SET007 Axioms: SET007+198.ax
%------------------------------------------------------------------------------
% File : SET007+198 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Natural 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 : nat_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 36 ( 2 unt; 0 def)
% Number of atoms : 172 ( 28 equ)
% Maximal formula atoms : 14 ( 4 avg)
% Number of connectives : 164 ( 28 ~; 6 |; 44 &)
% ( 6 <=>; 80 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 22 ( 20 usr; 1 prp; 0-3 aty)
% Number of functors : 23 ( 23 usr; 8 con; 0-4 aty)
% Number of variables : 63 ( 59 !; 4 ?)
% 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(rc1_nat_2,axiom,
? [A] :
( m1_subset_1(A,k5_numbers)
& ~ v1_xboole_0(A)
& v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal2(A)
& v1_xreal_0(A)
& v2_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_xcmplx_0(A)
& v1_int_1(A)
& ~ v1_realset1(A) ) ).
fof(rc2_nat_2,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v4_ordinal2(A)
& v1_xreal_0(A)
& v2_xreal_0(A)
& ~ v3_xreal_0(A)
& v1_xcmplx_0(A)
& v1_int_1(A)
& ~ v1_realset1(A) ) ).
fof(t1_nat_2,axiom,
$true ).
fof(t2_nat_2,axiom,
! [A] :
( v1_xreal_0(A)
=> ! [B] :
( v1_xreal_0(B)
=> ~ ( r1_xreal_0(np__0,A)
& ~ r1_xreal_0(B,np__0)
& r1_xreal_0(B,k7_xcmplx_0(A,k2_xcmplx_0(k1_int_1(k7_xcmplx_0(A,B)),np__1))) ) ) ) ).
fof(t3_nat_2,axiom,
$true ).
fof(t4_nat_2,axiom,
! [A] :
( v4_ordinal2(A)
=> k3_nat_1(np__0,A) = np__0 ) ).
fof(t5_nat_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> k3_nat_1(A,A) = np__1 ) ).
fof(t6_nat_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> k3_nat_1(A,np__1) = A ) ).
fof(t7_nat_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ! [D] :
( v4_ordinal2(D)
=> ( ( r1_xreal_0(A,B)
& r1_xreal_0(C,B)
& A = k2_xcmplx_0(k5_binarith(B,C),D) )
=> C = k2_xcmplx_0(k5_binarith(B,A),D) ) ) ) ) ) ).
fof(t8_nat_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r2_hidden(A,k2_finseq_1(B))
=> r2_hidden(k1_nat_1(k5_binarith(B,A),np__1),k2_finseq_1(B)) ) ) ) ).
fof(t9_nat_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( ~ r1_xreal_0(A,B)
=> k1_nat_1(k5_binarith(A,k2_xcmplx_0(B,np__1)),np__1) = k5_binarith(A,B) ) ) ) ).
fof(t10_nat_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r1_xreal_0(B,A)
=> k5_binarith(B,A) = np__0 ) ) ) ).
fof(t11_nat_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v4_ordinal2(B) )
=> ~ r1_xreal_0(A,k5_binarith(A,B)) ) ) ).
fof(t12_nat_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r1_xreal_0(B,A)
=> k3_power(np__2,A) = k3_xcmplx_0(k3_power(np__2,B),k3_series_1(np__2,k5_binarith(A,B))) ) ) ) ).
fof(t13_nat_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,A)
=> r1_nat_1(k3_series_1(np__2,B),k3_series_1(np__2,A)) ) ) ) ).
fof(t14_nat_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ~ ( ~ r1_xreal_0(B,np__0)
& k3_nat_1(A,B) = np__0
& r1_xreal_0(B,A) ) ) ) ).
fof(t15_nat_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r1_xreal_0(A,B)
=> ( r1_xreal_0(A,np__0)
| r1_xreal_0(np__1,k3_nat_1(B,A)) ) ) ) ) ).
fof(t16_nat_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( A != np__0
=> k3_nat_1(k2_xcmplx_0(B,A),A) = k1_nat_1(k3_nat_1(B,A),np__1) ) ) ) ).
fof(t17_nat_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ! [C] :
( v4_ordinal2(C)
=> ( ( r1_nat_1(A,B)
& r1_xreal_0(np__1,B)
& r1_xreal_0(np__1,C)
& r1_xreal_0(C,A) )
=> k3_nat_1(k5_binarith(B,C),A) = k6_xcmplx_0(k3_nat_1(B,A),np__1) ) ) ) ) ).
