ITP001 Axioms: ITP035^7.ax


%------------------------------------------------------------------------------
% File     : ITP035^7 : 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    : listRange.ax [Gau19]
%          : HL4035^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   48 (  11 unt;  28 typ;   0 def)
%            Number of atoms       :   33 (  11 equ;   1 cnn)
%            Maximal formula atoms :    4 (   0 avg)
%            Number of connectives :  157 (   1   ~;   1   |;   3   &; 137   @)
%                                         (   9 <=>;   6  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   6 avg; 137 nst)
%            Number of types       :    3 (   2 usr)
%            Number of type conns  :   46 (  46   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   28 (  26 usr;   2 con; 0-3 aty)
%            Number of variables   :   56 (   2   ^  42   !;   1   ?;  56   :)
%                                         (  11  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
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(tyop_2Enum_2Enum,type,
    tyop_2Enum_2Enum: $tType ).

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

thf(c_2Earithmetic_2E_2B,type,
    c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Earithmetic_2E_2D,type,
    c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

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

thf(c_2Eprim__rec_2E_3C,type,
    c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).

thf(c_2Earithmetic_2E_3C_3D,type,
    c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $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_2Elist_2EALL__DISTINCT,type,
    c_2Elist_2EALL__DISTINCT: 
      !>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).

thf(c_2Earithmetic_2EBIT1,type,
    c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

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_2Elist_2EEL,type,
    c_2Elist_2EEL: 
      !>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Elist_2Elist @ A_27a ) > A_27a ) ).

thf(c_2Elist_2EGENLIST,type,
    c_2Elist_2EGENLIST: 
      !>[A_27a: $tType] : ( ( tyop_2Enum_2Enum > A_27a ) > tyop_2Enum_2Enum > ( tyop_2Elist_2Elist @ A_27a ) ) ).

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

thf(c_2Elist_2ELENGTH,type,
    c_2Elist_2ELENGTH: 
      !>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > tyop_2Enum_2Enum ) ).

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_2Earithmetic_2ENUMERAL,type,
    c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Earithmetic_2EZERO,type,
    c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).

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

thf(c_2ElistRange_2ElistRangeINC,type,
    c_2ElistRange_2ElistRangeINC: tyop_2Enum_2Enum > tyop_2Enum_2Enum > ( tyop_2Elist_2Elist @ tyop_2Enum_2Enum ) ).

thf(c_2ElistRange_2ElistRangeLHI,type,
    c_2ElistRange_2ElistRangeLHI: tyop_2Enum_2Enum > tyop_2Enum_2Enum > ( tyop_2Elist_2Elist @ tyop_2Enum_2Enum ) ).

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_2ElistRange_2ElistRangeINC__def,axiom,
    ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2ElistRange_2ElistRangeINC @ V0m @ V1n )
      = ( c_2Elist_2EGENLIST @ tyop_2Enum_2Enum
        @ ^ [V2i: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B @ V0m @ V2i )
        @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2E_2B @ V1n @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0m ) ) ) ).

thf(thm_2ElistRange_2ElistRangeLHI__def,axiom,
    ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2ElistRange_2ElistRangeLHI @ V0m @ V1n )
      = ( c_2Elist_2EGENLIST @ tyop_2Enum_2Enum
        @ ^ [V2i: tyop_2Enum_2Enum] : ( c_2Earithmetic_2E_2B @ V0m @ V2i )
        @ ( c_2Earithmetic_2E_2D @ V1n @ V0m ) ) ) ).

thf(thm_2ElistRange_2ElistRangeINC__SING,axiom,
    ! [V0m: tyop_2Enum_2Enum] :
      ( ( c_2ElistRange_2ElistRangeINC @ V0m @ V0m )
      = ( c_2Elist_2ECONS @ tyop_2Enum_2Enum @ V0m @ ( c_2Elist_2ENIL @ tyop_2Enum_2Enum ) ) ) ).

thf(thm_2ElistRange_2EMEM__listRangeINC,axiom,
    ! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum] :
      ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V0x @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2ElistRange_2ElistRangeINC @ V2m @ V1n ) ) )
    <=> ( ( c_2Earithmetic_2E_3C_3D @ V2m @ V0x )
        & ( c_2Earithmetic_2E_3C_3D @ V0x @ V1n ) ) ) ).

