TPTP Problem File: COM074_5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : COM074_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 168
% 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_168 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 182 ( 41 unt; 48 typ; 0 def)
% Number of atoms : 341 ( 72 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 251 ( 44 ~; 18 |; 11 &)
% ( 33 <=>; 145 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 22 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 21 ( 15 >; 6 *; 0 +; 0 <<)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-2 aty)
% Number of functors : 35 ( 35 usr; 8 con; 0-4 aty)
% Number of variables : 379 ( 341 !; 0 ?; 379 :)
% ( 38 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:20:58
%------------------------------------------------------------------------------
%----Should-be-implicit typings (7)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_List_Olist,type,
list: $tType > $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_PresArith_Oatom,type,
atom: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tc_prod,type,
product_prod: ( $tType * $tType ) > $tType ).
%----Explicit typings (41)
tff(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
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_Groups_Ogroup__add,type,
group_add:
!>[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_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
tff(sy_c_Big__Operators_Olinorder__class_OMin,type,
big_linorder_Min:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(B,fun(A,C))) ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : fun(A,fun(B,A)) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_ListVector_Oiprod,type,
iprod:
!>[A: $tType] : fun(list(A),fun(list(A),A)) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olist_Olist__case,type,
list_case:
!>[T: $tType,A: $tType] : ( ( T * fun(A,fun(list(A),T)) ) > fun(list(A),T) ) ).
tff(sy_c_List_Oset,type,
set:
!>[A: $tType] : ( list(A) > fun(A,bool) ) ).
tff(sy_c_List_Otl,type,
tl:
!>[A: $tType] : fun(list(A),list(A)) ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > fun(A,bool) ) ).
tff(sy_c_PresArith_Oatom_OLe,type,
c_PresArith_Oatom_OLe: fun(int,fun(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))) ) > fun(atom,T) ) ).
tff(sy_c_PresArith_Oatom_Oatom__size,type,
atom_size: atom > nat ).
tff(sy_c_PresArith_Odivisor,type,
divisor: atom > int ).
tff(sy_c_PresArith_Olbounds,type,
lbounds: list(atom) > list(product_prod(int,list(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_fEx,type,
fEx:
!>[A: $tType] : fun(fun(A,bool),bool) ).
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_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,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_ks____,type,
ks: list(int) ).
tff(sy_v_l____,type,
l: int ).
tff(sy_v_xs,type,
xs: list(int) ).
%----Relevant facts (99)
tff(fact_0_norm,axiom,
! [X4: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X4),set(atom,as)))
=> ( divisor(X4) != zero_zero(int) ) ) ).
tff(fact_1_diff__self,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),A2) = zero_zero(A) ) ).
tff(fact_2__096lbounds_Aas_A_061_A_091_093_096,axiom,
lbounds(as) = nil(product_prod(int,list(int))) ).
tff(fact_3_less__bin__lemma,axiom,
! [La: int,K: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),La))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),K),La)),zero_zero(int))) ) ).
tff(fact_4_atom_Osimps_I1_J,axiom,
! [List2: list(int),Int2: int,List1: list(int),Int1: int] :
( ( aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int1),List1) = aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int2),List2) )
<=> ( ( Int1 = Int2 )
& ( List1 = List2 ) ) ) ).
tff(fact_5_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B1: A,Aa: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Aa),B1))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),Aa),B1)),zero_zero(A))) ) ) ).
tff(fact_6_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B1: A,Aa: A] :
( pp(aa(A,bool,ord_less_eq(A,Aa),B1))
<=> pp(aa(A,bool,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),Aa),B1)),zero_zero(A))) ) ) ).
tff(fact_7_atom_Osimps_I10_J,axiom,
! [A: $tType,List1: list(int),Int1: 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))] : aa(atom,A,atom_case(A,F1,F2,F3),aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int1),List1)) = aa(list(int),A,aa(int,fun(list(int),A),F1,Int1),List1) ).
tff(fact_8_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : pp(aa(A,bool,ord_less_eq(A,X),X)) ) ).
