SET007 Axioms: SET007+82.ax
%------------------------------------------------------------------------------
% File : SET007+82 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Convergent Sequences and the Limit of Sequences
% 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 : seq_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 52 ( 13 unt; 0 def)
% Number of atoms : 327 ( 23 equ)
% Maximal formula atoms : 16 ( 6 avg)
% Number of connectives : 299 ( 24 ~; 4 |; 166 &)
% ( 9 <=>; 96 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 18 ( 16 usr; 1 prp; 0-3 aty)
% Number of functors : 29 ( 29 usr; 5 con; 0-4 aty)
% Number of variables : 85 ( 75 !; 10 ?)
% 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_seq_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A)
& v3_seq_2(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A)
& v1_seq_2(A)
& v2_seq_2(A) ) ) ).
fof(cc2_seq_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A)
& v1_seq_2(A)
& v2_seq_2(A) )
=> ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A)
& v3_seq_2(A) ) ) ).
fof(t1_seq_2,axiom,
$true ).
fof(t2_seq_2,axiom,
$true ).
fof(t3_seq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ( ~ r1_xreal_0(A,np__0)
=> ( ~ r1_xreal_0(k7_xcmplx_0(A,np__2),np__0)
& ~ r1_xreal_0(k7_xcmplx_0(A,np__4),np__0) ) ) ) ).
fof(t4_seq_2,axiom,
$true ).
fof(t5_seq_2,axiom,
$true ).
fof(t6_seq_2,axiom,
$true ).
fof(t7_seq_2,axiom,
$true ).
fof(t8_seq_2,axiom,
$true ).
fof(t9_seq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( ( ~ r1_xreal_0(B,k4_xcmplx_0(A))
& ~ r1_xreal_0(A,B) )
<=> ~ r1_xreal_0(A,k18_complex1(B)) ) ) ) ).
fof(t10_seq_2,axiom,
$true ).
fof(t11_seq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( A != np__0
& B != np__0
& k18_complex1(k6_xcmplx_0(k5_xcmplx_0(A),k5_xcmplx_0(B))) != k6_real_1(k18_complex1(k6_xcmplx_0(A,B)),k4_real_1(k18_complex1(A),k18_complex1(B))) ) ) ) ).
fof(d1_seq_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ( v1_seq_2(A)
<=> ? [B] :
( v1_xreal_0(B)
& ! [C] :
~ ( r2_hidden(C,k1_relat_1(A))
& r1_xreal_0(B,k1_seq_1(A,C)) ) ) ) ) ).
fof(d2_seq_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ( v2_seq_2(A)
<=> ? [B] :
( v1_xreal_0(B)
& ! [C] :
~ ( r2_hidden(C,k1_relat_1(A))
& r1_xreal_0(k1_seq_1(A,C),B) ) ) ) ) ).
fof(d3_seq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v1_seq_2(A)
<=> ? [B] :
( v1_xreal_0(B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(B,k2_seq_1(k5_numbers,k1_numbers,A,C)) ) ) ) ) ).
fof(d4_seq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v2_seq_2(A)
<=> ? [B] :
( v1_xreal_0(B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,C),B) ) ) ) ) ).
fof(d5_seq_2,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A)
& v1_seq_1(A) )
=> ( v3_seq_2(A)
<=> ( v1_seq_2(A)
& v2_seq_2(A) ) ) ) ).
fof(t12_seq_2,axiom,
$true ).
fof(t13_seq_2,axiom,
$true ).
fof(t14_seq_2,axiom,
$true ).
fof(t15_seq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v3_seq_2(A)
<=> ? [B] :
( v1_xreal_0(B)
& ~ r1_xreal_0(B,np__0)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ~ r1_xreal_0(B,k18_complex1(k2_seq_1(k5_numbers,k1_numbers,A,C))) ) ) ) ) ).
fof(t16_seq_2,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)
=> ? [C] :
( v1_xreal_0(C)
& ~ r1_xreal_0(C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(D,B)
& r1_xreal_0(C,k18_complex1(k2_seq_1(k5_numbers,k1_numbers,A,D))) ) ) ) ) ) ).
fof(d6_seq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v4_seq_2(A)
<=> ? [B] :
( v1_xreal_0(B)
& ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& r1_xreal_0(D,E)
& r1_xreal_0(C,k18_complex1(k6_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,E),B))) ) ) ) ) ) ) ) ).
fof(d7_seq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v4_seq_2(A)
=> ! [B] :
( v1_xreal_0(B)
=> ( B = k1_seq_2(A)
<=> ! [C] :
( v1_xreal_0(C)
=> ~ ( ~ r1_xreal_0(C,np__0)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ? [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
& r1_xreal_0(D,E)
& r1_xreal_0(C,k18_complex1(k6_xcmplx_0(k2_seq_1(k5_numbers,k1_numbers,A,E),B))) ) ) ) ) ) ) ) ) ).
fof(t17_seq_2,axiom,
$true ).
fof(t18_seq_2,axiom,
$true ).
fof(t19_seq_2,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) )
=> ( ( v4_seq_2(A)
& v4_seq_2(B) )
=> v4_seq_2(k9_seq_1(A,B)) ) ) ) ).
fof(t20_seq_2,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) )
=> ( ( v4_seq_2(A)
& v4_seq_2(B) )
=> k2_seq_2(k9_seq_1(A,B)) = k3_real_1(k2_seq_2(A),k2_seq_2(B)) ) ) ) ).