thf(thm_2ElistRange_2ElistRangeINC__CONS,axiom,
    ! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
      ( ( c_2Earithmetic_2E_3C_3D @ V1m @ V0n )
     => ( ( c_2ElistRange_2ElistRangeINC @ V1m @ V0n )
        = ( c_2Elist_2ECONS @ tyop_2Enum_2Enum @ V1m @ ( c_2ElistRange_2ElistRangeINC @ ( c_2Earithmetic_2E_2B @ V1m @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V0n ) ) ) ) ).

thf(thm_2ElistRange_2ElistRangeINC__EMPTY,axiom,
    ! [V0n: tyop_2Enum_2Enum,V1m: tyop_2Enum_2Enum] :
      ( ( c_2Eprim__rec_2E_3C @ V0n @ V1m )
     => ( ( c_2ElistRange_2ElistRangeINC @ V1m @ V0n )
        = ( c_2Elist_2ENIL @ tyop_2Enum_2Enum ) ) ) ).

thf(thm_2ElistRange_2ElistRangeLHI__EQ,axiom,
    ! [V0m: tyop_2Enum_2Enum] :
      ( ( c_2ElistRange_2ElistRangeLHI @ V0m @ V0m )
      = ( c_2Elist_2ENIL @ tyop_2Enum_2Enum ) ) ).

thf(thm_2ElistRange_2EMEM__listRangeLHI,axiom,
    ! [V0x: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum,V2m: tyop_2Enum_2Enum] :
      ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V0x @ ( c_2Elist_2ELIST__TO__SET @ tyop_2Enum_2Enum @ ( c_2ElistRange_2ElistRangeLHI @ V2m @ V1n ) ) )
    <=> ( ( c_2Earithmetic_2E_3C_3D @ V2m @ V0x )
        & ( c_2Eprim__rec_2E_3C @ V0x @ V1n ) ) ) ).

thf(thm_2ElistRange_2ElistRangeLHI__EMPTY,axiom,
    ! [V0lo: tyop_2Enum_2Enum,V1hi: tyop_2Enum_2Enum] :
      ( ( c_2Earithmetic_2E_3C_3D @ V1hi @ V0lo )
     => ( ( c_2ElistRange_2ElistRangeLHI @ V0lo @ V1hi )
        = ( c_2Elist_2ENIL @ tyop_2Enum_2Enum ) ) ) ).

thf(thm_2ElistRange_2ElistRangeLHI__CONS,axiom,
    ! [V0lo: tyop_2Enum_2Enum,V1hi: tyop_2Enum_2Enum] :
      ( ( c_2Eprim__rec_2E_3C @ V0lo @ V1hi )
     => ( ( c_2ElistRange_2ElistRangeLHI @ V0lo @ V1hi )
        = ( c_2Elist_2ECONS @ tyop_2Enum_2Enum @ V0lo @ ( c_2ElistRange_2ElistRangeLHI @ ( c_2Earithmetic_2E_2B @ V0lo @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ V1hi ) ) ) ) ).

thf(thm_2ElistRange_2ElistRangeLHI__ALL__DISTINCT,axiom,
    ! [V0lo: tyop_2Enum_2Enum,V1hi: tyop_2Enum_2Enum] : ( c_2Elist_2EALL__DISTINCT @ tyop_2Enum_2Enum @ ( c_2ElistRange_2ElistRangeLHI @ V0lo @ V1hi ) ) ).

thf(thm_2ElistRange_2ELENGTH__listRangeLHI,axiom,
    ! [V0lo: tyop_2Enum_2Enum,V1hi: tyop_2Enum_2Enum] :
      ( ( c_2Elist_2ELENGTH @ tyop_2Enum_2Enum @ ( c_2ElistRange_2ElistRangeLHI @ V0lo @ V1hi ) )
      = ( c_2Earithmetic_2E_2D @ V1hi @ V0lo ) ) ).

thf(thm_2ElistRange_2EEL__listRangeLHI,axiom,
    ! [V0lo: tyop_2Enum_2Enum,V1i: tyop_2Enum_2Enum,V2hi: tyop_2Enum_2Enum] :
      ( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2E_2B @ V0lo @ V1i ) @ V2hi )
     => ( ( c_2Elist_2EEL @ tyop_2Enum_2Enum @ V1i @ ( c_2ElistRange_2ElistRangeLHI @ V0lo @ V2hi ) )
        = ( c_2Earithmetic_2E_2B @ V0lo @ V1i ) ) ) ).

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