TPTP Problem File: SWW493_5.p
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%------------------------------------------------------------------------------
% File : SWW493_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra line 156
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fta_156 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 138 ( 14 unt; 21 typ; 0 def)
% Number of atoms : 449 ( 37 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 371 ( 39 ~; 13 |; 31 &)
% ( 33 <=>; 255 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 14 ( 9 >; 5 *; 0 +; 0 <<)
% Number of predicates : 16 ( 15 usr; 0 prp; 1-3 aty)
% Number of functors : 4 ( 4 usr; 2 con; 0-4 aty)
% Number of variables : 468 ( 414 !; 37 ?; 468 :)
% ( 17 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:16:07
%------------------------------------------------------------------------------
%----Should-be-implicit typings (3)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_RealDef_Oreal,type,
real: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (18)
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_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
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_member,type,
member:
!>[A: $tType] : ( ( A * fun(A,bool) ) > $o ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_P,type,
p: real > $o ).
%----Relevant facts (100)
tff(fact_0_ex,axiom,
? [X11: real] : p(X11) ).
tff(fact_1_bz,axiom,
? [Z: real] :
! [X1: real] :
( p(X1)
=> ord_less(real,X1,Z) ) ).
tff(fact_2_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X4: A] : ~ ord_less(A,X4,X4) ) ).
tff(fact_3_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ( X3 != Y1 )
<=> ( ord_less(A,X3,Y1)
| ord_less(A,Y1,X3) ) ) ) ).
tff(fact_4_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ~ ord_less(A,X3,Y1)
<=> ( ord_less(A,Y1,X3)
| ( X3 = Y1 ) ) ) ) ).
tff(fact_5_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X4: A] :
( ord_less(A,X4,Y2)
| ( X4 = Y2 )
| ord_less(A,Y2,X4) ) ) ).
tff(fact_6_linorder__antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X3: A,Y1: A] :
( ~ ord_less(A,Y1,X3)
=> ( ~ ord_less(A,X3,Y1)
<=> ( X3 = Y1 ) ) ) ) ).
tff(fact_7_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X4: A] :
( ( X4 != Y2 )
=> ( ~ ord_less(A,X4,Y2)
=> ord_less(A,Y2,X4) ) ) ) ).
tff(fact_8_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y2: A,X4: A] :
( ( X4 != Y2 )
=> ( ~ ord_less(A,X4,Y2)
=> ord_less(A,Y2,X4) ) ) ) ).
tff(fact_9_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X4: A] :
( ord_less(A,X4,Y2)
=> ( X4 != Y2 ) ) ) ).
tff(fact_10_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X4: A] :
( ord_less(A,X4,Y2)
=> ~ ord_less(A,Y2,X4) ) ) ).
tff(fact_11_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X4: A] :
( ord_less(A,X4,Y2)
=> ~ ord_less(A,Y2,X4) ) ) ).
tff(fact_12_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X4: A] :
( ord_less(A,X4,Y2)
=> ( X4 != Y2 ) ) ) ).
tff(fact_13_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X4: A] :
( ord_less(A,X4,Y2)
=> ( Y2 != X4 ) ) ) ).
tff(fact_14_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X4: A] :
( ~ ord_less(A,X4,Y2)
=> ( ( X4 != Y2 )
=> ord_less(A,Y2,X4) ) ) ) ).
tff(fact_15_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X4: A] :
( ord_less(A,X4,Y2)
=> ~ ord_less(A,Y2,X4) ) ) ).
tff(fact_16_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z2: A,Y2: A,X4: A] :
( ord_less(A,X4,Y2)
=> ( ord_less(A,Y2,Z2)
=> ord_less(A,X4,Z2) ) ) ) ).
tff(fact_17_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A3: A] :
( ord_less(A,A3,B2)
=> ( ( B2 = C2 )
=> ord_less(A,A3,C2) ) ) ) ).
