ITP001 Axioms: ITP056^5.ax


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% 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    :   27 (   1 unt;   8 typ;   0 def)
%            Number of atoms       :  259 (   3 equ;   0 cnn)
%            Maximal formula atoms :   31 (   9 avg)
%            Number of connectives :  537 (   0   ~;   0   |;   8   &; 477   @)
%                                         (   8 <=>;  44  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   32 (  15 avg; 477 nst)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :   20 (  20   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   19 (  18 usr;  10 con; 0-4 aty)
%            Number of variables   :   92 (   2   ^  88   !;   2   ?;  92   :)
% SPC      : TH0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Etransfer_2EFUN__REL,type,
    c_2Etransfer_2EFUN__REL: del > del > del > del > $i ).

thf(mem_c_2Etransfer_2EFUN__REL,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( 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 ) ) ) ) ) ).

thf(tp_c_2Etransfer_2EPAIR__REL,type,
    c_2Etransfer_2EPAIR__REL: del > del > del > del > $i ).

thf(mem_c_2Etransfer_2EPAIR__REL,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( 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 ) ) ) ) ) ).

thf(tp_c_2Etransfer_2Ebi__unique,type,
    c_2Etransfer_2Ebi__unique: del > del > $i ).

thf(mem_c_2Etransfer_2Ebi__unique,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Etransfer_2Ebi__unique @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ bool ) ) ).

thf(tp_c_2Etransfer_2Ebitotal,type,
    c_2Etransfer_2Ebitotal: del > del > $i ).

thf(mem_c_2Etransfer_2Ebitotal,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Etransfer_2Ebitotal @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ bool ) ) ).

thf(tp_c_2Etransfer_2Eleft__unique,type,
    c_2Etransfer_2Eleft__unique: del > del > $i ).

thf(mem_c_2Etransfer_2Eleft__unique,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Etransfer_2Eleft__unique @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ bool ) ) ).

thf(tp_c_2Etransfer_2Eright__unique,type,
    c_2Etransfer_2Eright__unique: del > del > $i ).

thf(mem_c_2Etransfer_2Eright__unique,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Etransfer_2Eright__unique @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ bool ) ) ).

thf(tp_c_2Etransfer_2Esurj,type,
    c_2Etransfer_2Esurj: del > del > $i ).

thf(mem_c_2Etransfer_2Esurj,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Etransfer_2Esurj @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ bool ) ) ).

thf(tp_c_2Etransfer_2Etotal,type,
    c_2Etransfer_2Etotal: del > del > $i ).

thf(mem_c_2Etransfer_2Etotal,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Etransfer_2Etotal @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ bool ) ) ).

thf(ax_thm_2Etransfer_2Eright__unique__def,axiom,
    ! [A_27a: del,A_27b: del,V0R: $i] :
      ( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
     => ( ( p @ ( ap @ ( c_2Etransfer_2Eright__unique @ A_27a @ A_27b ) @ V0R ) )
      <=> ! [V1a: $i] :
            ( ( mem @ V1a @ A_27a )
           => ! [V2b1: $i] :
                ( ( mem @ V2b1 @ A_27b )
               => ! [V3b2: $i] :
                    ( ( mem @ V3b2 @ A_27b )
                   => ( ( ( p @ ( ap @ ( ap @ V0R @ V1a ) @ V2b1 ) )
                        & ( p @ ( ap @ ( ap @ V0R @ V1a ) @ V3b2 ) ) )
                     => ( V2b1 = V3b2 ) ) ) ) ) ) ) ).

thf(ax_thm_2Etransfer_2Eleft__unique__def,axiom,
    ! [A_27a: del,A_27b: del,V0R: $i] :
      ( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
     => ( ( p @ ( ap @ ( c_2Etransfer_2Eleft__unique @ A_27a @ A_27b ) @ V0R ) )
      <=> ! [V1a1: $i] :
            ( ( mem @ V1a1 @ A_27a )
           => ! [V2a2: $i] :
                ( ( mem @ V2a2 @ A_27a )
               => ! [V3b: $i] :
                    ( ( mem @ V3b @ A_27b )
                   => ( ( ( p @ ( ap @ ( ap @ V0R @ V1a1 ) @ V3b ) )
                        & ( p @ ( ap @ ( ap @ V0R @ V2a2 ) @ V3b ) ) )
                     => ( V1a1 = V2a2 ) ) ) ) ) ) ) ).

