ITP001 Axioms: ITP017^7.ax
%------------------------------------------------------------------------------
% File : ITP017^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : poset.ax [Gau19]
% : HL4017^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 68 ( 20 unt; 28 typ; 0 def)
% Number of atoms : 109 ( 12 equ; 1 cnn)
% Maximal formula atoms : 8 ( 1 avg)
% Number of connectives : 698 ( 1 ~; 2 |; 64 &; 577 @)
% ( 21 <=>; 33 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 11 avg; 577 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 341 ( 341 >; 0 *; 0 +; 0 <<)
% Number of symbols : 29 ( 27 usr; 1 con; 0-5 aty)
% Number of variables : 216 ( 9 ^ 170 !; 13 ?; 216 :)
% ( 24 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Eposet_2Ebottom,type,
c_2Eposet_2Ebottom:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) ) > A_27a > $o ) ).
thf(c_2Eposet_2Ecarrier,type,
c_2Eposet_2Ecarrier:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) ) > A_27a > $o ) ).
thf(c_2Eposet_2Echain,type,
c_2Eposet_2Echain:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) ) > ( A_27a > $o ) > $o ) ).
thf(c_2Eposet_2Ecomplete,type,
c_2Eposet_2Ecomplete:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) ) > $o ) ).
thf(c_2Eposet_2Econtinuous,type,
c_2Eposet_2Econtinuous:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) ) > ( A_27a > A_27a ) > $o ) ).
thf(c_2Eposet_2Edown__continuous,type,
c_2Eposet_2Edown__continuous:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) ) > ( A_27a > A_27a ) > $o ) ).
thf(c_2Eposet_2Efunction,type,
c_2Eposet_2Efunction:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > ( A_27a > A_27b ) > $o ) ).
thf(c_2Eposet_2Egfp,type,
c_2Eposet_2Egfp:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) ) > ( A_27a > A_27a ) > A_27a > $o ) ).
thf(c_2Eposet_2Eglb,type,
c_2Eposet_2Eglb:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Eposet_2Elfp,type,
c_2Eposet_2Elfp:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) ) > ( A_27a > A_27a ) > A_27a > $o ) ).
thf(c_2Eposet_2Elub,type,
c_2Eposet_2Elub:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Eposet_2Emonotonic,type,
c_2Eposet_2Emonotonic:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) ) > ( A_27a > A_27a ) > $o ) ).
thf(c_2Eposet_2Epointwise__lift,type,
c_2Eposet_2Epointwise__lift:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( tyop_2Epair_2Eprod @ ( A_27b > $o ) @ ( A_27b > A_27b > $o ) ) > ( tyop_2Epair_2Eprod @ ( ( A_27a > A_27b ) > $o ) @ ( ( A_27a > A_27b ) > ( A_27a > A_27b ) > $o ) ) ) ).
thf(c_2Eposet_2Eposet,type,
c_2Eposet_2Eposet:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) ) > $o ) ).
thf(c_2Eposet_2Erelation,type,
c_2Eposet_2Erelation:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) ) > A_27a > A_27a > $o ) ).
thf(c_2Eposet_2Etop,type,
c_2Eposet_2Etop:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) ) > A_27a > $o ) ).
thf(c_2Eposet_2Eup__continuous,type,
c_2Eposet_2Eup__continuous:
!>[A_27a: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) ) > ( A_27a > A_27a ) > $o ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Eposet_2Efunction__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0a: A_27a > $o,V1b: A_27b > $o,V2f: A_27a > A_27b] :
( ( c_2Eposet_2Efunction @ A_27a @ A_27b @ V0a @ V1b @ V2f )
<=> ! [V3x: A_27a] :
( ( V0a @ V3x )
=> ( V1b @ ( V2f @ V3x ) ) ) ) ).