tff(fact_18_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A3: A] :
( ( A3 = B2 )
=> ( ord_less(A,B2,C2)
=> ord_less(A,A3,C2) ) ) ) ).
tff(fact_19_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A3: A] :
( ord_less(A,A3,B2)
=> ~ ord_less(A,B2,A3) ) ) ).
tff(fact_20_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Pa: bool,Y1: A,X3: A] :
( ord_less(A,X3,Y1)
=> ( ord_less(A,Y1,X3)
=> pp(Pa) ) ) ) ).
tff(fact_21_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,B1: B,F1: fun(B,A),A1: A] :
( ord_less(A,A1,aa(B,A,F1,B1))
=> ( ord_less(B,B1,C1)
=> ( ! [X2: B,Y: B] :
( ord_less(B,X2,Y)
=> ord_less(A,aa(B,A,F1,X2),aa(B,A,F1,Y)) )
=> ord_less(A,A1,aa(B,A,F1,C1)) ) ) ) ) ).
tff(fact_22_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C1: B,B1: B,F1: fun(B,A),A1: A] :
( ( A1 = aa(B,A,F1,B1) )
=> ( ord_less(B,B1,C1)
=> ( ! [X2: B,Y: B] :
( ord_less(B,X2,Y)
=> ord_less(A,aa(B,A,F1,X2),aa(B,A,F1,Y)) )
=> ord_less(A,A1,aa(B,A,F1,C1)) ) ) ) ) ).
tff(fact_23_order__less__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,F1: fun(A,B),B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ( ord_less(B,aa(A,B,F1,B1),C1)
=> ( ! [X2: A,Y: A] :
( ord_less(A,X2,Y)
=> ord_less(B,aa(A,B,F1,X2),aa(A,B,F1,Y)) )
=> ord_less(B,aa(A,B,F1,A1),C1) ) ) ) ) ).
tff(fact_24_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C1: B,F1: fun(A,B),B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ( ( aa(A,B,F1,B1) = C1 )
=> ( ! [X2: A,Y: A] :
( ord_less(A,X2,Y)
=> ord_less(B,aa(A,B,F1,X2),aa(A,B,F1,Y)) )
=> ord_less(B,aa(A,B,F1,A1),C1) ) ) ) ) ).
tff(fact_25_dense,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Y2: A,X4: A] :
( ord_less(A,X4,Y2)
=> ? [Z: A] :
( ord_less(A,X4,Z)
& ord_less(A,Z,Y2) ) ) ) ).
tff(fact_26_lt__ex,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [X4: A] :
? [Y: A] : ord_less(A,Y,X4) ) ).
tff(fact_27_gt__ex,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [X3: A] :
? [X11: A] : ord_less(A,X3,X11) ) ).
tff(fact_28_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [Y1: A,X3: A,F1: fun(A,B)] :
( order_strict_mono(A,B,F1)
=> ( ord_less(A,X3,Y1)
=> ord_less(B,aa(A,B,F1,X3),aa(A,B,F1,Y1)) ) ) ) ).
tff(fact_29_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [Y1: A,X3: A,F1: fun(A,B)] :
( order_strict_mono(A,B,F1)
=> ( ord_less(B,aa(A,B,F1,X3),aa(A,B,F1,Y1))
<=> ord_less(A,X3,Y1) ) ) ) ).
tff(fact_30_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [Y1: A,X3: A,F1: fun(A,B)] :
( order_strict_mono(A,B,F1)
=> ( ( aa(A,B,F1,X3) = aa(A,B,F1,Y1) )
<=> ( X3 = Y1 ) ) ) ) ).
tff(fact_31_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F1: fun(A,B)] :
( ! [X2: A,Y: A] :
( ord_less(A,X2,Y)
=> ord_less(B,aa(A,B,F1,X2),aa(A,B,F1,Y)) )
=> order_strict_mono(A,B,F1) ) ) ).
tff(fact_32_strict__mono__mono,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F1: fun(A,B)] :
( order_strict_mono(A,B,F1)
=> order_mono(A,B,F1) ) ) ).
