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    :   35 (  12 unt;  12 typ;   0 def)
%            Number of atoms       :  259 (   7 equ)
%            Maximal formula atoms :   16 (   7 avg)
%            Number of connectives :   68 (   0   ~;  20   |;  29   &)
%                                         (  17 <=>;   2  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   4 avg)
%            Maximal term depth    :    2 (   1 avg)
%            Number of FOOLs       :  168 ( 168 fml;   0 var)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :    8 (   6   >;   2   *;   0   +;   0  <<)
%            Number of predicates  :   13 (  11 usr;   6 prp; 0-2 aty)
%            Number of functors    :   12 (  12 usr;   6 con; 0-2 aty)
%            Number of variables   :   30 (  30   !;   0   ?;  30   :)
% SPC      : TF0_SAT_EQU_NAR

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

tff(mem_c_2Emarker_2E_3A_2D,axiom,
    mem(c_2Emarker_2E_3A_2D,arr(ind,arr(bool,bool))) ).

tff(stp_fo_c_2Emarker_2E_3A_2D,type,
    fo__c_2Emarker_2E_3A_2D: ( tp__i * tp__o ) > tp__o ).

tff(stp_eq_fo_c_2Emarker_2E_3A_2D,axiom,
    ! [X0: tp__i,X1: tp__o] : ( inj__o(fo__c_2Emarker_2E_3A_2D(X0,X1)) = ap(ap(c_2Emarker_2E_3A_2D,inj__i(X0)),inj__o(X1)) ) ).

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

tff(mem_c_2Emarker_2EAC,axiom,
    mem(c_2Emarker_2EAC,arr(bool,arr(bool,bool))) ).

tff(stp_fo_c_2Emarker_2EAC,type,
    fo__c_2Emarker_2EAC: ( tp__o * tp__o ) > tp__o ).

tff(stp_eq_fo_c_2Emarker_2EAC,axiom,
    ! [X0: tp__o,X1: tp__o] : ( inj__o(fo__c_2Emarker_2EAC(X0,X1)) = ap(ap(c_2Emarker_2EAC,inj__o(X0)),inj__o(X1)) ) ).

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

tff(mem_c_2Emarker_2EAbbrev,axiom,
    mem(c_2Emarker_2EAbbrev,arr(bool,bool)) ).

tff(stp_fo_c_2Emarker_2EAbbrev,type,
    fo__c_2Emarker_2EAbbrev: tp__o > tp__o ).

tff(stp_eq_fo_c_2Emarker_2EAbbrev,axiom,
    ! [X0: tp__o] : ( inj__o(fo__c_2Emarker_2EAbbrev(X0)) = ap(c_2Emarker_2EAbbrev,inj__o(X0)) ) ).

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

tff(mem_c_2Emarker_2ECong,axiom,
    mem(c_2Emarker_2ECong,arr(bool,bool)) ).

tff(stp_fo_c_2Emarker_2ECong,type,
    fo__c_2Emarker_2ECong: tp__o > tp__o ).

tff(stp_eq_fo_c_2Emarker_2ECong,axiom,
    ! [X0: tp__o] : ( inj__o(fo__c_2Emarker_2ECong(X0)) = ap(c_2Emarker_2ECong,inj__o(X0)) ) ).

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

tff(mem_c_2Emarker_2EIfCases,axiom,
    mem(c_2Emarker_2EIfCases,bool) ).

tff(stp_fo_c_2Emarker_2EIfCases,type,
    fo__c_2Emarker_2EIfCases: tp__o ).

tff(stp_eq_fo_c_2Emarker_2EIfCases,axiom,
    inj__o(fo__c_2Emarker_2EIfCases) = c_2Emarker_2EIfCases ).

