TPTP Problem File: SWW494_5.p
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%------------------------------------------------------------------------------
% File : SWW494_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra line 162
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fta_162 [Bla13]
% Status : Theorem
% Rating : 0.00 v6.4.0
% Syntax : Number of formulae : 154 ( 28 unt; 32 typ; 0 def)
% Number of atoms : 347 ( 38 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 254 ( 29 ~; 8 |; 16 &)
% ( 23 <=>; 178 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 31 ( 16 >; 15 *; 0 +; 0 <<)
% Number of predicates : 19 ( 18 usr; 1 prp; 0-6 aty)
% Number of functors : 12 ( 12 usr; 5 con; 0-4 aty)
% Number of variables : 385 ( 339 !; 20 ?; 385 :)
% ( 26 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:16:16
%------------------------------------------------------------------------------
%----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 (29)
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Otop,type,
top:
!>[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_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_Lubs_OisLub,type,
isLub:
!>[A: $tType] : ( ( fun(A,bool) * fun(A,bool) * A ) > $o ) ).
tff(sy_c_Lubs_OisUb,type,
isUb:
!>[A: $tType] : ( ( fun(A,bool) * fun(A,bool) ) > fun(A,bool) ) ).
tff(sy_c_Lubs_OleastP,type,
leastP:
!>[A: $tType] : ( ( fun(A,bool) * A ) > $o ) ).
tff(sy_c_Lubs_Osetge,type,
setge:
!>[A: $tType] : ( ( A * fun(A,bool) ) > $o ) ).
tff(sy_c_Lubs_Osetle,type,
setle:
!>[A: $tType] : ( ( fun(A,bool) * A ) > $o ) ).
tff(sy_c_Lubs_Oubs,type,
ubs:
!>[A: $tType] : ( ( fun(A,bool) * fun(A,bool) ) > 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_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).
tff(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
tff(sy_c_Predicate_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( fun(B,fun(B,bool)) * fun(A,B) * A * A ) > $o ) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
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: fun(real,bool) ).
tff(sy_v_Y____,type,
y: real ).
tff(sy_v_thesis____,type,
thesis: $o ).
tff(sy_v_x____,type,
x: real ).
%----Relevant facts (100)
tff(fact_0_x,axiom,
pp(aa(real,bool,p,x)) ).
tff(fact_1_ex,axiom,
? [X11: real] : pp(aa(real,bool,p,X11)) ).
tff(fact_2__096EX_At_O_AisLub_AUNIV_A_ICollect_AP_J_At_096,axiom,
? [X11: real] : isLub(real,top_top(fun(real,bool)),collect(real,p),X11) ).
tff(fact_3_thx,axiom,
? [X1: real] : member(real,X1,collect(real,p)) ).
tff(fact_4_bz,axiom,
? [Z1: real] :
! [X3: real] :
( pp(aa(real,bool,p,X3))
=> ord_less(real,X3,Z1) ) ).
tff(fact_5_thY,axiom,
? [X11: real] : pp(aa(real,bool,isUb(real,top_top(fun(real,bool)),collect(real,p)),X11)) ).
tff(fact_6_top1I,axiom,
! [A: $tType,Xa: A] : pp(aa(A,bool,top_top(fun(A,bool)),Xa)) ).
tff(fact_7_UNIV__I,axiom,
! [A: $tType,Xa: A] : member(A,Xa,top_top(fun(A,bool))) ).
tff(fact_8_iso__tuple__UNIV__I,axiom,
! [A: $tType,Xa: A] : member(A,Xa,top_top(fun(A,bool))) ).
tff(fact_9_top__apply,axiom,
! [B: $tType,A: $tType] :
( top(A)
=> ! [Xa: B] : aa(B,A,top_top(fun(B,A)),Xa) = top_top(A) ) ).
tff(fact_10_real__isLub__unique,axiom,
! [Y1: real,Xa: real,S: fun(real,bool),R: fun(real,bool)] :
( isLub(real,R,S,Xa)
=> ( isLub(real,R,S,Y1)
=> ( Xa = Y1 ) ) ) ).
tff(fact_11_isLubD1a,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xa: A,S: fun(A,bool),R: fun(A,bool)] :
( isLub(A,R,S,Xa)
=> member(A,Xa,R) ) ) ).