tff(fact_9_iprod__left__diff__distrib,axiom,
! [A: $tType] :
( ring(A)
=> ! [Zs: list(A),Ys: list(A),Xs: list(A)] : aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),minus_minus(list(A)),Xs),Ys)),Zs) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Xs),Zs)),aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Ys),Zs)) ) ).
tff(fact_10_less__int__def,axiom,
! [W: int,Z1: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z1),W))
<=> ( pp(aa(int,bool,ord_less_eq(int,Z1),W))
& ( Z1 != W ) ) ) ).
tff(fact_11_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),zero_zero(int))) ).
tff(fact_12_iprod__Nil2,axiom,
! [A: $tType] :
( ring(A)
=> ! [Xs: list(A)] : aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Xs),nil(A)) = zero_zero(A) ) ).
tff(fact_13_iprod__Nil,axiom,
! [A: $tType] :
( ring(A)
=> ! [Ys: list(A)] : aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),nil(A)),Ys) = zero_zero(A) ) ).
tff(fact_14_list__diff__Nil2,axiom,
! [A: $tType] :
( group_add(A)
=> ! [Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),minus_minus(list(A)),Xs),nil(A)) = Xs ) ).
tff(fact_15_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X2: A] :
( ( zero_zero(A) = X2 )
<=> ( X2 = zero_zero(A) ) ) ) ).
tff(fact_16_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ pp(aa(A,bool,ord_less_eq(A,X),Y))
=> pp(aa(A,bool,ord_less_eq(A,Y),X)) ) ) ).
tff(fact_17_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [X2: A,G: fun(A,B),F: fun(A,B)] :
( pp(aa(fun(A,B),bool,ord_less_eq(fun(A,B),F),G))
=> pp(aa(B,bool,ord_less_eq(B,aa(A,B,F,X2)),aa(A,B,G,X2))) ) ) ).
tff(fact_18_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X: A] :
( pp(aa(A,bool,ord_less_eq(A,X),Y))
=> ( pp(aa(A,bool,ord_less_eq(A,Y),Z))
=> pp(aa(A,bool,ord_less_eq(A,X),Z)) ) ) ) ).
tff(fact_19_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,ord_less_eq(A,X),Y))
=> ( pp(aa(A,bool,ord_less_eq(A,Y),X))
=> ( X = Y ) ) ) ) ).
tff(fact_20_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A2: A] :
( pp(aa(A,bool,ord_less_eq(A,A2),B2))
=> ( ( B2 = C2 )
=> pp(aa(A,bool,ord_less_eq(A,A2),C2)) ) ) ) ).
tff(fact_21_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A2: A] :
( ( A2 = B2 )
=> ( pp(aa(A,bool,ord_less_eq(A,B2),C2))
=> pp(aa(A,bool,ord_less_eq(A,A2),C2)) ) ) ) ).
tff(fact_22_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [X2: A,Y2: A] :
( pp(aa(A,bool,ord_less_eq(A,Y2),X2))
=> ( pp(aa(A,bool,ord_less_eq(A,X2),Y2))
<=> ( X2 = Y2 ) ) ) ) ).
tff(fact_23_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [X2: A,G: fun(A,B),F: fun(A,B)] :
( pp(aa(fun(A,B),bool,ord_less_eq(fun(A,B),F),G))
=> pp(aa(B,bool,ord_less_eq(B,aa(A,B,F,X2)),aa(A,B,G,X2))) ) ) ).
tff(fact_24_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ( X = Y )
=> pp(aa(A,bool,ord_less_eq(A,X),Y)) ) ) ).
tff(fact_25_order__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X2: A] :
( ( X2 = Y2 )
<=> ( pp(aa(A,bool,ord_less_eq(A,X2),Y2))
& pp(aa(A,bool,ord_less_eq(A,Y2),X2)) ) ) ) ).
tff(fact_26_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,ord_less_eq(A,X),Y))
| pp(aa(A,bool,ord_less_eq(A,Y),X)) ) ) ).
tff(fact_27_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F: fun(A,B)] :
( pp(aa(fun(A,B),bool,ord_less_eq(fun(A,B),F),G))
<=> ! [X3: A] : pp(aa(B,bool,ord_less_eq(B,aa(A,B,F,X3)),aa(A,B,G,X3))) ) ) ).
