SET007 Axioms: SET007+172.ax


%------------------------------------------------------------------------------
% File     : SET007+172 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Two Programs for SCM. Part I - Preliminaries
% 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    : pre_ff [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   32 (   2 unit)
%            Number of atoms       :  169 (  45 equality)
%            Maximal formula depth :   23 (   8 average)
%            Number of connectives :  168 (  31 ~  ;   5  |;  59  &)
%                                         (   3 <=>;  70 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   13 (   1 propositional; 0-3 arity)
%            Number of functors    :   31 (   5 constant; 0-5 arity)
%            Number of variables   :   77 (   0 singleton;  74 !;   3 ?)
%            Maximal term depth    :    6 (   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(d1_pre_ff,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k3_pre_ff(A)
          <=> ? [C] :
                ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
                & m2_relset_1(C,k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers))
                & B = k1_pre_ff(k1_numbers,k1_numbers,k5_numbers,k5_numbers,k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,A))
                & k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,np__0) = k1_domain_1(k5_numbers,k5_numbers,np__0,np__1)
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,k1_nat_1(D,np__1)) = k1_domain_1(k5_numbers,k5_numbers,k2_pre_ff(k1_numbers,k1_numbers,k5_numbers,k5_numbers,k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,D)),k1_nat_1(k1_pre_ff(k1_numbers,k1_numbers,k5_numbers,k5_numbers,k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,D)),k2_pre_ff(k1_numbers,k1_numbers,k5_numbers,k5_numbers,k8_funct_2(k5_numbers,k12_mcart_1(k1_numbers,k1_numbers,k5_numbers,k5_numbers),C,D)))) ) ) ) ) ) )).

fof(t1_pre_ff,axiom,
    ( k3_pre_ff(np__0) = np__0
    & k3_pre_ff(np__1) = np__1
    & ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => k3_pre_ff(k1_nat_1(k1_nat_1(A,np__1),np__1)) = k1_nat_1(k3_pre_ff(A),k3_pre_ff(k1_nat_1(A,np__1))) ) )).

fof(t2_pre_ff,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => k5_int_1(A,np__1) = A ) )).

fof(t3_pre_ff,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ~ ( ~ r1_xreal_0(B,np__0)
              & k5_int_1(A,B) = np__0
              & r1_xreal_0(B,A) ) ) ) )).

fof(t4_pre_ff,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( r1_xreal_0(np__0,A)
           => ( r1_xreal_0(B,A)
              | k5_int_1(A,B) = np__0 ) ) ) ) )).

fof(t5_pre_ff,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ~ ( ~ r1_xreal_0(B,np__0)
                  & ~ r1_xreal_0(C,np__0)
                  & k5_int_1(k5_int_1(A,B),C) != k5_int_1(A,k3_xcmplx_0(B,C)) ) ) ) ) )).

fof(t6_pre_ff,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( k6_int_1(A,np__2) = np__0
        | k6_int_1(A,np__2) = np__1 ) ) )).

fof(t7_pre_ff,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( m2_subset_1(A,k1_numbers,k5_numbers)
       => m2_subset_1(k5_int_1(A,np__2),k1_numbers,k5_numbers) ) ) )).

fof(t8_pre_ff,axiom,(
    $true )).

fof(t9_pre_ff,axiom,(
    $true )).

fof(t10_pre_ff,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( r1_xreal_0(A,B)
               => ( r1_xreal_0(C,np__1)
                  | r1_xreal_0(k3_power(C,A),k3_power(C,B)) ) ) ) ) ) )).

fof(t11_pre_ff,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(B,A)
           => r1_xreal_0(k1_int_1(B),k1_int_1(A)) ) ) ) )).

fof(t12_pre_ff,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( r1_xreal_0(B,C)
               => ( r1_xreal_0(A,np__1)
                  | r1_xreal_0(B,np__0)
                  | r1_xreal_0(k5_power(A,B),k5_power(A,C)) ) ) ) ) ) )).

fof(t13_pre_ff,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ ( ~ r1_xreal_0(A,np__0)
          & k2_xcmplx_0(k1_int_1(k6_power(np__2,k2_nat_1(np__2,A))),np__1) = k1_int_1(k6_power(np__2,k1_nat_1(k2_nat_1(np__2,A),np__1))) ) ) )).

fof(t14_pre_ff,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => r1_xreal_0(k1_int_1(k6_power(np__2,k1_nat_1(k2_nat_1(np__2,A),np__1))),k2_xcmplx_0(k1_int_1(k6_power(np__2,k2_nat_1(np__2,A))),np__1)) ) ) )).

fof(t15_pre_ff,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => k1_int_1(k6_power(np__2,k2_nat_1(np__2,A))) = k1_int_1(k6_power(np__2,k1_nat_1(k2_nat_1(np__2,A),np__1))) ) ) )).

fof(t16_pre_ff,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => k2_xcmplx_0(k1_int_1(k6_power(np__2,A)),np__1) = k1_int_1(k6_power(np__2,k1_nat_1(k2_nat_1(np__2,A),np__1))) ) ) )).

