ITP001 Axioms: ITP091+5.ax


%------------------------------------------------------------------------------
% File     : ITP091+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    : EncodeVar+2.ax [Gau20]
%          : HL4091+5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   19 (   0 unt;   0 def)
%            Number of atoms       :   90 (   6 equ)
%            Maximal formula atoms :   10 (   4 avg)
%            Number of connectives :   71 (   0   ~;   0   |;   7   &)
%                                         (   8 <=>;  56  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   15 (   8 avg)
%            Maximal term depth    :    9 (   2 avg)
%            Number of predicates  :    4 (   3 usr;   0 prp; 1-2 aty)
%            Number of functors    :   36 (  36 usr;  12 con; 0-4 aty)
%            Number of variables   :   60 (  58   !;   2   ?)
% SPC      : FOF_SAT_RFO_SEQ

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(mem_c_2EEncodeVar_2Efixed__width,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => mem(c_2EEncodeVar_2Efixed__width(A_27a),arr(ty_2Enum_2Enum,arr(ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(A_27a,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,ty_2Elist_2Elist(bool)))))),bool))) ) ).

fof(mem_c_2EEncodeVar_2Eof__length,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => mem(c_2EEncodeVar_2Eof__length(A_27a),arr(ty_2Enum_2Enum,arr(ty_2Elist_2Elist(A_27a),bool))) ) ).

fof(ax_thm_2EEncodeVar_2Efixed__width__def,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0n] :
          ( mem(V0n,ty_2Enum_2Enum)
         => ! [V1c] :
              ( mem(V1c,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(A_27a,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,ty_2Elist_2Elist(bool)))))))
             => ( p(ap(ap(c_2EEncodeVar_2Efixed__width(A_27a),V0n),V1c))
              <=> ! [V2x] :
                    ( mem(V2x,A_27a)
                   => ( p(ap(ap(c_2ECoder_2Edomain(A_27a),V1c),V2x))
                     => ap(c_2Elist_2ELENGTH(bool),ap(ap(c_2ECoder_2Eencoder(A_27a),V1c),V2x)) = V0n ) ) ) ) ) ) ).

fof(ax_thm_2EEncodeVar_2Eof__length__def,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0l] :
          ( mem(V0l,ty_2Elist_2Elist(A_27a))
         => ! [V1n] :
              ( mem(V1n,ty_2Enum_2Enum)
             => ( p(ap(ap(c_2Ebool_2EIN(ty_2Elist_2Elist(A_27a)),V0l),ap(c_2EEncodeVar_2Eof__length(A_27a),V1n)))
              <=> ap(c_2Elist_2ELENGTH(A_27a),V0l) = V1n ) ) ) ) ).

fof(lameq_f2440,axiom,
    ! [A_27a,V1c] :
      ( mem(V1c,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(A_27a,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,ty_2Elist_2Elist(bool)))))))
     => ! [V0phi] :
          ( mem(V0phi,arr(A_27a,bool))
         => ! [V4w] : ap(f2440(A_27a,V1c,V0phi),V4w) = ap(V0phi,ap(ap(c_2ECoder_2Edecoder(A_27a),V1c),V4w)) ) ) ).

fof(conj_thm_2EEncodeVar_2Efixed__width__univ,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0phi] :
          ( mem(V0phi,arr(A_27a,bool))
         => ! [V1c] :
              ( mem(V1c,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(A_27a,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,ty_2Elist_2Elist(bool)))))))
             => ! [V2n] :
                  ( mem(V2n,ty_2Enum_2Enum)
                 => ( ( p(ap(c_2ECoder_2Ewf__coder(A_27a),V1c))
                      & p(ap(ap(c_2EEncodeVar_2Efixed__width(A_27a),V2n),V1c)) )
                   => ( ! [V3x] :
                          ( mem(V3x,A_27a)
                         => ( p(ap(ap(c_2ECoder_2Edomain(A_27a),V1c),V3x))
                           => p(ap(V0phi,V3x)) ) )
                    <=> p(ap(ap(c_2Ebool_2ERES__FORALL(ty_2Elist_2Elist(bool)),ap(c_2EEncodeVar_2Eof__length(bool),V2n)),f2440(A_27a,V1c,V0phi))) ) ) ) ) ) ) ).

