ITP001 Axioms: ITP105^7.ax


%------------------------------------------------------------------------------
% File     : ITP105^7 : 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    : veblen.ax [Gau19]
%          : HL4105^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   56 (   5 unt;  36 typ;   0 def)
%            Number of atoms       :   69 (   2 equ;   6 cnn)
%            Maximal formula atoms :   12 (   1 avg)
%            Number of connectives :  354 (   6   ~;   1   |;  12   &; 307   @)
%                                         (  12 <=>;  16  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (  10 avg; 307 nst)
%            Number of types       :    3 (   2 usr)
%            Number of type conns  :   93 (  93   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   36 (  34 usr;   3 con; 0-5 aty)
%            Number of variables   :  102 (   5   ^  67   !;   2   ?; 102   :)
%                                         (  28  !>;   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_2Enum_2E0,type,
    c_2Enum_2E0: tyop_2Enum_2Enum ).

thf(c_2Earithmetic_2E_3C_3D,type,
    c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $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_2EBIGINTER,type,
    c_2Epred__set_2EBIGINTER: 
      !>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $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_2EIMAGE,type,
    c_2Epred__set_2EIMAGE: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).

thf(c_2Ebool_2EIN,type,
    c_2Ebool_2EIN: 
      !>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).

thf(c_2Epred__set_2ESUBSET,type,
    c_2Epred__set_2ESUBSET: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).

thf(c_2Enum_2ESUC,type,
    c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Ebool_2ET,type,
    c_2Ebool_2ET: $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_2Eveblen_2Eclosed,type,
    c_2Eveblen_2Eclosed: 
      !>[A_27a: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) > $o ) ).

thf(c_2Eveblen_2Eclub,type,
    c_2Eveblen_2Eclub: 
      !>[A_27a: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) > $o ) ).

thf(c_2Eveblen_2Econtinuous,type,
    c_2Eveblen_2Econtinuous: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27b ) ) > $o ) ).

thf(c_2Eordinal_2EfromNat,type,
    c_2Eordinal_2EfromNat: 
      !>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).

thf(c_2Eordinal_2Eoleast,type,
    c_2Eordinal_2Eoleast: 
      !>[A_27a: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).

thf(c_2Eordinal_2EordSUC,type,
    c_2Eordinal_2EordSUC: 
      !>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).

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_2Eveblen_2Estrict__mono,type,
    c_2Eveblen_2Estrict__mono: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27b ) ) > $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_2Eveblen_2Eunbounded,type,
    c_2Eveblen_2Eunbounded: 
      !>[A_27a: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) > $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_2Eveblen_2Eclosed__def,axiom,
    ! [A_27a: $tType,V0A: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
      ( ( c_2Eveblen_2Eclosed @ A_27a @ V0A )
    <=> ! [V1g: tyop_2Enum_2Enum > ( tyop_2Eordinal_2Eordinal @ A_27a )] :
          ( ! [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( V1g @ V2n ) @ V0A )
         => ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a )
            @ ( c_2Eordinal_2Esup @ A_27a
              @ ( c_2Epred__set_2EGSPEC @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ tyop_2Enum_2Enum
                @ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ $o @ ( V1g @ V3n ) @ c_2Ebool_2ET ) ) )
            @ V0A ) ) ) ).

thf(thm_2Eveblen_2Eunbounded__def,axiom,
    ! [A_27a: $tType,V0A: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
      ( ( c_2Eveblen_2Eunbounded @ A_27a @ V0A )
    <=> ! [V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
        ? [V2b: tyop_2Eordinal_2Eordinal @ A_27a] :
          ( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V2b @ V0A )
          & ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ V2b ) ) ) ).

thf(thm_2Eveblen_2Eclub__def,axiom,
    ! [A_27a: $tType,V0A: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
      ( ( c_2Eveblen_2Eclub @ A_27a @ V0A )
    <=> ( ( c_2Eveblen_2Eclosed @ A_27a @ V0A )
        & ( c_2Eveblen_2Eunbounded @ A_27a @ V0A ) ) ) ).

thf(thm_2Eveblen_2Econtinuous__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27b )] :
      ( ( c_2Eveblen_2Econtinuous @ A_27a @ A_27b @ V0f )
    <=> ! [V1A: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
          ( ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V1A @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) )
         => ( ( V0f @ ( c_2Eordinal_2Esup @ A_27a @ V1A ) )
            = ( c_2Eordinal_2Esup @ A_27b @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) @ V0f @ V1A ) ) ) ) ) ).

thf(thm_2Eveblen_2Estrict__mono__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27b )] :
      ( ( c_2Eveblen_2Estrict__mono @ A_27a @ A_27b @ V0f )
    <=> ! [V1x: tyop_2Eordinal_2Eordinal @ A_27a,V2y: tyop_2Eordinal_2Eordinal @ A_27a] :
          ( ( c_2Eordinal_2Eordlt @ A_27a @ V1x @ V2y )
         => ( c_2Eordinal_2Eordlt @ A_27b @ ( V0f @ V1x ) @ ( V0f @ V2y ) ) ) ) ).

