ITP001 Axioms: ITP005^5.ax


%------------------------------------------------------------------------------
% File     : ITP005^5 : 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    : marker^2.ax [Gau20]
%          : HL4005^5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   25 (   0 unt;   7 typ;   0 def)
%            Number of atoms       :  196 (   2 equ;   0 cnn)
%            Maximal formula atoms :   34 (   7 avg)
%            Number of connectives :  320 (   0   ~;  20   |;  29   &; 235   @)
%                                         (  17 <=>;  19  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   16 (   8 avg; 235 nst)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :    2 (   2   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   18 (  16 usr;  15 con; 0-2 aty)
%            Number of variables   :   24 (   0   ^  24   !;   0   ?;  24   :)
% SPC      : TH0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Emarker_2E_3A_2D,type,
    c_2Emarker_2E_3A_2D: $i ).

thf(mem_c_2Emarker_2E_3A_2D,axiom,
    mem @ c_2Emarker_2E_3A_2D @ ( arr @ ind @ ( arr @ bool @ bool ) ) ).

thf(tp_c_2Emarker_2EAC,type,
    c_2Emarker_2EAC: $i ).

thf(mem_c_2Emarker_2EAC,axiom,
    mem @ c_2Emarker_2EAC @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).

thf(tp_c_2Emarker_2EAbbrev,type,
    c_2Emarker_2EAbbrev: $i ).

thf(mem_c_2Emarker_2EAbbrev,axiom,
    mem @ c_2Emarker_2EAbbrev @ ( arr @ bool @ bool ) ).

thf(tp_c_2Emarker_2ECong,type,
    c_2Emarker_2ECong: $i ).

thf(mem_c_2Emarker_2ECong,axiom,
    mem @ c_2Emarker_2ECong @ ( arr @ bool @ bool ) ).

thf(tp_c_2Emarker_2EIfCases,type,
    c_2Emarker_2EIfCases: $i ).

thf(mem_c_2Emarker_2EIfCases,axiom,
    mem @ c_2Emarker_2EIfCases @ bool ).

thf(tp_c_2Emarker_2Estmarker,type,
    c_2Emarker_2Estmarker: del > $i ).

thf(mem_c_2Emarker_2Estmarker,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Emarker_2Estmarker @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) ).

thf(tp_c_2Emarker_2Eunint,type,
    c_2Emarker_2Eunint: del > $i ).

thf(mem_c_2Emarker_2Eunint,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Emarker_2Eunint @ A_27a ) @ ( arr @ A_27a @ A_27a ) ) ).

thf(ax_thm_2Emarker_2Estmarker__def,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ( ( ap @ ( c_2Emarker_2Estmarker @ A_27a ) @ V0x )
        = V0x ) ) ).

thf(conj_thm_2Emarker_2Emove__left__conj,axiom,
    ! [V0p: $i] :
      ( ( mem @ V0p @ bool )
     => ! [V1q: $i] :
          ( ( mem @ V1q @ bool )
         => ! [V2m: $i] :
              ( ( mem @ V2m @ bool )
             => ( ( ( ( p @ V0p )
                    & ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) ) )
                <=> ( ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) )
                    & ( p @ V0p ) ) )
                & ( ( ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) )
                    & ( p @ V0p )
                    & ( p @ V1q ) )
                <=> ( ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) )
                    & ( p @ V0p )
                    & ( p @ V1q ) ) )
                & ( ( ( p @ V0p )
                    & ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) )
                    & ( p @ V1q ) )
                <=> ( ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) )
                    & ( p @ V0p )
                    & ( p @ V1q ) ) ) ) ) ) ) ).