tff(fact_12_Y,axiom,
! [X: real] :
( pp(aa(real,bool,p,X))
=> ord_less(real,X,y) ) ).
tff(fact_13_UNIV__def,axiom,
! [A: $tType] : top_top(fun(A,bool)) = collect(A,combk(bool,A,fTrue)) ).
tff(fact_14__096_B_Bthesis_O_A_I_B_Bx_AY_O_A_091_124_AP_Ax_059_A_B_Bx_O_AP_Ax_A_061_061_062_Ax_A_060_AY_A_124_093_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ( ? [X11: real] : pp(aa(real,bool,p,X11))
=> ! [Y4: real] :
~ ! [X3: real] :
( pp(aa(real,bool,p,X3))
=> ord_less(real,X3,Y4) ) ) ).
tff(fact_15_lemma__reals__complete,axiom,
! [S: fun(real,bool)] :
( ( ? [X3: real] : member(real,X3,S)
& ? [X12: real] : pp(aa(real,bool,isUb(real,top_top(fun(real,bool)),S),X12)) )
=> ? [X11: real] : isLub(real,top_top(fun(real,bool)),S,X11) ) ).
tff(fact_16_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ~ ord_less(A,X,X) ) ).
tff(fact_17_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,Xa: A] :
( ( Xa != Y1 )
<=> ( ord_less(A,Xa,Y1)
| ord_less(A,Y1,Xa) ) ) ) ).
tff(fact_18_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,Xa: A] :
( ~ ord_less(A,Xa,Y1)
<=> ( ord_less(A,Y1,Xa)
| ( Xa = Y1 ) ) ) ) ).
tff(fact_19_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_20_linorder__antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xa: A,Y1: A] :
( ~ ord_less(A,Y1,Xa)
=> ( ~ ord_less(A,Xa,Y1)
<=> ( Xa = Y1 ) ) ) ) ).
tff(fact_21_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_22_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ( X != Y ) ) ) ).
tff(fact_23_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_24_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_25_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ( X != Y ) ) ) ).
tff(fact_26_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ( Y != X ) ) ) ).
tff(fact_27_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Pa: bool,Y1: A,Xa: A] :
( ord_less(A,Xa,Y1)
=> ( ord_less(A,Y1,Xa)
=> pp(Pa) ) ) ) ).
tff(fact_28_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_29_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_30_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_31_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_32_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ~ ord_less(A,Y,X) ) ) ).
tff(fact_33_isUbD2a,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xa: A,S: fun(A,bool),R: fun(A,bool)] :
( pp(aa(A,bool,isUb(A,R,S),Xa))
=> member(A,Xa,R) ) ) ).
tff(fact_34_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_35_less__top,axiom,
! [A: $tType] :
( top(A)
=> ! [A2: A] :
( ( A2 != top_top(A) )
<=> ord_less(A,A2,top_top(A)) ) ) ).
tff(fact_36_not__top__less,axiom,
! [A: $tType] :
( top(A)
=> ! [A1: A] : ~ ord_less(A,top_top(A),A1) ) ).
tff(fact_37_isLub__isUb,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xa: A,S: fun(A,bool),R: fun(A,bool)] :
( isLub(A,R,S,Xa)
=> pp(aa(A,bool,isUb(A,R,S),Xa)) ) ) ).
tff(fact_38_reals__complete,axiom,
! [S: fun(real,bool)] :
( ? [X4: real] : member(real,X4,S)
=> ( ? [X12: real] : pp(aa(real,bool,isUb(real,top_top(fun(real,bool)),S),X12))
=> ? [X11: real] : isLub(real,top_top(fun(real,bool)),S,X11) ) ) ).
tff(fact_39_ubs__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [S: fun(A,bool),R: fun(A,bool)] : ubs(A,R,S) = collect(A,isUb(A,R,S)) ) ).
tff(fact_40_isLubI2,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xa: A,S: fun(A,bool),R: fun(A,bool)] :
( pp(aa(A,bool,isUb(A,R,S),Xa))
=> ( setge(A,Xa,collect(A,isUb(A,R,S)))
=> isLub(A,R,S,Xa) ) ) ) ).
tff(fact_41_isLubD3,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xa: A,S: fun(A,bool),R: fun(A,bool)] :
( isLub(A,R,S,Xa)
=> leastP(A,isUb(A,R,S),Xa) ) ) ).
