SET007 Axioms: SET007+869.ax


%------------------------------------------------------------------------------
% File     : SET007+869 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Lebesgue Integral of Simple Valued Function
% 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    : mesfunc3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   21 (   0 unit)
%            Number of atoms       :  267 (  53 equality)
%            Maximal formula depth :   29 (  15 average)
%            Number of connectives :  292 (  46 ~  ;   4  |; 115  &)
%                                         (   3 <=>; 124 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   20 (   0 propositional; 1-5 arity)
%            Number of functors    :   29 (  10 constant; 0-4 arity)
%            Number of variables   :  113 (   0 singleton; 106 !;   7 ?)
%            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(t1_mesfunc3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k6_supinf_1)
                & m2_relset_1(C,k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k6_supinf_1) )
             => ! [D] :
                  ( m2_finseq_1(D,k6_supinf_1)
                 => ! [E] :
                      ( m2_finseq_1(E,k6_supinf_1)
                     => ( ( k5_finsop_1(D) = k2_finseq_1(A)
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ~ ( r2_hidden(F,k5_finsop_1(D))
                                  & ! [G] :
                                      ( m2_finseq_1(G,k6_supinf_1)
                                     => ~ ( k5_finsop_1(G) = k2_finseq_1(B)
                                          & k2_seq_1(k5_numbers,k6_supinf_1,D,F) = k15_rvsum_1(G)
                                          & ! [H] :
                                              ( m2_subset_1(H,k1_numbers,k5_numbers)
                                             => ( r2_hidden(H,k5_finsop_1(G))
                                               => k2_seq_1(k5_numbers,k6_supinf_1,G,H) = k2_seq_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k6_supinf_1,C,k4_tarski(F,H)) ) ) ) ) ) )
                          & k5_finsop_1(E) = k2_finseq_1(B)
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ~ ( r2_hidden(F,k5_finsop_1(E))
                                  & ! [G] :
                                      ( m2_finseq_1(G,k6_supinf_1)
                                     => ~ ( k5_finsop_1(G) = k2_finseq_1(A)
                                          & k2_seq_1(k5_numbers,k6_supinf_1,E,F) = k15_rvsum_1(G)
                                          & ! [H] :
                                              ( m2_subset_1(H,k1_numbers,k5_numbers)
                                             => ( r2_hidden(H,k5_finsop_1(G))
                                               => k2_seq_1(k5_numbers,k6_supinf_1,G,H) = k2_seq_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k6_supinf_1,C,k4_tarski(H,F)) ) ) ) ) ) ) )
                       => k15_rvsum_1(D) = k15_rvsum_1(E) ) ) ) ) ) ) )).

fof(t2_mesfunc3,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k3_supinf_1)
     => ! [B] :
          ( m2_finseq_1(B,k6_supinf_1)
         => ( A = B
           => k5_convfun1(A) = k15_rvsum_1(B) ) ) ) )).

fof(t3_mesfunc3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(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)
                & m2_relset_1(C,A,k3_supinf_1) )
             => ~ ( r2_mesfunc2(A,B,C)
                  & ! [D] :
                      ( ( v1_prob_2(D)
                        & m2_finseq_1(D,B) )
                     => ! [E] :
                          ( m2_finseq_1(E,k3_supinf_1)
                         => ~ ( k1_relat_1(C) = k3_tarski(k2_relat_1(D))
                              & k5_finsop_1(D) = k5_finsop_1(E)
                              & ! [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                 => ( r2_hidden(F,k5_finsop_1(D))
                                   => ! [G] :
                                        ( r2_hidden(G,k1_funct_1(D,F))
                                       => k4_mesfunc1(A,C,G) = k4_mesfunc1(k5_numbers,E,F) ) ) )
                              & ! [F] :
                                  ~ ( r2_hidden(F,k1_relat_1(C))
                                    & ! [G] :
                                        ( m2_finseq_1(G,k3_supinf_1)
                                       => ~ ( k5_finsop_1(G) = k5_finsop_1(E)
                                            & ! [H] :
                                                ( m2_subset_1(H,k1_numbers,k5_numbers)
                                               => ( r2_hidden(H,k5_finsop_1(G))
                                                 => k4_mesfunc1(k5_numbers,G,H) = k2_extreal1(k4_mesfunc1(k5_numbers,E,H),k4_mesfunc1(A,k3_mesfunc2(k1_funct_1(D,H),A),F)) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t4_mesfunc3,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,A)
     => ( v1_prob_2(B)
      <=> ! [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
           => ! [D] :
                ( m2_subset_1(D,k1_numbers,k5_numbers)
               => ( ( r2_hidden(C,k5_finsop_1(B))
                    & r2_hidden(D,k5_finsop_1(B)) )
                 => ( C = D
                    | r1_xboole_0(k1_funct_1(B,C),k1_funct_1(B,D)) ) ) ) ) ) ) )).

