:: SCMYCIEL semantic presentation
begin
theorem Aux1: :: SCMYCIEL:1
for x, X being set holds not [x,X] in X
proof
let x, X be set ; ::_thesis: not [x,X] in X
assume A: [x,X] in X ; ::_thesis: contradiction
B: X in {x,X} by TARSKI:def_2;
{x,X} in {{x,X},{x}} by TARSKI:def_2;
hence contradiction by A, B, XREGULAR:7; ::_thesis: verum
end;
theorem Aux2: :: SCMYCIEL:2
for x, X being set holds [x,X] <> X
proof
let x, X be set ; ::_thesis: [x,X] <> X
assume [x,X] = X ; ::_thesis: contradiction
then {x,X} in X by TARSKI:def_2;
hence contradiction by TARSKI:def_2; ::_thesis: verum
end;
theorem Aux3: :: SCMYCIEL:3
for x, X being set holds [x,X] <> x
proof
let x, X be set ; ::_thesis: [x,X] <> x
assume [x,X] = x ; ::_thesis: contradiction
then {x,X} in x by TARSKI:def_2;
hence contradiction by TARSKI:def_2; ::_thesis: verum
end;
theorem Aux4: :: SCMYCIEL:4
for x1, y1, x2, y2, X being set st x1 in X & x2 in X & {x1,[y1,X]} = {x2,[y2,X]} holds
( x1 = x2 & y1 = y2 )
proof
let x1, y1, x2, y2, X be set ; ::_thesis: ( x1 in X & x2 in X & {x1,[y1,X]} = {x2,[y2,X]} implies ( x1 = x2 & y1 = y2 ) )
assume that
Ax1: x1 in X and
Ax2: x2 in X ; ::_thesis: ( not {x1,[y1,X]} = {x2,[y2,X]} or ( x1 = x2 & y1 = y2 ) )
assume A: {x1,[y1,X]} = {x2,[y2,X]} ; ::_thesis: ( x1 = x2 & y1 = y2 )
percases ( ( x1 = x2 & [y1,X] = [y2,X] ) or ( x1 = x2 & [y1,X] = x2 ) or ( x1 = [y2,X] & [y1,X] = x2 ) or ( x1 = [y2,X] & [y1,X] = [y2,X] ) ) by A, ZFMISC_1:6;
suppose ( x1 = x2 & [y1,X] = [y2,X] ) ; ::_thesis: ( x1 = x2 & y1 = y2 )
hence ( x1 = x2 & y1 = y2 ) by XTUPLE_0:1; ::_thesis: verum
end;
suppose ( x1 = x2 & [y1,X] = x2 ) ; ::_thesis: ( x1 = x2 & y1 = y2 )
hence ( x1 = x2 & y1 = y2 ) by Aux1, Ax2; ::_thesis: verum
end;
suppose ( x1 = [y2,X] & [y1,X] = x2 ) ; ::_thesis: ( x1 = x2 & y1 = y2 )
hence ( x1 = x2 & y1 = y2 ) by Aux1, Ax2; ::_thesis: verum
end;
suppose ( x1 = [y2,X] & [y1,X] = [y2,X] ) ; ::_thesis: ( x1 = x2 & y1 = y2 )
hence ( x1 = x2 & y1 = y2 ) by Aux1, Ax1; ::_thesis: verum
end;
end;
end;
theorem card3: :: SCMYCIEL:5
for X, v being set st 3 c= card X holds
ex v1, v2 being set st
( v1 in X & v2 in X & v1 <> v & v2 <> v & v1 <> v2 )
proof
let X, v be set ; ::_thesis: ( 3 c= card X implies ex v1, v2 being set st
( v1 in X & v2 in X & v1 <> v & v2 <> v & v1 <> v2 ) )
assume 3 c= card X ; ::_thesis: ex v1, v2 being set st
( v1 in X & v2 in X & v1 <> v & v2 <> v & v1 <> v2 )
then consider x, y, z being set such that
C: x in X and
D: y in X and
E: z in X and
F: x <> y and
G: x <> z and
H: y <> z by PENCIL_1:5;
( ( v <> x & v <> y & v <> z ) or v = x or v = y or v = z ) ;
hence ex v1, v2 being set st
( v1 in X & v2 in X & v1 <> v & v2 <> v & v1 <> v2 ) by C, D, E, F, G, H; ::_thesis: verum
end;
theorem Singletons0: :: SCMYCIEL:6
for x being set holds singletons {x} = {{x}}
proof
let x be set ; ::_thesis: singletons {x} = {{x}}
A: {x} c= {x} ;
thus singletons {x} c= {{x}} :: according to XBOOLE_0:def_10 ::_thesis: {{x}} c= singletons {x}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in singletons {x} or a in {{x}} )
assume a in singletons {x} ; ::_thesis: a in {{x}}
then consider f being Subset of {x} such that
A: a = f and
B: f is 1 -element ;
( f = {} or f = {x} ) by ZFMISC_1:33;
hence a in {{x}} by A, B, TARSKI:def_1; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in {{x}} or a in singletons {x} )
assume a in {{x}} ; ::_thesis: a in singletons {x}
then a = {x} by TARSKI:def_1;
hence a in singletons {x} by A; ::_thesis: verum
end;
registration
cluster Relation-like NAT -defined Function-like finite V37() FinSequence-like FinSubsequence-like for set ;
existence
not for b1 being FinSequence holds b1 is V37()
proof
reconsider f = <*{}*> as FinSequence ;
take f ; ::_thesis: f is V37()
now__::_thesis:_for_x_being_set_st_x_in_rng_f_holds_
x_is_finite
let x be set ; ::_thesis: ( x in rng f implies x is finite )
assume x in rng f ; ::_thesis: x is finite
then x in {{}} by FINSEQ_1:39;
hence x is finite ; ::_thesis: verum
end;
hence f is V37() by FINSET_1:def_2; ::_thesis: verum
end;
end;
theorem Part0: :: SCMYCIEL:7
for X being non empty finite set
for P being a_partition of X st card P < card X holds
ex p, x, y being set st
( p in P & x in p & y in p & x <> y )
proof
let X be non empty finite set ; ::_thesis: for P being a_partition of X st card P < card X holds
ex p, x, y being set st
( p in P & x in p & y in p & x <> y )
let P be a_partition of X; ::_thesis: ( card P < card X implies ex p, x, y being set st
( p in P & x in p & y in p & x <> y ) )
assume A: card P < card X ; ::_thesis: ex p, x, y being set st
( p in P & x in p & y in p & x <> y )
Aa: card P in card X by A, NAT_1:44;
consider x, y being set such that
C: x in X and
D: y in X and
E: x <> y and
F: (proj P) . x = (proj P) . y by Aa, FINSEQ_4:65;
take p = (proj P) . x; ::_thesis: ex x, y being set st
( p in P & x in p & y in p & x <> y )
take x ; ::_thesis: ex y being set st
( p in P & x in p & y in p & x <> y )
take y ; ::_thesis: ( p in P & x in p & y in p & x <> y )
thus p in P by C, FUNCT_2:5; ::_thesis: ( x in p & y in p & x <> y )
thus ( x in p & y in p ) by C, D, F, EQREL_1:def_9; ::_thesis: x <> y
thus x <> y by E; ::_thesis: verum
end;
registration
cluster Vertices {{}} -> empty ;
correctness
coherence
union {{}} is empty ;
by ZFMISC_1:25;
end;
theorem SingleVertices: :: SCMYCIEL:8
for x being set holds union {{},{x}} = {x}
proof
let x be set ; ::_thesis: union {{},{x}} = {x}
{x} = union (bool {x}) by ZFMISC_1:81;
hence union {{},{x}} = {x} by ZFMISC_1:24; ::_thesis: verum
end;
theorem BSPACEdef9: :: SCMYCIEL:9
for X being set
for s being Subset of X st s is 1 -element holds
ex x being set st
( x in X & s = {x} )
proof
let X be set ; ::_thesis: for s being Subset of X st s is 1 -element holds
ex x being set st
( x in X & s = {x} )
let s be Subset of X; ::_thesis: ( s is 1 -element implies ex x being set st
( x in X & s = {x} ) )
assume s is 1 -element ; ::_thesis: ex x being set st
( x in X & s = {x} )
then ( s is trivial & not s is empty ) ;
then consider x being Element of s such that
A: s = {x} by SUBSET_1:46;
take x ; ::_thesis: ( x in X & s = {x} )
x in s by A;
hence x in X ; ::_thesis: s = {x}
thus s = {x} by A; ::_thesis: verum
end;
theorem McopyV: :: SCMYCIEL:10
for X being set holds card { {X,[x,X]} where x is Element of X : x in X } = card X
proof
let X be set ; ::_thesis: card { {X,[x,X]} where x is Element of X : x in X } = card X
set uG = X;
set A = { {X,[x,X]} where x is Element of X : x in X } ;
deffunc H1( set ) -> set = {X,[$1,X]};
consider f being Function such that
B: dom f = X and
D: for x being set st x in X holds
f . x = H1(x) from FUNCT_1:sch_3();
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_dom_f_&_x2_in_dom_f_&_f_._x1_=_f_._x2_holds_
x1_=_x2
let x1, x2 be set ; ::_thesis: ( x1 in dom f & x2 in dom f & f . x1 = f . x2 implies x1 = x2 )
assume that
A1: x1 in dom f and
B1: x2 in dom f and
C1: f . x1 = f . x2 ; ::_thesis: x1 = x2
( H1(x1) = f . x1 & H1(x2) = f . x2 ) by A1, B1, B, D;
then ( [x1,X] = X or [x1,X] = [x2,X] ) by C1, ZFMISC_1:6;
hence x1 = x2 by Aux2, XTUPLE_0:1; ::_thesis: verum
end;
then X: f is one-to-one by FUNCT_1:def_4;
Y: rng f = { {X,[x,X]} where x is Element of X : x in X }
proof
thus rng f c= { {X,[x,X]} where x is Element of X : x in X } :: according to XBOOLE_0:def_10 ::_thesis: { {X,[x,X]} where x is Element of X : x in X } c= rng f
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f or y in { {X,[x,X]} where x is Element of X : x in X } )
assume y in rng f ; ::_thesis: y in { {X,[x,X]} where x is Element of X : x in X }
then consider a being set such that
A1: a in dom f and
B1: f . a = y by FUNCT_1:def_3;
y = {X,[a,X]} by A1, B1, B, D;
hence y in { {X,[x,X]} where x is Element of X : x in X } by A1, B; ::_thesis: verum
end;
thus { {X,[x,X]} where x is Element of X : x in X } c= rng f ::_thesis: verum
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { {X,[x,X]} where x is Element of X : x in X } or a in rng f )
assume a in { {X,[x,X]} where x is Element of X : x in X } ; ::_thesis: a in rng f
then consider x being Element of X such that
A1: a = {X,[x,X]} and
B1: x in X ;
f . x = a by A1, B1, D;
hence a in rng f by B, B1, FUNCT_1:def_3; ::_thesis: verum
end;
end;
{ {X,[x,X]} where x is Element of X : x in X } ,X are_equipotent by B, X, Y, WELLORD2:def_4;
hence card { {X,[x,X]} where x is Element of X : x in X } = card X by CARD_1:5; ::_thesis: verum
end;
definition
let G be set ;
func PairsOf G -> Subset of G means :LEdges: :: SCMYCIEL:def 1
for e being set holds
( e in it iff ( e in G & card e = 2 ) );
existence
ex b1 being Subset of G st
for e being set holds
( e in b1 iff ( e in G & card e = 2 ) )
proof
defpred S1[ set ] means card $1 = 2;
consider X being Subset of G such that
A: for x being set holds
( x in X iff ( x in G & S1[x] ) ) from SUBSET_1:sch_1();
take X ; ::_thesis: for e being set holds
( e in X iff ( e in G & card e = 2 ) )
thus for e being set holds
( e in X iff ( e in G & card e = 2 ) ) by A; ::_thesis: verum
end;
uniqueness
for b1, b2 being Subset of G st ( for e being set holds
( e in b1 iff ( e in G & card e = 2 ) ) ) & ( for e being set holds
( e in b2 iff ( e in G & card e = 2 ) ) ) holds
b1 = b2
proof
let it1, it2 be Subset of G; ::_thesis: ( ( for e being set holds
( e in it1 iff ( e in G & card e = 2 ) ) ) & ( for e being set holds
( e in it2 iff ( e in G & card e = 2 ) ) ) implies it1 = it2 )
assume that
A: for e being set holds
( e in it1 iff ( e in G & card e = 2 ) ) and
B: for e being set holds
( e in it2 iff ( e in G & card e = 2 ) ) ; ::_thesis: it1 = it2
now__::_thesis:_for_x_being_set_holds_
(_x_in_it1_iff_x_in_it2_)
let x be set ; ::_thesis: ( x in it1 iff x in it2 )
( x in it2 iff ( x in G & card x = 2 ) ) by B;
hence ( x in it1 iff x in it2 ) by A; ::_thesis: verum
end;
hence it1 = it2 by TARSKI:1; ::_thesis: verum
end;
end;
:: deftheorem LEdges defines PairsOf SCMYCIEL:def_1_:_
for G being set
for b2 being Subset of G holds
( b2 = PairsOf G iff for e being set holds
( e in b2 iff ( e in G & card e = 2 ) ) );
theorem SG4: :: SCMYCIEL:11
for X, e being set st e in PairsOf X holds
ex x, y being set st
( x <> y & x in union X & y in union X & e = {x,y} )
proof
let G, e be set ; ::_thesis: ( e in PairsOf G implies ex x, y being set st
( x <> y & x in union G & y in union G & e = {x,y} ) )
assume A: e in PairsOf G ; ::_thesis: ex x, y being set st
( x <> y & x in union G & y in union G & e = {x,y} )
card e = 2 by A, LEdges;
then consider x, y being set such that
D: x <> y and
E: e = {x,y} by CARD_2:60;
( x in e & y in e ) by E, TARSKI:def_2;
then ( x in union G & y in union G ) by A, TARSKI:def_4;
hence ex x, y being set st
( x <> y & x in union G & y in union G & e = {x,y} ) by D, E; ::_thesis: verum
end;
theorem SG4a: :: SCMYCIEL:12
for X, x, y being set st x <> y & {x,y} in X holds
{x,y} in PairsOf X
proof
let X, x, y be set ; ::_thesis: ( x <> y & {x,y} in X implies {x,y} in PairsOf X )
assume that
A: x <> y and
B: {x,y} in X ; ::_thesis: {x,y} in PairsOf X
card {x,y} = 2 by A, CARD_2:57;
hence {x,y} in PairsOf X by B, LEdges; ::_thesis: verum
end;
theorem SG5: :: SCMYCIEL:13
for X, x, y being set st {x,y} in PairsOf X holds
( x <> y & x in union X & y in union X )
proof
let G, a, b be set ; ::_thesis: ( {a,b} in PairsOf G implies ( a <> b & a in union G & b in union G ) )
assume {a,b} in PairsOf G ; ::_thesis: ( a <> b & a in union G & b in union G )
then consider x, y being set such that
C: x <> y and
A: ( x in union G & y in union G ) and
B: {a,b} = {x,y} by SG4;
( ( a = x & b = y ) or ( a = y & b = x ) ) by B, ZFMISC_1:6;
hence ( a <> b & a in union G & b in union G ) by A, C; ::_thesis: verum
end;
theorem SG6e: :: SCMYCIEL:14
for G, H being set st G c= H holds
PairsOf G c= PairsOf H
proof
let G, H be set ; ::_thesis: ( G c= H implies PairsOf G c= PairsOf H )
assume A: G c= H ; ::_thesis: PairsOf G c= PairsOf H
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in PairsOf G or e in PairsOf H )
assume AA: e in PairsOf G ; ::_thesis: e in PairsOf H
E: card e = 2 by AA, LEdges;
e in G by AA;
hence e in PairsOf H by A, E, LEdges; ::_thesis: verum
end;
theorem MnewE: :: SCMYCIEL:15
for X being finite set holds card { {x,[y,(union X)]} where x, y is Element of union X : {x,y} in PairsOf X } = 2 * (card (PairsOf X))
proof
let G be finite set ; ::_thesis: card { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in PairsOf G } = 2 * (card (PairsOf G))
set Y = union G;
set A = { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in PairsOf G } ;
set EG = PairsOf G;
set uG = union G;
set s = canFS (PairsOf G);
Aa: len (canFS (PairsOf G)) = card (PairsOf G) by UPROOTS:3;
Ac: rng (canFS (PairsOf G)) = PairsOf G by FUNCT_2:def_3;
defpred S1[ set , set ] means for a, b being set st $1 = {a,b} holds
$2 = {{a,[b,(union G)]},{b,[a,(union G)]}};
P0: for x, y1, y2 being set st x in PairsOf G & S1[x,y1] & S1[x,y2] holds
y1 = y2
proof
let x, v1, v2 be set ; ::_thesis: ( x in PairsOf G & S1[x,v1] & S1[x,v2] implies v1 = v2 )
assume that
A1: x in PairsOf G and
B1: S1[x,v1] and
C1: S1[x,v2] ; ::_thesis: v1 = v2
consider x1, y1 being set such that
x1 <> y1 and
x1 in union G and
y1 in union G and
F1: x = {x1,y1} by A1, SG4;
v2 = {{x1,[y1,(union G)]},{y1,[x1,(union G)]}} by F1, C1;
hence v1 = v2 by F1, B1; ::_thesis: verum
end;
P1: for x being set st x in PairsOf G holds
ex y being set st S1[x,y]
proof
let x be set ; ::_thesis: ( x in PairsOf G implies ex y being set st S1[x,y] )
assume x in PairsOf G ; ::_thesis: ex y being set st S1[x,y]
then consider x1, y1 being set such that
x1 <> y1 and
x1 in union G and
y1 in union G and
F1: x = {x1,y1} by SG4;
take y = {{x1,[y1,(union G)]},{y1,[x1,(union G)]}}; ::_thesis: S1[x,y]
let a, b be set ; ::_thesis: ( x = {a,b} implies y = {{a,[b,(union G)]},{b,[a,(union G)]}} )
assume x = {a,b} ; ::_thesis: y = {{a,[b,(union G)]},{b,[a,(union G)]}}
then ( ( a = x1 & b = y1 ) or ( a = y1 & b = x1 ) ) by F1, ZFMISC_1:6;
hence y = {{a,[b,(union G)]},{b,[a,(union G)]}} ; ::_thesis: verum
end;
consider f being Function such that
A: dom f = PairsOf G and
B: for x being set st x in PairsOf G holds
S1[x,f . x] from FUNCT_1:sch_2(P0, P1);
now__::_thesis:_for_y_being_set_st_y_in_rng_(f_*_(canFS_(PairsOf_G)))_holds_
y_is_finite
let y be set ; ::_thesis: ( y in rng (f * (canFS (PairsOf G))) implies y is finite )
assume y in rng (f * (canFS (PairsOf G))) ; ::_thesis: y is finite
then y in rng f by FUNCT_1:14;
then consider x being set such that
A1: x in dom f and
B1: y = f . x by FUNCT_1:def_3;
consider x1, y1 being set such that
x1 <> y1 and
x1 in union G and
y1 in union G and
F1: x = {x1,y1} by A1, A, SG4;
y = {{x1,[y1,(union G)]},{y1,[x1,(union G)]}} by F1, B1, A1, A, B;
hence y is finite ; ::_thesis: verum
end;
then reconsider S = f * (canFS (PairsOf G)) as V37() FinSequence by Ac, A, FINSEQ_1:16, FINSET_1:def_2;
Ca: dom S = dom (canFS (PairsOf G)) by Ac, A, RELAT_1:27;
deffunc H1( set ) -> Element of omega = card (S . $1);
consider L being FinSequence of NAT such that
C: len L = len S and
D: for j being Nat st j in dom L holds
L . j = H1(j) from FINSEQ_2:sch_1();
Ea: dom S = dom L by C, FINSEQ_3:29;
Eb: for i being Nat st i in dom S holds
( S . i is finite & L . i = card (S . i) ) by Ea, D;
now__::_thesis:_for_x,_y_being_set_st_x_<>_y_holds_
not_S_._x_meets_S_._y
let x, y be set ; ::_thesis: ( x <> y implies not S . b1 meets S . b2 )
assume A11: x <> y ; ::_thesis: not S . b1 meets S . b2
percases ( ( x in dom S & y in dom S ) or not x in dom S or not y in dom S ) ;
supposethat S1x: x in dom S and
S1y: y in dom S ; ::_thesis: not S . b1 meets S . b2
A1: ( x in dom (canFS (PairsOf G)) & (canFS (PairsOf G)) . x in dom f ) by S1x, FUNCT_1:11;
B1: ( y in dom (canFS (PairsOf G)) & (canFS (PairsOf G)) . y in dom f ) by S1y, FUNCT_1:11;
consider x1, y1 being set such that
x1 <> y1 and
D1x: ( x1 in union G & y1 in union G ) and
E1x: (canFS (PairsOf G)) . x = {x1,y1} by A1, A, SG4;
consider x2, y2 being set such that
x2 <> y2 and
D1y: ( x2 in union G & y2 in union G ) and
E1y: (canFS (PairsOf G)) . y = {x2,y2} by B1, A, SG4;
F1x: S . x = f . ((canFS (PairsOf G)) . x) by S1x, FUNCT_1:12;
F1y: S . y = f . ((canFS (PairsOf G)) . y) by S1y, FUNCT_1:12;
G1x: S . x = {{x1,[y1,(union G)]},{y1,[x1,(union G)]}} by E1x, F1x, A1, A, B;
G1y: S . y = {{x2,[y2,(union G)]},{y2,[x2,(union G)]}} by E1y, F1y, B1, A, B;
assume S . x meets S . y ; ::_thesis: contradiction
then consider a being set such that
Jx: a in S . x and
Jy: a in S . y by XBOOLE_0:3;
Kx: ( a = {x1,[y1,(union G)]} or a = {y1,[x1,(union G)]} ) by Jx, G1x, TARSKI:def_2;
Ky: ( a = {x2,[y2,(union G)]} or a = {y2,[x2,(union G)]} ) by Jy, G1y, TARSKI:def_2;
( ( x1 = x2 & y1 = y2 ) or ( x1 = y2 & y1 = x2 ) or ( y1 = x2 & x1 = y2 ) ) by D1x, D1y, Kx, Ky, Aux4;
hence contradiction by E1x, E1y, A11, A1, B1, FUNCT_1:def_4; ::_thesis: verum
end;
suppose ( not x in dom S or not y in dom S ) ; ::_thesis: S . b1 misses S . b2
then ( S . x = {} or S . y = {} ) by FUNCT_1:def_2;
hence S . x misses S . y by XBOOLE_1:65; ::_thesis: verum
end;
end;
end;
then Ec: S is disjoint_valued by PROB_2:def_2;
Union S = union (rng S) ;
then E: card (union (rng S)) = Sum L by Ea, Eb, Ec, DIST_1:17;
Fa: dom ((len L) |-> 2) = Seg (len L) by FUNCOP_1:13
.= dom L by FINSEQ_1:def_3 ;
now__::_thesis:_for_j_being_Nat_st_j_in_dom_L_holds_
L_._j_=_((len_L)_|->_2)_._j
let j be Nat; ::_thesis: ( j in dom L implies L . j = ((len L) |-> 2) . j )
assume A1: j in dom L ; ::_thesis: L . j = ((len L) |-> 2) . j
C1: S . j = f . ((canFS (PairsOf G)) . j) by A1, Ea, FUNCT_1:12;
consider x, y being set such that
D1: x <> y and
x in union G and
y in union G and
G1: (canFS (PairsOf G)) . j = {x,y} by SG4, A1, Ea, Ca, Ac, FUNCT_1:3;
H1: f . ((canFS (PairsOf G)) . j) = {{x,[y,(union G)]},{y,[x,(union G)]}} by G1, B, A1, Ea, Ca, Ac, FUNCT_1:3;
I1: now__::_thesis:_not_{x,[y,(union_G)]}_=_{y,[x,(union_G)]}
assume {x,[y,(union G)]} = {y,[x,(union G)]} ; ::_thesis: contradiction
then ( x = y or x = [x,(union G)] ) by ZFMISC_1:6;
hence contradiction by D1, Aux3; ::_thesis: verum
end;
J1: j in Seg (len L) by A1, FINSEQ_1:def_3;
thus L . j = card (S . j) by A1, D
.= 2 by I1, H1, C1, CARD_2:57
.= ((len L) |-> 2) . j by J1, FINSEQ_2:57 ; ::_thesis: verum
end;
then F: L = (len L) |-> 2 by Fa, FINSEQ_1:13;
G: len L = card (PairsOf G) by C, Ca, Aa, FINSEQ_3:29;
union (rng S) = { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in PairsOf G }
proof
thus union (rng S) c= { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in PairsOf G } :: according to XBOOLE_0:def_10 ::_thesis: { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in PairsOf G } c= union (rng S)
