SET007 Axioms: SET007+584.ax


%------------------------------------------------------------------------------
% File     : SET007+584 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Gauges
% 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    : jordan8 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   25 (   1 unit)
%            Number of atoms       :  225 (  24 equality)
%            Maximal formula depth :   31 (  10 average)
%            Number of connectives :  242 (  42 ~  ;   4  |; 109  &)
%                                         (   3 <=>;  84 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   26 (   1 propositional; 0-3 arity)
%            Number of functors    :   42 (   7 constant; 0-4 arity)
%            Number of variables   :   75 (   0 singleton;  75 !;   0 ?)
%            Maximal term depth    :    7 (   2 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_jordan8,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
        & m1_subset_1(B,k5_numbers) )
     => ( v1_relat_1(k1_jordan8(A,B))
        & ~ v3_relat_1(k1_jordan8(A,B))
        & v1_funct_1(k1_jordan8(A,B))
        & v1_finset_1(k1_jordan8(A,B))
        & v1_finseq_1(k1_jordan8(A,B))
        & v1_matrix_1(k1_jordan8(A,B))
        & v3_goboard1(k1_jordan8(A,B))
        & v4_goboard1(k1_jordan8(A,B)) ) ) )).

fof(fc2_jordan8,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & ~ v1_sppol_1(A)
        & ~ v2_sppol_1(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
        & m1_subset_1(B,k5_numbers) )
     => ( v1_relat_1(k1_jordan8(A,B))
        & ~ v3_relat_1(k1_jordan8(A,B))
        & v1_funct_1(k1_jordan8(A,B))
        & v1_finset_1(k1_jordan8(A,B))
        & v1_finseq_1(k1_jordan8(A,B))
        & v1_matrix_1(k1_jordan8(A,B))
        & v3_goboard1(k1_jordan8(A,B))
        & v4_goboard1(k1_jordan8(A,B))
        & v5_goboard1(k1_jordan8(A,B))
        & v6_goboard1(k1_jordan8(A,B)) ) ) )).

fof(t1_jordan8,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,A)
     => ( r1_xreal_0(np__2,k3_finseq_1(B))
       => k16_finseq_1(A,B,np__2) = k10_finseq_1(k4_finseq_4(k5_numbers,A,B,np__1),k4_finseq_4(k5_numbers,A,B,np__2)) ) ) )).

fof(t2_jordan8,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B,C] :
          ( m2_finseq_1(C,B)
         => ( r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(C))
           => k16_finseq_1(B,C,k1_nat_1(A,np__1)) = k7_finseq_1(k16_finseq_1(B,C,A),k9_finseq_1(k4_finseq_4(k5_numbers,B,C,k1_nat_1(A,np__1)))) ) ) ) )).

fof(t3_jordan8,axiom,(
    ! [A,B] :
      ( ( v1_matrix_1(B)
        & m2_finseq_1(B,k3_finseq_2(A)) )
     => r1_goboard1(A,k6_finseq_1(A),B) ) )).

fof(t4_jordan8,axiom,(
    $true )).

fof(t5_jordan8,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m2_finseq_1(C,B)
             => ! [D] :
                  ( ( v1_matrix_1(D)
                    & m2_finseq_1(D,k3_finseq_2(B)) )
                 => ( r1_goboard1(B,C,D)
                   => r1_goboard1(B,k1_rfinseq(B,C,A),D) ) ) ) ) ) )).

fof(t6_jordan8,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B,C] :
          ( m2_finseq_1(C,B)
         => ! [D] :
              ( ( v1_matrix_1(D)
                & m2_finseq_1(D,k3_finseq_2(B)) )
             => ~ ( r1_xreal_0(np__1,A)
                  & r1_xreal_0(k1_nat_1(A,np__1),k3_finseq_1(C))
                  & r1_goboard1(B,C,D)
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ! [G] :
                              ( m2_subset_1(G,k1_numbers,k5_numbers)
                             => ! [H] :
                                  ( m2_subset_1(H,k1_numbers,k5_numbers)
                                 => ~ ( r2_hidden(k4_tarski(E,F),k2_matrix_1(D))
                                      & k4_finseq_4(k5_numbers,B,C,A) = k3_matrix_1(B,D,E,F)
                                      & r2_hidden(k4_tarski(G,H),k2_matrix_1(D))
                                      & k4_finseq_4(k5_numbers,B,C,k1_nat_1(A,np__1)) = k3_matrix_1(B,D,G,H)
                                      & ~ ( ~ ( E = G
                                              & k1_nat_1(F,np__1) = H )
                                          & ~ ( k1_nat_1(E,np__1) = G
                                              & F = H )
                                          & ~ ( E = k1_nat_1(G,np__1)
                                              & F = H )
                                          & ~ ( E = G
                                              & F = k1_nat_1(H,np__1) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t7_jordan8,axiom,(
    ! [A] :
      ( ( ~ v3_relat_1(A)
        & v1_matrix_1(A)
        & v3_goboard1(A)
        & v4_goboard1(A)
        & v5_goboard1(A)
        & v6_goboard1(A)
        & m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_finseq_1(B,u1_struct_0(k15_euclid(np__2))) )
         => ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),B,A)
           => ( v2_goboard5(B)
              & v1_topreal1(B) ) ) ) ) )).

