SET007 Axioms: SET007+540.ax


%------------------------------------------------------------------------------
% File     : SET007+540 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Full Trees
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : bintree2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   39 (   0 unit)
%            Number of atoms       :  256 (  34 equality)
%            Maximal formula depth :   14 (   7 average)
%            Number of connectives :  272 (  55 ~  ;   0  |; 124  &)
%                                         (   4 <=>;  89 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   30 (   0 propositional; 1-3 arity)
%            Number of functors    :   44 (  11 constant; 0-3 arity)
%            Number of variables   :   87 (   0 singleton;  79 !;   8 ?)
%            Maximal term depth    :    5 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(fc1_bintree2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_relat_1(k1_bintree2(A,B))
        & v1_funct_1(k1_bintree2(A,B))
        & v2_funct_1(k1_bintree2(A,B))
        & v1_funct_2(k1_bintree2(A,B),k2_trees_2(A,B),k5_numbers) ) ) )).

fof(fc2_bintree2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree1(A)
        & m1_subset_1(B,k5_numbers) )
     => v1_finset_1(k2_trees_2(A,B)) ) )).

fof(cc1_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_trees_2(A)
        & v2_trees_9(A)
        & v1_bintree1(A) ) ) )).

fof(rc1_bintree2,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v1_trees_1(A)
      & v1_trees_2(A)
      & v2_trees_9(A)
      & v1_bintree1(A)
      & v1_bintree2(A) ) )).

fof(fc3_bintree2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_relat_1(k2_bintree2(A,B))
        & v1_funct_1(k2_bintree2(A,B))
        & v2_funct_1(k2_bintree2(A,B))
        & v1_finset_1(k2_bintree2(A,B))
        & v1_finseq_1(k2_bintree2(A,B)) ) ) )).

fof(t1_bintree2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ( r2_hidden(B,k3_finseq_2(A))
           => r2_hidden(k7_relat_1(B,k2_finseq_1(C)),k3_finseq_2(A)) ) ) ) )).

fof(t2_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree1(A) )
     => ! [B] :
          ( m1_trees_1(B,A)
         => m2_finseq_1(B,k6_margrel1) ) ) )).

fof(t3_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( A = k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1))
       => v1_bintree1(A) ) ) )).

fof(t4_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( A = k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1))
       => k3_trees_1(A) = k1_xboole_0 ) ) )).

fof(t5_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_bintree2(C,A)
             => ( r2_hidden(C,k2_trees_2(A,B))
               => m2_finseq_2(C,k6_margrel1,k4_finseq_2(B,k6_margrel1)) ) ) ) ) )).

fof(t6_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( ! [B] :
            ( m1_trees_1(B,A)
           => k1_trees_2(A,B) = k7_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k5_numbers)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__0)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__1))) )
       => k3_trees_1(A) = k1_xboole_0 ) ) )).

fof(t7_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( ! [B] :
            ( m1_trees_1(B,A)
           => k1_trees_2(A,B) = k7_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k5_numbers)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__0)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__1))) )
       => v1_bintree1(A) ) ) )).

fof(t8_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( A = k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1))
      <=> ! [B] :
            ( m1_trees_1(B,A)
           => k1_trees_2(A,B) = k7_domain_1(k1_zfmisc_1(k2_zfmisc_1(k5_numbers,k5_numbers)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__0)),k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__1))) ) ) ) )).

fof(d1_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree1(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_trees_2(A,B),k5_numbers)
                & m2_relset_1(C,k2_trees_2(A,B),k5_numbers) )
             => ( C = k1_bintree2(A,B)
              <=> ! [D] :
                    ( m1_bintree2(D,A)
                   => ( r2_hidden(D,k2_trees_2(A,B))
                     => ! [E] :
                          ( m2_finseq_2(E,k6_margrel1,k4_finseq_2(B,k6_margrel1))
                         => ( E = k4_finseq_5(k6_margrel1,D)
                           => k1_funct_1(C,D) = k1_nat_1(k9_binarith(B,E),np__1) ) ) ) ) ) ) ) ) )).

fof(d2_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( v1_bintree2(A)
      <=> A = k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1)) ) ) )).

fof(t9_bintree2,axiom,
    ( ~ v1_xboole_0(k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1)))
    & v1_trees_1(k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1))) )).

fof(t10_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( A = k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1))
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => r2_hidden(k5_euclid(B),k2_trees_2(A,B)) ) ) ) )).

fof(t11_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A) )
     => ( A = k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1))
       => ! [B] :
            ( ( ~ v1_xboole_0(B)
              & m2_subset_1(B,k1_numbers,k5_numbers) )
           => ! [C] :
                ( m2_finseq_2(C,k6_margrel1,k4_finseq_2(B,k6_margrel1))
               => r2_hidden(C,k2_trees_2(A,B)) ) ) ) ) )).

fof(t12_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => r1_tarski(k2_finseq_1(k3_series_1(np__2,B)),k2_relat_1(k1_bintree2(A,B))) ) ) )).

fof(d3_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => k2_bintree2(A,B) = k2_funct_1(k1_bintree2(A,B)) ) ) )).

fof(t13_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => k1_funct_1(k1_bintree2(A,B),k5_euclid(B)) = np__1 ) ) )).

fof(t14_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m2_finseq_2(C,k6_margrel1,k4_finseq_2(B,k6_margrel1))
             => ( C = k5_euclid(B)
               => k1_funct_1(k1_bintree2(A,B),k6_binarith(B,C)) = k3_series_1(np__2,B) ) ) ) ) )).

