ITP001 Axioms: ITP056+5.ax


%------------------------------------------------------------------------------
% File     : ITP056+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    : transfer+2.ax [Gau20]
%          : HL4056+5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   21 (   0 unt;   0 def)
%            Number of atoms       :  133 (   5 equ)
%            Maximal formula atoms :   13 (   6 avg)
%            Number of connectives :  112 (   0   ~;   0   |;   8   &)
%                                         (   8 <=>;  96  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (  11 avg)
%            Maximal term depth    :    6 (   2 avg)
%            Number of predicates  :    4 (   3 usr;   0 prp; 1-2 aty)
%            Number of functors    :   16 (  16 usr;   1 con; 0-4 aty)
%            Number of variables   :   98 (  96   !;   2   ?)
% SPC      : FOF_SAT_RFO_SEQ

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(mem_c_2Etransfer_2EFUN__REL,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => ! [A_27c] :
              ( ne(A_27c)
             => ! [A_27d] :
                  ( ne(A_27d)
                 => mem(c_2Etransfer_2EFUN__REL(A_27a,A_27b,A_27c,A_27d),arr(arr(A_27a,arr(A_27b,bool)),arr(arr(A_27c,arr(A_27d,bool)),arr(arr(A_27a,A_27c),arr(arr(A_27b,A_27d),bool))))) ) ) ) ) ).

fof(mem_c_2Etransfer_2EPAIR__REL,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => ! [A_27c] :
              ( ne(A_27c)
             => ! [A_27d] :
                  ( ne(A_27d)
                 => mem(c_2Etransfer_2EPAIR__REL(A_27a,A_27b,A_27c,A_27d),arr(arr(A_27a,arr(A_27b,bool)),arr(arr(A_27c,arr(A_27d,bool)),arr(ty_2Epair_2Eprod(A_27a,A_27c),arr(ty_2Epair_2Eprod(A_27b,A_27d),bool))))) ) ) ) ) ).

fof(mem_c_2Etransfer_2Ebi__unique,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => mem(c_2Etransfer_2Ebi__unique(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),bool)) ) ) ).

fof(mem_c_2Etransfer_2Ebitotal,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => mem(c_2Etransfer_2Ebitotal(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),bool)) ) ) ).

fof(mem_c_2Etransfer_2Eleft__unique,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => mem(c_2Etransfer_2Eleft__unique(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),bool)) ) ) ).

fof(mem_c_2Etransfer_2Eright__unique,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => mem(c_2Etransfer_2Eright__unique(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),bool)) ) ) ).

fof(mem_c_2Etransfer_2Esurj,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => mem(c_2Etransfer_2Esurj(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),bool)) ) ) ).

fof(mem_c_2Etransfer_2Etotal,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => mem(c_2Etransfer_2Etotal(A_27a,A_27b),arr(arr(A_27a,arr(A_27b,bool)),bool)) ) ) ).

fof(ax_thm_2Etransfer_2Eright__unique__def,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => ! [V0R] :
              ( mem(V0R,arr(A_27a,arr(A_27b,bool)))
             => ( p(ap(c_2Etransfer_2Eright__unique(A_27a,A_27b),V0R))
              <=> ! [V1a] :
                    ( mem(V1a,A_27a)
                   => ! [V2b1] :
                        ( mem(V2b1,A_27b)
                       => ! [V3b2] :
                            ( mem(V3b2,A_27b)
                           => ( ( p(ap(ap(V0R,V1a),V2b1))
                                & p(ap(ap(V0R,V1a),V3b2)) )
                             => V2b1 = V3b2 ) ) ) ) ) ) ) ) ).

fof(ax_thm_2Etransfer_2Eleft__unique__def,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => ! [V0R] :
              ( mem(V0R,arr(A_27a,arr(A_27b,bool)))
             => ( p(ap(c_2Etransfer_2Eleft__unique(A_27a,A_27b),V0R))
              <=> ! [V1a1] :
                    ( mem(V1a1,A_27a)
                   => ! [V2a2] :
                        ( mem(V2a2,A_27a)
                       => ! [V3b] :
                            ( mem(V3b,A_27b)
                           => ( ( p(ap(ap(V0R,V1a1),V3b))
                                & p(ap(ap(V0R,V2a2),V3b)) )
                             => V1a1 = V2a2 ) ) ) ) ) ) ) ) ).