fof(t8_jordan8,axiom,(
    ! [A] :
      ( ( ~ v3_relat_1(A)
        & v1_matrix_1(A)
        & v3_goboard1(A)
        & v4_goboard1(A)
        & v5_goboard1(A)
        & v6_goboard1(A)
        & m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m2_finseq_1(B,u1_struct_0(k15_euclid(np__2))) )
         => ~ ( r1_xreal_0(np__2,k3_finseq_1(B))
              & r1_goboard1(u1_struct_0(k15_euclid(np__2)),B,A)
              & v5_seqm_3(B) ) ) ) )).

fof(t9_jordan8,axiom,(
    ! [A] :
      ( ( ~ v3_relat_1(A)
        & v1_matrix_1(A)
        & v3_goboard1(A)
        & v4_goboard1(A)
        & v5_goboard1(A)
        & v6_goboard1(A)
        & m2_finseq_1(A,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m2_finseq_1(C,u1_struct_0(k15_euclid(np__2))) )
             => ( ( r1_goboard1(u1_struct_0(k15_euclid(np__2)),C,A)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ! [G] :
                                  ( m2_subset_1(G,k1_numbers,k5_numbers)
                                 => ( ( r2_hidden(k4_tarski(D,E),k2_matrix_1(A))
                                      & r2_hidden(k4_tarski(F,G),k2_matrix_1(A))
                                      & k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),C,k3_finseq_1(C)) = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),A,D,E)
                                      & B = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),A,F,G) )
                                   => k3_real_1(k18_complex1(k5_real_1(F,D)),k18_complex1(k5_real_1(G,E))) = np__1 ) ) ) ) ) )
               => ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ~ ( r2_hidden(k4_tarski(D,E),k2_matrix_1(A))
                              & B = k3_matrix_1(u1_struct_0(k15_euclid(np__2)),A,D,E) ) ) )
                  | r1_goboard1(u1_struct_0(k15_euclid(np__2)),k8_finseq_1(u1_struct_0(k15_euclid(np__2)),C,k12_finseq_1(u1_struct_0(k15_euclid(np__2)),B)),A) ) ) ) ) ) )).

fof(t10_jordan8,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] :
                  ( ( ~ v3_relat_1(D)
                    & v1_matrix_1(D)
                    & v3_goboard1(D)
                    & v4_goboard1(D)
                    & v5_goboard1(D)
                    & v6_goboard1(D)
                    & m2_finseq_1(D,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
                 => ( r1_xreal_0(np__1,C)
                   => ( r1_xreal_0(k3_finseq_1(D),k1_nat_1(A,B))
                      | r1_xreal_0(k1_matrix_1(D),C)
                      | r1_xboole_0(k3_goboard5(D,A,C),k3_goboard5(D,k1_nat_1(A,B),C))
                      | r1_xreal_0(B,np__1) ) ) ) ) ) ) )).

fof(t11_jordan8,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( v2_sppol_1(A)
      <=> r1_xreal_0(k20_pscomp_1(A),k18_pscomp_1(A)) ) ) )).

fof(t12_jordan8,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v6_compts_1(A,k15_euclid(np__2))
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ( v1_sppol_1(A)
      <=> r1_xreal_0(k19_pscomp_1(A),k21_pscomp_1(A)) ) ) )).