tff(fact_42_isLubI1,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xa: A,S: fun(A,bool),R: fun(A,bool)] :
( leastP(A,isUb(A,R,S),Xa)
=> isLub(A,R,S,Xa) ) ) ).
tff(fact_43_isLub__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xa: A,S: fun(A,bool),R: fun(A,bool)] :
( isLub(A,R,S,Xa)
<=> leastP(A,isUb(A,R,S),Xa) ) ) ).
tff(fact_44_UNIV__eq__I,axiom,
! [A: $tType,A3: fun(A,bool)] :
( ! [X1: A] : member(A,X1,A3)
=> ( top_top(fun(A,bool)) = A3 ) ) ).
tff(fact_45_UNIV__witness,axiom,
! [A: $tType] :
? [X1: A] : member(A,X1,top_top(fun(A,bool))) ).
tff(fact_46_leastP__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xa: A,Pa: fun(A,bool)] :
( leastP(A,Pa,Xa)
<=> ( pp(aa(A,bool,Pa,Xa))
& setge(A,Xa,collect(A,Pa)) ) ) ) ).
tff(fact_47_leastPD1,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xa: A,Pa: fun(A,bool)] :
( leastP(A,Pa,Xa)
=> pp(aa(A,bool,Pa,Xa)) ) ) ).
tff(fact_48_leastPD2,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xa: A,Pa: fun(A,bool)] :
( leastP(A,Pa,Xa)
=> setge(A,Xa,collect(A,Pa)) ) ) ).
tff(fact_49_isLub__ubs,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xa: A,S: fun(A,bool),R: fun(A,bool)] :
( isLub(A,R,S,Xa)
=> setge(A,Xa,ubs(A,R,S)) ) ) ).
tff(fact_50_posreal__complete,axiom,
! [Pa: fun(real,bool)] :
( ? [X3: real] : member(real,X3,Pa)
=> ( ? [Y3: real] :
! [X1: real] :
( member(real,X1,Pa)
=> ord_less(real,X1,Y3) )
=> ? [S1: real] :
! [Y3: real] :
( ? [X2: real] :
( member(real,X2,Pa)
& ord_less(real,Y3,X2) )
<=> ord_less(real,Y3,S1) ) ) ) ).
tff(fact_51_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,B2: B,F: fun(B,A),A2: A] :
( ord_less(A,A2,aa(B,A,F,B2))
=> ( ord_less(B,B2,C1)
=> ( ! [X1: B,Y2: B] :
( ord_less(B,X1,Y2)
=> ord_less(A,aa(B,A,F,X1),aa(B,A,F,Y2)) )
=> ord_less(A,A2,aa(B,A,F,C1)) ) ) ) ) ).
tff(fact_52_ord__eq__less__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(B,B2,C1)
=> ( ! [X1: B,Y2: B] :
( ord_less(B,X1,Y2)
=> ord_less(A,aa(B,A,F,X1),aa(B,A,F,Y2)) )
=> ord_less(A,A2,aa(B,A,F,C1)) ) ) ) ) ).
tff(fact_53_order__less__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [C1: B,F: fun(A,B),B2: A,A2: A] :
( ord_less(A,A2,B2)
=> ( ord_less(B,aa(A,B,F,B2),C1)
=> ( ! [X1: A,Y2: A] :
( ord_less(A,X1,Y2)
=> ord_less(B,aa(A,B,F,X1),aa(A,B,F,Y2)) )
=> ord_less(B,aa(A,B,F,A2),C1) ) ) ) ) ).
tff(fact_54_gt__ex,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Xa: A] :
? [X11: A] : ord_less(A,Xa,X11) ) ).
tff(fact_55_lt__ex,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [X: A] :
? [Y2: A] : ord_less(A,Y2,X) ) ).
tff(fact_56_dense,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ? [Z1: A] :
( ord_less(A,X,Z1)
& ord_less(A,Z1,Y) ) ) ) ).
tff(fact_57_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ord(B)
& ord(A) )
=> ! [C1: B,F: fun(A,B),B2: A,A2: A] :
( ord_less(A,A2,B2)
=> ( ( aa(A,B,F,B2) = C1 )
=> ( ! [X1: A,Y2: A] :
( ord_less(A,X1,Y2)
=> ord_less(B,aa(A,B,F,X1),aa(A,B,F,Y2)) )
=> ord_less(B,aa(A,B,F,A2),C1) ) ) ) ) ).
