ITP001 Axioms: ITP017^5.ax
%------------------------------------------------------------------------------
% File : ITP017^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : poset^2.ax [Gau20]
% : HL4017^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 67 ( 0 unt; 17 typ; 0 def)
% Number of atoms : 1237 ( 9 equ; 0 cnn)
% Maximal formula atoms : 76 ( 18 avg)
% Number of connectives : 2263 ( 0 ~; 1 |; 75 &;2020 @)
% ( 16 <=>; 151 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 16 avg;2020 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 19 ( 19 >; 0 *; 0 +; 0 <<)
% Number of symbols : 32 ( 31 usr; 14 con; 0-2 aty)
% Number of variables : 195 ( 9 ^ 174 !; 12 ?; 195 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Eposet_2Ebottom,type,
c_2Eposet_2Ebottom: del > $i ).
thf(mem_c_2Eposet_2Ebottom,axiom,
! [A_27a: del] : ( mem @ ( c_2Eposet_2Ebottom @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Eposet_2Ecarrier,type,
c_2Eposet_2Ecarrier: del > $i ).
thf(mem_c_2Eposet_2Ecarrier,axiom,
! [A_27a: del] : ( mem @ ( c_2Eposet_2Ecarrier @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Eposet_2Echain,type,
c_2Eposet_2Echain: del > $i ).
thf(mem_c_2Eposet_2Echain,axiom,
! [A_27a: del] : ( mem @ ( c_2Eposet_2Echain @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Eposet_2Ecomplete,type,
c_2Eposet_2Ecomplete: del > $i ).
thf(mem_c_2Eposet_2Ecomplete,axiom,
! [A_27a: del] : ( mem @ ( c_2Eposet_2Ecomplete @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ bool ) ) ).
thf(tp_c_2Eposet_2Econtinuous,type,
c_2Eposet_2Econtinuous: del > $i ).
thf(mem_c_2Eposet_2Econtinuous,axiom,
! [A_27a: del] : ( mem @ ( c_2Eposet_2Econtinuous @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Eposet_2Edown__continuous,type,
c_2Eposet_2Edown__continuous: del > $i ).
thf(mem_c_2Eposet_2Edown__continuous,axiom,
! [A_27a: del] : ( mem @ ( c_2Eposet_2Edown__continuous @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Eposet_2Efunction,type,
c_2Eposet_2Efunction: del > del > $i ).
thf(mem_c_2Eposet_2Efunction,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eposet_2Efunction @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) ) ) ) ).
thf(tp_c_2Eposet_2Egfp,type,
c_2Eposet_2Egfp: del > $i ).
thf(mem_c_2Eposet_2Egfp,axiom,
! [A_27a: del] : ( mem @ ( c_2Eposet_2Egfp @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Eposet_2Eglb,type,
c_2Eposet_2Eglb: del > $i ).
thf(mem_c_2Eposet_2Eglb,axiom,
! [A_27a: del] : ( mem @ ( c_2Eposet_2Eglb @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Eposet_2Elfp,type,
c_2Eposet_2Elfp: del > $i ).
thf(mem_c_2Eposet_2Elfp,axiom,
! [A_27a: del] : ( mem @ ( c_2Eposet_2Elfp @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Eposet_2Elub,type,
c_2Eposet_2Elub: del > $i ).
thf(mem_c_2Eposet_2Elub,axiom,
! [A_27a: del] : ( mem @ ( c_2Eposet_2Elub @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Eposet_2Emonotonic,type,
c_2Eposet_2Emonotonic: del > $i ).
thf(mem_c_2Eposet_2Emonotonic,axiom,
! [A_27a: del] : ( mem @ ( c_2Eposet_2Emonotonic @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Eposet_2Epointwise__lift,type,
c_2Eposet_2Epointwise__lift: del > del > $i ).
thf(mem_c_2Eposet_2Epointwise__lift,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eposet_2Epointwise__lift @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27b @ bool ) @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) ) @ ( ty_2Epair_2Eprod @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) ) ) ) ) ) ).