fof(lameq_f2441,axiom,
    ! [A_27a,V0phi] :
      ( mem(V0phi,arr(ty_2Elist_2Elist(A_27a),bool))
     => ! [V2w] : ap(f2441(A_27a,V0phi),V2w) = ap(V0phi,V2w) ) ).

fof(lameq_f2442,axiom,
    ! [A_27a,V3x] :
      ( mem(V3x,A_27a)
     => ! [V0phi] :
          ( mem(V0phi,arr(ty_2Elist_2Elist(A_27a),bool))
         => ! [V4w] : ap(f2442(A_27a,V3x,V0phi),V4w) = ap(V0phi,ap(ap(c_2Elist_2ECONS(A_27a),V3x),V4w)) ) ) ).

fof(conj_thm_2EEncodeVar_2Eof__length__univ__suc,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0phi] :
          ( mem(V0phi,arr(ty_2Elist_2Elist(A_27a),bool))
         => ! [V1n] :
              ( mem(V1n,ty_2Enum_2Enum)
             => ( p(ap(ap(c_2Ebool_2ERES__FORALL(ty_2Elist_2Elist(A_27a)),ap(c_2EEncodeVar_2Eof__length(A_27a),ap(c_2Enum_2ESUC,V1n))),f2441(A_27a,V0phi)))
              <=> ! [V3x] :
                    ( mem(V3x,A_27a)
                   => p(ap(ap(c_2Ebool_2ERES__FORALL(ty_2Elist_2Elist(A_27a)),ap(c_2EEncodeVar_2Eof__length(A_27a),V1n)),f2442(A_27a,V3x,V0phi))) ) ) ) ) ) ).

fof(lameq_f2443,axiom,
    ! [A_27a,V0phi] :
      ( mem(V0phi,arr(ty_2Elist_2Elist(A_27a),bool))
     => ! [V1w] : ap(f2443(A_27a,V0phi),V1w) = ap(V0phi,V1w) ) ).

fof(conj_thm_2EEncodeVar_2Eof__length__univ__zero,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0phi] :
          ( mem(V0phi,arr(ty_2Elist_2Elist(A_27a),bool))
         => ( p(ap(ap(c_2Ebool_2ERES__FORALL(ty_2Elist_2Elist(A_27a)),ap(c_2EEncodeVar_2Eof__length(A_27a),c_2Enum_2E0)),f2443(A_27a,V0phi)))
          <=> p(ap(V0phi,c_2Elist_2ENIL(A_27a))) ) ) ) ).

fof(conj_thm_2EEncodeVar_2Efixed__width__exists,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0phi] :
          ( mem(V0phi,arr(A_27a,bool))
         => ! [V1c] :
              ( mem(V1c,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(A_27a,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,ty_2Elist_2Elist(bool)))))))
             => ! [V2n] :
                  ( mem(V2n,ty_2Enum_2Enum)
                 => ( ( p(ap(c_2ECoder_2Ewf__coder(A_27a),V1c))
                      & p(ap(ap(c_2EEncodeVar_2Efixed__width(A_27a),V2n),V1c)) )
                   => ( ? [V3x] :
                          ( mem(V3x,A_27a)
                          & p(ap(ap(c_2ECoder_2Edomain(A_27a),V1c),V3x))
                          & p(ap(V0phi,V3x)) )
                    <=> p(ap(ap(c_2Ebool_2ERES__EXISTS(ty_2Elist_2Elist(bool)),ap(c_2EEncodeVar_2Eof__length(bool),V2n)),f2440(A_27a,V1c,V0phi))) ) ) ) ) ) ) ).

fof(conj_thm_2EEncodeVar_2Eof__length__exists__suc,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0phi] :
          ( mem(V0phi,arr(ty_2Elist_2Elist(A_27a),bool))
         => ! [V1n] :
              ( mem(V1n,ty_2Enum_2Enum)
             => ( p(ap(ap(c_2Ebool_2ERES__EXISTS(ty_2Elist_2Elist(A_27a)),ap(c_2EEncodeVar_2Eof__length(A_27a),ap(c_2Enum_2ESUC,V1n))),f2441(A_27a,V0phi)))
              <=> ? [V3x] :
                    ( mem(V3x,A_27a)
                    & p(ap(ap(c_2Ebool_2ERES__EXISTS(ty_2Elist_2Elist(A_27a)),ap(c_2EEncodeVar_2Eof__length(A_27a),V1n)),f2442(A_27a,V3x,V0phi))) ) ) ) ) ) ).