tff(fact_33_less__induct,axiom,
! [A: $tType] :
( wellorder(A)
=> ! [A1: A,Pa: fun(A,bool)] :
( ! [X2: A] :
( ! [Y3: A] :
( ord_less(A,Y3,X2)
=> pp(aa(A,bool,Pa,Y3)) )
=> pp(aa(A,bool,Pa,X2)) )
=> pp(aa(A,bool,Pa,A1)) ) ) ).
tff(fact_34_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [Y1: A,X3: A,F1: fun(A,B)] :
( order_strict_mono(A,B,F1)
=> ( ord_less_eq(B,aa(A,B,F1,X3),aa(A,B,F1,Y1))
<=> ord_less_eq(A,X3,Y1) ) ) ) ).
tff(fact_35_posreal__complete,axiom,
! [Pa: fun(real,bool)] :
( ? [X1: real] : member(real,X1,Pa)
=> ( ? [Y3: real] :
! [X2: real] :
( member(real,X2,Pa)
=> ord_less(real,X2,Y3) )
=> ? [S1: real] :
! [Y3: real] :
( ? [X: real] :
( member(real,X,Pa)
& ord_less(real,Y3,X) )
<=> ord_less(real,Y3,S1) ) ) ) ).
tff(fact_36_linordered__field__no__lb,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X1: A] :
? [Y: A] : ord_less(A,Y,X1) ) ).
tff(fact_37_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X4: A] : ord_less_eq(A,X4,X4) ) ).
tff(fact_38_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X4: A] :
( ~ ord_less_eq(A,X4,Y2)
=> ord_less_eq(A,Y2,X4) ) ) ).
tff(fact_39_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [X3: A,G: fun(A,B),F1: fun(A,B)] :
( ord_less_eq(fun(A,B),F1,G)
=> ord_less_eq(B,aa(A,B,F1,X3),aa(A,B,G,X3)) ) ) ).
tff(fact_40_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z2: A,Y2: A,X4: A] :
( ord_less_eq(A,X4,Y2)
=> ( ord_less_eq(A,Y2,Z2)
=> ord_less_eq(A,X4,Z2) ) ) ) ).
tff(fact_41_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X4: A] :
( ord_less_eq(A,X4,Y2)
=> ( ord_less_eq(A,Y2,X4)
=> ( X4 = Y2 ) ) ) ) ).
tff(fact_42_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A3: A] :
( ord_less_eq(A,A3,B2)
=> ( ( B2 = C2 )
=> ord_less_eq(A,A3,C2) ) ) ) ).
tff(fact_43_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B2: A,A3: A] :
( ( A3 = B2 )
=> ( ord_less_eq(A,B2,C2)
=> ord_less_eq(A,A3,C2) ) ) ) ).
tff(fact_44_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [X3: A,Y1: A] :
( ord_less_eq(A,Y1,X3)
=> ( ord_less_eq(A,X3,Y1)
<=> ( X3 = Y1 ) ) ) ) ).
tff(fact_45_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [X3: A,G: fun(A,B),F1: fun(A,B)] :
( ord_less_eq(fun(A,B),F1,G)
=> ord_less_eq(B,aa(A,B,F1,X3),aa(A,B,G,X3)) ) ) ).
tff(fact_46_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X4: A] :
( ( X4 = Y2 )
=> ord_less_eq(A,X4,Y2) ) ) ).
tff(fact_47_monoD,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [Y1: A,X3: A,F1: fun(A,B)] :
( order_mono(A,B,F1)
=> ( ord_less_eq(A,X3,Y1)
=> ord_less_eq(B,aa(A,B,F1,X3),aa(A,B,F1,Y1)) ) ) ) ).
tff(fact_48_order__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X3: A] :
( ( X3 = Y1 )
<=> ( ord_less_eq(A,X3,Y1)
& ord_less_eq(A,Y1,X3) ) ) ) ).