thf(tp_c_2Eposet_2Eposet,type,
c_2Eposet_2Eposet: del > $i ).
thf(mem_c_2Eposet_2Eposet,axiom,
! [A_27a: del] : ( mem @ ( c_2Eposet_2Eposet @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ bool ) ) ).
thf(tp_c_2Eposet_2Erelation,type,
c_2Eposet_2Erelation: del > $i ).
thf(mem_c_2Eposet_2Erelation,axiom,
! [A_27a: del] : ( mem @ ( c_2Eposet_2Erelation @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Eposet_2Etop,type,
c_2Eposet_2Etop: del > $i ).
thf(mem_c_2Eposet_2Etop,axiom,
! [A_27a: del] : ( mem @ ( c_2Eposet_2Etop @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Eposet_2Eup__continuous,type,
c_2Eposet_2Eup__continuous: del > $i ).
thf(mem_c_2Eposet_2Eup__continuous,axiom,
! [A_27a: del] : ( mem @ ( c_2Eposet_2Eup__continuous @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ bool ) ) ) ).
thf(ax_thm_2Eposet_2Efunction__def,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ ( arr @ A_27a @ bool ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( arr @ A_27b @ bool ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Efunction @ A_27a @ A_27b ) @ V0a ) @ V1b ) @ V2f ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V0a @ V3x ) )
=> ( p @ ( ap @ V1b @ ( ap @ V2f @ V3x ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eposet_2Eposet__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) )
<=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V0s @ V2x ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V0s @ V3x ) )
=> ( p @ ( ap @ ( ap @ V1r @ V3x ) @ V3x ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ V0s @ V4x ) )
& ( p @ ( ap @ V0s @ V5y ) )
& ( p @ ( ap @ ( ap @ V1r @ V4x ) @ V5y ) )
& ( p @ ( ap @ ( ap @ V1r @ V5y ) @ V4x ) ) )
=> ( V4x = V5y ) ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27a )
=> ! [V8z: $i] :
( ( mem @ V8z @ A_27a )
=> ( ( ( p @ ( ap @ V0s @ V6x ) )
& ( p @ ( ap @ V0s @ V7y ) )
& ( p @ ( ap @ V0s @ V8z ) )
& ( p @ ( ap @ ( ap @ V1r @ V6x ) @ V7y ) )
& ( p @ ( ap @ ( ap @ V1r @ V7y ) @ V8z ) ) )
=> ( p @ ( ap @ ( ap @ V1r @ V6x ) @ V8z ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eposet_2Ecarrier__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Eposet_2Ecarrier @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) )
= V0s ) ) ) ).
thf(ax_thm_2Eposet_2Erelation__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Eposet_2Erelation @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) )
= V1r ) ) ) ).