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in union (rng S) or a in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in PairsOf G } )
assume a in union (rng S) ; ::_thesis: a in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in PairsOf G }
then consider YY being set such that
A1: a in YY and
B1: YY in rng S by TARSKI:def_4;
consider b being set such that
C1: b in dom S and
D1a: YY = S . b by B1, FUNCT_1:def_3;
D1b: S . b = f . ((canFS (PairsOf G)) . b) by C1, FUNCT_1:12;
C1a: (canFS (PairsOf G)) . b in PairsOf G by C1, Ca, Ac, FUNCT_1:3;
consider x, y being set such that
x <> y and
E1: x in union G and
F1: y in union G and
G1: (canFS (PairsOf G)) . b = {x,y} by SG4, C1, Ca, Ac, FUNCT_1:3;
f . ((canFS (PairsOf G)) . b) = {{x,[y,(union G)]},{y,[x,(union G)]}} by G1, B, C1, Ca, Ac, FUNCT_1:3;
then ( a = {x,[y,(union G)]} or a = {y,[x,(union G)]} ) by A1, D1a, D1b, TARSKI:def_2;
hence a in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in PairsOf G } by G1, C1a, E1, F1; ::_thesis: verum
end;
thus { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in PairsOf G } c= union (rng S) ::_thesis: verum
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in PairsOf G } or a in union (rng S) )
assume a in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in PairsOf G } ; ::_thesis: a in union (rng S)
then consider x, y being Element of union G such that
A1: a = {x,[y,(union G)]} and
B1: {x,y} in PairsOf G ;
consider c being set such that
c in dom (canFS (PairsOf G)) and
D1: (canFS (PairsOf G)) . c = {x,y} by B1, Ac, FUNCT_1:def_3;
rng S = rng f by A, Ac, RELAT_1:28;
then E1: f . ((canFS (PairsOf G)) . c) in rng S by A, D1, B1, FUNCT_1:3;
f . ((canFS (PairsOf G)) . c) = {{x,[y,(union G)]},{y,[x,(union G)]}} by D1, B1, B;
then a in f . ((canFS (PairsOf G)) . c) by A1, TARSKI:def_2;
hence a in union (rng S) by E1, TARSKI:def_4; ::_thesis: verum
end;
end;
hence card { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in PairsOf G } = 2 * (card (PairsOf G)) by E, F, G, RVSUM_1:80; ::_thesis: verum
end;
theorem :: SCMYCIEL:16
for X being finite set holds card { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } = 2 * (card (PairsOf X))
proof
let X be finite set ; ::_thesis: card { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } = 2 * (card (PairsOf X))
set Y = union X;
set B = { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } ;
set A = { {x,[y,(union X)]} where x, y is Element of union X : {x,y} in PairsOf X } ;
percases ( { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } is empty or not { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } is empty ) ;
supposeS1: { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } is empty ; ::_thesis: card { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } = 2 * (card (PairsOf X))
now__::_thesis:__{__{x,[y,(union_X)]}_where_x,_y_is_Element_of_union_X_:_{x,y}_in_PairsOf_X__}__is_empty
assume not { {x,[y,(union X)]} where x, y is Element of union X : {x,y} in PairsOf X } is empty ; ::_thesis: contradiction
then consider a being set such that
A1: a in { {x,[y,(union X)]} where x, y is Element of union X : {x,y} in PairsOf X } by XBOOLE_0:def_1;
consider x, y being Element of union X such that
a = {x,[y,(union X)]} and
C1: {x,y} in PairsOf X by A1;
[x,y] in { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } by C1;
hence contradiction by S1; ::_thesis: verum
end;
hence card { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } = 2 * (card (PairsOf X)) by S1, MnewE; ::_thesis: verum
end;
supposeS1: not { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } is empty ; ::_thesis: card { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } = 2 * (card (PairsOf X))
then consider b being set such that
Aa1: b in { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } by XBOOLE_0:def_1;
consider x, y being Element of union X such that
b = [x,y] and
Ca1: {x,y} in PairsOf X by Aa1;
Ea1: x in {x,y} by TARSKI:def_2;
S1a: union X <> {} by Ca1, Ea1, TARSKI:def_4;
defpred S1[ set , set ] means for a, b being Element of union X st a in union X & b in union X & $1 = {a,[b,(union X)]} holds
$2 = [a,b];
P: for c being set st c in { {x,[y,(union X)]} where x, y is Element of union X : {x,y} in PairsOf X } holds
ex d being set st
( d in { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } & S1[c,d] )
proof
let c be set ; ::_thesis: ( c in { {x,[y,(union X)]} where x, y is Element of union X : {x,y} in PairsOf X } implies ex d being set st
( d in { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } & S1[c,d] ) )
assume c in { {x,[y,(union X)]} where x, y is Element of union X : {x,y} in PairsOf X } ; ::_thesis: ex d being set st
( d in { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } & S1[c,d] )
then consider x, y being Element of union X such that
B2: c = {x,[y,(union X)]} and
C2: {x,y} in PairsOf X ;
take d = [x,y]; ::_thesis: ( d in { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } & S1[c,d] )
thus d in { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } by C2; ::_thesis: S1[c,d]
thus S1[c,d] ::_thesis: verum
proof
let a, b be Element of union X; ::_thesis: ( a in union X & b in union X & c = {a,[b,(union X)]} implies d = [a,b] )
assume A3: ( a in union X & b in union X ) ; ::_thesis: ( not c = {a,[b,(union X)]} or d = [a,b] )
assume c = {a,[b,(union X)]} ; ::_thesis: d = [a,b]
then ( a = x & b = y ) by B2, A3, Aux4;
hence d = [a,b] ; ::_thesis: verum
end;
end;
consider f being Function of { {x,[y,(union X)]} where x, y is Element of union X : {x,y} in PairsOf X } , { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } such that
A1: for c being set st c in { {x,[y,(union X)]} where x, y is Element of union X : {x,y} in PairsOf X } holds
S1[c,f . c] from FUNCT_2:sch_1(P);
domf: dom f = { {x,[y,(union X)]} where x, y is Element of union X : {x,y} in PairsOf X } by S1, FUNCT_2:def_1;
B1: f is one-to-one
proof
let c1, c2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not c1 in dom f or not c2 in dom f or not f . c1 = f . c2 or c1 = c2 )
assume that
A2: c1 in dom f and
B2: c2 in dom f and
C2: f . c1 = f . c2 ; ::_thesis: c1 = c2
consider x1, y1 being Element of union X such that
E2: c1 = {x1,[y1,(union X)]} and
{x1,y1} in PairsOf X by A2, domf;
consider x2, y2 being Element of union X such that
G2: c2 = {x2,[y2,(union X)]} and
{x2,y2} in PairsOf X by B2, domf;
I2: f . c1 = [x1,y1] by A1, A2, domf, S1a, E2;
J2: f . c2 = [x2,y2] by A1, B2, domf, S1a, G2;
( x1 = x2 & y1 = y2 ) by C2, I2, J2, XTUPLE_0:1;
hence c1 = c2 by E2, G2; ::_thesis: verum
end;
C1a: rng f = { [x,y] where x, y is Element of union X : {x,y} in PairsOf X }
proof
thus rng f c= { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } :: according to XBOOLE_0:def_10 ::_thesis: { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } c= rng f
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in rng f or b in { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } )
assume b in rng f ; ::_thesis: b in { [x,y] where x, y is Element of union X : {x,y} in PairsOf X }
then consider a being set such that
A2: a in dom f and
B2: b = f . a by FUNCT_1:def_3;
consider x, y being Element of union X such that
C2: a = {x,[y,(union X)]} and
D2: {x,y} in PairsOf X by A2, domf;
F2: b = [x,y] by B2, A2, A1, domf, S1a, C2;
thus b in { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } by F2, D2; ::_thesis: verum
end;
thus { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } c= rng f ::_thesis: verum
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } or b in rng f )
assume A2: b in { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } ; ::_thesis: b in rng f
consider x, y being Element of union X such that
B2: b = [x,y] and
C2: {x,y} in PairsOf X by A2;
set a = {x,[y,(union X)]};
D2: {x,[y,(union X)]} in { {x,[y,(union X)]} where x, y is Element of union X : {x,y} in PairsOf X } by C2;
F2: f . {x,[y,(union X)]} = b by D2, A1, B2, S1a;
thus b in rng f by D2, F2, domf, FUNCT_1:3; ::_thesis: verum
end;
end;
C1: f is onto by C1a, FUNCT_2:def_3;
thus card { [x,y] where x, y is Element of union X : {x,y} in PairsOf X } = card { {x,[y,(union X)]} where x, y is Element of union X : {x,y} in PairsOf X } by B1, C1, S1, EULER_1:11
.= 2 * (card (PairsOf X)) by MnewE ; ::_thesis: verum
end;
end;
end;
registration
let X be finite set ;
cluster PairsOf X -> finite ;
coherence
PairsOf X is finite ;
end;
definition
let X be set ;
attrX is void means :Lvoid: :: SCMYCIEL:def 2
X = {{}};
end;
:: deftheorem Lvoid defines void SCMYCIEL:def_2_:_
for X being set holds
( X is void iff X = {{}} );
registration
cluster void for set ;
existence
ex b1 being set st b1 is void by Lvoid;
end;
registration
cluster void -> finite for set ;
coherence
for b1 being set st b1 is void holds
b1 is finite by Lvoid;
end;
registration
let G be void set ;
cluster Vertices G -> empty ;
coherence
union G is empty
proof
G = {{}} by Lvoid;
hence union G is empty ; ::_thesis: verum
end;
end;
theorem VoidGE: :: SCMYCIEL:17
for X being set st X is void holds
PairsOf X = {}
proof
let G be set ; ::_thesis: ( G is void implies PairsOf G = {} )
assume A: G is void ; ::_thesis: PairsOf G = {}
assume PairsOf G <> {} ; ::_thesis: contradiction
then consider x being set such that
B: x in PairsOf G by XBOOLE_0:def_1;
D: card x = 2 by B, LEdges;
G = {{}} by A, Lvoid;
then x = {} by B, TARSKI:def_1;
hence contradiction by D; ::_thesis: verum
end;
theorem uVoid1: :: SCMYCIEL:18
for X being set holds
( not union X = {} or X = {} or X = {{}} )
proof
let X be set ; ::_thesis: ( not union X = {} or X = {} or X = {{}} )
assume A: union X = {} ; ::_thesis: ( X = {} or X = {{}} )
assume X <> {} ; ::_thesis: X = {{}}
then consider x being set such that
B: x in X by XBOOLE_0:def_1;
thus X = {{}} ::_thesis: verum
proof
thus X c= {{}} :: according to XBOOLE_0:def_10 ::_thesis: {{}} c= X
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in X or a in {{}} )
assume a in X ; ::_thesis: a in {{}}
then a = {} by A, ORDERS_1:6;
hence a in {{}} by TARSKI:def_1; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in {{}} or a in X )
assume a in {{}} ; ::_thesis: a in X
then a = {} by TARSKI:def_1;
hence a in X by B, A, ORDERS_1:6; ::_thesis: verum
end;
end;
definition
let X be set ;
attrX is pairfree means :Ledgeless: :: SCMYCIEL:def 3
PairsOf X is empty ;
end;
:: deftheorem Ledgeless defines pairfree SCMYCIEL:def_3_:_
for X being set holds
( X is pairfree iff PairsOf X is empty );
theorem GsingleE: :: SCMYCIEL:19
for X, x being set st card (union X) = 1 holds
X is pairfree
proof
let G, x be set ; ::_thesis: ( card (union G) = 1 implies G is pairfree )
assume A: card (union G) = 1 ; ::_thesis: G is pairfree
assume not G is pairfree ; ::_thesis: contradiction
then PairsOf G <> {} by Ledgeless;
then consider e being set such that
C: e in PairsOf G by XBOOLE_0:def_1;
consider x, y being set such that
D: x <> y and
E: x in union G and
F: y in union G and
e = {x,y} by C, SG4;
consider w being set such that
H: union G = {w} by A, CARD_2:42;
x = w by E, H, TARSKI:def_1;
hence contradiction by D, F, H, TARSKI:def_1; ::_thesis: verum
end;
CSGLem1: for X being set holds union { V where V is finite Subset of X : card V <= 2 } = X
proof
let X be set ; ::_thesis: union { V where V is finite Subset of X : card V <= 2 } = X
set G = { V where V is finite Subset of X : card V <= 2 } ;
thus union { V where V is finite Subset of X : card V <= 2 } c= X :: according to XBOOLE_0:def_10 ::_thesis: X c= union { V where V is finite Subset of X : card V <= 2 }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union { V where V is finite Subset of X : card V <= 2 } or x in X )
assume x in union { V where V is finite Subset of X : card V <= 2 } ; ::_thesis: x in X
then consider a being set such that
Ax: x in a and
Ay: a in { V where V is finite Subset of X : card V <= 2 } by TARSKI:def_4;
consider V being finite Subset of X such that
A2: ( a = V & card V <= 2 ) by Ay;
thus x in X by Ax, A2; ::_thesis: verum
end;
thus X c= union { V where V is finite Subset of X : card V <= 2 } ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in union { V where V is finite Subset of X : card V <= 2 } )
A2a: card {x} = 1 by CARD_1:30;
B2: x in {x} by TARSKI:def_1;
assume x in X ; ::_thesis: x in union { V where V is finite Subset of X : card V <= 2 }
then {x} c= X by ZFMISC_1:31;
then {x} in { V where V is finite Subset of X : card V <= 2 } by A2a;
hence x in union { V where V is finite Subset of X : card V <= 2 } by B2, TARSKI:def_4; ::_thesis: verum
end;
end;
registration
cluster non empty finite-membered for set ;
existence
ex b1 being set st
( b1 is finite-membered & not b1 is empty )
proof
take {{}} ; ::_thesis: ( {{}} is finite-membered & not {{}} is empty )
thus ( {{}} is finite-membered & not {{}} is empty ) ; ::_thesis: verum
end;
end;
registration
let X be finite-membered set ;
let Y be set ;
clusterX /\ Y -> finite-membered ;
coherence
X /\ Y is finite-membered
proof
let x be set ; :: according to FINSET_1:def_6 ::_thesis: ( not x in X /\ Y or x is finite )
assume x in X /\ Y ; ::_thesis: x is finite
then x in X by XBOOLE_0:def_4;
hence x is finite ; ::_thesis: verum
end;
clusterX \ Y -> finite-membered ;
coherence
X \ Y is finite-membered ;
end;
begin
definition
let n be Nat;
let X be set ;
attrX is n -at_most_dimensional means :Lnatmost: :: SCMYCIEL:def 4
for x being set st x in X holds
card x c= n + 1;
end;
:: deftheorem Lnatmost defines -at_most_dimensional SCMYCIEL:def_4_:_
for n being Nat
for X being set holds
( X is n -at_most_dimensional iff for x being set st x in X holds
card x c= n + 1 );
registration
let n be Nat;
clustern -at_most_dimensional -> finite-membered for set ;
correctness
coherence
for b1 being set st b1 is n -at_most_dimensional holds
b1 is finite-membered ;
proof
let X be set ; ::_thesis: ( X is n -at_most_dimensional implies X is finite-membered )
assume A: X is n -at_most_dimensional ; ::_thesis: X is finite-membered
thus X is finite-membered ::_thesis: verum
proof
let x be set ; :: according to FINSET_1:def_6 ::_thesis: ( not x in X or x is finite )
assume x in X ; ::_thesis: x is finite
then card x c= n + 1 by A, Lnatmost;
hence x is finite ; ::_thesis: verum
end;
end;
end;
Void0: for n being Nat holds {{}} is n -at_most_dimensional
proof
let n be Nat; ::_thesis: {{}} is n -at_most_dimensional
set E = {{}};
thus {{}} is n -at_most_dimensional ::_thesis: verum
proof
let x be set ; :: according to SCMYCIEL:def_4 ::_thesis: ( x in {{}} implies card x c= n + 1 )
assume x in {{}} ; ::_thesis: card x c= n + 1
then x = {} by TARSKI:def_1;
hence card x c= n + 1 ; ::_thesis: verum
end;
end;
registration
let n be Nat;
cluster non empty subset-closed n -at_most_dimensional for set ;
existence
ex b1 being set st
( b1 is n -at_most_dimensional & b1 is subset-closed & not b1 is empty )
proof
set E = {{}};
take {{}} ; ::_thesis: ( {{}} is n -at_most_dimensional & {{}} is subset-closed & not {{}} is empty )
thus ( {{}} is n -at_most_dimensional & {{}} is subset-closed & not {{}} is empty ) by Void0; ::_thesis: verum
end;
end;
theorem SG1: :: SCMYCIEL:20
for G being non empty subset-closed set holds {} in G
proof
let G be non empty subset-closed set ; ::_thesis: {} in G
consider z being set such that
A2: z in G by XBOOLE_0:def_1;
{} c= z by XBOOLE_1:2;
hence {} in G by A2, CLASSES1:def_1; ::_thesis: verum
end;
theorem Lnatmost1: :: SCMYCIEL:21
for n being natural number
for X being b1 -at_most_dimensional set
for x being Element of X st x in X holds
card x <= n + 1 by Lnatmost, NAT_1:39;
registration
let n be Nat;
let X, Y be n -at_most_dimensional set ;
clusterX \/ Y -> n -at_most_dimensional ;
coherence
X \/ Y is n -at_most_dimensional
proof
let x be set ; :: according to SCMYCIEL:def_4 ::_thesis: ( x in X \/ Y implies card x c= n + 1 )
assume A: x in X \/ Y ; ::_thesis: card x c= n + 1
( x in X or x in Y ) by A, XBOOLE_0:def_3;
hence card x c= n + 1 by Lnatmost; ::_thesis: verum
end;
end;
registration
let n be Nat;
let X be n -at_most_dimensional set ;
let Y be set ;
clusterX /\ Y -> n -at_most_dimensional ;
coherence
X /\ Y is n -at_most_dimensional
proof
let x be set ; :: according to SCMYCIEL:def_4 ::_thesis: ( x in X /\ Y implies card x c= n + 1 )
assume x in X /\ Y ; ::_thesis: card x c= n + 1
then x in X by XBOOLE_0:def_4;
hence card x c= n + 1 by Lnatmost; ::_thesis: verum
end;
clusterX \ Y -> n -at_most_dimensional ;
coherence
X \ Y is n -at_most_dimensional
proof
let x be set ; :: according to SCMYCIEL:def_4 ::_thesis: ( x in X \ Y implies card x c= n + 1 )
assume x in X \ Y ; ::_thesis: card x c= n + 1
hence card x c= n + 1 by Lnatmost; ::_thesis: verum
end;
end;
registration
let n be Nat;
let X be n -at_most_dimensional set ;
cluster -> n -at_most_dimensional for Element of bool X;
correctness
coherence
for b1 being Subset of X holds b1 is n -at_most_dimensional ;
proof
let Y be Subset of X; ::_thesis: Y is n -at_most_dimensional
let x be set ; :: according to SCMYCIEL:def_4 ::_thesis: ( x in Y implies card x c= n + 1 )
assume x in Y ; ::_thesis: card x c= n + 1
hence card x c= n + 1 by Lnatmost; ::_thesis: verum
end;
end;
definition
let s be set ;
attrs is SimpleGraph-like means :LSGlike: :: SCMYCIEL:def 5
( s is 1 -at_most_dimensional & s is subset-closed & not s is empty );
end;
:: deftheorem LSGlike defines SimpleGraph-like SCMYCIEL:def_5_:_
for s being set holds
( s is SimpleGraph-like iff ( s is 1 -at_most_dimensional & s is subset-closed & not s is empty ) );
registration
cluster SimpleGraph-like -> non empty subset-closed 1 -at_most_dimensional for set ;
correctness
coherence
for b1 being set st b1 is SimpleGraph-like holds
( b1 is 1 -at_most_dimensional & b1 is subset-closed & not b1 is empty );
by LSGlike;
cluster non empty subset-closed 1 -at_most_dimensional -> SimpleGraph-like for set ;
correctness
coherence
for b1 being set st b1 is 1 -at_most_dimensional & b1 is subset-closed & not b1 is empty holds
b1 is SimpleGraph-like ;
by LSGlike;
end;
theorem eSG1: :: SCMYCIEL:22
{{}} is SimpleGraph-like
proof
{{}} is 1 -at_most_dimensional by Void0;
hence {{}} is SimpleGraph-like ; ::_thesis: verum
end;
registration
cluster{{}} -> SimpleGraph-like ;
correctness
coherence
{{}} is SimpleGraph-like ;
by eSG1;
end;
registration
cluster SimpleGraph-like for set ;
existence
ex b1 being set st b1 is SimpleGraph-like by eSG1;
end;
definition
mode SimpleGraph is SimpleGraph-like set ;
end;
registration
cluster non empty finite-membered V233() V267() subset-closed void 1 -at_most_dimensional SimpleGraph-like for set ;