fof(t5_mesfunc3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( ( ~ v1_xboole_0(C)
            & v1_prob_1(C,A)
            & v1_measure1(C,A)
            & m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
         => ! [D] :
              ( ( v1_prob_2(D)
                & m2_finseq_1(D,C) )
             => ! [E] :
                  ( m2_finseq_1(E,C)
                 => ( ( k5_finsop_1(E) = k5_finsop_1(D)
                      & ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ( r2_hidden(F,k5_finsop_1(E))
                           => k1_funct_1(E,F) = k3_xboole_0(B,k1_funct_1(D,F)) ) ) )
                   => ( v1_prob_2(E)
                      & m2_finseq_1(E,C) ) ) ) ) ) ) )).

fof(t6_mesfunc3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B,C] :
          ( m2_finseq_1(C,A)
         => ! [D] :
              ( m2_finseq_1(D,A)
             => ( ( k5_finsop_1(D) = k5_finsop_1(C)
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( r2_hidden(E,k5_finsop_1(D))
                       => k1_funct_1(D,E) = k3_xboole_0(B,k1_funct_1(C,E)) ) ) )
               => k3_tarski(k2_relat_1(D)) = k3_xboole_0(B,k3_tarski(k2_relat_1(C))) ) ) ) ) )).

fof(t7_mesfunc3,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,A)
     => ! [C] :
          ( m2_subset_1(C,k1_numbers,k5_numbers)
         => ( r2_hidden(C,k5_finsop_1(B))
           => ( r1_tarski(k1_funct_1(B,C),k3_tarski(k2_relat_1(B)))
              & k3_xboole_0(k1_funct_1(B,C),k3_tarski(k2_relat_1(B))) = k1_funct_1(B,C) ) ) ) ) )).

fof(t8_mesfunc3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(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] :
              ( m3_measure1(C,A,B)
             => ! [D] :
                  ( ( v1_prob_2(D)
                    & m2_finseq_1(D,B) )
                 => k5_finsop_1(D) = k5_finsop_1(k5_finseqop(B,k3_supinf_1,D,C)) ) ) ) ) )).

fof(t9_mesfunc3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(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] :
              ( m3_measure1(C,A,B)
             => ! [D] :
                  ( ( v1_prob_2(D)
                    & m2_finseq_1(D,B) )
                 => k4_mesfunc1(B,C,k3_tarski(k2_relat_1(D))) = k5_convfun1(k5_finseqop(B,k3_supinf_1,D,C)) ) ) ) ) )).

fof(t10_mesfunc3,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k3_supinf_1)
     => ! [B] :
          ( m2_finseq_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ( ( k5_finsop_1(A) = k5_finsop_1(B)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k5_finsop_1(B))
                       => k4_mesfunc1(k5_numbers,B,D) = k2_extreal1(C,k4_mesfunc1(k5_numbers,A,D)) ) ) )
               => ( ( ~ ( C != k4_supinf_1
                        & C != k5_supinf_1 )
                    & ? [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                        & r2_hidden(D,k5_finsop_1(A))
                        & r1_supinf_1(k1_supinf_2,k4_mesfunc1(k5_numbers,A,D)) )
                    & ? [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                        & r2_hidden(D,k5_finsop_1(A))
                        & r1_supinf_1(k4_mesfunc1(k5_numbers,A,D),k1_supinf_2) ) )
                  | k5_convfun1(B) = k2_extreal1(C,k5_convfun1(A)) ) ) ) ) ) )).

