SET007 Axioms: SET007+45.ax


%------------------------------------------------------------------------------
% File     : SET007+45 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Basic Properties of Real Numbers
% 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    : real_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  115 (  92 unit)
%            Number of atoms       :  176 (  13 equality)
%            Maximal formula depth :   15 (   2 average)
%            Number of connectives :   76 (  15 ~  ;   1  |;  24  &)
%                                         (   2 <=>;  34 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    8 (   1 propositional; 0-2 arity)
%            Number of functors    :   15 (   2 constant; 0-2 arity)
%            Number of variables   :   42 (   0 singleton;  41 !;   1 ?)
%            Maximal term depth    :    3 (   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(cc1_real_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ( v1_xreal_0(A)
        & v1_xcmplx_0(A) ) ) )).

fof(t1_real_1,axiom,(
    $true )).

fof(t2_real_1,axiom,(
    $true )).

fof(t3_real_1,axiom,(
    $true )).

fof(t4_real_1,axiom,(
    $true )).

fof(t5_real_1,axiom,(
    $true )).

fof(t6_real_1,axiom,(
    $true )).

fof(t7_real_1,axiom,(
    $true )).

fof(t8_real_1,axiom,(
    $true )).

fof(t9_real_1,axiom,(
    $true )).

fof(t10_real_1,axiom,(
    $true )).

fof(t11_real_1,axiom,(
    $true )).

fof(t12_real_1,axiom,(
    $true )).

fof(t13_real_1,axiom,(
    $true )).

fof(t14_real_1,axiom,(
    $true )).

fof(t15_real_1,axiom,(
    $true )).

fof(t16_real_1,axiom,(
    $true )).

fof(t17_real_1,axiom,(
    $true )).

fof(t18_real_1,axiom,(
    $true )).

fof(t19_real_1,axiom,(
    $true )).

fof(t20_real_1,axiom,(
    $true )).

fof(t21_real_1,axiom,(
    $true )).

fof(t22_real_1,axiom,(
    $true )).

fof(t23_real_1,axiom,(
    $true )).

fof(t24_real_1,axiom,(
    $true )).

fof(t25_real_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => k6_xcmplx_0(A,np__0) = A ) )).

fof(t26_real_1,axiom,(
    k1_real_1(np__0) = np__0 )).

fof(d1_real_1,axiom,(
    $true )).

fof(d2_real_1,axiom,(
    $true )).

fof(d3_real_1,axiom,(
    $true )).

fof(d4_real_1,axiom,(
    $true )).

fof(d5_real_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ( r1_xreal_0(A,B)
          <=> ~ ( r1_xreal_0(B,A)
                & B != A ) ) ) ) )).

fof(t27_real_1,axiom,(
    $true )).

fof(t28_real_1,axiom,(
    $true )).

fof(t29_real_1,axiom,(
    $true )).

fof(t30_real_1,axiom,(
    $true )).

fof(t31_real_1,axiom,(
    $true )).

fof(t32_real_1,axiom,(
    $true )).

fof(t33_real_1,axiom,(
    $true )).

fof(t34_real_1,axiom,(
    $true )).

fof(t35_real_1,axiom,(
    $true )).

fof(t36_real_1,axiom,(
    $true )).

fof(t37_real_1,axiom,(
    $true )).

fof(t38_real_1,axiom,(
    $true )).

fof(t39_real_1,axiom,(
    $true )).

fof(t40_real_1,axiom,(
    $true )).

fof(t41_real_1,axiom,(
    $true )).

fof(t42_real_1,axiom,(
    $true )).

fof(t43_real_1,axiom,(
    $true )).

fof(t44_real_1,axiom,(
    $true )).

fof(t45_real_1,axiom,(
    $true )).

fof(t46_real_1,axiom,(
    $true )).

fof(t47_real_1,axiom,(
    $true )).

fof(t48_real_1,axiom,(
    $true )).

fof(t49_real_1,axiom,(
    $true )).

fof(t50_real_1,axiom,(
    $true )).

fof(t51_real_1,axiom,(
    $true )).

fof(t52_real_1,axiom,(
    $true )).

fof(t53_real_1,axiom,(
    $true )).

fof(t54_real_1,axiom,(
    $true )).

fof(t55_real_1,axiom,(
    $true )).

fof(t56_real_1,axiom,(
    $true )).

fof(t57_real_1,axiom,(
    $true )).

fof(t58_real_1,axiom,(
    $true )).

fof(t59_real_1,axiom,(
    $true )).

fof(t60_real_1,axiom,(
    $true )).

fof(t61_real_1,axiom,(
    $true )).

fof(t62_real_1,axiom,(
    $true )).

fof(t63_real_1,axiom,(
    $true )).

fof(t64_real_1,axiom,(
    $true )).

