SET007 Axioms: SET007+721.ax


%------------------------------------------------------------------------------
% File     : SET007+721 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Fibonacci 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    : fib_num [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   18 (   4 unit)
%            Number of atoms       :   97 (  19 equality)
%            Maximal formula depth :   14 (   6 average)
%            Number of connectives :   89 (  10 ~  ;   2  |;  27  &)
%                                         (   1 <=>;  49 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   11 (   0 propositional; 1-3 arity)
%            Number of functors    :   28 (   8 constant; 0-4 arity)
%            Number of variables   :   38 (   0 singleton;  38 !;   0 ?)
%            Maximal term depth    :    5 (   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(t1_fib_num,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k6_nat_1(A,B) = k6_nat_1(A,k1_nat_1(B,A)) ) ) )).

fof(t2_fib_num,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)
             => ( k6_nat_1(A,B) = np__1
               => k6_nat_1(A,k2_nat_1(B,C)) = k6_nat_1(A,C) ) ) ) ) )).

fof(t3_fib_num,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ~ ( ~ r1_xreal_0(A,np__0)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ~ ( ~ r1_xreal_0(B,np__0)
                  & ~ r1_xreal_0(k7_xcmplx_0(np__1,B),np__0)
                  & r1_xreal_0(k7_xcmplx_0(np__1,B),A) ) ) ) ) )).

fof(t4_fib_num,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k3_pre_ff(k1_nat_1(A,k1_nat_1(B,np__1))) = k1_nat_1(k2_nat_1(k3_pre_ff(B),k3_pre_ff(A)),k2_nat_1(k3_pre_ff(k1_nat_1(B,np__1)),k3_pre_ff(k1_nat_1(A,np__1)))) ) ) )).

fof(t5_fib_num,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k6_nat_1(k3_pre_ff(A),k3_pre_ff(B)) = k3_pre_ff(k6_nat_1(A,B)) ) ) )).

fof(t6_fib_num,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( v1_xreal_0(C)
             => ! [D] :
                  ( v1_xreal_0(D)
                 => ( r1_xreal_0(np__0,k1_quin_1(B,C,D))
                   => ( B = np__0
                      | ( k2_xcmplx_0(k2_xcmplx_0(k3_xcmplx_0(B,k5_square_1(A)),k3_xcmplx_0(C,A)),D) = np__0
                      <=> ( A = k7_xcmplx_0(k6_xcmplx_0(k4_xcmplx_0(C),k8_square_1(k1_quin_1(B,C,D))),k3_xcmplx_0(np__2,B))
                          | A = k7_xcmplx_0(k2_xcmplx_0(k4_xcmplx_0(C),k8_square_1(k1_quin_1(B,C,D))),k3_xcmplx_0(np__2,B)) ) ) ) ) ) ) ) ) )).

fof(d1_fib_num,axiom,(
    k1_fib_num = k7_xcmplx_0(k2_xcmplx_0(np__1,k9_square_1(np__5)),np__2) )).

fof(d2_fib_num,axiom,(
    k2_fib_num = k7_xcmplx_0(k6_xcmplx_0(np__1,k9_square_1(np__5)),np__2) )).

fof(t7_fib_num,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k3_pre_ff(A) = k7_xcmplx_0(k6_xcmplx_0(k3_power(k1_fib_num,A),k3_power(k2_fib_num,A)),k9_square_1(np__5)) ) )).

fof(t8_fib_num,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ r1_xreal_0(np__1,k18_complex1(k6_xcmplx_0(k3_pre_ff(A),k7_xcmplx_0(k3_power(k1_fib_num,A),k9_square_1(np__5))))) ) )).

fof(t9_fib_num,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => k2_seq_1(k5_numbers,k1_numbers,A,C) = k2_seq_1(k5_numbers,k1_numbers,B,C) )
           => A = B ) ) ) )).

fof(t10_fib_num,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( v2_relat_1(B)
               => k11_seq_1(k19_seq_1(A,B),k19_seq_1(B,C)) = k19_seq_1(A,C) ) ) ) ) )).

fof(t11_fib_num,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( k2_seq_1(k5_numbers,k1_numbers,k19_seq_1(A,B),C) = k7_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,B,C))
                & k2_seq_1(k5_numbers,k1_numbers,k19_seq_1(A,B),C) = k3_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,C),k5_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,B,C))) ) ) ) ) )).

fof(t12_fib_num,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => k2_seq_1(k5_numbers,k1_numbers,A,B) = k7_xcmplx_0(k3_pre_ff(k1_nat_1(B,np__1)),k3_pre_ff(B)) )
       => ( v4_seq_2(A)
          & k2_seq_2(A) = k1_fib_num ) ) ) )).

fof(s1_fib_num,axiom,
    ( ( p1_s1_fib_num(np__0)
      & p1_s1_fib_num(np__1)
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( ( p1_s1_fib_num(A)
              & p1_s1_fib_num(k1_nat_1(A,np__1)) )
           => p1_s1_fib_num(k1_nat_1(A,np__2)) ) ) )
   => ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => p1_s1_fib_num(A) ) )).

fof(s2_fib_num,axiom,
    ( ( ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ( p1_s2_fib_num(A,B)
               => p1_s2_fib_num(B,A) ) ) )
      & ! [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)
                   => ~ ( ~ r1_xreal_0(A,B)
                        & ~ r1_xreal_0(A,C)
                        & ~ p1_s2_fib_num(B,C) ) ) )
           => ! [B] :
                ( m2_subset_1(B,k1_numbers,k5_numbers)
               => ( r1_xreal_0(B,A)
                 => p1_s2_fib_num(A,B) ) ) ) ) )
   => ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => p1_s2_fib_num(A,B) ) ) )).

fof(dt_k1_fib_num,axiom,(
    v1_xreal_0(k1_fib_num) )).

fof(dt_k2_fib_num,axiom,(
    v1_xreal_0(k2_fib_num) )).
%------------------------------------------------------------------------------