ITP001 Axioms: ITP105^5.ax


%------------------------------------------------------------------------------
% File     : ITP105^5 : TPTP v9.0.0. 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    : veblen^2.ax [Gau20]
%          : HL4105^5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   23 (   0 unt;   5 typ;   0 def)
%            Number of atoms       :  421 (   1 equ;   0 cnn)
%            Maximal formula atoms :   88 (  18 avg)
%            Number of connectives :  661 (   5   ~;   0   |;  12   &; 599   @)
%                                         (   5 <=>;  40  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (  15 avg; 599 nst)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :    7 (   7   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   36 (  35 usr;  30 con; 0-2 aty)
%            Number of variables   :   65 (   5   ^  59   !;   1   ?;  65   :)
% SPC      : TH0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Eveblen_2Eclosed,type,
    c_2Eveblen_2Eclosed: del > $i ).

thf(mem_c_2Eveblen_2Eclosed,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eveblen_2Eclosed @ A_27a ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ bool ) ) ).

thf(tp_c_2Eveblen_2Eclub,type,
    c_2Eveblen_2Eclub: del > $i ).

thf(mem_c_2Eveblen_2Eclub,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eveblen_2Eclub @ A_27a ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ bool ) ) ).

thf(tp_c_2Eveblen_2Econtinuous,type,
    c_2Eveblen_2Econtinuous: del > del > $i ).

thf(mem_c_2Eveblen_2Econtinuous,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eveblen_2Econtinuous @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ bool ) ) ).

thf(tp_c_2Eveblen_2Estrict__mono,type,
    c_2Eveblen_2Estrict__mono: del > del > $i ).

thf(mem_c_2Eveblen_2Estrict__mono,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Eveblen_2Estrict__mono @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ bool ) ) ).

thf(tp_c_2Eveblen_2Eunbounded,type,
    c_2Eveblen_2Eunbounded: del > $i ).

thf(mem_c_2Eveblen_2Eunbounded,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eveblen_2Eunbounded @ A_27a ) @ ( arr @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ bool ) ) ).

thf(conj_thm_2Eveblen_2Ebetter__induction,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( ( p @ ( ap @ V0P @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) )
          & ! [V1a: $i] :
              ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( p @ ( ap @ V0P @ V1a ) )
               => ( p @ ( ap @ V0P @ ( ap @ ( c_2Eordinal_2EordSUC @ A_27a ) @ V1a ) ) ) ) )
          & ! [V2a: $i] :
              ( ( mem @ V2a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
             => ( ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ ( c_2Eordinal_2EfromNat @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V2a ) )
                  & ! [V3b: $i] :
                      ( ( mem @ V3b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                     => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V3b ) @ V2a ) )
                       => ( p @ ( ap @ V0P @ V3b ) ) ) ) )
               => ( p @ ( ap @ V0P @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ ( ap @ ( c_2Eordinal_2Epreds @ A_27a ) @ V2a ) ) ) ) ) ) )
       => ! [V4a: $i] :
            ( ( mem @ V4a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ( p @ ( ap @ V0P @ V4a ) ) ) ) ) ).

thf(ax_thm_2Eveblen_2Eclosed__def,axiom,
    ! [A_27a: del,V0A: $i] :
      ( ( mem @ V0A @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( c_2Eveblen_2Eclosed @ A_27a ) @ V0A ) )
      <=> ! [V1g: $i] :
            ( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
           => ( ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ V0A ) )
             => ( p
                @ ( ap
                  @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                    @ ( ap @ ( c_2Eordinal_2Esup @ A_27a )
                      @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ty_2Enum_2Enum )
                        @ ( lam @ ty_2Enum_2Enum
                          @ ^ [V3n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ( ap @ V1g @ V3n ) ) @ c_2Ebool_2ET ) ) ) ) )
                  @ V0A ) ) ) ) ) ) ).

thf(ax_thm_2Eveblen_2Eunbounded__def,axiom,
    ! [A_27a: del,V0A: $i] :
      ( ( mem @ V0A @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( c_2Eveblen_2Eunbounded @ A_27a ) @ V0A ) )
      <=> ! [V1a: $i] :
            ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ? [V2b: $i] :
                ( ( mem @ V2b @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                & ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V2b ) @ V0A ) )
                & ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ V2b ) ) ) ) ) ) ).

thf(ax_thm_2Eveblen_2Eclub__def,axiom,
    ! [A_27a: del,V0A: $i] :
      ( ( mem @ V0A @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ( ( p @ ( ap @ ( c_2Eveblen_2Eclub @ A_27a ) @ V0A ) )
      <=> ( ( p @ ( ap @ ( c_2Eveblen_2Eclosed @ A_27a ) @ V0A ) )
          & ( p @ ( ap @ ( c_2Eveblen_2Eunbounded @ A_27a ) @ V0A ) ) ) ) ) ).