tff(fact_49_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X4: A] :
( ord_less_eq(A,X4,Y2)
| ord_less_eq(A,Y2,X4) ) ) ).
tff(fact_50_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F1: fun(A,B)] :
( ord_less_eq(fun(A,B),F1,G)
<=> ! [X: A] : ord_less_eq(B,aa(A,B,F1,X),aa(A,B,G,X)) ) ) ).
tff(fact_51_order__le__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z2: A,Y2: A,X4: A] :
( ord_less_eq(A,X4,Y2)
=> ( ord_less(A,Y2,Z2)
=> ord_less(A,X4,Z2) ) ) ) ).
tff(fact_52_order__less__le__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z2: A,Y2: A,X4: A] :
( ord_less(A,X4,Y2)
=> ( ord_less_eq(A,Y2,Z2)
=> ord_less(A,X4,Z2) ) ) ) ).
tff(fact_53_order__le__neq__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( ord_less_eq(A,A3,B2)
=> ( ( A3 != B2 )
=> ord_less(A,A3,B2) ) ) ) ).
tff(fact_54_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X4: A] :
( ord_less_eq(A,X4,Y2)
=> ( ord_less(A,X4,Y2)
| ( X4 = Y2 ) ) ) ) ).
tff(fact_55_linorder__antisym__conv2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ord_less_eq(A,X3,Y1)
=> ( ~ ord_less(A,X3,Y1)
<=> ( X3 = Y1 ) ) ) ) ).
tff(fact_56_order__less__imp__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X4: A] :
( ord_less(A,X4,Y2)
=> ord_less_eq(A,X4,Y2) ) ) ).
tff(fact_57_leD,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X4: A,Y2: A] :
( ord_less_eq(A,Y2,X4)
=> ~ ord_less(A,X4,Y2) ) ) ).
tff(fact_58_order__neq__le__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A3: A] :
( ( A3 != B2 )
=> ( ord_less_eq(A,A3,B2)
=> ord_less(A,A3,B2) ) ) ) ).
tff(fact_59_linorder__antisym__conv1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ~ ord_less(A,X3,Y1)
=> ( ord_less_eq(A,X3,Y1)
<=> ( X3 = Y1 ) ) ) ) ).
tff(fact_60_not__leE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X4: A,Y2: A] :
( ~ ord_less_eq(A,Y2,X4)
=> ord_less(A,X4,Y2) ) ) ).
tff(fact_61_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X4: A] :
( ~ ord_less(A,X4,Y2)
=> ord_less_eq(A,Y2,X4) ) ) ).
tff(fact_62_order__le__less,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X3: A] :
( ord_less_eq(A,X3,Y1)
<=> ( ord_less(A,X3,Y1)
| ( X3 = Y1 ) ) ) ) ).
tff(fact_63_less__le__not__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X3: A] :
( ord_less(A,X3,Y1)
<=> ( ord_less_eq(A,X3,Y1)
& ~ ord_less_eq(A,Y1,X3) ) ) ) ).
tff(fact_64_order__less__le,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X3: A] :
( ord_less(A,X3,Y1)
<=> ( ord_less_eq(A,X3,Y1)
& ( X3 != Y1 ) ) ) ) ).
tff(fact_65_linorder__le__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X4: A] :
( ord_less_eq(A,X4,Y2)
| ord_less(A,Y2,X4) ) ) ).
tff(fact_66_linorder__not__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ~ ord_less_eq(A,X3,Y1)
<=> ord_less(A,Y1,X3) ) ) ).
tff(fact_67_linorder__not__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X3: A] :
( ~ ord_less(A,X3,Y1)
<=> ord_less_eq(A,Y1,X3) ) ) ).
tff(fact_68_monoI,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F1: fun(A,B)] :
( ! [X2: A,Y: A] :
( ord_less_eq(A,X2,Y)
=> ord_less_eq(B,aa(A,B,F1,X2),aa(A,B,F1,Y)) )
=> order_mono(A,B,F1) ) ) ).