tff(fact_28_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( ( X != Y )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).
tff(fact_29_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
tff(fact_30_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z)) ) ) ) ).
tff(fact_31_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
=> ( ( B2 = C2 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).
tff(fact_32_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A2: A] :
( ( A2 = B2 )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).
tff(fact_33_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ).
tff(fact_34_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [P1: bool,Y2: A,X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X2))
=> pp(P1) ) ) ) ).
tff(fact_35_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( Y != X ) ) ) ).
tff(fact_36_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( X != Y ) ) ) ).
tff(fact_37_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
tff(fact_38_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
tff(fact_39_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( X != Y ) ) ) ).
tff(fact_40_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ( X != Y )
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).
tff(fact_41_linorder__antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: A,Y2: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X2))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
<=> ( X2 = Y2 ) ) ) ) ).
tff(fact_42_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
| ( X = Y )
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
tff(fact_43_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X2))
| ( X2 = Y2 ) ) ) ) ).
tff(fact_44_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ( X2 != Y2 )
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X2)) ) ) ) ).
tff(fact_45_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X)) ) ).
tff(fact_46_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [D: A,C1: A,B1: A,Aa: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Aa),B1) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C1),D) )
=> ( ( Aa = B1 )
<=> ( C1 = D ) ) ) ) ).
tff(fact_47_order__le__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X: A] :
( pp(aa(A,bool,ord_less_eq(A,X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z)) ) ) ) ).
tff(fact_48_order__less__le__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> ( pp(aa(A,bool,ord_less_eq(A,Y),Z))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z)) ) ) ) ).
tff(fact_49_order__le__neq__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A2: A] :
( pp(aa(A,bool,ord_less_eq(A,A2),B2))
=> ( ( A2 != B2 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).
tff(fact_50_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,ord_less_eq(A,X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
| ( X = Y ) ) ) ) ).
tff(fact_51_linorder__antisym__conv2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( pp(aa(A,bool,ord_less_eq(A,X2),Y2))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
<=> ( X2 = Y2 ) ) ) ) ).
tff(fact_52_order__less__imp__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> pp(aa(A,bool,ord_less_eq(A,X),Y)) ) ) ).
tff(fact_53_leD,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( pp(aa(A,bool,ord_less_eq(A,Y),X))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).
tff(fact_54_order__neq__le__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A2: A] :
( ( A2 != B2 )
=> ( pp(aa(A,bool,ord_less_eq(A,A2),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).
tff(fact_55_linorder__antisym__conv1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
=> ( pp(aa(A,bool,ord_less_eq(A,X2),Y2))
<=> ( X2 = Y2 ) ) ) ) ).
tff(fact_56_not__leE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ pp(aa(A,bool,ord_less_eq(A,Y),X))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).
tff(fact_57_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
=> pp(aa(A,bool,ord_less_eq(A,Y),X)) ) ) ).
tff(fact_58_order__le__less,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X2: A] :
( pp(aa(A,bool,ord_less_eq(A,X2),Y2))
<=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
| ( X2 = Y2 ) ) ) ) ).
tff(fact_59_less__le__not__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
<=> ( pp(aa(A,bool,ord_less_eq(A,X2),Y2))
& ~ pp(aa(A,bool,ord_less_eq(A,Y2),X2)) ) ) ) ).
tff(fact_60_order__less__le,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
<=> ( pp(aa(A,bool,ord_less_eq(A,X2),Y2))
& ( X2 != Y2 ) ) ) ) ).
tff(fact_61_linorder__le__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( pp(aa(A,bool,ord_less_eq(A,X),Y))
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).
tff(fact_62_linorder__not__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ~ pp(aa(A,bool,ord_less_eq(A,X2),Y2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X2)) ) ) ).
tff(fact_63_linorder__not__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
<=> pp(aa(A,bool,ord_less_eq(A,Y2),X2)) ) ) ).
tff(fact_64_right__minus__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B1: A,Aa: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Aa),B1) = zero_zero(A) )
<=> ( Aa = B1 ) ) ) ).
