SET007 Axioms: SET007+203.ax
%------------------------------------------------------------------------------
% File : SET007+203 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Some Elementary Notions of the Theory of Petri Nets
% 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 : net_1 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 73 ( 22 unt; 0 def)
% Number of atoms : 285 ( 44 equ)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 239 ( 27 ~; 7 |; 80 &)
% ( 24 <=>; 101 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 12 ( 10 usr; 1 prp; 0-3 aty)
% Number of functors : 23 ( 23 usr; 1 con; 0-3 aty)
% Number of variables : 132 ( 125 !; 7 ?)
% 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_net_1,axiom,
? [A] :
( l1_net_1(A)
& v1_net_1(A) ) ).
fof(fc1_net_1,axiom,
( v1_net_1(g1_net_1(k1_xboole_0,k1_xboole_0,k1_xboole_0))
& v2_net_1(g1_net_1(k1_xboole_0,k1_xboole_0,k1_xboole_0)) ) ).
fof(rc2_net_1,axiom,
? [A] :
( l1_net_1(A)
& v1_net_1(A)
& v2_net_1(A) ) ).
fof(d1_net_1,axiom,
! [A] :
( l1_net_1(A)
=> ( v2_net_1(A)
<=> ( r1_xboole_0(u1_net_1(A),u2_net_1(A))
& r1_tarski(u3_net_1(A),k2_xboole_0(k2_zfmisc_1(u1_net_1(A),u2_net_1(A)),k2_zfmisc_1(u2_net_1(A),u1_net_1(A)))) ) ) ) ).
fof(d2_net_1,axiom,
! [A] :
( l1_net_1(A)
=> k1_net_1(A) = k2_xboole_0(u1_net_1(A),u2_net_1(A)) ) ).
fof(t1_net_1,axiom,
$true ).
fof(t2_net_1,axiom,
$true ).
fof(t3_net_1,axiom,
$true ).
fof(t4_net_1,axiom,
! [A] :
( l1_net_1(A)
=> r1_tarski(u1_net_1(A),k1_net_1(A)) ) ).
fof(t5_net_1,axiom,
! [A] :
( l1_net_1(A)
=> r1_tarski(u2_net_1(A),k1_net_1(A)) ) ).
fof(t6_net_1,axiom,
! [A,B] :
( l1_net_1(B)
=> ( r2_hidden(A,k1_net_1(B))
<=> ( r2_hidden(A,u1_net_1(B))
| r2_hidden(A,u2_net_1(B)) ) ) ) ).
fof(t7_net_1,axiom,
! [A,B] :
( l1_net_1(B)
=> ~ ( k1_net_1(B) != k1_xboole_0
& m1_subset_1(A,k1_net_1(B))
& ~ m1_subset_1(A,u1_net_1(B))
& ~ m1_subset_1(A,u2_net_1(B)) ) ) ).
fof(t8_net_1,axiom,
! [A,B] :
( l1_net_1(B)
=> ( m1_subset_1(A,u1_net_1(B))
=> ( u1_net_1(B) = k1_xboole_0
| m1_subset_1(A,k1_net_1(B)) ) ) ) ).
fof(t9_net_1,axiom,
! [A,B] :
( l1_net_1(B)
=> ( m1_subset_1(A,u2_net_1(B))
=> ( u2_net_1(B) = k1_xboole_0
| m1_subset_1(A,k1_net_1(B)) ) ) ) ).
fof(t10_net_1,axiom,
$true ).
fof(t11_net_1,axiom,
! [A,B] :
( ( v2_net_1(B)
& l1_net_1(B) )
=> ~ ( r2_hidden(A,u1_net_1(B))
& r2_hidden(A,u2_net_1(B)) ) ) ).