thf(ax_thm_2Eveblen_2Econtinuous__def,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) )
     => ( ( p @ ( ap @ ( c_2Eveblen_2Econtinuous @ A_27a @ A_27b ) @ V0f ) )
      <=> ! [V1A: $i] :
            ( ( mem @ V1A @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
           => ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ V1A ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) )
             => ( ( ap @ V0f @ ( ap @ ( c_2Eordinal_2Esup @ A_27a ) @ V1A ) )
                = ( ap @ ( c_2Eordinal_2Esup @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) @ V0f ) @ V1A ) ) ) ) ) ) ) ).

thf(ax_thm_2Eveblen_2Estrict__mono__def,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27b ) ) )
     => ( ( p @ ( ap @ ( c_2Eveblen_2Estrict__mono @ A_27a @ A_27b ) @ V0f ) )
      <=> ! [V1x: $i] :
            ( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
           => ! [V2y: $i] :
                ( ( mem @ V2y @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
               => ( ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1x ) @ V2y ) )
                 => ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27b ) @ ( ap @ V0f @ V1x ) ) @ ( ap @ V0f @ V2y ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eveblen_2Enrange__IN__Uinf,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ A_27b ) )
     => ( p
        @ ( ap
          @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) )
            @ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ ty_2Enum_2Enum )
              @ ( lam @ ty_2Enum_2Enum
                @ ^ [V1n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ ( ap @ V0f @ V1n ) ) @ c_2Ebool_2ET ) ) ) )
          @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) ) ) ).

thf(conj_thm_2Eveblen_2Eincreasing,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( ( p @ ( ap @ ( c_2Eveblen_2Estrict__mono @ A_27a @ A_27a ) @ V0f ) )
              & ( p @ ( ap @ ( c_2Eveblen_2Econtinuous @ A_27a @ A_27a ) @ V0f ) ) )
           => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ V0f @ V1x ) ) @ V1x ) ) ) ) ) ).

thf(conj_thm_2Eveblen_2Eclubs__exist,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
     => ( ( ( p @ ( ap @ ( c_2Eveblen_2Estrict__mono @ A_27a @ A_27a ) @ V0f ) )
          & ( p @ ( ap @ ( c_2Eveblen_2Econtinuous @ A_27a @ A_27a ) @ V0f ) ) )
       => ( p @ ( ap @ ( c_2Eveblen_2Eclub @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ V0f ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ) ) ).

thf(conj_thm_2Eveblen_2Emono__natI,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
     => ( ! [V1n: tp__ty_2Enum_2Enum] :
            ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ V0f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
       => ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
            ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
           => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) ) ) ) ).

thf(conj_thm_2Eveblen_2Esup__mem__INTER,axiom,
    ! [A_27a: del,V0A: $i] :
      ( ( mem @ V0A @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) )
         => ( ( ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( c_2Eveblen_2Eclub @ A_27a ) @ ( ap @ V0A @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
              & ! [V3n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ V0A @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) @ ( ap @ V0A @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
              & ! [V4n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) @ ( ap @ V0A @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) )
              & ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
                  ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V5m ) ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) )
                 => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V5m ) ) ) ) ) )
           => ( p
              @ ( ap
                @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                  @ ( ap @ ( c_2Eordinal_2Esup @ A_27a )
                    @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ ty_2Enum_2Enum )
                      @ ( lam @ ty_2Enum_2Enum
                        @ ^ [V7n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ( ap @ V1f @ V7n ) ) @ c_2Ebool_2ET ) ) ) ) )
                @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
                  @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ty_2Enum_2Enum )
                    @ ( lam @ ty_2Enum_2Enum
                      @ ^ [V8n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ bool ) @ ( ap @ V0A @ V8n ) ) @ c_2Ebool_2ET ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eveblen_2Eoleast__leq,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) )
     => ! [V1a: $i] :
          ( ( mem @ V1a @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
         => ( ( p @ ( ap @ V0P @ V1a ) )
           => ~ ( p @ ( ap @ ( ap @ ( c_2Eordinal_2Eordlt @ A_27a ) @ V1a ) @ ( ap @ ( c_2Eordinal_2Eoleast @ A_27a ) @ V0P ) ) ) ) ) ) ).

thf(conj_thm_2Eveblen_2Eclub__INTER,axiom,
    ! [A_27a: del,V0A: $i] :
      ( ( mem @ V0A @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) ) )
     => ( ( ! [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( c_2Eveblen_2Eclub @ A_27a ) @ ( ap @ V0A @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
          & ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Eordinal_2Eordinal @ A_27a ) ) @ ( ap @ V0A @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ V0A @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) )
       => ( p
          @ ( ap @ ( c_2Eveblen_2Eclub @ A_27a )
            @ ( ap @ ( c_2Epred__set_2EBIGINTER @ ( ty_2Eordinal_2Eordinal @ A_27a ) )
              @ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ ty_2Enum_2Enum )
                @ ( lam @ ty_2Enum_2Enum
                  @ ^ [V3n: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ ( ty_2Eordinal_2Eordinal @ A_27a ) @ bool ) @ bool ) @ ( ap @ V0A @ V3n ) ) @ c_2Ebool_2ET ) ) ) ) ) ) ) ) ).

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