fof(t11_mesfunc3,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k6_supinf_1)
     => m2_finseq_1(A,k3_supinf_1) ) )).

fof(d1_mesfunc3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(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)
                & m2_relset_1(C,A,k3_supinf_1) )
             => ! [D] :
                  ( ( v1_prob_2(D)
                    & m2_finseq_1(D,B) )
                 => ! [E] :
                      ( m2_finseq_1(E,k3_supinf_1)
                     => ( r1_mesfunc3(A,B,C,D,E)
                      <=> ( k1_relat_1(C) = k3_tarski(k2_relat_1(D))
                          & k5_finsop_1(D) = k5_finsop_1(E)
                          & ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ( r2_hidden(F,k5_finsop_1(D))
                               => ! [G] :
                                    ( r2_hidden(G,k1_funct_1(D,F))
                                   => k4_mesfunc1(A,C,G) = k4_mesfunc1(k5_numbers,E,F) ) ) ) ) ) ) ) ) ) ) )).

fof(t12_mesfunc3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(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)
                & m2_relset_1(C,A,k3_supinf_1) )
             => ~ ( r2_mesfunc2(A,B,C)
                  & ! [D] :
                      ( ( v1_prob_2(D)
                        & m2_finseq_1(D,B) )
                     => ! [E] :
                          ( m2_finseq_1(E,k3_supinf_1)
                         => ~ r1_mesfunc3(A,B,C,D,E) ) ) ) ) ) ) )).

fof(t13_mesfunc3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(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_prob_2(C)
                & m2_finseq_1(C,B) )
             => ? [D] :
                  ( v1_prob_2(D)
                  & m2_finseq_1(D,B)
                  & k3_tarski(k2_relat_1(C)) = k3_tarski(k2_relat_1(D))
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( r2_hidden(E,k5_finsop_1(D))
                       => ( k1_funct_1(D,E) != k1_xboole_0
                          & ? [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                              & r2_hidden(F,k5_finsop_1(C))
                              & k1_funct_1(C,F) = k1_funct_1(D,E) ) ) ) ) ) ) ) ) )).

fof(t14_mesfunc3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(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)
                & m2_relset_1(C,A,k3_supinf_1) )
             => ~ ( r2_mesfunc2(A,B,C)
                  & ! [D] :
                      ( r2_hidden(D,k1_relat_1(C))
                     => r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,C,D)) )
                  & ! [D] :
                      ( ( v1_prob_2(D)
                        & m2_finseq_1(D,B) )
                     => ! [E] :
                          ( m2_finseq_1(E,k3_supinf_1)
                         => ~ ( r1_mesfunc3(A,B,C,D,E)
                              & k4_mesfunc1(k5_numbers,E,np__1) = k1_supinf_2
                              & ! [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                 => ( ( r1_xreal_0(np__2,F)
                                      & r2_hidden(F,k5_finsop_1(E)) )
                                   => ( ~ r1_supinf_1(k4_mesfunc1(k5_numbers,E,F),k1_supinf_2)
                                      & ~ r1_supinf_1(k4_supinf_1,k4_mesfunc1(k5_numbers,E,F)) ) ) ) ) ) ) ) ) ) ) )).

fof(t15_mesfunc3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(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)
                & m2_relset_1(C,A,k3_supinf_1) )
             => ! [D] :
                  ( ( v1_prob_2(D)
                    & m2_finseq_1(D,B) )
                 => ! [E] :
                      ( m2_finseq_1(E,k3_supinf_1)
                     => ! [F] :
                          ( m1_subset_1(F,A)
                         => ~ ( r1_mesfunc3(A,B,C,D,E)
                              & r2_hidden(F,k1_relat_1(C))
                              & ! [G] :
                                  ( m2_finseq_1(G,k3_supinf_1)
                                 => ~ ( k5_finsop_1(G) = k5_finsop_1(E)
                                      & ! [H] :
                                          ( m2_subset_1(H,k1_numbers,k5_numbers)
                                         => ( r2_hidden(H,k5_finsop_1(G))
                                           => k4_mesfunc1(k5_numbers,G,H) = k2_extreal1(k4_mesfunc1(k5_numbers,E,H),k4_mesfunc1(A,k3_mesfunc2(k1_funct_1(D,H),A),F)) ) )
                                      & k4_mesfunc1(A,C,F) = k5_convfun1(G) ) ) ) ) ) ) ) ) ) )).