existence
ex b1 being SimpleGraph st b1 is void
proof
reconsider G = {{}} as SimpleGraph ;
take G ; ::_thesis: G is void
thus G is void by Lvoid; ::_thesis: verum
end;
cluster non empty finite finite-membered V233() V267() subset-closed 1 -at_most_dimensional SimpleGraph-like for set ;
existence
ex b1 being SimpleGraph st b1 is finite by eSG1;
end;
notation
let G be set ;
synonym Vertices G for union G;
synonym Edges G for PairsOf G;
end;
notation
let X be set ;
synonym edgeless X for pairfree ;
end;
theorem FinSG: :: SCMYCIEL:23
for G being SimpleGraph st Vertices G is finite holds
G is finite
proof
let G be SimpleGraph; ::_thesis: ( Vertices G is finite implies G is finite )
assume A: Vertices G is finite ; ::_thesis: G is finite
G c= bool (Vertices G) by ZFMISC_1:82;
hence G is finite by A; ::_thesis: verum
end;
theorem Vertices0: :: SCMYCIEL:24
for G being SimpleGraph
for x being set holds
( x in Vertices G iff {x} in G )
proof
let G be SimpleGraph; ::_thesis: for x being set holds
( x in Vertices G iff {x} in G )
let x be set ; ::_thesis: ( x in Vertices G iff {x} in G )
thus ( x in Vertices G implies {x} in G ) ::_thesis: ( {x} in G implies x in Vertices G )
proof
assume x in Vertices G ; ::_thesis: {x} in G
then consider y being set such that
A1: x in y and
B1: y in G by TARSKI:def_4;
{x} c= y by A1, ZFMISC_1:31;
hence {x} in G by B1, CLASSES1:def_1; ::_thesis: verum
end;
x in {x} by TARSKI:def_1;
hence ( {x} in G implies x in Vertices G ) by TARSKI:def_4; ::_thesis: verum
end;
theorem SingleVertex: :: SCMYCIEL:25
for x being set holds {{},{x}} is SimpleGraph
proof
let x be set ; ::_thesis: {{},{x}} is SimpleGraph
set H = {{},{x}};
B: {{},{x}} is 1 -at_most_dimensional
proof
let a be set ; :: according to SCMYCIEL:def_4 ::_thesis: ( a in {{},{x}} implies card a c= 1 + 1 )
assume A1: a in {{},{x}} ; ::_thesis: card a c= 1 + 1
percases ( a = {} or a = {x} ) by A1, TARSKI:def_2;
suppose a = {} ; ::_thesis: card a c= 1 + 1
hence card a c= 1 + 1 ; ::_thesis: verum
end;
suppose a = {x} ; ::_thesis: card a c= 1 + 1
then card a = 1 by CARD_1:30;
hence card a c= 1 + 1 by NAT_1:39; ::_thesis: verum
end;
end;
end;
{{},{x}} is subset-closed
proof
let X, Y be set ; :: according to CLASSES1:def_1 ::_thesis: ( not X in {{},{x}} or not Y c= X or Y in {{},{x}} )
assume that
A1: X in {{},{x}} and
B1: Y c= X ; ::_thesis: Y in {{},{x}}
percases ( X = {} or X = {x} ) by A1, TARSKI:def_2;
suppose X = {} ; ::_thesis: Y in {{},{x}}
then Y = {} by B1;
hence Y in {{},{x}} by TARSKI:def_2; ::_thesis: verum
end;
supposeS1: X = {x} ; ::_thesis: Y in {{},{x}}
percases ( Y = {} or Y = {x} ) by S1, B1, ZFMISC_1:33;
suppose Y = {} ; ::_thesis: Y in {{},{x}}
hence Y in {{},{x}} by TARSKI:def_2; ::_thesis: verum
end;
suppose Y = {x} ; ::_thesis: Y in {{},{x}}
hence Y in {{},{x}} by TARSKI:def_2; ::_thesis: verum
end;
end;
end;
end;
end;
hence {{},{x}} is SimpleGraph by B; ::_thesis: verum
end;
definition
let X be finite finite-membered set ;
func order X -> Nat equals :: SCMYCIEL:def 6
card (union X);
coherence
card (union X) is Nat ;
end;
:: deftheorem defines order SCMYCIEL:def_6_:_
for X being finite finite-membered set holds order X = card (union X);
definition
let X be finite set ;
func size X -> Nat equals :: SCMYCIEL:def 7
card (PairsOf X);
coherence
card (PairsOf X) is Nat ;
end;
:: deftheorem defines size SCMYCIEL:def_7_:_
for X being finite set holds size X = card (PairsOf X);
theorem Lorder1: :: SCMYCIEL:26
for G being finite SimpleGraph holds order G <= card G
proof
let G be finite SimpleGraph; ::_thesis: order G <= card G
set uG = union G;
A: card (singletons (union G)) = card (union G) by BSPACE:41;
singletons (union G) c= G
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in singletons (union G) or x in G )
assume x in singletons (union G) ; ::_thesis: x in G
then consider f being Subset of (union G) such that
B: x = f and
C: f is 1 -element ;
consider a being set such that
D: a in union G and
E: f = {a} by C, BSPACEdef9;
consider y being set such that
F: a in y and
G: y in G by D, TARSKI:def_4;
{a} c= y by F, ZFMISC_1:31;
hence x in G by G, E, B, CLASSES1:def_1; ::_thesis: verum
end;
hence order G <= card G by A, NAT_1:43; ::_thesis: verum
end;
definition
let G be SimpleGraph;
mode Vertex of G is Element of Vertices G;
mode Edge of G is Element of Edges G;
end;
theorem SG0: :: SCMYCIEL:27
for G being SimpleGraph holds G = ({{}} \/ (singletons (Vertices G))) \/ (Edges G)
proof
let G be SimpleGraph; ::_thesis: G = ({{}} \/ (singletons (Vertices G))) \/ (Edges G)
thus G c= ({{}} \/ (singletons (Vertices G))) \/ (Edges G) :: according to XBOOLE_0:def_10 ::_thesis: ({{}} \/ (singletons (Vertices G))) \/ (Edges G) c= G
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in G or x in ({{}} \/ (singletons (Vertices G))) \/ (Edges G) )
assume A: x in G ; ::_thesis: x in ({{}} \/ (singletons (Vertices G))) \/ (Edges G)
reconsider v = x as finite set by A;
B: card v <= 1 + 1 by A, Lnatmost1;
percases ( card v = 0 or card v = 1 or card v = 2 ) by B, NAT_1:26;
suppose card v = 0 ; ::_thesis: x in ({{}} \/ (singletons (Vertices G))) \/ (Edges G)
then v = {} ;
then v in {{}} by TARSKI:def_1;
then v in {{}} \/ (singletons (Vertices G)) by XBOOLE_0:def_3;
hence x in ({{}} \/ (singletons (Vertices G))) \/ (Edges G) by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose card v = 1 ; ::_thesis: x in ({{}} \/ (singletons (Vertices G))) \/ (Edges G)
then consider a being set such that
A1: v = {a} by CARD_2:42;
B1: a in v by A1, TARSKI:def_1;
C1: a in union G by B1, A, TARSKI:def_4;
reconsider v = v as Subset of (Vertices G) by C1, A1, ZFMISC_1:31;
v in singletons (Vertices G) by A1;
then v in {{}} \/ (singletons (Vertices G)) by XBOOLE_0:def_3;
hence x in ({{}} \/ (singletons (Vertices G))) \/ (Edges G) by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose card v = 2 ; ::_thesis: x in ({{}} \/ (singletons (Vertices G))) \/ (Edges G)
then v in Edges G by A, LEdges;
hence x in ({{}} \/ (singletons (Vertices G))) \/ (Edges G) by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
thus ({{}} \/ (singletons (Vertices G))) \/ (Edges G) c= G ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ({{}} \/ (singletons (Vertices G))) \/ (Edges G) or x in G )
assume x in ({{}} \/ (singletons (Vertices G))) \/ (Edges G) ; ::_thesis: x in G
then A1: ( x in {{}} \/ (singletons (Vertices G)) or x in Edges G ) by XBOOLE_0:def_3;
percases ( x in {{}} or x in singletons (Vertices G) or x in Edges G ) by A1, XBOOLE_0:def_3;
supposeS1: x in {{}} ; ::_thesis: x in G
consider z being set such that
A2: z in G by XBOOLE_0:def_1;
B2: {} c= z by XBOOLE_1:2;
x = {} by S1, TARSKI:def_1;
hence x in G by B2, A2, CLASSES1:def_1; ::_thesis: verum
end;
suppose x in singletons (Vertices G) ; ::_thesis: x in G
then consider f being Subset of (Vertices G) such that
A2: x = f and
B2: f is 1 -element ;
consider v being set such that
C2: v in Vertices G and
D2: f = {v} by B2, BSPACEdef9;
thus x in G by A2, C2, D2, Vertices0; ::_thesis: verum
end;
suppose x in Edges G ; ::_thesis: x in G
hence x in G ; ::_thesis: verum
end;
end;
end;
end;
theorem VoidGV: :: SCMYCIEL:28
for G being SimpleGraph st Vertices G = {} holds
G is void by uVoid1, Lvoid;
theorem SG2: :: SCMYCIEL:29
for G being SimpleGraph
for x being set st x in G & x <> {} & ( for y being set holds
( not x = {y} or not y in Vertices G ) ) holds
x in Edges G
proof
let G be SimpleGraph; ::_thesis: for x being set st x in G & x <> {} & ( for y being set holds
( not x = {y} or not y in Vertices G ) ) holds
x in Edges G
let x be set ; ::_thesis: ( x in G & x <> {} & ( for y being set holds
( not x = {y} or not y in Vertices G ) ) implies x in Edges G )
assume that
A: x in G and
B: x <> {} ; ::_thesis: ( ex y being set st
( x = {y} & y in Vertices G ) or x in Edges G )
x in ({{}} \/ (singletons (Vertices G))) \/ (Edges G) by A, SG0;
then ( x in {{}} \/ (singletons (Vertices G)) or x in Edges G ) by XBOOLE_0:def_3;
then C: ( x in {{}} or x in singletons (Vertices G) or x in Edges G ) by XBOOLE_0:def_3;
percases ( x in singletons (Vertices G) or x in Edges G ) by C, B, TARSKI:def_1;
suppose x in singletons (Vertices G) ; ::_thesis: ( ex y being set st
( x = {y} & y in Vertices G ) or x in Edges G )
then consider f being Subset of (Vertices G) such that
A2: x = f and
B2: f is 1 -element ;
consider v being set such that
C2: v in Vertices G and
D2: f = {v} by B2, BSPACEdef9;
thus ( ex y being set st
( x = {y} & y in Vertices G ) or x in Edges G ) by D2, C2, A2; ::_thesis: verum
end;
suppose x in Edges G ; ::_thesis: ( ex y being set st
( x = {y} & y in Vertices G ) or x in Edges G )
hence ( ex y being set st
( x = {y} & y in Vertices G ) or x in Edges G ) ; ::_thesis: verum
end;
end;
end;
theorem :: SCMYCIEL:30
for G being SimpleGraph
for x being set st Vertices G = {x} holds
G = {{},{x}}
proof
let G be SimpleGraph; ::_thesis: for x being set st Vertices G = {x} holds
G = {{},{x}}
let a be set ; ::_thesis: ( Vertices G = {a} implies G = {{},{a}} )
assume A: Vertices G = {a} ; ::_thesis: G = {{},{a}}
B: now__::_thesis:_not_Edges_G_<>_{}
assume Edges G <> {} ; ::_thesis: contradiction
then consider e being set such that
C: e in Edges G by XBOOLE_0:def_1;
consider x, y being set such that
D: x <> y and
E: x in Vertices G and
F: y in Vertices G and
e = {x,y} by C, SG4;
x = a by E, A, TARSKI:def_1;
hence contradiction by D, F, A, TARSKI:def_1; ::_thesis: verum
end;
C: singletons (Vertices G) = {{a}} by A, Singletons0;
thus G = ({{}} \/ (singletons (Vertices G))) \/ (Edges G) by SG0
.= {{},{a}} by C, B, ENUMSET1:1 ; ::_thesis: verum
end;
theorem size0SG: :: SCMYCIEL:31
for X being set ex G being SimpleGraph st
( G is edgeless & Vertices G = X )
proof
let X be set ; ::_thesis: ex G being SimpleGraph st
( G is edgeless & Vertices G = X )
set G = {{}} \/ (singletons X);
A: {{}} \/ (singletons X) is subset-closed
proof
let x, y be set ; :: according to CLASSES1:def_1 ::_thesis: ( not x in {{}} \/ (singletons X) or not y c= x or y in {{}} \/ (singletons X) )
assume that
A1: x in {{}} \/ (singletons X) and
B1: y c= x ; ::_thesis: y in {{}} \/ (singletons X)
percases ( x in {{}} or x in singletons X ) by A1, XBOOLE_0:def_3;
suppose x in {{}} ; ::_thesis: y in {{}} \/ (singletons X)
then x = {} by TARSKI:def_1;
then y = {} by B1;
then y in {{}} by TARSKI:def_1;
hence y in {{}} \/ (singletons X) by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose x in singletons X ; ::_thesis: y in {{}} \/ (singletons X)
then consider f being Subset of X such that
A2: x = f and
B2: f is 1 -element ;
consider v being set such that
v in X and
D2: f = {v} by B2, BSPACEdef9;
percases ( y = {} or y = {v} ) by B1, A2, D2, ZFMISC_1:33;
suppose y = {} ; ::_thesis: y in {{}} \/ (singletons X)
then y in {{}} by TARSKI:def_1;
hence y in {{}} \/ (singletons X) by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose y = {v} ; ::_thesis: y in {{}} \/ (singletons X)
hence y in {{}} \/ (singletons X) by A1, D2, A2; ::_thesis: verum
end;
end;
end;
end;
end;
B: {{}} \/ (singletons X) is 1 -at_most_dimensional
proof
let x be set ; :: according to SCMYCIEL:def_4 ::_thesis: ( x in {{}} \/ (singletons X) implies card x c= 1 + 1 )
assume Aa: x in {{}} \/ (singletons X) ; ::_thesis: card x c= 1 + 1
percases ( x in {{}} or x in singletons X ) by Aa, XBOOLE_0:def_3;
suppose x in {{}} ; ::_thesis: card x c= 1 + 1
then x = {} by TARSKI:def_1;
hence card x c= 1 + 1 ; ::_thesis: verum
end;
suppose x in singletons X ; ::_thesis: card x c= 1 + 1
then consider f being Subset of X such that
A2: x = f and
B2: f is 1 -element ;
consider v being set such that
v in X and
D2: f = {v} by B2, BSPACEdef9;
card x = 1 by A2, D2, CARD_1:30;
hence card x c= 1 + 1 by NAT_1:39; ::_thesis: verum
end;
end;
end;
reconsider G = {{}} \/ (singletons X) as SimpleGraph by A, B;
take G ; ::_thesis: ( G is edgeless & Vertices G = X )
now__::_thesis:_not_Edges_G_<>_{}
assume Edges G <> {} ; ::_thesis: contradiction
then consider e being set such that
A: e in Edges G by XBOOLE_0:def_1;
B: ( e in G & card e = 2 ) by A, LEdges;
percases ( e in {{}} or e in singletons X ) by A, XBOOLE_0:def_3;
suppose e in {{}} ; ::_thesis: contradiction
hence contradiction by B, CARD_1:27, TARSKI:def_1; ::_thesis: verum
end;
suppose e in singletons X ; ::_thesis: contradiction
then consider f being Subset of X such that
A2: e = f and
B2: f is 1 -element ;
consider v being set such that
v in X and
D2: f = {v} by B2, BSPACEdef9;
thus contradiction by B, A2, D2, CARD_1:30; ::_thesis: verum
end;
end;
end;
hence G is edgeless by Ledgeless; ::_thesis: Vertices G = X
thus Vertices G = X ::_thesis: verum
proof
thus Vertices G c= X :: according to XBOOLE_0:def_10 ::_thesis: X c= Vertices G
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Vertices G or x in X )
assume x in Vertices G ; ::_thesis: x in X
then consider y being set such that
A1: x in y and
B1: y in G by TARSKI:def_4;
percases ( y in {{}} or y in singletons X ) by B1, XBOOLE_0:def_3;
suppose y in {{}} ; ::_thesis: x in X
hence x in X by A1, TARSKI:def_1; ::_thesis: verum
end;
suppose y in singletons X ; ::_thesis: x in X
then consider f being Subset of X such that
A2: y = f and
f is 1 -element ;
thus x in X by A2, A1; ::_thesis: verum
end;
end;
end;
thus X c= Vertices G ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in Vertices G )
assume x in X ; ::_thesis: x in Vertices G
then reconsider f = {x} as Subset of X by ZFMISC_1:31;
f is 1 -element ;
then {x} in singletons X ;
then {x} in G by XBOOLE_0:def_3;
hence x in Vertices G by Vertices0; ::_thesis: verum
end;
end;
end;
definition
let G be SimpleGraph;
let v be Element of Vertices G;
func Adjacent v -> Subset of (Vertices G) means :Ladj: :: SCMYCIEL:def 8
for x being Element of Vertices G holds
( x in it iff {v,x} in Edges G );
existence
ex b1 being Subset of (Vertices G) st
for x being Element of Vertices G holds
( x in b1 iff {v,x} in Edges G )
proof
set A = { x where x is Element of Vertices G : {v,x} in Edges G } ;
{ x where x is Element of Vertices G : {v,x} in Edges G } c= Vertices G
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { x where x is Element of Vertices G : {v,x} in Edges G } or a in Vertices G )
assume a in { x where x is Element of Vertices G : {v,x} in Edges G } ; ::_thesis: a in Vertices G
then consider x being Element of Vertices G such that
A1: a = x and
B1: {v,x} in Edges G ;
thus a in Vertices G by A1, B1, SG5; ::_thesis: verum
end;
then reconsider A = { x where x is Element of Vertices G : {v,x} in Edges G } as Subset of (Vertices G) ;
take A ; ::_thesis: for x being Element of Vertices G holds
( x in A iff {v,x} in Edges G )
let x be Element of Vertices G; ::_thesis: ( x in A iff {v,x} in Edges G )
hereby ::_thesis: ( {v,x} in Edges G implies x in A )
assume x in A ; ::_thesis: {v,x} in Edges G
then consider a being Element of Vertices G such that
A1: x = a and
B1: {v,a} in Edges G ;
thus {v,x} in Edges G by A1, B1; ::_thesis: verum
end;
thus ( {v,x} in Edges G implies x in A ) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being Subset of (Vertices G) st ( for x being Element of Vertices G holds
( x in b1 iff {v,x} in Edges G ) ) & ( for x being Element of Vertices G holds
( x in b2 iff {v,x} in Edges G ) ) holds
b1 = b2
proof
let A1, A2 be Subset of (Vertices G); ::_thesis: ( ( for x being Element of Vertices G holds
( x in A1 iff {v,x} in Edges G ) ) & ( for x being Element of Vertices G holds
( x in A2 iff {v,x} in Edges G ) ) implies A1 = A2 )
assume that
A1: for x being Element of Vertices G holds
( x in A1 iff {v,x} in Edges G ) and
A2: for x being Element of Vertices G holds
( x in A2 iff {v,x} in Edges G ) ; ::_thesis: A1 = A2
thus A1 c= A2 :: according to XBOOLE_0:def_10 ::_thesis: A2 c= A1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A1 or x in A2 )
assume B1: x in A1 ; ::_thesis: x in A2
then {v,x} in Edges G by A1;
hence x in A2 by A2, B1; ::_thesis: verum
end;
thus A2 c= A1 ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A2 or x in A1 )
assume B1: x in A2 ; ::_thesis: x in A1
then {v,x} in Edges G by A2;
hence x in A1 by A1, B1; ::_thesis: verum
end;
end;
end;
:: deftheorem Ladj defines Adjacent SCMYCIEL:def_8_:_
for G being SimpleGraph
for v being Element of Vertices G
for b3 being Subset of (Vertices G) holds
( b3 = Adjacent v iff for x being Element of Vertices G holds
( x in b3 iff {v,x} in Edges G ) );
definition
let X be set ;
mode SimpleGraph of X -> SimpleGraph means :LSGofX: :: SCMYCIEL:def 9
Vertices it = X;
existence
ex b1 being SimpleGraph st Vertices b1 = X
proof
consider G being SimpleGraph such that
G is edgeless and
A: Vertices G = X by size0SG;
take G ; ::_thesis: Vertices G = X
thus Vertices G = X by A; ::_thesis: verum
end;
end;
:: deftheorem LSGofX defines SimpleGraph SCMYCIEL:def_9_:_
for X being set
for b2 being SimpleGraph holds
( b2 is SimpleGraph of X iff Vertices b2 = X );
definition
let X be set ;
func CompleteSGraph X -> SimpleGraph of X equals :: SCMYCIEL:def 10
{ V where V is finite Subset of X : card V <= 2 } ;
coherence
{ V where V is finite Subset of X : card V <= 2 } is SimpleGraph of X
proof
set G = { V where V is finite Subset of X : card V <= 2 } ;
A: { V where V is finite Subset of X : card V <= 2 } is subset-closed
proof
let x, y be set ; :: according to CLASSES1:def_1 ::_thesis: ( not x in { V where V is finite Subset of X : card V <= 2 } or not y c= x or y in { V where V is finite Subset of X : card V <= 2 } )
assume that
A1: x in { V where V is finite Subset of X : card V <= 2 } and
B1: y c= x ; ::_thesis: y in { V where V is finite Subset of X : card V <= 2 }
consider V being finite Subset of X such that
C1: x = V and
D1: card V <= 2 by A1;
reconsider y1 = y as finite Subset of X by C1, B1, XBOOLE_1:1;
card y1 <= card V by B1, C1, NAT_1:43;
then card y1 <= 2 by D1, XXREAL_0:2;
hence y in { V where V is finite Subset of X : card V <= 2 } ; ::_thesis: verum
end;
B: { V where V is finite Subset of X : card V <= 2 } is 1 -at_most_dimensional
proof
let x be set ; :: according to SCMYCIEL:def_4 ::_thesis: ( x in { V where V is finite Subset of X : card V <= 2 } implies card x c= 1 + 1 )
assume x in { V where V is finite Subset of X : card V <= 2 } ; ::_thesis: card x c= 1 + 1
then consider V being finite Subset of X such that
C1: x = V and
D1: card V <= 2 ;
thus card x c= 1 + 1 by C1, D1, NAT_1:39; ::_thesis: verum
end;
Z1: {} c= X by XBOOLE_1:2;
card {} <= 2 ;
then {} in { V where V is finite Subset of X : card V <= 2 } by Z1;
then reconsider G = { V where V is finite Subset of X : card V <= 2 } as SimpleGraph by A, B;
