SET007 Axioms: SET007+776.ax
%------------------------------------------------------------------------------
% File : SET007+776 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : The Class of Series-Parallel Graphs. Part II
% 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 : neckla_2 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 22 ( 5 unt; 0 def)
% Number of atoms : 126 ( 12 equ)
% Maximal formula atoms : 34 ( 5 avg)
% Number of connectives : 117 ( 13 ~; 3 |; 58 &)
% ( 6 <=>; 37 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 26 ( 24 usr; 1 prp; 0-3 aty)
% Number of functors : 19 ( 19 usr; 8 con; 0-6 aty)
% Number of variables : 38 ( 36 !; 2 ?)
% 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_neckla_2,axiom,
! [A] :
( v4_ordinal2(A)
=> ! [B] :
( m1_subset_1(B,k4_classes1(A))
=> v1_finset_1(B) ) ) ).
fof(cc2_neckla_2,axiom,
! [A] :
( m1_subset_1(A,k13_classes2)
=> v1_finset_1(A) ) ).
fof(cc3_neckla_2,axiom,
! [A] :
( ( v3_ordinal1(A)
& v1_finset_1(A) )
=> ( v1_ordinal1(A)
& v2_ordinal1(A)
& v3_ordinal1(A)
& v4_ordinal2(A)
& v1_xcmplx_0(A)
& v1_card_1(A)
& v1_xreal_0(A)
& ~ v3_xreal_0(A) ) ) ).
fof(rc1_neckla_2,axiom,
? [A] :
( l1_orders_2(A)
& ~ v3_struct_0(A)
& v1_orders_2(A)
& v6_group_1(A)
& v1_neckla_2(A) ) ).
fof(fc1_neckla_2,axiom,
~ v1_xboole_0(k3_neckla_2) ).
fof(fc2_neckla_2,axiom,
~ v1_xboole_0(k4_neckla_2) ).
fof(t1_neckla_2,axiom,
! [A] :
( ( ~ v1_xboole_0(A)
& v1_classes2(A) )
=> ! [B,C] :
( ( r2_hidden(B,A)
& r2_hidden(C,A) )
=> ! [D] :
( m2_relset_1(D,B,C)
=> r2_hidden(D,A) ) ) ) ).
fof(t2_neckla_2,axiom,
u1_orders_2(k4_necklace(np__4)) = k4_enumset1(k4_tarski(np__0,np__1),k4_tarski(np__1,np__0),k4_tarski(np__1,np__2),k4_tarski(np__2,np__1),k4_tarski(np__2,np__3),k4_tarski(np__3,np__2)) ).
fof(t3_neckla_2,axiom,
! [A] :
( r2_hidden(A,k13_classes2)
=> v1_finset_1(A) ) ).
fof(d1_neckla_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& l1_orders_2(A) )
=> ( v1_neckla_2(A)
<=> ~ r3_necklace(k4_necklace(np__4),A) ) ) ).
fof(d2_neckla_2,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ! [C] :
( ( v1_orders_2(C)
& l1_orders_2(C) )
=> ( C = k1_neckla_2(A,B)
<=> ( u1_struct_0(C) = k2_xboole_0(u1_struct_0(A),u1_struct_0(B))
& u1_orders_2(C) = k2_xboole_0(u1_orders_2(A),u1_orders_2(B)) ) ) ) ) ) ).
fof(d3_neckla_2,axiom,
! [A] :
( l1_orders_2(A)
=> ! [B] :
( l1_orders_2(B)
=> ! [C] :
( ( v1_orders_2(C)
& l1_orders_2(C) )
=> ( C = k2_neckla_2(A,B)
<=> ( u1_struct_0(C) = k2_xboole_0(u1_struct_0(A),u1_struct_0(B))
& u1_orders_2(C) = k2_xboole_0(k2_xboole_0(k2_xboole_0(u1_orders_2(A),u1_orders_2(B)),k2_zfmisc_1(u1_struct_0(A),u1_struct_0(B))),k2_zfmisc_1(u1_struct_0(B),u1_struct_0(A))) ) ) ) ) ) ).
fof(d4_neckla_2,axiom,
! [A] :
( A = k3_neckla_2
<=> ! [B] :
( r2_hidden(B,A)
<=> ? [C] :
( v1_orders_2(C)
& l1_orders_2(C)
& B = C
& r2_hidden(u1_struct_0(C),k13_classes2) ) ) ) ).
