TPTP Problem File: COM118_5.p
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%------------------------------------------------------------------------------
% File : COM118_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 283
% 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_283 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 158 ( 20 unt; 39 typ; 0 def)
% Number of atoms : 365 ( 49 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 286 ( 40 ~; 8 |; 13 &)
% ( 34 <=>; 191 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 27 ( 18 >; 9 *; 0 +; 0 <<)
% Number of predicates : 19 ( 18 usr; 0 prp; 1-3 aty)
% Number of functors : 18 ( 18 usr; 4 con; 0-4 aty)
% Number of variables : 352 ( 322 !; 5 ?; 352 :)
% ( 25 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:25:37
%------------------------------------------------------------------------------
%----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 (34)
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_Rings_Osemiring__1,type,
semiring_1:
!>[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_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
tff(sy_c_Groups_Ocomm__monoid,type,
comm_monoid:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups_Osemigroup,type,
semigroup:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Int_Oiszero,type,
iszero:
!>[A: $tType] : ( A > $o ) ).
tff(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : fun(A,bool) ).
tff(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
tff(sy_c_List_Oset,type,
set:
!>[A: $tType] : ( list(A) > fun(A,bool) ) ).
tff(sy_c_Nat_Osemiring__1__class_ONats,type,
semiring_1_Nats:
!>[A: $tType] : fun(A,bool) ).
tff(sy_c_Nat__Transfer_Otsub,type,
nat_tsub: ( int * int ) > int ).
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_Oasubst,type,
asubst: ( int * list(int) * atom ) > atom ).
tff(sy_c_PresArith_Odivisor,type,
divisor: atom > int ).
tff(sy_c_PresArith_Ohd__coeff,type,
hd_coeff: atom > int ).
tff(sy_c_QEpres__Mirabelle__iocckttzyp_Ohd__coeffs1,type,
qEpres1907640072oeffs1: list(atom) > list(atom) ).
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_member,type,
member:
!>[A: $tType] : ( ( A * fun(A,bool) ) > $o ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_a,type,
a: atom ).
tff(sy_v_as,type,
as: list(atom) ).
%----Relevant facts (99)
tff(fact_0_div0,axiom,
! [X1: atom] :
( member(atom,X1,set(atom,as))
=> ( divisor(X1) != zero_zero(int) ) ) ).
tff(fact_1_a,axiom,
member(atom,a,set(atom,qEpres1907640072oeffs1(as))) ).
tff(fact_2_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X3: A] :
( ( zero_zero(A) = X3 )
<=> ( X3 = zero_zero(A) ) ) ) ).
tff(fact_3_of__int__0,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A,zero_zero(int)) = zero_zero(A) ) ) ).
tff(fact_4_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z2: int] :
( ( zero_zero(A) = ring_1_of_int(A,Z2) )
<=> ( Z2 = zero_zero(int) ) ) ) ).
tff(fact_5_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z2: int] :
( ( ring_1_of_int(A,Z2) = zero_zero(A) )
<=> ( Z2 = zero_zero(int) ) ) ) ).
tff(fact_6_divisor__asubst,axiom,
! [A1: atom,Ks: list(int),I: int] : divisor(asubst(I,Ks,A1)) = divisor(A1) ).
tff(fact_7_sgn0,axiom,
! [A: $tType] :
( sgn_if(A)
=> ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).
tff(fact_8_Nats__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> member(A,zero_zero(A),semiring_1_Nats(A)) ) ).
tff(fact_9_iszero__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> iszero(A,zero_zero(A)) ) ).
tff(fact_10_iszero__def,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Z2: A] :
( iszero(A,Z2)
<=> ( Z2 = zero_zero(A) ) ) ) ).
tff(fact_11_of__int__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z2: int,W1: int] :
( ( ring_1_of_int(A,W1) = ring_1_of_int(A,Z2) )
<=> ( W1 = Z2 ) ) ) ).
tff(fact_12_hd0,axiom,
! [X1: atom] :
( member(atom,X1,set(atom,as))
=> ( hd_coeff(X1) != zero_zero(int) ) ) ).
tff(fact_13_sgn__sgn,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] : sgn_sgn(A,sgn_sgn(A,A1)) = sgn_sgn(A,A1) ) ).
tff(fact_14_sgn__0__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Aa: A] :
( ( sgn_sgn(A,Aa) = zero_zero(A) )
<=> ( Aa = zero_zero(A) ) ) ) ).