fof(t12_net_1,axiom,
! [A,B,C] :
( ( v2_net_1(C)
& l1_net_1(C) )
=> ( ( r2_hidden(k4_tarski(A,B),u3_net_1(C))
& r2_hidden(A,u2_net_1(C)) )
=> r2_hidden(B,u1_net_1(C)) ) ) ).
fof(t13_net_1,axiom,
! [A,B,C] :
( ( v2_net_1(C)
& l1_net_1(C) )
=> ( ( r2_hidden(k4_tarski(A,B),u3_net_1(C))
& r2_hidden(B,u2_net_1(C)) )
=> r2_hidden(A,u1_net_1(C)) ) ) ).
fof(t14_net_1,axiom,
! [A,B,C] :
( ( v2_net_1(C)
& l1_net_1(C) )
=> ( ( r2_hidden(k4_tarski(A,B),u3_net_1(C))
& r2_hidden(A,u1_net_1(C)) )
=> r2_hidden(B,u2_net_1(C)) ) ) ).
fof(t15_net_1,axiom,
! [A,B,C] :
( ( v2_net_1(C)
& l1_net_1(C) )
=> ( ( r2_hidden(k4_tarski(A,B),u3_net_1(C))
& r2_hidden(B,u1_net_1(C)) )
=> r2_hidden(A,u2_net_1(C)) ) ) ).
fof(d3_net_1,axiom,
$true ).
fof(d4_net_1,axiom,
$true ).
fof(d5_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B,C] :
( r1_net_1(A,B,C)
<=> ( r2_hidden(k4_tarski(C,B),u3_net_1(A))
& r2_hidden(B,u2_net_1(A)) ) ) ) ).
fof(d6_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B,C] :
( r2_net_1(A,B,C)
<=> ( r2_hidden(k4_tarski(B,C),u3_net_1(A))
& r2_hidden(B,u2_net_1(A)) ) ) ) ).
fof(d7_net_1,axiom,
! [A] :
( l1_net_1(A)
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> ! [C] :
( C = k2_net_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ( r2_hidden(D,k1_net_1(A))
& r2_hidden(k4_tarski(D,B),u3_net_1(A)) ) ) ) ) ) ).
fof(d8_net_1,axiom,
! [A] :
( l1_net_1(A)
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> ! [C] :
( C = k3_net_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ( r2_hidden(D,k1_net_1(A))
& r2_hidden(k4_tarski(B,D),u3_net_1(A)) ) ) ) ) ) ).
fof(t16_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> r1_tarski(k2_net_1(A,B),k1_net_1(A)) ) ) ).
fof(t17_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> r1_tarski(k2_net_1(A,B),k1_net_1(A)) ) ) ).
fof(t18_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> r1_tarski(k3_net_1(A,B),k1_net_1(A)) ) ) ).
fof(t19_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> r1_tarski(k3_net_1(A,B),k1_net_1(A)) ) ) ).
fof(t20_net_1,axiom,
! [A,B] :
( ( v2_net_1(B)
& l1_net_1(B) )
=> ! [C] :
( m1_subset_1(C,k1_net_1(B))
=> ( r2_hidden(C,u2_net_1(B))
=> ( r2_hidden(A,k2_net_1(B,C))
<=> r1_net_1(B,C,A) ) ) ) ) ).
fof(t21_net_1,axiom,
! [A,B] :
( ( v2_net_1(B)
& l1_net_1(B) )
=> ! [C] :
( m1_subset_1(C,k1_net_1(B))
=> ( r2_hidden(C,u2_net_1(B))
=> ( r2_hidden(A,k3_net_1(B,C))
<=> r2_net_1(B,C,A) ) ) ) ) ).
fof(d9_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> ( k1_net_1(A) != k1_xboole_0
=> ! [C] :
( C = k4_net_1(A,B)
<=> ( ( r2_hidden(B,u1_net_1(A))
=> C = k1_tarski(B) )
& ( r2_hidden(B,u2_net_1(A))
=> C = k2_net_1(A,B) ) ) ) ) ) ) ).