tff(fact_69_dense__le,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Z2: A,Y2: A] :
( ! [X2: A] :
( ord_less(A,X2,Y2)
=> ord_less_eq(A,X2,Z2) )
=> ord_less_eq(A,Y2,Z2) ) ) ).
tff(fact_70_dense__le__bounded,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Z2: A,Y2: A,X4: A] :
( ord_less(A,X4,Y2)
=> ( ! [W: A] :
( ord_less(A,X4,W)
=> ( ord_less(A,W,Y2)
=> ord_less_eq(A,W,Z2) ) )
=> ord_less_eq(A,Y2,Z2) ) ) ) ).
tff(fact_71_order__less__le__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,F1: fun(A,B),B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ( ord_less_eq(B,aa(A,B,F1,B1),C1)
=> ( ! [X2: A,Y: A] :
( ord_less(A,X2,Y)
=> ord_less(B,aa(A,B,F1,X2),aa(A,B,F1,Y)) )
=> ord_less(B,aa(A,B,F1,A1),C1) ) ) ) ) ).
tff(fact_72_order__le__less__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,F1: fun(A,B),B1: A,A1: A] :
( ord_less_eq(A,A1,B1)
=> ( ord_less(B,aa(A,B,F1,B1),C1)
=> ( ! [X2: A,Y: A] :
( ord_less_eq(A,X2,Y)
=> ord_less_eq(B,aa(A,B,F1,X2),aa(A,B,F1,Y)) )
=> ord_less(B,aa(A,B,F1,A1),C1) ) ) ) ) ).
tff(fact_73_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F1: fun(A,B)] :
( ord_less(fun(A,B),F1,G)
<=> ( ord_less_eq(fun(A,B),F1,G)
& ~ ord_less_eq(fun(A,B),G,F1) ) ) ) ).
tff(fact_74_le__funI,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F1: fun(A,B)] :
( ! [X2: A] : ord_less_eq(B,aa(A,B,F1,X2),aa(A,B,G,X2))
=> ord_less_eq(fun(A,B),F1,G) ) ) ).
tff(fact_75_mem__def,axiom,
! [A: $tType,A2: fun(A,bool),X3: A] :
( member(A,X3,A2)
<=> pp(aa(A,bool,A2,X3)) ) ).
tff(fact_76_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,B1: B,F1: fun(B,A),A1: A] :
( ord_less_eq(A,A1,aa(B,A,F1,B1))
=> ( ord_less(B,B1,C1)
=> ( ! [X2: B,Y: B] :
( ord_less(B,X2,Y)
=> ord_less(A,aa(B,A,F1,X2),aa(B,A,F1,Y)) )
=> ord_less(A,A1,aa(B,A,F1,C1)) ) ) ) ) ).
tff(fact_77_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,B1: B,F1: fun(B,A),A1: A] :
( ord_less(A,A1,aa(B,A,F1,B1))
=> ( ord_less_eq(B,B1,C1)
=> ( ! [X2: B,Y: B] :
( ord_less_eq(B,X2,Y)
=> ord_less_eq(A,aa(B,A,F1,X2),aa(B,A,F1,Y)) )
=> ord_less(A,A1,aa(B,A,F1,C1)) ) ) ) ) ).
tff(fact_78_linordered__field__no__ub,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X1: A] :
? [X11: A] : ord_less(A,X1,X11) ) ).
tff(fact_79_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C1: B,F1: fun(A,B),B1: A,A1: A] :
( ord_less_eq(A,A1,B1)
=> ( ( aa(A,B,F1,B1) = C1 )
=> ( ! [X2: A,Y: A] :
( ord_less_eq(A,X2,Y)
=> ord_less_eq(B,aa(A,B,F1,X2),aa(A,B,F1,Y)) )
=> ord_less_eq(B,aa(A,B,F1,A1),C1) ) ) ) ) ).