thf(ax_thm_2Etransfer_2Ebi__unique__def,axiom,
    ! [A_27a: del,A_27b: del,V0R: $i] :
      ( ( 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 ) ) ) ) ) ).

thf(ax_thm_2Etransfer_2Etotal__def,axiom,
    ! [A_27a: del,A_27b: del,V0R: $i] :
      ( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
     => ( ( p @ ( ap @ ( c_2Etransfer_2Etotal @ A_27a @ A_27b ) @ V0R ) )
      <=> ! [V1x: $i] :
            ( ( mem @ V1x @ A_27a )
           => ? [V2y: $i] :
                ( ( mem @ V2y @ A_27b )
                & ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) ) ) ) ) ) ).

thf(ax_thm_2Etransfer_2Esurj__def,axiom,
    ! [A_27a: del,A_27b: del,V0R: $i] :
      ( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
     => ( ( p @ ( ap @ ( c_2Etransfer_2Esurj @ A_27a @ A_27b ) @ V0R ) )
      <=> ! [V1y: $i] :
            ( ( mem @ V1y @ A_27b )
           => ? [V2x: $i] :
                ( ( mem @ V2x @ A_27a )
                & ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V1y ) ) ) ) ) ) ).

thf(ax_thm_2Etransfer_2Ebitotal__def,axiom,
    ! [A_27a: del,A_27b: del,V0R: $i] :
      ( ( 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 ) ) ) ) ) ).

thf(ax_thm_2Etransfer_2EFUN__REL__def,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0AB: $i] :
      ( ( mem @ V0AB @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
     => ! [V1CD: $i] :
          ( ( mem @ V1CD @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) )
         => ! [V2f: $i] :
              ( ( mem @ V2f @ ( arr @ A_27a @ A_27c ) )
             => ! [V3g: $i] :
                  ( ( 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: $i] :
                        ( ( mem @ V4a @ A_27a )
                       => ! [V5b: $i] :
                            ( ( mem @ V5b @ A_27b )
                           => ( ( p @ ( ap @ ( ap @ V0AB @ V4a ) @ V5b ) )
                             => ( p @ ( ap @ ( ap @ V1CD @ ( ap @ V2f @ V4a ) ) @ ( ap @ V3g @ V5b ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Etransfer_2EFUN__REL__COMB,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0AB: $i] :
      ( ( mem @ V0AB @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
     => ! [V1CD: $i] :
          ( ( mem @ V1CD @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) )
         => ! [V2f: $i] :
              ( ( mem @ V2f @ ( arr @ A_27a @ A_27c ) )
             => ! [V3g: $i] :
                  ( ( mem @ V3g @ ( arr @ A_27b @ A_27d ) )
                 => ! [V4a: $i] :
                      ( ( mem @ V4a @ A_27a )
                     => ! [V5b: $i] :
                          ( ( 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 ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Etransfer_2EFUN__REL__ABS,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0AB: $i] :
      ( ( mem @ V0AB @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
     => ! [V1CD: $i] :
          ( ( mem @ V1CD @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) )
         => ! [V2f: $i] :
              ( ( mem @ V2f @ ( arr @ A_27a @ A_27c ) )
             => ! [V3g: $i] :
                  ( ( mem @ V3g @ ( arr @ A_27b @ A_27d ) )
                 => ( ! [V4a: $i] :
                        ( ( mem @ V4a @ A_27a )
                       => ! [V5b: $i] :
                            ( ( 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 )
                          @ ( lam @ A_27a
                            @ ^ [V6a: $i] : ( ap @ V2f @ V6a ) ) )
                        @ ( lam @ A_27b
                          @ ^ [V7b: $i] : ( ap @ V3g @ V7b ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Etransfer_2EFUN__REL__EQ2,axiom,
    ! [A_27a: del,A_27b: del] :
      ( ( 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 ) ) ) ).

thf(ax_thm_2Etransfer_2EPAIR__REL__def,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0AB: $i] :
      ( ( mem @ V0AB @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
     => ! [V1CD: $i] :
          ( ( mem @ V1CD @ ( arr @ A_27c @ ( arr @ A_27d @ bool ) ) )
         => ! [V2a: $i] :
              ( ( mem @ V2a @ A_27a )
             => ! [V3c: $i] :
                  ( ( mem @ V3c @ A_27c )
                 => ! [V4b: $i] :
                      ( ( mem @ V4b @ A_27b )
                     => ! [V5d: $i] :
                          ( ( 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 ) ) ) ) ) ) ) ) ) ) ).

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