fof(t22_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> ~ ( k1_net_1(A) != k1_xboole_0
& k4_net_1(A,B) != k1_tarski(B)
& k4_net_1(A,B) != k2_net_1(A,B) ) ) ) ).
fof(t23_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> ( k1_net_1(A) != k1_xboole_0
=> r1_tarski(k4_net_1(A,B),k1_net_1(A)) ) ) ) ).
fof(t24_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> ( k1_net_1(A) != k1_xboole_0
=> r1_tarski(k4_net_1(A,B),k1_net_1(A)) ) ) ) ).
fof(d10_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> ( k1_net_1(A) != k1_xboole_0
=> ! [C] :
( C = k5_net_1(A,B)
<=> ( ( r2_hidden(B,u1_net_1(A))
=> C = k1_tarski(B) )
& ( r2_hidden(B,u2_net_1(A))
=> C = k3_net_1(A,B) ) ) ) ) ) ) ).
fof(t25_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> ~ ( k1_net_1(A) != k1_xboole_0
& k5_net_1(A,B) != k1_tarski(B)
& k5_net_1(A,B) != k3_net_1(A,B) ) ) ) ).
fof(t26_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> ( k1_net_1(A) != k1_xboole_0
=> r1_tarski(k5_net_1(A,B),k1_net_1(A)) ) ) ) ).
fof(t27_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> ( k1_net_1(A) != k1_xboole_0
=> r1_tarski(k5_net_1(A,B),k1_net_1(A)) ) ) ) ).
fof(d11_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> k6_net_1(A,B) = k2_xboole_0(k4_net_1(A,B),k5_net_1(A,B)) ) ) ).
fof(d12_net_1,axiom,
! [A] :
( l1_net_1(A)
=> ! [B] :
( m1_subset_1(B,u2_net_1(A))
=> ! [C] :
( C = k7_net_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ( r2_hidden(D,u1_net_1(A))
& r2_hidden(k4_tarski(D,B),u3_net_1(A)) ) ) ) ) ) ).
fof(d13_net_1,axiom,
! [A] :
( l1_net_1(A)
=> ! [B] :
( m1_subset_1(B,u2_net_1(A))
=> ! [C] :
( C = k8_net_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ( r2_hidden(D,u1_net_1(A))
& r2_hidden(k4_tarski(B,D),u3_net_1(A)) ) ) ) ) ) ).
fof(d14_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B,C] :
( C = k9_net_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ( r1_tarski(D,k1_net_1(A))
& ? [E] :
( m1_subset_1(E,k1_net_1(A))
& r2_hidden(E,B)
& D = k4_net_1(A,E) ) ) ) ) ) ).
fof(d15_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B,C] :
( C = k10_net_1(A,B)
<=> ! [D] :
( r2_hidden(D,C)
<=> ( r1_tarski(D,k1_net_1(A))
& ? [E] :
( m1_subset_1(E,k1_net_1(A))
& r2_hidden(E,B)
& D = k5_net_1(A,E) ) ) ) ) ) ).
fof(t28_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> ! [C] :
( ( r1_tarski(C,k1_net_1(A))
& r2_hidden(B,C) )
=> ( k1_net_1(A) = k1_xboole_0
| r2_hidden(k4_net_1(A,B),k9_net_1(A,C)) ) ) ) ) ).
fof(t29_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_net_1(A))
=> ! [C] :
( ( r1_tarski(C,k1_net_1(A))
& r2_hidden(B,C) )
=> ( k1_net_1(A) = k1_xboole_0
| r2_hidden(k5_net_1(A,B),k10_net_1(A,C)) ) ) ) ) ).
fof(d16_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] : k11_net_1(A,B) = k3_tarski(k9_net_1(A,B)) ) ).
fof(d17_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] : k12_net_1(A,B) = k3_tarski(k10_net_1(A,B)) ) ).