fof(t16_mesfunc3,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k3_supinf_1)
     => ! [B] :
          ( m2_finseq_1(B,k6_supinf_1)
         => ( A = B
           => k5_convfun1(A) = k15_rvsum_1(B) ) ) ) )).

fof(t17_mesfunc3,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k3_supinf_1)
     => ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r2_hidden(B,k5_finsop_1(A))
             => r1_supinf_1(k1_supinf_2,k4_mesfunc1(k5_numbers,A,B)) ) )
       => ( ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ~ ( r2_hidden(B,k5_finsop_1(A))
                  & k4_mesfunc1(k5_numbers,A,B) = k4_supinf_1 ) )
          | k5_convfun1(A) = k4_supinf_1 ) ) ) )).

fof(d2_mesfunc3,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(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] :
              ( m3_measure1(C,A,B)
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,A,k3_supinf_1) )
                 => ( ( r2_mesfunc2(A,B,D)
                      & ! [E] :
                          ( r2_hidden(E,k1_relat_1(D))
                         => r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,D,E)) ) )
                   => ( k1_relat_1(D) = k1_xboole_0
                      | ! [E] :
                          ( m1_subset_1(E,k3_supinf_1)
                         => ( E = k1_mesfunc3(A,B,C,D)
                          <=> ? [F] :
                                ( v1_prob_2(F)
                                & m2_finseq_1(F,B)
                                & ? [G] :
                                    ( m2_finseq_1(G,k3_supinf_1)
                                    & ? [H] :
                                        ( m2_finseq_1(H,k3_supinf_1)
                                        & r1_mesfunc3(A,B,D,F,G)
                                        & k4_mesfunc1(k5_numbers,G,np__1) = k1_supinf_2
                                        & ! [I] :
                                            ( m2_subset_1(I,k1_numbers,k5_numbers)
                                           => ( ( r1_xreal_0(np__2,I)
                                                & r2_hidden(I,k5_finsop_1(G)) )
                                             => ( ~ r1_supinf_1(k4_mesfunc1(k5_numbers,G,I),k1_supinf_2)
                                                & ~ r1_supinf_1(k4_supinf_1,k4_mesfunc1(k5_numbers,G,I)) ) ) )
                                        & k5_finsop_1(H) = k5_finsop_1(F)
                                        & ! [I] :
                                            ( m2_subset_1(I,k1_numbers,k5_numbers)
                                           => ( r2_hidden(I,k5_finsop_1(H))
                                             => k4_mesfunc1(k5_numbers,H,I) = k2_extreal1(k4_mesfunc1(k5_numbers,G,I),k4_mesfunc1(k5_numbers,k5_finseqop(B,k3_supinf_1,F,C),I)) ) )
                                        & E = k5_convfun1(H) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t18_mesfunc3,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k3_supinf_1)
     => ! [B] :
          ( m1_subset_1(B,k3_supinf_1)
         => ! [C] :
              ( m1_subset_1(C,k3_supinf_1)
             => ( ( C = k3_finseq_1(A)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k5_finsop_1(A))
                       => k4_mesfunc1(k5_numbers,A,D) = B ) ) )
               => k5_convfun1(A) = k2_extreal1(C,B) ) ) ) ) )).

fof(dt_k1_mesfunc3,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & v1_prob_1(B,A)
        & v1_measure1(B,A)
        & m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
        & m3_measure1(C,A,B)
        & v1_funct_1(D)
        & m1_relset_1(D,A,k3_supinf_1) )
     => m1_subset_1(k1_mesfunc3(A,B,C,D),k3_supinf_1) ) )).
%------------------------------------------------------------------------------