TPTP Problem File: NUM966_5.p
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%------------------------------------------------------------------------------
% File : NUM966_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 95
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_95 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 157 ( 21 unt; 34 typ; 0 def)
% Number of atoms : 347 ( 49 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 244 ( 20 ~; 6 |; 11 &)
% ( 31 <=>; 176 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 17 ( 12 >; 5 *; 0 +; 0 <<)
% Number of predicates : 23 ( 22 usr; 0 prp; 1-3 aty)
% Number of functors : 10 ( 10 usr; 3 con; 0-4 aty)
% Number of variables : 377 ( 346 !; 1 ?; 377 :)
% ( 30 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:24:13
%------------------------------------------------------------------------------
%----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 (31)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1,type,
ring_1:
!>[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_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim1727834104eiling:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim1804426504_field:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[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_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Osemigroup,type,
semigroup:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
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_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).
tff(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( fun(A,B) > $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_t____,type,
t: int ).
%----Relevant facts (99)
tff(fact_0_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ord_less_eq(A,X,X) ) ).
tff(fact_1_one__le__floor,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X1: A] :
( ord_less_eq(int,one_one(int),archim2085082626_floor(A,X1))
<=> ord_less_eq(A,one_one(A),X1) ) ) ).
tff(fact_2_ceiling__le__one,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X1: A] :
( ord_less_eq(int,archimedean_ceiling(A,X1),one_one(int))
<=> ord_less_eq(A,X1,one_one(A)) ) ) ).
tff(fact_3_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X1: A] :
( ( one_one(A) = X1 )
<=> ( X1 = one_one(A) ) ) ) ).
tff(fact_4_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_5_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,X,Y)
| ord_less_eq(A,Y,X) ) ) ).
tff(fact_6_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_7_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ( X = Y )
=> ord_less_eq(A,X,Y) ) ) ).
tff(fact_8_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_9_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_10_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C: A,B1: A,A1: A] :
( ( A1 = B1 )
=> ( ord_less_eq(A,B1,C)
=> ord_less_eq(A,A1,C) ) ) ) ).
tff(fact_11_floor__one,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ( archim2085082626_floor(A,one_one(A)) = one_one(int) ) ) ).
tff(fact_12_ceiling__one,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).
tff(fact_13_ceiling__mono,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X: A,Y: A] :
( ord_less_eq(A,Y,X)
=> ord_less_eq(int,archimedean_ceiling(A,Y),archimedean_ceiling(A,X)) ) ) ).
tff(fact_14_floor__mono,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,X,Y)
=> ord_less_eq(int,archim2085082626_floor(A,X),archim2085082626_floor(A,Y)) ) ) ).
tff(fact_15_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ ord_less_eq(A,X,Y)
=> ord_less_eq(A,Y,X) ) ) ).
tff(fact_16_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_17_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X: A] :
( ord_less_eq(A,X,Y)
=> ( ord_less_eq(A,Y,Z)
=> ord_less_eq(A,X,Z) ) ) ) ).
tff(fact_18_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,X,Y)
=> ( ord_less_eq(A,Y,X)
=> ( X = Y ) ) ) ) ).
tff(fact_19_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C: A,B1: A,A1: A] :
( ord_less_eq(A,A1,B1)
=> ( ( B1 = C )
=> ord_less_eq(A,A1,C) ) ) ) ).
tff(fact_20_le__funI,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F: fun(A,B)] :
( ! [X2: A] : ord_less_eq(B,aa(A,B,F,X2),aa(A,B,G,X2))
=> ord_less_eq(fun(A,B),F,G) ) ) ).
tff(fact_21_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,B2: B,F: fun(B,A),A2: A] :
( ord_less_eq(A,A2,aa(B,A,F,B2))
=> ( ord_less_eq(B,B2,C1)
=> ( ! [X2: B,Y2: B] :
( ord_less_eq(B,X2,Y2)
=> ord_less_eq(A,aa(B,A,F,X2),aa(B,A,F,Y2)) )
=> ord_less_eq(A,A2,aa(B,A,F,C1)) ) ) ) ) ).
