ITP001 Axioms: ITP103^7.ax


%------------------------------------------------------------------------------
% File     : ITP103^7 : 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    : tc.ax [Gau19]
%          : HL4103^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   81 (  20 unt;  40 typ;   0 def)
%            Number of atoms       :  194 (  37 equ;   3 cnn)
%            Maximal formula atoms :   13 (   2 avg)
%            Number of connectives :  612 (   3   ~;   6   |;  17   &; 563   @)
%                                         (  13 <=>;  10  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   20 (   7 avg; 563 nst)
%            Number of types       :    2 (   1 usr)
%            Number of type conns  :  240 ( 240   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   41 (  39 usr;   1 con; 0-7 aty)
%            Number of variables   :  177 (   0   ^ 131   !;   5   ?; 177   :)
%                                         (  41  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Efinite__map_2Efmap,type,
    tyop_2Efinite__map_2Efmap: $tType > $tType > $tType ).

thf(tyop_2Elist_2Elist,type,
    tyop_2Elist_2Elist: $tType > $tType ).

thf(tyop_2Emin_2Ebool,type,
    tyop_2Emin_2Ebool: $tType ).

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

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

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_2Ebool_2ECOND,type,
    c_2Ebool_2ECOND: 
      !>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).

thf(c_2Elist_2ECONS,type,
    c_2Elist_2ECONS: 
      !>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).

thf(c_2Epred__set_2EEMPTY,type,
    c_2Epred__set_2EEMPTY: 
      !>[A_27a: $tType] : ( A_27a > $o ) ).

thf(c_2Erelation_2EEMPTY__REL,type,
    c_2Erelation_2EEMPTY__REL: 
      !>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).

thf(c_2Efinite__map_2EFAPPLY,type,
    c_2Efinite__map_2EFAPPLY: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > A_27a > A_27b ) ).

thf(c_2Efinite__map_2EFDOM,type,
    c_2Efinite__map_2EFDOM: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > A_27a > $o ) ).

thf(c_2Epred__set_2EFINITE,type,
    c_2Epred__set_2EFINITE: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Etc_2EFMAP__TO__RELN,type,
    c_2Etc_2EFMAP__TO__RELN: 
      !>[A_27a: $tType] : ( ( tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o ) ) > A_27a > A_27a > $o ) ).

thf(c_2Efinite__map_2EFUN__FMAP,type,
    c_2Efinite__map_2EFUN__FMAP: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) ) ).

thf(c_2Ecombin_2EI,type,
    c_2Ecombin_2EI: 
      !>[A_27a: $tType] : ( A_27a > A_27a ) ).

thf(c_2Ebool_2EIN,type,
    c_2Ebool_2EIN: 
      !>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).

thf(c_2Epred__set_2EINSERT,type,
    c_2Epred__set_2EINSERT: 
      !>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).

thf(c_2Elist_2ELIST__TO__SET,type,
    c_2Elist_2ELIST__TO__SET: 
      !>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27a > $o ) ).

thf(c_2Elist_2ENIL,type,
    c_2Elist_2ENIL: 
      !>[A_27a: $tType] : ( tyop_2Elist_2Elist @ A_27a ) ).

thf(c_2Erelation_2EO,type,
    c_2Erelation_2EO: 
      !>[A_27g: $tType,A_27h: $tType,A_27k: $tType] : ( ( A_27h > A_27k > $o ) > ( A_27g > A_27h > $o ) > A_27g > A_27k > $o ) ).

thf(c_2Erelation_2ERDOM,type,
    c_2Erelation_2ERDOM: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > A_27a > $o ) ).

thf(c_2Etc_2ERELN__TO__FMAP,type,
    c_2Etc_2ERELN__TO__FMAP: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o ) ) ) ).

thf(c_2Erelation_2ERTC,type,
    c_2Erelation_2ERTC: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).

thf(c_2Erelation_2ERUNION,type,
    c_2Erelation_2ERUNION: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( A_27a > A_27b > $o ) > A_27a > A_27b > $o ) ).

thf(c_2Epred__set_2ESUBSET,type,
    c_2Epred__set_2ESUBSET: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).

thf(c_2Erelation_2ETC,type,
    c_2Erelation_2ETC: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).

thf(c_2Etc_2ETC__ITER,type,
    c_2Etc_2ETC__ITER: 
      !>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o ) ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o ) ) ) ).

thf(c_2Etc_2ETC__MOD,type,
    c_2Etc_2ETC__MOD: 
      !>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).

thf(c_2Epred__set_2EUNION,type,
    c_2Epred__set_2EUNION: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).