Vertices G = X by CSGLem1;
hence { V where V is finite Subset of X : card V <= 2 } is SimpleGraph of X by LSGofX; ::_thesis: verum
end;
end;
:: deftheorem defines CompleteSGraph SCMYCIEL:def_10_:_
for X being set holds CompleteSGraph X = { V where V is finite Subset of X : card V <= 2 } ;
theorem CSGdef: :: SCMYCIEL:32
for G being SimpleGraph st ( for x, y being set st x in Vertices G & y in Vertices G holds
{x,y} in G ) holds
G = CompleteSGraph (Vertices G)
proof
let G be SimpleGraph; ::_thesis: ( ( for x, y being set st x in Vertices G & y in Vertices G holds
{x,y} in G ) implies G = CompleteSGraph (Vertices G) )
assume A: for x, y being set st x in Vertices G & y in Vertices G holds
{x,y} in G ; ::_thesis: G = CompleteSGraph (Vertices G)
set C = { V where V is finite Subset of (Vertices G) : card V <= 2 } ;
{ V where V is finite Subset of (Vertices G) : card V <= 2 } = G
proof
thus { V where V is finite Subset of (Vertices G) : card V <= 2 } c= G :: according to XBOOLE_0:def_10 ::_thesis: G c= { V where V is finite Subset of (Vertices G) : card V <= 2 }
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { V where V is finite Subset of (Vertices G) : card V <= 2 } or a in G )
assume a in { V where V is finite Subset of (Vertices G) : card V <= 2 } ; ::_thesis: a in G
then consider V being finite Subset of (Vertices G) such that
B1: a = V and
C1: card V <= 2 ;
percases ( card V = 0 or card V = 1 or card V = 2 ) by C1, NAT_1:26;
suppose card V = 0 ; ::_thesis: a in G
then V = {} ;
hence a in G by B1, SG1; ::_thesis: verum
end;
suppose card V = 1 ; ::_thesis: a in G
then consider c being set such that
B2: V = {c} by CARD_2:42;
c in V by B2, TARSKI:def_1;
then {c,c} in G by A;
hence a in G by B2, B1, ENUMSET1:29; ::_thesis: verum
end;
suppose card V = 2 ; ::_thesis: a in G
then consider c, d being set such that
c <> d and
B2: V = {c,d} by CARD_2:60;
( c in V & d in V ) by B2, TARSKI:def_2;
hence a in G by A, B2, B1; ::_thesis: verum
end;
end;
end;
thus G c= { V where V is finite Subset of (Vertices G) : card V <= 2 } ::_thesis: verum
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in G or a in { V where V is finite Subset of (Vertices G) : card V <= 2 } )
assume A1: a in G ; ::_thesis: a in { V where V is finite Subset of (Vertices G) : card V <= 2 }
then reconsider aa = a as finite set ;
B1: card aa <= 1 + 1 by A1, Lnatmost1;
a c= union G by A1, ZFMISC_1:74;
hence a in { V where V is finite Subset of (Vertices G) : card V <= 2 } by B1; ::_thesis: verum
end;
end;
hence G = CompleteSGraph (Vertices G) ; ::_thesis: verum
end;
registration
let X be finite set ;
cluster CompleteSGraph X -> finite ;
correctness
coherence
CompleteSGraph X is finite ;
proof
set G = CompleteSGraph X;
CompleteSGraph X c= bool X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in CompleteSGraph X or x in bool X )
assume x in CompleteSGraph X ; ::_thesis: x in bool X
then consider V being finite Subset of X such that
A: x = V and
card V <= 2 ;
thus x in bool X by A; ::_thesis: verum
end;
hence CompleteSGraph X is finite ; ::_thesis: verum
end;
end;
theorem CSG1a: :: SCMYCIEL:33
for X, x being set st x in X holds
{x} in CompleteSGraph X
proof
let X, x be set ; ::_thesis: ( x in X implies {x} in CompleteSGraph X )
assume A: x in X ; ::_thesis: {x} in CompleteSGraph X
B: {x} c= X by A, ZFMISC_1:31;
C: card {x} = 1 by CARD_1:30;
thus {x} in CompleteSGraph X by C, B; ::_thesis: verum
end;
theorem CSG1: :: SCMYCIEL:34
for X, x, y being set st x in X & y in X holds
{x,y} in CompleteSGraph X
proof
let X be set ; ::_thesis: for x, y being set st x in X & y in X holds
{x,y} in CompleteSGraph X
let x, y be set ; ::_thesis: ( x in X & y in X implies {x,y} in CompleteSGraph X )
assume that
A: x in X and
Aa: y in X ; ::_thesis: {x,y} in CompleteSGraph X
B: {x,y} c= X by A, Aa, ZFMISC_1:32;
C: card {x,y} <= 2 by CARD_2:50;
thus {x,y} in CompleteSGraph X by C, B; ::_thesis: verum
end;
theorem eCSG0: :: SCMYCIEL:35
CompleteSGraph {} = {{}}
proof
for x, y being set st x in union {{}} & y in union {{}} holds
{x,y} in {{}} ;
hence CompleteSGraph {} = {{}} by CSGdef; ::_thesis: verum
end;
theorem P1: :: SCMYCIEL:36
for x being set holds CompleteSGraph {x} = {{},{x}}
proof
let x be set ; ::_thesis: CompleteSGraph {x} = {{},{x}}
thus CompleteSGraph {x} c= {{},{x}} :: according to XBOOLE_0:def_10 ::_thesis: {{},{x}} c= CompleteSGraph {x}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in CompleteSGraph {x} or a in {{},{x}} )
assume a in CompleteSGraph {x} ; ::_thesis: a in {{},{x}}
then consider V being finite Subset of {x} such that
A: a = V and
card V <= 2 ;
( a = {} or a = {x} ) by A, ZFMISC_1:33;
hence a in {{},{x}} by TARSKI:def_2; ::_thesis: verum
end;
Aa: {x} = Vertices (CompleteSGraph {x}) by CSGLem1;
Ab: x in {x} by TARSKI:def_1;
thus {{},{x}} c= CompleteSGraph {x} ::_thesis: verum
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in {{},{x}} or a in CompleteSGraph {x} )
assume a in {{},{x}} ; ::_thesis: a in CompleteSGraph {x}
then ( a = {} or a = {x} ) by TARSKI:def_2;
hence a in CompleteSGraph {x} by Aa, Ab, SG1, Vertices0; ::_thesis: verum
end;
end;
theorem P2: :: SCMYCIEL:37
for x, y being set holds CompleteSGraph {x,y} = {{},{x},{y},{x,y}}
proof
let x, y be set ; ::_thesis: CompleteSGraph {x,y} = {{},{x},{y},{x,y}}
thus CompleteSGraph {x,y} c= {{},{x},{y},{x,y}} :: according to XBOOLE_0:def_10 ::_thesis: {{},{x},{y},{x,y}} c= CompleteSGraph {x,y}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in CompleteSGraph {x,y} or a in {{},{x},{y},{x,y}} )
assume a in CompleteSGraph {x,y} ; ::_thesis: a in {{},{x},{y},{x,y}}
then consider V being finite Subset of {x,y} such that
A: a = V and
card V <= 2 ;
( a = {} or a = {x} or a = {y} or a = {x,y} ) by A, ZFMISC_1:36;
hence a in {{},{x},{y},{x,y}} by ENUMSET1:def_2; ::_thesis: verum
end;
Aa: {x,y} = Vertices (CompleteSGraph {x,y}) by CSGLem1;
Ab: x in {x,y} by TARSKI:def_2;
Ac: y in {x,y} by TARSKI:def_2;
Ad: card {x,y} <= 2 by CARD_2:50;
Ae: {x,y} c= {x,y} ;
thus {{},{x},{y},{x,y}} c= CompleteSGraph {x,y} ::_thesis: verum
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in {{},{x},{y},{x,y}} or a in CompleteSGraph {x,y} )
assume a in {{},{x},{y},{x,y}} ; ::_thesis: a in CompleteSGraph {x,y}
then ( a = {} or a = {x} or a = {y} or a = {x,y} ) by ENUMSET1:def_2;
hence a in CompleteSGraph {x,y} by Aa, Ab, Ac, Ad, Ae, SG1, Vertices0; ::_thesis: verum
end;
end;
theorem :: SCMYCIEL:38
for X, Y being set st X c= Y holds
CompleteSGraph X c= CompleteSGraph Y
proof
let X, Y be set ; ::_thesis: ( X c= Y implies CompleteSGraph X c= CompleteSGraph Y )
assume A: X c= Y ; ::_thesis: CompleteSGraph X c= CompleteSGraph Y
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in CompleteSGraph X or a in CompleteSGraph Y )
assume a in CompleteSGraph X ; ::_thesis: a in CompleteSGraph Y
then consider V being finite Subset of X such that
A1: a = V and
B1: card V <= 2 ;
V is Subset of Y by A, XBOOLE_1:1;
hence a in CompleteSGraph Y by A1, B1; ::_thesis: verum
end;
theorem CSGsingle: :: SCMYCIEL:39
for G being SimpleGraph
for x being set st x in Vertices G holds
CompleteSGraph {x} c= G
proof
let G be SimpleGraph; ::_thesis: for x being set st x in Vertices G holds
CompleteSGraph {x} c= G
let x be set ; ::_thesis: ( x in Vertices G implies CompleteSGraph {x} c= G )
assume A: x in Vertices G ; ::_thesis: CompleteSGraph {x} c= G
B: CompleteSGraph {x} = {{},{x}} by P1;
C: {} in G by SG1;
D: {x} in G by A, Vertices0;
thus CompleteSGraph {x} c= G by B, C, D, ZFMISC_1:32; ::_thesis: verum
end;
registration
let G be SimpleGraph;
cluster finite-membered 1 -at_most_dimensional SimpleGraph-like for Element of bool G;
existence
ex b1 being Subset of G st b1 is SimpleGraph-like
proof
G c= G ;
then reconsider H = G as Subset of G ;
take H ; ::_thesis: H is SimpleGraph-like
thus H is SimpleGraph-like ; ::_thesis: verum
end;
end;
definition
let G be SimpleGraph;
mode Subgraph of G is SimpleGraph-like Subset of G;
end;
CisSG: for G being SimpleGraph holds (CompleteSGraph (Vertices G)) \ (Edges G) is SimpleGraph
proof
let G be SimpleGraph; ::_thesis: (CompleteSGraph (Vertices G)) \ (Edges G) is SimpleGraph
set CSGVG = CompleteSGraph (Vertices G);
set C = (CompleteSGraph (Vertices G)) \ (Edges G);
Z1: {} in CompleteSGraph (Vertices G) by SG1;
now__::_thesis:_not_{}_in_Edges_G
assume {} in Edges G ; ::_thesis: contradiction
then consider x, y being set such that
( x <> y & x in Vertices G & y in Vertices G ) and
A1: {} = {x,y} by SG4;
thus contradiction by A1; ::_thesis: verum
end;
then reconsider C = (CompleteSGraph (Vertices G)) \ (Edges G) as non empty set by Z1, XBOOLE_0:def_5;
C is subset-closed
proof
let X, Y be set ; :: according to CLASSES1:def_1 ::_thesis: ( not X in C or not Y c= X or Y in C )
assume that
A1: X in C and
B1: Y c= X ; ::_thesis: Y in C
assume Y nin C ; ::_thesis: contradiction
then C1: ( Y nin CompleteSGraph (Vertices G) or Y in Edges G ) by XBOOLE_0:def_5;
D1: ( X in CompleteSGraph (Vertices G) & not X in Edges G ) by A1, XBOOLE_0:def_5;
E1: Y in Edges G by B1, C1, D1, CLASSES1:def_1;
F1: card Y = 2 by E1, LEdges;
reconsider X = X as finite set by A1;
G1: card X <= 1 + 1 by A1, Lnatmost1;
H1: 2 <= card X by F1, B1, NAT_1:43;
card X = 2 by G1, H1, XXREAL_0:1;
hence contradiction by D1, C1, B1, F1, CARD_FIN:1; ::_thesis: verum
end;
hence (CompleteSGraph (Vertices G)) \ (Edges G) is SimpleGraph ; ::_thesis: verum
end;
Compl1: for G being SimpleGraph holds Vertices G = Vertices ((CompleteSGraph (Vertices G)) \ (Edges G))
proof
let G be SimpleGraph; ::_thesis: Vertices G = Vertices ((CompleteSGraph (Vertices G)) \ (Edges G))
set CG = (CompleteSGraph (Vertices G)) \ (Edges G);
Aa: (CompleteSGraph (Vertices G)) \ (Edges G) is SimpleGraph by CisSG;
now__::_thesis:_for_a_being_set_holds_
(_(_a_in_Vertices_G_implies_a_in_Vertices_((CompleteSGraph_(Vertices_G))_\_(Edges_G))_)_&_(_a_in_Vertices_((CompleteSGraph_(Vertices_G))_\_(Edges_G))_implies_a_in_Vertices_G_)_)
let a be set ; ::_thesis: ( ( a in Vertices G implies a in Vertices ((CompleteSGraph (Vertices G)) \ (Edges G)) ) & ( a in Vertices ((CompleteSGraph (Vertices G)) \ (Edges G)) implies a in Vertices G ) )
hereby ::_thesis: ( a in Vertices ((CompleteSGraph (Vertices G)) \ (Edges G)) implies a in Vertices G )
assume a in Vertices G ; ::_thesis: a in Vertices ((CompleteSGraph (Vertices G)) \ (Edges G))
then A1: {a} in CompleteSGraph (Vertices G) by CSG1a;
now__::_thesis:_not_{a}_in_Edges_G
assume {a} in Edges G ; ::_thesis: contradiction
then {a,a} in Edges G by ENUMSET1:29;
hence contradiction by SG5; ::_thesis: verum
end;
then {a} in (CompleteSGraph (Vertices G)) \ (Edges G) by A1, XBOOLE_0:def_5;
hence a in Vertices ((CompleteSGraph (Vertices G)) \ (Edges G)) by Aa, Vertices0; ::_thesis: verum
end;
assume a in Vertices ((CompleteSGraph (Vertices G)) \ (Edges G)) ; ::_thesis: a in Vertices G
then {a} in (CompleteSGraph (Vertices G)) \ (Edges G) by Aa, Vertices0;
then a in Vertices (CompleteSGraph (Vertices G)) by Vertices0;
hence a in Vertices G by CSGLem1; ::_thesis: verum
end;
hence Vertices G = Vertices ((CompleteSGraph (Vertices G)) \ (Edges G)) by TARSKI:1; ::_thesis: verum
end;
Compl1a: for G being SimpleGraph
for x, y being set st x <> y & x in Vertices G & y in Vertices G holds
( {x,y} in Edges G iff {x,y} nin Edges ((CompleteSGraph (Vertices G)) \ (Edges G)) )
proof
let G be SimpleGraph; ::_thesis: for x, y being set st x <> y & x in Vertices G & y in Vertices G holds
( {x,y} in Edges G iff {x,y} nin Edges ((CompleteSGraph (Vertices G)) \ (Edges G)) )
let x, y be set ; ::_thesis: ( x <> y & x in Vertices G & y in Vertices G implies ( {x,y} in Edges G iff {x,y} nin Edges ((CompleteSGraph (Vertices G)) \ (Edges G)) ) )
assume that
A: x <> y and
B: x in Vertices G and
C: y in Vertices G ; ::_thesis: ( {x,y} in Edges G iff {x,y} nin Edges ((CompleteSGraph (Vertices G)) \ (Edges G)) )
set CG = (CompleteSGraph (Vertices G)) \ (Edges G);
thus ( {x,y} in Edges G implies {x,y} nin Edges ((CompleteSGraph (Vertices G)) \ (Edges G)) ) by XBOOLE_0:def_5; ::_thesis: ( {x,y} nin Edges ((CompleteSGraph (Vertices G)) \ (Edges G)) implies {x,y} in Edges G )
assume D: {x,y} nin Edges ((CompleteSGraph (Vertices G)) \ (Edges G)) ; ::_thesis: {x,y} in Edges G
assume E: {x,y} nin Edges G ; ::_thesis: contradiction
{x,y} in CompleteSGraph (Vertices G) by B, C, CSG1;
then {x,y} in (CompleteSGraph (Vertices G)) \ (Edges G) by E, XBOOLE_0:def_5;
hence contradiction by D, A, SG4a; ::_thesis: verum
end;
Compl2: for G, CG being SimpleGraph st CG = (CompleteSGraph (Vertices G)) \ (Edges G) holds
(CompleteSGraph (Vertices CG)) \ (Edges CG) = G
proof
let G, CG be SimpleGraph; ::_thesis: ( CG = (CompleteSGraph (Vertices G)) \ (Edges G) implies (CompleteSGraph (Vertices CG)) \ (Edges CG) = G )
assume AAa: CG = (CompleteSGraph (Vertices G)) \ (Edges G) ; ::_thesis: (CompleteSGraph (Vertices CG)) \ (Edges CG) = G
set CCG = (CompleteSGraph (Vertices CG)) \ (Edges CG);
A: Vertices G = Vertices CG by AAa, Compl1;
Aa: Vertices CG = Vertices ((CompleteSGraph (Vertices CG)) \ (Edges CG)) by Compl1;
(CompleteSGraph (Vertices CG)) \ (Edges CG) is SimpleGraph by CisSG;
then B: (CompleteSGraph (Vertices CG)) \ (Edges CG) = ({{}} \/ (singletons (Vertices ((CompleteSGraph (Vertices CG)) \ (Edges CG))))) \/ (Edges ((CompleteSGraph (Vertices CG)) \ (Edges CG))) by SG0;
D: G = ({{}} \/ (singletons (Vertices G))) \/ (Edges G) by SG0;
now__::_thesis:_for_a_being_set_holds_
(_(_a_in_Edges_((CompleteSGraph_(Vertices_CG))_\_(Edges_CG))_implies_a_in_Edges_G_)_&_(_a_in_Edges_G_implies_a_in_Edges_((CompleteSGraph_(Vertices_CG))_\_(Edges_CG))_)_)
let a be set ; ::_thesis: ( ( a in Edges ((CompleteSGraph (Vertices CG)) \ (Edges CG)) implies a in Edges G ) & ( a in Edges G implies a in Edges ((CompleteSGraph (Vertices CG)) \ (Edges CG)) ) )
hereby ::_thesis: ( a in Edges G implies a in Edges ((CompleteSGraph (Vertices CG)) \ (Edges CG)) )
assume S1: a in Edges ((CompleteSGraph (Vertices CG)) \ (Edges CG)) ; ::_thesis: a in Edges G
then consider x, y being set such that
A0: x <> y and
A1: ( x in Vertices ((CompleteSGraph (Vertices CG)) \ (Edges CG)) & y in Vertices ((CompleteSGraph (Vertices CG)) \ (Edges CG)) ) and
B1: a = {x,y} by SG4;
{x,y} nin Edges CG by A0, Aa, S1, B1, A1, Compl1a;
hence a in Edges G by A0, Aa, A, A1, B1, AAa, Compl1a; ::_thesis: verum
end;
assume S1: a in Edges G ; ::_thesis: a in Edges ((CompleteSGraph (Vertices CG)) \ (Edges CG))
then consider x, y being set such that
A0: x <> y and
A1: ( x in Vertices G & y in Vertices G ) and
B1: a = {x,y} by SG4;
{x,y} nin Edges CG by A0, S1, B1, A1, AAa, Compl1a;
hence a in Edges ((CompleteSGraph (Vertices CG)) \ (Edges CG)) by A0, A, A1, B1, Compl1a; ::_thesis: verum
end;
hence (CompleteSGraph (Vertices CG)) \ (Edges CG) = G by A, Aa, B, D, TARSKI:1; ::_thesis: verum
end;
definition
let G be SimpleGraph;
func Complement G -> SimpleGraph equals :: SCMYCIEL:def 11
(CompleteSGraph (Vertices G)) \ (Edges G);
correctness
coherence
(CompleteSGraph (Vertices G)) \ (Edges G) is SimpleGraph;
by CisSG;
involutiveness
for b1, b2 being SimpleGraph st b1 = (CompleteSGraph (Vertices b2)) \ (Edges b2) holds
b2 = (CompleteSGraph (Vertices b1)) \ (Edges b1) by Compl2;
end;
:: deftheorem defines Complement SCMYCIEL:def_11_:_
for G being SimpleGraph holds Complement G = (CompleteSGraph (Vertices G)) \ (Edges G);
theorem :: SCMYCIEL:40
for G being SimpleGraph holds Vertices G = Vertices (Complement G) by Compl1;
theorem :: SCMYCIEL:41
for G being SimpleGraph
for x, y being set st x <> y & x in Vertices G & y in Vertices G holds
( {x,y} in Edges G iff {x,y} nin Edges (Complement G) ) by Compl1a;
begin
definition
let G be SimpleGraph;
let L be set ;
funcG SubgraphInducedBy L -> Subset of G equals :: SCMYCIEL:def 12
G /\ (bool L);
coherence
G /\ (bool L) is Subset of G by XBOOLE_1:17;
end;
:: deftheorem defines SubgraphInducedBy SCMYCIEL:def_12_:_
for G being SimpleGraph
for L being set holds G SubgraphInducedBy L = G /\ (bool L);
registration
let G be SimpleGraph;
let L be set ;
clusterG SubgraphInducedBy L -> SimpleGraph-like ;
coherence
G SubgraphInducedBy L is SimpleGraph-like
proof
set S = G /\ (bool L);
Aa: {} in G by SG1;
{} c= L by XBOOLE_1:2;
then reconsider S = G /\ (bool L) as non empty set by Aa, XBOOLE_0:def_4;
S is subset-closed by XBOOLE_0:def_4, CLASSES1:def_1, XBOOLE_1:1;
hence G SubgraphInducedBy L is SimpleGraph-like ; ::_thesis: verum
end;
end;
theorem :: SCMYCIEL:42
for G being SimpleGraph holds G = G SubgraphInducedBy (Vertices G)
proof
let G be SimpleGraph; ::_thesis: G = G SubgraphInducedBy (Vertices G)
thus G c= G SubgraphInducedBy (Vertices G) :: according to XBOOLE_0:def_10 ::_thesis: G SubgraphInducedBy (Vertices G) c= G
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in G or x in G SubgraphInducedBy (Vertices G) )
assume A1: x in G ; ::_thesis: x in G SubgraphInducedBy (Vertices G)
B1: x c= union G by A1, ZFMISC_1:74;
thus x in G SubgraphInducedBy (Vertices G) by A1, B1, XBOOLE_0:def_4; ::_thesis: verum
end;
thus G SubgraphInducedBy (Vertices G) c= G ; ::_thesis: verum
end;
theorem Sub3a: :: SCMYCIEL:43
for G being SimpleGraph
for L being set holds G SubgraphInducedBy L = G SubgraphInducedBy (L /\ (Vertices G))
proof
let G be SimpleGraph; ::_thesis: for L being set holds G SubgraphInducedBy L = G SubgraphInducedBy (L /\ (Vertices G))
let L be set ; ::_thesis: G SubgraphInducedBy L = G SubgraphInducedBy (L /\ (Vertices G))
thus G SubgraphInducedBy L c= G SubgraphInducedBy (L /\ (Vertices G)) :: according to XBOOLE_0:def_10 ::_thesis: G SubgraphInducedBy (L /\ (Vertices G)) c= G SubgraphInducedBy L
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in G SubgraphInducedBy L or x in G SubgraphInducedBy (L /\ (Vertices G)) )
assume A1: x in G SubgraphInducedBy L ; ::_thesis: x in G SubgraphInducedBy (L /\ (Vertices G))
then C1a: x in bool L by XBOOLE_0:def_4;
D1: x c= Vertices G by A1, ZFMISC_1:74;
E1: x c= L /\ (Vertices G) by C1a, D1, XBOOLE_1:19;
thus x in G SubgraphInducedBy (L /\ (Vertices G)) by A1, E1, XBOOLE_0:def_4; ::_thesis: verum
end;
thus G SubgraphInducedBy (L /\ (Vertices G)) c= G SubgraphInducedBy L ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in G SubgraphInducedBy (L /\ (Vertices G)) or x in G SubgraphInducedBy L )
assume A1: x in G SubgraphInducedBy (L /\ (Vertices G)) ; ::_thesis: x in G SubgraphInducedBy L
then x in bool (L /\ (Vertices G)) by XBOOLE_0:def_4;
then D1: x c= L by XBOOLE_1:18;
thus x in G SubgraphInducedBy L by A1, D1, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
registration
