TPTP Problem File: COM049_5.p
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%------------------------------------------------------------------------------
% File : COM049_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 102
% 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_102 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 192 ( 40 unt; 48 typ; 0 def)
% Number of atoms : 336 ( 83 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 227 ( 35 ~; 16 |; 9 &)
% ( 28 <=>; 139 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 11 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 29 ( 17 >; 12 *; 0 +; 0 <<)
% Number of predicates : 21 ( 20 usr; 0 prp; 1-6 aty)
% Number of functors : 25 ( 25 usr; 5 con; 0-5 aty)
% Number of variables : 414 ( 362 !; 6 ?; 414 :)
% ( 46 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:17:18
%------------------------------------------------------------------------------
%----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 (43)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Osgn__if,type,
sgn_if:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Olattice,type,
lattice:
!>[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_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[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_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(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
tff(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( ( A * A ) > 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_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_Predicate_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( fun(B,fun(B,bool)) * fun(A,B) * A * A ) > $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: atom > 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_fdisj,type,
fdisj: 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) ).
%----Relevant facts (98)
tff(fact_0_norm,axiom,
! [X1: atom] :
( pp(aa(fun(atom,bool),bool,aa(atom,fun(fun(atom,bool),bool),member(atom),X1),set(atom,as)))
=> ( divisor(X1) != zero_zero(int) ) ) ).
tff(fact_1_Un__iff,axiom,
! [A: $tType,B3: fun(A,bool),A3: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),sup_sup(fun(A,bool),A3,B3)))
<=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A3))
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B3)) ) ) ).
tff(fact_2_UnE,axiom,
! [A: $tType,B3: fun(A,bool),A3: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),sup_sup(fun(A,bool),A3,B3)))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A3))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B3)) ) ) ).
tff(fact_3_sup1E,axiom,
! [A: $tType,X: A,B3: fun(A,bool),A3: fun(A,bool)] :
( pp(aa(A,bool,sup_sup(fun(A,bool),A3,B3),X))
=> ( ~ pp(aa(A,bool,A3,X))
=> pp(aa(A,bool,B3,X)) ) ) ).
tff(fact_4_sup1CI,axiom,
! [A: $tType,A3: fun(A,bool),X: A,B3: fun(A,bool)] :
( ( ~ pp(aa(A,bool,B3,X))
=> pp(aa(A,bool,A3,X)) )
=> pp(aa(A,bool,sup_sup(fun(A,bool),A3,B3),X)) ) ).
tff(fact_5_UnCI,axiom,
! [A: $tType,A3: fun(A,bool),B3: fun(A,bool),C1: A] :
( ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B3))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A3)) )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),sup_sup(fun(A,bool),A3,B3))) ) ).
tff(fact_6_sup_Oleft__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A2: A] : sup_sup(A,A2,sup_sup(A,A2,B2)) = sup_sup(A,A2,B2) ) ).
tff(fact_7_sup__left__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Y1: A,X3: A] : sup_sup(A,X3,sup_sup(A,X3,Y1)) = sup_sup(A,X3,Y1) ) ).
tff(fact_8_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( lattice(B)
=> ! [G: fun(A,B),F: fun(A,B),X1: A] : aa(A,B,sup_sup(fun(A,B),F,G),X1) = sup_sup(B,aa(A,B,F,X1),aa(A,B,G,X1)) ) ).
tff(fact_9_sup__apply,axiom,
! [A: $tType,B: $tType] :
( lattice(A)
=> ! [X: B,G: fun(B,A),F: fun(B,A)] : aa(B,A,sup_sup(fun(B,A),F,G),X) = sup_sup(A,aa(B,A,F,X),aa(B,A,G,X)) ) ).
tff(fact_10_less__supI1,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A2: A,X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),A2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),sup_sup(A,A2,B2))) ) ) ).
tff(fact_11_less__supI2,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A2: A,B2: A,X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),B2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),sup_sup(A,A2,B2))) ) ) ).
tff(fact_12_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X: A] :
( ( zero_zero(A) = X )
<=> ( X = zero_zero(A) ) ) ) ).
tff(fact_13_sup__assoc,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Z2: A,Y1: A,X3: A] : sup_sup(A,sup_sup(A,X3,Y1),Z2) = sup_sup(A,X3,sup_sup(A,Y1,Z2)) ) ).
