%------------------------------------------------------------------------------
% File     : SET007+199 : TPTP v8.2.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Euler's Theorem and Small Fermat's Theorem
% 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    : euler_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   40 (  15 unt;   0 def)
%            Number of atoms       :  137 (  32 equ)
%            Maximal formula atoms :   10 (   3 avg)
%            Number of connectives :   98 (   1   ~;  14   |;  11   &)
%                                         (   2 <=>;  70  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   12 (   5 avg)
%            Maximal term depth    :    4 (   1 avg)
%            Number of predicates  :   13 (  11 usr;   1 prp; 0-3 aty)
%            Number of functors    :   19 (  19 usr;   4 con; 0-4 aty)
%            Number of variables   :   65 (  65   !;   0   ?)
% 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_euler_2,axiom,
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_int_2(A,B)
          <=> r2_int_2(A,B) ) ) ) ).

fof(t2_euler_2,axiom,
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_int_1(B)
         => ( r1_xreal_0(np__1,k3_xcmplx_0(A,B))
           => ( r1_xreal_0(A,np__1)
              | r1_xreal_0(np__1,B) ) ) ) ) ).

fof(t3_euler_2,axiom,
    $true ).

fof(t4_euler_2,axiom,
    $true ).

fof(t5_euler_2,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)
             => ( ( r2_int_2(A,C)
                  & r2_int_2(B,C) )
               => ( A = np__0
                  | B = np__0
                  | C = np__0
                  | r2_int_2(C,k4_nat_1(k2_nat_1(A,B),C)) ) ) ) ) ) ).

fof(t6_euler_2,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)
                 => ( ( r2_int_2(A,C)
                      & r2_int_2(D,A)
                      & C = k4_nat_1(k2_nat_1(D,B),A) )
                   => ( r1_xreal_0(A,np__1)
                      | B = np__0
                      | r2_int_2(A,B) ) ) ) ) ) ) ).

fof(t7_euler_2,axiom,
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k4_nat_1(k4_nat_1(A,B),B) = k4_nat_1(A,B) ) ) ).

fof(t8_euler_2,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)
             => k4_nat_1(k1_nat_1(A,B),C) = k4_nat_1(k1_nat_1(k4_nat_1(A,C),k4_nat_1(B,C)),C) ) ) ) ).

fof(t9_euler_2,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)
             => k4_nat_1(k2_nat_1(A,B),C) = k4_nat_1(k2_nat_1(A,k4_nat_1(B,C)),C) ) ) ) ).

fof(t10_euler_2,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)
             => k4_nat_1(k2_nat_1(A,B),C) = k4_nat_1(k2_nat_1(k4_nat_1(A,C),B),C) ) ) ) ).

fof(t11_euler_2,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)
             => k4_nat_1(k2_nat_1(A,B),C) = k4_nat_1(k2_nat_1(k4_nat_1(A,C),k4_nat_1(B,C)),C) ) ) ) ).

fof(t12_euler_2,axiom,
    $true ).

fof(t13_euler_2,axiom,
    $true ).

fof(t14_euler_2,axiom,
    $true ).

fof(t15_euler_2,axiom,
    $true ).

fof(t16_euler_2,axiom,
    $true ).

fof(t17_euler_2,axiom,
    $true ).

fof(t18_euler_2,axiom,
    $true ).

fof(t19_euler_2,axiom,
    $true ).

fof(t20_euler_2,axiom,
    $true ).

fof(t21_euler_2,axiom,
    $true ).

fof(t22_euler_2,axiom,
    $true ).

fof(t23_euler_2,axiom,
    $true ).

fof(t24_euler_2,axiom,
    $true ).

fof(t25_euler_2,axiom,
    ! [A] :
      ( m1_trees_4(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_trees_4(B,k1_numbers,k5_numbers)
         => ( r1_rfinseq(A,B)
           => k10_wsierp_1(A) = k10_wsierp_1(B) ) ) ) ).