thf(ax_thm_2Eposet_2Etop__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Etop @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2x ) )
<=> ( ( p @ ( ap @ V0s @ V2x ) )
& ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ V0s @ V3y ) )
=> ( p @ ( ap @ ( ap @ V1r @ V3y ) @ V2x ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eposet_2Ebottom__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Ebottom @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2x ) )
<=> ( ( p @ ( ap @ V0s @ V2x ) )
& ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ V0s @ V3y ) )
=> ( p @ ( ap @ ( ap @ V1r @ V2x ) @ V3y ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eposet_2Echain__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2c: $i] :
( ( mem @ V2c @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Echain @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2c ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ V0s @ V3x ) )
& ( p @ ( ap @ V0s @ V4y ) )
& ( p @ ( ap @ V2c @ V3x ) )
& ( p @ ( ap @ V2c @ V4y ) ) )
=> ( ( p @ ( ap @ ( ap @ V1r @ V3x ) @ V4y ) )
| ( p @ ( ap @ ( ap @ V1r @ V4y ) @ V3x ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eposet_2Elub__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( arr @ A_27a @ bool ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elub @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2p ) @ V3x ) )
<=> ( ( p @ ( ap @ V0s @ V3x ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ V0s @ V4y ) )
& ( p @ ( ap @ V2p @ V4y ) ) )
=> ( p @ ( ap @ ( ap @ V1r @ V4y ) @ V3x ) ) ) )
& ! [V5z: $i] :
( ( mem @ V5z @ A_27a )
=> ( ( ( p @ ( ap @ V0s @ V5z ) )
& ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ( ( ( p @ ( ap @ V0s @ V6y ) )
& ( p @ ( ap @ V2p @ V6y ) ) )
=> ( p @ ( ap @ ( ap @ V1r @ V6y ) @ V5z ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V1r @ V3x ) @ V5z ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eposet_2Eglb__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( arr @ A_27a @ bool ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Eglb @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2p ) @ V3x ) )
<=> ( ( p @ ( ap @ V0s @ V3x ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ V0s @ V4y ) )
& ( p @ ( ap @ V2p @ V4y ) ) )
=> ( p @ ( ap @ ( ap @ V1r @ V3x ) @ V4y ) ) ) )
& ! [V5z: $i] :
( ( mem @ V5z @ A_27a )
=> ( ( ( p @ ( ap @ V0s @ V5z ) )
& ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ( ( ( p @ ( ap @ V0s @ V6y ) )
& ( p @ ( ap @ V2p @ V6y ) ) )
=> ( p @ ( ap @ ( ap @ V1r @ V5z ) @ V6y ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V1r @ V5z ) @ V3x ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eposet_2Ecomplete__def,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
=> ( ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) )
<=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ A_27a @ bool ) )
=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elub @ A_27a ) @ V0p ) @ V1c ) @ V2x ) ) )
& ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Eglb @ A_27a ) @ V0p ) @ V1c ) @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Eposet_2Eposet__nonempty,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) )
=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V0s @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Eposet_2Eposet__refl,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) )
& ( p @ ( ap @ V0s @ V2x ) ) )
=> ( p @ ( ap @ ( ap @ V1r @ V2x ) @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Eposet_2Eposet__antisym,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) )
& ( p @ ( ap @ V0s @ V2x ) )
& ( p @ ( ap @ V0s @ V3y ) )
& ( p @ ( ap @ ( ap @ V1r @ V2x ) @ V3y ) )
& ( p @ ( ap @ ( ap @ V1r @ V3y ) @ V2x ) ) )
=> ( V2x = V3y ) ) ) ) ) ) ).
thf(conj_thm_2Eposet_2Eposet__trans,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4z: $i] :
( ( mem @ V4z @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) )
& ( p @ ( ap @ V0s @ V2x ) )
& ( p @ ( ap @ V0s @ V3y ) )
& ( p @ ( ap @ V0s @ V4z ) )
& ( p @ ( ap @ ( ap @ V1r @ V2x ) @ V3y ) )
& ( p @ ( ap @ ( ap @ V1r @ V3y ) @ V4z ) ) )
=> ( p @ ( ap @ ( ap @ V1r @ V2x ) @ V4z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eposet_2Elub__pred,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( arr @ A_27a @ bool ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p
@ ( ap
@ ( ap @ ( ap @ ( c_2Eposet_2Elub @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) )
@ ( lam @ A_27a
@ ^ [V4j: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0s @ V4j ) ) @ ( ap @ V2p @ V4j ) ) ) )
@ V3x ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elub @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2p ) @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Eposet_2Eglb__pred,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2p: $i] :
( ( mem @ V2p @ ( arr @ A_27a @ bool ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p
@ ( ap
@ ( ap @ ( ap @ ( c_2Eposet_2Eglb @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) )
@ ( lam @ A_27a
@ ^ [V4j: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0s @ V4j ) ) @ ( ap @ V2p @ V4j ) ) ) )
@ V3x ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Eglb @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2p ) @ V3x ) ) ) ) ) ) ) ).
