_I1_J,axiom,
! [A: $tType,P1: fun(A,bool)] : filter(A,P1,nil(A)) = nil(A) ).
tff(fact_19_list_Osimps_I3_J,axiom,
! [A: $tType,List: list(A),A3: A] : cons(A,A3,List) != nil(A) ).
tff(fact_20_list_Osimps_I2_J,axiom,
! [A: $tType,List: list(A),A3: A] : nil(A) != cons(A,A3,List) ).
tff(fact_21_set__subset__Cons,axiom,
! [A: $tType,Xa1: A,Xsa: list(A)] : ord_less_eq(fun(A,bool),set(A,Xsa),set(A,cons(A,Xa1,Xsa))) ).
tff(fact_22_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_23_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_24_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [Xa1: A] :
( ( zero_zero(A) = Xa1 )
<=> ( Xa1 = zero_zero(A) ) ) ) ).
tff(fact_25_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ~ ord_less_eq(A,X3,Y1)
=> ord_less_eq(A,Y1,X3) ) ) ).
tff(fact_26_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [Xa1: 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,Xa1),aa(A,B,G,Xa1)) ) ) ).
tff(fact_27_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X3: A] :
( ord_less_eq(A,X3,Y1)
=> ( ord_less_eq(A,Y1,Z)
=> ord_less_eq(A,X3,Z) ) ) ) ).
tff(fact_28_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X3: A] :
( ord_less_eq(A,X3,Y1)
=> ( ord_less_eq(A,Y1,X3)
=> ( X3 = Y1 ) ) ) ) ).
tff(fact_29_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A2: A] :
( ord_less_eq(A,A2,B2)
=> ( ( B2 = C2 )
=> ord_less_eq(A,A2,C2) ) ) ) ).
tff(fact_30_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A2: A] :
( ( A2 = B2 )
=> ( ord_less_eq(A,B2,C2)
=> ord_less_eq(A,A2,C2) ) ) ) ).
tff(fact_31_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [Xa1: A,Y: A] :
( ord_less_eq(A,Y,Xa1)
=> ( ord_less_eq(A,Xa1,Y)
<=> ( Xa1 = Y ) ) ) ) ).
tff(fact_32_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [Xa1: 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,Xa1),aa(A,B,G,Xa1)) ) ) ).
tff(fact_33_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X3: A] :
( ( X3 = Y1 )
=> ord_less_eq(A,X3,Y1) ) ) ).
tff(fact_34_order__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,Xa1: A] :
( ( Xa1 = Y )
<=> ( ord_less_eq(A,Xa1,Y)
& ord_less_eq(A,Y,Xa1) ) ) ) ).
tff(fact_35_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ord_less_eq(A,X3,Y1)
| ord_less_eq(A,Y1,X3) ) ) ).
tff(fact_36_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_37_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ~ ord_less(A,X3,Y1)
=> ( ( X3 != Y1 )
=> ord_less(A,Y1,X3) ) ) ) ).
tff(fact_38_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X3: A] :
( ord_less(A,X3,Y1)
=> ~ ord_less(A,Y1,X3) ) ) ).
tff(fact_39_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X3: A] :
( ord_less(A,X3,Y1)
=> ( ord_less(A,Y1,Z)
=> ord_less(A,X3,Z) ) ) ) ).
tff(fact_40_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A2: A] :
( ord_less(A,A2,B2)
=> ( ( B2 = C2 )
=> ord_less(A,A2,C2) ) ) ) ).
tff(fact_41_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A2: A] :
( ( A2 = B2 )
=> ( ord_less(A,B2,C2)
=> ord_less(A,A2,C2) ) ) ) ).
tff(fact_42_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A2: A] :
( ord_less(A,A2,B2)
=> ~ ord_less(A,B2,A2) ) ) ).
tff(fact_43_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [P1: bool,Y: A,Xa1: A] :
( ord_less(A,Xa1,Y)
=> ( ord_less(A,Y,Xa1)
=> pp(P1) ) ) ) ).
tff(fact_44_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X3: A] :
( ord_less(A,X3,Y1)
=> ( Y1 != X3 ) ) ) ).
tff(fact_45_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X3: A] :
( ord_less(A,X3,Y1)
=> ( X3 != Y1 ) ) ) ).
tff(fact_46_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X3: A] :
( ord_less(A,X3,Y1)
=> ~ ord_less(A,Y1,X3) ) ) ).
tff(fact_47_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X3: A] :
( ord_less(A,X3,Y1)
=> ~ ord_less(A,Y1,X3) ) ) ).
tff(fact_48_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X3: A] :
( ord_less(A,X3,Y1)
=> ( X3 != Y1 ) ) ) ).
tff(fact_49_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ( X3 != Y1 )
=> ( ~ ord_less(A,X3,Y1)
=> ord_less(A,Y1,X3) ) ) ) ).
tff(fact_50_linorder__antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xa1: A,Y: A] :
( ~ ord_less(A,Y,Xa1)
=> ( ~ ord_less(A,Xa1,Y)
<=> ( Xa1 = Y ) ) ) ) ).
