SET007 Axioms: SET007+721.ax
%------------------------------------------------------------------------------
% File : SET007+721 : TPTP v9.0.0. 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 unt; 0 def)
% Number of atoms : 97 ( 19 equ)
% Maximal formula atoms : 16 ( 5 avg)
% Number of connectives : 89 ( 10 ~; 2 |; 27 &)
% ( 1 <=>; 49 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 11 ( 10 usr; 0 prp; 1-3 aty)
% Number of functors : 28 ( 28 usr; 8 con; 0-4 aty)
% Number of variables : 38 ( 38 !; 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_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) ).
%------------------------------------------------------------------------------