tff(fact_14_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Z2: A,Y1: A,X3: A] : sup_sup(A,sup_sup(A,X3,Y1),Z2) = sup_sup(A,X3,sup_sup(A,Y1,Z2)) ) ).
tff(fact_15_sup_Oassoc,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,B2: A,A2: A] : sup_sup(A,sup_sup(A,A2,B2),C2) = sup_sup(A,A2,sup_sup(A,B2,C2)) ) ).
tff(fact_16_sup__left__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Z2: A,Y1: A,X3: A] : sup_sup(A,X3,sup_sup(A,Y1,Z2)) = sup_sup(A,Y1,sup_sup(A,X3,Z2)) ) ).
tff(fact_17_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Z2: A,Y1: A,X3: A] : sup_sup(A,X3,sup_sup(A,Y1,Z2)) = sup_sup(A,Y1,sup_sup(A,X3,Z2)) ) ).
tff(fact_18_sup_Oleft__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [C2: A,A2: A,B2: A] : sup_sup(A,B2,sup_sup(A,A2,C2)) = sup_sup(A,A2,sup_sup(A,B2,C2)) ) ).
tff(fact_19_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Y1: A,X3: A] : sup_sup(A,X3,sup_sup(A,X3,Y1)) = sup_sup(A,X3,Y1) ) ).
tff(fact_20_sup__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Y1: A,X3: A] : sup_sup(A,X3,Y1) = sup_sup(A,Y1,X3) ) ).
tff(fact_21_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Y1: A,X3: A] : sup_sup(A,X3,Y1) = sup_sup(A,Y1,X3) ) ).
tff(fact_22_sup_Ocommute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B2: A,A2: A] : sup_sup(A,A2,B2) = sup_sup(A,B2,A2) ) ).
tff(fact_23_sup__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X3: A] : sup_sup(A,X3,X3) = X3 ) ).
tff(fact_24_sup_Oidem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A2: A] : sup_sup(A,A2,A2) = A2 ) ).
tff(fact_25_UnI2,axiom,
! [A: $tType,A3: fun(A,bool),B3: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B3))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),sup_sup(fun(A,bool),A3,B3))) ) ).
tff(fact_26_UnI1,axiom,
! [A: $tType,B3: fun(A,bool),A3: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A3))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),sup_sup(fun(A,bool),A3,B3))) ) ).
tff(fact_27_sup1I2,axiom,
! [A: $tType,A3: fun(A,bool),X: A,B3: fun(A,bool)] :
( pp(aa(A,bool,B3,X))
=> pp(aa(A,bool,sup_sup(fun(A,bool),A3,B3),X)) ) ).
tff(fact_28_sup1I1,axiom,
! [A: $tType,B3: fun(A,bool),X: A,A3: fun(A,bool)] :
( pp(aa(A,bool,A3,X))
=> pp(aa(A,bool,sup_sup(fun(A,bool),A3,B3),X)) ) ).
tff(fact_29_ball__Un,axiom,
! [A: $tType,P1: fun(A,bool),B3: fun(A,bool),A3: fun(A,bool)] :
( ! [X4: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),sup_sup(fun(A,bool),A3,B3)))
=> pp(aa(A,bool,P1,X4)) )
<=> ( ! [X4: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),A3))
=> pp(aa(A,bool,P1,X4)) )
& ! [X4: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),B3))
=> pp(aa(A,bool,P1,X4)) ) ) ) ).
tff(fact_30_bex__Un,axiom,
! [A: $tType,P1: fun(A,bool),B3: fun(A,bool),A3: fun(A,bool)] :
( ? [X4: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),sup_sup(fun(A,bool),A3,B3)))
& pp(aa(A,bool,P1,X4)) )
<=> ( ? [X4: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),A3))
& pp(aa(A,bool,P1,X4)) )
| ? [X4: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),B3))
& pp(aa(A,bool,P1,X4)) ) ) ) ).
tff(fact_31_Un__assoc,axiom,
! [A: $tType,C3: fun(A,bool),B3: fun(A,bool),A3: fun(A,bool)] : sup_sup(fun(A,bool),sup_sup(fun(A,bool),A3,B3),C3) = sup_sup(fun(A,bool),A3,sup_sup(fun(A,bool),B3,C3)) ).
