SET007 Axioms: SET007+131.ax
%------------------------------------------------------------------------------
% File : SET007+131 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Divisibility of Integers and Integer Relatively Primes
% 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 : int_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 65 ( 16 unt; 0 def)
% Number of atoms : 246 ( 41 equ)
% Maximal formula atoms : 18 ( 3 avg)
% Number of connectives : 201 ( 20 ~; 5 |; 47 &)
% ( 12 <=>; 117 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of predicates : 14 ( 12 usr; 1 prp; 0-3 aty)
% Number of functors : 20 ( 20 usr; 8 con; 0-2 aty)
% Number of variables : 98 ( 98 !; 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_int_2,axiom,
$true ).
fof(t2_int_2,axiom,
$true ).
fof(t3_int_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_nat_1(np__0,A)
<=> A = np__0 ) ) ).
fof(t4_int_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( A = np__0
| B = np__0 )
<=> k5_nat_1(A,B) = np__0 ) ) ) ).
fof(t5_int_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( ( A = np__0
& B = np__0 )
<=> k6_nat_1(A,B) = np__0 ) ) ) ).
fof(t6_int_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> k2_nat_1(A,B) = k2_nat_1(k5_nat_1(A,B),k6_nat_1(A,B)) ) ) ).
fof(t7_int_2,axiom,
$true ).
fof(t8_int_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( m2_subset_1(k1_real_1(A),k1_numbers,k5_numbers)
<=> A = np__0 ) ) ).
fof(t9_int_2,axiom,
~ m2_subset_1(k1_real_1(np__1),k1_numbers,k5_numbers) ).
fof(t10_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ( r2_int_1(np__0,A)
<=> A = np__0 ) ) ).
fof(t11_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ( r2_int_1(A,k4_xcmplx_0(A))
& r2_int_1(k4_xcmplx_0(A),A) ) ) ).
fof(t12_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( r2_int_1(A,B)
=> r2_int_1(A,k3_xcmplx_0(B,C)) ) ) ) ) ).
fof(t13_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( r2_int_1(A,B)
& r2_int_1(B,C) )
=> r2_int_1(A,C) ) ) ) ) ).
fof(t14_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( ( r2_int_1(A,B)
=> r2_int_1(A,k4_xcmplx_0(B)) )
& ( r2_int_1(A,k4_xcmplx_0(B))
=> r2_int_1(A,B) )
& ( r2_int_1(A,B)
=> r2_int_1(k4_xcmplx_0(A),B) )
& ( r2_int_1(k4_xcmplx_0(A),B)
=> r2_int_1(A,B) )
& ( r2_int_1(A,B)
=> r2_int_1(k4_xcmplx_0(A),k4_xcmplx_0(B)) )
& ( r2_int_1(k4_xcmplx_0(A),k4_xcmplx_0(B))
=> r2_int_1(A,B) )
& ( r2_int_1(A,k4_xcmplx_0(B))
=> r2_int_1(k4_xcmplx_0(A),B) )
& ( r2_int_1(k4_xcmplx_0(A),B)
=> r2_int_1(A,k4_xcmplx_0(B)) ) ) ) ) ).
fof(t15_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ~ ( r2_int_1(A,B)
& r2_int_1(B,A)
& A != B
& A != k4_xcmplx_0(B) ) ) ) ).
fof(t16_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ( r2_int_1(A,np__0)
& r2_int_1(np__1,A)
& r2_int_1(k1_real_1(np__1),A) ) ) ).
fof(t17_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ~ ( ( r2_int_1(A,np__1)
| r2_int_1(A,k1_real_1(np__1)) )
& A != np__1
& A != k1_real_1(np__1) ) ) ).
fof(t18_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ( ( A = np__1
| A = k1_real_1(np__1) )
=> ( r2_int_1(A,np__1)
& r2_int_1(A,k1_real_1(np__1)) ) ) ) ).
fof(t19_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( r1_int_1(A,B,C)
<=> r2_int_1(C,k6_xcmplx_0(A,B)) ) ) ) ) ).
fof(t20_int_2,axiom,
! [A] :
( v1_int_1(A)
=> m2_subset_1(k1_int_2(A),k1_numbers,k5_numbers) ) ).
fof(t21_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( r2_int_1(A,B)
<=> r1_nat_1(k1_int_2(A),k1_int_2(B)) ) ) ) ).
fof(d1_int_2,axiom,
$true ).
fof(d2_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> k2_int_2(A,B) = k5_nat_1(k1_int_2(A),k1_int_2(B)) ) ) ).
fof(t22_int_2,axiom,
$true ).
fof(t23_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> m2_subset_1(k2_int_2(A,B),k1_numbers,k5_numbers) ) ) ).
fof(t24_int_2,axiom,
$true ).
fof(t25_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> r2_int_1(A,k2_int_2(A,B)) ) ) ).
fof(t26_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> r2_int_1(A,k2_int_2(B,A)) ) ) ).
fof(t27_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( r2_int_1(A,C)
& r2_int_1(B,C) )
=> r2_int_1(k2_int_2(A,B),C) ) ) ) ) ).
fof(d3_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> k3_int_2(A,B) = k6_nat_1(k1_int_2(A),k1_int_2(B)) ) ) ).
fof(t28_int_2,axiom,
$true ).
fof(t29_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> m2_subset_1(k3_int_2(A,B),k1_numbers,k5_numbers) ) ) ).
fof(t30_int_2,axiom,
$true ).
fof(t31_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> r2_int_1(k3_int_2(A,B),A) ) ) ).