fof(ax_thm_2Etransfer_2Ebi__unique__def,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => ! [V0R] :
              ( mem(V0R,arr(A_27a,arr(A_27b,bool)))
             => ( p(ap(c_2Etransfer_2Ebi__unique(A_27a,A_27b),V0R))
              <=> ( p(ap(c_2Etransfer_2Eleft__unique(A_27a,A_27b),V0R))
                  & p(ap(c_2Etransfer_2Eright__unique(A_27a,A_27b),V0R)) ) ) ) ) ) ).

fof(ax_thm_2Etransfer_2Etotal__def,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => ! [V0R] :
              ( mem(V0R,arr(A_27a,arr(A_27b,bool)))
             => ( p(ap(c_2Etransfer_2Etotal(A_27a,A_27b),V0R))
              <=> ! [V1x] :
                    ( mem(V1x,A_27a)
                   => ? [V2y] :
                        ( mem(V2y,A_27b)
                        & p(ap(ap(V0R,V1x),V2y)) ) ) ) ) ) ) ).

fof(ax_thm_2Etransfer_2Esurj__def,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => ! [V0R] :
              ( mem(V0R,arr(A_27a,arr(A_27b,bool)))
             => ( p(ap(c_2Etransfer_2Esurj(A_27a,A_27b),V0R))
              <=> ! [V1y] :
                    ( mem(V1y,A_27b)
                   => ? [V2x] :
                        ( mem(V2x,A_27a)
                        & p(ap(ap(V0R,V2x),V1y)) ) ) ) ) ) ) ).

fof(ax_thm_2Etransfer_2Ebitotal__def,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => ! [V0R] :
              ( mem(V0R,arr(A_27a,arr(A_27b,bool)))
             => ( p(ap(c_2Etransfer_2Ebitotal(A_27a,A_27b),V0R))
              <=> ( p(ap(c_2Etransfer_2Etotal(A_27a,A_27b),V0R))
                  & p(ap(c_2Etransfer_2Esurj(A_27a,A_27b),V0R)) ) ) ) ) ) ).

fof(ax_thm_2Etransfer_2EFUN__REL__def,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => ! [A_27c] :
              ( ne(A_27c)
             => ! [A_27d] :
                  ( ne(A_27d)
                 => ! [V0AB] :
                      ( mem(V0AB,arr(A_27a,arr(A_27b,bool)))
                     => ! [V1CD] :
                          ( mem(V1CD,arr(A_27c,arr(A_27d,bool)))
                         => ! [V2f] :
                              ( mem(V2f,arr(A_27a,A_27c))
                             => ! [V3g] :
                                  ( mem(V3g,arr(A_27b,A_27d))
                                 => ( p(ap(ap(ap(ap(c_2Etransfer_2EFUN__REL(A_27a,A_27b,A_27c,A_27d),V0AB),V1CD),V2f),V3g))
                                  <=> ! [V4a] :
                                        ( mem(V4a,A_27a)
                                       => ! [V5b] :
                                            ( mem(V5b,A_27b)
                                           => ( p(ap(ap(V0AB,V4a),V5b))
                                             => p(ap(ap(V1CD,ap(V2f,V4a)),ap(V3g,V5b))) ) ) ) ) ) ) ) ) ) ) ) ) ).

fof(conj_thm_2Etransfer_2EFUN__REL__COMB,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => ! [A_27c] :
              ( ne(A_27c)
             => ! [A_27d] :
                  ( ne(A_27d)
                 => ! [V0AB] :
                      ( mem(V0AB,arr(A_27a,arr(A_27b,bool)))
                     => ! [V1CD] :
                          ( mem(V1CD,arr(A_27c,arr(A_27d,bool)))
                         => ! [V2f] :
                              ( mem(V2f,arr(A_27a,A_27c))
                             => ! [V3g] :
                                  ( mem(V3g,arr(A_27b,A_27d))
                                 => ! [V4a] :
                                      ( mem(V4a,A_27a)
                                     => ! [V5b] :
                                          ( mem(V5b,A_27b)
                                         => ( ( p(ap(ap(ap(ap(c_2Etransfer_2EFUN__REL(A_27a,A_27b,A_27c,A_27d),V0AB),V1CD),V2f),V3g))
                                              & p(ap(ap(V0AB,V4a),V5b)) )
                                           => p(ap(ap(V1CD,ap(V2f,V4a)),ap(V3g,V5b))) ) ) ) ) ) ) ) ) ) ) ) ).

