TPTP Problem File: SWW495_5.p
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%------------------------------------------------------------------------------
% File : SWW495_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra line 170
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fta_170 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.25 v7.1.0, 0.33 v6.4.0
% Syntax : Number of formulae : 147 ( 33 unt; 27 typ; 0 def)
% Number of atoms : 320 ( 42 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 239 ( 39 ~; 10 |; 10 &)
% ( 20 <=>; 160 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 19 ( 10 >; 9 *; 0 +; 0 <<)
% Number of predicates : 13 ( 12 usr; 0 prp; 1-4 aty)
% Number of functors : 13 ( 13 usr; 8 con; 0-4 aty)
% Number of variables : 327 ( 304 !; 6 ?; 327 :)
% ( 17 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:16:41
%------------------------------------------------------------------------------
%----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 (24)
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_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) * A ) > $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_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
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_L____,type,
l: real ).
tff(sy_v_P,type,
p: fun(real,bool) ).
tff(sy_v_Y____,type,
y1: real ).
tff(sy_v_x____,type,
x: real ).
tff(sy_v_y____,type,
y: real ).
tff(sy_v_z____,type,
z: real ).
%----Relevant facts (100)
tff(fact_0_x,axiom,
pp(aa(real,bool,p,x)) ).
tff(fact_1_bz,axiom,
? [Z1: real] :
! [X4: real] :
( pp(aa(real,bool,p,X4))
=> ord_less(real,X4,Z1) ) ).
tff(fact_2_L_H,axiom,
! [X4: real] :
( pp(aa(real,bool,p,X4))
=> ord_less_eq(real,X4,l) ) ).
tff(fact_3_z_I2_J,axiom,
ord_less(real,y,z) ).
tff(fact_4_thx,axiom,
? [X: real] : member(real,X,collect(real,p)) ).
tff(fact_5__096L_A_060_061_AY_096,axiom,
ord_less_eq(real,l,y1) ).
tff(fact_6_Y,axiom,
! [X2: real] :
( pp(aa(real,bool,p,X2))
=> ord_less(real,X2,y1) ) ).
tff(fact_7__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,
~ ( ? [X1: real] : pp(aa(real,bool,p,X1))
=> ! [Y2: real] :
~ ! [X4: real] :
( pp(aa(real,bool,p,X4))
=> ord_less(real,X4,Y2) ) ) ).
tff(fact_8_xY,axiom,
ord_less(real,x,y1) ).
tff(fact_9_L,axiom,
isLub(real,top_top(fun(real,bool)),collect(real,p),l) ).
tff(fact_10_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A] : ~ ord_less(A,X2,X2) ) ).
tff(fact_11_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Ya: A,Xa: A] :
( ( Xa != Ya )
<=> ( ord_less(A,Xa,Ya)
| ord_less(A,Ya,Xa) ) ) ) ).
tff(fact_12_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Ya: A,Xa: A] :
( ~ ord_less(A,Xa,Ya)
<=> ( ord_less(A,Ya,Xa)
| ( Xa = Ya ) ) ) ) ).
tff(fact_13_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_14_linorder__antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xa: A,Ya: A] :
( ~ ord_less(A,Ya,Xa)
=> ( ~ ord_less(A,Xa,Ya)
<=> ( Xa = Ya ) ) ) ) ).
tff(fact_15_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_16_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_17_ex,axiom,
? [X1: real] : pp(aa(real,bool,p,X1)) ).
tff(fact_18_z_I1_J,axiom,
pp(aa(real,bool,p,z)) ).
tff(fact_19_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A] : ord_less_eq(A,X2,X2) ) ).
tff(fact_20_Y_H,axiom,
! [X4: real] :
( pp(aa(real,bool,p,X4))
=> ord_less_eq(real,X4,y1) ) ).
tff(fact_21__096EX_At_O_AisLub_AUNIV_A_ICollect_AP_J_At_096,axiom,
? [X1: real] : isLub(real,top_top(fun(real,bool)),collect(real,p),X1) ).