fof(t32_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> r2_int_1(k3_int_2(A,B),B) ) ) ).
fof(t33_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( r2_int_1(C,A)
& r2_int_1(C,B) )
=> r2_int_1(C,k3_int_2(A,B)) ) ) ) ) ).
fof(t34_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( ( A = np__0
| B = np__0 )
<=> k2_int_2(A,B) = np__0 ) ) ) ).
fof(t35_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( ( A = np__0
& B = np__0 )
<=> k3_int_2(A,B) = np__0 ) ) ) ).
fof(d4_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ( r1_int_2(A,B)
<=> k3_int_2(A,B) = np__1 ) ) ) ).
fof(t36_int_2,axiom,
$true ).
fof(t37_int_2,axiom,
$true ).
fof(t38_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ~ ( ~ ( A = np__0
& B = np__0 )
& ! [C] :
( v1_int_1(C)
=> ! [D] :
( v1_int_1(D)
=> ~ ( A = k3_xcmplx_0(k3_int_2(A,B),C)
& B = k3_xcmplx_0(k3_int_2(A,B),D)
& r1_int_2(C,D) ) ) ) ) ) ) ).
fof(t39_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( r1_int_2(A,B)
=> ( k3_int_2(k3_xcmplx_0(C,A),k3_xcmplx_0(C,B)) = k1_int_2(C)
& k3_int_2(k3_xcmplx_0(C,A),k3_xcmplx_0(B,C)) = k1_int_2(C)
& k3_int_2(k3_xcmplx_0(A,C),k3_xcmplx_0(C,B)) = k1_int_2(C)
& k3_int_2(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) = k1_int_2(C) ) ) ) ) ) ).
fof(t40_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( r2_int_1(A,k3_xcmplx_0(B,C))
& r1_int_2(B,A) )
=> r2_int_1(A,C) ) ) ) ) ).
fof(t41_int_2,axiom,
! [A] :
( v1_int_1(A)
=> ! [B] :
( v1_int_1(B)
=> ! [C] :
( v1_int_1(C)
=> ( ( r1_int_2(A,B)
& r1_int_2(C,B) )
=> r1_int_2(k3_xcmplx_0(A,C),B) ) ) ) ) ).
fof(d5_int_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( v1_int_2(A)
<=> ( ~ r1_xreal_0(A,np__1)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r1_nat_1(B,A)
& B != np__1
& B != A ) ) ) ) ) ).
fof(d6_int_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_int_2(A,B)
<=> k6_nat_1(A,B) = np__1 ) ) ) ).
fof(t42_int_2,axiom,
$true ).
fof(t43_int_2,axiom,
$true ).
fof(t44_int_2,axiom,
v1_int_2(np__2) ).
fof(t45_int_2,axiom,
$true ).
fof(t46_int_2,axiom,
~ v1_int_2(np__4) ).
fof(t47_int_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( v1_int_2(A)
& v1_int_2(B)
& ~ r2_int_2(A,B)
& A != B ) ) ) ).
fof(t48_int_2,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ~ ( r1_xreal_0(np__2,A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( v1_int_2(B)
& r1_nat_1(B,A) ) ) ) ) ).
fof(s1_int_2,axiom,
( ( p1_s1_int_2(f1_s1_int_2)
& ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(f1_s1_int_2,A)
& p1_s1_int_2(A) )
=> p1_s1_int_2(k1_nat_1(A,np__1)) ) ) )
=> ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(f1_s1_int_2,A)
=> p1_s1_int_2(A) ) ) ) ).
fof(s2_int_2,axiom,
( ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(f1_s2_int_2,A)
& ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r1_xreal_0(f1_s2_int_2,B)
=> ( r1_xreal_0(A,B)
| p1_s2_int_2(B) ) ) ) )
=> p1_s2_int_2(A) ) )
=> ! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ( r1_xreal_0(f1_s2_int_2,A)
=> p1_s2_int_2(A) ) ) ) ).
fof(symmetry_r1_int_2,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_int_1(B) )
=> ( r1_int_2(A,B)
=> r1_int_2(B,A) ) ) ).
fof(symmetry_r2_int_2,axiom,
! [A,B] :
( ( m1_subset_1(A,k5_numbers)
& m1_subset_1(B,k5_numbers) )
=> ( r2_int_2(A,B)
=> r2_int_2(B,A) ) ) ).
fof(dt_k1_int_2,axiom,
! [A] :
( v1_int_1(A)
=> m2_subset_1(k1_int_2(A),k1_numbers,k5_numbers) ) ).
fof(projectivity_k1_int_2,axiom,
! [A] :
( v1_int_1(A)
=> k1_int_2(k1_int_2(A)) = k1_int_2(A) ) ).
fof(redefinition_k1_int_2,axiom,
! [A] :
( v1_int_1(A)
=> k1_int_2(A) = k16_complex1(A) ) ).
fof(dt_k2_int_2,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_int_1(B) )
=> v1_int_1(k2_int_2(A,B)) ) ).
fof(commutativity_k2_int_2,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_int_1(B) )
=> k2_int_2(A,B) = k2_int_2(B,A) ) ).
fof(dt_k3_int_2,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_int_1(B) )
=> v1_int_1(k3_int_2(A,B)) ) ).
fof(commutativity_k3_int_2,axiom,
! [A,B] :
( ( v1_int_1(A)
& v1_int_1(B) )
=> k3_int_2(A,B) = k3_int_2(B,A) ) ).
%------------------------------------------------------------------------------