fof(d1_euler_2,axiom,
    ! [A] :
      ( m1_trees_4(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_trees_4(C,k1_numbers,k5_numbers)
             => ( C = k2_euler_2(A,B)
              <=> ( k3_finseq_1(C) = k3_finseq_1(A)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(A))
                       => k3_wsierp_1(C,D) = k4_nat_1(k3_wsierp_1(A,D),B) ) ) ) ) ) ) ) ).

fof(t26_euler_2,axiom,
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_trees_4(B,k1_numbers,k5_numbers)
         => ( A != np__0
           => k4_nat_1(k10_wsierp_1(k2_euler_2(B,A)),A) = k4_nat_1(k10_wsierp_1(B),A) ) ) ) ).

fof(t27_euler_2,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)
             => ( ( k4_nat_1(k2_nat_1(A,C),B) = k4_nat_1(C,B)
                  & r2_int_2(B,C) )
               => ( A = np__0
                  | r1_xreal_0(B,np__1)
                  | C = np__0
                  | k4_nat_1(A,B) = np__1 ) ) ) ) ) ).

fof(t28_euler_2,axiom,
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_trees_4(B,k1_numbers,k5_numbers)
         => k2_euler_2(k2_euler_2(B,A),A) = k2_euler_2(B,A) ) ) ).

fof(t29_euler_2,axiom,
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_trees_4(C,k1_numbers,k5_numbers)
             => k2_euler_2(k1_euler_2(A,k2_euler_2(C,B)),B) = k2_euler_2(k1_euler_2(A,C),B) ) ) ) ).

fof(t30_euler_2,axiom,
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m1_trees_4(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_trees_4(C,k1_numbers,k5_numbers)
             => k2_euler_2(k4_wsierp_1(k1_numbers,k5_numbers,B,C),A) = k4_wsierp_1(k1_numbers,k5_numbers,k2_euler_2(B,A),k2_euler_2(C,A)) ) ) ) ).

fof(t31_euler_2,axiom,
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m1_trees_4(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m1_trees_4(D,k1_numbers,k5_numbers)
                 => k2_euler_2(k1_euler_2(A,k4_wsierp_1(k1_numbers,k5_numbers,C,D)),B) = k4_wsierp_1(k1_numbers,k5_numbers,k2_euler_2(k1_euler_2(A,C),B),k2_euler_2(k1_euler_2(A,D),B)) ) ) ) ) ).

fof(t32_euler_2,axiom,
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_int_2(A,B)
           => ( A = np__0
              | B = np__0
              | ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => r2_int_2(k3_euler_2(A,C),B) ) ) ) ) ) ).

fof(t33_euler_2,axiom,
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_int_2(A,B)
           => ( A = np__0
              | r1_xreal_0(B,np__1)
              | k4_nat_1(k3_euler_2(A,k1_euler_1(B)),B) = np__1 ) ) ) ) ).

fof(t34_euler_2,axiom,
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( v1_int_2(B)
              & r2_int_2(A,B) )
           => ( A = np__0
              | k4_nat_1(k3_euler_2(A,B),B) = k4_nat_1(A,B) ) ) ) ) ).

fof(dt_k1_euler_2,axiom,
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_finseq_1(B,k5_numbers) )
     => m1_trees_4(k1_euler_2(A,B),k1_numbers,k5_numbers) ) ).

fof(redefinition_k1_euler_2,axiom,
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_finseq_1(B,k5_numbers) )
     => k1_euler_2(A,B) = k9_rvsum_1(A,B) ) ).

fof(dt_k2_euler_2,axiom,
    ! [A,B] :
      ( ( m1_finseq_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m1_trees_4(k2_euler_2(A,B),k1_numbers,k5_numbers) ) ).

fof(dt_k3_euler_2,axiom,
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_subset_1(k3_euler_2(A,B),k1_numbers,k5_numbers) ) ).

fof(redefinition_k3_euler_2,axiom,
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k3_euler_2(A,B) = k2_newton(A,B) ) ).

%------------------------------------------------------------------------------