tff(fact_22__096_B_Bthesis_O_A_I_B_BL_O_AisLub_AUNIV_A_ICollect_AP_J_AL_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [L: real] : ~ isLub(real,top_top(fun(real,bool)),collect(real,p),L) ).
tff(fact_23_top__greatest,axiom,
! [A: $tType] :
( top(A)
=> ! [A3: A] : ord_less_eq(A,A3,top_top(A)) ) ).
tff(fact_24_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_25_top__unique,axiom,
! [A: $tType] :
( top(A)
=> ! [A2: A] :
( ord_less_eq(A,top_top(A),A2)
<=> ( A2 = top_top(A) ) ) ) ).
tff(fact_26_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_27_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_28_order__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [Ya: A,Xa: A] :
( ( Xa = Ya )
<=> ( ord_less_eq(A,Xa,Ya)
& ord_less_eq(A,Ya,Xa) ) ) ) ).
tff(fact_29_top__le,axiom,
! [A: $tType] :
( top(A)
=> ! [A3: A] :
( ord_less_eq(A,top_top(A),A3)
=> ( A3 = top_top(A) ) ) ) ).
tff(fact_30_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X2: A] :
( ( X2 = Y1 )
=> ord_less_eq(A,X2,Y1) ) ) ).
tff(fact_31_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_32_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [Xa: A,Ya: A] :
( ord_less_eq(A,Ya,Xa)
=> ( ord_less_eq(A,Xa,Ya)
<=> ( Xa = Ya ) ) ) ) ).
tff(fact_33_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B3: A,A3: A] :
( ( A3 = B3 )
=> ( ord_less_eq(A,B3,C2)
=> ord_less_eq(A,A3,C2) ) ) ) ).
tff(fact_34_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B3: A,A3: A] :
( ord_less_eq(A,A3,B3)
=> ( ( B3 = C2 )
=> ord_less_eq(A,A3,C2) ) ) ) ).
tff(fact_35_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_36_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_37_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_38_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_39_less__top,axiom,
! [A: $tType] :
( top(A)
=> ! [A2: A] :
( ( A2 != top_top(A) )
<=> ord_less(A,A2,top_top(A)) ) ) ).
tff(fact_40_not__top__less,axiom,
! [A: $tType] :
( top(A)
=> ! [A3: A] : ~ ord_less(A,top_top(A),A3) ) ).
tff(fact_41_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_42_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_43_order__le__neq__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B3: A,A3: A] :
( ord_less_eq(A,A3,B3)
=> ( ( A3 != B3 )
=> ord_less(A,A3,B3) ) ) ) ).
tff(fact_44_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_45_linorder__antisym__conv2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Ya: A,Xa: A] :
( ord_less_eq(A,Xa,Ya)
=> ( ~ ord_less(A,Xa,Ya)
<=> ( Xa = Ya ) ) ) ) ).
tff(fact_46_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_47_leD,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: A,Y1: A] :
( ord_less_eq(A,Y1,X2)
=> ~ ord_less(A,X2,Y1) ) ) ).
tff(fact_48_order__neq__le__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B3: A,A3: A] :
( ( A3 != B3 )
=> ( ord_less_eq(A,A3,B3)
=> ord_less(A,A3,B3) ) ) ) ).
tff(fact_49_linorder__antisym__conv1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Ya: A,Xa: A] :
( ~ ord_less(A,Xa,Ya)
=> ( ord_less_eq(A,Xa,Ya)
<=> ( Xa = Ya ) ) ) ) ).
tff(fact_50_not__leE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: A,Y1: A] :
( ~ ord_less_eq(A,Y1,X2)
=> ord_less(A,X2,Y1) ) ) ).
tff(fact_51_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X2: A] :
( ~ ord_less(A,X2,Y1)
=> ord_less_eq(A,Y1,X2) ) ) ).
tff(fact_52_order__le__less,axiom,
! [A: $tType] :
( order(A)
=> ! [Ya: A,Xa: A] :
( ord_less_eq(A,Xa,Ya)
<=> ( ord_less(A,Xa,Ya)
| ( Xa = Ya ) ) ) ) ).
