%------------------------------------------------------------------------------
% File     : ITP209_3 : 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 (   5 unt;  10 typ;   0 def)
%            Number of atoms       :   19 (  12 equ)
%            Maximal formula atoms :    3 (   1 avg)
%            Number of connectives :    8 (   0   ~;   0   |;   1   &)
%                                         (   1 <=>;   6  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    7 (   4 avg)
%            Maximal term depth    :    5 (   2 avg)
%            Number of types       :    4 (   3 usr)
%            Number of type conns  :    5 (   3   >;   2   *;   0   +;   0  <<)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :    6 (   6 usr;   4 con; 0-2 aty)
%            Number of variables   :   35 (  35   !;   0   ?;  35   :)
% SPC      : TX0_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 14:22:46.591
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
tff(ty_n_t__fun_Itf__a_Mt__fun_Itf__a_Mtf__a_J_J,type,
    fun_a_fun_a_a: $tType ).

tff(ty_n_t__fun_Itf__a_Mtf__a_J,type,
    fun_a_a: $tType ).

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

% Explicit typings (7)
tff(sy_c_Syntax__Match_Oac__operator_001tf__a,type,
    syntax_ac_operator_a: fun_a_fun_a_a > $o ).

tff(sy_c_aa_001tf__a_001t__fun_Itf__a_Mtf__a_J,type,
    aa_a_fun_a_a: ( fun_a_fun_a_a * a ) > fun_a_a ).

tff(sy_c_aa_001tf__a_001tf__a,type,
    aa_a_a: ( fun_a_a * a ) > a ).

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 (10)
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_ac__operator_Ointro,axiom,
    ! [Fa: fun_a_fun_a_a] :
      ( ! [A: a,B: a,C: a] : ( aa_a_a(aa_a_fun_a_a(Fa,aa_a_a(aa_a_fun_a_a(Fa,A),B)),C) = aa_a_a(aa_a_fun_a_a(Fa,A),aa_a_a(aa_a_fun_a_a(Fa,B),C)) )
     => ( ! [A: a,B: a] : ( aa_a_a(aa_a_fun_a_a(Fa,A),B) = aa_a_a(aa_a_fun_a_a(Fa,B),A) )
       => syntax_ac_operator_a(Fa) ) ) ).

% ac_operator.intro
tff(fact_5_ac__operator_Ocommute,axiom,
    ! [Fa: fun_a_fun_a_a,Aa: a,Ba: a] :
      ( syntax_ac_operator_a(Fa)
     => ( aa_a_a(aa_a_fun_a_a(Fa,Aa),Ba) = aa_a_a(aa_a_fun_a_a(Fa,Ba),Aa) ) ) ).

% ac_operator.commute
tff(fact_6_ac__operator_Oleft__assoc,axiom,
    ! [Fa: fun_a_fun_a_a,Aa: a,Ba: a,Ca: a] :
      ( syntax_ac_operator_a(Fa)
     => ( aa_a_a(aa_a_fun_a_a(Fa,Aa),aa_a_a(aa_a_fun_a_a(Fa,Ba),Ca)) = aa_a_a(aa_a_fun_a_a(Fa,aa_a_a(aa_a_fun_a_a(Fa,Aa),Ba)),Ca) ) ) ).

% ac_operator.left_assoc
tff(fact_7_ac__operator_Oright__assoc,axiom,
    ! [Fa: fun_a_fun_a_a,Aa: a,Ba: a,Ca: a] :
      ( syntax_ac_operator_a(Fa)
     => ( aa_a_a(aa_a_fun_a_a(Fa,aa_a_a(aa_a_fun_a_a(Fa,Aa),Ba)),Ca) = aa_a_a(aa_a_fun_a_a(Fa,Aa),aa_a_a(aa_a_fun_a_a(Fa,Ba),Ca)) ) ) ).

% ac_operator.right_assoc
tff(fact_8_ac__operator_Oleft__commute,axiom,
    ! [Fa: fun_a_fun_a_a,Aa: a,Ba: a,Ca: a] :
      ( syntax_ac_operator_a(Fa)
     => ( aa_a_a(aa_a_fun_a_a(Fa,Aa),aa_a_a(aa_a_fun_a_a(Fa,Ba),Ca)) = aa_a_a(aa_a_fun_a_a(Fa,Ba),aa_a_a(aa_a_fun_a_a(Fa,Aa),Ca)) ) ) ).

% ac_operator.left_commute
tff(fact_9_ac__operator__def,axiom,
    ! [Fa: fun_a_fun_a_a] :
      ( syntax_ac_operator_a(Fa)
    <=> ( ! [A2: a,B2: a,C2: a] : ( aa_a_a(aa_a_fun_a_a(Fa,aa_a_a(aa_a_fun_a_a(Fa,A2),B2)),C2) = aa_a_a(aa_a_fun_a_a(Fa,A2),aa_a_a(aa_a_fun_a_a(Fa,B2),C2)) )
        & ! [A2: a,B2: a] : ( aa_a_a(aa_a_fun_a_a(Fa,A2),B2) = aa_a_a(aa_a_fun_a_a(Fa,B2),A2) ) ) ) ).

% 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) ).

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