tff(fact_22_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C1: B,B2: B,F: fun(B,A),A2: A] :
( ( A2 = aa(B,A,F,B2) )
=> ( ord_less_eq(B,B2,C1)
=> ( ! [X2: B,Y2: B] :
( ord_less_eq(B,X2,Y2)
=> ord_less_eq(A,aa(B,A,F,X2),aa(B,A,F,Y2)) )
=> ord_less_eq(A,A2,aa(B,A,F,C1)) ) ) ) ) ).
tff(fact_23_order__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,F: fun(A,B),B2: A,A2: A] :
( ord_less_eq(A,A2,B2)
=> ( ord_less_eq(B,aa(A,B,F,B2),C1)
=> ( ! [X2: A,Y2: A] :
( ord_less_eq(A,X2,Y2)
=> ord_less_eq(B,aa(A,B,F,X2),aa(A,B,F,Y2)) )
=> ord_less_eq(B,aa(A,B,F,A2),C1) ) ) ) ) ).
tff(fact_24_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C1: B,F: fun(A,B),B2: A,A2: A] :
( ord_less_eq(A,A2,B2)
=> ( ( aa(A,B,F,B2) = C1 )
=> ( ! [X2: A,Y2: A] :
( ord_less_eq(A,X2,Y2)
=> ord_less_eq(B,aa(A,B,F,X2),aa(A,B,F,Y2)) )
=> ord_less_eq(B,aa(A,B,F,A2),C1) ) ) ) ) ).
tff(fact_25_ceiling__le__iff,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Z1: int,X1: A] :
( ord_less_eq(int,archimedean_ceiling(A,X1),Z1)
<=> ord_less_eq(A,X1,ring_1_of_int(A,Z1)) ) ) ).
tff(fact_26_le__floor__iff,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X1: A,Z1: int] :
( ord_less_eq(int,Z1,archim2085082626_floor(A,X1))
<=> ord_less_eq(A,ring_1_of_int(A,Z1),X1) ) ) ).
tff(fact_27_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [Y1: A,X1: A,F: fun(A,B)] :
( order_strict_mono(A,B,F)
=> ( ord_less_eq(B,aa(A,B,F,X1),aa(A,B,F,Y1))
<=> ord_less_eq(A,X1,Y1) ) ) ) ).
tff(fact_28_semigroup_Oassoc,axiom,
! [A: $tType,C1: A,B2: A,A2: 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,A2),B2)),C1) = aa(A,A,aa(A,fun(A,A),F,A2),aa(A,A,aa(A,fun(A,A),F,B2),C1)) ) ) ).
tff(fact_29_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [Y1: A,X1: A,F: fun(A,B)] :
( order_strict_mono(A,B,F)
=> ( ( aa(A,B,F,X1) = aa(A,B,F,Y1) )
<=> ( X1 = Y1 ) ) ) ) ).
tff(fact_30_floor__of__int,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Z: int] : archim2085082626_floor(A,ring_1_of_int(A,Z)) = Z ) ).
tff(fact_31_ceiling__of__int,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Z: int] : archimedean_ceiling(A,ring_1_of_int(A,Z)) = Z ) ).
tff(fact_32_of__int__floor__le,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X: A] : ord_less_eq(A,ring_1_of_int(A,archim2085082626_floor(A,X)),X) ) ).
tff(fact_33_le__of__int__ceiling,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X: A] : ord_less_eq(A,X,ring_1_of_int(A,archimedean_ceiling(A,X))) ) ).
tff(fact_34_of__int__1,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A,one_one(int)) = one_one(A) ) ) ).
tff(fact_35_of__int__le__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z1: int,W: int] :
( ord_less_eq(A,ring_1_of_int(A,W),ring_1_of_int(A,Z1))
<=> ord_less_eq(int,W,Z1) ) ) ).
tff(fact_36_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_37_ex__le__of__int,axiom,
! [A: $tType] :
( archim1804426504_field(A)
=> ! [X: A] :
? [Z2: int] : ord_less_eq(A,X,ring_1_of_int(A,Z2)) ) ).