tff(fact_53_less__le__not__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Ya: A,Xa: A] :
( ord_less(A,Xa,Ya)
<=> ( ord_less_eq(A,Xa,Ya)
& ~ ord_less_eq(A,Ya,Xa) ) ) ) ).
tff(fact_54_order__less__le,axiom,
! [A: $tType] :
( order(A)
=> ! [Ya: A,Xa: A] :
( ord_less(A,Xa,Ya)
<=> ( ord_less_eq(A,Xa,Ya)
& ( Xa != Ya ) ) ) ) ).
tff(fact_55_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_56_linorder__not__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Ya: A,Xa: A] :
( ~ ord_less_eq(A,Xa,Ya)
<=> ord_less(A,Ya,Xa) ) ) ).
tff(fact_57_linorder__not__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Ya: A,Xa: A] :
( ~ ord_less(A,Xa,Ya)
<=> ord_less_eq(A,Ya,Xa) ) ) ).
tff(fact_58_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_59_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X2: A] :
( ord_less(A,X2,Y1)
=> ~ ord_less(A,Y1,X2) ) ) ).
tff(fact_60_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_61_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B3: A,A3: A] :
( ord_less(A,A3,B3)
=> ( ( B3 = C2 )
=> ord_less(A,A3,C2) ) ) ) ).
tff(fact_62_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C2: A,B3: A,A3: A] :
( ( A3 = B3 )
=> ( ord_less(A,B3,C2)
=> ord_less(A,A3,C2) ) ) ) ).
tff(fact_63_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B3: A,A3: A] :
( ord_less(A,A3,B3)
=> ~ ord_less(A,B3,A3) ) ) ).
tff(fact_64_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Pa: bool,Ya: A,Xa: A] :
( ord_less(A,Xa,Ya)
=> ( ord_less(A,Ya,Xa)
=> pp(Pa) ) ) ) ).
tff(fact_65_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X2: A] :
( ord_less(A,X2,Y1)
=> ( Y1 != X2 ) ) ) ).
tff(fact_66_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X2: A] :
( ord_less(A,X2,Y1)
=> ( X2 != Y1 ) ) ) ).
tff(fact_67_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_68_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_69_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X2: A] :
( ord_less(A,X2,Y1)
=> ( X2 != Y1 ) ) ) ).
tff(fact_70_thY,axiom,
? [X1: real] : isUb(real,top_top(fun(real,bool)),collect(real,p),X1) ).
tff(fact_71_top1I,axiom,
! [A: $tType,Xa: A] : pp(aa(A,bool,top_top(fun(A,bool)),Xa)) ).
tff(fact_72_UNIV__I,axiom,
! [A: $tType,Xa: A] : member(A,Xa,top_top(fun(A,bool))) ).
tff(fact_73_iso__tuple__UNIV__I,axiom,
! [A: $tType,Xa: A] : member(A,Xa,top_top(fun(A,bool))) ).
tff(fact_74_order__le__less__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(B,B2,C1)
=> ( ! [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,A2,aa(B,A,F,C1)) ) ) ) ) ).
tff(fact_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X: A] : aa(A,B,F,X) = aa(A,B,G,X)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),Xa: A] :
( member(A,Xa,A1)
<=> pp(aa(A,bool,A1,Xa)) ) ).
tff(fact_77_Collect__def,axiom,
! [A: $tType,Pa: fun(A,bool)] : collect(A,Pa) = Pa ).
tff(fact_78_order__less__le__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_eq(B,B2,C1)
=> ( ! [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,A2,aa(B,A,F,C1)) ) ) ) ) ).
tff(fact_79_order__le__less__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(B,aa(A,B,F,B2),C1)
=> ( ! [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,A2),C1) ) ) ) ) ).
tff(fact_80_order__less__le__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_eq(B,aa(A,B,F,B2),C1)
=> ( ! [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,A2),C1) ) ) ) ) ).
tff(fact_81_subset__UNIV,axiom,
! [A: $tType,A1: fun(A,bool)] : ord_less_eq(fun(A,bool),A1,top_top(fun(A,bool))) ).
