TPTP Problem File: COM026_5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : COM026_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Quantifier elimination for Presburger arithmetic line 21
% 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_21 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 154 ( 19 unt; 34 typ; 0 def)
% Number of atoms : 363 ( 63 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 282 ( 39 ~; 8 |; 13 &)
% ( 34 <=>; 188 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 18 ( 12 >; 6 *; 0 +; 0 <<)
% Number of predicates : 18 ( 17 usr; 0 prp; 1-3 aty)
% Number of functors : 14 ( 14 usr; 4 con; 0-4 aty)
% Number of variables : 354 ( 324 !; 5 ?; 354 :)
% ( 25 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:15:59
%------------------------------------------------------------------------------
%----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 (29)
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_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_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
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_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_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_Ohd__coeff,type,
hd_coeff: fun(atom,int) ).
tff(sy_c_PresArith_Ozlcms,type,
zlcms: list(int) > int ).
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_as,type,
as: list(atom) ).
%----Relevant facts (98)
tff(fact_0__C0_C,axiom,
! [X4: atom] :
( member(atom,X4,set(atom,as))
=> ( aa(atom,int,hd_coeff,X4) != zero_zero(int) ) ) ).
tff(fact_1_of__int__0__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int] :
( ord_less(A,zero_zero(A),ring_1_of_int(A,Z))
<=> ord_less(int,zero_zero(int),Z) ) ) ).
tff(fact_2_of__int__less__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int] :
( ord_less(A,ring_1_of_int(A,Z),zero_zero(A))
<=> ord_less(int,Z,zero_zero(int)) ) ) ).
tff(fact_3_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A] : ~ ord_less(A,X2,X2) ) ).
tff(fact_4_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ( X1 != Y1 )
<=> ( ord_less(A,X1,Y1)
| ord_less(A,Y1,X1) ) ) ) ).
tff(fact_5_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ~ ord_less(A,X1,Y1)
<=> ( ord_less(A,Y1,X1)
| ( X1 = Y1 ) ) ) ) ).
tff(fact_6_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
| ( X2 = Y2 )
| ord_less(A,Y2,X2) ) ) ).
tff(fact_7_linorder__antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X1: A,Y1: A] :
( ~ ord_less(A,Y1,X1)
=> ( ~ ord_less(A,X1,Y1)
<=> ( X1 = Y1 ) ) ) ) ).
tff(fact_8_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ( X2 != Y2 )
=> ( ~ ord_less(A,X2,Y2)
=> ord_less(A,Y2,X2) ) ) ) ).
tff(fact_9_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y2: A,X2: A] :
( ( X2 != Y2 )
=> ( ~ ord_less(A,X2,Y2)
=> ord_less(A,Y2,X2) ) ) ) ).
tff(fact_10_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ( X2 != Y2 ) ) ) ).
tff(fact_11_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ~ ord_less(A,Y2,X2) ) ) ).
tff(fact_12_of__int__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z: int,W: int] :
( ( ring_1_of_int(A,W) = ring_1_of_int(A,Z) )
<=> ( W = Z ) ) ) ).
tff(fact_13_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z: int] :
( ( ring_1_of_int(A,Z) = zero_zero(A) )
<=> ( Z = zero_zero(int) ) ) ) ).
tff(fact_14_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ring_char_0(A)
=> ! [Z: int] :
( ( zero_zero(A) = ring_1_of_int(A,Z) )
<=> ( Z = zero_zero(int) ) ) ) ).
tff(fact_15_of__int__0,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A,zero_zero(int)) = zero_zero(A) ) ) ).
tff(fact_16_of__int__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int,W: int] :
( ord_less(A,ring_1_of_int(A,W),ring_1_of_int(A,Z))
<=> ord_less(int,W,Z) ) ) ).
tff(fact_17_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X1: A] :
( ( zero_zero(A) = X1 )
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_18_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ~ ord_less(A,X2,Y2)
=> ( ( X2 != Y2 )
=> ord_less(A,Y2,X2) ) ) ) ).
tff(fact_19_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ~ ord_less(A,Y2,X2) ) ) ).
tff(fact_20_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z1: A,Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ( ord_less(A,Y2,Z1)
=> ord_less(A,X2,Z1) ) ) ) ).
