%------------------------------------------------------------------------------
% File     : ITP209_4 : TPTP v8.2.0. 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    :   23 (   7 unt;  10 typ;   0 def)
%            Number of atoms       :   21 (  12 equ)
%            Maximal formula atoms :    3 (   1 avg)
%            Number of connectives :    8 (   0   ~;   0   |;   1   &)
%                                         (   1 <=>;   6  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    8 (   5 avg)
%            Maximal term depth    :    6 (   2 avg)
%            Number of types       :    2 (   1 usr)
%            Number of type conns  :    9 (   5   >;   4   *;   0   +;   0  <<)
%            Number of predicates  :    4 (   3 usr;   0 prp; 2-4 aty)
%            Number of functors    :    6 (   6 usr;   4 con; 0-4 aty)
%            Number of variables   :   56 (  49   !;   0   ?;  56   :)
%                                         (   7  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TX1_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 14:22:46.654
%------------------------------------------------------------------------------
% Could-be-implicit typings (2)
tff(ty_t_fun,type,
    fun: ( $tType * $tType ) > $tType ).

tff(ty_tf_a,type,
    a: $tType ).

% Explicit typings (8)
tff(sy_c_Syntax__Match_Oac__operator,type,
    syntax_ac_operator: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).

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

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

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_v_a,type,
    a2: a ).

tff(sy_v_b,type,
    b: a ).

tff(sy_v_c,type,
    c: a ).

tff(sy_v_f,type,
    f: fun(a,fun(a,a)) ).

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

% ac_operator_axioms
tff(fact_1_commute,axiom,
    ! [Aa: a,Ba: a] : aa(a,a,aa(a,fun(a,a),f,Aa),Ba) = aa(a,a,aa(a,fun(a,a),f,Ba),Aa) ).

% commute
tff(fact_2_left__commute,axiom,
    ! [Aa: a,Ba: a,Ca: a] : aa(a,a,aa(a,fun(a,a),f,Aa),aa(a,a,aa(a,fun(a,a),f,Ba),Ca)) = aa(a,a,aa(a,fun(a,a),f,Ba),aa(a,a,aa(a,fun(a,a),f,Aa),Ca)) ).

% left_commute
tff(fact_3_right__assoc,axiom,
    ! [Aa: a,Ba: a,Ca: a] : aa(a,a,aa(a,fun(a,a),f,aa(a,a,aa(a,fun(a,a),f,Aa),Ba)),Ca) = aa(a,a,aa(a,fun(a,a),f,Aa),aa(a,a,aa(a,fun(a,a),f,Ba),Ca)) ).

% right_assoc
tff(fact_4_syntax__nomatch__def,axiom,
    ! [B: $tType,C: $tType,Pat: B,Obj: C] : syntax2379306206330768139omatch(B,C,Pat,Obj) ).

% syntax_nomatch_def
tff(fact_5_syntax__fo__nomatch__def,axiom,
    ! [B: $tType,C: $tType,Pat: B,Obj: C] : syntax7388354845996824322omatch(B,C,Pat,Obj) ).

% syntax_fo_nomatch_def
tff(fact_6_ac__operator_Ointro,axiom,
    ! [B: $tType,Fa: fun(B,fun(B,B))] :
      ( ! [A2: B,B2: B,C2: B] : aa(B,B,aa(B,fun(B,B),Fa,aa(B,B,aa(B,fun(B,B),Fa,A2),B2)),C2) = aa(B,B,aa(B,fun(B,B),Fa,A2),aa(B,B,aa(B,fun(B,B),Fa,B2),C2))
     => ( ! [A2: B,B2: B] : aa(B,B,aa(B,fun(B,B),Fa,A2),B2) = aa(B,B,aa(B,fun(B,B),Fa,B2),A2)
       => syntax_ac_operator(B,Fa) ) ) ).

% ac_operator.intro
tff(fact_7_ac__operator_Ocommute,axiom,
    ! [B: $tType,Fa: fun(B,fun(B,B)),Aa: B,Ba: B] :
      ( syntax_ac_operator(B,Fa)
     => ( aa(B,B,aa(B,fun(B,B),Fa,Aa),Ba) = aa(B,B,aa(B,fun(B,B),Fa,Ba),Aa) ) ) ).

% ac_operator.commute
tff(fact_8_ac__operator_Oleft__assoc,axiom,
    ! [B: $tType,Fa: fun(B,fun(B,B)),Aa: B,Ba: B,Ca: B] :
      ( syntax_ac_operator(B,Fa)
     => ( aa(B,B,aa(B,fun(B,B),Fa,Aa),aa(B,B,aa(B,fun(B,B),Fa,Ba),Ca)) = aa(B,B,aa(B,fun(B,B),Fa,aa(B,B,aa(B,fun(B,B),Fa,Aa),Ba)),Ca) ) ) ).

% ac_operator.left_assoc
tff(fact_9_ac__operator_Oright__assoc,axiom,
    ! [B: $tType,Fa: fun(B,fun(B,B)),Aa: B,Ba: B,Ca: B] :
      ( syntax_ac_operator(B,Fa)
     => ( aa(B,B,aa(B,fun(B,B),Fa,aa(B,B,aa(B,fun(B,B),Fa,Aa),Ba)),Ca) = aa(B,B,aa(B,fun(B,B),Fa,Aa),aa(B,B,aa(B,fun(B,B),Fa,Ba),Ca)) ) ) ).

% ac_operator.right_assoc
tff(fact_10_ac__operator_Oleft__commute,axiom,
    ! [B: $tType,Fa: fun(B,fun(B,B)),Aa: B,Ba: B,Ca: B] :
      ( syntax_ac_operator(B,Fa)
     => ( aa(B,B,aa(B,fun(B,B),Fa,Aa),aa(B,B,aa(B,fun(B,B),Fa,Ba),Ca)) = aa(B,B,aa(B,fun(B,B),Fa,Ba),aa(B,B,aa(B,fun(B,B),Fa,Aa),Ca)) ) ) ).

% ac_operator.left_commute
tff(fact_11_ac__operator__def,axiom,
    ! [B: $tType,Fa: fun(B,fun(B,B))] :
      ( syntax_ac_operator(B,Fa)
    <=> ( ! [A3: B,B3: B,C3: B] : aa(B,B,aa(B,fun(B,B),Fa,aa(B,B,aa(B,fun(B,B),Fa,A3),B3)),C3) = aa(B,B,aa(B,fun(B,B),Fa,A3),aa(B,B,aa(B,fun(B,B),Fa,B3),C3))
        & ! [A3: B,B3: B] : aa(B,B,aa(B,fun(B,B),Fa,A3),B3) = aa(B,B,aa(B,fun(B,B),Fa,B3),A3) ) ) ).

% ac_operator_def

% Conjectures (1)
tff(conj_0,conjecture,
    aa(a,a,aa(a,fun(a,a),f,aa(a,a,aa(a,fun(a,a),f,a2),b)),c) = aa(a,a,aa(a,fun(a,a),f,aa(a,a,aa(a,fun(a,a),f,a2),c)),b) ).

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