fof(lameq_f978,axiom,
    ! [A_27c,A_27a,V2f] :
      ( mem(V2f,arr(A_27a,A_27c))
     => ! [V6a] : ap(f978(A_27c,A_27a,V2f),V6a) = ap(V2f,V6a) ) ).

fof(lameq_f979,axiom,
    ! [A_27d,A_27b,V3g] :
      ( mem(V3g,arr(A_27b,A_27d))
     => ! [V7b] : ap(f979(A_27d,A_27b,V3g),V7b) = ap(V3g,V7b) ) ).

fof(conj_thm_2Etransfer_2EFUN__REL__ABS,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => ! [A_27c] :
              ( ne(A_27c)
             => ! [A_27d] :
                  ( ne(A_27d)
                 => ! [V0AB] :
                      ( mem(V0AB,arr(A_27a,arr(A_27b,bool)))
                     => ! [V1CD] :
                          ( mem(V1CD,arr(A_27c,arr(A_27d,bool)))
                         => ! [V2f] :
                              ( mem(V2f,arr(A_27a,A_27c))
                             => ! [V3g] :
                                  ( mem(V3g,arr(A_27b,A_27d))
                                 => ( ! [V4a] :
                                        ( mem(V4a,A_27a)
                                       => ! [V5b] :
                                            ( mem(V5b,A_27b)
                                           => ( p(ap(ap(V0AB,V4a),V5b))
                                             => p(ap(ap(V1CD,ap(V2f,V4a)),ap(V3g,V5b))) ) ) )
                                   => p(ap(ap(ap(ap(c_2Etransfer_2EFUN__REL(A_27a,A_27b,A_27c,A_27d),V0AB),V1CD),f978(A_27c,A_27a,V2f)),f979(A_27d,A_27b,V3g))) ) ) ) ) ) ) ) ) ) ).

fof(conj_thm_2Etransfer_2EFUN__REL__EQ2,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => ap(ap(c_2Etransfer_2EFUN__REL(A_27a,A_27a,A_27b,A_27b),c_2Emin_2E_3D(A_27a)),c_2Emin_2E_3D(A_27b)) = c_2Emin_2E_3D(arr(A_27a,A_27b)) ) ) ).

fof(ax_thm_2Etransfer_2EPAIR__REL__def,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [A_27b] :
          ( ne(A_27b)
         => ! [A_27c] :
              ( ne(A_27c)
             => ! [A_27d] :
                  ( ne(A_27d)
                 => ! [V0AB] :
                      ( mem(V0AB,arr(A_27a,arr(A_27b,bool)))
                     => ! [V1CD] :
                          ( mem(V1CD,arr(A_27c,arr(A_27d,bool)))
                         => ! [V2a] :
                              ( mem(V2a,A_27a)
                             => ! [V3c] :
                                  ( mem(V3c,A_27c)
                                 => ! [V4b] :
                                      ( mem(V4b,A_27b)
                                     => ! [V5d] :
                                          ( mem(V5d,A_27d)
                                         => ( p(ap(ap(ap(ap(c_2Etransfer_2EPAIR__REL(A_27a,A_27b,A_27c,A_27d),V0AB),V1CD),ap(ap(c_2Epair_2E_2C(A_27a,A_27c),V2a),V3c)),ap(ap(c_2Epair_2E_2C(A_27b,A_27d),V4b),V5d)))
                                          <=> ( p(ap(ap(V0AB,V2a),V4b))
                                              & p(ap(ap(V1CD,V3c),V5d)) ) ) ) ) ) ) ) ) ) ) ) ) ).

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