TPTP Problem File: NUM931_5.p
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%------------------------------------------------------------------------------
% File : NUM931_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 35
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_35 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.25 v7.1.0, 0.67 v6.4.0
% Syntax : Number of formulae : 143 ( 19 unt; 26 typ; 0 def)
% Number of atoms : 358 ( 37 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 281 ( 40 ~; 8 |; 13 &)
% ( 34 <=>; 186 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 13 ( 8 >; 5 *; 0 +; 0 <<)
% Number of predicates : 15 ( 14 usr; 0 prp; 1-3 aty)
% Number of functors : 10 ( 10 usr; 3 con; 0-4 aty)
% Number of variables : 323 ( 300 !; 4 ?; 323 :)
% ( 19 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:22:46
%------------------------------------------------------------------------------
%----Should-be-implicit typings (3)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (23)
tff(sy_cl_Groups_Oone,type,
one:
!>[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_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_cl_Archimedean__Field_Ofloor__ceiling,type,
archim1727834104eiling:
!>[A: $tType] : $o ).
tff(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim2085082626_floor:
!>[A: $tType] : ( A > int ) ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
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_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_pp,type,
pp: bool > $o ).
tff(sy_v_m,type,
m: int ).
%----Relevant facts (98)
tff(fact_0__096_126_A1_A_060_061_Am_096,axiom,
~ ord_less_eq(int,one_one(int),m) ).
tff(fact_1_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A] : ord_less_eq(A,X2,X2) ) ).
tff(fact_2_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq(int,zero_zero(int),zero_zero(int)) ).
tff(fact_3_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_4_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_5_Nat__Transfer_Otransfer__nat__int__function__closures_I6_J,axiom,
ord_less_eq(int,zero_zero(int),one_one(int)) ).
tff(fact_6_zero__le__floor,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X3: A] :
( ord_less_eq(int,zero_zero(int),archim2085082626_floor(A,X3))
<=> ord_less_eq(A,zero_zero(A),X3) ) ) ).
tff(fact_7_ceiling__le__zero,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X3: A] :
( ord_less_eq(int,archimedean_ceiling(A,X3),zero_zero(int))
<=> ord_less_eq(A,X3,zero_zero(A)) ) ) ).
tff(fact_8_not__one__le__zero,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ~ ord_less_eq(A,one_one(A),zero_zero(A)) ) ).
tff(fact_9_zero__le__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ord_less_eq(A,zero_zero(A),one_one(A)) ) ).
tff(fact_10_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_11_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_12_floor__zero,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ( archim2085082626_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).
tff(fact_13_ceiling__zero,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).
tff(fact_14_floor__one,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ( archim2085082626_floor(A,one_one(A)) = one_one(int) ) ) ).
tff(fact_15_ceiling__one,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).
tff(fact_16_one__le__floor,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X3: A] :
( ord_less_eq(int,one_one(int),archim2085082626_floor(A,X3))
<=> ord_less_eq(A,one_one(A),X3) ) ) ).
tff(fact_17_ceiling__le__one,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X3: A] :
( ord_less_eq(int,archimedean_ceiling(A,X3),one_one(int))
<=> ord_less_eq(A,X3,one_one(A)) ) ) ).
tff(fact_18_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X3: A] :
( ( one_one(A) = X3 )
<=> ( X3 = one_one(A) ) ) ) ).
tff(fact_19_ceiling__mono,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X2: A,Y1: A] :
( ord_less_eq(A,Y1,X2)
=> ord_less_eq(int,archimedean_ceiling(A,Y1),archimedean_ceiling(A,X2)) ) ) ).
tff(fact_20_floor__mono,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Y1: A,X2: A] :
( ord_less_eq(A,X2,Y1)
=> ord_less_eq(int,archim2085082626_floor(A,X2),archim2085082626_floor(A,Y1)) ) ) ).
tff(fact_21_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
tff(fact_22_one__neq__zero,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( one_one(A) != zero_zero(A) ) ) ).
tff(fact_23_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X3: A] :
( ( zero_zero(A) = X3 )
<=> ( X3 = zero_zero(A) ) ) ) ).
tff(fact_24_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_25_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_26_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X2: A] :
( ord_less_eq(A,X2,Y1)
=> ( ord_less_eq(A,Y1,Z)
=> ord_less_eq(A,X2,Z) ) ) ) ).
tff(fact_27_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_28_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_29_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_30_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_31_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_32_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X2: A] :
( ( X2 = Y1 )
=> ord_less_eq(A,X2,Y1) ) ) ).
tff(fact_33_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_34_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_35_ceiling__less__one,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X3: A] :
( ord_less(int,archimedean_ceiling(A,X3),one_one(int))
<=> ord_less_eq(A,X3,zero_zero(A)) ) ) ).
tff(fact_36_zero__less__floor,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X3: A] :
( ord_less(int,zero_zero(int),archim2085082626_floor(A,X3))
<=> ord_less_eq(A,one_one(A),X3) ) ) ).
tff(fact_37_one__le__ceiling,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X3: A] :
( ord_less_eq(int,one_one(int),archimedean_ceiling(A,X3))
<=> ord_less(A,zero_zero(A),X3) ) ) ).
