ITP001 Axioms: ITP017^7.ax


%------------------------------------------------------------------------------
% File     : ITP017^7 : TPTP v9.0.0. 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 ) ) ) ).

%------------------------------------------------------------------------------