tff(fact_58_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y: A,X: A] :
( ( X != Y )
=> ( ~ ord_less(A,X,Y)
=> ord_less(A,Y,X) ) ) ) ).
tff(fact_59_in__inv__imagep,axiom,
! [A: $tType,B: $tType,Y1: B,Xa: B,F: fun(B,A),R1: fun(A,fun(A,bool))] :
( inv_imagep(A,B,R1,F,Xa,Y1)
<=> pp(aa(A,bool,aa(A,fun(A,bool),R1,aa(B,A,F,Xa)),aa(B,A,F,Y1))) ) ).
tff(fact_60_isLubD1,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xa: A,S: fun(A,bool),R: fun(A,bool)] :
( isLub(A,R,S,Xa)
=> setle(A,S,Xa) ) ) ).
tff(fact_61_isUb__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xa: A,S: fun(A,bool),R: fun(A,bool)] :
( pp(aa(A,bool,isUb(A,R,S),Xa))
<=> ( setle(A,S,Xa)
& member(A,Xa,R) ) ) ) ).
tff(fact_62_isUbI,axiom,
! [A: $tType] :
( ord(A)
=> ! [R: fun(A,bool),Xa: A,S: fun(A,bool)] :
( setle(A,S,Xa)
=> ( member(A,Xa,R)
=> pp(aa(A,bool,isUb(A,R,S),Xa)) ) ) ) ).
tff(fact_63_isUbD2,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xa: A,S: fun(A,bool),R: fun(A,bool)] :
( pp(aa(A,bool,isUb(A,R,S),Xa))
=> setle(A,S,Xa) ) ) ).
tff(fact_64_isLub__le__isUb,axiom,
! [A: $tType] :
( ord(A)
=> ! [Y1: A,Xa: A,S: fun(A,bool),R: fun(A,bool)] :
( isLub(A,R,S,Xa)
=> ( pp(aa(A,bool,isUb(A,R,S),Y1))
=> ord_less_eq(A,Xa,Y1) ) ) ) ).
tff(fact_65_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [Y1: A,Xa: A,F: fun(A,B)] :
( order_strict_mono(A,B,F)
=> ( ord_less(B,aa(A,B,F,Xa),aa(A,B,F,Y1))
<=> ord_less(A,Xa,Y1) ) ) ) ).
tff(fact_66_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [Y1: A,Xa: A,F: fun(A,B)] :
( order_strict_mono(A,B,F)
=> ( ord_less(A,Xa,Y1)
=> ord_less(B,aa(A,B,F,Xa),aa(A,B,F,Y1)) ) ) ) ).
tff(fact_67_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ord_less_eq(A,X,X) ) ).
tff(fact_68_setleD,axiom,
! [A: $tType] :
( ord(A)
=> ! [Y1: A,Xa: A,S: fun(A,bool)] :
( setle(A,S,Xa)
=> ( member(A,Y1,S)
=> ord_less_eq(A,Y1,Xa) ) ) ) ).
tff(fact_69_setle__def,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xa: A,S: fun(A,bool)] :
( setle(A,S,Xa)
<=> ! [X2: A] :
( member(A,X2,S)
=> ord_less_eq(A,X2,Xa) ) ) ) ).
tff(fact_70_isUbD,axiom,
! [A: $tType] :
( ord(A)
=> ! [Y1: A,Xa: A,S: fun(A,bool),R: fun(A,bool)] :
( pp(aa(A,bool,isUb(A,R,S),Xa))
=> ( member(A,Y1,S)
=> ord_less_eq(A,Y1,Xa) ) ) ) ).
tff(fact_71_isLubD2,axiom,
! [A: $tType] :
( ord(A)
=> ! [Y1: A,Xa: A,S: fun(A,bool),R: fun(A,bool)] :
( isLub(A,R,S,Xa)
=> ( member(A,Y1,S)
=> ord_less_eq(A,Y1,Xa) ) ) ) ).
tff(fact_72_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)
<=> ! [X2: A] : ord_less_eq(B,aa(A,B,F,X2),aa(A,B,G,X2)) ) ) ).