tff(fact_38_floor__le__zero,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X3: A] :
( ord_less_eq(int,archim2085082626_floor(A,X3),zero_zero(int))
<=> ord_less(A,X3,one_one(A)) ) ) ).
tff(fact_39_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_40_order__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_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_eq(A,A1,aa(B,A,F,C)) ) ) ) ) ).
tff(fact_41_ord__eq__le__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_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_eq(A,A1,aa(B,A,F,C)) ) ) ) ) ).
tff(fact_42_order__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_eq(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_eq(B,aa(A,B,F,A1),C) ) ) ) ) ).
tff(fact_43_floor__less__zero,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X3: A] :
( ord_less(int,archim2085082626_floor(A,X3),zero_zero(int))
<=> ord_less(A,X3,zero_zero(A)) ) ) ).
tff(fact_44_zero__less__ceiling,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X3: A] :
( ord_less(int,zero_zero(int),archimedean_ceiling(A,X3))
<=> ord_less(A,zero_zero(A),X3) ) ) ).
tff(fact_45_floor__less__one,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X3: A] :
( ord_less(int,archim2085082626_floor(A,X3),one_one(int))
<=> ord_less(A,X3,one_one(A)) ) ) ).
tff(fact_46_one__less__ceiling,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X3: A] :
( ord_less(int,one_one(int),archimedean_ceiling(A,X3))
<=> ord_less(A,one_one(A),X3) ) ) ).
tff(fact_47_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A] : ~ ord_less(A,X2,X2) ) ).
tff(fact_48_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_49_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_50_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_51_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_52_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_53_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_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,A2: A] :
( ord_less(A,A2,B2)
=> ~ ord_less(A,B2,A2) ) ) ).
tff(fact_61_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_62_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_63_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X2: A] :
( ord_less(A,X2,Y1)
=> ( ord_less(A,Y1,Z)
=> ord_less(A,X2,Z) ) ) ) ).
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_ceiling__less__cancel,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Y1: A,X2: A] :
( ord_less(int,archimedean_ceiling(A,X2),archimedean_ceiling(A,Y1))
=> ord_less(A,X2,Y1) ) ) ).
tff(fact_66_floor__less__cancel,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Y1: A,X2: A] :
( ord_less(int,archim2085082626_floor(A,X2),archim2085082626_floor(A,Y1))
=> ord_less(A,X2,Y1) ) ) ).
tff(fact_67_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_68_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_69_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_70_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_71_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_72_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_73_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_74_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_75_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X2: A] :
( ~ ord_less(A,X2,Y1)
=> ord_less_eq(A,Y1,X2) ) ) ).
tff(fact_76_not__leE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: A,Y1: A] :
( ~ ord_less_eq(A,Y1,X2)
=> ord_less(A,X2,Y1) ) ) ).
tff(fact_77_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_78_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_79_leD,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: A,Y1: A] :
( ord_less_eq(A,Y1,X2)
=> ~ ord_less(A,X2,Y1) ) ) ).
tff(fact_80_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_81_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_82_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_83_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_84_order__less__le__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X2: A] :
( ord_less(A,X2,Y1)
=> ( ord_less_eq(A,Y1,Z)
=> ord_less(A,X2,Z) ) ) ) ).
tff(fact_85_order__le__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X2: A] :
( ord_less_eq(A,X2,Y1)
=> ( ord_less(A,Y1,Z)
=> ord_less(A,X2,Z) ) ) ) ).
tff(fact_86_not__one__less__zero,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ~ ord_less(A,one_one(A),zero_zero(A)) ) ).
tff(fact_87_zero__less__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ord_less(A,zero_zero(A),one_one(A)) ) ).
tff(fact_88_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ord_less_eq(int,one_one(int),Z2)
<=> ord_less(int,zero_zero(int),Z2) ) ).
tff(fact_89_minf_I8_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z1: A] :
! [X1: A] :
( ord_less(A,X1,Z1)
=> ~ ord_less_eq(A,T,X1) ) ) ).
tff(fact_90_pinf_I8_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z1: A] :
! [X1: A] :
( ord_less(A,Z1,X1)
=> ord_less_eq(A,T,X1) ) ) ).
tff(fact_91_minf_I6_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z1: A] :
! [X1: A] :
( ord_less(A,X1,Z1)
=> ord_less_eq(A,X1,T) ) ) ).
tff(fact_92_less__int__def,axiom,
! [W: int,Z2: int] :
( ord_less(int,Z2,W)
<=> ( ord_less_eq(int,Z2,W)
& ( Z2 != W ) ) ) ).
tff(fact_93_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_94_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_95_pinf_I6_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z1: A] :
! [X1: A] :
( ord_less(A,Z1,X1)
=> ~ ord_less_eq(A,X1,T) ) ) ).
tff(fact_96_dense__le,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Z: A,Y1: A] :
( ! [X: A] :
( ord_less(A,X,Y1)
=> ord_less_eq(A,X,Z) )
=> ord_less_eq(A,Y1,Z) ) ) ).
tff(fact_97_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) ) ) ) ) ).
%----Arities (16)
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___Orderings_Oord,axiom,
ord(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_eq(int,m,zero_zero(int)) ).
%------------------------------------------------------------------------------