fof(d5_neckla_2,axiom,
! [A] :
( m1_subset_1(A,k1_zfmisc_1(k3_neckla_2))
=> ( A = k4_neckla_2
<=> ( ! [B] :
( ( v1_orders_2(B)
& l1_orders_2(B) )
=> ( ( v1_realset1(u1_struct_0(B))
& r2_hidden(u1_struct_0(B),k13_classes2) )
=> ( v1_xboole_0(u1_struct_0(B))
| r2_hidden(B,A) ) ) )
& ! [B] :
( ( v1_orders_2(B)
& l1_orders_2(B) )
=> ! [C] :
( ( v1_orders_2(C)
& l1_orders_2(C) )
=> ( ( r1_xboole_0(u1_struct_0(B),u1_struct_0(C))
& r2_hidden(B,A)
& r2_hidden(C,A) )
=> ( r2_hidden(k1_neckla_2(B,C),A)
& r2_hidden(k2_neckla_2(B,C),A) ) ) ) )
& ! [B] :
( m1_subset_1(B,k1_zfmisc_1(k3_neckla_2))
=> ( ( ! [C] :
( ( v1_orders_2(C)
& l1_orders_2(C) )
=> ( ( v1_realset1(u1_struct_0(C))
& r2_hidden(u1_struct_0(C),k13_classes2) )
=> ( v1_xboole_0(u1_struct_0(C))
| r2_hidden(C,B) ) ) )
& ! [C] :
( ( v1_orders_2(C)
& l1_orders_2(C) )
=> ! [D] :
( ( v1_orders_2(D)
& l1_orders_2(D) )
=> ( ( r1_xboole_0(u1_struct_0(C),u1_struct_0(D))
& r2_hidden(C,B)
& r2_hidden(D,B) )
=> ( r2_hidden(k1_neckla_2(C,D),B)
& r2_hidden(k2_neckla_2(C,D),B) ) ) ) ) )
=> r1_tarski(A,B) ) ) ) ) ) ).
fof(t4_neckla_2,axiom,
! [A] :
( r2_hidden(A,k4_neckla_2)
=> ( ~ v3_struct_0(A)
& v1_orders_2(A)
& v6_group_1(A)
& l1_orders_2(A) ) ) ).
fof(t5_neckla_2,axiom,
! [A] :
( l1_orders_2(A)
=> ( r2_hidden(A,k4_neckla_2)
=> r2_hidden(u1_struct_0(A),k13_classes2) ) ) ).
fof(t6_neckla_2,axiom,
! [A] :
~ ( r2_hidden(A,k4_neckla_2)
& ~ ( ~ v3_struct_0(A)
& v1_orders_2(A)
& v3_realset2(A)
& l1_orders_2(A) )
& ! [B] :
( ( v1_orders_2(B)
& l1_orders_2(B) )
=> ! [C] :
( ( v1_orders_2(C)
& l1_orders_2(C) )
=> ~ ( r1_xboole_0(u1_struct_0(B),u1_struct_0(C))
& r2_hidden(B,k4_neckla_2)
& r2_hidden(C,k4_neckla_2)
& ( A = k1_neckla_2(B,C)
| A = k2_neckla_2(B,C) ) ) ) ) ) ).
fof(t7_neckla_2,axiom,
! [A] :
( ( ~ v3_struct_0(A)
& v1_orders_2(A)
& l1_orders_2(A) )
=> ( r2_hidden(A,k4_neckla_2)
=> v1_neckla_2(A) ) ) ).
fof(dt_k1_neckla_2,axiom,
! [A,B] :
( ( l1_orders_2(A)
& l1_orders_2(B) )
=> ( v1_orders_2(k1_neckla_2(A,B))
& l1_orders_2(k1_neckla_2(A,B)) ) ) ).
fof(dt_k2_neckla_2,axiom,
! [A,B] :
( ( l1_orders_2(A)
& l1_orders_2(B) )
=> ( v1_orders_2(k2_neckla_2(A,B))
& l1_orders_2(k2_neckla_2(A,B)) ) ) ).
fof(dt_k3_neckla_2,axiom,
$true ).
fof(dt_k4_neckla_2,axiom,
m1_subset_1(k4_neckla_2,k1_zfmisc_1(k3_neckla_2)) ).
%------------------------------------------------------------------------------