tff(fact_65_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B1: A,Aa: A] :
( ( Aa = B1 )
<=> ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Aa),B1) = zero_zero(A) ) ) ) ).
tff(fact_66_diff__0__right,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A2: A] : aa(A,A,aa(A,fun(A,A),minus_minus(A),A2),zero_zero(A)) = A2 ) ).
tff(fact_67_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [D: A,C1: A,B1: A,Aa: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Aa),B1) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C1),D) )
=> ( pp(aa(A,bool,ord_less_eq(A,Aa),B1))
<=> pp(aa(A,bool,ord_less_eq(A,C1),D)) ) ) ) ).
tff(fact_68_diff__eq__diff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [D: A,C1: A,B1: A,Aa: A] :
( ( aa(A,A,aa(A,fun(A,A),minus_minus(A),Aa),B1) = aa(A,A,aa(A,fun(A,A),minus_minus(A),C1),D) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Aa),B1))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C1),D)) ) ) ) ).
tff(fact_69_tl_Osimps_I1_J,axiom,
! [A: $tType] : aa(list(A),list(A),tl(A),nil(A)) = nil(A) ).
tff(fact_70_iprod0__if__coeffs0,axiom,
! [A: $tType] :
( ring(A)
=> ! [Xsa: list(A),Cs: list(A)] :
( ! [X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),set(A,Cs)))
=> ( X1 = zero_zero(A) ) )
=> ( aa(list(A),A,aa(list(A),fun(list(A),A),iprod(A),Cs),Xsa) = zero_zero(A) ) ) ) ).
tff(fact_71_transfer__nat__int__set__cong,axiom,
! [P2: fun(int,bool),P1: fun(int,bool)] :
( ! [X1: int] :
( pp(aa(int,bool,ord_less_eq(int,zero_zero(int)),X1))
=> ( pp(aa(int,bool,P1,X1))
<=> pp(aa(int,bool,P2,X1)) ) )
=> ( collect(int,aa(fun(int,bool),fun(int,bool),aa(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),aa(fun(int,bool),fun(int,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),ord_less_eq(int,zero_zero(int)))),P1)) = collect(int,aa(fun(int,bool),fun(int,bool),aa(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),combs(int,bool,bool),aa(fun(int,bool),fun(int,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj),ord_less_eq(int,zero_zero(int)))),P2)) ) ) ).
tff(fact_72_atom_Osize_I1_J,axiom,
! [List: list(int),Int: int] : atom_size(aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int),List)) = zero_zero(nat) ).
tff(fact_73_list_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,F2: fun(B,fun(list(B),A)),F1: A] : aa(list(B),A,list_case(A,B,F1,F2),nil(B)) = F1 ).
tff(fact_74_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),A1))
<=> pp(aa(A,bool,A1,X2)) ) ).
tff(fact_75_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_76_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,B1: B,F: fun(B,A),Aa: A] :
( pp(aa(A,bool,ord_less_eq(A,Aa),aa(B,A,F,B1)))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B1),C1))
=> ( ! [X1: B,Y1: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X1),Y1))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X1)),aa(B,A,F,Y1))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Aa),aa(B,A,F,C1))) ) ) ) ) ).
tff(fact_77_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,B1: B,F: fun(B,A),Aa: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Aa),aa(B,A,F,B1)))
=> ( pp(aa(B,bool,ord_less_eq(B,B1),C1))
=> ( ! [X1: B,Y1: B] :
( pp(aa(B,bool,ord_less_eq(B,X1),Y1))
=> pp(aa(A,bool,ord_less_eq(A,aa(B,A,F,X1)),aa(B,A,F,Y1))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Aa),aa(B,A,F,C1))) ) ) ) ) ).
tff(fact_78_order__le__less__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,F: fun(A,B),B1: A,Aa: A] :
( pp(aa(A,bool,ord_less_eq(A,Aa),B1))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,B1)),C1))
=> ( ! [X1: A,Y1: A] :
( pp(aa(A,bool,ord_less_eq(A,X1),Y1))
=> pp(aa(B,bool,ord_less_eq(B,aa(A,B,F,X1)),aa(A,B,F,Y1))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,Aa)),C1)) ) ) ) ) ).