thf(conj_thm_2Eposet_2Ecomplete__up,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) )
=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elub @ A_27a ) @ V0p ) @ V1c ) @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Eposet_2Ecomplete__down,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) )
=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Eglb @ A_27a ) @ V0p ) @ V1c ) @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Eposet_2Ecomplete__top,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ V0p ) )
& ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) ) )
=> ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Eposet_2Etop @ A_27a ) @ V0p ) @ V1x ) ) ) ) ) ).
thf(conj_thm_2Eposet_2Ecomplete__bottom,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
=> ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ V0p ) )
& ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) ) )
=> ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Eposet_2Ebottom @ A_27a ) @ V0p ) @ V1x ) ) ) ) ) ).
thf(ax_thm_2Eposet_2Epointwise__lift__def,axiom,
! [A_27a: del,A_27b: del,V0t: $i] :
( ( mem @ V0t @ ( arr @ A_27a @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2r: $i] :
( ( mem @ V2r @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Eposet_2Epointwise__lift @ A_27a @ A_27b ) @ V0t ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27b @ bool ) @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) ) @ V1s ) @ V2r ) )
= ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) ) ) @ ( ap @ ( ap @ ( c_2Eposet_2Efunction @ A_27a @ A_27b ) @ V0t ) @ V1s ) )
@ ( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V3f: $i] :
( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V4g: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V0t @ V5x ) ) @ ( ap @ ( ap @ V2r @ ( ap @ V3f @ V5x ) ) @ ( ap @ V4g @ V5x ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eposet_2Ecomplete__pointwise,axiom,
! [A_27a: del,A_27b: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) )
=> ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ ( arr @ A_27b @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Eposet_2Epointwise__lift @ A_27b @ A_27a ) @ V1t ) @ V0p ) ) ) ) ) ) ).
thf(ax_thm_2Eposet_2Emonotonic__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Emonotonic @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2f ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ V0s @ V3x ) )
& ( p @ ( ap @ V0s @ V4y ) )
& ( p @ ( ap @ ( ap @ V1r @ V3x ) @ V4y ) ) )
=> ( p @ ( ap @ ( ap @ V1r @ ( ap @ V2f @ V3x ) ) @ ( ap @ V2f @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eposet_2Eup__continuous__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Eup__continuous @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2f ) )
<=> ! [V3c: $i] :
( ( mem @ V3c @ ( arr @ A_27a @ bool ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Echain @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V3c ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elub @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V3c ) @ V4x ) ) )
=> ( p
@ ( ap
@ ( ap @ ( ap @ ( c_2Eposet_2Elub @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) )
@ ( lam @ A_27a
@ ^ [V5y: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V6z: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0s @ V6z ) ) @ ( ap @ V3c @ V6z ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V5y ) @ ( ap @ V2f @ V6z ) ) ) ) ) ) )
@ ( ap @ V2f @ V4x ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eposet_2Edown__continuous__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Edown__continuous @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2f ) )
<=> ! [V3c: $i] :
( ( mem @ V3c @ ( arr @ A_27a @ bool ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Echain @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V3c ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Eglb @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V3c ) @ V4x ) ) )
=> ( p
@ ( ap
@ ( ap @ ( ap @ ( c_2Eposet_2Eglb @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) )
@ ( lam @ A_27a
@ ^ [V5y: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V6z: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0s @ V6z ) ) @ ( ap @ V3c @ V6z ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V5y ) @ ( ap @ V2f @ V6z ) ) ) ) ) ) )
@ ( ap @ V2f @ V4x ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eposet_2Econtinuous__def,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Econtinuous @ A_27a ) @ V0p ) @ V1f ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Eposet_2Eup__continuous @ A_27a ) @ V0p ) @ V1f ) )
& ( p @ ( ap @ ( ap @ ( c_2Eposet_2Edown__continuous @ A_27a ) @ V0p ) @ V1f ) ) ) ) ) ) ).