thf(thm_2Eposet_2Eposet__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o] :
( ( c_2Eposet_2Eposet @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) )
<=> ( ? [V2x: A_27a] : ( V0s @ V2x )
& ! [V3x: A_27a] :
( ( V0s @ V3x )
=> ( V1r @ V3x @ V3x ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ( ( V0s @ V4x )
& ( V0s @ V5y )
& ( V1r @ V4x @ V5y )
& ( V1r @ V5y @ V4x ) )
=> ( V4x = V5y ) )
& ! [V6x: A_27a,V7y: A_27a,V8z: A_27a] :
( ( ( V0s @ V6x )
& ( V0s @ V7y )
& ( V0s @ V8z )
& ( V1r @ V6x @ V7y )
& ( V1r @ V7y @ V8z ) )
=> ( V1r @ V6x @ V8z ) ) ) ) ).
thf(thm_2Eposet_2Ecarrier__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o] :
( ( c_2Eposet_2Ecarrier @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) )
= V0s ) ).
thf(thm_2Eposet_2Erelation__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o] :
( ( c_2Eposet_2Erelation @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) )
= V1r ) ).
thf(thm_2Eposet_2Etop__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o,V2x: A_27a] :
( ( c_2Eposet_2Etop @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) @ V2x )
<=> ( ( V0s @ V2x )
& ! [V3y: A_27a] :
( ( V0s @ V3y )
=> ( V1r @ V3y @ V2x ) ) ) ) ).
thf(thm_2Eposet_2Ebottom__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o,V2x: A_27a] :
( ( c_2Eposet_2Ebottom @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) @ V2x )
<=> ( ( V0s @ V2x )
& ! [V3y: A_27a] :
( ( V0s @ V3y )
=> ( V1r @ V2x @ V3y ) ) ) ) ).
thf(thm_2Eposet_2Echain__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o,V2c: A_27a > $o] :
( ( c_2Eposet_2Echain @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) @ V2c )
<=> ! [V3x: A_27a,V4y: A_27a] :
( ( ( V0s @ V3x )
& ( V0s @ V4y )
& ( V2c @ V3x )
& ( V2c @ V4y ) )
=> ( ( V1r @ V3x @ V4y )
| ( V1r @ V4y @ V3x ) ) ) ) ).
thf(thm_2Eposet_2Elub__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o,V2p: A_27a > $o,V3x: A_27a] :
( ( c_2Eposet_2Elub @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) @ V2p @ V3x )
<=> ( ( V0s @ V3x )
& ! [V4y: A_27a] :
( ( ( V0s @ V4y )
& ( V2p @ V4y ) )
=> ( V1r @ V4y @ V3x ) )
& ! [V5z: A_27a] :
( ( ( V0s @ V5z )
& ! [V6y: A_27a] :
( ( ( V0s @ V6y )
& ( V2p @ V6y ) )
=> ( V1r @ V6y @ V5z ) ) )
=> ( V1r @ V3x @ V5z ) ) ) ) ).
thf(thm_2Eposet_2Eglb__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o,V2p: A_27a > $o,V3x: A_27a] :
( ( c_2Eposet_2Eglb @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) @ V2p @ V3x )
<=> ( ( V0s @ V3x )
& ! [V4y: A_27a] :
( ( ( V0s @ V4y )
& ( V2p @ V4y ) )
=> ( V1r @ V3x @ V4y ) )
& ! [V5z: A_27a] :
( ( ( V0s @ V5z )
& ! [V6y: A_27a] :
( ( ( V0s @ V6y )
& ( V2p @ V6y ) )
=> ( V1r @ V5z @ V6y ) ) )
=> ( V1r @ V5z @ V3x ) ) ) ) ).
thf(thm_2Eposet_2Ecomplete__def,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o )] :
( ( c_2Eposet_2Ecomplete @ A_27a @ V0p )
<=> ! [V1c: A_27a > $o] :
( ? [V2x: A_27a] : ( c_2Eposet_2Elub @ A_27a @ V0p @ V1c @ V2x )
& ? [V3x: A_27a] : ( c_2Eposet_2Eglb @ A_27a @ V0p @ V1c @ V3x ) ) ) ).