thf(thm_2Eveblen_2Ebetter__induction,axiom,
    ! [A_27a: $tType,V0P: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
      ( ( ( V0P @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
        & ! [V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
            ( ( V0P @ V1a )
           => ( V0P @ ( c_2Eordinal_2EordSUC @ A_27a @ V1a ) ) )
        & ! [V2a: tyop_2Eordinal_2Eordinal @ A_27a] :
            ( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V2a )
              & ! [V3b: tyop_2Eordinal_2Eordinal @ A_27a] :
                  ( ( c_2Eordinal_2Eordlt @ A_27a @ V3b @ V2a )
                 => ( V0P @ V3b ) ) )
           => ( V0P @ ( c_2Eordinal_2Esup @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V2a ) ) ) ) )
     => ! [V4a: tyop_2Eordinal_2Eordinal @ A_27a] : ( V0P @ V4a ) ) ).

thf(thm_2Eveblen_2Enrange__IN__Uinf,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Enum_2Enum > A_27b] :
      ( c_2Ecardinal_2Ecardleq @ A_27b @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a )
      @ ( c_2Epred__set_2EGSPEC @ A_27b @ tyop_2Enum_2Enum
        @ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ A_27b @ $o @ ( V0f @ V1n ) @ c_2Ebool_2ET ) )
      @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) ) ).

thf(thm_2Eveblen_2Eincreasing,axiom,
    ! [A_27a: $tType,V0f: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ),V1x: tyop_2Eordinal_2Eordinal @ A_27a] :
      ( ( ( c_2Eveblen_2Estrict__mono @ A_27a @ A_27a @ V0f )
        & ( c_2Eveblen_2Econtinuous @ A_27a @ A_27a @ V0f ) )
     => ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( V0f @ V1x ) @ V1x ) ) ) ).

thf(thm_2Eveblen_2Eclubs__exist,axiom,
    ! [A_27a: $tType,V0f: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a )] :
      ( ( ( c_2Eveblen_2Estrict__mono @ A_27a @ A_27a @ V0f )
        & ( c_2Eveblen_2Econtinuous @ A_27a @ A_27a @ V0f ) )
     => ( c_2Eveblen_2Eclub @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0f @ ( c_2Epred__set_2EUNIV @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ).

thf(thm_2Eveblen_2Emono__natI,axiom,
    ! [A_27a: $tType,V0f: tyop_2Enum_2Enum > ( tyop_2Eordinal_2Eordinal @ A_27a )] :
      ( ! [V1n: tyop_2Enum_2Enum] : ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( V0f @ ( c_2Enum_2ESUC @ V1n ) ) @ ( V0f @ V1n ) ) )
     => ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
          ( ( c_2Earithmetic_2E_3C_3D @ V2m @ V3n )
         => ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( V0f @ V3n ) @ ( V0f @ V2m ) ) ) ) ) ).

thf(thm_2Eveblen_2Esup__mem__INTER,axiom,
    ! [A_27a: $tType,V0f: tyop_2Enum_2Enum > ( tyop_2Eordinal_2Eordinal @ A_27a ),V1A: tyop_2Enum_2Enum > ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
      ( ( ! [V2n: tyop_2Enum_2Enum] : ( c_2Eveblen_2Eclub @ A_27a @ ( V1A @ V2n ) )
        & ! [V3n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( V1A @ ( c_2Enum_2ESUC @ V3n ) ) @ ( V1A @ V3n ) )
        & ! [V4n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( V0f @ V4n ) @ ( V1A @ V4n ) )
        & ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
            ( ( c_2Earithmetic_2E_3C_3D @ V5m @ V6n )
           => ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( V0f @ V6n ) @ ( V0f @ V5m ) ) ) ) )
     => ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a )
        @ ( c_2Eordinal_2Esup @ A_27a
          @ ( c_2Epred__set_2EGSPEC @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ tyop_2Enum_2Enum
            @ ^ [V7n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ $o @ ( V0f @ V7n ) @ c_2Ebool_2ET ) ) )
        @ ( c_2Epred__set_2EBIGINTER @ ( tyop_2Eordinal_2Eordinal @ A_27a )
          @ ( c_2Epred__set_2EGSPEC @ ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) @ tyop_2Enum_2Enum
            @ ^ [V8n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) @ $o @ ( V1A @ V8n ) @ c_2Ebool_2ET ) ) ) ) ) ).

thf(thm_2Eveblen_2Eoleast__leq,axiom,
    ! [A_27a: $tType,V0P: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
      ( ( V0P @ V1a )
     => ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ ( c_2Eordinal_2Eoleast @ A_27a @ V0P ) ) ) ) ).

thf(thm_2Eveblen_2Eclub__INTER,axiom,
    ! [A_27a: $tType,V0A: tyop_2Enum_2Enum > ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
      ( ( ! [V1n: tyop_2Enum_2Enum] : ( c_2Eveblen_2Eclub @ A_27a @ ( V0A @ V1n ) )
        & ! [V2n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( V0A @ ( c_2Enum_2ESUC @ V2n ) ) @ ( V0A @ V2n ) ) )
     => ( c_2Eveblen_2Eclub @ A_27a
        @ ( c_2Epred__set_2EBIGINTER @ ( tyop_2Eordinal_2Eordinal @ A_27a )
          @ ( c_2Epred__set_2EGSPEC @ ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) @ tyop_2Enum_2Enum
            @ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) @ $o @ ( V0A @ V3n ) @ c_2Ebool_2ET ) ) ) ) ) ).

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