tff(fact_38_strict__mono__mono,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B)] :
( order_strict_mono(A,B,F)
=> order_mono(A,B,F) ) ) ).
tff(fact_39_floor__correct,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X: A] :
( ord_less_eq(A,ring_1_of_int(A,archim2085082626_floor(A,X)),X)
& ord_less(A,X,ring_1_of_int(A,plus_plus(int,archim2085082626_floor(A,X),one_one(int)))) ) ) ).
tff(fact_40_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,A2: A,B2: A] :
( ( plus_plus(A,B2,A2) = plus_plus(A,C1,A2) )
<=> ( B2 = C1 ) ) ) ).
tff(fact_41_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,B2: A,A2: A] :
( ( plus_plus(A,A2,B2) = plus_plus(A,A2,C1) )
<=> ( B2 = C1 ) ) ) ).
tff(fact_42_add__le__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B2: A,A2: A,C1: A] :
( ord_less_eq(A,plus_plus(A,C1,A2),plus_plus(A,C1,B2))
<=> ord_less_eq(A,A2,B2) ) ) ).
tff(fact_43_add__le__cancel__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B2: A,C1: A,A2: A] :
( ord_less_eq(A,plus_plus(A,A2,C1),plus_plus(A,B2,C1))
<=> ord_less_eq(A,A2,B2) ) ) ).
tff(fact_44_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B2: A,A2: A,C1: A] :
( ord_less(A,plus_plus(A,C1,A2),plus_plus(A,C1,B2))
<=> ord_less(A,A2,B2) ) ) ).
tff(fact_45_add__less__cancel__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B2: A,C1: A,A2: A] :
( ord_less(A,plus_plus(A,A2,C1),plus_plus(A,B2,C1))
<=> ord_less(A,A2,B2) ) ) ).
tff(fact_46_of__int__less__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Z1: int,W: int] :
( ord_less(A,ring_1_of_int(A,W),ring_1_of_int(A,Z1))
<=> ord_less(int,W,Z1) ) ) ).
tff(fact_47_zle__add1__eq__le,axiom,
! [Z1: int,W: int] :
( ord_less(int,W,plus_plus(int,Z1,one_one(int)))
<=> ord_less_eq(int,W,Z1) ) ).
tff(fact_48_of__int__add,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Z: int,W1: int] : ring_1_of_int(A,plus_plus(int,W1,Z)) = plus_plus(A,ring_1_of_int(A,W1),ring_1_of_int(A,Z)) ) ).
tff(fact_49_floor__less__one,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X1: A] :
( ord_less(int,archim2085082626_floor(A,X1),one_one(int))
<=> ord_less(A,X1,one_one(A)) ) ) ).
tff(fact_50_one__less__ceiling,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X1: A] :
( ord_less(int,one_one(int),archimedean_ceiling(A,X1))
<=> ord_less(A,one_one(A),X1) ) ) ).
tff(fact_51_floor__add__of__int,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Z: int,X: A] : archim2085082626_floor(A,plus_plus(A,X,ring_1_of_int(A,Z))) = plus_plus(int,archim2085082626_floor(A,X),Z) ) ).
tff(fact_52_ceiling__add__of__int,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Z: int,X: A] : archimedean_ceiling(A,plus_plus(A,X,ring_1_of_int(A,Z))) = plus_plus(int,archimedean_ceiling(A,X),Z) ) ).
tff(fact_53_floor__add__one,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X: A] : archim2085082626_floor(A,plus_plus(A,X,one_one(A))) = plus_plus(int,archim2085082626_floor(A,X),one_one(int)) ) ).
tff(fact_54_ceiling__add__one,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X: A] : archimedean_ceiling(A,plus_plus(A,X,one_one(A))) = plus_plus(int,archimedean_ceiling(A,X),one_one(int)) ) ).
