ITP001 Axioms: ITP104^7.ax


%------------------------------------------------------------------------------
% File     : ITP104^7 : 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    : ucord.ax [Gau19]
%          : HL4104^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   36 (   7 unt;  22 typ;   0 def)
%            Number of atoms       :   38 (   3 equ;   4 cnn)
%            Maximal formula atoms :    5 (   1 avg)
%            Number of connectives :  191 (   4   ~;   1   |;   1   &; 177   @)
%                                         (   7 <=>;   1  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   13 (   7 avg; 177 nst)
%            Number of types       :    3 (   2 usr)
%            Number of type conns  :   69 (  69   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   22 (  20 usr;   1 con; 0-4 aty)
%            Number of variables   :   41 (   2   ^  23   !;   1   ?;  41   :)
%                                         (  15  !>;   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_2Enum_2Enum,type,
    tyop_2Enum_2Enum: $tType ).

thf(tyop_2Eordinal_2Eordinal,type,
    tyop_2Eordinal_2Eordinal: $tType > $tType ).

thf(tyop_2Epair_2Eprod,type,
    tyop_2Epair_2Eprod: $tType > $tType > $tType ).

thf(tyop_2Esum_2Esum,type,
    tyop_2Esum_2Esum: $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_2Epred__set_2EGSPEC,type,
    c_2Epred__set_2EGSPEC: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).

thf(c_2Epred__set_2EUNIV,type,
    c_2Epred__set_2EUNIV: 
      !>[A_27a: $tType] : ( A_27a > $o ) ).

thf(c_2Ebool_2E_5C_2F,type,
    c_2Ebool_2E_5C_2F: $o > $o > $o ).

thf(c_2Ecardinal_2Ecardleq,type,
    c_2Ecardinal_2Ecardleq: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).

thf(c_2Epred__set_2Ecountable,type,
    c_2Epred__set_2Ecountable: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Eucord_2Eomega1,type,
    c_2Eucord_2Eomega1: 
      !>[A_27a: $tType] : ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) ).

thf(c_2Eordinal_2Eordlt,type,
    c_2Eordinal_2Eordlt: 
      !>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) ).

thf(c_2Eordinal_2Epreds,type,
    c_2Eordinal_2Epreds: 
      !>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) ).

thf(c_2Eordinal_2Esup,type,
    c_2Eordinal_2Esup: 
      !>[A_27a: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).

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_2Eucord_2Eomega1__def,axiom,
    ! [A_27a: $tType] :
      ( ( c_2Eucord_2Eomega1 @ A_27a )
      = ( c_2Eordinal_2Esup @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) )
        @ ( c_2Epred__set_2EGSPEC @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) )
          @ ^ [V0a: tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) )] : ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ $o @ V0a @ ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( c_2Eordinal_2Epreds @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) @ V0a ) ) ) ) ) ) ).

thf(thm_2Eucord_2Eucinf__uncountable,axiom,
    ! [A_27a: $tType] : ( (~) @ ( c_2Epred__set_2Ecountable @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) ) ) ) ).

thf(thm_2Eucord_2EUnum__cardlt__ucinf,axiom,
    ! [A_27a: $tType] : ( (~) @ ( c_2Ecardinal_2Ecardleq @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ).

thf(thm_2Eucord_2EUnum__cardle__ucinf,axiom,
    ! [A_27a: $tType] : ( c_2Ecardinal_2Ecardleq @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) ) ) ).

thf(thm_2Eucord_2Eucord__sup__exists__lemma,axiom,
    ! [A_27a: $tType] :
      ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) )
      @ ( c_2Epred__set_2EGSPEC @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) )
        @ ^ [V0a: tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) )] : ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ $o @ V0a @ ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( c_2Eordinal_2Epreds @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) @ V0a ) ) ) )
      @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) ) ) ).

thf(thm_2Eucord_2Ex__lt__omega1__countable,axiom,
    ! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) )] :
      ( ( c_2Eordinal_2Eordlt @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) @ V0x @ ( c_2Eucord_2Eomega1 @ A_27a ) )
      = ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( c_2Eordinal_2Epreds @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) @ V0x ) ) ) ).

thf(thm_2Eucord_2Eomega1__not__countable,axiom,
    ! [A_27a: $tType] : ( (~) @ ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( c_2Eordinal_2Epreds @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) @ ( c_2Eucord_2Eomega1 @ A_27a ) ) ) ) ).

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