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

thf(c_2Etc_2E_5E_7C,type,
    c_2Etc_2E_5E_7C: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > A_27a > A_27a > $o ) ).

thf(c_2Etc_2E_5E_7C_5E,type,
    c_2Etc_2E_5E_7C_5E: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > A_27a > A_27a > $o ) ).

thf(c_2Efinite__map_2Eo__f,type,
    c_2Efinite__map_2Eo__f: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27b > A_27c ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27b ) > ( tyop_2Efinite__map_2Efmap @ A_27a @ A_27c ) ) ).

thf(c_2Etc_2EsubTC,type,
    c_2Etc_2EsubTC: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > A_27a > A_27a > $o ) ).

thf(c_2Etc_2E_7C_5E,type,
    c_2Etc_2E_7C_5E: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > A_27a > A_27a > $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_2Etc_2EDRESTR,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2a: A_27a,V3b: A_27a] :
      ( ( c_2Etc_2E_5E_7C @ A_27a @ V0R @ V1s @ V2a @ V3b )
    <=> ( ( c_2Ebool_2EIN @ A_27a @ V2a @ V1s )
        & ( V0R @ V2a @ V3b ) ) ) ).

thf(thm_2Etc_2ERRESTR,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2a: A_27a,V3b: A_27a] :
      ( ( c_2Etc_2E_7C_5E @ A_27a @ V0R @ V1s @ V2a @ V3b )
    <=> ( ( c_2Ebool_2EIN @ A_27a @ V3b @ V1s )
        & ( V0R @ V2a @ V3b ) ) ) ).

thf(thm_2Etc_2EBRESTR,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o] :
      ( ( c_2Etc_2E_5E_7C_5E @ A_27a @ V0R @ V1s )
      = ( c_2Etc_2E_7C_5E @ A_27a @ ( c_2Etc_2E_5E_7C @ A_27a @ V0R @ V1s ) @ V1s ) ) ).

thf(thm_2Etc_2EsubTC,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2x: A_27a,V3y: A_27a] :
      ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s @ V2x @ V3y )
    <=> ( ( V0R @ V2x @ V3y )
        | ? [V4a: A_27a,V5b: A_27a] :
            ( ( c_2Erelation_2ERTC @ A_27a @ ( c_2Etc_2E_5E_7C_5E @ A_27a @ V0R @ V1s ) @ V4a @ V5b )
            & ( c_2Ebool_2EIN @ A_27a @ V4a @ V1s )
            & ( c_2Ebool_2EIN @ A_27a @ V5b @ V1s )
            & ( V0R @ V2x @ V4a )
            & ( V0R @ V5b @ V3y ) ) ) ) ).

thf(thm_2Etc_2EFMAP__TO__RELN,axiom,
    ! [A_27a: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o ),V1x: A_27a] :
      ( ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V0f @ V1x )
      = ( c_2Ebool_2ECOND @ ( A_27a > $o ) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( A_27a > $o ) @ V0f ) ) @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( A_27a > $o ) @ V0f @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).

thf(thm_2Etc_2ERELN__TO__FMAP,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
      ( ( c_2Etc_2ERELN__TO__FMAP @ A_27a @ V0R )
      = ( c_2Efinite__map_2EFUN__FMAP @ A_27a @ ( A_27a > $o ) @ V0R @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) ) ) ).

thf(thm_2Etc_2ETC__MOD,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1rx: A_27a > $o,V2ra: A_27a > $o] :
      ( ( c_2Etc_2ETC__MOD @ A_27a @ V0x @ V1rx @ V2ra )
      = ( c_2Ebool_2ECOND @ ( A_27a > $o ) @ ( c_2Ebool_2EIN @ A_27a @ V0x @ V2ra ) @ ( c_2Epred__set_2EUNION @ A_27a @ V2ra @ V1rx ) @ V2ra ) ) ).

thf(thm_2Etc_2ETC__ITER,axiom,
    ! [A_27a: $tType] :
      ( ! [V0r: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o )] :
          ( ( c_2Etc_2ETC__ITER @ A_27a @ ( c_2Elist_2ENIL @ A_27a ) @ V0r )
          = V0r )
      & ! [V1x: A_27a,V2d: tyop_2Elist_2Elist @ A_27a,V3r: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o )] :
          ( ( c_2Etc_2ETC__ITER @ A_27a @ ( c_2Elist_2ECONS @ A_27a @ V1x @ V2d ) @ V3r )
          = ( c_2Etc_2ETC__ITER @ A_27a @ V2d @ ( c_2Efinite__map_2Eo__f @ A_27a @ ( A_27a > $o ) @ ( A_27a > $o ) @ ( c_2Etc_2ETC__MOD @ A_27a @ V1x @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( A_27a > $o ) @ V3r @ V1x ) ) @ V3r ) ) ) ) ).

thf(thm_2Etc_2EDRESTR__IN,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2a: A_27a] :
      ( ( c_2Etc_2E_5E_7C @ A_27a @ V0R @ V1s @ V2a )
      = ( c_2Ebool_2ECOND @ ( A_27a > $o ) @ ( c_2Ebool_2EIN @ A_27a @ V2a @ V1s ) @ ( V0R @ V2a ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).