tff(fact_55_less__int__def,axiom,
! [W: int,Z1: int] :
( ord_less(int,Z1,W)
<=> ( ord_less_eq(int,Z1,W)
& ( Z1 != W ) ) ) ).
tff(fact_56_add1__zle__eq,axiom,
! [Z1: int,W: int] :
( ord_less_eq(int,plus_plus(int,W,one_one(int)),Z1)
<=> ord_less(int,W,Z1) ) ).
tff(fact_57_zless__imp__add1__zle,axiom,
! [Z: int,W1: int] :
( ord_less(int,W1,Z)
=> ord_less_eq(int,plus_plus(int,W1,one_one(int)),Z) ) ).
tff(fact_58_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ ord_less(A,X,Y)
=> ( ( X != Y )
=> ord_less(A,Y,X) ) ) ) ).
tff(fact_59_add__less__imp__less__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B1: A,A1: A,C: A] :
( ord_less(A,plus_plus(A,C,A1),plus_plus(A,C,B1))
=> ord_less(A,A1,B1) ) ) ).
tff(fact_60_add__less__imp__less__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B1: A,C: A,A1: A] :
( ord_less(A,plus_plus(A,A1,C),plus_plus(A,B1,C))
=> ord_less(A,A1,B1) ) ) ).
tff(fact_61_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,A1: A,B1: A] :
( ( plus_plus(A,B1,A1) = plus_plus(A,C,A1) )
=> ( B1 = C ) ) ) ).
tff(fact_62_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C: A,B1: A,A1: A] :
( ( plus_plus(A,A1,B1) = plus_plus(A,A1,C) )
=> ( B1 = C ) ) ) ).
tff(fact_63_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,B1: A,A1: A] :
( ( plus_plus(A,A1,B1) = plus_plus(A,A1,C) )
=> ( B1 = C ) ) ) ).
tff(fact_64_floor__less__cancel,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Y: A,X: A] :
( ord_less(int,archim2085082626_floor(A,X),archim2085082626_floor(A,Y))
=> ord_less(A,X,Y) ) ) ).
tff(fact_65_ceiling__less__cancel,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Y: A,X: A] :
( ord_less(int,archimedean_ceiling(A,X),archimedean_ceiling(A,Y))
=> ord_less(A,X,Y) ) ) ).
tff(fact_66_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ~ ord_less(A,Y,X) ) ) ).
tff(fact_67_add__le__less__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [D: A,C: A,B1: A,A1: A] :
( ord_less_eq(A,A1,B1)
=> ( ord_less(A,C,D)
=> ord_less(A,plus_plus(A,A1,C),plus_plus(A,B1,D)) ) ) ) ).
tff(fact_68_add__less__le__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [D: A,C: A,B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ( ord_less_eq(A,C,D)
=> ord_less(A,plus_plus(A,A1,C),plus_plus(A,B1,D)) ) ) ) ).
tff(fact_69_add__strict__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [D: A,C: A,B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ( ord_less(A,C,D)
=> ord_less(A,plus_plus(A,A1,C),plus_plus(A,B1,D)) ) ) ) ).
tff(fact_70_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X: A] :
( ord_less(A,X,Y)
=> ( ord_less(A,Y,Z)
=> ord_less(A,X,Z) ) ) ) ).
tff(fact_71_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C: A,B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ( ( B1 = C )
=> ord_less(A,A1,C) ) ) ) ).
tff(fact_72_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C: A,B1: A,A1: A] :
( ( A1 = B1 )
=> ( ord_less(A,B1,C)
=> ord_less(A,A1,C) ) ) ) ).
tff(fact_73_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ~ ord_less(A,B1,A1) ) ) ).
tff(fact_74_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_75_add__strict__left__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [C: A,B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ord_less(A,plus_plus(A,C,A1),plus_plus(A,C,B1)) ) ) ).
tff(fact_76_add__strict__right__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [C: A,B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ord_less(A,plus_plus(A,A1,C),plus_plus(A,B1,C)) ) ) ).
tff(fact_77_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ( Y != X ) ) ) ).
tff(fact_78_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ( X != Y ) ) ) ).