tff(fact_15_semigroup_Oassoc,axiom,
! [A: $tType,C: A,B1: A,Aa: A,F: fun(A,fun(A,A))] :
( semigroup(A,F)
=> ( aa(A,A,aa(A,fun(A,A),F,aa(A,A,aa(A,fun(A,A),F,Aa),B1)),C) = aa(A,A,aa(A,fun(A,A),F,Aa),aa(A,A,aa(A,fun(A,A),F,B1),C)) ) ) ).
tff(fact_16_comm__monoid_Ocomm__neutral,axiom,
! [A: $tType,Aa: A,Z2: A,F: fun(A,fun(A,A))] :
( comm_monoid(A,F,Z2)
=> ( aa(A,A,aa(A,fun(A,A),F,Aa),Z2) = Aa ) ) ).
tff(fact_17_Ints__of__int,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Z2: int] : member(A,ring_1_of_int(A,Z2),ring_1_Ints(A)) ) ).
tff(fact_18_sgn__greater,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Aa: A] :
( ord_less(A,zero_zero(A),sgn_sgn(A,Aa))
<=> ord_less(A,zero_zero(A),Aa) ) ) ).
tff(fact_19_sgn__less,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Aa: A] :
( ord_less(A,sgn_sgn(A,Aa),zero_zero(A))
<=> ord_less(A,Aa,zero_zero(A)) ) ) ).
tff(fact_20_of__int__le__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( ord_less_eq(A,ring_1_of_int(A,Z2),zero_zero(A))
<=> ord_less_eq(int,Z2,zero_zero(int)) ) ) ).
tff(fact_21_of__int__0__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( ord_less_eq(A,zero_zero(A),ring_1_of_int(A,Z2))
<=> ord_less_eq(int,zero_zero(int),Z2) ) ) ).
tff(fact_22_of__int__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int,W1: int] :
( ord_less(A,ring_1_of_int(A,W1),ring_1_of_int(A,Z2))
<=> ord_less(int,W1,Z2) ) ) ).
tff(fact_23_of__int__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int,W1: int] :
( ord_less_eq(A,ring_1_of_int(A,W1),ring_1_of_int(A,Z2))
<=> ord_less_eq(int,W1,Z2) ) ) ).
tff(fact_24_of__int__0__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( ord_less(A,zero_zero(A),ring_1_of_int(A,Z2))
<=> ord_less(int,zero_zero(int),Z2) ) ) ).
tff(fact_25_of__int__less__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z2: int] :
( ord_less(A,ring_1_of_int(A,Z2),zero_zero(A))
<=> ord_less(int,Z2,zero_zero(int)) ) ) ).
tff(fact_26_less__int__def,axiom,
! [W1: int,Z2: int] :
( ord_less(int,Z2,W1)
<=> ( ord_less_eq(int,Z2,W1)
& ( Z2 != W1 ) ) ) ).
tff(fact_27_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y1: A,X2: A] :
( ( X2 != Y1 )
=> ( ~ ord_less(A,X2,Y1)
=> ord_less(A,Y1,X2) ) ) ) ).
tff(fact_28_Ints__0,axiom,
! [A: $tType] :
( ring_1(A)
=> member(A,zero_zero(A),ring_1_Ints(A)) ) ).
tff(fact_29_Ints__cases,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Q: A] :
( member(A,Q,ring_1_Ints(A))
=> ~ ! [Z: int] : Q != ring_1_of_int(A,Z) ) ) ).
tff(fact_30_Ints__induct,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [P: fun(A,bool),Q: A] :
( member(A,Q,ring_1_Ints(A))
=> ( ! [Z: int] : pp(aa(A,bool,P,ring_1_of_int(A,Z)))
=> pp(aa(A,bool,P,Q)) ) ) ) ).
tff(fact_31_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A] : ord_less_eq(A,X2,X2) ) ).
tff(fact_32_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq(int,zero_zero(int),zero_zero(int)) ).
tff(fact_33_conj__le__cong,axiom,
! [P1: bool,P: bool,X3: int] :
( ( ord_less_eq(int,zero_zero(int),X3)
=> ( pp(P)
<=> pp(P1) ) )
=> ( ( ord_less_eq(int,zero_zero(int),X3)
& pp(P) )
<=> ( ord_less_eq(int,zero_zero(int),X3)
& pp(P1) ) ) ) ).
tff(fact_34_imp__le__cong,axiom,
! [P1: bool,P: bool,X3: int] :
( ( ord_less_eq(int,zero_zero(int),X3)
=> ( pp(P)
<=> pp(P1) ) )
=> ( ( ord_less_eq(int,zero_zero(int),X3)
=> pp(P) )
<=> ( ord_less_eq(int,zero_zero(int),X3)
=> pp(P1) ) ) ) ).
tff(fact_35_order__le__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z1: A,Y1: A,X2: A] :
( ord_less_eq(A,X2,Y1)
=> ( ord_less(A,Y1,Z1)
=> ord_less(A,X2,Z1) ) ) ) ).