let G be finite SimpleGraph;
let L be set ;
clusterG SubgraphInducedBy L -> finite ;
coherence
G SubgraphInducedBy L is finite ;
end;
registration
let G be SimpleGraph;
let L be finite set ;
clusterG SubgraphInducedBy L -> finite ;
coherence
G SubgraphInducedBy L is finite ;
end;
theorem Sub0b: :: SCMYCIEL:44
for G, H being SimpleGraph st G c= H holds
G c= H SubgraphInducedBy (Vertices G)
proof
let G, H be SimpleGraph; ::_thesis: ( G c= H implies G c= H SubgraphInducedBy (Vertices G) )
assume A: G c= H ; ::_thesis: G c= H SubgraphInducedBy (Vertices G)
set L = Vertices G;
let g be set ; :: according to TARSKI:def_3 ::_thesis: ( not g in G or g in H SubgraphInducedBy (Vertices G) )
assume A1: g in G ; ::_thesis: g in H SubgraphInducedBy (Vertices G)
g c= Vertices G by A1, ZFMISC_1:74;
hence g in H SubgraphInducedBy (Vertices G) by A1, A, XBOOLE_0:def_4; ::_thesis: verum
end;
Sub1: for G being SimpleGraph
for L, x being set st x in Vertices (G SubgraphInducedBy L) holds
x in L
proof
let G be SimpleGraph; ::_thesis: for L, x being set st x in Vertices (G SubgraphInducedBy L) holds
x in L
let L be set ; ::_thesis: for x being set st x in Vertices (G SubgraphInducedBy L) holds
x in L
set S = G /\ (bool L);
let x be set ; ::_thesis: ( x in Vertices (G SubgraphInducedBy L) implies x in L )
assume A: x in Vertices (G SubgraphInducedBy L) ; ::_thesis: x in L
consider Y being set such that
B: x in Y and
C: Y in G /\ (bool L) by A, TARSKI:def_4;
set y = Y;
Y in bool L by C, XBOOLE_0:def_4;
hence x in L by B; ::_thesis: verum
end;
Sub3: for G being SimpleGraph
for L, x being set st x in L & x in Vertices G holds
x in Vertices (G SubgraphInducedBy L)
proof
let G be SimpleGraph; ::_thesis: for L, x being set st x in L & x in Vertices G holds
x in Vertices (G SubgraphInducedBy L)
let L, x be set ; ::_thesis: ( x in L & x in Vertices G implies x in Vertices (G SubgraphInducedBy L) )
assume that
A: x in L and
B: x in Vertices G ; ::_thesis: x in Vertices (G SubgraphInducedBy L)
C: {x} in G by B, Vertices0;
D: {x} c= L by A, ZFMISC_1:31;
E: {x} in G SubgraphInducedBy L by C, D, XBOOLE_0:def_4;
thus x in Vertices (G SubgraphInducedBy L) by E, Vertices0; ::_thesis: verum
end;
theorem Sub5: :: SCMYCIEL:45
for G being SimpleGraph
for L being set holds Vertices (G SubgraphInducedBy L) = (Vertices G) /\ L
proof
let G be SimpleGraph; ::_thesis: for L being set holds Vertices (G SubgraphInducedBy L) = (Vertices G) /\ L
let L be set ; ::_thesis: Vertices (G SubgraphInducedBy L) = (Vertices G) /\ L
set S = G SubgraphInducedBy L;
set uS = union (G SubgraphInducedBy L);
set uG = union G;
union (G /\ (bool L)) c= (union G) /\ (union (bool L)) by ZFMISC_1:79;
hence union (G SubgraphInducedBy L) c= (union G) /\ L by ZFMISC_1:81; :: according to XBOOLE_0:def_10 ::_thesis: (Vertices G) /\ L c= Vertices (G SubgraphInducedBy L)
thus (union G) /\ L c= union (G SubgraphInducedBy L) ::_thesis: verum
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (union G) /\ L or a in union (G SubgraphInducedBy L) )
assume a in (union G) /\ L ; ::_thesis: a in union (G SubgraphInducedBy L)
then ( a in union G & a in L ) by XBOOLE_0:def_4;
hence a in union (G SubgraphInducedBy L) by Sub3; ::_thesis: verum
end;
end;
Sub0c: for G being SimpleGraph
for L being set st L c= Vertices G holds
Vertices (G SubgraphInducedBy L) = L
proof
let G be SimpleGraph; ::_thesis: for L being set st L c= Vertices G holds
Vertices (G SubgraphInducedBy L) = L
let L be set ; ::_thesis: ( L c= Vertices G implies Vertices (G SubgraphInducedBy L) = L )
assume A: L c= union G ; ::_thesis: Vertices (G SubgraphInducedBy L) = L
thus Vertices (G SubgraphInducedBy L) = (Vertices G) /\ L by Sub5
.= L by A, XBOOLE_1:28 ; ::_thesis: verum
end;
Sub6: for G being SimpleGraph
for L, x, y being set st x in L & y in L & {x,y} in G holds
{x,y} in G SubgraphInducedBy L
proof
let G be SimpleGraph; ::_thesis: for L, x, y being set st x in L & y in L & {x,y} in G holds
{x,y} in G SubgraphInducedBy L
let L, x, y be set ; ::_thesis: ( x in L & y in L & {x,y} in G implies {x,y} in G SubgraphInducedBy L )
assume that
A: x in L and
B: y in L and
C: {x,y} in G ; ::_thesis: {x,y} in G SubgraphInducedBy L
{x,y} c= L by A, B, ZFMISC_1:32;
hence {x,y} in G SubgraphInducedBy L by C, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem SingleSub: :: SCMYCIEL:46
for G being SimpleGraph
for x being set st x in Vertices G holds
G SubgraphInducedBy {x} = {{},{x}}
proof
let G be SimpleGraph; ::_thesis: for x being set st x in Vertices G holds
G SubgraphInducedBy {x} = {{},{x}}
let x be set ; ::_thesis: ( x in Vertices G implies G SubgraphInducedBy {x} = {{},{x}} )
assume A: x in Vertices G ; ::_thesis: G SubgraphInducedBy {x} = {{},{x}}
set Gx = G SubgraphInducedBy {x};
thus G SubgraphInducedBy {x} c= {{},{x}} :: according to XBOOLE_0:def_10 ::_thesis: {{},{x}} c= G SubgraphInducedBy {x}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in G SubgraphInducedBy {x} or a in {{},{x}} )
assume a in G SubgraphInducedBy {x} ; ::_thesis: a in {{},{x}}
then a in bool {x} by XBOOLE_0:def_4;
then ( a = {} or a = {x} ) by ZFMISC_1:33;
hence a in {{},{x}} by TARSKI:def_2; ::_thesis: verum
end;
thus {{},{x}} c= G SubgraphInducedBy {x} ::_thesis: verum
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in {{},{x}} or a in G SubgraphInducedBy {x} )
A1: ( {} in G & {x} in G ) by SG1, A, Vertices0;
assume a in {{},{x}} ; ::_thesis: a in G SubgraphInducedBy {x}
then B1: ( a = {} or a = {x} ) by TARSKI:def_2;
then a c= {x} by ZFMISC_1:33;
hence a in G SubgraphInducedBy {x} by A1, B1, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
begin
definition
let G be SimpleGraph;
attrG is clique means :Lclique: :: SCMYCIEL:def 13
G = CompleteSGraph (Vertices G);
end;
:: deftheorem Lclique defines clique SCMYCIEL:def_13_:_
for G being SimpleGraph holds
( G is clique iff G = CompleteSGraph (Vertices G) );
theorem Lclique1: :: SCMYCIEL:47
for G being SimpleGraph st ( for x, y being set st x <> y & x in Vertices G & y in Vertices G holds
{x,y} in Edges G ) holds
G is clique
proof
let G be SimpleGraph; ::_thesis: ( ( for x, y being set st x <> y & x in Vertices G & y in Vertices G holds
{x,y} in Edges G ) implies G is clique )
assume A: for x, y being set st x <> y & x in Vertices G & y in Vertices G holds
{x,y} in Edges G ; ::_thesis: G is clique
now__::_thesis:_for_x,_y_being_set_st_x_in_Vertices_G_&_y_in_Vertices_G_holds_
{x,y}_in_G
let x, y be set ; ::_thesis: ( x in Vertices G & y in Vertices G implies {b1,b2} in G )
assume that
A1: x in Vertices G and
B1: y in Vertices G ; ::_thesis: {b1,b2} in G
percases ( x <> y or x = y ) ;
suppose x <> y ; ::_thesis: {b1,b2} in G
then {x,y} in Edges G by A1, B1, A;
hence {x,y} in G ; ::_thesis: verum
end;
suppose x = y ; ::_thesis: {b1,b2} in G
then {x,y} = {x} by ENUMSET1:29;
hence {x,y} in G by A1, Vertices0; ::_thesis: verum
end;
end;
end;
then G = CompleteSGraph (Vertices G) by CSGdef;
hence G is clique by Lclique; ::_thesis: verum
end;
theorem eclique: :: SCMYCIEL:48
{{}} is clique
proof
thus {{}} = CompleteSGraph (Vertices {{}}) by eCSG0; :: according to SCMYCIEL:def_13 ::_thesis: verum
end;
registration
cluster non empty finite-membered V233() V267() subset-closed 1 -at_most_dimensional SimpleGraph-like clique for set ;
existence
ex b1 being SimpleGraph st b1 is clique by eclique;
let G be SimpleGraph;
cluster non empty finite-membered V233() V267() subset-closed 1 -at_most_dimensional SimpleGraph-like clique for Element of bool G;
existence
ex b1 being Subgraph of G st b1 is clique
proof
{} in G by SG1;
then reconsider S = {{}} as Subgraph of G by ZFMISC_1:31;
take S ; ::_thesis: S is clique
thus S is clique by eclique; ::_thesis: verum
end;
end;
definition
let G be SimpleGraph;
mode Clique of G is clique Subgraph of G;
end;
theorem cliqueCSG0: :: SCMYCIEL:49
for X being set holds CompleteSGraph X is clique
proof
let X be set ; ::_thesis: CompleteSGraph X is clique
CompleteSGraph X = CompleteSGraph (Vertices (CompleteSGraph X)) by CSGLem1;
hence CompleteSGraph X is clique by Lclique; ::_thesis: verum
end;
registration
let X be set ;
cluster CompleteSGraph X -> clique ;
correctness
coherence
CompleteSGraph X is clique ;
by cliqueCSG0;
end;
theorem SingleClique: :: SCMYCIEL:50
for G being SimpleGraph
for x being set st x in Vertices G holds
{{},{x}} is Clique of G
proof
let G be SimpleGraph; ::_thesis: for x being set st x in Vertices G holds
{{},{x}} is Clique of G
let x be set ; ::_thesis: ( x in Vertices G implies {{},{x}} is Clique of G )
assume A: x in Vertices G ; ::_thesis: {{},{x}} is Clique of G
set C = CompleteSGraph {x};
B: CompleteSGraph {x} = {{},{x}} by P1;
thus {{},{x}} is Clique of G by B, A, CSGsingle; ::_thesis: verum
end;
theorem Cliqueon2: :: SCMYCIEL:51
for G being SimpleGraph
for x, y being set st x in Vertices G & y in Vertices G & {x,y} in G holds
{{},{x},{y},{x,y}} is Clique of G
proof
let G be SimpleGraph; ::_thesis: for x, y being set st x in Vertices G & y in Vertices G & {x,y} in G holds
{{},{x},{y},{x,y}} is Clique of G
let x, y be set ; ::_thesis: ( x in Vertices G & y in Vertices G & {x,y} in G implies {{},{x},{y},{x,y}} is Clique of G )
assume that
A: x in Vertices G and
B: y in Vertices G and
AB: {x,y} in G ; ::_thesis: {{},{x},{y},{x,y}} is Clique of G
set C = CompleteSGraph {x,y};
D: CompleteSGraph {x,y} = {{},{x},{y},{x,y}} by P2;
CompleteSGraph {x,y} c= G
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in CompleteSGraph {x,y} or a in G )
assume A1: a in CompleteSGraph {x,y} ; ::_thesis: a in G
percases ( a = {} or a = {x} or a = {y} or a = {x,y} ) by A1, D, ENUMSET1:def_2;
suppose a = {} ; ::_thesis: a in G
hence a in G by SG1; ::_thesis: verum
end;
suppose a = {x} ; ::_thesis: a in G
hence a in G by A, Vertices0; ::_thesis: verum
end;
suppose a = {y} ; ::_thesis: a in G
hence a in G by B, Vertices0; ::_thesis: verum
end;
suppose a = {x,y} ; ::_thesis: a in G
hence a in G by AB; ::_thesis: verum
end;
end;
end;
hence {{},{x},{y},{x,y}} is Clique of G by P2; ::_thesis: verum
end;
registration
let G be SimpleGraph;
cluster non empty finite finite-membered V233() V267() subset-closed 1 -at_most_dimensional SimpleGraph-like clique for Element of bool G;
existence
ex b1 being Clique of G st b1 is finite
proof
{} in G by SG1;
then reconsider C = {{}} as Subgraph of G by ZFMISC_1:31;
C is clique by eclique;
hence ex b1 being Clique of G st b1 is finite ; ::_thesis: verum
end;
end;
theorem CliqueS: :: SCMYCIEL:52
for G being SimpleGraph
for x being set st x in union G holds
ex C being finite Clique of G st Vertices C = {x}
proof
let G be SimpleGraph; ::_thesis: for x being set st x in union G holds
ex C being finite Clique of G st Vertices C = {x}
let x be set ; ::_thesis: ( x in union G implies ex C being finite Clique of G st Vertices C = {x} )
assume A: x in union G ; ::_thesis: ex C being finite Clique of G st Vertices C = {x}
set C = CompleteSGraph {x};
B: CompleteSGraph {x} = {{},{x}} by P1;
reconsider C = CompleteSGraph {x} as finite Clique of G by A, B, SingleClique;
take C ; ::_thesis: Vertices C = {x}
thus Vertices C = {x} by B, SingleVertices; ::_thesis: verum
end;
theorem Clique2a: :: SCMYCIEL:53
for C being clique SimpleGraph
for u, v being set st u in Vertices C & v in Vertices C holds
{u,v} in C
proof
let C be clique SimpleGraph; ::_thesis: for u, v being set st u in Vertices C & v in Vertices C holds
{u,v} in C
let u, v be set ; ::_thesis: ( u in Vertices C & v in Vertices C implies {u,v} in C )
assume that
A: u in Vertices C and
B: v in Vertices C ; ::_thesis: {u,v} in C
C = CompleteSGraph (Vertices C) by Lclique;
hence {u,v} in C by A, B, CSG1; ::_thesis: verum
end;
definition
let G be SimpleGraph;
attrG is with_finite_clique# means :Lwfcno: :: SCMYCIEL:def 14
ex C being finite Clique of G st
for D being finite Clique of G holds order D <= order C;
end;
:: deftheorem Lwfcno defines with_finite_clique# SCMYCIEL:def_14_:_
for G being SimpleGraph holds
( G is with_finite_clique# iff ex C being finite Clique of G st
for D being finite Clique of G holds order D <= order C );
registration
cluster non empty finite-membered V233() V267() subset-closed 1 -at_most_dimensional SimpleGraph-like with_finite_clique# for set ;
existence
ex b1 being SimpleGraph st b1 is with_finite_clique#
proof
take G = the void SimpleGraph; ::_thesis: G is with_finite_clique#
{} in G by SG1;
then {{}} c= G by ZFMISC_1:31;
then reconsider C = {{}} as finite Clique of G by eclique;
take C ; :: according to SCMYCIEL:def_14 ::_thesis: for D being finite Clique of G holds order D <= order C
let D be finite Clique of G; ::_thesis: order D <= order C
union D c= union G by ZFMISC_1:77;
hence order D <= order C ; ::_thesis: verum
end;
end;
registration
cluster finite SimpleGraph-like -> with_finite_clique# for set ;
coherence
for b1 being SimpleGraph st b1 is finite holds
b1 is with_finite_clique#
proof
let G be SimpleGraph; ::_thesis: ( G is finite implies G is with_finite_clique# )
assume G is finite ; ::_thesis: G is with_finite_clique#
then reconsider R9 = G as finite SimpleGraph ;
defpred S1[ Nat] means ex c being finite Clique of G st order c = c1;
A1: for k being Nat st S1[k] holds
k <= card R9
proof
let k be Nat; ::_thesis: ( S1[k] implies k <= card R9 )
assume S1[k] ; ::_thesis: k <= card R9
then consider c being finite Clique of G such that
C2: order c = k ;
D2: card c <= card R9 by NAT_1:43;
order c <= card c by Lorder1;
hence k <= card R9 by C2, D2, XXREAL_0:2; ::_thesis: verum
end;
{} in G by SG1;
then {{}} c= G by ZFMISC_1:31;
then reconsider E = {{}} as finite Clique of G by eclique;
order E = 0 ;
then A2: ex k being Nat st S1[k] ;
consider k being Nat such that
A3: S1[k] and
A4: for n being Nat st S1[n] holds
n <= k from NAT_1:sch_6(A1, A2);
consider c being finite Clique of G such that
A5: order c = k by A3;
for D being finite Clique of G holds order D <= order c by A5, A4;
hence G is with_finite_clique# by Lwfcno; ::_thesis: verum
end;
end;
definition
let G be with_finite_clique# SimpleGraph;
func clique# G -> Nat means :Lcliqueno: :: SCMYCIEL:def 15
( ex C being finite Clique of G st order C = it & ( for T being finite Clique of G holds order T <= it ) );
existence
ex b1 being Nat st
( ex C being finite Clique of G st order C = b1 & ( for T being finite Clique of G holds order T <= b1 ) )
proof
consider C being finite Clique of G such that
A: for D being finite Clique of G holds order D <= order C by Lwfcno;
take itt = order C; ::_thesis: ( ex C being finite Clique of G st order C = itt & ( for T being finite Clique of G holds order T <= itt ) )
thus ex A being finite Clique of G st order A = itt ; ::_thesis: for T being finite Clique of G holds order T <= itt
let T be finite Clique of G; ::_thesis: order T <= itt
thus order T <= itt by A; ::_thesis: verum
end;
uniqueness
for b1, b2 being Nat st ex C being finite Clique of G st order C = b1 & ( for T being finite Clique of G holds order T <= b1 ) & ex C being finite Clique of G st order C = b2 & ( for T being finite Clique of G holds order T <= b2 ) holds
b1 = b2
proof
let it1, it2 be Nat; ::_thesis: ( ex C being finite Clique of G st order C = it1 & ( for T being finite Clique of G holds order T <= it1 ) & ex C being finite Clique of G st order C = it2 & ( for T being finite Clique of G holds order T <= it2 ) implies it1 = it2 )
assume that
A1: ex C being finite Clique of G st order C = it1 and
B1: for T being finite Clique of G holds order T <= it1 and
A2: ex C being finite Clique of G st order C = it2 and
B2: for T being finite Clique of G holds order T <= it2 ; ::_thesis: it1 = it2
consider C1 being finite Clique of G such that
C1: order C1 = it1 by A1;
consider C2 being finite Clique of G such that
D1: order C2 = it2 by A2;
( it1 <= it2 & it2 <= it1 ) by B1, B2, C1, D1;
hence it1 = it2 by XXREAL_0:1; ::_thesis: verum
end;
end;
:: deftheorem Lcliqueno defines clique# SCMYCIEL:def_15_:_
for G being with_finite_clique# SimpleGraph
for b2 being Nat holds
( b2 = clique# G iff ( ex C being finite Clique of G st order C = b2 & ( for T being finite Clique of G holds order T <= b2 ) ) );
theorem Cno0: :: SCMYCIEL:54
for G being with_finite_clique# SimpleGraph st clique# G = 0 holds
Vertices G = {}
proof
let G be with_finite_clique# SimpleGraph; ::_thesis: ( clique# G = 0 implies Vertices G = {} )
assume Aa: clique# G = 0 ; ::_thesis: Vertices G = {}
assume Vertices G <> {} ; ::_thesis: contradiction
then consider v being set such that
Aa1: v in Vertices G by XBOOLE_0:def_1;
consider D being finite Clique of G such that
B1: Vertices D = {v} by Aa1, CliqueS;
order D <= 0 by Aa, Lcliqueno;
hence contradiction by B1; ::_thesis: verum
end;
theorem :: SCMYCIEL:55
for G being void SimpleGraph holds clique# G = 0
proof
let G be void SimpleGraph; ::_thesis: clique# G = 0
assume A: clique# G <> 0 ; ::_thesis: contradiction
consider C being finite Clique of G such that
B: order C = clique# G by Lcliqueno;
Vertices C c= Vertices G by ZFMISC_1:77;
hence contradiction by A, B; ::_thesis: verum
end;
theorem Th8: :: SCMYCIEL:56
for G being SimpleGraph
for x, y being set st {x,y} in G holds
G SubgraphInducedBy {x,y} is Clique of G
proof
let G be SimpleGraph; ::_thesis: for x, y being set st {x,y} in G holds
G SubgraphInducedBy {x,y} is Clique of G
let x, y be set ; ::_thesis: ( {x,y} in G implies G SubgraphInducedBy {x,y} is Clique of G )
assume C: {x,y} in G ; ::_thesis: G SubgraphInducedBy {x,y} is Clique of G
set S = G SubgraphInducedBy {x,y};
now__::_thesis:_for_a,_b_being_set_st_a_in_union_(G_SubgraphInducedBy_{x,y})_&_b_in_union_(G_SubgraphInducedBy_{x,y})_holds_
{a,b}_in_G_SubgraphInducedBy_{x,y}
let a, b be set ; ::_thesis: ( a in union (G SubgraphInducedBy {x,y}) & b in union (G SubgraphInducedBy {x,y}) implies {b1,b2} in G SubgraphInducedBy {x,y} )
assume that
B1: a in union (G SubgraphInducedBy {x,y}) and
C1: b in union (G SubgraphInducedBy {x,y}) ; ::_thesis: {b1,b2} in G SubgraphInducedBy {x,y}
D1: ( a in {x,y} & b in {x,y} ) by B1, C1, Sub1;
then E1: ( ( a = x or a = y ) & ( b = x or b = y ) ) by TARSKI:def_2;
percases ( a = b or a <> b ) ;
suppose a = b ; ::_thesis: {b1,b2} in G SubgraphInducedBy {x,y}
then {a,b} = {a} by ENUMSET1:29;
hence {a,b} in G SubgraphInducedBy {x,y} by B1, Vertices0; ::_thesis: verum
end;
suppose a <> b ; ::_thesis: {b1,b2} in G SubgraphInducedBy {x,y}
hence {a,b} in G SubgraphInducedBy {x,y} by D1, E1, C, Sub6; ::_thesis: verum
end;
end;
end;
then G SubgraphInducedBy {x,y} = CompleteSGraph (Vertices (G SubgraphInducedBy {x,y})) by CSGdef;
hence G SubgraphInducedBy {x,y} is Clique of G ; ::_thesis: verum