tff(fact_32_Un__left__commute,axiom,
! [A: $tType,C3: fun(A,bool),B3: fun(A,bool),A3: fun(A,bool)] : sup_sup(fun(A,bool),A3,sup_sup(fun(A,bool),B3,C3)) = sup_sup(fun(A,bool),B3,sup_sup(fun(A,bool),A3,C3)) ).
tff(fact_33_Un__left__absorb,axiom,
! [A: $tType,B3: fun(A,bool),A3: fun(A,bool)] : sup_sup(fun(A,bool),A3,sup_sup(fun(A,bool),A3,B3)) = sup_sup(fun(A,bool),A3,B3) ).
tff(fact_34_Un__commute,axiom,
! [A: $tType,B3: fun(A,bool),A3: fun(A,bool)] : sup_sup(fun(A,bool),A3,B3) = sup_sup(fun(A,bool),B3,A3) ).
tff(fact_35_Un__def,axiom,
! [A: $tType,B3: fun(A,bool),A3: fun(A,bool)] : sup_sup(fun(A,bool),A3,B3) = collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fdisj,combc(A,fun(A,bool),bool,member(A),A3)),combc(A,fun(A,bool),bool,member(A),B3))) ).
tff(fact_36_Un__absorb,axiom,
! [A: $tType,A3: fun(A,bool)] : sup_sup(fun(A,bool),A3,A3) = A3 ).
tff(fact_37_sup__Un__eq,axiom,
! [A: $tType,S: fun(A,bool),R2: fun(A,bool),X1: A] :
( pp(aa(A,bool,sup_sup(fun(A,bool),combc(A,fun(A,bool),bool,member(A),R2),combc(A,fun(A,bool),bool,member(A),S)),X1))
<=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),sup_sup(fun(A,bool),R2,S))) ) ).
tff(fact_38_Collect__disj__eq,axiom,
! [A: $tType,Q1: fun(A,bool),P1: fun(A,bool)] : collect(A,combs(A,bool,bool,combb(bool,fun(bool,bool),A,fdisj,P1),Q1)) = sup_sup(fun(A,bool),collect(A,P1),collect(A,Q1)) ).
tff(fact_39_of__int__less__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z1: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),ring_1_of_int(A,Z1)),zero_zero(A)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z1),zero_zero(int))) ) ) ).
tff(fact_40_of__int__0__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z1: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),ring_1_of_int(A,Z1)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z1)) ) ) ).
tff(fact_41_Sup__fin_Oidem,axiom,
! [A: $tType] :
( lattice(A)
=> ! [X3: A] : sup_sup(A,X3,X3) = X3 ) ).
tff(fact_42_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y1))
=> ( ( X3 != Y1 )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y1),X3)) ) ) ) ).
tff(fact_43_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y1))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y1),X3)) ) ) ).
tff(fact_44_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z2: A,Y1: A,X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y1))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y1),Z2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z2)) ) ) ) ).
tff(fact_45_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_46_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_47_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_48_of__int__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z1: int,W: int] :
( ( ring_1_of_int(A,W) = ring_1_of_int(A,Z1) )
<=> ( W = Z1 ) ) ) ).
tff(fact_49_of__int__0,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A,zero_zero(int)) = zero_zero(A) ) ) ).
tff(fact_50_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z1: int] :
( ( zero_zero(A) = ring_1_of_int(A,Z1) )
<=> ( Z1 = zero_zero(int) ) ) ) ).
tff(fact_51_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z1: int] :
( ( ring_1_of_int(A,Z1) = zero_zero(A) )
<=> ( Z1 = zero_zero(int) ) ) ) ).
tff(fact_52_of__int__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z1: int,W: int] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),ring_1_of_int(A,W)),ring_1_of_int(A,Z1)))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z1)) ) ) ).
tff(fact_53_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X3: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),X3)) ) ).
tff(fact_54_linorder__neq__iff,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_55_not__less__iff__gr__or__eq,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,aa(A,fun(A,bool),ord_less(A),Y),X))
| ( X = Y ) ) ) ) ).