fof(d2_pre_ff,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( A = np__0
             => ( B = k5_pre_ff(A)
              <=> B = np__0 ) )
            & ( A != np__0
             => ( B = k5_pre_ff(A)
              <=> ? [C] :
                    ( m2_subset_1(C,k1_numbers,k5_numbers)
                    & ? [D] :
                        ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,k3_finseq_2(k5_numbers))
                        & m2_relset_1(D,k5_numbers,k3_finseq_2(k5_numbers))
                        & k1_nat_1(C,np__1) = A
                        & B = k4_finseq_4(k5_numbers,k5_numbers,k4_pre_ff(D,C),A)
                        & k4_pre_ff(D,np__0) = k12_finseq_1(k5_numbers,np__1)
                        & ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ( ! [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                 => ( k1_nat_1(E,np__2) = k2_nat_1(np__2,F)
                                   => k4_pre_ff(D,k1_nat_1(E,np__1)) = k8_finseq_1(k5_numbers,k4_pre_ff(D,E),k12_finseq_1(k5_numbers,k4_finseq_4(k5_numbers,k5_numbers,k4_pre_ff(D,E),F))) ) )
                              & ! [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                 => ( k1_nat_1(E,np__2) = k1_nat_1(k2_nat_1(np__2,F),np__1)
                                   => k4_pre_ff(D,k1_nat_1(E,np__1)) = k8_finseq_1(k5_numbers,k4_pre_ff(D,E),k12_finseq_1(k5_numbers,k1_nat_1(k4_finseq_4(k5_numbers,k5_numbers,k4_pre_ff(D,E),F),k4_finseq_4(k5_numbers,k5_numbers,k4_pre_ff(D,E),k1_nat_1(F,np__1))))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t17_pre_ff,axiom,
    ( k5_pre_ff(np__0) = np__0
    & k5_pre_ff(np__1) = np__1
    & ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( k5_pre_ff(k2_nat_1(np__2,A)) = k5_pre_ff(A)
          & k5_pre_ff(k1_nat_1(k2_nat_1(np__2,A),np__1)) = k1_nat_1(k5_pre_ff(A),k5_pre_ff(k1_nat_1(A,np__1))) ) ) )).

fof(t18_pre_ff,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( A != np__0
              & A = k2_nat_1(np__2,B)
              & r1_xreal_0(A,B) ) ) ) )).

fof(t19_pre_ff,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( A = k1_nat_1(k2_nat_1(np__2,B),np__1)
              & r1_xreal_0(A,B) ) ) ) )).

fof(t20_pre_ff,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => B = k1_nat_1(k2_nat_1(A,k5_pre_ff(np__0)),k2_nat_1(B,k5_pre_ff(k1_nat_1(np__0,np__1)))) ) ) )).

fof(t21_pre_ff,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( ( A = k1_nat_1(k2_nat_1(np__2,B),np__1)
                          & k5_pre_ff(E) = k1_nat_1(k2_nat_1(C,k5_pre_ff(A)),k2_nat_1(D,k5_pre_ff(k1_nat_1(A,np__1)))) )
                       => k5_pre_ff(E) = k1_nat_1(k2_nat_1(C,k5_pre_ff(B)),k2_nat_1(k1_nat_1(D,C),k5_pre_ff(k1_nat_1(B,np__1)))) ) ) ) ) ) ) )).

fof(t22_pre_ff,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( ( A = k2_nat_1(np__2,B)
                          & k5_pre_ff(E) = k1_nat_1(k2_nat_1(C,k5_pre_ff(A)),k2_nat_1(D,k5_pre_ff(k1_nat_1(A,np__1)))) )
                       => k5_pre_ff(E) = k1_nat_1(k2_nat_1(k1_nat_1(C,D),k5_pre_ff(B)),k2_nat_1(D,k5_pre_ff(k1_nat_1(B,np__1)))) ) ) ) ) ) ) )).

fof(dt_k1_pre_ff,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(A))
        & ~ v1_xboole_0(D)
        & m1_subset_1(D,k1_zfmisc_1(B))
        & m1_subset_1(E,k12_mcart_1(A,B,C,D)) )
     => m2_subset_1(k1_pre_ff(A,B,C,D,E),A,C) ) )).

fof(redefinition_k1_pre_ff,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(A))
        & ~ v1_xboole_0(D)
        & m1_subset_1(D,k1_zfmisc_1(B))
        & m1_subset_1(E,k12_mcart_1(A,B,C,D)) )
     => k1_pre_ff(A,B,C,D,E) = k1_mcart_1(E) ) )).

fof(dt_k2_pre_ff,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(A))
        & ~ v1_xboole_0(D)
        & m1_subset_1(D,k1_zfmisc_1(B))
        & m1_subset_1(E,k12_mcart_1(A,B,C,D)) )
     => m2_subset_1(k2_pre_ff(A,B,C,D,E),B,D) ) )).

fof(redefinition_k2_pre_ff,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & ~ v1_xboole_0(B)
        & ~ v1_xboole_0(C)
        & m1_subset_1(C,k1_zfmisc_1(A))
        & ~ v1_xboole_0(D)
        & m1_subset_1(D,k1_zfmisc_1(B))
        & m1_subset_1(E,k12_mcart_1(A,B,C,D)) )
     => k2_pre_ff(A,B,C,D,E) = k2_mcart_1(E) ) )).

fof(dt_k3_pre_ff,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => m2_subset_1(k3_pre_ff(A),k1_numbers,k5_numbers) ) )).

fof(dt_k4_pre_ff,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k3_finseq_2(k5_numbers))
        & m1_relset_1(A,k5_numbers,k3_finseq_2(k5_numbers))
        & m1_subset_1(B,k5_numbers) )
     => m2_finseq_1(k4_pre_ff(A,B),k5_numbers) ) )).

fof(redefinition_k4_pre_ff,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k3_finseq_2(k5_numbers))
        & m1_relset_1(A,k5_numbers,k3_finseq_2(k5_numbers))
        & m1_subset_1(B,k5_numbers) )
     => k4_pre_ff(A,B) = k1_funct_1(A,B) ) )).

fof(dt_k5_pre_ff,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => m2_subset_1(k5_pre_ff(A),k1_numbers,k5_numbers) ) )).
%------------------------------------------------------------------------------