thf(thm_2Eposet_2Epointwise__lift__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: A_27a > $o,V1s: A_27b > $o,V2r: A_27b > A_27b > $o] :
( ( c_2Eposet_2Epointwise__lift @ A_27a @ A_27b @ V0t @ ( c_2Epair_2E_2C @ ( A_27b > $o ) @ ( A_27b > A_27b > $o ) @ V1s @ V2r ) )
= ( c_2Epair_2E_2C @ ( ( A_27a > A_27b ) > $o ) @ ( ( A_27a > A_27b ) > ( A_27a > A_27b ) > $o ) @ ( c_2Eposet_2Efunction @ A_27a @ A_27b @ V0t @ V1s )
@ ^ [V3f: A_27a > A_27b,V4g: A_27a > A_27b] :
( c_2Ebool_2E_21 @ A_27a
@ ^ [V5x: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( V0t @ V5x ) @ ( V2r @ ( V3f @ V5x ) @ ( V4g @ V5x ) ) ) ) ) ) ).
thf(thm_2Eposet_2Emonotonic__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o,V2f: A_27a > A_27a] :
( ( c_2Eposet_2Emonotonic @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) @ V2f )
<=> ! [V3x: A_27a,V4y: A_27a] :
( ( ( V0s @ V3x )
& ( V0s @ V4y )
& ( V1r @ V3x @ V4y ) )
=> ( V1r @ ( V2f @ V3x ) @ ( V2f @ V4y ) ) ) ) ).
thf(thm_2Eposet_2Eup__continuous__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o,V2f: A_27a > A_27a] :
( ( c_2Eposet_2Eup__continuous @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) @ V2f )
<=> ! [V3c: A_27a > $o,V4x: A_27a] :
( ( ( c_2Eposet_2Echain @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) @ V3c )
& ( c_2Eposet_2Elub @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) @ V3c @ V4x ) )
=> ( c_2Eposet_2Elub @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r )
@ ^ [V5y: A_27a] :
( c_2Ebool_2E_3F @ A_27a
@ ^ [V6z: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2E_2F_5C @ ( V0s @ V6z ) @ ( V3c @ V6z ) ) @ ( c_2Emin_2E_3D @ A_27a @ V5y @ ( V2f @ V6z ) ) ) )
@ ( V2f @ V4x ) ) ) ) ).
thf(thm_2Eposet_2Edown__continuous__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o,V2f: A_27a > A_27a] :
( ( c_2Eposet_2Edown__continuous @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) @ V2f )
<=> ! [V3c: A_27a > $o,V4x: A_27a] :
( ( ( c_2Eposet_2Echain @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) @ V3c )
& ( c_2Eposet_2Eglb @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) @ V3c @ V4x ) )
=> ( c_2Eposet_2Eglb @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r )
@ ^ [V5y: A_27a] :
( c_2Ebool_2E_3F @ A_27a
@ ^ [V6z: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2E_2F_5C @ ( V0s @ V6z ) @ ( V3c @ V6z ) ) @ ( c_2Emin_2E_3D @ A_27a @ V5y @ ( V2f @ V6z ) ) ) )
@ ( V2f @ V4x ) ) ) ) ).
thf(thm_2Eposet_2Econtinuous__def,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ),V1f: A_27a > A_27a] :
( ( c_2Eposet_2Econtinuous @ A_27a @ V0p @ V1f )
<=> ( ( c_2Eposet_2Eup__continuous @ A_27a @ V0p @ V1f )
& ( c_2Eposet_2Edown__continuous @ A_27a @ V0p @ V1f ) ) ) ).
thf(thm_2Eposet_2Elfp__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o,V2f: A_27a > A_27a,V3x: A_27a] :
( ( c_2Eposet_2Elfp @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) @ V2f @ V3x )
<=> ( ( V0s @ V3x )
& ( ( V2f @ V3x )
= V3x )
& ! [V4y: A_27a] :
( ( ( V0s @ V4y )
& ( V1r @ ( V2f @ V4y ) @ V4y ) )
=> ( V1r @ V3x @ V4y ) ) ) ) ).