tff(fact_79_order__less__le__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,F: fun(A,B),B1: A,Aa: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Aa),B1))
=> ( pp(aa(B,bool,ord_less_eq(B,aa(A,B,F,B1)),C1))
=> ( ! [X1: A,Y1: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X1),Y1))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,X1)),aa(A,B,F,Y1))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,Aa)),C1)) ) ) ) ) ).
tff(fact_80_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F: fun(A,B)] :
( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less(fun(A,B)),F),G))
<=> ( pp(aa(fun(A,B),bool,ord_less_eq(fun(A,B),F),G))
& ~ pp(aa(fun(A,B),bool,ord_less_eq(fun(A,B),G),F)) ) ) ) ).
tff(fact_81_atom_Osize_I4_J,axiom,
! [List: list(int),Int: int] : size_size(atom,aa(list(int),atom,aa(int,fun(list(int),atom),c_PresArith_Oatom_OLe,Int),List)) = zero_zero(nat) ).
tff(fact_82_le0,axiom,
! [N: nat] : pp(aa(nat,bool,ord_less_eq(nat,zero_zero(nat)),N)) ).
tff(fact_83_less__zeroE,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).
tff(fact_84_diff__self__eq__0,axiom,
! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),M) = zero_zero(nat) ).
tff(fact_85_less__nat__zero__code,axiom,
! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).
tff(fact_86_zero__less__diff,axiom,
! [M1: nat,Na: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Na),M1)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),Na)) ) ).
tff(fact_87_diff__0__eq__0,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),zero_zero(nat)),N) = zero_zero(nat) ).
tff(fact_88_neq0__conv,axiom,
! [Na: nat] :
( ( Na != zero_zero(nat) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Na)) ) ).
tff(fact_89_diff__is__0__eq,axiom,
! [Na: nat,M1: nat] :
( ( aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M1),Na) = zero_zero(nat) )
<=> pp(aa(nat,bool,ord_less_eq(nat,M1),Na)) ) ).
tff(fact_90_le__0__eq,axiom,
! [Na: nat] :
( pp(aa(nat,bool,ord_less_eq(nat,Na),zero_zero(nat)))
<=> ( Na = zero_zero(nat) ) ) ).
tff(fact_91_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : pp(aa(nat,bool,ord_less_eq(nat,zero_zero(nat)),N)) ).
tff(fact_92_le__refl,axiom,
! [N: nat] : pp(aa(nat,bool,ord_less_eq(nat,N),N)) ).
tff(fact_93_diff__le__self,axiom,
! [N: nat,M: nat] : pp(aa(nat,bool,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M),N)),M)) ).
tff(fact_94_nat__le__linear,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,ord_less_eq(nat,M),N))
| pp(aa(nat,bool,ord_less_eq(nat,N),M)) ) ).
tff(fact_95_nat__less__le,axiom,
! [Na: nat,M1: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),Na))
<=> ( pp(aa(nat,bool,ord_less_eq(nat,M1),Na))
& ( M1 != Na ) ) ) ).
tff(fact_96_le__eq__less__or__eq,axiom,
! [Na: nat,M1: nat] :
( pp(aa(nat,bool,ord_less_eq(nat,M1),Na))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M1),Na))
| ( M1 = Na ) ) ) ).
tff(fact_97_eq__imp__le,axiom,
! [N: nat,M: nat] :
( ( M = N )
=> pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ).
tff(fact_98_less__imp__le__nat,axiom,
! [N: nat,M: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
=> pp(aa(nat,bool,ord_less_eq(nat,M),N)) ) ).
%----Arities (20)
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___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(int) ).
tff(arity_Int_Oint___Orderings_Opreorder,axiom,
preorder(int) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Groups_Ogroup__add,axiom,
group_add(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_Int_Oint___Rings_Oring,axiom,
ring(int) ).
tff(arity_Nat_Onat___Orderings_Opreorder,axiom,
preorder(nat) ).
tff(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
tff(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
tff(arity_Nat_Onat___Orderings_Oord,axiom,
ord(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
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_fEx_1_1_U,axiom,
! [A: $tType,X: A,P: fun(A,bool)] :
( ~ pp(aa(A,bool,P,X))
| pp(aa(fun(A,bool),bool,fEx(A),P)) ) ).