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)
<=> ! [X4: A] : ord_less_eq(B,aa(A,B,F,X4),aa(A,B,G,X4)) ) ) ).
tff(fact_37_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X2: A] :
( ord_less_eq(A,X2,Y1)
| ord_less_eq(A,Y1,X2) ) ) ).
tff(fact_38_order__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X3: A] :
( ( X3 = Y2 )
<=> ( ord_less_eq(A,X3,Y2)
& ord_less_eq(A,Y2,X3) ) ) ) ).
tff(fact_39_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X2: A] :
( ( X2 = Y1 )
=> ord_less_eq(A,X2,Y1) ) ) ).
tff(fact_40_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [X3: 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,X3),aa(A,B,G,X3)) ) ) ).
tff(fact_41_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [X3: A,Y2: A] :
( ord_less_eq(A,Y2,X3)
=> ( ord_less_eq(A,X3,Y2)
<=> ( X3 = Y2 ) ) ) ) ).
tff(fact_42_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A1: A] :
( ( A1 = B2 )
=> ( ord_less_eq(A,B2,C2)
=> ord_less_eq(A,A1,C2) ) ) ) ).
tff(fact_43_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A1: A] :
( ord_less_eq(A,A1,B2)
=> ( ( B2 = C2 )
=> ord_less_eq(A,A1,C2) ) ) ) ).
tff(fact_44_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X2: A] :
( ord_less_eq(A,X2,Y1)
=> ( ord_less_eq(A,Y1,X2)
=> ( X2 = Y1 ) ) ) ) ).
tff(fact_45_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z1: A,Y1: A,X2: A] :
( ord_less_eq(A,X2,Y1)
=> ( ord_less_eq(A,Y1,Z1)
=> ord_less_eq(A,X2,Z1) ) ) ) ).
tff(fact_46_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [X3: 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,X3),aa(A,B,G,X3)) ) ) ).
tff(fact_47_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X2: A] :
( ~ ord_less_eq(A,X2,Y1)
=> ord_less_eq(A,Y1,X2) ) ) ).
tff(fact_48_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A] : ~ ord_less(A,X2,X2) ) ).
tff(fact_49_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X3: A] :
( ( X3 != Y2 )
<=> ( ord_less(A,X3,Y2)
| ord_less(A,Y2,X3) ) ) ) ).
tff(fact_50_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X3: A] :
( ~ ord_less(A,X3,Y2)
<=> ( ord_less(A,Y2,X3)
| ( X3 = Y2 ) ) ) ) ).
tff(fact_51_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X2: A] :
( ord_less(A,X2,Y1)
| ( X2 = Y1 )
| ord_less(A,Y1,X2) ) ) ).
tff(fact_52_linorder__antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X3: A,Y2: A] :
( ~ ord_less(A,Y2,X3)
=> ( ~ ord_less(A,X3,Y2)
<=> ( X3 = Y2 ) ) ) ) ).
tff(fact_53_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X2: A] :
( ( X2 != Y1 )
=> ( ~ ord_less(A,X2,Y1)
=> ord_less(A,Y1,X2) ) ) ) ).
tff(fact_54_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X2: A] :
( ord_less(A,X2,Y1)
=> ( X2 != Y1 ) ) ) ).
tff(fact_55_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X2: A] :
( ord_less(A,X2,Y1)
=> ~ ord_less(A,Y1,X2) ) ) ).
tff(fact_56_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X2: A] :
( ord_less(A,X2,Y1)
=> ~ ord_less(A,Y1,X2) ) ) ).
tff(fact_57_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X2: A] :
( ord_less(A,X2,Y1)
=> ( X2 != Y1 ) ) ) ).
tff(fact_58_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X2: A] :
( ord_less(A,X2,Y1)
=> ( Y1 != X2 ) ) ) ).
tff(fact_59_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [P: bool,Y2: A,X3: A] :
( ord_less(A,X3,Y2)
=> ( ord_less(A,Y2,X3)
=> pp(P) ) ) ) ).
tff(fact_60_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A1: A] :
( ord_less(A,A1,B2)
=> ~ ord_less(A,B2,A1) ) ) ).
tff(fact_61_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A1: A] :
( ( A1 = B2 )
=> ( ord_less(A,B2,C2)
=> ord_less(A,A1,C2) ) ) ) ).