thf(thm_2Etc_2EDRESTR__EMPTY,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
      ( ( c_2Etc_2E_5E_7C @ A_27a @ V0R @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
      = ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) ).

thf(thm_2Etc_2EDRESTR__RDOM,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
      ( ( c_2Etc_2E_5E_7C @ A_27a @ V0R @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) )
      = V0R ) ).

thf(thm_2Etc_2EREMPTY__RRESTR,axiom,
    ! [A_27a: $tType,V0s: A_27a > $o] :
      ( ( c_2Etc_2E_7C_5E @ A_27a @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) @ V0s )
      = ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) ).

thf(thm_2Etc_2EO__REMPTY__O,axiom,
    ! [A_27a: $tType] :
      ( ! [V0R: A_27a > A_27a > $o] :
          ( ( c_2Erelation_2EO @ A_27a @ A_27a @ A_27a @ V0R @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) )
          = ( c_2Erelation_2EEMPTY__REL @ A_27a ) )
      & ! [V1R: A_27a > A_27a > $o] :
          ( ( c_2Erelation_2EO @ A_27a @ A_27a @ A_27a @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) @ V1R )
          = ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) ) ).

thf(thm_2Etc_2EsubTC__thm,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o] :
      ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s )
      = ( c_2Erelation_2ERUNION @ A_27a @ A_27a @ V0R @ ( c_2Erelation_2EO @ A_27a @ A_27a @ A_27a @ V0R @ ( c_2Erelation_2EO @ A_27a @ A_27a @ A_27a @ ( c_2Etc_2E_5E_7C @ A_27a @ ( c_2Erelation_2ERTC @ A_27a @ ( c_2Etc_2E_5E_7C_5E @ A_27a @ V0R @ V1s ) ) @ V1s ) @ V0R ) ) ) ) ).

thf(thm_2Etc_2EsubTC__EMPTY,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
      ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
      = V0R ) ).

thf(thm_2Etc_2ERTC__INSERT,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2a: A_27a,V3w: A_27a,V4z: A_27a] :
      ( ( c_2Erelation_2ERTC @ A_27a @ ( c_2Etc_2E_5E_7C_5E @ A_27a @ V0R @ ( c_2Epred__set_2EINSERT @ A_27a @ V2a @ V1s ) ) @ V3w @ V4z )
    <=> ( ( c_2Erelation_2ERTC @ A_27a @ ( c_2Etc_2E_5E_7C_5E @ A_27a @ V0R @ V1s ) @ V3w @ V4z )
        | ( ( ( V2a = V3w )
            | ? [V5x: A_27a] :
                ( ( c_2Ebool_2EIN @ A_27a @ V5x @ V1s )
                & ( c_2Erelation_2ERTC @ A_27a @ ( c_2Etc_2E_5E_7C_5E @ A_27a @ V0R @ V1s ) @ V3w @ V5x )
                & ( V0R @ V5x @ V2a ) ) )
          & ( ( V2a = V4z )
            | ? [V6y: A_27a] :
                ( ( c_2Ebool_2EIN @ A_27a @ V6y @ V1s )
                & ( V0R @ V2a @ V6y )
                & ( c_2Erelation_2ERTC @ A_27a @ ( c_2Etc_2E_5E_7C_5E @ A_27a @ V0R @ V1s ) @ V6y @ V4z ) ) ) ) ) ) ).

thf(thm_2Etc_2EsubTC__INSERT,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2q: A_27a,V3x: A_27a,V4y: A_27a] :
      ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Epred__set_2EINSERT @ A_27a @ V2q @ V1s ) @ V3x @ V4y )
    <=> ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s @ V3x @ V4y )
        | ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s @ V3x @ V2q )
          & ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s @ V2q @ V4y ) ) ) ) ).

thf(thm_2Etc_2EsubTC__RDOM,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
      ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) )
      = ( c_2Erelation_2ETC @ A_27a @ V0R ) ) ).

thf(thm_2Etc_2EsubTC__INSERT__COR,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2x: A_27a,V3a: A_27a] :
      ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ V1s ) @ V3a )
      = ( c_2Ebool_2ECOND @ ( A_27a > $o ) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s @ V3a ) ) @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s @ V3a ) @ ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s @ V2x ) ) @ ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s @ V3a ) ) ) ).

thf(thm_2Etc_2ERDOM__SUBSET__FDOM,axiom,
    ! [A_27a: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o )] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V0f ) ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( A_27a > $o ) @ V0f ) ) ).

thf(thm_2Etc_2EFINITE__RDOM,axiom,
    ! [A_27a: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o )] : ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V0f ) ) ) ).