tff(fact_80_order__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,F1: fun(A,B),B1: A,A1: A] :
( ord_less_eq(A,A1,B1)
=> ( ord_less_eq(B,aa(A,B,F1,B1),C1)
=> ( ! [X2: A,Y: A] :
( ord_less_eq(A,X2,Y)
=> ord_less_eq(B,aa(A,B,F1,X2),aa(A,B,F1,Y)) )
=> ord_less_eq(B,aa(A,B,F1,A1),C1) ) ) ) ) ).
tff(fact_81_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,B1: B,F1: fun(B,A),A1: A] :
( ord_less_eq(A,A1,aa(B,A,F1,B1))
=> ( ord_less_eq(B,B1,C1)
=> ( ! [X2: B,Y: B] :
( ord_less_eq(B,X2,Y)
=> ord_less_eq(A,aa(B,A,F1,X2),aa(B,A,F1,Y)) )
=> ord_less_eq(A,A1,aa(B,A,F1,C1)) ) ) ) ) ).
tff(fact_82_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C1: B,B1: B,F1: fun(B,A),A1: A] :
( ( A1 = aa(B,A,F1,B1) )
=> ( ord_less_eq(B,B1,C1)
=> ( ! [X2: B,Y: B] :
( ord_less_eq(B,X2,Y)
=> ord_less_eq(A,aa(B,A,F1,X2),aa(B,A,F1,Y)) )
=> ord_less_eq(A,A1,aa(B,A,F1,C1)) ) ) ) ) ).
tff(fact_83_less__eq__real__def,axiom,
! [Y1: real,X3: real] :
( ord_less_eq(real,X3,Y1)
<=> ( ord_less(real,X3,Y1)
| ( X3 = Y1 ) ) ) ).
tff(fact_84_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_85_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_86_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_87_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_88_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ord(C)
=> ! [F: D] :
? [Z: C] :
! [X1: C] :
( ord_less(C,X1,Z)
=> ( F = F ) ) ) ).
tff(fact_89_pinf_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Q1: fun(A,bool),Q: fun(A,bool),P: fun(A,bool),Pa: fun(A,bool)] :
( ? [Z1: A] :
! [X2: A] :
( ord_less(A,Z1,X2)
=> ( pp(aa(A,bool,Pa,X2))
<=> pp(aa(A,bool,P,X2)) ) )
=> ( ? [Z1: A] :
! [X2: A] :
( ord_less(A,Z1,X2)
=> ( pp(aa(A,bool,Q,X2))
<=> pp(aa(A,bool,Q1,X2)) ) )
=> ? [Z: A] :
! [X1: A] :
( ord_less(A,Z,X1)
=> ( ( pp(aa(A,bool,Pa,X1))
| pp(aa(A,bool,Q,X1)) )
<=> ( pp(aa(A,bool,P,X1))
| pp(aa(A,bool,Q1,X1)) ) ) ) ) ) ) ).
tff(fact_90_pinf_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Q1: fun(A,bool),Q: fun(A,bool),P: fun(A,bool),Pa: fun(A,bool)] :
( ? [Z1: A] :
! [X2: A] :
( ord_less(A,Z1,X2)
=> ( pp(aa(A,bool,Pa,X2))
<=> pp(aa(A,bool,P,X2)) ) )
=> ( ? [Z1: A] :
! [X2: A] :
( ord_less(A,Z1,X2)
=> ( pp(aa(A,bool,Q,X2))
<=> pp(aa(A,bool,Q1,X2)) ) )
=> ? [Z: A] :
! [X1: A] :
( ord_less(A,Z,X1)
=> ( ( pp(aa(A,bool,Pa,X1))
& pp(aa(A,bool,Q,X1)) )
<=> ( pp(aa(A,bool,P,X1))
& pp(aa(A,bool,Q1,X1)) ) ) ) ) ) ) ).