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

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

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

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

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

tff(conj_thm_2Emarker_2Emove__left__conj,axiom,
    ! [V0p: tp__o,V1q: tp__o,V2m: tp__o] :
      ( ( ( p(inj__o(V0p))
          & p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m))) )
      <=> ( p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m)))
          & p(inj__o(V0p)) ) )
      & ( ( p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m)))
          & p(inj__o(V0p))
          & p(inj__o(V1q)) )
      <=> ( p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m)))
          & p(inj__o(V0p))
          & p(inj__o(V1q)) ) )
      & ( ( p(inj__o(V0p))
          & p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m)))
          & p(inj__o(V1q)) )
      <=> ( p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m)))
          & p(inj__o(V0p))
          & p(inj__o(V1q)) ) ) ) ).

tff(conj_thm_2Emarker_2Emove__right__conj,axiom,
    ! [V0p: tp__o,V1q: tp__o,V2m: tp__o] :
      ( ( ( p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m)))
          & p(inj__o(V0p)) )
      <=> ( p(inj__o(V0p))
          & p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m))) ) )
      & ( ( p(inj__o(V0p))
          & p(inj__o(V1q))
          & p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m))) )
      <=> ( p(inj__o(V0p))
          & p(inj__o(V1q))
          & p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m))) ) )
      & ( ( p(inj__o(V0p))
          & p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m)))
          & p(inj__o(V1q)) )
      <=> ( p(inj__o(V0p))
          & p(inj__o(V1q))
          & p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m))) ) ) ) ).

tff(conj_thm_2Emarker_2Emove__left__disj,axiom,
    ! [V0p: tp__o,V1q: tp__o,V2m: tp__o] :
      ( ( ( p(inj__o(V0p))
          | p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m))) )
      <=> ( p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m)))
          | p(inj__o(V0p)) ) )
      & ( ( p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m)))
          | p(inj__o(V0p))
          | p(inj__o(V1q)) )
      <=> ( p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m)))
          | p(inj__o(V0p))
          | p(inj__o(V1q)) ) )
      & ( ( p(inj__o(V0p))
          | p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m)))
          | p(inj__o(V1q)) )
      <=> ( p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m)))
          | p(inj__o(V0p))
          | p(inj__o(V1q)) ) ) ) ).

tff(conj_thm_2Emarker_2Emove__right__disj,axiom,
    ! [V0p: tp__o,V1q: tp__o,V2m: tp__o] :
      ( ( ( p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m)))
          | p(inj__o(V0p)) )
      <=> ( p(inj__o(V0p))
          | p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m))) ) )
      & ( ( p(inj__o(V0p))
          | p(inj__o(V1q))
          | p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m))) )
      <=> ( p(inj__o(V0p))
          | p(inj__o(V1q))
          | p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m))) ) )
      & ( ( p(inj__o(V0p))
          | p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m)))
          | p(inj__o(V1q)) )
      <=> ( p(inj__o(V0p))
          | p(inj__o(V1q))
          | p(ap(c_2Emarker_2Estmarker(bool),inj__o(V2m))) ) ) ) ).

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

tff(ax_thm_2Emarker_2EAbbrev__def,axiom,
    ! [V0x: tp__o] :
      ( p(ap(c_2Emarker_2EAbbrev,inj__o(V0x)))
    <=> p(inj__o(V0x)) ) ).

tff(ax_thm_2Emarker_2EIfCases__def,axiom,
    ( p(inj__o(fo__c_2Emarker_2EIfCases))
  <=> $true ) ).

tff(ax_thm_2Emarker_2EAC__DEF,axiom,
    ! [V0b1: tp__o,V1b2: tp__o] :
      ( p(ap(ap(c_2Emarker_2EAC,inj__o(V0b1)),inj__o(V1b2)))
    <=> ( p(inj__o(V0b1))
        & p(inj__o(V1b2)) ) ) ).

tff(ax_thm_2Emarker_2ECong__def,axiom,
    ! [V0x: tp__o] :
      ( p(ap(c_2Emarker_2ECong,inj__o(V0x)))
    <=> p(inj__o(V0x)) ) ).

tff(ax_thm_2Emarker_2Elabel__def,axiom,
    ! [V0lab: tp__i,V1argument: tp__o] :
      ( p(ap(ap(c_2Emarker_2E_3A_2D,inj__i(V0lab)),inj__o(V1argument)))
    <=> p(inj__o(V1argument)) ) ).

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