thf(thm_2Etc_2EFDOM__RDOM,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
      ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) )
     => ( ( c_2Efinite__map_2EFDOM @ A_27a @ ( A_27a > $o ) @ ( c_2Etc_2ERELN__TO__FMAP @ A_27a @ V0R ) )
        = ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) ) ) ).

thf(thm_2Etc_2ERELN__TO__FMAP__TO__RELN__ID,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
      ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) )
     => ( ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ ( c_2Etc_2ERELN__TO__FMAP @ A_27a @ V0R ) )
        = V0R ) ) ).

thf(thm_2Etc_2ERDOM__subTC,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o] :
      ( ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s ) )
      = ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) ) ).

thf(thm_2Etc_2ENOT__IN__RDOM,axiom,
    ! [A_27a: $tType,V0Q: A_27a > A_27a > $o,V1x: A_27a] :
      ( ( ( V0Q @ V1x )
        = ( c_2Epred__set_2EEMPTY @ A_27a ) )
    <=> ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0Q ) ) ) ) ).

thf(thm_2Etc_2ETC__MOD__EMPTY__ID,axiom,
    ! [A_27a: $tType,V0x: A_27a,V1ra: A_27a > $o] :
      ( ( c_2Etc_2ETC__MOD @ A_27a @ V0x @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
      = ( c_2Ecombin_2EI @ ( A_27a > $o ) ) ) ).

thf(thm_2Etc_2EI__o__f,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efinite__map_2Efmap @ A_27a @ A_27b] :
      ( ( c_2Efinite__map_2Eo__f @ A_27a @ A_27b @ A_27b @ ( c_2Ecombin_2EI @ A_27b ) @ V0f )
      = V0f ) ).

thf(thm_2Etc_2EsubTC__MAX__RDOM,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2x: A_27a] :
      ( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) ) )
     => ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ V1s ) )
        = ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s ) ) ) ).

thf(thm_2Etc_2EsubTC__SUPERSET__RDOM,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o] :
      ( ( c_2Epred__set_2EFINITE @ A_27a @ V1s )
     => ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) @ V1s ) )
        = ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) ) ) ) ).

thf(thm_2Etc_2EsubTC__FDOM,axiom,
    ! [A_27a: $tType,V0g: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o ),V1R: A_27a > A_27a > $o] :
      ( ( ( c_2Etc_2EsubTC @ A_27a @ V1R @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V1R ) )
        = ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V0g ) )
     => ( ( c_2Etc_2EsubTC @ A_27a @ V1R @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( A_27a > $o ) @ V0g ) )
        = ( c_2Etc_2EsubTC @ A_27a @ V1R @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V1R ) ) ) ) ).

thf(thm_2Etc_2ESUBSET__FDOM__LEM,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2f: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o )] :
      ( ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s )
        = ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V2f ) )
     => ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( A_27a > $o ) @ V2f ) ) ) ).

thf(thm_2Etc_2EsubTC__FDOM__RDOM,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1f: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o )] :
      ( ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( A_27a > $o ) @ V1f ) )
        = ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V1f ) )
     => ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Erelation_2ERDOM @ A_27a @ A_27a @ V0R ) )
        = ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V1f ) ) ) ).

thf(thm_2Etc_2ETC__MOD__LEM,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o,V2x: A_27a,V3f: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o )] :
      ( ( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Efinite__map_2EFDOM @ A_27a @ ( A_27a > $o ) @ V3f ) )
        & ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ V1s )
          = ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V3f ) ) )
     => ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ V1s ) )
        = ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ ( c_2Efinite__map_2Eo__f @ A_27a @ ( A_27a > $o ) @ ( A_27a > $o ) @ ( c_2Etc_2ETC__MOD @ A_27a @ V2x @ ( c_2Efinite__map_2EFAPPLY @ A_27a @ ( A_27a > $o ) @ V3f @ V2x ) ) @ V3f ) ) ) ) ).

thf(thm_2Etc_2ETC__ITER__THM,axiom,
    ! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1d: tyop_2Elist_2Elist @ A_27a,V2f: tyop_2Efinite__map_2Efmap @ A_27a @ ( A_27a > $o ),V3s: A_27a > $o] :
      ( ( ( ( c_2Epred__set_2EUNION @ A_27a @ V3s @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V1d ) )
          = ( c_2Efinite__map_2EFDOM @ A_27a @ ( A_27a > $o ) @ V2f ) )
        & ( ( c_2Etc_2EsubTC @ A_27a @ V0R @ V3s )
          = ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ V2f ) ) )
     => ( ( c_2Erelation_2ETC @ A_27a @ V0R )
        = ( c_2Etc_2EFMAP__TO__RELN @ A_27a @ ( c_2Etc_2ETC__ITER @ A_27a @ V1d @ V2f ) ) ) ) ).

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