fof(d1_jordan8,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_matrix_1(C)
                & m2_finseq_1(C,k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) )
             => ( C = k1_jordan8(A,B)
              <=> ( k3_finseq_1(C) = k1_nat_1(k1_card_4(np__2,B),np__3)
                  & k3_finseq_1(C) = k1_matrix_1(C)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r2_hidden(k4_tarski(D,E),k2_matrix_1(C))
                           => k3_matrix_1(u1_struct_0(k15_euclid(np__2)),C,D,E) = k23_euclid(k3_real_1(k18_pscomp_1(A),k4_real_1(k6_real_1(k5_real_1(k20_pscomp_1(A),k18_pscomp_1(A)),k1_card_4(np__2,B)),k5_real_1(D,np__2))),k3_real_1(k21_pscomp_1(A),k4_real_1(k6_real_1(k5_real_1(k19_pscomp_1(A),k21_pscomp_1(A)),k1_card_4(np__2,B)),k5_real_1(E,np__2)))) ) ) ) ) ) ) ) ) )).

fof(t13_jordan8,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => r1_xreal_0(np__4,k3_finseq_1(k1_jordan8(B,A))) ) ) )).

fof(t14_jordan8,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
             => ( ( r1_xreal_0(np__1,A)
                  & r1_xreal_0(A,k3_finseq_1(k1_jordan8(C,B))) )
               => k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(C,B),np__2,A)) = k18_pscomp_1(C) ) ) ) ) )).

fof(t15_jordan8,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
             => ( ( r1_xreal_0(np__1,A)
                  & r1_xreal_0(A,k3_finseq_1(k1_jordan8(C,B))) )
               => k21_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(C,B),k5_binarith(k3_finseq_1(k1_jordan8(C,B)),np__1),A)) = k20_pscomp_1(C) ) ) ) ) )).

fof(t16_jordan8,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
             => ( ( r1_xreal_0(np__1,A)
                  & r1_xreal_0(A,k3_finseq_1(k1_jordan8(C,B))) )
               => k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(C,B),A,np__2)) = k21_pscomp_1(C) ) ) ) ) )).

fof(t17_jordan8,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
             => ( ( r1_xreal_0(np__1,A)
                  & r1_xreal_0(A,k3_finseq_1(k1_jordan8(C,B))) )
               => k22_euclid(k3_matrix_1(u1_struct_0(k15_euclid(np__2)),k1_jordan8(C,B),A,k5_binarith(k3_finseq_1(k1_jordan8(C,B)),np__1))) = k19_pscomp_1(C) ) ) ) ) )).

fof(t18_jordan8,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v6_compts_1(C,k15_euclid(np__2))
                & ~ v1_sppol_1(C)
                & ~ v2_sppol_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
             => ( r1_xreal_0(A,k3_finseq_1(k1_jordan8(C,B)))
               => r1_xboole_0(k3_goboard5(k1_jordan8(C,B),A,k3_finseq_1(k1_jordan8(C,B))),C) ) ) ) ) )).

fof(t19_jordan8,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v6_compts_1(C,k15_euclid(np__2))
                & ~ v1_sppol_1(C)
                & ~ v2_sppol_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
             => ( r1_xreal_0(A,k3_finseq_1(k1_jordan8(C,B)))
               => r1_xboole_0(k3_goboard5(k1_jordan8(C,B),k3_finseq_1(k1_jordan8(C,B)),A),C) ) ) ) ) )).

fof(t20_jordan8,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v6_compts_1(C,k15_euclid(np__2))
                & ~ v1_sppol_1(C)
                & ~ v2_sppol_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
             => ( r1_xreal_0(A,k3_finseq_1(k1_jordan8(C,B)))
               => r1_xboole_0(k3_goboard5(k1_jordan8(C,B),A,np__0),C) ) ) ) ) )).

fof(t21_jordan8,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & v6_compts_1(C,k15_euclid(np__2))
                & ~ v1_sppol_1(C)
                & ~ v2_sppol_1(C)
                & m1_subset_1(C,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
             => ( r1_xreal_0(A,k3_finseq_1(k1_jordan8(C,B)))
               => r1_xboole_0(k3_goboard5(k1_jordan8(C,B),np__0,A),C) ) ) ) ) )).

fof(dt_k1_jordan8,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
        & m1_subset_1(B,k5_numbers) )
     => ( v1_matrix_1(k1_jordan8(A,B))
        & m2_finseq_1(k1_jordan8(A,B),k3_finseq_2(u1_struct_0(k15_euclid(np__2)))) ) ) )).
%------------------------------------------------------------------------------