thf(conj_thm_2Emarker_2Emove__right__conj,axiom,
    ! [V0p: $i] :
      ( ( mem @ V0p @ bool )
     => ! [V1q: $i] :
          ( ( mem @ V1q @ bool )
         => ! [V2m: $i] :
              ( ( mem @ V2m @ bool )
             => ( ( ( ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) )
                    & ( p @ V0p ) )
                <=> ( ( p @ V0p )
                    & ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) ) ) )
                & ( ( ( p @ V0p )
                    & ( p @ V1q )
                    & ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) ) )
                <=> ( ( p @ V0p )
                    & ( p @ V1q )
                    & ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) ) ) )
                & ( ( ( p @ V0p )
                    & ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) )
                    & ( p @ V1q ) )
                <=> ( ( p @ V0p )
                    & ( p @ V1q )
                    & ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Emarker_2Emove__left__disj,axiom,
    ! [V0p: $i] :
      ( ( mem @ V0p @ bool )
     => ! [V1q: $i] :
          ( ( mem @ V1q @ bool )
         => ! [V2m: $i] :
              ( ( mem @ V2m @ bool )
             => ( ( ( ( p @ V0p )
                    | ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) ) )
                <=> ( ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) )
                    | ( p @ V0p ) ) )
                & ( ( ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) )
                    | ( p @ V0p )
                    | ( p @ V1q ) )
                <=> ( ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) )
                    | ( p @ V0p )
                    | ( p @ V1q ) ) )
                & ( ( ( p @ V0p )
                    | ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) )
                    | ( p @ V1q ) )
                <=> ( ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) )
                    | ( p @ V0p )
                    | ( p @ V1q ) ) ) ) ) ) ) ).

thf(conj_thm_2Emarker_2Emove__right__disj,axiom,
    ! [V0p: $i] :
      ( ( mem @ V0p @ bool )
     => ! [V1q: $i] :
          ( ( mem @ V1q @ bool )
         => ! [V2m: $i] :
              ( ( mem @ V2m @ bool )
             => ( ( ( ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) )
                    | ( p @ V0p ) )
                <=> ( ( p @ V0p )
                    | ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) ) ) )
                & ( ( ( p @ V0p )
                    | ( p @ V1q )
                    | ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) ) )
                <=> ( ( p @ V0p )
                    | ( p @ V1q )
                    | ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) ) ) )
                & ( ( ( p @ V0p )
                    | ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) )
                    | ( p @ V1q ) )
                <=> ( ( p @ V0p )
                    | ( p @ V1q )
                    | ( p @ ( ap @ ( c_2Emarker_2Estmarker @ bool ) @ V2m ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Emarker_2Eunint__def,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ( ( ap @ ( c_2Emarker_2Eunint @ A_27a ) @ V0x )
        = V0x ) ) ).

thf(ax_thm_2Emarker_2EAbbrev__def,axiom,
    ! [V0x: $i] :
      ( ( mem @ V0x @ bool )
     => ( ( p @ ( ap @ c_2Emarker_2EAbbrev @ V0x ) )
      <=> ( p @ V0x ) ) ) ).

thf(ax_thm_2Emarker_2EIfCases__def,axiom,
    ( ( p @ c_2Emarker_2EIfCases )
  <=> $true ) ).

thf(ax_thm_2Emarker_2EAC__DEF,axiom,
    ! [V0b1: $i] :
      ( ( mem @ V0b1 @ bool )
     => ! [V1b2: $i] :
          ( ( mem @ V1b2 @ bool )
         => ( ( p @ ( ap @ ( ap @ c_2Emarker_2EAC @ V0b1 ) @ V1b2 ) )
          <=> ( ( p @ V0b1 )
              & ( p @ V1b2 ) ) ) ) ) ).

thf(ax_thm_2Emarker_2ECong__def,axiom,
    ! [V0x: $i] :
      ( ( mem @ V0x @ bool )
     => ( ( p @ ( ap @ c_2Emarker_2ECong @ V0x ) )
      <=> ( p @ V0x ) ) ) ).

thf(ax_thm_2Emarker_2Elabel__def,axiom,
    ! [V0lab: tp__i,V1argument: $i] :
      ( ( mem @ V1argument @ bool )
     => ( ( p @ ( ap @ ( ap @ c_2Emarker_2E_3A_2D @ ( inj__i @ V0lab ) ) @ V1argument ) )
      <=> ( p @ V1argument ) ) ) ).

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