SET007 Axioms: SET007+160.ax


%------------------------------------------------------------------------------
% File     : SET007+160 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Properties of Caratheodor's Measure
% 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    : measure4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   44 (   6 unit)
%            Number of atoms       :  244 (  29 equality)
%            Maximal formula depth :   18 (   8 average)
%            Number of connectives :  219 (  19 ~  ;   0  |; 103  &)
%                                         (   6 <=>;  91 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   23 (   1 propositional; 0-3 arity)
%            Number of functors    :   33 (   9 constant; 0-5 arity)
%            Number of variables   :  133 (   0 singleton; 128 !;   5 ?)
%            Maximal term depth    :    4 (   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_measure4,axiom,(
    ! [A,B] :
      ( m1_measure4(B,A)
     => ( ~ v1_xboole_0(k3_measure4(A,B))
        & v1_prob_1(k3_measure4(A,B),A)
        & v1_finsub_1(k3_measure4(A,B))
        & v2_finsub_1(k3_measure4(A,B))
        & v1_measure1(k3_measure4(A,B),A) ) ) )).

fof(t1_measure4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ( ( r1_supinf_1(k1_supinf_2,A)
                  & r1_supinf_1(k1_supinf_2,B)
                  & r1_supinf_1(k1_supinf_2,C) )
               => k2_supinf_2(k2_supinf_2(A,B),C) = k2_supinf_2(A,k2_supinf_2(B,C)) ) ) ) ) )).

fof(t2_measure4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ~ ( A != k5_supinf_1
                  & A != k4_supinf_1
                  & ~ ( r1_supinf_1(k2_supinf_2(B,A),C)
                    <=> r1_supinf_1(B,k4_supinf_2(C,A)) ) ) ) ) ) )).

fof(t3_measure4,axiom,(
    $true )).

fof(t4_measure4,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,B)
            & m2_relset_1(C,k5_numbers,B) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k5_numbers,B)
                & m2_relset_1(D,k5_numbers,B) )
             => ! [E] :
                  ( m2_subset_1(E,k1_zfmisc_1(A),B)
                 => ( ! [F] :
                        ( m2_subset_1(F,k1_numbers,k5_numbers)
                       => k8_funct_2(k5_numbers,B,D,F) = k3_xboole_0(E,k8_funct_2(k5_numbers,B,C,F)) )
                   => k3_tarski(k2_relat_1(D)) = k3_xboole_0(E,k3_tarski(k2_relat_1(C))) ) ) ) ) ) )).

fof(t5_measure4,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,B)
            & m2_relset_1(C,k5_numbers,B) )
         => ! [D] :
              ( ( v1_funct_1(D)
                & v1_funct_2(D,k5_numbers,B)
                & m2_relset_1(D,k5_numbers,B) )
             => ( ( k8_funct_2(k5_numbers,B,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k2_xboole_0(k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,D,E)) ) )
               => ! [E] :
                    ( ( v1_funct_1(E)
                      & v1_funct_2(E,k5_numbers,B)
                      & m2_relset_1(E,k5_numbers,B) )
                   => ( ( k8_funct_2(k5_numbers,B,E,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
                        & ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => k8_funct_2(k5_numbers,B,E,k1_nat_1(F,np__1)) = k4_xboole_0(k8_funct_2(k5_numbers,B,C,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,D,F)) ) )
                     => k3_tarski(k2_relat_1(C)) = k3_tarski(k2_relat_1(E)) ) ) ) ) ) ) )).

fof(t6_measure4,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(k1_zfmisc_1(A))
      & v1_prob_1(k1_zfmisc_1(A),A)
      & v1_measure1(k1_zfmisc_1(A),A)
      & m1_subset_1(k1_zfmisc_1(A),k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(t7_measure4,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k3_supinf_1)
     => ! [C] :
          ( m1_subset_1(C,k3_supinf_1)
         => ? [D] :
              ( v1_funct_1(D)
              & v1_funct_2(D,k1_zfmisc_1(A),k3_supinf_1)
              & m2_relset_1(D,k1_zfmisc_1(A),k3_supinf_1)
              & ! [E] :
                  ( m1_subset_1(E,k1_zfmisc_1(A))
                 => ( ( E = k1_xboole_0
                     => k1_measure1(k1_zfmisc_1(A),D,E) = B )
                    & ( E != k1_xboole_0
                     => k1_measure1(k1_zfmisc_1(A),D,E) = C ) ) ) ) ) ) )).