tff(fact_79_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ~ ord_less(A,Y,X) ) ) ).
tff(fact_80_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ~ ord_less(A,Y,X) ) ) ).
tff(fact_81_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ( X != Y ) ) ) ).
tff(fact_82_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ( X != Y )
=> ( ~ ord_less(A,X,Y)
=> ord_less(A,Y,X) ) ) ) ).
tff(fact_83_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_84_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C: A,B1: A,A1: A] : plus_plus(A,plus_plus(A,A1,B1),C) = plus_plus(A,A1,plus_plus(A,B1,C)) ) ).
tff(fact_85_zless__add1__eq,axiom,
! [Z1: int,W: int] :
( ord_less(int,W,plus_plus(int,Z1,one_one(int)))
<=> ( ord_less(int,W,Z1)
| ( W = Z1 ) ) ) ).
tff(fact_86_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
| ( X = Y )
| ord_less(A,Y,X) ) ) ).
tff(fact_87_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_88_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_89_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ~ ord_less(A,X,X) ) ).
tff(fact_90_floor__less__iff,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [Z1: int,X1: A] :
( ord_less(int,archim2085082626_floor(A,X1),Z1)
<=> ord_less(A,X1,ring_1_of_int(A,Z1)) ) ) ).
tff(fact_91_less__ceiling__iff,axiom,
! [A: $tType] :
( archim1727834104eiling(A)
=> ! [X1: A,Z1: int] :
( ord_less(int,Z1,archimedean_ceiling(A,X1))
<=> ord_less(A,ring_1_of_int(A,Z1),X1) ) ) ).
tff(fact_92_add__le__imp__le__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B1: A,A1: A,C: A] :
( ord_less_eq(A,plus_plus(A,C,A1),plus_plus(A,C,B1))
=> ord_less_eq(A,A1,B1) ) ) ).
tff(fact_93_add__le__imp__le__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B1: A,C: A,A1: A] :
( ord_less_eq(A,plus_plus(A,A1,C),plus_plus(A,B1,C))
=> ord_less_eq(A,A1,B1) ) ) ).
tff(fact_94_add__mono,axiom,
! [A: $tType] :
( ordere779506340up_add(A)
=> ! [D: A,C: A,B1: A,A1: A] :
( ord_less_eq(A,A1,B1)
=> ( ord_less_eq(A,C,D)
=> ord_less_eq(A,plus_plus(A,A1,C),plus_plus(A,B1,D)) ) ) ) ).
tff(fact_95_add__left__mono,axiom,
! [A: $tType] :
( ordere779506340up_add(A)
=> ! [C: A,B1: A,A1: A] :
( ord_less_eq(A,A1,B1)
=> ord_less_eq(A,plus_plus(A,C,A1),plus_plus(A,C,B1)) ) ) ).
tff(fact_96_add__right__mono,axiom,
! [A: $tType] :
( ordere779506340up_add(A)
=> ! [C: A,B1: A,A1: A] :
( ord_less_eq(A,A1,B1)
=> ord_less_eq(A,plus_plus(A,A1,C),plus_plus(A,B1,C)) ) ) ).
tff(fact_97_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_98_order__le__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X: A] :
( ord_less_eq(A,X,Y)
=> ( ord_less(A,Y,Z)
=> ord_less(A,X,Z) ) ) ) ).
%----Arities (21)
tff(arity_fun___Orderings_Opreorder,axiom,
! [T_1: $tType,T_2: $tType] :
( preorder(T_2)
=> preorder(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oorder,axiom,
! [T_1: $tType,T_2: $tType] :
( order(T_2)
=> order(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oord,axiom,
! [T_1: $tType,T_2: $tType] :
( ord(T_2)
=> ord(fun(T_1,T_2)) ) ).
tff(arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(int) ).
tff(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(int) ).
tff(arity_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(int) ).
tff(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(int) ).
tff(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(int) ).
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___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___Rings_Oring__1,axiom,
ring_1(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,one_one(int),t) ).
%------------------------------------------------------------------------------