tff(fact_62_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A1: A] :
( ord_less(A,A1,B2)
=> ( ( B2 = C2 )
=> ord_less(A,A1,C2) ) ) ) ).
tff(fact_63_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z1: A,Y1: A,X2: A] :
( ord_less(A,X2,Y1)
=> ( ord_less(A,Y1,Z1)
=> ord_less(A,X2,Z1) ) ) ) ).
tff(fact_64_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X2: A] :
( ord_less(A,X2,Y1)
=> ~ ord_less(A,Y1,X2) ) ) ).
tff(fact_65_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X2: A] :
( ~ ord_less(A,X2,Y1)
=> ( ( X2 != Y1 )
=> ord_less(A,Y1,X2) ) ) ) ).
tff(fact_66_linorder__not__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X3: A] :
( ~ ord_less(A,X3,Y2)
<=> ord_less_eq(A,Y2,X3) ) ) ).
tff(fact_67_linorder__not__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X3: A] :
( ~ ord_less_eq(A,X3,Y2)
<=> ord_less(A,Y2,X3) ) ) ).
tff(fact_68_linorder__le__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X2: A] :
( ord_less_eq(A,X2,Y1)
| ord_less(A,Y1,X2) ) ) ).
tff(fact_69_order__less__le,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X3: A] :
( ord_less(A,X3,Y2)
<=> ( ord_less_eq(A,X3,Y2)
& ( X3 != Y2 ) ) ) ) ).
tff(fact_70_less__le__not__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X3: A] :
( ord_less(A,X3,Y2)
<=> ( ord_less_eq(A,X3,Y2)
& ~ ord_less_eq(A,Y2,X3) ) ) ) ).
tff(fact_71_order__le__less,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X3: A] :
( ord_less_eq(A,X3,Y2)
<=> ( ord_less(A,X3,Y2)
| ( X3 = Y2 ) ) ) ) ).
tff(fact_72_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X2: A] :
( ~ ord_less(A,X2,Y1)
=> ord_less_eq(A,Y1,X2) ) ) ).
tff(fact_73_not__leE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: A,Y1: A] :
( ~ ord_less_eq(A,Y1,X2)
=> ord_less(A,X2,Y1) ) ) ).
tff(fact_74_mem__def,axiom,
! [A: $tType,A2: fun(A,bool),X3: A] :
( member(A,X3,A2)
<=> pp(aa(A,bool,A2,X3)) ) ).
tff(fact_75_linorder__antisym__conv1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X3: A] :
( ~ ord_less(A,X3,Y2)
=> ( ord_less_eq(A,X3,Y2)
<=> ( X3 = Y2 ) ) ) ) ).
tff(fact_76_order__neq__le__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A1: A] :
( ( A1 != B2 )
=> ( ord_less_eq(A,A1,B2)
=> ord_less(A,A1,B2) ) ) ) ).
tff(fact_77_leD,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: A,Y1: A] :
( ord_less_eq(A,Y1,X2)
=> ~ ord_less(A,X2,Y1) ) ) ).
tff(fact_78_order__less__imp__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X2: A] :
( ord_less(A,X2,Y1)
=> ord_less_eq(A,X2,Y1) ) ) ).
tff(fact_79_linorder__antisym__conv2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X3: A] :
( ord_less_eq(A,X3,Y2)
=> ( ~ ord_less(A,X3,Y2)
<=> ( X3 = Y2 ) ) ) ) ).
tff(fact_80_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X2: A] :
( ord_less_eq(A,X2,Y1)
=> ( ord_less(A,X2,Y1)
| ( X2 = Y1 ) ) ) ) ).
tff(fact_81_order__le__neq__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A1: A] :
( ord_less_eq(A,A1,B2)
=> ( ( A1 != B2 )
=> ord_less(A,A1,B2) ) ) ) ).
tff(fact_82_order__less__le__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z1: A,Y1: A,X2: A] :
( ord_less(A,X2,Y1)
=> ( ord_less_eq(A,Y1,Z1)
=> ord_less(A,X2,Z1) ) ) ) ).
tff(fact_83_minf_I8_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z: A] :
! [X1: A] :
( ord_less(A,X1,Z)
=> ~ ord_less_eq(A,T,X1) ) ) ).
tff(fact_84_pinf_I8_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z: A] :
! [X1: A] :
( ord_less(A,Z,X1)
=> ord_less_eq(A,T,X1) ) ) ).
tff(fact_85_minf_I6_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z: A] :
! [X1: A] :
( ord_less(A,X1,Z)
=> ord_less_eq(A,X1,T) ) ) ).