fof(t8_measure4,axiom,(
    ! [A] :
    ? [B] :
      ( v1_funct_1(B)
      & v1_funct_2(B,k1_zfmisc_1(A),k3_supinf_1)
      & m2_relset_1(B,k1_zfmisc_1(A),k3_supinf_1)
      & ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => k1_measure1(k1_zfmisc_1(A),B,C) = k1_supinf_2 ) ) )).

fof(t9_measure4,axiom,(
    $true )).

fof(t10_measure4,axiom,(
    $true )).

fof(t11_measure4,axiom,(
    ! [A] :
    ? [B] :
      ( v1_funct_1(B)
      & v1_funct_2(B,k1_zfmisc_1(A),k3_supinf_1)
      & m2_relset_1(B,k1_zfmisc_1(A),k3_supinf_1)
      & v6_supinf_2(B,k1_zfmisc_1(A))
      & k1_funct_1(B,k1_xboole_0) = k1_supinf_2
      & ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(A))
             => ( r1_tarski(C,D)
               => ( r1_supinf_1(k1_measure1(k1_zfmisc_1(A),B,C),k1_measure1(k1_zfmisc_1(A),B,D))
                  & ! [E] :
                      ( ( v1_funct_1(E)
                        & v1_funct_2(E,k5_numbers,k1_zfmisc_1(A))
                        & m2_relset_1(E,k5_numbers,k1_zfmisc_1(A)) )
                     => r1_supinf_1(k1_measure1(k1_zfmisc_1(A),B,k5_setfam_1(A,k1_measure4(A,E))),k19_supinf_2(k2_measure4(k5_numbers,k1_zfmisc_1(A),k3_supinf_1,E,B))) ) ) ) ) ) ) )).

fof(d1_measure4,axiom,(
    $true )).

fof(d2_measure4,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k1_zfmisc_1(A),k3_supinf_1)
        & m2_relset_1(B,k1_zfmisc_1(A),k3_supinf_1) )
     => ( m1_measure4(B,A)
      <=> ( v6_supinf_2(B,k1_zfmisc_1(A))
          & k1_funct_1(B,k1_xboole_0) = k1_supinf_2
          & ! [C] :
              ( m1_subset_1(C,k1_zfmisc_1(A))
             => ! [D] :
                  ( m1_subset_1(D,k1_zfmisc_1(A))
                 => ( r1_tarski(C,D)
                   => ( r1_supinf_1(k1_measure1(k1_zfmisc_1(A),B,C),k1_measure1(k1_zfmisc_1(A),B,D))
                      & ! [E] :
                          ( ( v1_funct_1(E)
                            & v1_funct_2(E,k5_numbers,k1_zfmisc_1(A))
                            & m2_relset_1(E,k5_numbers,k1_zfmisc_1(A)) )
                         => r1_supinf_1(k1_measure1(k1_zfmisc_1(A),B,k5_setfam_1(A,k1_measure4(A,E))),k19_supinf_2(k2_measure4(k5_numbers,k1_zfmisc_1(A),k3_supinf_1,E,B))) ) ) ) ) ) ) ) ) )).

fof(d3_measure4,axiom,(
    ! [A,B] :
      ( m1_measure4(B,A)
     => ! [C] :
          ( ( ~ v1_xboole_0(C)
            & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ( C = k3_measure4(A,B)
          <=> ! [D] :
                ( m1_subset_1(D,k1_zfmisc_1(A))
               => ( r2_hidden(D,C)
                <=> ! [E] :
                      ( m1_subset_1(E,k1_zfmisc_1(A))
                     => ! [F] :
                          ( m1_subset_1(F,k1_zfmisc_1(A))
                         => ( ( r1_tarski(E,D)
                              & r1_tarski(F,k4_xboole_0(A,D)) )
                           => r1_supinf_1(k2_supinf_2(k1_measure1(k1_zfmisc_1(A),B,E),k1_measure1(k1_zfmisc_1(A),B,F)),k1_measure1(k1_zfmisc_1(A),B,k4_subset_1(A,E,F))) ) ) ) ) ) ) ) ) )).

fof(t12_measure4,axiom,(
    ! [A,B] :
      ( m1_measure4(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(A))
             => r1_supinf_1(k1_measure1(k1_zfmisc_1(A),B,k4_subset_1(A,C,D)),k2_supinf_2(k1_measure1(k1_zfmisc_1(A),B,C),k1_measure1(k1_zfmisc_1(A),B,D))) ) ) ) )).

fof(t13_measure4,axiom,(
    $true )).