fof(t21_seq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( v4_seq_2(B)
=> v4_seq_2(k14_seq_1(B,A)) ) ) ) ).
fof(t22_seq_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_numbers)
& m2_relset_1(B,k5_numbers,k1_numbers) )
=> ( v4_seq_2(B)
=> k2_seq_2(k14_seq_1(B,A)) = k3_xcmplx_0(A,k2_seq_2(B)) ) ) ) ).
fof(t23_seq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v4_seq_2(A)
=> v4_seq_2(k17_seq_1(A)) ) ) ).
fof(t24_seq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v4_seq_2(A)
=> k2_seq_2(k17_seq_1(A)) = k1_real_1(k2_seq_2(A)) ) ) ).
fof(t25_seq_2,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) )
=> ( ( v4_seq_2(A)
& v4_seq_2(B) )
=> v4_seq_2(k10_seq_1(A,B)) ) ) ) ).
fof(t26_seq_2,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) )
=> ( ( v4_seq_2(A)
& v4_seq_2(B) )
=> k2_seq_2(k10_seq_1(A,B)) = k5_real_1(k2_seq_2(A),k2_seq_2(B)) ) ) ) ).
fof(t27_seq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( v4_seq_2(A)
=> v3_seq_2(A) ) ) ).
fof(t28_seq_2,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) )
=> ( ( v4_seq_2(A)
& v4_seq_2(B) )
=> v4_seq_2(k11_seq_1(A,B)) ) ) ) ).
fof(t29_seq_2,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) )
=> ( ( v4_seq_2(A)
& v4_seq_2(B) )
=> k2_seq_2(k11_seq_1(A,B)) = k4_real_1(k2_seq_2(A),k2_seq_2(B)) ) ) ) ).
fof(t30_seq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ~ ( v4_seq_2(A)
& k2_seq_2(A) != np__0
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ? [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
& r1_xreal_0(B,C)
& r1_xreal_0(k18_complex1(k2_seq_1(k5_numbers,k1_numbers,A,C)),k6_real_1(k18_complex1(k2_seq_2(A)),np__2)) ) ) ) ) ).
fof(t31_seq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,A,B)) ) )
=> r1_xreal_0(np__0,k2_seq_2(A)) ) ) ).
fof(t32_seq_2,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) )
=> ( ( v4_seq_2(A)
& v4_seq_2(B)
& ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,B,C)) ) )
=> r1_xreal_0(k2_seq_2(A),k2_seq_2(B)) ) ) ) ).
fof(t33_seq_2,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) )
=> ( ( v4_seq_2(A)
& v4_seq_2(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,D),k2_seq_1(k5_numbers,k1_numbers,C,D))
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,D),k2_seq_1(k5_numbers,k1_numbers,B,D)) ) )
& k2_seq_2(A) = k2_seq_2(B) )
=> v4_seq_2(C) ) ) ) ) ).
fof(t34_seq_2,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) )
=> ( ( v4_seq_2(A)
& v4_seq_2(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,D),k2_seq_1(k5_numbers,k1_numbers,C,D))
& r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,D),k2_seq_1(k5_numbers,k1_numbers,B,D)) ) )
& k2_seq_2(A) = k2_seq_2(B) )
=> k2_seq_2(C) = k2_seq_2(A) ) ) ) ) ).
fof(t35_seq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(A)
& v2_relat_1(A) )
=> ( k2_seq_2(A) = np__0
| v4_seq_2(k18_seq_1(A)) ) ) ) ).
fof(t36_seq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m2_relset_1(A,k5_numbers,k1_numbers) )
=> ( ( v4_seq_2(A)
& v2_relat_1(A) )
=> ( k2_seq_2(A) = np__0
| k2_seq_2(k18_seq_1(A)) = k2_real_1(k2_seq_2(A)) ) ) ) ).
fof(t37_seq_2,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) )
=> ( ( v4_seq_2(A)
& v4_seq_2(B)
& v2_relat_1(B) )
=> ( k2_seq_2(B) = np__0
| v4_seq_2(k19_seq_1(A,B)) ) ) ) ) ).
fof(t38_seq_2,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) )
=> ( ( v4_seq_2(A)
& v4_seq_2(B)
& v2_relat_1(B) )
=> ( k2_seq_2(B) = np__0
| k2_seq_2(k19_seq_1(A,B)) = k6_real_1(k2_seq_2(A),k2_seq_2(B)) ) ) ) ) ).
fof(t39_seq_2,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) )
=> ( ( v4_seq_2(A)
& v3_seq_2(B)
& k2_seq_2(A) = np__0 )
=> v4_seq_2(k11_seq_1(A,B)) ) ) ) ).
fof(t40_seq_2,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) )
=> ( ( v4_seq_2(A)
& v3_seq_2(B)
& k2_seq_2(A) = np__0 )
=> k2_seq_2(k11_seq_1(A,B)) = np__0 ) ) ) ).
fof(dt_k1_seq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> v1_xreal_0(k1_seq_2(A)) ) ).
fof(dt_k2_seq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> m1_subset_1(k2_seq_2(A),k1_numbers) ) ).
fof(redefinition_k2_seq_2,axiom,
! [A] :
( ( v1_funct_1(A)
& v1_funct_2(A,k5_numbers,k1_numbers)
& m1_relset_1(A,k5_numbers,k1_numbers) )
=> k2_seq_2(A) = k1_seq_2(A) ) ).
%------------------------------------------------------------------------------