fof(t15_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => k1_funct_1(k2_bintree2(A,B),np__1) = k5_euclid(B) ) ) )).

fof(t16_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m2_finseq_2(C,k6_margrel1,k4_finseq_2(B,k6_margrel1))
             => ( C = k5_euclid(B)
               => k1_funct_1(k2_bintree2(A,B),k3_series_1(np__2,B)) = k6_binarith(B,C) ) ) ) ) )).

fof(t17_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r2_hidden(C,k2_finseq_1(k3_series_1(np__2,B)))
               => k1_funct_1(k2_bintree2(A,B),C) = k4_finseq_5(k6_margrel1,k1_binari_3(B,k5_binarith(C,np__1))) ) ) ) ) )).

fof(t18_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_card_1(k2_trees_2(A,B)) = k3_series_1(np__2,B) ) ) )).

fof(t19_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => k3_finseq_1(k2_bintree2(A,B)) = k3_series_1(np__2,B) ) ) )).

fof(t20_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => k4_finseq_1(k2_bintree2(A,B)) = k2_finseq_1(k3_series_1(np__2,B)) ) ) )).

fof(t21_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => k2_relat_1(k2_bintree2(A,B)) = k2_trees_2(A,B) ) ) )).

fof(t22_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => k1_funct_1(k2_bintree2(A,np__1),np__1) = k13_binarith(k5_numbers,np__0) ) )).

fof(t23_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => k1_funct_1(k2_bintree2(A,np__1),np__2) = k13_binarith(k5_numbers,np__1) ) )).

fof(t24_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_subset_1(B,k1_numbers,k5_numbers) )
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m2_subset_1(C,k1_numbers,k5_numbers) )
             => ( r1_xreal_0(C,k3_series_1(np__2,k1_nat_1(B,np__1)))
               => ! [D] :
                    ( m2_finseq_2(D,k6_margrel1,k4_finseq_2(B,k6_margrel1))
                   => ( D = k1_funct_1(k2_bintree2(A,B),k3_nat_1(k1_nat_1(C,np__1),np__2))
                     => k1_funct_1(k2_bintree2(A,k1_nat_1(B,np__1)),C) = k7_finseq_1(D,k13_binarith(k5_numbers,k4_nat_1(k1_nat_1(C,np__1),np__2))) ) ) ) ) ) ) )).

fof(s1_bintree2,axiom,
    ( ! [A] :
        ( m1_subset_1(A,f1_s1_bintree2)
       => ? [B] :
            ( m1_subset_1(B,f1_s1_bintree2)
            & ? [C] :
                ( m1_subset_1(C,f1_s1_bintree2)
                & p1_s1_bintree2(A,B,C) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v3_trees_2(A)
        & v2_bintree1(A)
        & m3_trees_2(A,f1_s1_bintree2)
        & k1_relat_1(A) = k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1))
        & k1_funct_1(A,k1_xboole_0) = f2_s1_bintree2
        & ! [B] :
            ( m1_trees_1(B,k1_relat_1(A))
           => p1_s1_bintree2(k3_trees_2(f1_s1_bintree2,A,B),k1_funct_1(A,k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__0))),k1_funct_1(A,k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__1)))) ) ) )).

fof(s2_bintree2,axiom,
    ( ( ! [A] :
          ( m1_subset_1(A,f1_s2_bintree2)
         => ? [B] :
              ( m1_subset_1(B,f1_s2_bintree2)
              & p1_s2_bintree2(A,B) ) )
      & ! [A] :
          ( m1_subset_1(A,f1_s2_bintree2)
         => ? [B] :
              ( m1_subset_1(B,f1_s2_bintree2)
              & p2_s2_bintree2(A,B) ) ) )
   => ? [A] :
        ( v1_funct_1(A)
        & v3_trees_2(A)
        & v2_bintree1(A)
        & m3_trees_2(A,f1_s2_bintree2)
        & k1_relat_1(A) = k3_finseq_2(k7_domain_1(k5_numbers,np__0,np__1))
        & k1_funct_1(A,k1_xboole_0) = f2_s2_bintree2
        & ! [B] :
            ( m1_trees_1(B,k1_relat_1(A))
           => ( p1_s2_bintree2(k3_trees_2(f1_s2_bintree2,A,B),k1_funct_1(A,k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__0))))
              & p2_s2_bintree2(k3_trees_2(f1_s2_bintree2,A,B),k1_funct_1(A,k8_finseq_1(k5_numbers,B,k13_binarith(k5_numbers,np__1)))) ) ) ) )).

fof(dt_m1_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree1(A) )
     => ! [B] :
          ( m1_bintree2(B,A)
         => m2_finseq_1(B,k6_margrel1) ) ) )).

fof(existence_m1_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree1(A) )
     => ? [B] : m1_bintree2(B,A) ) )).

fof(redefinition_m1_bintree2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree1(A) )
     => ! [B] :
          ( m1_bintree2(B,A)
        <=> m1_subset_1(B,A) ) ) )).

fof(dt_k1_bintree2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree1(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_funct_1(k1_bintree2(A,B))
        & v1_funct_2(k1_bintree2(A,B),k2_trees_2(A,B),k5_numbers)
        & m2_relset_1(k1_bintree2(A,B),k2_trees_2(A,B),k5_numbers) ) ) )).

fof(dt_k2_bintree2,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v1_trees_1(A)
        & v1_bintree2(A)
        & ~ v1_xboole_0(B)
        & m1_subset_1(B,k5_numbers) )
     => m1_trees_4(k2_bintree2(A,B),A,k2_trees_2(A,B)) ) )).
%------------------------------------------------------------------------------