ITP001 Axioms: ITP051^7.ax


%------------------------------------------------------------------------------
% File     : ITP051^7 : 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    : readerMonad.ax [Gau19]
%          : HL4051^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   38 (  19 unt;  16 typ;   0 def)
%            Number of atoms       :   25 (  17 equ;   1 cnn)
%            Maximal formula atoms :    2 (   0 avg)
%            Number of connectives :  199 (   1   ~;   1   |;   2   &; 187   @)
%                                         (   7 <=>;   1  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   6 avg; 187 nst)
%            Number of types       :    2 (   1 usr)
%            Number of type conns  :   89 (  89   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   17 (  15 usr;   1 con; 0-8 aty)
%            Number of variables   :  110 (   1   ^  87   !;   1   ?; 110   :)
%                                         (  21  !>;   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(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_2EreaderMonad_2EBIND,type,
    c_2EreaderMonad_2EBIND: 
      !>[A_27a: $tType,A_27b: $tType,A_27s: $tType] : ( ( A_27s > A_27a ) > ( A_27a > A_27s > A_27b ) > A_27s > A_27b ) ).

thf(c_2EreaderMonad_2EFMAP,type,
    c_2EreaderMonad_2EFMAP: 
      !>[A_27a: $tType,A_27b: $tType,A_27s: $tType] : ( ( A_27a > A_27b ) > ( A_27s > A_27a ) > A_27s > A_27b ) ).

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

thf(c_2EreaderMonad_2EJOIN,type,
    c_2EreaderMonad_2EJOIN: 
      !>[A_27a: $tType,A_27s: $tType] : ( ( A_27s > A_27s > A_27a ) > A_27s > A_27a ) ).

thf(c_2EreaderMonad_2EMCOMPOSE,type,
    c_2EreaderMonad_2EMCOMPOSE: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27s: $tType] : ( ( A_27a > A_27s > A_27b ) > ( A_27b > A_27s > A_27c ) > A_27a > A_27s > A_27c ) ).

thf(c_2EreaderMonad_2EUNIT,type,
    c_2EreaderMonad_2EUNIT: 
      !>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > A_27a ) ).

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

thf(c_2Ecombin_2Eo,type,
    c_2Ecombin_2Eo: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27c > A_27b ) > ( A_27a > A_27c ) > A_27a > A_27b ) ).

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_2EreaderMonad_2EBIND__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27s: $tType,V0M: A_27s > A_27a,V1f: A_27a > A_27s > A_27b,V2s: A_27s] :
      ( ( c_2EreaderMonad_2EBIND @ A_27a @ A_27b @ A_27s @ V0M @ V1f @ V2s )
      = ( V1f @ ( V0M @ V2s ) @ V2s ) ) ).

thf(thm_2EreaderMonad_2EUNIT__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1s: A_27b] :
      ( ( c_2EreaderMonad_2EUNIT @ A_27a @ A_27b @ V0x @ V1s )
      = V0x ) ).

thf(thm_2EreaderMonad_2EMCOMPOSE__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27s: $tType,V0f1: A_27a > A_27s > A_27b,V1f2: A_27b > A_27s > A_27c,V2a: A_27a] :
      ( ( c_2EreaderMonad_2EMCOMPOSE @ A_27a @ A_27b @ A_27c @ A_27s @ V0f1 @ V1f2 @ V2a )
      = ( c_2EreaderMonad_2EBIND @ A_27b @ A_27c @ A_27s @ ( V0f1 @ V2a ) @ V1f2 ) ) ).

thf(thm_2EreaderMonad_2EFMAP__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27s: $tType,V0f: A_27a > A_27b,V1M1: A_27s > A_27a] :
      ( ( c_2EreaderMonad_2EFMAP @ A_27a @ A_27b @ A_27s @ V0f @ V1M1 )
      = ( c_2Ecombin_2Eo @ A_27s @ A_27b @ A_27a @ V0f @ V1M1 ) ) ).

thf(thm_2EreaderMonad_2EJOIN__def,axiom,
    ! [A_27a: $tType,A_27s: $tType,V0MM: A_27s > A_27s > A_27a,V1s: A_27s] :
      ( ( c_2EreaderMonad_2EJOIN @ A_27a @ A_27s @ V0MM @ V1s )
      = ( V0MM @ V1s @ V1s ) ) ).

thf(thm_2EreaderMonad_2EBIND__UNITR,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0m: A_27a > A_27b] :
      ( ( c_2EreaderMonad_2EBIND @ A_27b @ A_27b @ A_27a @ V0m @ ( c_2EreaderMonad_2EUNIT @ A_27b @ A_27a ) )
      = V0m ) ).