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,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(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,aa(B,fun(A,C),aa(fun(A,fun(B,C)),fun(B,fun(A,C)),combc(A,B,C),P),Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
tff(help_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,aa(A,fun(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,aa(fun(A,B),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,B),fun(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_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 ) ) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ( X != Y )
| pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y)) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
pp(aa(int,bool,ord_less_eq(int,big_linorder_Min(int,collect(int,aa(fun(int,fun(list(int),bool)),fun(int,bool),aa(fun(fun(list(int),bool),bool),fun(fun(int,fun(list(int),bool)),fun(int,bool)),combb(fun(list(int),bool),bool,int),fEx(list(int))),aa(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),bool)),aa(fun(fun(list(int),fun(int,bool)),fun(list(int),bool)),fun(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),bool))),combb(fun(list(int),fun(int,bool)),fun(list(int),bool),int),aa(fun(fun(int,bool),bool),fun(fun(list(int),fun(int,bool)),fun(list(int),bool)),combb(fun(int,bool),bool,list(int)),fEx(int))),aa(fun(list(int),fun(int,bool)),fun(int,fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),fun(fun(list(int),fun(int,bool)),fun(int,fun(list(int),fun(int,bool)))),combc(int,fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))),aa(fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),aa(fun(fun(list(int),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool)))),fun(fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(int,fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))))),combb(fun(list(int),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,bool))),int),combs(list(int),fun(int,bool),fun(int,bool))),aa(fun(int,fun(list(int),fun(int,fun(bool,bool)))),fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool)))),aa(fun(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool)))),fun(fun(int,fun(list(int),fun(int,fun(bool,bool)))),fun(int,fun(list(int),fun(fun(int,bool),fun(int,bool))))),combb(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool))),int),aa(fun(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool))),fun(fun(list(int),fun(int,fun(bool,bool))),fun(list(int),fun(fun(int,bool),fun(int,bool)))),combb(fun(int,fun(bool,bool)),fun(fun(int,bool),fun(int,bool)),list(int)),combs(int,bool,bool))),aa(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),fun(int,fun(bool,bool)))),aa(fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool)))),fun(fun(int,fun(list(int),fun(int,bool))),fun(int,fun(list(int),fun(int,fun(bool,bool))))),combb(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool))),int),aa(fun(fun(int,bool),fun(int,fun(bool,bool))),fun(fun(list(int),fun(int,bool)),fun(list(int),fun(int,fun(bool,bool)))),combb(fun(int,bool),fun(int,fun(bool,bool)),list(int)),aa(fun(bool,fun(bool,bool)),fun(fun(int,bool),fun(int,fun(bool,bool))),combb(bool,fun(bool,bool),int),fconj))),aa(fun(list(int),fun(int,int)),fun(int,fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),fun(fun(list(int),fun(int,int)),fun(int,fun(list(int),fun(int,bool)))),combc(int,fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))),aa(fun(int,fun(fun(int,int),fun(int,bool))),fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),aa(fun(fun(fun(int,int),fun(int,bool)),fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool)))),fun(fun(int,fun(fun(int,int),fun(int,bool))),fun