fof(conj_thm_2EEncodeVar_2Eof__length__exists__zero,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0phi] :
          ( mem(V0phi,arr(ty_2Elist_2Elist(A_27a),bool))
         => ( p(ap(ap(c_2Ebool_2ERES__EXISTS(ty_2Elist_2Elist(A_27a)),ap(c_2EEncodeVar_2Eof__length(A_27a),c_2Enum_2E0)),f2443(A_27a,V0phi)))
          <=> p(ap(V0phi,c_2Elist_2ENIL(A_27a))) ) ) ) ).

fof(conj_thm_2EEncodeVar_2Efixed__width__unit,axiom,
    ! [V0p] :
      ( mem(V0p,arr(ty_2Eone_2Eone,bool))
     => p(ap(ap(c_2EEncodeVar_2Efixed__width(ty_2Eone_2Eone),c_2Enum_2E0),ap(c_2ECoder_2Eunit__coder,V0p))) ) ).

fof(conj_thm_2EEncodeVar_2Efixed__width__bool,axiom,
    ! [V0p] :
      ( mem(V0p,arr(bool,bool))
     => p(ap(ap(c_2EEncodeVar_2Efixed__width(bool),ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,c_2Earithmetic_2EZERO))),ap(c_2ECoder_2Ebool__coder,V0p))) ) ).

fof(conj_thm_2EEncodeVar_2Efixed__width__prod,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => ! [V0c1] :
              ( mem(V0c1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(A_27a,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,ty_2Elist_2Elist(bool)))))))
             => ! [V1c2] :
                  ( mem(V1c2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(A_27b,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27b,ty_2Elist_2Elist(bool)))))))
                 => ! [V2n1] :
                      ( mem(V2n1,ty_2Enum_2Enum)
                     => ! [V3n2] :
                          ( mem(V3n2,ty_2Enum_2Enum)
                         => ( ( p(ap(ap(c_2EEncodeVar_2Efixed__width(A_27a),V2n1),V0c1))
                              & p(ap(ap(c_2EEncodeVar_2Efixed__width(A_27b),V3n2),V1c2)) )
                           => p(ap(ap(c_2EEncodeVar_2Efixed__width(ty_2Epair_2Eprod(A_27a,A_27b)),ap(ap(c_2Earithmetic_2E_2B,V2n1),V3n2)),ap(ap(c_2ECoder_2Eprod__coder(A_27a,A_27b,A_27a,A_27b),V0c1),V1c2))) ) ) ) ) ) ) ) ).

fof(conj_thm_2EEncodeVar_2Efixed__width__sum,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => ! [V0c1] :
              ( mem(V0c1,ty_2Epair_2Eprod(arr(A_27a,bool),ty_2Epair_2Eprod(arr(A_27a,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27a,ty_2Elist_2Elist(bool)))))))
             => ! [V1c2] :
                  ( mem(V1c2,ty_2Epair_2Eprod(arr(A_27b,bool),ty_2Epair_2Eprod(arr(A_27b,ty_2Elist_2Elist(bool)),arr(ty_2Elist_2Elist(bool),ty_2Eoption_2Eoption(ty_2Epair_2Eprod(A_27b,ty_2Elist_2Elist(bool)))))))
                 => ! [V2n] :
                      ( mem(V2n,ty_2Enum_2Enum)
                     => ( ( p(ap(ap(c_2EEncodeVar_2Efixed__width(A_27a),V2n),V0c1))
                          & p(ap(ap(c_2EEncodeVar_2Efixed__width(A_27b),V2n),V1c2)) )
                       => p(ap(ap(c_2EEncodeVar_2Efixed__width(ty_2Esum_2Esum(A_27a,A_27b)),ap(c_2Enum_2ESUC,V2n)),ap(ap(c_2ECoder_2Esum__coder(A_27a,A_27b,A_27a,A_27b),V0c1),V1c2))) ) ) ) ) ) ) ).

fof(conj_thm_2EEncodeVar_2Efixed__width__bnum,axiom,
    ! [V0m] :
      ( mem(V0m,ty_2Enum_2Enum)
     => ! [V1p] :
          ( mem(V1p,arr(ty_2Enum_2Enum,bool))
         => p(ap(ap(c_2EEncodeVar_2Efixed__width(ty_2Enum_2Enum),V0m),ap(ap(c_2ECoder_2Ebnum__coder,V0m),V1p))) ) ) ).

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