fof(t30_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B,C] :
( r1_tarski(C,k1_net_1(A))
=> ( k1_net_1(A) = k1_xboole_0
| ( r2_hidden(B,k11_net_1(A,C))
<=> ? [D] :
( m1_subset_1(D,k1_net_1(A))
& r2_hidden(D,C)
& r2_hidden(B,k4_net_1(A,D)) ) ) ) ) ) ).
fof(t31_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B,C] :
( r1_tarski(C,k1_net_1(A))
=> ( k1_net_1(A) = k1_xboole_0
| ( r2_hidden(B,k12_net_1(A,C))
<=> ? [D] :
( m1_subset_1(D,k1_net_1(A))
& r2_hidden(D,C)
& r2_hidden(B,k5_net_1(A,D)) ) ) ) ) ) ).
fof(t32_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_net_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_net_1(A))
=> ( k1_net_1(A) != k1_xboole_0
=> ( r2_hidden(C,k11_net_1(A,B))
<=> ~ ( ~ ( r2_hidden(C,B)
& r2_hidden(C,u1_net_1(A)) )
& ! [D] :
( m1_subset_1(D,k1_net_1(A))
=> ~ ( r2_hidden(D,B)
& r2_hidden(D,u2_net_1(A))
& r1_net_1(A,D,C) ) ) ) ) ) ) ) ) ).
fof(t33_net_1,axiom,
! [A] :
( ( v2_net_1(A)
& l1_net_1(A) )
=> ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k1_net_1(A)))
=> ! [C] :
( m1_subset_1(C,k1_net_1(A))
=> ( k1_net_1(A) != k1_xboole_0
=> ( r2_hidden(C,k12_net_1(A,B))
<=> ~ ( ~ ( r2_hidden(C,B)
& r2_hidden(C,u1_net_1(A)) )
& ! [D] :
( m1_subset_1(D,k1_net_1(A))
=> ~ ( r2_hidden(D,B)
& r2_hidden(D,u2_net_1(A))
& r2_net_1(A,D,C) ) ) ) ) ) ) ) ) ).
fof(dt_l1_net_1,axiom,
$true ).
fof(existence_l1_net_1,axiom,
? [A] : l1_net_1(A) ).
fof(abstractness_v1_net_1,axiom,
! [A] :
( l1_net_1(A)
=> ( v1_net_1(A)
=> A = g1_net_1(u1_net_1(A),u2_net_1(A),u3_net_1(A)) ) ) ).
fof(dt_k1_net_1,axiom,
$true ).
fof(dt_k2_net_1,axiom,
$true ).
fof(dt_k3_net_1,axiom,
$true ).
fof(dt_k4_net_1,axiom,
$true ).
fof(dt_k5_net_1,axiom,
$true ).
fof(dt_k6_net_1,axiom,
$true ).
fof(dt_k7_net_1,axiom,
$true ).
fof(dt_k8_net_1,axiom,
$true ).
fof(dt_k9_net_1,axiom,
$true ).
fof(dt_k10_net_1,axiom,
$true ).
fof(dt_k11_net_1,axiom,
$true ).
fof(dt_k12_net_1,axiom,
$true ).
fof(dt_u1_net_1,axiom,
$true ).
fof(dt_u2_net_1,axiom,
$true ).
fof(dt_u3_net_1,axiom,
! [A] :
( l1_net_1(A)
=> v1_relat_1(u3_net_1(A)) ) ).
fof(dt_g1_net_1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ( v1_net_1(g1_net_1(A,B,C))
& l1_net_1(g1_net_1(A,B,C)) ) ) ).
fof(free_g1_net_1,axiom,
! [A,B,C] :
( v1_relat_1(C)
=> ! [D,E,F] :
( g1_net_1(A,B,C) = g1_net_1(D,E,F)
=> ( A = D
& B = E
& C = F ) ) ) ).
%------------------------------------------------------------------------------