(int,fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))))),combb(fun(fun(int,int),fun(int,bool)),fun(fun(list(int),fun(int,int)),fun(list(int),fun(int,bool))),int),combb(fun(int,int),fun(int,bool),list(int))),aa(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool))),aa(fun(fun(int,bool),fun(fun(int,int),fun(int,bool))),fun(fun(int,fun(int,bool)),fun(int,fun(fun(int,int),fun(int,bool)))),combb(fun(int,bool),fun(fun(int,int),fun(int,bool)),int),combb(int,bool,int)),fequal(int)))),aa(fun(list(int),int),fun(list(int),fun(int,int)),aa(fun(int,fun(int,int)),fun(fun(list(int),int),fun(list(int),fun(int,int))),combb(int,fun(int,int),list(int)),minus_minus(int)),aa(list(int),fun(list(int),int),aa(fun(list(int),fun(list(int),int)),fun(list(int),fun(list(int),int)),combc(list(int),list(int),int),aa(fun(list(int),list(int)),fun(list(int),fun(list(int),int)),aa(fun(list(int),fun(list(int),int)),fun(fun(list(int),list(int)),fun(list(int),fun(list(int),int))),combb(list(int),fun(list(int),int),list(int)),iprod(int)),tl(int))),xs))))))),aa(fun(atom,bool),fun(list(int),fun(int,bool)),aa(fun(list(int),fun(fun(atom,bool),fun(int,bool))),fun(fun(atom,bool),fun(list(int),fun(int,bool))),combc(list(int),fun(atom,bool),fun(int,bool)),aa(fun(list(int),fun(int,fun(fun(atom,bool),bool))),fun(list(int),fun(fun(atom,bool),fun(int,bool))),aa(fun(fun(int,fun(fun(atom,bool),bool)),fun(fun(atom,bool),fun(int,bool))),fun(fun(list(int),fun(int,fun(fun(atom,bool),bool))),fun(list(int),fun(fun(atom,bool),fun(int,bool)))),combb(fun(int,fun(fun(atom,bool),bool)),fun(fun(atom,bool),fun(int,bool)),list(int)),combc(int,fun(atom,bool),bool)),aa(fun(list(int),fun(int,atom)),fun(list(int),fun(int,fun(fun(atom,bool),bool))),aa(fun(fun(int,atom),fun(int,fun(fun(atom,bool),bool))),fun(fun(list(int),fun(int,atom)),fun(list(int),fun(int,fun(fun(atom,bool),bool)))),combb(fun(int,atom),fun(int,fun(fun(atom,bool),bool)),list(int)),aa(fun(atom,fun(fun(atom,bool),bool)),fun(fun(int,atom),fun(int,fun(fun(atom,bool),bool))),combb(atom,fun(fun(atom,bool),bool),int),member(atom))),aa(fun(int,fun(list(int),atom)),fun(list(int),fun(int,atom)),combc(int,list(int),atom),c_PresArith_Oatom_OLe)))),collect(atom,aa(fun(atom,bool),fun(atom,bool),aa(fun(atom,fun(bool,bool)),fun(fun(atom,bool),fun(atom,bool)),combs(atom,bool,bool),aa(fun(atom,bool),fun(atom,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(atom,bool),fun(atom,fun(bool,bool))),combb(bool,fun(bool,bool),atom),fconj),aa(fun(atom,bool),fun(atom,bool),aa(fun(atom,fun(fun(atom,bool),bool)),fun(fun(atom,bool),fun(atom,bool)),combc(atom,fun(atom,bool),bool),member(atom)),set(atom,as)))),atom_case(bool,aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),list_case(bool,int,fFalse,aa(fun(int,bool),fun(int,fun(list(int),bool)),aa(fun(bool,fun(list(int),bool)),fun(fun(int,bool),fun(int,fun(list(int),bool))),combb(bool,fun(list(int),bool),int),combk(bool,list(int))),aa(int,fun(int,bool),aa(fun(int,fun(int,bool)),fun(int,fun(int,bool)),combc(int,int,bool),ord_less(int)),zero_zero(int))))),aa(fun(int,fun(list(int),bool)),fun(int,fun(int,fun(list(int),bool))),combk(fun(int,fun(list(int),bool)),int),aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),aa(bool,fun(list(int),bool),combk(bool,list(int)),fFalse))),aa(fun(int,fun(list(int),bool)),fun(int,fun(int,fun(list(int),bool))),combk(fun(int,fun(list(int),bool)),int),aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),aa(bool,fun(list(int),bool),combk(bool,list(int)),fFalse))))))))))))),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(list(int),int,aa(list(int),fun(list(int),int),iprod(int),ks),xs)),l))) ).
%------------------------------------------------------------------------------