tff(fact_51_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ord_less(A,X3,Y1)
| ( X3 = Y1 )
| ord_less(A,Y1,X3) ) ) ).
tff(fact_52_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,Xa1: A] :
( ~ ord_less(A,Xa1,Y)
<=> ( ord_less(A,Y,Xa1)
| ( Xa1 = Y ) ) ) ) ).
tff(fact_53_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,Xa1: A] :
( ( Xa1 != Y )
<=> ( ord_less(A,Xa1,Y)
| ord_less(A,Y,Xa1) ) ) ) ).
tff(fact_54_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X3: A] : ~ ord_less(A,X3,X3) ) ).
tff(fact_55_not__Cons__self2,axiom,
! [A: $tType,Xs1: list(A),X3: A] : cons(A,X3,Xs1) != Xs1 ).
tff(fact_56_not__Cons__self,axiom,
! [A: $tType,X3: A,Xs1: list(A)] : Xs1 != cons(A,X3,Xs1) ).
tff(fact_57_order__le__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X3: A] :
( ord_less_eq(A,X3,Y1)
=> ( ord_less(A,Y1,Z)
=> ord_less(A,X3,Z) ) ) ) ).
tff(fact_58_order__less__le__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X3: A] :
( ord_less(A,X3,Y1)
=> ( ord_less_eq(A,Y1,Z)
=> ord_less(A,X3,Z) ) ) ) ).
tff(fact_59_order__le__neq__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A2: A] :
( ord_less_eq(A,A2,B2)
=> ( ( A2 != B2 )
=> ord_less(A,A2,B2) ) ) ) ).
tff(fact_60_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X3: A] :
( ord_less_eq(A,X3,Y1)
=> ( ord_less(A,X3,Y1)
| ( X3 = Y1 ) ) ) ) ).
tff(fact_61_linorder__antisym__conv2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,Xa1: A] :
( ord_less_eq(A,Xa1,Y)
=> ( ~ ord_less(A,Xa1,Y)
<=> ( Xa1 = Y ) ) ) ) ).
tff(fact_62_order__less__imp__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X3: A] :
( ord_less(A,X3,Y1)
=> ord_less_eq(A,X3,Y1) ) ) ).
tff(fact_63_leD,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X3: A,Y1: A] :
( ord_less_eq(A,Y1,X3)
=> ~ ord_less(A,X3,Y1) ) ) ).
tff(fact_64_order__neq__le__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A2: A] :
( ( A2 != B2 )
=> ( ord_less_eq(A,A2,B2)
=> ord_less(A,A2,B2) ) ) ) ).
tff(fact_65_linorder__antisym__conv1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,Xa1: A] :
( ~ ord_less(A,Xa1,Y)
=> ( ord_less_eq(A,Xa1,Y)
<=> ( Xa1 = Y ) ) ) ) ).
tff(fact_66_not__leE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X3: A,Y1: A] :
( ~ ord_less_eq(A,Y1,X3)
=> ord_less(A,X3,Y1) ) ) ).
tff(fact_67_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ~ ord_less(A,X3,Y1)
=> ord_less_eq(A,Y1,X3) ) ) ).
tff(fact_68_order__le__less,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,Xa1: A] :
( ord_less_eq(A,Xa1,Y)
<=> ( ord_less(A,Xa1,Y)
| ( Xa1 = Y ) ) ) ) ).
tff(fact_69_less__le__not__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,Xa1: A] :
( ord_less(A,Xa1,Y)
<=> ( ord_less_eq(A,Xa1,Y)
& ~ ord_less_eq(A,Y,Xa1) ) ) ) ).
tff(fact_70_order__less__le,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,Xa1: A] :
( ord_less(A,Xa1,Y)
<=> ( ord_less_eq(A,Xa1,Y)
& ( Xa1 != Y ) ) ) ) ).
tff(fact_71_linorder__le__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ord_less_eq(A,X3,Y1)
| ord_less(A,Y1,X3) ) ) ).
tff(fact_72_linorder__not__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,Xa1: A] :
( ~ ord_less_eq(A,Xa1,Y)
<=> ord_less(A,Y,Xa1) ) ) ).
tff(fact_73_linorder__not__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,Xa1: A] :
( ~ ord_less(A,Xa1,Y)
<=> ord_less_eq(A,Y,Xa1) ) ) ).
tff(fact_74_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X1: A] : aa(A,B,F,X1) = aa(A,B,G,X1)
=> ( F = G ) ) ).
tff(fact_75_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),Xa1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa1),A1))
<=> pp(aa(A,bool,A1,Xa1)) ) ).
tff(fact_76_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_77_set__ConsD,axiom,
! [A: $tType,Xsa: list(A),Xa1: A,Y: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y),set(A,cons(A,Xa1,Xsa))))
=> ( ( Y = Xa1 )
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y),set(A,Xsa))) ) ) ).