thf(thm_2EreaderMonad_2EBIND__UNITL,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0x: A_27c,V1f: A_27c > A_27a > A_27b] :
      ( ( c_2EreaderMonad_2EBIND @ A_27c @ A_27b @ A_27a @ ( c_2EreaderMonad_2EUNIT @ A_27c @ A_27a @ V0x ) @ V1f )
      = ( V1f @ V0x ) ) ).

thf(thm_2EreaderMonad_2EMCOMPOSE__LEFT__ID,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0g: A_27a > A_27b > A_27c] :
      ( ( c_2EreaderMonad_2EMCOMPOSE @ A_27a @ A_27a @ A_27c @ A_27b @ ( c_2EreaderMonad_2EUNIT @ A_27a @ A_27b ) @ V0g )
      = V0g ) ).

thf(thm_2EreaderMonad_2EMCOMPOSE__RIGHT__ID,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b > A_27c] :
      ( ( c_2EreaderMonad_2EMCOMPOSE @ A_27a @ A_27c @ A_27c @ A_27b @ V0f @ ( c_2EreaderMonad_2EUNIT @ A_27c @ A_27b ) )
      = V0f ) ).

thf(thm_2EreaderMonad_2EMCOMPOSE__ASSOC,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,V0h: A_27e > A_27b > A_27c,V1g: A_27d > A_27b > A_27e,V2f: A_27a > A_27b > A_27d] :
      ( ( c_2EreaderMonad_2EMCOMPOSE @ A_27a @ A_27d @ A_27c @ A_27b @ V2f @ ( c_2EreaderMonad_2EMCOMPOSE @ A_27d @ A_27e @ A_27c @ A_27b @ V1g @ V0h ) )
      = ( c_2EreaderMonad_2EMCOMPOSE @ A_27a @ A_27e @ A_27c @ A_27b @ ( c_2EreaderMonad_2EMCOMPOSE @ A_27a @ A_27d @ A_27e @ A_27b @ V2f @ V1g ) @ V0h ) ) ).

thf(thm_2EreaderMonad_2EFMAP__ID,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0M: A_27a > A_27b] :
      ( ( ( c_2EreaderMonad_2EFMAP @ A_27b @ A_27b @ A_27a
          @ ^ [V1x: A_27b] : V1x
          @ V0M )
        = V0M )
      & ( ( c_2EreaderMonad_2EFMAP @ A_27b @ A_27b @ A_27a @ ( c_2Ecombin_2EI @ A_27b ) @ V0M )
        = V0M ) ) ).

thf(thm_2EreaderMonad_2EFMAP__o,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0g: A_27b > A_27d,V1f: A_27d > A_27c] :
      ( ( c_2EreaderMonad_2EFMAP @ A_27b @ A_27c @ A_27a @ ( c_2Ecombin_2Eo @ A_27b @ A_27c @ A_27d @ V1f @ V0g ) )
      = ( c_2Ecombin_2Eo @ ( A_27a > A_27b ) @ ( A_27a > A_27c ) @ ( A_27a > A_27d ) @ ( c_2EreaderMonad_2EFMAP @ A_27d @ A_27c @ A_27a @ V1f ) @ ( c_2EreaderMonad_2EFMAP @ A_27b @ A_27d @ A_27a @ V0g ) ) ) ).

thf(thm_2EreaderMonad_2EFMAP__BIND,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27c > A_27b,V1M: A_27a > A_27c] :
      ( ( c_2EreaderMonad_2EFMAP @ A_27c @ A_27b @ A_27a @ V0f @ V1M )
      = ( c_2EreaderMonad_2EBIND @ A_27c @ A_27b @ A_27a @ V1M @ ( c_2Ecombin_2Eo @ A_27c @ ( A_27a > A_27b ) @ A_27b @ ( c_2EreaderMonad_2EUNIT @ A_27b @ A_27a ) @ V0f ) ) ) ).

thf(thm_2EreaderMonad_2EBIND__JOIN,axiom,
    ! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27c > A_27a > A_27b,V1M: A_27a > A_27c] :
      ( ( c_2EreaderMonad_2EBIND @ A_27c @ A_27b @ A_27a @ V1M @ V0f )
      = ( c_2EreaderMonad_2EJOIN @ A_27b @ A_27a @ ( c_2EreaderMonad_2EFMAP @ A_27c @ ( A_27a > A_27b ) @ A_27a @ V0f @ V1M ) ) ) ).

thf(thm_2EreaderMonad_2EJOIN__BIND,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0M: A_27a > A_27a > A_27b] :
      ( ( c_2EreaderMonad_2EJOIN @ A_27b @ A_27a @ V0M )
      = ( c_2EreaderMonad_2EBIND @ ( A_27a > A_27b ) @ A_27b @ A_27a @ V0M @ ( c_2Ecombin_2EI @ ( A_27a > A_27b ) ) ) ) ).

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