thf(thm_2Eposet_2Egfp__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o,V2f: A_27a > A_27a,V3x: A_27a] :
( ( c_2Eposet_2Egfp @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) @ V2f @ V3x )
<=> ( ( V0s @ V3x )
& ( ( V2f @ V3x )
= V3x )
& ! [V4y: A_27a] :
( ( ( V0s @ V4y )
& ( V1r @ V4y @ ( V2f @ V4y ) ) )
=> ( V1r @ V4y @ V3x ) ) ) ) ).
thf(thm_2Eposet_2Eposet__nonempty,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o] :
( ( c_2Eposet_2Eposet @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) )
=> ? [V2x: A_27a] : ( V0s @ V2x ) ) ).
thf(thm_2Eposet_2Eposet__refl,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o,V2x: A_27a] :
( ( ( c_2Eposet_2Eposet @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) )
& ( V0s @ V2x ) )
=> ( V1r @ V2x @ V2x ) ) ).
thf(thm_2Eposet_2Eposet__antisym,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o,V2x: A_27a,V3y: A_27a] :
( ( ( c_2Eposet_2Eposet @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) )
& ( V0s @ V2x )
& ( V0s @ V3y )
& ( V1r @ V2x @ V3y )
& ( V1r @ V3y @ V2x ) )
=> ( V2x = V3y ) ) ).
thf(thm_2Eposet_2Eposet__trans,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o,V2x: A_27a,V3y: A_27a,V4z: A_27a] :
( ( ( c_2Eposet_2Eposet @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) )
& ( V0s @ V2x )
& ( V0s @ V3y )
& ( V0s @ V4z )
& ( V1r @ V2x @ V3y )
& ( V1r @ V3y @ V4z ) )
=> ( V1r @ V2x @ V4z ) ) ).
thf(thm_2Eposet_2Elub__pred,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o,V2p: A_27a > $o,V3x: A_27a] :
( ( c_2Eposet_2Elub @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r )
@ ^ [V4j: A_27a] : ( c_2Ebool_2E_2F_5C @ ( V0s @ V4j ) @ ( V2p @ V4j ) )
@ V3x )
= ( c_2Eposet_2Elub @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) @ V2p @ V3x ) ) ).
thf(thm_2Eposet_2Eglb__pred,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: A_27a > A_27a > $o,V2p: A_27a > $o,V3x: A_27a] :
( ( c_2Eposet_2Eglb @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r )
@ ^ [V4j: A_27a] : ( c_2Ebool_2E_2F_5C @ ( V0s @ V4j ) @ ( V2p @ V4j ) )
@ V3x )
= ( c_2Eposet_2Eglb @ A_27a @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ) @ V0s @ V1r ) @ V2p @ V3x ) ) ).
thf(thm_2Eposet_2Ecomplete__up,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ),V1c: A_27a > $o] :
( ( c_2Eposet_2Ecomplete @ A_27a @ V0p )
=> ? [V2x: A_27a] : ( c_2Eposet_2Elub @ A_27a @ V0p @ V1c @ V2x ) ) ).
thf(thm_2Eposet_2Ecomplete__down,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ),V1c: A_27a > $o] :
( ( c_2Eposet_2Ecomplete @ A_27a @ V0p )
=> ? [V2x: A_27a] : ( c_2Eposet_2Eglb @ A_27a @ V0p @ V1c @ V2x ) ) ).
thf(thm_2Eposet_2Ecomplete__top,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o )] :
( ( ( c_2Eposet_2Eposet @ A_27a @ V0p )
& ( c_2Eposet_2Ecomplete @ A_27a @ V0p ) )
=> ? [V1x: A_27a] : ( c_2Eposet_2Etop @ A_27a @ V0p @ V1x ) ) ).
thf(thm_2Eposet_2Ecomplete__bottom,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o )] :
( ( ( c_2Eposet_2Eposet @ A_27a @ V0p )
& ( c_2Eposet_2Ecomplete @ A_27a @ V0p ) )
=> ? [V1x: A_27a] : ( c_2Eposet_2Ebottom @ A_27a @ V0p @ V1x ) ) ).
