%------------------------------------------------------------------------------
% File     : SET007+847 : TPTP v9.0.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Catalan 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    : catalan1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   25 (   3 unt;   0 def)
%            Number of atoms       :   71 (   9 equ)
%            Maximal formula atoms :    9 (   2 avg)
%            Number of connectives :   68 (  22   ~;   0   |;  17   &)
%                                         (   0 <=>;  29  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   10 (   4 avg)
%            Maximal term depth    :    6 (   2 avg)
%            Number of predicates  :   11 (  10 usr;   0 prp; 1-3 aty)
%            Number of functors    :   18 (  18 usr;   7 con; 0-2 aty)
%            Number of variables   :   22 (  22   !;   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(fc1_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => ( v4_ordinal2(k1_catalan1(A))
        & v1_xcmplx_0(k1_catalan1(A))
        & v1_xreal_0(k1_catalan1(A))
        & ~ v3_xreal_0(k1_catalan1(A))
        & v1_int_1(k1_catalan1(A)) ) ) ).

fof(fc2_catalan1,axiom,
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => ( ~ v1_xboole_0(k1_catalan1(A))
        & v4_ordinal2(k1_catalan1(A))
        & v1_xcmplx_0(k1_catalan1(A))
        & v1_xreal_0(k1_catalan1(A))
        & v2_xreal_0(k1_catalan1(A))
        & ~ v3_xreal_0(k1_catalan1(A))
        & v1_int_1(k1_catalan1(A)) ) ) ).

fof(t1_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__1)
       => r1_xreal_0(k5_binarith(A,np__1),k5_binarith(k3_xcmplx_0(np__2,A),np__3)) ) ) ).

fof(t2_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => ( r1_xreal_0(np__1,A)
       => r1_xreal_0(k5_binarith(A,np__1),k5_binarith(k3_xcmplx_0(np__2,A),np__2)) ) ) ).

fof(t3_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => ~ ( ~ r1_xreal_0(A,np__1)
          & r1_xreal_0(k5_binarith(k3_xcmplx_0(np__2,A),np__1),A) ) ) ).

fof(t4_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__1)
       => k1_nat_1(k5_binarith(A,np__2),np__1) = k5_binarith(A,np__1) ) ) ).

fof(t5_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => ~ ( ~ r1_xreal_0(A,np__1)
          & r1_xreal_0(k7_xcmplx_0(k6_xcmplx_0(k3_xcmplx_0(k3_xcmplx_0(np__4,A),A),k3_xcmplx_0(np__2,A)),k2_xcmplx_0(A,np__1)),np__1) ) ) ).

fof(t6_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => ~ ( ~ r1_xreal_0(A,np__1)
          & r1_xreal_0(k6_newton(k3_xcmplx_0(np__2,A)),k3_xcmplx_0(k3_xcmplx_0(k11_newton(k5_binarith(k3_xcmplx_0(np__2,A),np__2)),A),k2_xcmplx_0(A,np__1))) ) ) ).

fof(t7_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => ~ r1_xreal_0(np__4,k3_xcmplx_0(np__2,k6_xcmplx_0(np__2,k7_xcmplx_0(np__3,k2_xcmplx_0(A,np__1))))) ) ).

fof(d1_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => k1_catalan1(A) = k7_xcmplx_0(k7_binom(k5_binarith(A,np__1),k5_binarith(k3_xcmplx_0(np__2,A),np__2)),A) ) ).

fof(t8_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__1)
       => k1_catalan1(A) = k7_xcmplx_0(k11_newton(k5_binarith(k3_xcmplx_0(np__2,A),np__2)),k3_xcmplx_0(k11_newton(k5_binarith(A,np__1)),k6_newton(A))) ) ) ).

fof(t9_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__1)
       => k1_catalan1(A) = k6_xcmplx_0(k2_nat_1(np__4,k7_binom(k5_binarith(A,np__1),k5_binarith(k3_xcmplx_0(np__2,A),np__3))),k7_binom(k5_binarith(A,np__1),k5_binarith(k3_xcmplx_0(np__2,A),np__1))) ) ) ).

fof(t10_catalan1,axiom,
    k1_catalan1(np__0) = np__0 ).

fof(t11_catalan1,axiom,
    k1_catalan1(np__1) = np__1 ).

fof(t12_catalan1,axiom,
    k1_catalan1(np__2) = np__1 ).

fof(t13_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => v1_int_1(k1_catalan1(A)) ) ).

fof(t14_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => ( r1_xreal_0(np__1,A)
       => k1_catalan1(k2_xcmplx_0(A,np__1)) = k7_xcmplx_0(k6_newton(k3_xcmplx_0(np__2,A)),k3_xcmplx_0(k6_newton(A),k6_newton(k2_xcmplx_0(A,np__1)))) ) ) ).

fof(t15_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => ~ ( ~ r1_xreal_0(A,np__1)
          & r1_xreal_0(k1_catalan1(k2_xcmplx_0(A,np__1)),k1_catalan1(A)) ) ) ).

fof(t16_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => r1_xreal_0(k1_catalan1(A),k1_catalan1(k2_xcmplx_0(A,np__1))) ) ).

fof(t17_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => r1_xreal_0(np__0,k1_catalan1(A)) ) ).

fof(t18_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => m2_subset_1(k1_catalan1(A),k1_numbers,k5_numbers) ) ).

fof(t19_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ r1_xreal_0(A,np__0)
       => k1_catalan1(k2_xcmplx_0(A,np__1)) = k3_xcmplx_0(k3_xcmplx_0(np__2,k6_xcmplx_0(np__2,k7_xcmplx_0(np__3,k2_xcmplx_0(A,np__1)))),k1_catalan1(A)) ) ) ).

fof(t20_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => ~ ( ~ r1_xreal_0(A,np__0)
          & r1_xreal_0(k1_catalan1(A),np__0) ) ) ).

fof(t21_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => ~ ( ~ r1_xreal_0(A,np__0)
          & r1_xreal_0(k3_xcmplx_0(np__4,k1_catalan1(A)),k1_catalan1(k2_xcmplx_0(A,np__1))) ) ) ).

fof(dt_k1_catalan1,axiom,
    ! [A] :
      ( v4_ordinal2(A)
     => m1_subset_1(k1_catalan1(A),k1_numbers) ) ).

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