tff(fact_21_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B2: A,A2: A] :
( ord_less(A,A2,B2)
=> ( ( B2 = C1 )
=> ord_less(A,A2,C1) ) ) ) ).
tff(fact_22_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B2: A,A2: A] :
( ( A2 = B2 )
=> ( ord_less(A,B2,C1)
=> ord_less(A,A2,C1) ) ) ) ).
tff(fact_23_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_24_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [P: bool,Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ( ord_less(A,Y1,X1)
=> pp(P) ) ) ) ).
tff(fact_25_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ( Y2 != X2 ) ) ) ).
tff(fact_26_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ( X2 != Y2 ) ) ) ).
tff(fact_27_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ~ ord_less(A,Y2,X2) ) ) ).
tff(fact_28_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) )
<=> ! [X3: B] :
( member(B,X3,set(B,Xsa))
=> ( aa(B,A,F,X3) = aa(B,A,G,X3) ) ) ) ).
tff(fact_29_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C: B,B1: B,F: fun(B,A),A1: A] :
( ord_less(A,A1,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,A1,aa(B,A,F,C)) ) ) ) ) ).
tff(fact_30_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C: B,B1: B,F: fun(B,A),A1: A] :
( ( A1 = 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,A1,aa(B,A,F,C)) ) ) ) ) ).
tff(fact_31_order__less__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C: B,F: fun(A,B),B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ( ord_less(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,A1),C) ) ) ) ) ).
tff(fact_32_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C: B,F: fun(A,B),B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ( ( 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,A1),C) ) ) ) ) ).
tff(fact_33_dense,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ? [Z2: A] :
( ord_less(A,X2,Z2)
& ord_less(A,Z2,Y2) ) ) ) ).
tff(fact_34_lt__ex,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [X2: A] :
? [Y: A] : ord_less(A,Y,X2) ) ).
tff(fact_35_gt__ex,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [X1: A] :
? [X11: A] : ord_less(A,X1,X11) ) ).
tff(fact_36_sgn__less,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( ord_less(A,sgn_sgn(A,A1),zero_zero(A))
<=> ord_less(A,A1,zero_zero(A)) ) ) ).
tff(fact_37_sgn__sgn,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] : sgn_sgn(A,sgn_sgn(A,A2)) = sgn_sgn(A,A2) ) ).
tff(fact_38_sgn0,axiom,
! [A: $tType] :
( sgn_if(A)
=> ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).
tff(fact_39_sgn__greater,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( ord_less(A,zero_zero(A),sgn_sgn(A,A1))
<=> ord_less(A,zero_zero(A),A1) ) ) ).
tff(fact_40_sgn__0__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( ( sgn_sgn(A,A1) = zero_zero(A) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_41_zlcms0__iff,axiom,
! [Is: list(int)] :
( ( zlcms(Is) = zero_zero(int) )
<=> member(int,zero_zero(int),set(int,Is)) ) ).
tff(fact_42_ex__map__conv,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Ys: list(B)] :
( ? [Xs: list(A)] : Ys = map(A,B,F,Xs)
<=> ! [X3: B] :
( member(B,X3,set(B,Ys))
=> ? [Xa: A] : X3 = aa(A,B,F,Xa) ) ) ).
tff(fact_43_map__ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),Xsa: list(A)] :
( ! [X: A] :
( member(A,X,set(A,Xsa))
=> ( aa(A,B,F,X) = aa(A,B,G,X) ) )
=> ( map(A,B,F,Xsa) = map(A,B,G,Xsa) ) ) ).
tff(fact_44_map__idI,axiom,
! [A: $tType,F: fun(A,A),Xsa: list(A)] :
( ! [X: A] :
( member(A,X,set(A,Xsa))
=> ( aa(A,A,F,X) = X ) )
=> ( map(A,A,F,Xsa) = Xsa ) ) ).
tff(fact_45_map__cong,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B),Ys: list(A),Xsa: list(A)] :
( ( Xsa = Ys )
=> ( ! [X: A] :
( member(A,X,set(A,Ys))
=> ( aa(A,B,F,X) = aa(A,B,G,X) ) )
=> ( map(A,B,F,Xsa) = map(A,B,G,Ys) ) ) ) ).