fof(t65_real_1,axiom,(
    $true )).

fof(t66_real_1,axiom,(
    $true )).

fof(t67_real_1,axiom,(
    $true )).

fof(t68_real_1,axiom,(
    $true )).

fof(t69_real_1,axiom,(
    $true )).

fof(t70_real_1,axiom,(
    $true )).

fof(t71_real_1,axiom,(
    $true )).

fof(t72_real_1,axiom,(
    $true )).

fof(t73_real_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ~ r1_xreal_0(A,np__0)
               => ( ~ ( ~ r1_xreal_0(C,B)
                      & r1_xreal_0(k7_xcmplx_0(C,A),k7_xcmplx_0(B,A)) )
                  & ~ ( ~ r1_xreal_0(k7_xcmplx_0(C,A),k7_xcmplx_0(B,A))
                      & r1_xreal_0(C,B) ) ) ) ) ) ) )).

fof(t74_real_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ( ~ r1_xreal_0(np__0,A)
               => ( ~ ( ~ r1_xreal_0(C,B)
                      & r1_xreal_0(k7_xcmplx_0(B,A),k7_xcmplx_0(C,A)) )
                  & ~ ( ~ r1_xreal_0(k7_xcmplx_0(B,A),k7_xcmplx_0(C,A))
                      & r1_xreal_0(C,B) ) ) ) ) ) ) )).

fof(t75_real_1,axiom,(
    $true )).

fof(t76_real_1,axiom,(
    $true )).

fof(t77_real_1,axiom,(
    $true )).

fof(t78_real_1,axiom,(
    $true )).

fof(t79_real_1,axiom,(
    $true )).

fof(t80_real_1,axiom,(
    $true )).

fof(t81_real_1,axiom,(
    $true )).

fof(t82_real_1,axiom,(
    $true )).

fof(t83_real_1,axiom,(
    $true )).

fof(t84_real_1,axiom,(
    $true )).

fof(t85_real_1,axiom,(
    $true )).

fof(t86_real_1,axiom,(
    $true )).

fof(t87_real_1,axiom,(
    $true )).

fof(t88_real_1,axiom,(
    $true )).

fof(t89_real_1,axiom,(
    $true )).

fof(t90_real_1,axiom,(
    $true )).

fof(t91_real_1,axiom,(
    $true )).

fof(t92_real_1,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( ( ( r1_xreal_0(A,B)
                        & r1_xreal_0(C,D) )
                     => r1_xreal_0(k6_xcmplx_0(A,D),k6_xcmplx_0(B,C)) )
                    & ~ ( ( ( ~ r1_xreal_0(B,A)
                            & r1_xreal_0(C,D) )
                          | ( r1_xreal_0(A,B)
                            & ~ r1_xreal_0(D,C) ) )
                        & r1_xreal_0(k6_xcmplx_0(B,C),k6_xcmplx_0(A,D)) ) ) ) ) ) ) )).

fof(s1_real_1,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
      & ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r2_hidden(B,A)
          <=> p1_s1_real_1(B) ) ) ) )).

fof(dt_k1_real_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => m1_subset_1(k1_real_1(A),k1_numbers) ) )).

fof(involutiveness_k1_real_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k1_real_1(k1_real_1(A)) = A ) )).

fof(redefinition_k1_real_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k1_real_1(A) = k4_xcmplx_0(A) ) )).

fof(dt_k2_real_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => m1_subset_1(k2_real_1(A),k1_numbers) ) )).

fof(involutiveness_k2_real_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k2_real_1(k2_real_1(A)) = A ) )).

fof(redefinition_k2_real_1,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => k2_real_1(A) = k5_xcmplx_0(A) ) )).

fof(dt_k3_real_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k3_real_1(A,B),k1_numbers) ) )).

fof(commutativity_k3_real_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k3_real_1(A,B) = k3_real_1(B,A) ) )).

fof(redefinition_k3_real_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k3_real_1(A,B) = k2_xcmplx_0(A,B) ) )).

fof(dt_k4_real_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k4_real_1(A,B),k1_numbers) ) )).

fof(commutativity_k4_real_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k4_real_1(A,B) = k4_real_1(B,A) ) )).

fof(redefinition_k4_real_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k4_real_1(A,B) = k3_xcmplx_0(A,B) ) )).

fof(dt_k5_real_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k5_real_1(A,B),k1_numbers) ) )).

fof(redefinition_k5_real_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k5_real_1(A,B) = k6_xcmplx_0(A,B) ) )).

fof(dt_k6_real_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k6_real_1(A,B),k1_numbers) ) )).

fof(redefinition_k6_real_1,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => k6_real_1(A,B) = k7_xcmplx_0(A,B) ) )).
%------------------------------------------------------------------------------