ITP001 Axioms: ITP056^7.ax


%------------------------------------------------------------------------------
% File     : ITP056^7 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 syntactic export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : transfer.ax [Gau19]
%          : HL4056^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   37 (  12 unt;  19 typ;   0 def)
%            Number of atoms       :   30 (   4 equ;   1 cnn)
%            Maximal formula atoms :    3 (   0 avg)
%            Number of connectives :  169 (   1   ~;   1   |;   7   &; 138   @)
%                                         (  15 <=>;   7  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   21 (  10 avg; 138 nst)
%            Number of types       :    2 (   1 usr)
%            Number of type conns  :   92 (  92   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   20 (  18 usr;   1 con; 0-8 aty)
%            Number of variables   :  113 (   2   ^  83   !;   3   ?; 113   :)
%                                         (  25  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
    tyop_2Emin_2Ebool: $tType ).

thf(tyop_2Emin_2Efun,type,
    tyop_2Emin_2Efun: $tType > $tType > $tType ).

thf(tyop_2Epair_2Eprod,type,
    tyop_2Epair_2Eprod: $tType > $tType > $tType ).

thf(c_2Ebool_2E_21,type,
    c_2Ebool_2E_21: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Epair_2E_2C,type,
    c_2Epair_2E_2C: 
      !>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).

thf(c_2Ebool_2E_2F_5C,type,
    c_2Ebool_2E_2F_5C: $o > $o > $o ).

thf(c_2Emin_2E_3D,type,
    c_2Emin_2E_3D: 
      !>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).

thf(c_2Emin_2E_3D_3D_3E,type,
    c_2Emin_2E_3D_3D_3E: $o > $o > $o ).

thf(c_2Ebool_2E_3F,type,
    c_2Ebool_2E_3F: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Etransfer_2EFUN__REL,type,
    c_2Etransfer_2EFUN__REL: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27b > $o ) > ( A_27c > A_27d > $o ) > ( A_27a > A_27c ) > ( A_27b > A_27d ) > $o ) ).

thf(c_2Etransfer_2EPAIR__REL,type,
    c_2Etransfer_2EPAIR__REL: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType] : ( ( A_27a > A_27b > $o ) > ( A_27c > A_27d > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) > ( tyop_2Epair_2Eprod @ A_27b @ A_27d ) > $o ) ).

thf(c_2Ebool_2E_5C_2F,type,
    c_2Ebool_2E_5C_2F: $o > $o > $o ).

thf(c_2Etransfer_2Ebi__unique,type,
    c_2Etransfer_2Ebi__unique: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > $o ) ).

thf(c_2Etransfer_2Ebitotal,type,
    c_2Etransfer_2Ebitotal: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > $o ) ).

thf(c_2Etransfer_2Eleft__unique,type,
    c_2Etransfer_2Eleft__unique: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > $o ) ).

thf(c_2Etransfer_2Eright__unique,type,
    c_2Etransfer_2Eright__unique: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > $o ) ).

thf(c_2Etransfer_2Esurj,type,
    c_2Etransfer_2Esurj: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > $o ) ).

thf(c_2Etransfer_2Etotal,type,
    c_2Etransfer_2Etotal: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > $o ) ).

thf(c_2Ebool_2E_7E,type,
    c_2Ebool_2E_7E: $o > $o ).

thf(logicdef_2E_2F_5C,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
    <=> ( V0
        & V1 ) ) ).

thf(logicdef_2E_5C_2F,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
    <=> ( V0
        | V1 ) ) ).

thf(logicdef_2E_7E,axiom,
    ! [V0: $o] :
      ( ( c_2Ebool_2E_7E @ V0 )
    <=> ( (~) @ V0 ) ) ).

thf(logicdef_2E_3D_3D_3E,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
    <=> ( V0
       => V1 ) ) ).

thf(logicdef_2E_3D,axiom,
    ! [A_27a: $tType,V0: A_27a,V1: A_27a] :
      ( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
    <=> ( V0 = V1 ) ) ).

thf(quantdef_2E_21,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o] :
      ( ( c_2Ebool_2E_21 @ A_27a @ V0f )
    <=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).

thf(quantdef_2E_3F,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o] :
      ( ( c_2Ebool_2E_3F @ A_27a @ V0f )
    <=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).

thf(thm_2Etransfer_2Eright__unique__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
      ( ( c_2Etransfer_2Eright__unique @ A_27a @ A_27b @ V0R )
    <=> ! [V1a: A_27a,V2b1: A_27b,V3b2: A_27b] :
          ( ( ( V0R @ V1a @ V2b1 )
            & ( V0R @ V1a @ V3b2 ) )
         => ( V2b1 = V3b2 ) ) ) ).

