%------------------------------------------------------------------------------
% File     : ITP209^2 : TPTP v9.0.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Syntax_Match 00088_002922
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0023_Syntax_Match_00088_002922 [Des22]

% Status   : Theorem
% Rating   : 0.00 v8.1.0
% Syntax   : Number of formulae    :   21 (   8 unt;   8 typ;   0 def)
%            Number of atoms       :   26 (  15 equ;   0 cnn)
%            Maximal formula atoms :    3 (   2 avg)
%            Number of connectives :  104 (   0   ~;   0   |;   1   &;  97   @)
%                                         (   0 <=>;   6  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   5 avg)
%            Number of types       :    2 (   1 usr)
%            Number of type conns  :   21 (  21   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    9 (   7 usr;   4 con; 0-4 aty)
%            Number of variables   :   54 (   5   ^;  44   !;   0   ?;  54   :)
%                                         (   5  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 14:22:46.652
%------------------------------------------------------------------------------
% Could-be-implicit typings (1)
thf(ty_tf_a,type,
    a: $tType ).

% Explicit typings (7)
thf(sy_c_Syntax__Match_Oac__operator,type,
    syntax_ac_operator: 
      !>[A: $tType] : ( ( A > A > A ) > $o ) ).

thf(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
    syntax7388354845996824322omatch: 
      !>[A: $tType,B: $tType] : ( A > B > $o ) ).

thf(sy_c_Syntax__Match_Osyntax__nomatch,type,
    syntax2379306206330768139omatch: 
      !>[A: $tType,B: $tType] : ( A > B > $o ) ).

thf(sy_v_a,type,
    a2: a ).

thf(sy_v_b,type,
    b: a ).

thf(sy_v_c,type,
    c: a ).

thf(sy_v_f,type,
    f: a > a > a ).

% Relevant facts (12)
thf(fact_0_ac__operator__axioms,axiom,
    syntax_ac_operator @ a @ f ).

% ac_operator_axioms
thf(fact_1_commute,axiom,
    ! [A2: a,B2: a] :
      ( ( f @ A2 @ B2 )
      = ( f @ B2 @ A2 ) ) ).

% commute
thf(fact_2_left__commute,axiom,
    ! [A2: a,B2: a,C: a] :
      ( ( f @ A2 @ ( f @ B2 @ C ) )
      = ( f @ B2 @ ( f @ A2 @ C ) ) ) ).

% left_commute
thf(fact_3_right__assoc,axiom,
    ! [A2: a,B2: a,C: a] :
      ( ( f @ ( f @ A2 @ B2 ) @ C )
      = ( f @ A2 @ ( f @ B2 @ C ) ) ) ).

% right_assoc
thf(fact_4_syntax__nomatch__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( syntax2379306206330768139omatch @ A @ B )
      = ( ^ [Pat: A,Obj: B] : $true ) ) ).

% syntax_nomatch_def
thf(fact_5_syntax__fo__nomatch__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( syntax7388354845996824322omatch @ A @ B )
      = ( ^ [Pat: A,Obj: B] : $true ) ) ).

% syntax_fo_nomatch_def
thf(fact_6_ac__operator_Ointro,axiom,
    ! [A: $tType,F: A > A > A] :
      ( ! [A3: A,B3: A,C2: A] :
          ( ( F @ ( F @ A3 @ B3 ) @ C2 )
          = ( F @ A3 @ ( F @ B3 @ C2 ) ) )
     => ( ! [A3: A,B3: A] :
            ( ( F @ A3 @ B3 )
            = ( F @ B3 @ A3 ) )
       => ( syntax_ac_operator @ A @ F ) ) ) ).

% ac_operator.intro
thf(fact_7_ac__operator_Ocommute,axiom,
    ! [A: $tType,F: A > A > A,A2: A,B2: A] :
      ( ( syntax_ac_operator @ A @ F )
     => ( ( F @ A2 @ B2 )
        = ( F @ B2 @ A2 ) ) ) ).

% ac_operator.commute
thf(fact_8_ac__operator_Oleft__assoc,axiom,
    ! [A: $tType,F: A > A > A,A2: A,B2: A,C: A] :
      ( ( syntax_ac_operator @ A @ F )
     => ( ( F @ A2 @ ( F @ B2 @ C ) )
        = ( F @ ( F @ A2 @ B2 ) @ C ) ) ) ).

% ac_operator.left_assoc
thf(fact_9_ac__operator_Oright__assoc,axiom,
    ! [A: $tType,F: A > A > A,A2: A,B2: A,C: A] :
      ( ( syntax_ac_operator @ A @ F )
     => ( ( F @ ( F @ A2 @ B2 ) @ C )
        = ( F @ A2 @ ( F @ B2 @ C ) ) ) ) ).

% ac_operator.right_assoc
thf(fact_10_ac__operator_Oleft__commute,axiom,
    ! [A: $tType,F: A > A > A,A2: A,B2: A,C: A] :
      ( ( syntax_ac_operator @ A @ F )
     => ( ( F @ A2 @ ( F @ B2 @ C ) )
        = ( F @ B2 @ ( F @ A2 @ C ) ) ) ) ).

% ac_operator.left_commute
thf(fact_11_ac__operator__def,axiom,
    ! [A: $tType] :
      ( ( syntax_ac_operator @ A )
      = ( ^ [F2: A > A > A] :
            ( ! [A4: A,B4: A,C3: A] :
                ( ( F2 @ ( F2 @ A4 @ B4 ) @ C3 )
                = ( F2 @ A4 @ ( F2 @ B4 @ C3 ) ) )
            & ! [A4: A,B4: A] :
                ( ( F2 @ A4 @ B4 )
                = ( F2 @ B4 @ A4 ) ) ) ) ) ).

% ac_operator_def

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( f @ ( f @ a2 @ b ) @ c )
    = ( f @ ( f @ a2 @ c ) @ b ) ) ).

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