tff(fact_46_zlcms__pos,axiom,
! [Is: list(int)] :
( ! [X: int] :
( member(int,X,set(int,Is))
=> ( X != zero_zero(int) ) )
=> ord_less(int,zero_zero(int),zlcms(Is)) ) ).
tff(fact_47_sgn__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] :
( ord_less(A,zero_zero(A),A2)
=> ( sgn_sgn(A,A2) = one_one(A) ) ) ) ).
tff(fact_48_sgn__1__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( ( sgn_sgn(A,A1) = one_one(A) )
<=> ord_less(A,zero_zero(A),A1) ) ) ).
tff(fact_49_elem__le__zlcms,axiom,
! [I: int,Is: list(int)] :
( ! [X: int] :
( member(int,X,set(int,Is))
=> ( X != zero_zero(int) ) )
=> ( member(int,I,set(int,Is))
=> ord_less_eq(int,I,zlcms(Is)) ) ) ).
tff(fact_50_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A] : ord_less_eq(A,X2,X2) ) ).
tff(fact_51_of__int__1,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A,one_one(int)) = one_one(A) ) ) ).
tff(fact_52_of__int__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int,W: int] :
( ord_less_eq(A,ring_1_of_int(A,W),ring_1_of_int(A,Z))
<=> ord_less_eq(int,W,Z) ) ) ).
tff(fact_53_of__int__0__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int] :
( ord_less_eq(A,zero_zero(A),ring_1_of_int(A,Z))
<=> ord_less_eq(int,zero_zero(int),Z) ) ) ).
tff(fact_54_of__int__le__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z: int] :
( ord_less_eq(A,ring_1_of_int(A,Z),zero_zero(A))
<=> ord_less_eq(int,Z,zero_zero(int)) ) ) ).
tff(fact_55_less__int__def,axiom,
! [W: int,Z: int] :
( ord_less(int,Z,W)
<=> ( ord_less_eq(int,Z,W)
& ( Z != W ) ) ) ).
tff(fact_56_not__one__le__zero,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ~ ord_less_eq(A,one_one(A),zero_zero(A)) ) ).
tff(fact_57_zero__le__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ord_less_eq(A,zero_zero(A),one_one(A)) ) ).
tff(fact_58_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X1: A] :
( ( one_one(A) = X1 )
<=> ( X1 = one_one(A) ) ) ) ).
tff(fact_59_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)
<=> ! [X3: A] : ord_less_eq(B,aa(A,B,F,X3),aa(A,B,G,X3)) ) ) ).
tff(fact_60_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ord_less_eq(A,X2,Y2)
| ord_less_eq(A,Y2,X2) ) ) ).
tff(fact_61_order__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ( X1 = Y1 )
<=> ( ord_less_eq(A,X1,Y1)
& ord_less_eq(A,Y1,X1) ) ) ) ).
tff(fact_62_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X2: A] :
( ( X2 = Y2 )
=> ord_less_eq(A,X2,Y2) ) ) ).
tff(fact_63_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [X1: 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,X1),aa(A,B,G,X1)) ) ) ).
tff(fact_64_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [X1: A,Y1: A] :
( ord_less_eq(A,Y1,X1)
=> ( ord_less_eq(A,X1,Y1)
<=> ( X1 = Y1 ) ) ) ) ).
tff(fact_65_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B2: A,A2: A] :
( ( A2 = B2 )
=> ( ord_less_eq(A,B2,C1)
=> ord_less_eq(A,A2,C1) ) ) ) ).
tff(fact_66_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C1: A,B2: A,A2: A] :
( ord_less_eq(A,A2,B2)
=> ( ( B2 = C1 )
=> ord_less_eq(A,A2,C1) ) ) ) ).
tff(fact_67_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X2: A] :
( ord_less_eq(A,X2,Y2)
=> ( ord_less_eq(A,Y2,X2)
=> ( X2 = Y2 ) ) ) ) ).
tff(fact_68_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z1: A,Y2: A,X2: A] :
( ord_less_eq(A,X2,Y2)
=> ( ord_less_eq(A,Y2,Z1)
=> ord_less_eq(A,X2,Z1) ) ) ) ).
tff(fact_69_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [X1: 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,X1),aa(A,B,G,X1)) ) ) ).