tff(fact_86_pinf_I6_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z: A] :
! [X1: A] :
( ord_less(A,Z,X1)
=> ~ ord_less_eq(A,X1,T) ) ) ).
tff(fact_87_dense__le,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Z1: A,Y1: A] :
( ! [X: A] :
( ord_less(A,X,Y1)
=> ord_less_eq(A,X,Z1) )
=> ord_less_eq(A,Y1,Z1) ) ) ).
tff(fact_88_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_89_le__funI,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F: fun(A,B)] :
( ! [X: A] : ord_less_eq(B,aa(A,B,F,X),aa(A,B,G,X))
=> ord_less_eq(fun(A,B),F,G) ) ) ).
tff(fact_90_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C: B,B1: B,F: fun(B,A),Aa: A] :
( ord_less_eq(A,Aa,aa(B,A,F,B1))
=> ( ord_less(B,B1,C)
=> ( ! [X: B,Y: B] :
( ord_less(B,X,Y)
=> ord_less(A,aa(B,A,F,X),aa(B,A,F,Y)) )
=> ord_less(A,Aa,aa(B,A,F,C)) ) ) ) ) ).
tff(fact_91_dense__le__bounded,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Z1: A,Y1: A,X2: A] :
( ord_less(A,X2,Y1)
=> ( ! [W: A] :
( ord_less(A,X2,W)
=> ( ord_less(A,W,Y1)
=> ord_less_eq(A,W,Z1) ) )
=> ord_less_eq(A,Y1,Z1) ) ) ) ).
tff(fact_92_order__less__le__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C: B,F: fun(A,B),B1: A,Aa: A] :
( ord_less(A,Aa,B1)
=> ( ord_less_eq(B,aa(A,B,F,B1),C)
=> ( ! [X: A,Y: A] :
( ord_less(A,X,Y)
=> ord_less(B,aa(A,B,F,X),aa(A,B,F,Y)) )
=> ord_less(B,aa(A,B,F,Aa),C) ) ) ) ) ).
tff(fact_93_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C: B,B1: B,F: fun(B,A),Aa: A] :
( ord_less(A,Aa,aa(B,A,F,B1))
=> ( ord_less_eq(B,B1,C)
=> ( ! [X: B,Y: B] :
( ord_less_eq(B,X,Y)
=> ord_less_eq(A,aa(B,A,F,X),aa(B,A,F,Y)) )
=> ord_less(A,Aa,aa(B,A,F,C)) ) ) ) ) ).
tff(fact_94_order__le__less__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C: B,F: fun(A,B),B1: A,Aa: A] :
( ord_less_eq(A,Aa,B1)
=> ( ord_less(B,aa(A,B,F,B1),C)
=> ( ! [X: A,Y: A] :
( ord_less_eq(A,X,Y)
=> ord_less_eq(B,aa(A,B,F,X),aa(A,B,F,Y)) )
=> ord_less(B,aa(A,B,F,Aa),C) ) ) ) ) ).
tff(fact_95_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
! [Y1: int,X2: int] :
( ord_less_eq(int,zero_zero(int),X2)
=> ( ord_less_eq(int,zero_zero(int),Y1)
=> ord_less_eq(int,zero_zero(int),nat_tsub(X2,Y1)) ) ) ).
tff(fact_96_minf_I11_J,axiom,
! [C1: $tType,D: $tType] :
( ord(C1)
=> ! [F1: D] :
? [Z: C1] :
! [X1: C1] :
( ord_less(C1,X1,Z)
=> ( F1 = F1 ) ) ) ).
tff(fact_97_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C: B,B1: B,F: fun(B,A),Aa: A] :
( ord_less(A,Aa,aa(B,A,F,B1))
=> ( ord_less(B,B1,C)
=> ( ! [X: B,Y: B] :
( ord_less(B,X,Y)
=> ord_less(A,aa(B,A,F,X),aa(B,A,F,Y)) )
=> ord_less(A,Aa,aa(B,A,F,C)) ) ) ) ) ).
tff(fact_98_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C: B,B1: B,F: fun(B,A),Aa: A] :
( ( Aa = aa(B,A,F,B1) )
=> ( ord_less(B,B1,C)
=> ( ! [X: B,Y: B] :
( ord_less(B,X,Y)
=> ord_less(A,aa(B,A,F,X),aa(B,A,F,Y)) )
=> ord_less(A,Aa,aa(B,A,F,C)) ) ) ) ) ).
%----Arities (17)
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___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Orderings_Opreorder,axiom,
preorder(int) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(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_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 (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (1)
tff(conj_0,conjecture,
divisor(a) != zero_zero(int) ).
%------------------------------------------------------------------------------