end;
theorem Cno2: :: SCMYCIEL:57
for G being with_finite_clique# SimpleGraph st Edges G <> {} holds
clique# G >= 2
proof
let G be with_finite_clique# SimpleGraph; ::_thesis: ( Edges G <> {} implies clique# G >= 2 )
assume A: Edges G <> {} ; ::_thesis: clique# G >= 2
consider e being set such that
B: e in Edges G by A, XBOOLE_0:def_1;
consider x, y being set such that
C: x <> y and
D: x in Vertices G and
E: y in Vertices G and
F: e = {x,y} by B, SG4;
reconsider S = G SubgraphInducedBy {x,y} as finite Clique of G by F, B, Th8;
G: Vertices S = (Vertices G) /\ {x,y} by Sub5;
H: {x,y} c= Vertices G by D, E, ZFMISC_1:32;
Vertices S = {x,y} by G, H, XBOOLE_1:28;
then order S = 2 by C, CARD_2:57;
hence clique# G >= 2 by Lcliqueno; ::_thesis: verum
end;
theorem CliqueSubno0: :: SCMYCIEL:58
for G, H being with_finite_clique# SimpleGraph st G c= H holds
clique# G <= clique# H
proof
let G, H be with_finite_clique# SimpleGraph; ::_thesis: ( G c= H implies clique# G <= clique# H )
assume A: G c= H ; ::_thesis: clique# G <= clique# H
consider D being finite Clique of G such that
C: order D = clique# G by Lcliqueno;
D is Clique of H by A, XBOOLE_1:1;
hence clique# G <= clique# H by C, Lcliqueno; ::_thesis: verum
end;
theorem cliqueCSG: :: SCMYCIEL:59
for X being finite set holds clique# (CompleteSGraph X) = card X
proof
let X be finite set ; ::_thesis: clique# (CompleteSGraph X) = card X
set G = CompleteSGraph X;
B: order (CompleteSGraph X) = card X by CSGLem1;
C: CompleteSGraph X c= CompleteSGraph X ;
for T being finite Clique of (CompleteSGraph X) holds order T <= order (CompleteSGraph X) by NAT_1:43, ZFMISC_1:77;
hence clique# (CompleteSGraph X) = card X by B, C, Lcliqueno; ::_thesis: verum
end;
definition
let G be SimpleGraph;
let P be a_partition of Vertices G;
attrP is Clique-wise means :LCliquewise: :: SCMYCIEL:def 16
for x being set st x in P holds
G SubgraphInducedBy x is Clique of G;
end;
:: deftheorem LCliquewise defines Clique-wise SCMYCIEL:def_16_:_
for G being SimpleGraph
for P being a_partition of Vertices G holds
( P is Clique-wise iff for x being set st x in P holds
G SubgraphInducedBy x is Clique of G );
registration
let G be SimpleGraph;
clusterV267() Clique-wise for a_partition of Vertices G;
correctness
existence
ex b1 being a_partition of Vertices G st b1 is Clique-wise ;
proof
set VG = Vertices G;
percases ( Vertices G is empty or not Vertices G is empty ) ;
suppose Vertices G is empty ; ::_thesis: ex b1 being a_partition of Vertices G st b1 is Clique-wise
then reconsider S = {} as a_partition of Vertices G by EQREL_1:45;
take S ; ::_thesis: S is Clique-wise
for x being set st x in S holds
G SubgraphInducedBy x is Clique of G ;
hence S is Clique-wise by LCliquewise; ::_thesis: verum
end;
suppose not Vertices G is empty ; ::_thesis: ex b1 being a_partition of Vertices G st b1 is Clique-wise
then reconsider cRp1 = Vertices G as non empty set ;
set S = SmallestPartition (Vertices G);
A3: SmallestPartition (Vertices G) = { {x} where x is Element of cRp1 : verum } by EQREL_1:37;
take SmallestPartition (Vertices G) ; ::_thesis: SmallestPartition (Vertices G) is Clique-wise
now__::_thesis:_for_z_being_set_st_z_in_SmallestPartition_(Vertices_G)_holds_
G_SubgraphInducedBy_z_is_Clique_of_G
let z be set ; ::_thesis: ( z in SmallestPartition (Vertices G) implies G SubgraphInducedBy z is Clique of G )
assume A5: z in SmallestPartition (Vertices G) ; ::_thesis: G SubgraphInducedBy z is Clique of G
consider x being Element of cRp1 such that
B2: z = {x} and
verum by A3, A5;
G SubgraphInducedBy z = {{},{x}} by B2, SingleSub;
hence G SubgraphInducedBy z is Clique of G by SingleClique; ::_thesis: verum
end;
hence SmallestPartition (Vertices G) is Clique-wise by LCliquewise; ::_thesis: verum
end;
end;
end;
end;
definition
let G be SimpleGraph;
mode Clique-partition of G is Clique-wise a_partition of Vertices G;
end;
registration
let G be void SimpleGraph;
cluster empty -> Clique-wise for a_partition of Vertices G;
correctness
coherence
for b1 being a_partition of Vertices G st b1 is empty holds
b1 is Clique-wise ;
proof
let P be a_partition of Vertices G; ::_thesis: ( P is empty implies P is Clique-wise )
assume P is empty ; ::_thesis: P is Clique-wise
for x being set st x in P holds
G SubgraphInducedBy x is Clique of G ;
hence P is Clique-wise by LCliquewise; ::_thesis: verum
end;
end;
definition
let G be SimpleGraph;
attrG is with_finite_cliquecover# means :Lwfclicov: :: SCMYCIEL:def 17
ex C being Clique-partition of G st C is finite ;
end;
:: deftheorem Lwfclicov defines with_finite_cliquecover# SCMYCIEL:def_17_:_
for G being SimpleGraph holds
( G is with_finite_cliquecover# iff ex C being Clique-partition of G st C is finite );
registration
cluster finite SimpleGraph-like -> with_finite_cliquecover# for set ;
correctness
coherence
for b1 being SimpleGraph st b1 is finite holds
b1 is with_finite_cliquecover# ;
proof
let G be SimpleGraph; ::_thesis: ( G is finite implies G is with_finite_cliquecover# )
assume A1: G is finite ; ::_thesis: G is with_finite_cliquecover#
set VG = Vertices G;
percases ( Vertices G is empty or not Vertices G is empty ) ;
suppose Vertices G is empty ; ::_thesis: G is with_finite_cliquecover#
then reconsider S = {} as a_partition of Vertices G by EQREL_1:45;
for x being set st x in S holds
G SubgraphInducedBy x is Clique of G ;
then reconsider S = S as Clique-partition of G by LCliquewise;
take S ; :: according to SCMYCIEL:def_17 ::_thesis: S is finite
thus S is finite ; ::_thesis: verum
end;
supposeA2: not Vertices G is empty ; ::_thesis: G is with_finite_cliquecover#
reconsider cRp1 = Vertices G as non empty finite set by A2, A1;
set S = SmallestPartition (Vertices G);
deffunc H1( set ) -> set = {c1};
defpred S1[ set ] means verum;
A3: SmallestPartition (Vertices G) = { H1(x) where x is Element of cRp1 : S1[x] } by EQREL_1:37;
A4: { H1(x) where x is Element of cRp1 : S1[x] } is finite from PRE_CIRC:sch_1();
now__::_thesis:_for_z_being_set_st_z_in_SmallestPartition_(Vertices_G)_holds_
G_SubgraphInducedBy_z_is_Clique_of_G
let z be set ; ::_thesis: ( z in SmallestPartition (Vertices G) implies G SubgraphInducedBy z is Clique of G )
assume A5: z in SmallestPartition (Vertices G) ; ::_thesis: G SubgraphInducedBy z is Clique of G
consider x being Element of Vertices G such that
B2: z = {x} and
verum by A5, A3;
G SubgraphInducedBy z = {{},{x}} by B2, A2, SingleSub;
hence G SubgraphInducedBy z is Clique of G by A2, SingleClique; ::_thesis: verum
end;
then reconsider S = SmallestPartition (Vertices G) as Clique-partition of G by LCliquewise;
take S ; :: according to SCMYCIEL:def_17 ::_thesis: S is finite
thus S is finite by A4; ::_thesis: verum
end;
end;
end;
end;
registration
let G be with_finite_cliquecover# SimpleGraph;
cluster finite V267() Clique-wise for a_partition of Vertices G;
correctness
existence
ex b1 being Clique-partition of G st b1 is finite ;
by Lwfclicov;
end;
registration
let G be with_finite_cliquecover# SimpleGraph;
let S be Subset of (Vertices G);
clusterG SubgraphInducedBy S -> with_finite_cliquecover# ;
correctness
coherence
G SubgraphInducedBy S is with_finite_cliquecover# ;
proof
set H = G SubgraphInducedBy S;
consider C being Clique-partition of G such that
A: C is finite by Lwfclicov;
reconsider VH = Vertices (G SubgraphInducedBy S) as Subset of (Vertices G) by ZFMISC_1:77;
reconsider D = C | VH as a_partition of Vertices (G SubgraphInducedBy S) ;
now__::_thesis:_for_p_being_set_st_p_in_D_holds_
(G_SubgraphInducedBy_S)_SubgraphInducedBy_p_is_Clique_of_(G_SubgraphInducedBy_S)
let p be set ; ::_thesis: ( p in D implies (G SubgraphInducedBy S) SubgraphInducedBy p is Clique of (G SubgraphInducedBy S) )
assume A1: p in D ; ::_thesis: (G SubgraphInducedBy S) SubgraphInducedBy p is Clique of (G SubgraphInducedBy S)
set Hp = (G SubgraphInducedBy S) SubgraphInducedBy p;
now__::_thesis:_for_x,_y_being_set_st_x_in_union_((G_SubgraphInducedBy_S)_SubgraphInducedBy_p)_&_y_in_union_((G_SubgraphInducedBy_S)_SubgraphInducedBy_p)_holds_
{x,y}_in_(G_SubgraphInducedBy_S)_SubgraphInducedBy_p
let x, y be set ; ::_thesis: ( x in union ((G SubgraphInducedBy S) SubgraphInducedBy p) & y in union ((G SubgraphInducedBy S) SubgraphInducedBy p) implies {x,y} in (G SubgraphInducedBy S) SubgraphInducedBy p )
assume that
B2: x in union ((G SubgraphInducedBy S) SubgraphInducedBy p) and
C2: y in union ((G SubgraphInducedBy S) SubgraphInducedBy p) ; ::_thesis: {x,y} in (G SubgraphInducedBy S) SubgraphInducedBy p
consider c being Element of C such that
D2: p = c /\ VH and
c meets VH by A1;
G2: x in p by B2, Sub1;
H2: y in p by C2, Sub1;
I2a: x in VH by D2, G2, XBOOLE_0:def_4;
I2aa: y in VH by D2, H2, XBOOLE_0:def_4;
set Gc = G SubgraphInducedBy c;
I2: G SubgraphInducedBy c is Clique of G by I2a, LCliquewise;
I2b: G SubgraphInducedBy c = CompleteSGraph (Vertices (G SubgraphInducedBy c)) by I2, Lclique;
( x in c & y in c ) by G2, H2, D2, XBOOLE_0:def_4;
then ( x in Vertices (G SubgraphInducedBy c) & y in Vertices (G SubgraphInducedBy c) ) by Sub3;
then F2aa: {x,y} in G SubgraphInducedBy c by I2b, CSG1;
( x in S & y in S ) by I2a, I2aa, Sub1;
then {x,y} c= S by ZFMISC_1:32;
then F2: {x,y} in G SubgraphInducedBy S by F2aa, XBOOLE_0:def_4;
{x,y} c= p by G2, H2, ZFMISC_1:32;
hence {x,y} in (G SubgraphInducedBy S) SubgraphInducedBy p by F2, XBOOLE_0:def_4; ::_thesis: verum
end;
then (G SubgraphInducedBy S) SubgraphInducedBy p = CompleteSGraph (Vertices ((G SubgraphInducedBy S) SubgraphInducedBy p)) by CSGdef;
hence (G SubgraphInducedBy S) SubgraphInducedBy p is Clique of (G SubgraphInducedBy S) ; ::_thesis: verum
end;
then reconsider D = D as Clique-partition of (G SubgraphInducedBy S) by LCliquewise;
take D ; :: according to SCMYCIEL:def_17 ::_thesis: D is finite
thus D is finite by A; ::_thesis: verum
end;
end;
definition
let G be with_finite_cliquecover# SimpleGraph;
func cliquecover# G -> Nat means :Lclicovno: :: SCMYCIEL:def 18
( ex C being finite Clique-partition of G st card C = it & ( for C being finite Clique-partition of G holds it <= card C ) );
existence
ex b1 being Nat st
( ex C being finite Clique-partition of G st card C = b1 & ( for C being finite Clique-partition of G holds b1 <= card C ) )
proof
defpred S1[ Nat] means ex C being finite Clique-partition of G st card C = $1;
consider C being Clique-partition of G such that
A1: C is finite by Lwfclicov;
card C = card C ;
then A2: ex k being Nat st S1[k] by A1;
consider n being Nat such that
A3: S1[n] and
A4: for k being Nat st S1[k] holds
n <= k from NAT_1:sch_5(A2);
take n ; ::_thesis: ( ex C being finite Clique-partition of G st card C = n & ( for C being finite Clique-partition of G holds n <= card C ) )
thus ex C being finite Clique-partition of G st card C = n by A3; ::_thesis: for C being finite Clique-partition of G holds n <= card C
let C be finite Clique-partition of G; ::_thesis: n <= card C
thus n <= card C by A4; ::_thesis: verum
end;
uniqueness
for b1, b2 being Nat st ex C being finite Clique-partition of G st card C = b1 & ( for C being finite Clique-partition of G holds b1 <= card C ) & ex C being finite Clique-partition of G st card C = b2 & ( for C being finite Clique-partition of G holds b2 <= card C ) holds
b1 = b2
proof
let it1, it2 be Nat; ::_thesis: ( ex C being finite Clique-partition of G st card C = it1 & ( for C being finite Clique-partition of G holds it1 <= card C ) & ex C being finite Clique-partition of G st card C = it2 & ( for C being finite Clique-partition of G holds it2 <= card C ) implies it1 = it2 )
assume that
A1: ex C being finite Clique-partition of G st card C = it1 and
A1a: for C being finite Clique-partition of G holds it1 <= card C and
A2: ex C being finite Clique-partition of G st card C = it2 and
A2a: for C being finite Clique-partition of G holds it2 <= card C ; ::_thesis: it1 = it2
consider C1 being finite Clique-partition of G such that
B1: card C1 = it1 by A1;
consider C2 being finite Clique-partition of G such that
B2: card C2 = it2 by A2;
( it1 <= card C2 & it2 <= card C1 ) by A1a, A2a;
hence it1 = it2 by B1, B2, XXREAL_0:1; ::_thesis: verum
end;
end;
:: deftheorem Lclicovno defines cliquecover# SCMYCIEL:def_18_:_
for G being with_finite_cliquecover# SimpleGraph
for b2 being Nat holds
( b2 = cliquecover# G iff ( ex C being finite Clique-partition of G st card C = b2 & ( for C being finite Clique-partition of G holds b2 <= card C ) ) );
begin
definition
let G be SimpleGraph;
let S be Subset of (Vertices G);
attrS is stable means :Lstable: :: SCMYCIEL:def 19
for x, y being set st x <> y & x in S & y in S holds
{x,y} nin G;
end;
:: deftheorem Lstable defines stable SCMYCIEL:def_19_:_
for G being SimpleGraph
for S being Subset of (Vertices G) holds
( S is stable iff for x, y being set st x <> y & x in S & y in S holds
{x,y} nin G );
theorem stable0: :: SCMYCIEL:60
for G being SimpleGraph holds {} (Vertices G) is stable
proof
let G be SimpleGraph; ::_thesis: {} (Vertices G) is stable
let x, y be set ; :: according to SCMYCIEL:def_19 ::_thesis: ( x <> y & x in {} (Vertices G) & y in {} (Vertices G) implies {x,y} nin G )
assume that
x <> y and
A: x in {} (Vertices G) and
y in {} (Vertices G) ; ::_thesis: {x,y} nin G
thus {x,y} nin G by A; ::_thesis: verum
end;
theorem stable1: :: SCMYCIEL:61
for G being SimpleGraph
for S being Subset of (Vertices G)
for v being set st S = {v} holds
S is stable
proof
let G be SimpleGraph; ::_thesis: for S being Subset of (Vertices G)
for v being set st S = {v} holds
S is stable
let S be Subset of (Vertices G); ::_thesis: for v being set st S = {v} holds
S is stable
let v be set ; ::_thesis: ( S = {v} implies S is stable )
assume A: S = {v} ; ::_thesis: S is stable
let x, y be set ; :: according to SCMYCIEL:def_19 ::_thesis: ( x <> y & x in S & y in S implies {x,y} nin G )
assume that
B: x <> y and
C: ( x in S & y in S ) ; ::_thesis: {x,y} nin G
( x = v & y = v ) by A, C, TARSKI:def_1;
hence {x,y} nin G by B; ::_thesis: verum
end;
registration
let G be SimpleGraph;
cluster trivial -> stable for Element of bool (Vertices G);
coherence
for b1 being Subset of (Vertices G) st b1 is trivial holds
b1 is stable
proof
let S be Subset of (Vertices G); ::_thesis: ( S is trivial implies S is stable )
assume A: S is trivial ; ::_thesis: S is stable
percases ( S is empty or ex c being set st S = {c} ) by A, ZFMISC_1:131;
suppose S is empty ; ::_thesis: S is stable
then S = {} (Vertices G) ;
hence S is stable by stable0; ::_thesis: verum
end;
suppose ex c being set st S = {c} ; ::_thesis: S is stable
then consider c being set such that
A1: S = {c} ;
thus S is stable by A1, stable1; ::_thesis: verum
end;
end;
end;
end;
registration
let G be SimpleGraph;
cluster stable for Element of bool (Vertices G);
existence
ex b1 being Subset of (Vertices G) st b1 is stable
proof
take {} (Vertices G) ; ::_thesis: {} (Vertices G) is stable
thus {} (Vertices G) is stable ; ::_thesis: verum
end;
end;
definition
let G be SimpleGraph;
mode StableSet of G is stable Subset of (Vertices G);
end;
theorem Th14: :: SCMYCIEL:62
for G being SimpleGraph
for x, y being set st x in Vertices G & y in Vertices G & {x,y} nin G holds
{x,y} is StableSet of G
proof
let G be SimpleGraph; ::_thesis: for x, y being set st x in Vertices G & y in Vertices G & {x,y} nin G holds
{x,y} is StableSet of G
let x, y be set ; ::_thesis: ( x in Vertices G & y in Vertices G & {x,y} nin G implies {x,y} is StableSet of G )
assume that
A: x in Vertices G and
B: y in Vertices G and
C: {x,y} nin G ; ::_thesis: {x,y} is StableSet of G
reconsider S = {x,y} as Subset of (Vertices G) by A, B, ZFMISC_1:32;
S is stable
proof
let a, b be set ; :: according to SCMYCIEL:def_19 ::_thesis: ( a <> b & a in S & b in S implies {a,b} nin G )
assume that
A1: a <> b and
B1: a in S and
C1: b in S ; ::_thesis: {a,b} nin G
( ( a = x or a = y ) & ( b = x or b = y ) ) by B1, C1, TARSKI:def_2;
hence {a,b} nin G by C, A1; ::_thesis: verum
end;
hence {x,y} is StableSet of G ; ::_thesis: verum
end;
theorem Th19: :: SCMYCIEL:63
for G being with_finite_clique# SimpleGraph st clique# G = 1 holds
Vertices G is StableSet of G
proof
let R be with_finite_clique# SimpleGraph; ::_thesis: ( clique# R = 1 implies Vertices R is StableSet of R )
assume A1: clique# R = 1 ; ::_thesis: Vertices R is StableSet of R
set cR = Vertices R;
A2a: Vertices R c= Vertices R ;
now__::_thesis:_for_a,_b_being_set_st_a_<>_b_&_a_in_Vertices_R_&_b_in_Vertices_R_holds_
not_{a,b}_in_R
let a, b be set ; ::_thesis: ( a <> b & a in Vertices R & b in Vertices R implies not {a,b} in R )
assume A3: ( a <> b & a in Vertices R & b in Vertices R ) ; ::_thesis: not {a,b} in R
A3a: {a,b} c= Vertices R by A3, ZFMISC_1:32;
assume {a,b} in R ; ::_thesis: contradiction
then reconsider H = R SubgraphInducedBy {a,b} as finite Clique of R by Th8;
Vertices H = {a,b} by A3a, Sub0c;
then order H = 2 by A3, CARD_2:57;
hence contradiction by A1, Lcliqueno; ::_thesis: verum
end;
hence Vertices R is StableSet of R by A2a, Lstable; ::_thesis: verum
end;
registration
let G be SimpleGraph;
cluster finite stable for Element of bool (Vertices G);
existence
ex b1 being StableSet of G st b1 is finite
proof
take {} (Vertices G) ; ::_thesis: {} (Vertices G) is finite
thus {} (Vertices G) is finite ; ::_thesis: verum
end;
end;
theorem Th16: :: SCMYCIEL:64
for G being SimpleGraph
for A being StableSet of G
for B being Subset of A holds B is StableSet of G
proof
let R be SimpleGraph; ::_thesis: for A being StableSet of R
for B being Subset of A holds B is StableSet of R
let A be StableSet of R; ::_thesis: for B being Subset of A holds B is StableSet of R
let B be Subset of A; ::_thesis: B is StableSet of R
set VR = Vertices R;
reconsider BB = B as Subset of (Vertices R) by XBOOLE_1:1;
BB is stable
proof
let x, y be set ; :: according to SCMYCIEL:def_19 ::_thesis: ( x <> y & x in BB & y in BB implies {x,y} nin R )
assume A1: ( x <> y & x in BB & y in BB ) ; ::_thesis: {x,y} nin R
thus {x,y} nin R by A1, Lstable; ::_thesis: verum
end;
hence B is StableSet of R ; ::_thesis: verum
end;
definition
let G be SimpleGraph;
let P be a_partition of Vertices G;
attrP is StableSet-wise means :LStableSetwise: :: SCMYCIEL:def 20
for x being set st x in P holds
x is StableSet of G;
end;
:: deftheorem LStableSetwise defines StableSet-wise SCMYCIEL:def_20_:_
for G being SimpleGraph
for P being a_partition of Vertices G holds
( P is StableSet-wise iff for x being set st x in P holds
x is StableSet of G );
theorem Coloring1: :: SCMYCIEL:65
for G being SimpleGraph holds SmallestPartition (Vertices G) is StableSet-wise
proof
let G be SimpleGraph; ::_thesis: SmallestPartition (Vertices G) is StableSet-wise