tff(fact_78_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_79_filter__False,axiom,
! [A: $tType,P1: fun(A,bool),Xsa: list(A)] :
( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),set(A,Xsa)))
=> ~ pp(aa(A,bool,P1,X1)) )
=> ( filter(A,P1,Xsa) = nil(A) ) ) ).
tff(fact_80_filter__True,axiom,
! [A: $tType,P1: fun(A,bool),Xsa: list(A)] :
( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),set(A,Xsa)))
=> pp(aa(A,bool,P1,X1)) )
=> ( filter(A,P1,Xsa) = Xsa ) ) ).
tff(fact_81_filter__cong,axiom,
! [A: $tType,Q1: fun(A,bool),P1: fun(A,bool),Ys: list(A),Xsa: list(A)] :
( ( Xsa = Ys )
=> ( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),set(A,Ys)))
=> ( pp(aa(A,bool,P1,X1))
<=> pp(aa(A,bool,Q1,X1)) ) )
=> ( filter(A,P1,Xsa) = filter(A,Q1,Ys) ) ) ) ).
tff(fact_82_map__cong,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),Ys: list(A),Xsa: list(A)] :
( ( Xsa = Ys )
=> ( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),set(A,Ys)))
=> ( aa(A,B,F,X1) = aa(A,B,G,X1) ) )
=> ( map(A,B,F,Xsa) = map(A,B,G,Ys) ) ) ) ).
tff(fact_83_map__idI,axiom,
! [A: $tType,F: fun(A,A),Xsa: list(A)] :
( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),set(A,Xsa)))
=> ( aa(A,A,F,X1) = X1 ) )
=> ( map(A,A,F,Xsa) = Xsa ) ) ).
tff(fact_84_map__ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),Xsa: list(A)] :
( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),set(A,Xsa)))
=> ( aa(A,B,F,X1) = aa(A,B,G,X1) ) )
=> ( map(A,B,F,Xsa) = map(A,B,G,Xsa) ) ) ).
tff(fact_85_ex__map__conv,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Ys: list(B)] :
( ? [Xs: list(A)] : Ys = 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,Ys)))
=> ? [Xa2: A] : X2 = aa(A,B,F,Xa2) ) ) ).
tff(fact_86_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F: fun(A,B)] :
( ord_less(fun(A,B),F,G)
<=> ( ord_less_eq(fun(A,B),F,G)
& ~ ord_less_eq(fun(A,B),G,F) ) ) ) ).
tff(fact_87_zlcms__pos,axiom,
! [Is: list(int)] :
( ! [X1: int] :
( pp(aa(fun(int,bool),bool,aa(int,fun(fun(int,bool),bool),member(int),X1),set(int,Is)))
=> ( X1 != zero_zero(int) ) )
=> ord_less(int,zero_zero(int),zlcms(Is)) ) ).
tff(fact_88_elem__le__zlcms,axiom,
! [I: int,Is: list(int)] :
( ! [X1: int] :
( pp(aa(fun(int,bool),bool,aa(int,fun(fun(int,bool),bool),member(int),X1),set(int,Is)))
=> ( X1 != 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_89_equalityI,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B1)
=> ( ord_less_eq(fun(A,bool),B1,A1)
=> ( A1 = B1 ) ) ) ).
tff(fact_90_subsetD,axiom,
! [A: $tType,C1: A,B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B1)
=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B1)) ) ) ).
tff(fact_91_subset__refl,axiom,
! [A: $tType,A1: fun(A,bool)] : ord_less_eq(fun(A,bool),A1,A1) ).
tff(fact_92_psubset__eq,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less(fun(A,bool),A1,B1)
<=> ( ord_less_eq(fun(A,bool),A1,B1)
& ( A1 != B1 ) ) ) ).
tff(fact_93_set__eq__subset,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ( A1 = B1 )
<=> ( ord_less_eq(fun(A,bool),A1,B1)
& ord_less_eq(fun(A,bool),B1,A1) ) ) ).
tff(fact_94_subset__iff__psubset__eq,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B1)
<=> ( ord_less(fun(A,bool),A1,B1)
| ( A1 = B1 ) ) ) ).
tff(fact_95_equalityD1,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ( A1 = B1 )
=> ord_less_eq(fun(A,bool),A1,B1) ) ).
tff(fact_96_equalityD2,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ( A1 = B1 )
=> ord_less_eq(fun(A,bool),B1,A1) ) ).
tff(fact_97_psubset__imp__subset,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less(fun(A,bool),A1,B1)
=> ord_less_eq(fun(A,bool),A1,B1) ) ).
tff(fact_98_in__mono,axiom,
! [A: $tType,Xa1: A,B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B1)
=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa1),A1))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa1),B1)) ) ) ).
%----Arities (12)
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___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,
? [X: int] :
( ord_less_eq(int,zero_zero(int),X)
& ord_less(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))))
& ! [Xa: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),Xa),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(Xa,cons(int,X,xs)) ) ) ).
%------------------------------------------------------------------------------