tff(fact_73_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_74_order__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,Xa: A] :
( ( Xa = Y1 )
<=> ( ord_less_eq(A,Xa,Y1)
& ord_less_eq(A,Y1,Xa) ) ) ) ).
tff(fact_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X1: A] : aa(A,B,F,X1) = aa(A,B,G,X1)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A3: fun(A,bool),Xa: A] :
( member(A,Xa,A3)
<=> pp(aa(A,bool,A3,Xa)) ) ).
tff(fact_77_Collect__def,axiom,
! [A: $tType,Pa: fun(A,bool)] : collect(A,Pa) = Pa ).
tff(fact_78_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [Y1: A,Xa: A,F: fun(A,B)] :
( order_strict_mono(A,B,F)
=> ( ( aa(A,B,F,Xa) = aa(A,B,F,Y1) )
<=> ( Xa = Y1 ) ) ) ) ).
tff(fact_79_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [Y1: A,Xa: A,F: fun(A,B)] :
( order_strict_mono(A,B,F)
=> ( ord_less_eq(B,aa(A,B,F,Xa),aa(A,B,F,Y1))
<=> ord_less_eq(A,Xa,Y1) ) ) ) ).
tff(fact_80_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ( X = Y )
=> ord_less_eq(A,X,Y) ) ) ).
tff(fact_81_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [Xa: 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,Xa),aa(A,B,G,Xa)) ) ) ).
tff(fact_82_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [Xa: A,Y1: A] :
( ord_less_eq(A,Y1,Xa)
=> ( ord_less_eq(A,Xa,Y1)
<=> ( Xa = Y1 ) ) ) ) ).
tff(fact_83_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_84_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_85_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_86_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_87_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [Xa: 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,Xa),aa(A,B,G,Xa)) ) ) ).
tff(fact_88_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_89_subset__UNIV,axiom,
! [A: $tType,A3: fun(A,bool)] : ord_less_eq(fun(A,bool),A3,top_top(fun(A,bool))) ).
tff(fact_90_top__le,axiom,
! [A: $tType] :
( top(A)
=> ! [A1: A] :
( ord_less_eq(A,top_top(A),A1)
=> ( A1 = top_top(A) ) ) ) ).
tff(fact_91_top__unique,axiom,
! [A: $tType] :
( top(A)
=> ! [A2: A] :
( ord_less_eq(A,top_top(A),A2)
<=> ( A2 = top_top(A) ) ) ) ).
tff(fact_92_top__greatest,axiom,
! [A: $tType] :
( top(A)
=> ! [A1: A] : ord_less_eq(A,A1,top_top(A)) ) ).
tff(fact_93_linorder__not__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,Xa: A] :
( ~ ord_less(A,Xa,Y1)
<=> ord_less_eq(A,Y1,Xa) ) ) ).
tff(fact_94_linorder__not__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,Xa: A] :
( ~ ord_less_eq(A,Xa,Y1)
<=> ord_less(A,Y1,Xa) ) ) ).
tff(fact_95_linorder__le__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,X,Y)
| ord_less(A,Y,X) ) ) ).
tff(fact_96_order__less__le,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,Xa: A] :
( ord_less(A,Xa,Y1)
<=> ( ord_less_eq(A,Xa,Y1)
& ( Xa != Y1 ) ) ) ) ).
tff(fact_97_less__le__not__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,Xa: A] :
( ord_less(A,Xa,Y1)
<=> ( ord_less_eq(A,Xa,Y1)
& ~ ord_less_eq(A,Y1,Xa) ) ) ) ).
tff(fact_98_order__le__less,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,Xa: A] :
( ord_less_eq(A,Xa,Y1)
<=> ( ord_less(A,Xa,Y1)
| ( Xa = Y1 ) ) ) ) ).
tff(fact_99_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ ord_less(A,X,Y)
=> ord_less_eq(A,Y,X) ) ) ).
%----Arities (15)
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_Otop,axiom,
! [T_1: $tType,T_2: $tType] :
( top(T_2)
=> top(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_Otop,axiom,
top(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___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 (5)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,combk(A,B,P),Q) = P ).
tff(help_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
%----Conjectures (2)
tff(conj_0,hypothesis,
! [L: real] :
( isLub(real,top_top(fun(real,bool)),collect(real,p),L)
=> thesis ) ).
tff(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------