tff(fact_70_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ~ ord_less_eq(A,X2,Y2)
=> ord_less_eq(A,Y2,X2) ) ) ).
tff(fact_71_order__le__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z1: A,Y2: A,X2: A] :
( ord_less_eq(A,X2,Y2)
=> ( ord_less(A,Y2,Z1)
=> ord_less(A,X2,Z1) ) ) ) ).
tff(fact_72_order__less__le__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z1: A,Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ( ord_less_eq(A,Y2,Z1)
=> ord_less(A,X2,Z1) ) ) ) ).
tff(fact_73_mem__def,axiom,
! [A: $tType,A3: fun(A,bool),X1: A] :
( member(A,X1,A3)
<=> pp(aa(A,bool,A3,X1)) ) ).
tff(fact_74_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_75_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X2: A] :
( ord_less_eq(A,X2,Y2)
=> ( ord_less(A,X2,Y2)
| ( X2 = Y2 ) ) ) ) ).
tff(fact_76_linorder__antisym__conv2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
=> ( ~ ord_less(A,X1,Y1)
<=> ( X1 = Y1 ) ) ) ) ).
tff(fact_77_order__less__imp__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ord_less_eq(A,X2,Y2) ) ) ).
tff(fact_78_leD,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: A,Y2: A] :
( ord_less_eq(A,Y2,X2)
=> ~ ord_less(A,X2,Y2) ) ) ).
tff(fact_79_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_80_linorder__antisym__conv1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ~ ord_less(A,X1,Y1)
=> ( ord_less_eq(A,X1,Y1)
<=> ( X1 = Y1 ) ) ) ) ).
tff(fact_81_not__leE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: A,Y2: A] :
( ~ ord_less_eq(A,Y2,X2)
=> ord_less(A,X2,Y2) ) ) ).
tff(fact_82_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ~ ord_less(A,X2,Y2)
=> ord_less_eq(A,Y2,X2) ) ) ).
tff(fact_83_order__le__less,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
<=> ( ord_less(A,X1,Y1)
| ( X1 = Y1 ) ) ) ) ).
tff(fact_84_less__le__not__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
<=> ( ord_less_eq(A,X1,Y1)
& ~ ord_less_eq(A,Y1,X1) ) ) ) ).
tff(fact_85_order__less__le,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
<=> ( ord_less_eq(A,X1,Y1)
& ( X1 != Y1 ) ) ) ) ).
tff(fact_86_linorder__le__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ord_less_eq(A,X2,Y2)
| ord_less(A,Y2,X2) ) ) ).
tff(fact_87_linorder__not__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ~ ord_less_eq(A,X1,Y1)
<=> ord_less(A,Y1,X1) ) ) ).
tff(fact_88_linorder__not__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ~ ord_less(A,X1,Y1)
<=> ord_less_eq(A,Y1,X1) ) ) ).
tff(fact_89_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
tff(fact_90_one__neq__zero,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( one_one(A) != zero_zero(A) ) ) ).
tff(fact_91_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ord_less_eq(int,one_one(int),Z)
<=> ord_less(int,zero_zero(int),Z) ) ).
tff(fact_92_not__one__less__zero,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ~ ord_less(A,one_one(A),zero_zero(A)) ) ).
tff(fact_93_zero__less__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ord_less(A,zero_zero(A),one_one(A)) ) ).
tff(fact_94_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C: B,B1: B,F: fun(B,A),A1: A] :
( ord_less_eq(A,A1,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,A1,aa(B,A,F,C)) ) ) ) ) ).
tff(fact_95_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C: B,B1: B,F: fun(B,A),A1: A] :
( ord_less(A,A1,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,A1,aa(B,A,F,C)) ) ) ) ) ).
tff(fact_96_order__le__less__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C: B,F: fun(A,B),B1: A,A1: A] :
( ord_less_eq(A,A1,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,A1),C) ) ) ) ) ).
tff(fact_97_order__less__le__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C: B,F: fun(A,B),B1: A,A1: A] :
( ord_less(A,A1,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,A1),C) ) ) ) ) ).
%----Arities (19)
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__semidom,axiom,
linordered_semidom(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___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___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,
ord_less(int,zero_zero(int),zlcms(map(atom,int,hd_coeff,as))) ).
%------------------------------------------------------------------------------