thf(thm_2Eposet_2Ecomplete__pointwise,axiom,
! [A_27a: $tType,A_27b: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ),V1t: A_27b > $o] :
( ( c_2Eposet_2Ecomplete @ A_27a @ V0p )
=> ( c_2Eposet_2Ecomplete @ ( A_27b > A_27a ) @ ( c_2Eposet_2Epointwise__lift @ A_27b @ A_27a @ V1t @ V0p ) ) ) ).
thf(thm_2Eposet_2Elfp__unique,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ),V1f: A_27a > A_27a,V2x: A_27a,V3x_27: A_27a] :
( ( ( c_2Eposet_2Eposet @ A_27a @ V0p )
& ( c_2Eposet_2Elfp @ A_27a @ V0p @ V1f @ V2x )
& ( c_2Eposet_2Elfp @ A_27a @ V0p @ V1f @ V3x_27 ) )
=> ( V2x = V3x_27 ) ) ).
thf(thm_2Eposet_2Egfp__unique,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ),V1f: A_27a > A_27a,V2x: A_27a,V3x_27: A_27a] :
( ( ( c_2Eposet_2Eposet @ A_27a @ V0p )
& ( c_2Eposet_2Egfp @ A_27a @ V0p @ V1f @ V2x )
& ( c_2Eposet_2Egfp @ A_27a @ V0p @ V1f @ V3x_27 ) )
=> ( V2x = V3x_27 ) ) ).
thf(thm_2Eposet_2Eknaster__tarski__lfp,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ),V1f: A_27a > A_27a] :
( ( ( c_2Eposet_2Eposet @ A_27a @ V0p )
& ( c_2Eposet_2Ecomplete @ A_27a @ V0p )
& ( c_2Eposet_2Efunction @ A_27a @ A_27a @ ( c_2Eposet_2Ecarrier @ A_27a @ V0p ) @ ( c_2Eposet_2Ecarrier @ A_27a @ V0p ) @ V1f )
& ( c_2Eposet_2Emonotonic @ A_27a @ V0p @ V1f ) )
=> ? [V2x: A_27a] : ( c_2Eposet_2Elfp @ A_27a @ V0p @ V1f @ V2x ) ) ).
thf(thm_2Eposet_2Eknaster__tarski__gfp,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ),V1f: A_27a > A_27a] :
( ( ( c_2Eposet_2Eposet @ A_27a @ V0p )
& ( c_2Eposet_2Ecomplete @ A_27a @ V0p )
& ( c_2Eposet_2Efunction @ A_27a @ A_27a @ ( c_2Eposet_2Ecarrier @ A_27a @ V0p ) @ ( c_2Eposet_2Ecarrier @ A_27a @ V0p ) @ V1f )
& ( c_2Eposet_2Emonotonic @ A_27a @ V0p @ V1f ) )
=> ? [V2x: A_27a] : ( c_2Eposet_2Egfp @ A_27a @ V0p @ V1f @ V2x ) ) ).
thf(thm_2Eposet_2Eknaster__tarski,axiom,
! [A_27a: $tType,V0p: tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( A_27a > A_27a > $o ),V1f: A_27a > A_27a] :
( ( ( c_2Eposet_2Eposet @ A_27a @ V0p )
& ( c_2Eposet_2Ecomplete @ A_27a @ V0p )
& ( c_2Eposet_2Efunction @ A_27a @ A_27a @ ( c_2Eposet_2Ecarrier @ A_27a @ V0p ) @ ( c_2Eposet_2Ecarrier @ A_27a @ V0p ) @ V1f )
& ( c_2Eposet_2Emonotonic @ A_27a @ V0p @ V1f ) )
=> ( ? [V2x: A_27a] : ( c_2Eposet_2Elfp @ A_27a @ V0p @ V1f @ V2x )
& ? [V3x: A_27a] : ( c_2Eposet_2Egfp @ A_27a @ V0p @ V1f @ V3x ) ) ) ).
%------------------------------------------------------------------------------