tff(fact_82_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_83_UNIV__def,axiom,
! [A: $tType] : top_top(fun(A,bool)) = collect(A,combk(bool,A,fTrue)) ).
tff(fact_84_isLub__le__isUb,axiom,
! [A: $tType] :
( ord(A)
=> ! [Ya: A,Xa: A,S: fun(A,bool),R: fun(A,bool)] :
( isLub(A,R,S,Xa)
=> ( isUb(A,R,S,Ya)
=> ord_less_eq(A,Xa,Ya) ) ) ) ).
tff(fact_85_isLub__isUb,axiom,
! [A: $tType] :
( ord(A)
=> ! [Xa: A,S: fun(A,bool),R: fun(A,bool)] :
( isLub(A,R,S,Xa)
=> isUb(A,R,S,Xa) ) ) ).
tff(fact_86_subsetD,axiom,
! [A: $tType,C1: A,B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B1)
=> ( member(A,C1,A1)
=> member(A,C1,B1) ) ) ).
tff(fact_87_equalityI,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B1)
=> ( ord_less_eq(fun(A,bool),B1,A1)
=> ( A1 = B1 ) ) ) ).
tff(fact_88_subset__refl,axiom,
! [A: $tType,A1: fun(A,bool)] : ord_less_eq(fun(A,bool),A1,A1) ).
tff(fact_89_psubset__eq,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less(fun(A,bool),A1,B1)
<=> ( ord_less_eq(fun(A,bool),A1,B1)
& ( A1 != B1 ) ) ) ).
tff(fact_90_set__eq__subset,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ( A1 = B1 )
<=> ( ord_less_eq(fun(A,bool),A1,B1)
& ord_less_eq(fun(A,bool),B1,A1) ) ) ).
tff(fact_91_subset__iff__psubset__eq,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B1)
<=> ( ord_less(fun(A,bool),A1,B1)
| ( A1 = B1 ) ) ) ).
tff(fact_92_rev__predicate1D,axiom,
! [A: $tType,Q1: fun(A,bool),Xa: A,Pa: fun(A,bool)] :
( pp(aa(A,bool,Pa,Xa))
=> ( ord_less_eq(fun(A,bool),Pa,Q1)
=> pp(aa(A,bool,Q1,Xa)) ) ) ).
tff(fact_93_equalityD1,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ( A1 = B1 )
=> ord_less_eq(fun(A,bool),A1,B1) ) ).
tff(fact_94_equalityD2,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ( A1 = B1 )
=> ord_less_eq(fun(A,bool),B1,A1) ) ).
tff(fact_95_psubset__imp__subset,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less(fun(A,bool),A1,B1)
=> ord_less_eq(fun(A,bool),A1,B1) ) ).
tff(fact_96_in__mono,axiom,
! [A: $tType,Xa: A,B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less_eq(fun(A,bool),A1,B1)
=> ( member(A,Xa,A1)
=> member(A,Xa,B1) ) ) ).
tff(fact_97_predicate1D,axiom,
! [A: $tType,Xa: A,Q1: fun(A,bool),Pa: fun(A,bool)] :
( ord_less_eq(fun(A,bool),Pa,Q1)
=> ( pp(aa(A,bool,Pa,Xa))
=> pp(aa(A,bool,Q1,Xa)) ) ) ).
tff(fact_98_set__rev__mp,axiom,
! [A: $tType,B1: fun(A,bool),A1: fun(A,bool),Xa: A] :
( member(A,Xa,A1)
=> ( ord_less_eq(fun(A,bool),A1,B1)
=> member(A,Xa,B1) ) ) ).
tff(fact_99_psubset__subset__trans,axiom,
! [A: $tType,C: fun(A,bool),B1: fun(A,bool),A1: fun(A,bool)] :
( ord_less(fun(A,bool),A1,B1)
=> ( ord_less_eq(fun(A,bool),B1,C)
=> ord_less(fun(A,bool),A1,C) ) ) ).
%----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_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___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 (1)
tff(conj_0,conjecture,
ord_less(real,y,l) ).
%------------------------------------------------------------------------------