fof(t14_measure4,axiom,(
    ! [A,B] :
      ( m1_measure4(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( r2_hidden(C,k3_measure4(A,B))
          <=> ! [D] :
                ( m1_subset_1(D,k1_zfmisc_1(A))
               => ! [E] :
                    ( m1_subset_1(E,k1_zfmisc_1(A))
                   => ( ( r1_tarski(D,C)
                        & r1_tarski(E,k4_xboole_0(A,C)) )
                     => k2_supinf_2(k1_measure1(k1_zfmisc_1(A),B,D),k1_measure1(k1_zfmisc_1(A),B,E)) = k1_measure1(k1_zfmisc_1(A),B,k4_subset_1(A,D,E)) ) ) ) ) ) ) )).

fof(t15_measure4,axiom,(
    ! [A,B] :
      ( m1_measure4(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ! [D] :
              ( m1_subset_1(D,k1_zfmisc_1(A))
             => ( ( r2_hidden(C,k3_measure4(A,B))
                  & r2_hidden(D,k3_measure4(A,B))
                  & r1_xboole_0(D,C) )
               => k1_measure1(k1_zfmisc_1(A),B,k4_subset_1(A,C,D)) = k2_supinf_2(k1_measure1(k1_zfmisc_1(A),B,C),k1_measure1(k1_zfmisc_1(A),B,D)) ) ) ) ) )).

fof(t16_measure4,axiom,(
    ! [A,B,C] :
      ( m1_measure4(C,B)
     => ( r2_hidden(A,k3_measure4(B,C))
       => r2_hidden(k4_xboole_0(B,A),k3_measure4(B,C)) ) ) )).

fof(t17_measure4,axiom,(
    ! [A,B,C,D] :
      ( m1_measure4(D,A)
     => ( ( r2_hidden(B,k3_measure4(A,D))
          & r2_hidden(C,k3_measure4(A,D)) )
       => r2_hidden(k2_xboole_0(B,C),k3_measure4(A,D)) ) ) )).

fof(t18_measure4,axiom,(
    ! [A,B,C,D] :
      ( m1_measure4(D,A)
     => ( ( r2_hidden(B,k3_measure4(A,D))
          & r2_hidden(C,k3_measure4(A,D)) )
       => r2_hidden(k3_xboole_0(B,C),k3_measure4(A,D)) ) ) )).

fof(t19_measure4,axiom,(
    ! [A,B,C,D] :
      ( m1_measure4(D,A)
     => ( ( r2_hidden(B,k3_measure4(A,D))
          & r2_hidden(C,k3_measure4(A,D)) )
       => r2_hidden(k4_xboole_0(B,C),k3_measure4(A,D)) ) ) )).

fof(t20_measure4,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & v1_prob_1(B,A)
        & v1_measure1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k5_numbers,B)
            & m2_relset_1(C,k5_numbers,B) )
         => ! [D] :
              ( m2_subset_1(D,k1_zfmisc_1(A),B)
             => ? [E] :
                  ( v1_funct_1(E)
                  & v1_funct_2(E,k5_numbers,B)
                  & m2_relset_1(E,k5_numbers,B)
                  & ! [F] :
                      ( m2_subset_1(F,k1_numbers,k5_numbers)
                     => k8_funct_2(k5_numbers,B,E,F) = k3_measure1(A,B,D,k8_funct_2(k5_numbers,B,C,F)) ) ) ) ) ) )).

fof(t21_measure4,axiom,(
    ! [A,B] :
      ( m1_measure4(B,A)
     => ( ~ v1_xboole_0(k3_measure4(A,B))
        & v1_prob_1(k3_measure4(A,B),A)
        & v1_measure1(k3_measure4(A,B),A)
        & m1_subset_1(k3_measure4(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) )).

fof(d4_measure4,axiom,(
    ! [A,B] :
      ( m1_measure4(B,A)
     => ! [C] :
          ( ( v1_funct_1(C)
            & v1_funct_2(C,k3_measure4(A,B),k3_supinf_1)
            & m2_relset_1(C,k3_measure4(A,B),k3_supinf_1) )
         => ( C = k5_measure4(A,B)
          <=> ! [D] :
                ( m1_subset_1(D,k1_zfmisc_1(A))
               => ( r2_hidden(D,k3_measure4(A,B))
                 => k1_funct_1(C,D) = k1_measure1(k1_zfmisc_1(A),B,D) ) ) ) ) ) )).

fof(t22_measure4,axiom,(
    ! [A,B] :
      ( m1_measure4(B,A)
     => m1_measure1(k5_measure4(A,B),A,k3_measure4(A,B)) ) )).