set C = SmallestPartition (Vertices G);
let c be set ; :: according to SCMYCIEL:def_20 ::_thesis: ( c in SmallestPartition (Vertices G) implies c is StableSet of G )
assume A: c in SmallestPartition (Vertices G) ; ::_thesis: c is StableSet of G
consider a being set such that
a in Vertices G and
Ab: c = Class ((id (Vertices G)),a) by A, EQREL_1:def_3;
reconsider cc = c as Subset of (Vertices G) by A;
Z: cc is stable
proof
let x, y be set ; :: according to SCMYCIEL:def_19 ::_thesis: ( x <> y & x in cc & y in cc implies {x,y} nin G )
assume that
A1: x <> y and
B1: x in cc and
C1: y in cc ; ::_thesis: {x,y} nin G
D1: [a,x] in id (Vertices G) by B1, Ab, RELAT_1:169;
E1: [a,y] in id (Vertices G) by C1, Ab, RELAT_1:169;
E1a: a = y by E1, RELAT_1:def_10;
thus {x,y} nin G by D1, E1a, A1, RELAT_1:def_10; ::_thesis: verum
end;
thus c is StableSet of G by Z; ::_thesis: verum
end;
registration
let G be SimpleGraph;
clusterV267() StableSet-wise for a_partition of Vertices G;
existence
ex b1 being a_partition of Vertices G st b1 is StableSet-wise
proof
take SmallestPartition (Vertices G) ; ::_thesis: SmallestPartition (Vertices G) is StableSet-wise
thus SmallestPartition (Vertices G) is StableSet-wise by Coloring1; ::_thesis: verum
end;
end;
definition
let G be SimpleGraph;
mode Coloring of G is StableSet-wise a_partition of Vertices G;
end;
definition
let G be SimpleGraph;
attrG is finitely_colorable means :Lfc: :: SCMYCIEL:def 21
ex C being Coloring of G st C is finite ;
end;
:: deftheorem Lfc defines finitely_colorable SCMYCIEL:def_21_:_
for G being SimpleGraph holds
( G is finitely_colorable iff ex C being Coloring of G st C is finite );
registration
cluster non empty finite-membered V233() V267() subset-closed 1 -at_most_dimensional SimpleGraph-like finitely_colorable for set ;
existence
ex b1 being SimpleGraph st b1 is finitely_colorable
proof
take G = the finite SimpleGraph; ::_thesis: G is finitely_colorable
SmallestPartition (Vertices G) is Coloring of G by Coloring1;
hence G is finitely_colorable by Lfc; ::_thesis: verum
end;
end;
registration
cluster finite SimpleGraph-like -> finitely_colorable for set ;
correctness
coherence
for b1 being SimpleGraph st b1 is finite holds
b1 is finitely_colorable ;
proof
let G be SimpleGraph; ::_thesis: ( G is finite implies G is finitely_colorable )
assume A: G is finite ; ::_thesis: G is finitely_colorable
SmallestPartition (Vertices G) is Coloring of G by Coloring1;
hence G is finitely_colorable by A, Lfc; ::_thesis: verum
end;
end;
registration
let G be finitely_colorable SimpleGraph;
cluster finite V267() StableSet-wise for a_partition of Vertices G;
existence
ex b1 being Coloring of G st b1 is finite by Lfc;
end;
theorem SGClique0: :: SCMYCIEL:66
for G being SimpleGraph
for S being Clique of G
for L being set st L c= Vertices S holds
G SubgraphInducedBy L is Clique of G
proof
let G be SimpleGraph; ::_thesis: for S being Clique of G
for L being set st L c= Vertices S holds
G SubgraphInducedBy L is Clique of G
let S be Clique of G; ::_thesis: for L being set st L c= Vertices S holds
G SubgraphInducedBy L is Clique of G
let L be set ; ::_thesis: ( L c= Vertices S implies G SubgraphInducedBy L is Clique of G )
assume AA: L c= Vertices S ; ::_thesis: G SubgraphInducedBy L is Clique of G
set g = G SubgraphInducedBy L;
now__::_thesis:_for_x,_y_being_set_st_x_in_union_(G_SubgraphInducedBy_L)_&_y_in_union_(G_SubgraphInducedBy_L)_holds_
{x,y}_in_G_SubgraphInducedBy_L
let x, y be set ; ::_thesis: ( x in union (G SubgraphInducedBy L) & y in union (G SubgraphInducedBy L) implies {x,y} in G SubgraphInducedBy L )
assume that
B1: x in union (G SubgraphInducedBy L) and
C1: y in union (G SubgraphInducedBy L) ; ::_thesis: {x,y} in G SubgraphInducedBy L
G1: x in L by B1, Sub1;
H1: y in L by C1, Sub1;
F1a: {x,y} in S by G1, H1, AA, Clique2a;
thus {x,y} in G SubgraphInducedBy L by G1, H1, F1a, Sub6; ::_thesis: verum
end;
then G SubgraphInducedBy L = CompleteSGraph (Vertices (G SubgraphInducedBy L)) by CSGdef;
hence G SubgraphInducedBy L is Clique of G ; ::_thesis: verum
end;
theorem Tsr0: :: SCMYCIEL:67
for G being SimpleGraph
for C being Coloring of G
for S being Subset of (Vertices G) holds C | S is Coloring of (G SubgraphInducedBy S)
proof
let G be SimpleGraph; ::_thesis: for C being Coloring of G
for S being Subset of (Vertices G) holds C | S is Coloring of (G SubgraphInducedBy S)
let C be Coloring of G; ::_thesis: for S being Subset of (Vertices G) holds C | S is Coloring of (G SubgraphInducedBy S)
let S be Subset of (Vertices G); ::_thesis: C | S is Coloring of (G SubgraphInducedBy S)
set g = G SubgraphInducedBy S;
A: Vertices (G SubgraphInducedBy S) = S /\ (Vertices G) by Sub5
.= S by XBOOLE_1:28 ;
reconsider CS = C | S as a_partition of Vertices (G SubgraphInducedBy S) by A;
CS is StableSet-wise
proof
let x be set ; :: according to SCMYCIEL:def_20 ::_thesis: ( x in CS implies x is StableSet of (G SubgraphInducedBy S) )
assume A1: x in CS ; ::_thesis: x is StableSet of (G SubgraphInducedBy S)
reconsider xx = x as Subset of (Vertices (G SubgraphInducedBy S)) by A1;
consider z being Element of C such that
B1: xx = z /\ S and
z meets S by A1;
xx is stable
proof
let x, y be set ; :: according to SCMYCIEL:def_19 ::_thesis: ( x <> y & x in xx & y in xx implies {x,y} nin G SubgraphInducedBy S )
assume that
A2: x <> y and
B2: x in xx and
C2: y in xx ; ::_thesis: {x,y} nin G SubgraphInducedBy S
assume D2a: {x,y} in G SubgraphInducedBy S ; ::_thesis: contradiction
E2: x in z by B1, B2, XBOOLE_0:def_4;
F2: y in z by C2, B1, XBOOLE_0:def_4;
z is StableSet of G by B1, B2, LStableSetwise;
hence contradiction by D2a, A2, E2, F2, Lstable; ::_thesis: verum
end;
hence x is StableSet of (G SubgraphInducedBy S) ; ::_thesis: verum
end;
hence C | S is Coloring of (G SubgraphInducedBy S) ; ::_thesis: verum
end;
registration
let G be finitely_colorable SimpleGraph;
let S be set ;
clusterG SubgraphInducedBy S -> finitely_colorable ;
coherence
G SubgraphInducedBy S is finitely_colorable
proof
consider C being Coloring of G such that
A: C is finite by Lfc;
reconsider C = C as finite Coloring of G by A;
reconsider SX = S /\ (Vertices G) as Subset of (Vertices G) by XBOOLE_1:17;
C: G SubgraphInducedBy SX = G SubgraphInducedBy S by Sub3a;
reconsider D = C | SX as Coloring of (G SubgraphInducedBy S) by Tsr0, C;
take D ; :: according to SCMYCIEL:def_21 ::_thesis: D is finite
thus D is finite ; ::_thesis: verum
end;
end;
definition
let G be finitely_colorable SimpleGraph;
func chromatic# G -> Nat means :Lchro: :: SCMYCIEL:def 22
( ex C being finite Coloring of G st card C = it & ( for C being finite Coloring of G holds it <= card C ) );
existence
ex b1 being Nat st
( ex C being finite Coloring of G st card C = b1 & ( for C being finite Coloring of G holds b1 <= card C ) )
proof
defpred S1[ Nat] means ex C being finite Coloring of G st card C = $1;
consider C being Coloring of G such that
A1: C is finite by Lfc;
card C = card C ;
then A2: ex k being Nat st S1[k] by A1;
consider n being Nat such that
A3: S1[n] and
A4: for k being Nat st S1[k] holds
n <= k from NAT_1:sch_5(A2);
take n ; ::_thesis: ( ex C being finite Coloring of G st card C = n & ( for C being finite Coloring of G holds n <= card C ) )
thus ex C being finite Coloring of G st card C = n by A3; ::_thesis: for C being finite Coloring of G holds n <= card C
let C be finite Coloring of G; ::_thesis: n <= card C
thus n <= card C by A4; ::_thesis: verum
end;
uniqueness
for b1, b2 being Nat st ex C being finite Coloring of G st card C = b1 & ( for C being finite Coloring of G holds b1 <= card C ) & ex C being finite Coloring of G st card C = b2 & ( for C being finite Coloring of G holds b2 <= card C ) holds
b1 = b2
proof
let it1, it2 be Nat; ::_thesis: ( ex C being finite Coloring of G st card C = it1 & ( for C being finite Coloring of G holds it1 <= card C ) & ex C being finite Coloring of G st card C = it2 & ( for C being finite Coloring of G holds it2 <= card C ) implies it1 = it2 )
assume that
A5: ex C being finite Coloring of G st card C = it1 and
A6: for C being finite Coloring of G holds it1 <= card C and
A7: ex C being finite Coloring of G st card C = it2 and
A8: for C being finite Coloring of G holds it2 <= card C ; ::_thesis: it1 = it2
consider C1 being finite Coloring of G such that
A9: card C1 = it1 by A5;
consider C2 being finite Coloring of G such that
A10: card C2 = it2 by A7;
( it1 <= card C2 & it2 <= card C1 ) by A6, A8;
hence it1 = it2 by A9, A10, XXREAL_0:1; ::_thesis: verum
end;
end;
:: deftheorem Lchro defines chromatic# SCMYCIEL:def_22_:_
for G being finitely_colorable SimpleGraph
for b2 being Nat holds
( b2 = chromatic# G iff ( ex C being finite Coloring of G st card C = b2 & ( for C being finite Coloring of G holds b2 <= card C ) ) );
theorem Subchro: :: SCMYCIEL:68
for G, H being finitely_colorable SimpleGraph st G c= H holds
chromatic# G <= chromatic# H
proof
let G, H be finitely_colorable SimpleGraph; ::_thesis: ( G c= H implies chromatic# G <= chromatic# H )
assume A: G c= H ; ::_thesis: chromatic# G <= chromatic# H
then reconsider S = Vertices G as Subset of (Vertices H) by ZFMISC_1:77;
set g = H SubgraphInducedBy S;
Aa: G c= H SubgraphInducedBy S by A, Sub0b;
consider C being finite Coloring of H such that
B: card C = chromatic# H by Lchro;
reconsider g = H SubgraphInducedBy S as finitely_colorable SimpleGraph ;
reconsider Cg = C | S as finite Coloring of g by Tsr0;
Ca: Vertices G = Vertices g by Sub0c;
Cb: G c= g
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in G or a in g )
assume a in G ; ::_thesis: a in g
then a in ({{}} \/ (singletons (Vertices G))) \/ (Edges G) by SG0;
then A1: ( a in {{}} \/ (singletons (Vertices G)) or a in Edges G ) by XBOOLE_0:def_3;
percases ( a in {{}} or a in singletons (Vertices G) or a in Edges G ) by A1, XBOOLE_0:def_3;
suppose a in {{}} ; ::_thesis: a in g
then a = {} by TARSKI:def_1;
hence a in g by SG1; ::_thesis: verum
end;
suppose a in singletons (Vertices G) ; ::_thesis: a in g
then a in {{}} \/ (singletons (Vertices g)) by Ca, XBOOLE_0:def_3;
then a in ({{}} \/ (singletons (Vertices g))) \/ (Edges g) by XBOOLE_0:def_3;
hence a in g by SG0; ::_thesis: verum
end;
suppose a in Edges G ; ::_thesis: a in g
then a in G ;
hence a in g by Aa; ::_thesis: verum
end;
end;
end;
reconsider Cg1 = Cg as a_partition of Vertices G ;
Cg1 is StableSet-wise
proof
let x be set ; :: according to SCMYCIEL:def_20 ::_thesis: ( x in Cg1 implies x is StableSet of G )
assume A1: x in Cg1 ; ::_thesis: x is StableSet of G
reconsider xx = x as Subset of (Vertices G) by A1;
reconsider xxx = x as Subset of (Vertices g) by A1;
xx is stable
proof
let x, y be set ; :: according to SCMYCIEL:def_19 ::_thesis: ( x <> y & x in xx & y in xx implies {x,y} nin G )
assume that
A2: x <> y and
B2: x in xx and
C2: y in xx ; ::_thesis: {x,y} nin G
D2: xxx is stable by A1, LStableSetwise;
assume {x,y} in G ; ::_thesis: contradiction
hence contradiction by Cb, A2, B2, C2, D2, Lstable; ::_thesis: verum
end;
hence x is StableSet of G ; ::_thesis: verum
end;
then D: card Cg >= chromatic# G by Lchro;
card C >= card (C | S) by MYCIELSK:8;
hence chromatic# G <= chromatic# H by D, B, XXREAL_0:2; ::_thesis: verum
end;
theorem chromaticCSG: :: SCMYCIEL:69
for X being finite set holds chromatic# (CompleteSGraph X) = card X
proof
let X be finite set ; ::_thesis: chromatic# (CompleteSGraph X) = card X
set n = card X;
set G = CompleteSGraph X;
set D = SmallestPartition X;
B: card (SmallestPartition X) = card X by TOPGEN_2:12;
D: Vertices (CompleteSGraph X) = X by CSGLem1;
reconsider D = SmallestPartition X as a_partition of Vertices (CompleteSGraph X) by CSGLem1;
Ca: D is StableSet-wise
proof
let x be set ; :: according to SCMYCIEL:def_20 ::_thesis: ( x in D implies x is StableSet of (CompleteSGraph X) )
assume AA: x in D ; ::_thesis: x is StableSet of (CompleteSGraph X)
then reconsider xx = x as Subset of (Vertices (CompleteSGraph X)) ;
xx is stable
proof
let x, y be set ; :: according to SCMYCIEL:def_19 ::_thesis: ( x <> y & x in xx & y in xx implies {x,y} nin CompleteSGraph X )
assume that
A1: x <> y and
B1: x in xx and
C1: y in xx ; ::_thesis: {x,y} nin CompleteSGraph X
not X is empty by AA;
then D = { {a} where a is Element of X : verum } by EQREL_1:37;
then consider a being Element of X such that
D1: xx = {a} and
verum by AA;
( a = x & y = a ) by D1, B1, C1, TARSKI:def_1;
hence {x,y} nin CompleteSGraph X by A1; ::_thesis: verum
end;
hence x is StableSet of (CompleteSGraph X) ; ::_thesis: verum
end;
for C being finite Coloring of (CompleteSGraph X) holds card X <= card C
proof
let C be finite Coloring of (CompleteSGraph X); ::_thesis: card X <= card C
assume Aa: card X > card C ; ::_thesis: contradiction
then not X is empty ;
then consider p, x, y being set such that
A1: p in C and
B1: x in p and
C1: y in p and
D1: x <> y by Aa, D, Part0;
E1: p is StableSet of (CompleteSGraph X) by A1, LStableSetwise;
reconsider p = p as Subset of (Vertices (CompleteSGraph X)) by A1;
F1: {x,y} nin CompleteSGraph X by E1, B1, C1, D1, Lstable;
p c= X by D;
hence contradiction by B1, C1, F1, CSG1; ::_thesis: verum
end;
hence chromatic# (CompleteSGraph X) = card X by B, Ca, Lchro; ::_thesis: verum
end;
theorem AdjCol: :: SCMYCIEL:70
for G being finitely_colorable SimpleGraph
for C being finite Coloring of G
for c being set st c in C & card C = chromatic# G holds
ex v being Element of Vertices G st
( v in c & ( for d being Element of C st d <> c holds
ex w being Element of Vertices G st
( w in Adjacent v & w in d ) ) )
proof
let G be finitely_colorable SimpleGraph; ::_thesis: for C being finite Coloring of G
for c being set st c in C & card C = chromatic# G holds
ex v being Element of Vertices G st
( v in c & ( for d being Element of C st d <> c holds
ex w being Element of Vertices G st
( w in Adjacent v & w in d ) ) )
let C be finite Coloring of G; ::_thesis: for c being set st c in C & card C = chromatic# G holds
ex v being Element of Vertices G st
( v in c & ( for d being Element of C st d <> c holds
ex w being Element of Vertices G st
( w in Adjacent v & w in d ) ) )
let c be set ; ::_thesis: ( c in C & card C = chromatic# G implies ex v being Element of Vertices G st
( v in c & ( for d being Element of C st d <> c holds
ex w being Element of Vertices G st
( w in Adjacent v & w in d ) ) ) )
assume that
A1: c in C and
A2: card C = chromatic# G ; ::_thesis: ex v being Element of Vertices G st
( v in c & ( for d being Element of C st d <> c holds
ex w being Element of Vertices G st
( w in Adjacent v & w in d ) ) )
assume A3: for v being Element of Vertices G holds
( not v in c or ex d being Element of C st
( d <> c & ( for w being Element of Vertices G holds
( not w in Adjacent v or not w in d ) ) ) ) ; ::_thesis: contradiction
set uG = Vertices G;
A4: union C = Vertices G by EQREL_1:def_4;
reconsider c = c as Subset of (Vertices G) by A1;
set Cc = C \ {c};
A6: c in {c} by TARSKI:def_1;
percases ( C \ {c} is empty or not C \ {c} is empty ) ;
supposeA7: C \ {c} is empty ; ::_thesis: contradiction
consider v being set such that
A8: v in c by A1, XBOOLE_0:def_1;
reconsider v = v as Element of Vertices G by A8;
consider d being Element of C such that
A9: d <> c and
for w being Element of Vertices G holds
( not w in Adjacent v or not w in d ) by A8, A3;
0 = (card C) - (card {c}) by A1, A7, CARD_1:27, EULER_1:4;
then 0 + 1 = ((card C) - 1) + 1 by CARD_1:30;
then consider x being set such that
A10: C = {x} by CARD_2:42;
( c = x & d = x ) by A1, A10, TARSKI:def_1;
hence contradiction by A9; ::_thesis: verum
end;
suppose not C \ {c} is empty ; ::_thesis: contradiction
then reconsider Cc = C \ {c} as non empty set ;
defpred S1[ set , set ] means for vv being Element of Vertices G st $1 = vv holds
( $2 <> c & $2 in C & ( for w being Element of Vertices G holds
( not w in Adjacent vv or not w in $2 ) ) );
A11: for e being set st e in c holds
ex u being set st S1[e,u]
proof
let v be set ; ::_thesis: ( v in c implies ex u being set st S1[v,u] )
assume A12: v in c ; ::_thesis: ex u being set st S1[v,u]
reconsider vv = v as Element of Vertices G by A12;
consider d being Element of C such that
A13: d <> c and
A14: for w being Element of Vertices G holds
( not w in Adjacent vv or not w in d ) by A12, A3;
take d ; ::_thesis: S1[v,d]
thus S1[v,d] by A13, A14, A1; ::_thesis: verum
end;
consider r being Function such that
A15: dom r = c and
A16: for e being set st e in c holds
S1[e,r . e] from CLASSES1:sch_1(A11);
defpred S2[ set ] means verum;
deffunc H1( set ) -> set = $1 \/ (r " {$1});
reconsider Cc = Cc as non empty finite set ;
set D = { H1(d) where d is Element of Cc : S2[d] } ;
consider d being set such that
A17: d in Cc by XBOOLE_0:def_1;
A18: d \/ (r " {d}) in { H1(d) where d is Element of Cc : S2[d] } by A17;
A19: { H1(d) where d is Element of Cc : S2[d] } c= bool (Vertices G)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { H1(d) where d is Element of Cc : S2[d] } or x in bool (Vertices G) )
assume x in { H1(d) where d is Element of Cc : S2[d] } ; ::_thesis: x in bool (Vertices G)
then consider d being Element of Cc such that
A20: x = d \/ (r " {d}) ;
A21: r " {d} c= c by A15, RELAT_1:132;
A22: r " {d} c= Vertices G by A21, XBOOLE_1:1;
d in C by XBOOLE_0:def_5;
then x c= Vertices G by A20, A22, XBOOLE_1:8;
hence x in bool (Vertices G) ; ::_thesis: verum
end;
A23: union { H1(d) where d is Element of Cc : S2[d] } = Vertices G
proof
thus union { H1(d) where d is Element of Cc : S2[d] } c= Vertices G :: according to XBOOLE_0:def_10 ::_thesis: Vertices G c= union { H1(d) where d is Element of Cc : S2[d] }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union { H1(d) where d is Element of Cc : S2[d] } or x in Vertices G )
assume x in union { H1(d) where d is Element of Cc : S2[d] } ; ::_thesis: x in Vertices G
then consider Y being set such that
A24: x in Y and
A25: Y in { H1(d) where d is Element of Cc : S2[d] } by TARSKI:def_4;
thus x in Vertices G by A24, A25, A19; ::_thesis: verum
end;
thus Vertices G c= union { H1(d) where d is Element of Cc : S2[d] } ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Vertices G or x in union { H1(d) where d is Element of Cc : S2[d] } )
assume A26: x in Vertices G ; ::_thesis: x in union { H1(d) where d is Element of Cc : S2[d] }
then consider d being set such that
A27: x in d and
A28: d in C by A4, TARSKI:def_4;
reconsider xp1 = x as Element of Vertices G by A26;
percases ( d = c or d <> c ) ;
supposeA29: d = c ; ::_thesis: x in union { H1(d) where d is Element of Cc : S2[d] }
then r . xp1 <> c by A27, A16;