tff(fact_56_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y1))
| ( X3 = Y1 )
| pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y1),X3)) ) ) ).
tff(fact_57_linorder__antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
<=> ( X = Y ) ) ) ) ).
tff(fact_58_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ( X3 != Y1 )
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y1))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y1),X3)) ) ) ) ).
tff(fact_59_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y1))
=> ( X3 != Y1 ) ) ) ).
tff(fact_60_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y1))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y1),X3)) ) ) ).
tff(fact_61_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y1))
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y1),X3)) ) ) ).
tff(fact_62_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y1))
=> ( X3 != Y1 ) ) ) ).
tff(fact_63_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y1))
=> ( Y1 != X3 ) ) ) ).
tff(fact_64_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [P1: bool,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))
=> pp(P1) ) ) ) ).
tff(fact_65_gt__ex,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [X: A] :
? [X11: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X11)) ) ).
tff(fact_66_lt__ex,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [X3: A] :
? [Y2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),X3)) ) ).
tff(fact_67_dense,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Y1: A,X3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y1))
=> ? [Z: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),Y1)) ) ) ) ).
tff(fact_68_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C1: B,F: fun(A,B),B1: A,A1: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A1),B1))
=> ( ( aa(A,B,F,B1) = C1 )
=> ( ! [X2: A,Y2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,X2)),aa(A,B,F,Y2))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,A1)),C1)) ) ) ) ) ).
tff(fact_69_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,B1: B,F: fun(B,A),A1: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A1),aa(B,A,F,B1)))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B1),C1))
=> ( ! [X2: B,Y2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X2)),aa(B,A,F,Y2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A1),aa(B,A,F,C1))) ) ) ) ) ).
tff(fact_70_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C1: B,B1: B,F: fun(B,A),A1: A] :
( ( A1 = aa(B,A,F,B1) )
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B1),C1))
=> ( ! [X2: B,Y2: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X2)),aa(B,A,F,Y2))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A1),aa(B,A,F,C1))) ) ) ) ) ).
tff(fact_71_order__less__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,F: fun(A,B),B1: A,A1: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A1),B1))
=> ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,B1)),C1))
=> ( ! [X2: A,Y2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y2))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,X2)),aa(A,B,F,Y2))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F,A1)),C1)) ) ) ) ) ).
tff(fact_72_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y1: A,X3: A] :
( ( X3 != Y1 )
=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y1))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y1),X3)) ) ) ) ).
tff(fact_73_in__inv__imagep,axiom,
! [A: $tType,B: $tType,Y: B,X: B,F: fun(B,A),R1: fun(A,fun(A,bool))] :
( inv_imagep(A,B,R1,F,X,Y)
<=> pp(aa(A,bool,aa(A,fun(A,bool),R1,aa(B,A,F,X)),aa(B,A,F,Y))) ) ).
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_mem__def,axiom,
! [A: $tType,A3: fun(A,bool),X: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X),A3))
<=> pp(aa(A,bool,A3,X)) ) ).
tff(fact_76_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_77_inv__imagep__def,axiom,
! [B: $tType,A: $tType,F: fun(A,B),R1: fun(B,fun(B,bool)),X1: A,Xa: A] :
( inv_imagep(B,A,R1,F,X1,Xa)
<=> pp(aa(B,bool,aa(B,fun(B,bool),R1,aa(A,B,F,X1)),aa(A,B,F,Xa))) ) ).
tff(fact_78_sgn__greater,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),sgn_sgn(A,A1)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A1)) ) ) ).
tff(fact_79_sgn__less,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),sgn_sgn(A,A1)),zero_zero(A)))
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A1),zero_zero(A))) ) ) ).
tff(fact_80_sgn__sgn,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] : sgn_sgn(A,sgn_sgn(A,A2)) = sgn_sgn(A,A2) ) ).
tff(fact_81_sgn0,axiom,
! [A: $tType] :
( sgn_if(A)
=> ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).
tff(fact_82_sgn__0__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( ( sgn_sgn(A,A1) = zero_zero(A) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_83_sgn__1__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( ( sgn_sgn(A,A1) = one_one(A) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A1)) ) ) ).