fof(t23_measure4,axiom,(
    ! [A,B] :
      ( m1_measure4(B,A)
     => m3_measure1(k5_measure4(A,B),A,k3_measure4(A,B)) ) )).

fof(t24_measure4,axiom,(
    ! [A,B] :
      ( m1_measure4(B,A)
     => ! [C] :
          ( m1_subset_1(C,k1_zfmisc_1(A))
         => ( k1_measure1(k1_zfmisc_1(A),B,C) = k1_supinf_2
           => r2_hidden(C,k3_measure4(A,B)) ) ) ) )).

fof(t25_measure4,axiom,(
    ! [A,B] :
      ( m1_measure4(B,A)
     => r1_measure3(A,k3_measure4(A,B),k7_measure4(A,B)) ) )).

fof(dt_m1_measure4,axiom,(
    ! [A,B] :
      ( m1_measure4(B,A)
     => ( v1_funct_1(B)
        & v1_funct_2(B,k1_zfmisc_1(A),k3_supinf_1)
        & m2_relset_1(B,k1_zfmisc_1(A),k3_supinf_1) ) ) )).

fof(existence_m1_measure4,axiom,(
    ! [A] :
    ? [B] : m1_measure4(B,A) )).

fof(dt_k1_measure4,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => m1_subset_1(k1_measure4(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) )).

fof(redefinition_k1_measure4,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
        & m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
     => k1_measure4(A,B) = k2_relat_1(B) ) )).

fof(dt_k2_measure4,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & v1_funct_1(D)
        & v1_funct_2(D,A,B)
        & m1_relset_1(D,A,B)
        & v1_funct_1(E)
        & v1_funct_2(E,B,C)
        & m1_relset_1(E,B,C) )
     => ( v1_funct_1(k2_measure4(A,B,C,D,E))
        & v1_funct_2(k2_measure4(A,B,C,D,E),A,C)
        & m2_relset_1(k2_measure4(A,B,C,D,E),A,C) ) ) )).

fof(redefinition_k2_measure4,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & v1_funct_1(D)
        & v1_funct_2(D,A,B)
        & m1_relset_1(D,A,B)
        & v1_funct_1(E)
        & v1_funct_2(E,B,C)
        & m1_relset_1(E,B,C) )
     => k2_measure4(A,B,C,D,E) = k5_relat_1(D,E) ) )).

fof(dt_k3_measure4,axiom,(
    ! [A,B] :
      ( m1_measure4(B,A)
     => ( ~ v1_xboole_0(k3_measure4(A,B))
        & m1_subset_1(k3_measure4(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) )).

fof(dt_k4_measure4,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v1_prob_1(B,A)
        & v1_measure1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & m1_measure2(C,A,B) )
     => m2_subset_1(k4_measure4(A,B,C),k1_zfmisc_1(A),B) ) )).

fof(redefinition_k4_measure4,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(B)
        & v1_prob_1(B,A)
        & v1_measure1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & m1_measure2(C,A,B) )
     => k4_measure4(A,B,C) = k3_tarski(C) ) )).

fof(dt_k5_measure4,axiom,(
    ! [A,B] :
      ( m1_measure4(B,A)
     => ( v1_funct_1(k5_measure4(A,B))
        & v1_funct_2(k5_measure4(A,B),k3_measure4(A,B),k3_supinf_1)
        & m2_relset_1(k5_measure4(A,B),k3_measure4(A,B),k3_supinf_1) ) ) )).

fof(dt_k6_measure4,axiom,(
    ! [A,B,C] :
      ( ( m1_measure4(B,A)
        & m1_subset_1(C,k3_measure4(A,B)) )
     => m1_subset_1(k6_measure4(A,B,C),k3_supinf_1) ) )).

fof(redefinition_k6_measure4,axiom,(
    ! [A,B,C] :
      ( ( m1_measure4(B,A)
        & m1_subset_1(C,k3_measure4(A,B)) )
     => k6_measure4(A,B,C) = k1_funct_1(B,C) ) )).

fof(dt_k7_measure4,axiom,(
    ! [A,B] :
      ( m1_measure4(B,A)
     => m3_measure1(k7_measure4(A,B),A,k3_measure4(A,B)) ) )).

fof(redefinition_k7_measure4,axiom,(
    ! [A,B] :
      ( m1_measure4(B,A)
     => k7_measure4(A,B) = k5_measure4(A,B) ) )).
%------------------------------------------------------------------------------