tff(fact_91_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ord(C)
=> ! [F: D] :
? [Z: C] :
! [X1: C] :
( ord_less(C,Z,X1)
=> ( F = F ) ) ) ).
tff(fact_92_minf_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z: A] :
! [X1: A] :
( ord_less(A,X1,Z)
=> ( X1 != T ) ) ) ).
tff(fact_93_minf_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Q1: fun(A,bool),Q: fun(A,bool),P: fun(A,bool),Pa: fun(A,bool)] :
( ? [Z1: A] :
! [X2: A] :
( ord_less(A,X2,Z1)
=> ( pp(aa(A,bool,Pa,X2))
<=> pp(aa(A,bool,P,X2)) ) )
=> ( ? [Z1: A] :
! [X2: A] :
( ord_less(A,X2,Z1)
=> ( pp(aa(A,bool,Q,X2))
<=> pp(aa(A,bool,Q1,X2)) ) )
=> ? [Z: A] :
! [X1: A] :
( ord_less(A,X1,Z)
=> ( ( pp(aa(A,bool,Pa,X1))
| pp(aa(A,bool,Q,X1)) )
<=> ( pp(aa(A,bool,P,X1))
| pp(aa(A,bool,Q1,X1)) ) ) ) ) ) ) ).
tff(fact_94_minf_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Q1: fun(A,bool),Q: fun(A,bool),P: fun(A,bool),Pa: fun(A,bool)] :
( ? [Z1: A] :
! [X2: A] :
( ord_less(A,X2,Z1)
=> ( pp(aa(A,bool,Pa,X2))
<=> pp(aa(A,bool,P,X2)) ) )
=> ( ? [Z1: A] :
! [X2: A] :
( ord_less(A,X2,Z1)
=> ( pp(aa(A,bool,Q,X2))
<=> pp(aa(A,bool,Q1,X2)) ) )
=> ? [Z: A] :
! [X1: A] :
( ord_less(A,X1,Z)
=> ( ( pp(aa(A,bool,Pa,X1))
& pp(aa(A,bool,Q,X1)) )
<=> ( pp(aa(A,bool,P,X1))
& pp(aa(A,bool,Q1,X1)) ) ) ) ) ) ) ).
tff(fact_95_pinf_I4_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z: A] :
! [X1: A] :
( ord_less(A,Z,X1)
=> ( X1 != T ) ) ) ).
tff(fact_96_minf_I7_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z: A] :
! [X1: A] :
( ord_less(A,X1,Z)
=> ~ ord_less(A,T,X1) ) ) ).
tff(fact_97_pinf_I5_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z: A] :
! [X1: A] :
( ord_less(A,Z,X1)
=> ~ ord_less(A,X1,T) ) ) ).
tff(fact_98_pinf_I3_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z: A] :
! [X1: A] :
( ord_less(A,Z,X1)
=> ( X1 != T ) ) ) ).
tff(fact_99_pinf_I7_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [T: A] :
? [Z: A] :
! [X1: A] :
( ord_less(A,Z,X1)
=> ord_less(A,T,X1) ) ) ).
%----Arities (14)
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_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) ).
tff(arity_RealDef_Oreal___Orderings_Odense__linorder,axiom,
dense_linorder(real) ).
tff(arity_RealDef_Oreal___Fields_Olinordered__field,axiom,
linordered_field(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom(real) ).
tff(arity_RealDef_Oreal___Orderings_Opreorder,axiom,
preorder(real) ).
tff(arity_RealDef_Oreal___Orderings_Olinorder,axiom,
linorder(real) ).
tff(arity_RealDef_Oreal___Orderings_Oorder,axiom,
order(real) ).
tff(arity_RealDef_Oreal___Orderings_Oord,axiom,
ord(real) ).
%----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,
? [S: real] :
! [Y: real] :
( ? [X: real] :
( p(X)
& ord_less(real,Y,X) )
<=> ord_less(real,Y,S) ) ).
%------------------------------------------------------------------------------