tff(fact_84_sgn__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
=> ( sgn_sgn(A,A2) = one_one(A) ) ) ) ).
tff(fact_85_of__int__1,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A,one_one(int)) = one_one(A) ) ) ).
tff(fact_86_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X: A] :
( ( one_one(A) = X )
<=> ( X = one_one(A) ) ) ) ).
tff(fact_87_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
tff(fact_88_one__neq__zero,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( one_one(A) != zero_zero(A) ) ) ).
tff(fact_89_zero__less__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).
tff(fact_90_not__one__less__zero,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),zero_zero(A))) ) ).
tff(fact_91_sgn__1__neg,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( ( sgn_sgn(A,A1) = uminus_uminus(A,one_one(A)) )
<=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A1),zero_zero(A))) ) ) ).
tff(fact_92_sgn__neg,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
=> ( sgn_sgn(A,A2) = uminus_uminus(A,one_one(A)) ) ) ) ).
tff(fact_93_neg__equal__iff__equal,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B1: A,A1: A] :
( ( uminus_uminus(A,A1) = uminus_uminus(A,B1) )
<=> ( A1 = B1 ) ) ) ).
tff(fact_94_compl__eq__compl__iff,axiom,
! [A: $tType] :
( boolean_algebra(A)
=> ! [Y: A,X: A] :
( ( uminus_uminus(A,X) = uminus_uminus(A,Y) )
<=> ( X = Y ) ) ) ).
tff(fact_95_neg__equal__zero,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ( uminus_uminus(A,A1) = A1 )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_96_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A1: A] :
( ( uminus_uminus(A,A1) = zero_zero(A) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_97_equal__neg__zero,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ( A1 = uminus_uminus(A,A1) )
<=> ( A1 = zero_zero(A) ) ) ) ).
%----Arities (29)
tff(arity_fun___Lattices_Osemilattice__sup,axiom,
! [T_1: $tType,T_2: $tType] :
( lattice(T_2)
=> semilattice_sup(fun(T_1,T_2)) ) ).
tff(arity_fun___Lattices_Oboolean__algebra,axiom,
! [T_1: $tType,T_2: $tType] :
( boolean_algebra(T_2)
=> boolean_algebra(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Opreorder,axiom,
! [T_1: $tType,T_2: $tType] :
( preorder(T_2)
=> preorder(fun(T_1,T_2)) ) ).
tff(arity_fun___Lattices_Olattice,axiom,
! [T_1: $tType,T_2: $tType] :
( lattice(T_2)
=> lattice(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_Olinordered__ab__group__add,axiom,
linord219039673up_add(int) ).
tff(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
tff(arity_Int_Oint___Lattices_Osemilattice__sup,axiom,
semilattice_sup(int) ).
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one(int) ).
tff(arity_Int_Oint___Orderings_Opreorder,axiom,
preorder(int) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Lattices_Olattice,axiom,
lattice(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___Int_Oring__char__0,axiom,
ring_char_0(int) ).
tff(arity_Int_Oint___Orderings_Oord,axiom,
ord(int) ).
tff(arity_Int_Oint___Groups_Osgn__if,axiom,
sgn_if(int) ).
tff(arity_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Groups_Oone,axiom,
one(int) ).
tff(arity_HOL_Obool___Lattices_Osemilattice__sup,axiom,
semilattice_sup(bool) ).
tff(arity_HOL_Obool___Lattices_Oboolean__algebra,axiom,
boolean_algebra(bool) ).
tff(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Lattices_Olattice,axiom,
lattice(bool) ).
tff(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
tff(arity_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
%----Helper facts (16)
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_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,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_fdisj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q))
| pp(P)
| 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,
set(atom,as) = sup_sup(fun(atom,bool),sup_sup(fun(atom,bool),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,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)),fTrue))),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)),fTrue)))))),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,aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),list_case(bool,int,fFalse,combb(bool,fun(list(int),bool),int,combk(bool,list(int)),aa(int,fun(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))))))),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,aa(fun(list(int),bool),fun(int,fun(list(int),bool)),combk(fun(list(int),bool),int),list_case(bool,int,fFalse,combb(bool,fun(list(int),bool),int,combk(bool,list(int)),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))))))) ).
%------------------------------------------------------------------------------