thf(thm_2Etransfer_2Eleft__unique__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
      ( ( c_2Etransfer_2Eleft__unique @ A_27a @ A_27b @ V0R )
    <=> ! [V1a1: A_27a,V2a2: A_27a,V3b: A_27b] :
          ( ( ( V0R @ V1a1 @ V3b )
            & ( V0R @ V2a2 @ V3b ) )
         => ( V1a1 = V2a2 ) ) ) ).

thf(thm_2Etransfer_2Ebi__unique__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
      ( ( c_2Etransfer_2Ebi__unique @ A_27a @ A_27b @ V0R )
    <=> ( ( c_2Etransfer_2Eleft__unique @ A_27a @ A_27b @ V0R )
        & ( c_2Etransfer_2Eright__unique @ A_27a @ A_27b @ V0R ) ) ) ).

thf(thm_2Etransfer_2Etotal__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
      ( ( c_2Etransfer_2Etotal @ A_27a @ A_27b @ V0R )
    <=> ! [V1x: A_27a] :
        ? [V2y: A_27b] : ( V0R @ V1x @ V2y ) ) ).

thf(thm_2Etransfer_2Esurj__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
      ( ( c_2Etransfer_2Esurj @ A_27a @ A_27b @ V0R )
    <=> ! [V1y: A_27b] :
        ? [V2x: A_27a] : ( V0R @ V2x @ V1y ) ) ).

thf(thm_2Etransfer_2Ebitotal__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
      ( ( c_2Etransfer_2Ebitotal @ A_27a @ A_27b @ V0R )
    <=> ( ( c_2Etransfer_2Etotal @ A_27a @ A_27b @ V0R )
        & ( c_2Etransfer_2Esurj @ A_27a @ A_27b @ V0R ) ) ) ).

thf(thm_2Etransfer_2EFUN__REL__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0AB: A_27a > A_27b > $o,V1CD: A_27c > A_27d > $o,V2f: A_27a > A_27c,V3g: A_27b > A_27d] :
      ( ( c_2Etransfer_2EFUN__REL @ A_27a @ A_27b @ A_27c @ A_27d @ V0AB @ V1CD @ V2f @ V3g )
    <=> ! [V4a: A_27a,V5b: A_27b] :
          ( ( V0AB @ V4a @ V5b )
         => ( V1CD @ ( V2f @ V4a ) @ ( V3g @ V5b ) ) ) ) ).

thf(thm_2Etransfer_2EPAIR__REL__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0AB: A_27a > A_27b > $o,V1CD: A_27c > A_27d > $o,V2a: A_27a,V3c: A_27c,V4b: A_27b,V5d: A_27d] :
      ( ( c_2Etransfer_2EPAIR__REL @ A_27a @ A_27b @ A_27c @ A_27d @ V0AB @ V1CD @ ( c_2Epair_2E_2C @ A_27a @ A_27c @ V2a @ V3c ) @ ( c_2Epair_2E_2C @ A_27b @ A_27d @ V4b @ V5d ) )
    <=> ( ( V0AB @ V2a @ V4b )
        & ( V1CD @ V3c @ V5d ) ) ) ).

thf(thm_2Etransfer_2EFUN__REL__COMB,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0g: A_27b > A_27d,V1f: A_27a > A_27c,V2b: A_27b,V3a: A_27a,V4CD: A_27c > A_27d > $o,V5AB: A_27a > A_27b > $o] :
      ( ( ( c_2Etransfer_2EFUN__REL @ A_27a @ A_27b @ A_27c @ A_27d @ V5AB @ V4CD @ V1f @ V0g )
        & ( V5AB @ V3a @ V2b ) )
     => ( V4CD @ ( V1f @ V3a ) @ ( V0g @ V2b ) ) ) ).

thf(thm_2Etransfer_2EFUN__REL__ABS,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0g: A_27b > A_27d,V1f: A_27a > A_27c,V2CD: A_27c > A_27d > $o,V3AB: A_27a > A_27b > $o] :
      ( ! [V4a: A_27a,V5b: A_27b] :
          ( ( V3AB @ V4a @ V5b )
         => ( V2CD @ ( V1f @ V4a ) @ ( V0g @ V5b ) ) )
     => ( c_2Etransfer_2EFUN__REL @ A_27a @ A_27b @ A_27c @ A_27d @ V3AB @ V2CD
        @ ^ [V6a: A_27a] : ( V1f @ V6a )
        @ ^ [V7b: A_27b] : ( V0g @ V7b ) ) ) ).

thf(thm_2Etransfer_2EFUN__REL__EQ2,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ( 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 @ ( A_27a > A_27b ) ) ) ).

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