thf(ax_thm_2Eposet_2Elfp__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elfp @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2f ) @ V3x ) )
<=> ( ( p @ ( ap @ V0s @ V3x ) )
& ( ( ap @ V2f @ V3x )
= V3x )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ V0s @ V4y ) )
& ( p @ ( ap @ ( ap @ V1r @ ( ap @ V2f @ V4y ) ) @ V4y ) ) )
=> ( p @ ( ap @ ( ap @ V1r @ V3x ) @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Eposet_2Egfp__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Egfp @ A_27a ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ V0s ) @ V1r ) ) @ V2f ) @ V3x ) )
<=> ( ( p @ ( ap @ V0s @ V3x ) )
& ( ( ap @ V2f @ V3x )
= V3x )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ V0s @ V4y ) )
& ( p @ ( ap @ ( ap @ V1r @ V4y ) @ ( ap @ V2f @ V4y ) ) ) )
=> ( p @ ( ap @ ( ap @ V1r @ V4y ) @ V3x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Eposet_2Elfp__unique,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3x_27: $i] :
( ( mem @ V3x_27 @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ V0p ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elfp @ A_27a ) @ V0p ) @ V1f ) @ V2x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elfp @ A_27a ) @ V0p ) @ V1f ) @ V3x_27 ) ) )
=> ( V2x = V3x_27 ) ) ) ) ) ) ).
thf(conj_thm_2Eposet_2Egfp__unique,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3x_27: $i] :
( ( mem @ V3x_27 @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ V0p ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Egfp @ A_27a ) @ V0p ) @ V1f ) @ V2x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Egfp @ A_27a ) @ V0p ) @ V1f ) @ V3x_27 ) ) )
=> ( V2x = V3x_27 ) ) ) ) ) ) ).
thf(conj_thm_2Eposet_2Eknaster__tarski__lfp,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ V0p ) )
& ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Efunction @ A_27a @ A_27a ) @ ( ap @ ( c_2Eposet_2Ecarrier @ A_27a ) @ V0p ) ) @ ( ap @ ( c_2Eposet_2Ecarrier @ A_27a ) @ V0p ) ) @ V1f ) )
& ( p @ ( ap @ ( ap @ ( c_2Eposet_2Emonotonic @ A_27a ) @ V0p ) @ V1f ) ) )
=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elfp @ A_27a ) @ V0p ) @ V1f ) @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Eposet_2Eknaster__tarski__gfp,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ V0p ) )
& ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Efunction @ A_27a @ A_27a ) @ ( ap @ ( c_2Eposet_2Ecarrier @ A_27a ) @ V0p ) ) @ ( ap @ ( c_2Eposet_2Ecarrier @ A_27a ) @ V0p ) ) @ V1f ) )
& ( p @ ( ap @ ( ap @ ( c_2Eposet_2Emonotonic @ A_27a ) @ V0p ) @ V1f ) ) )
=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Egfp @ A_27a ) @ V0p ) @ V1f ) @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Eposet_2Eknaster__tarski,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ( p @ ( ap @ ( c_2Eposet_2Eposet @ A_27a ) @ V0p ) )
& ( p @ ( ap @ ( c_2Eposet_2Ecomplete @ A_27a ) @ V0p ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Efunction @ A_27a @ A_27a ) @ ( ap @ ( c_2Eposet_2Ecarrier @ A_27a ) @ V0p ) ) @ ( ap @ ( c_2Eposet_2Ecarrier @ A_27a ) @ V0p ) ) @ V1f ) )
& ( p @ ( ap @ ( ap @ ( c_2Eposet_2Emonotonic @ A_27a ) @ V0p ) @ V1f ) ) )
=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Elfp @ A_27a ) @ V0p ) @ V1f ) @ V2x ) ) )
& ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eposet_2Egfp @ A_27a ) @ V0p ) @ V1f ) @ V3x ) ) ) ) ) ) ) ).
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