fof(t18_nat_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,A)
=> k3_nat_1(k3_series_1(np__2,A),k3_series_1(np__2,B)) = k3_series_1(np__2,k5_binarith(A,B)) ) ) ) ).
fof(t19_nat_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> k4_nat_1(k3_series_1(np__2,A),np__2) = np__0 ) ) ).
fof(t20_nat_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ r1_xreal_0(A,np__0)
=> ( k4_nat_1(A,np__2) = np__0
<=> k4_nat_1(k5_binarith(A,np__1),np__2) = np__1 ) ) ) ).
fof(t21_nat_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v4_ordinal2(A) )
=> ~ ( A != np__1
& r1_xreal_0(A,np__1) ) ) ).
fof(t22_nat_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( v4_ordinal2(B)
=> ( r1_xreal_0(A,B)
=> ( r1_xreal_0(k2_xcmplx_0(A,A),B)
| k3_nat_1(B,A) = np__1 ) ) ) ) ).
fof(t23_nat_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( v1_abian(A)
<=> k4_nat_1(A,np__2) = np__0 ) ) ).
fof(t24_nat_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ~ v1_abian(A)
<=> k4_nat_1(A,np__2) = np__1 ) ) ).
fof(t25_nat_2,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)
=> ( ( r1_xreal_0(np__1,C)
& r1_xreal_0(B,A)
& r1_nat_1(k2_nat_1(np__2,C),B) )
=> ( v1_abian(k3_nat_1(A,C))
<=> v1_abian(k3_nat_1(k5_binarith(A,B),C)) ) ) ) ) ) ).
fof(t26_nat_2,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)
=> ( r1_xreal_0(A,B)
=> r1_xreal_0(k3_nat_1(A,C),k3_nat_1(B,C)) ) ) ) ) ).
fof(t27_nat_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(B,k2_nat_1(np__2,A))
=> r1_xreal_0(k3_nat_1(k1_nat_1(B,np__1),np__2),A) ) ) ) ).
fof(t28_nat_2,axiom,
! [A] :
( ( v1_abian(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> k3_nat_1(A,np__2) = k3_nat_1(k1_nat_1(A,np__1),np__2) ) ).
fof(t29_nat_2,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)
=> k3_nat_1(k3_nat_1(A,B),C) = k3_nat_1(A,k2_nat_1(B,C)) ) ) ) ).
fof(d1_nat_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ( v1_realset1(A)
<=> ( A = np__0
| A = np__1 ) ) ) ).
fof(t30_nat_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ( ~ v1_realset1(A)
<=> ( ~ v1_xboole_0(A)
& A != np__1 ) ) ) ).
fof(t31_nat_2,axiom,
! [A] :
( ( v4_ordinal2(A)
& ~ v1_realset1(A) )
=> r1_xreal_0(np__2,A) ) ).
fof(s1_nat_2,axiom,
( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,A)
=> ( r1_xreal_0(f3_s1_nat_2,A)
| ! [B] :
( m1_subset_1(B,f1_s1_nat_2)
=> ? [C] :
( m1_subset_1(C,f1_s1_nat_2)
& p1_s1_nat_2(A,B,C) ) ) ) ) )
=> ? [A] :
( m2_finseq_1(A,f1_s1_nat_2)
& k3_finseq_1(A) = f3_s1_nat_2
& ( k4_finseq_4(k5_numbers,f1_s1_nat_2,A,np__1) = f2_s1_nat_2
| f3_s1_nat_2 = np__0 )
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(np__1,B)
=> ( r1_xreal_0(f3_s1_nat_2,B)
| p1_s1_nat_2(B,k4_finseq_4(k5_numbers,f1_s1_nat_2,A,B),k4_finseq_4(k5_numbers,f1_s1_nat_2,A,k1_nat_1(B,np__1))) ) ) ) ) ) ).
fof(s2_nat_2,axiom,
( ( p1_s2_nat_2(np__2)
& ! [A] :
( ( ~ v1_realset1(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> ( p1_s2_nat_2(A)
=> p1_s2_nat_2(k1_nat_1(A,np__1)) ) ) )
=> ! [A] :
( ( ~ v1_realset1(A)
& m2_subset_1(A,k1_numbers,k5_numbers) )
=> p1_s2_nat_2(A) ) ) ).
%------------------------------------------------------------------------------