then A30: not r . xp1 in {c} by TARSKI:def_1;
r . xp1 in C by A27, A29, A16;
then A31: r . xp1 in Cc by A30, XBOOLE_0:def_5;
r . xp1 in {(r . xp1)} by TARSKI:def_1;
then x in r " {(r . xp1)} by A27, A29, A15, FUNCT_1:def_7;
then A32: x in (r . xp1) \/ (r " {(r . xp1)}) by XBOOLE_0:def_3;
(r . xp1) \/ (r " {(r . xp1)}) in { H1(d) where d is Element of Cc : S2[d] } by A31;
hence x in union { H1(d) where d is Element of Cc : S2[d] } by A32, TARSKI:def_4; ::_thesis: verum
end;
suppose d <> c ; ::_thesis: x in union { H1(d) where d is Element of Cc : S2[d] }
then not d in {c} by TARSKI:def_1;
then d in Cc by A28, XBOOLE_0:def_5;
then A33: d \/ (r " {d}) in { H1(d) where d is Element of Cc : S2[d] } ;
x in d \/ (r " {d}) by A27, XBOOLE_0:def_3;
hence x in union { H1(d) where d is Element of Cc : S2[d] } by A33, TARSKI:def_4; ::_thesis: verum
end;
end;
end;
end;
A34: for A being Subset of (Vertices G) st A in { H1(d) where d is Element of Cc : S2[d] } holds
( A <> {} & ( for B being Subset of (Vertices G) holds
( not B in { H1(d) where d is Element of Cc : S2[d] } or A = B or A misses B ) ) )
proof
let A be Subset of (Vertices G); ::_thesis: ( A in { H1(d) where d is Element of Cc : S2[d] } implies ( A <> {} & ( for B being Subset of (Vertices G) holds
( not B in { H1(d) where d is Element of Cc : S2[d] } or A = B or A misses B ) ) ) )
assume A in { H1(d) where d is Element of Cc : S2[d] } ; ::_thesis: ( A <> {} & ( for B being Subset of (Vertices G) holds
( not B in { H1(d) where d is Element of Cc : S2[d] } or A = B or A misses B ) ) )
then consider da being Element of Cc such that
A35: A = da \/ (r " {da}) ;
A36: da in C by XBOOLE_0:def_5;
hence A <> {} by A35; ::_thesis: for B being Subset of (Vertices G) holds
( not B in { H1(d) where d is Element of Cc : S2[d] } or A = B or A misses B )
let B be Subset of (Vertices G); ::_thesis: ( not B in { H1(d) where d is Element of Cc : S2[d] } or A = B or A misses B )
assume B in { H1(d) where d is Element of Cc : S2[d] } ; ::_thesis: ( A = B or A misses B )
then consider db being Element of Cc such that
A37: B = db \/ (r " {db}) ;
A38: db in C by XBOOLE_0:def_5;
percases ( da = db or da <> db ) ;
suppose da = db ; ::_thesis: ( A = B or A misses B )
hence ( A = B or A misses B ) by A35, A37; ::_thesis: verum
end;
supposeA39: da <> db ; ::_thesis: ( A = B or A misses B )
then A40: da misses db by A36, A38, EQREL_1:def_4;
A41: r " {da} misses r " {db} by A39, FUNCT_1:71, ZFMISC_1:11;
assume A <> B ; ::_thesis: A misses B
assume A meets B ; ::_thesis: contradiction
then consider x being set such that
A42: x in A and
A43: x in B by XBOOLE_0:3;
percases ( ( x in da & x in db ) or ( x in da & x in r " {db} ) or ( x in r " {da} & x in db ) or ( x in r " {da} & x in r " {db} ) ) by A42, A43, A35, A37, XBOOLE_0:def_3;
suppose ( x in da & x in db ) ; ::_thesis: contradiction
hence contradiction by A40, XBOOLE_0:3; ::_thesis: verum
end;
supposethat A44: x in da and
A45: x in r " {db} ; ::_thesis: contradiction
A46: da <> c by A6, XBOOLE_0:def_5;
r " {db} c= c by A15, RELAT_1:132;
then da meets c by A44, A45, XBOOLE_0:3;
hence contradiction by A46, A36, A1, EQREL_1:def_4; ::_thesis: verum
end;
supposethat A47: x in r " {da} and
A48: x in db ; ::_thesis: contradiction
A49: db <> c by A6, XBOOLE_0:def_5;
r " {da} c= c by A15, RELAT_1:132;
then db meets c by A47, A48, XBOOLE_0:3;
hence contradiction by A49, A38, A1, EQREL_1:def_4; ::_thesis: verum
end;
suppose ( x in r " {da} & x in r " {db} ) ; ::_thesis: contradiction
hence contradiction by A41, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
end;
end;
reconsider D = { H1(d) where d is Element of Cc : S2[d] } as a_partition of Vertices G by A19, A23, A34, EQREL_1:def_4;
now__::_thesis:_for_x_being_set_st_x_in_D_holds_
x_is_StableSet_of_G
let x be set ; ::_thesis: ( x in D implies x is StableSet of G )
assume A50: x in D ; ::_thesis: x is StableSet of G
then reconsider S = x as Subset of (Vertices G) ;
consider d being Element of Cc such that
A51: x = d \/ (r " {d}) by A50;
A52: r " {d} c= c by A15, RELAT_1:132;
A53: d in C by XBOOLE_0:def_5;
A54: d is StableSet of G by A53, LStableSetwise;
A55: c is StableSet of G by A1, LStableSetwise;
S is stable
proof
let a, b be set ; :: according to SCMYCIEL:def_19 ::_thesis: ( a <> b & a in S & b in S implies {a,b} nin G )
assume that
A58: a <> b and
A56: a in S and
A57: b in S ; ::_thesis: {a,b} nin G
reconsider aa = a, bb = b as Vertex of G by A56, A57;
percases ( ( a in d & b in d ) or ( a in d & b in r " {d} ) or ( a in r " {d} & b in d ) or ( a in r " {d} & b in r " {d} ) ) by A56, A57, A51, XBOOLE_0:def_3;
suppose ( a in d & b in d ) ; ::_thesis: {a,b} nin G
hence not {a,b} in G by A54, A58, Lstable; ::_thesis: verum
end;
supposethat A59: a in d and
A60: b in r " {d} ; ::_thesis: {a,b} nin G
r . b in {d} by A60, FUNCT_1:def_7;
then r . b = d by TARSKI:def_1;
then not a in Adjacent bb by A59, A52, A60, A16;
then not {aa,bb} in Edges G by Ladj;
hence not {a,b} in G by A58, SG4a; ::_thesis: verum
end;
supposethat A61: a in r " {d} and
A62: b in d ; ::_thesis: {a,b} nin G
r . a in {d} by A61, FUNCT_1:def_7;
then r . a = d by TARSKI:def_1;
then not b in Adjacent aa by A62, A52, A61, A16;
then not {bb,aa} in Edges G by Ladj;
hence not {a,b} in G by A58, SG4a; ::_thesis: verum
end;
suppose ( a in r " {d} & b in r " {d} ) ; ::_thesis: {a,b} nin G
hence not {a,b} in G by A52, A55, A58, Lstable; ::_thesis: verum
end;
end;
end;
hence x is StableSet of G ; ::_thesis: verum
end;
then reconsider D = D as Coloring of G by LStableSetwise;
card Cc = (card C) - (card {c}) by A1, EULER_1:4;
then (card Cc) + 1 = ((card C) - 1) + 1 by CARD_1:30;
then A63: card Cc < card C by NAT_1:13;
deffunc H2( set ) -> set = $1 \/ (r " {$1});
consider s being Function such that
A64: dom s = Cc and
A65: for x being set st x in Cc holds
s . x = H2(x) from FUNCT_1:sch_3();
A66: rng s c= D
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng s or y in D )
assume y in rng s ; ::_thesis: y in D
then consider d being set such that
A67: d in dom s and
A68: y = s . d by FUNCT_1:def_3;
y = d \/ (r " {d}) by A64, A65, A67, A68;
hence y in D by A67, A64; ::_thesis: verum
end;
then reconsider s = s as Function of Cc,D by A64, FUNCT_2:2;
A69: s is one-to-one
proof
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom s or not x2 in dom s or not s . x1 = s . x2 or x1 = x2 )
assume that
A70: x1 in dom s and
A71: x2 in dom s and
A72: s . x1 = s . x2 ; ::_thesis: x1 = x2
A73: s . x1 = x1 \/ (r " {x1}) by A70, A65, A64;
A74: s . x2 = x2 \/ (r " {x2}) by A71, A65, A64;
thus x1 c= x2 :: according to XBOOLE_0:def_10 ::_thesis: x2 c= x1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in x1 or x in x2 )
assume A75: x in x1 ; ::_thesis: x in x2
then A76: x in s . x1 by A73, XBOOLE_0:def_3;
percases ( x in x2 or x in r " {x2} ) by A76, A72, A74, XBOOLE_0:def_3;
suppose x in x2 ; ::_thesis: x in x2
hence x in x2 ; ::_thesis: verum
end;
supposeA77: x in r " {x2} ; ::_thesis: x in x2
A78: r " {x2} c= dom r by RELAT_1:132;
A79: x1 in C by A64, A70, XBOOLE_0:def_5;
reconsider x1 = x1 as Subset of (Vertices G) by A64, A70;
x1 meets c by A78, A77, A15, A75, XBOOLE_0:3;
then x1 = c by A79, A1, EQREL_1:def_4;
hence x in x2 by A6, A64, A70, XBOOLE_0:def_5; ::_thesis: verum
end;
end;
end;
thus x2 c= x1 ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in x2 or x in x1 )
assume A80: x in x2 ; ::_thesis: x in x1
then A81: x in s . x2 by A74, XBOOLE_0:def_3;
percases ( x in x1 or x in r " {x1} ) by A81, A72, A73, XBOOLE_0:def_3;
suppose x in x1 ; ::_thesis: x in x1
hence x in x1 ; ::_thesis: verum
end;
supposeA82: x in r " {x1} ; ::_thesis: x in x1
A83: r " {x1} c= dom r by RELAT_1:132;
A84: x2 in C by A64, A71, XBOOLE_0:def_5;
reconsider x2 = x2 as Subset of (Vertices G) by A64, A71;
x2 meets c by A83, A82, A15, A80, XBOOLE_0:3;
then x2 = c by A84, A1, EQREL_1:def_4;
hence x in x1 by A6, A64, A71, XBOOLE_0:def_5; ::_thesis: verum
end;
end;
end;
end;
D c= rng s
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in D or x in rng s )
assume x in D ; ::_thesis: x in rng s
then consider d being Element of Cc such that
A85: x = d \/ (r " {d}) ;
s . d = d \/ (r " {d}) by A65;
hence x in rng s by A85, A64, FUNCT_1:def_3; ::_thesis: verum
end;
then D = rng s by A66, XBOOLE_0:def_10;
then s is onto by FUNCT_2:def_3;
then A86: card Cc = card D by A69, A18, EULER_1:11;
then D is finite ;
hence contradiction by A63, A86, A2, Lchro; ::_thesis: verum
end;
end;
end;
definition
let G be SimpleGraph;
attrG is with_finite_stability# means :Lwfstab: :: SCMYCIEL:def 23
ex A being finite StableSet of G st
for B being finite StableSet of G holds card B <= card A;
end;
:: deftheorem Lwfstab defines with_finite_stability# SCMYCIEL:def_23_:_
for G being SimpleGraph holds
( G is with_finite_stability# iff ex A being finite StableSet of G st
for B being finite StableSet of G holds card B <= card A );
registration
cluster finite SimpleGraph-like -> with_finite_stability# for set ;
correctness
coherence
for b1 being SimpleGraph st b1 is finite holds
b1 is with_finite_stability# ;
proof
let R be SimpleGraph; ::_thesis: ( R is finite implies R is with_finite_stability# )
assume R is finite ; ::_thesis: R is with_finite_stability#
then reconsider R9 = R as finite SimpleGraph ;
reconsider VR = Vertices R9 as finite set ;
defpred S1[ Nat] means ex A being finite StableSet of R9 st card A = c1;
A1: for k being Nat st S1[k] holds
k <= card VR by NAT_1:43;
( {} VR is StableSet of R & card {} = 0 ) ;
then A2: ex k being Nat st S1[k] ;
consider k being Nat such that
A3: S1[k] and
A4: for n being Nat st S1[n] holds
n <= k from NAT_1:sch_6(A1, A2);
consider S being finite StableSet of R such that
A5: card S = k by A3;
take S ; :: according to SCMYCIEL:def_23 ::_thesis: for B being finite StableSet of R holds card B <= card S
let T be finite StableSet of R; ::_thesis: card T <= card S
thus card T <= card S by A5, A4; ::_thesis: verum
end;
end;
registration
let G be with_finite_stability# SimpleGraph;
cluster stable -> finite for Element of bool (Vertices G);
correctness
coherence
for b1 being StableSet of G holds b1 is finite ;
proof
consider A being finite StableSet of G such that
A1: for B being finite StableSet of G holds card A >= card B by Lwfstab;
given B being StableSet of G such that A2: B is infinite ; ::_thesis: contradiction
consider C being finite Subset of B such that
A3: card C > card A by A2, DILWORTH:5;
C is StableSet of G by Th16;
hence contradiction by A1, A3; ::_thesis: verum
end;
end;
registration
cluster non empty finite-membered V233() V267() subset-closed non void 1 -at_most_dimensional SimpleGraph-like with_finite_stability# for set ;
existence
ex b1 being SimpleGraph st
( b1 is with_finite_stability# & not b1 is void )
proof
reconsider G = {{},{{}}} as SimpleGraph by SingleVertex;
set A = union G;
union G = {{}} by SingleVertices;
then not G is void ;
hence ex b1 being SimpleGraph st
( b1 is with_finite_stability# & not b1 is void ) ; ::_thesis: verum
end;
end;
definition
let G be with_finite_stability# SimpleGraph;
func stability# G -> Nat means :Lstabno: :: SCMYCIEL:def 24
( ex A being finite StableSet of G st card A = it & ( for T being finite StableSet of G holds card T <= it ) );
existence
ex b1 being Nat st
( ex A being finite StableSet of G st card A = b1 & ( for T being finite StableSet of G holds card T <= b1 ) )
proof
consider A being finite StableSet of G such that
A1: for B being finite StableSet of G holds card A >= card B by Lwfstab;
take itt = card A; ::_thesis: ( ex A being finite StableSet of G st card A = itt & ( for T being finite StableSet of G holds card T <= itt ) )
thus ex A being finite StableSet of G st card A = itt ; ::_thesis: for T being finite StableSet of G holds card T <= itt
let T be finite StableSet of G; ::_thesis: card T <= itt
thus card T <= itt by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being Nat st ex A being finite StableSet of G st card A = b1 & ( for T being finite StableSet of G holds card T <= b1 ) & ex A being finite StableSet of G st card A = b2 & ( for T being finite StableSet of G holds card T <= b2 ) holds
b1 = b2
proof
let it1, it2 be Nat; ::_thesis: ( ex A being finite StableSet of G st card A = it1 & ( for T being finite StableSet of G holds card T <= it1 ) & ex A being finite StableSet of G st card A = it2 & ( for T being finite StableSet of G holds card T <= it2 ) implies it1 = it2 )
assume that
A2: ex S1 being finite StableSet of G st card S1 = it1 and
A3: for T being finite StableSet of G holds card T <= it1 and
A4: ex S2 being finite StableSet of G st card S2 = it2 and
A5: for T being finite StableSet of G holds card T <= it2 ; ::_thesis: it1 = it2
consider S1 being finite StableSet of G such that
A6: card S1 = it1 by A2;
consider S2 being finite StableSet of G such that
A7: card S2 = it2 by A4;
( it1 <= it2 & it2 <= it1 ) by A3, A5, A6, A7;
hence it1 = it2 by XXREAL_0:1; ::_thesis: verum
end;
end;
:: deftheorem Lstabno defines stability# SCMYCIEL:def_24_:_
for G being with_finite_stability# SimpleGraph
for b2 being Nat holds
( b2 = stability# G iff ( ex A being finite StableSet of G st card A = b2 & ( for T being finite StableSet of G holds card T <= b2 ) ) );
registration
let G be non void with_finite_stability# SimpleGraph;
cluster stability# G -> positive ;
correctness
coherence
stability# G is positive ;
proof
Vertices G <> {} by VoidGV;
then consider v being set such that
A: v in Vertices G by XBOOLE_0:def_1;
reconsider S = {v} as finite Subset of (Vertices G) by A, ZFMISC_1:31;
card S <= stability# G by Lstabno;
hence stability# G is positive ; ::_thesis: verum
end;
end;
theorem Th21: :: SCMYCIEL:71
for G being with_finite_stability# SimpleGraph st stability# G = 1 holds
G is clique
proof
let R be with_finite_stability# SimpleGraph; ::_thesis: ( stability# R = 1 implies R is clique )
assume A1: stability# R = 1 ; ::_thesis: R is clique
set cR = Vertices R;
now__::_thesis:_for_a,_b_being_set_st_a_<>_b_&_a_in_Vertices_R_&_b_in_Vertices_R_holds_
not_{a,b}_nin_Edges_R
let a, b be set ; ::_thesis: ( a <> b & a in Vertices R & b in Vertices R implies not {a,b} nin Edges R )
assume that
A3: a <> b and
A2: ( a in Vertices R & b in Vertices R ) ; ::_thesis: not {a,b} nin Edges R
assume {a,b} nin Edges R ; ::_thesis: contradiction
then {a,b} nin R by A3, SG4a;
then A5: {a,b} is StableSet of R by A2, Th14;
card {a,b} = 2 by A3, CARD_2:57;
hence contradiction by A1, A5, Lstabno; ::_thesis: verum
end;
hence R is clique by Lclique1; ::_thesis: verum
end;
registration
cluster SimpleGraph-like with_finite_clique# with_finite_stability# -> finite for set ;
correctness
coherence
for b1 being SimpleGraph st b1 is with_finite_clique# & b1 is with_finite_stability# holds
b1 is finite ;
proof
let R be SimpleGraph; ::_thesis: ( R is with_finite_clique# & R is with_finite_stability# implies R is finite )
assume A1: R is with_finite_clique# ; ::_thesis: ( not R is with_finite_stability# or R is finite )
assume A2: R is with_finite_stability# ; ::_thesis: R is finite
assume A3: R is infinite ; ::_thesis: contradiction
set VR = Vertices R;
A3a: Vertices R is infinite by A3, FinSG;
A3bb: R c= R ;
reconsider R = R as with_finite_clique# with_finite_stability# SimpleGraph by A1, A2;
consider C being finite Clique of R such that
A4: order C = clique# R by Lcliqueno;
reconsider VC = Vertices C as finite Subset of (Vertices R) by ZFMISC_1:77;
consider An being finite StableSet of R such that
A5: card An = stability# R by Lstabno;
reconsider VAn = An as finite Subset of (Vertices R) ;
set h = clique# R;
set w = stability# R;
A6: 0 + 1 <= clique# R by A3a, Cno0, NAT_1:14;
not R is void by A3;
then A7: 0 + 1 <= stability# R by NAT_1:13;
percases ( clique# R = 1 or stability# R = 1 or ( clique# R > 1 & stability# R > 1 ) ) by A6, A7, XXREAL_0:1;
suppose clique# R = 1 ; ::_thesis: contradiction
then A9: Vertices R is StableSet of R by Th19;
consider Y being finite Subset of (Vertices R) such that
A10: card Y > stability# R by A3a, DILWORTH:5;
Y is StableSet of R by A9, Th16;
hence contradiction by A10, Lstabno; ::_thesis: verum
end;
suppose stability# R = 1 ; ::_thesis: contradiction
then A11: R is Clique of R by A3bb, Th21;
consider Y being finite Subset of (Vertices R) such that
A12: card Y > clique# R by A3a, DILWORTH:5;
A12a: R SubgraphInducedBy Y is Clique of R by A11, SGClique0;
order (R SubgraphInducedBy Y) = card Y by Sub0c;
hence contradiction by A12, A12a, Lcliqueno; ::_thesis: verum
end;
supposeA13: ( clique# R > 1 & stability# R > 1 ) ; ::_thesis: contradiction
set m = (max ((clique# R),(stability# R))) + 1;
reconsider m = (max ((clique# R),(stability# R))) + 1 as natural number ;
consider r being natural number such that
A14: for X being finite set
for P being a_partition of the_subsets_of_card (2,X) st card X >= r & card P = 2 holds
ex S being Subset of X st
( card S >= m & S is_homogeneous_for P ) by RAMSEY_1:17;
consider Y being finite Subset of (Vertices R) such that
A15: card Y > r by A3a, DILWORTH:5;
set X = (Y \/ VAn) \/ VC;
reconsider X = (Y \/ VAn) \/ VC as finite Subset of (Vertices R) ;
( Y c= Y \/ An & Y \/ An c= (Y \/ An) \/ VC ) by XBOOLE_1:7;
then A16: card Y <= card X by NAT_1:43, XBOOLE_1:1;
set A = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } ;
set B = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } ;
set E = the_subsets_of_card (2,X);
set P = { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } };
A17: { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } c= the_subsets_of_card (2,X)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } or x in the_subsets_of_card (2,X) )
assume x in { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } ; ::_thesis: x in the_subsets_of_card (2,X)
then consider xx, yy being Element of Vertices R such that
A18: {xx,yy} = x and
A19: xx <> yy and
A20: xx in X and
A21: yy in X and
{xx,yy} in Edges R ;
( x is Subset of X & card x = 2 ) by A18, A19, A20, A21, CARD_2:57, ZFMISC_1:32;
hence x in the_subsets_of_card (2,X) ; ::_thesis: verum
end;
A22: { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } c= the_subsets_of_card (2,X)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } or x in the_subsets_of_card (2,X) )
assume x in { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } ; ::_thesis: x in the_subsets_of_card (2,X)
then consider xx, yy being Element of Vertices R such that
A23: {xx,yy} = x and
A24: xx <> yy and
A25: xx in X and
A26: yy in X and
{xx,yy} nin Edges R ;
( x is Subset of X & card x = 2 ) by A23, A24, A25, A26, CARD_2:57, ZFMISC_1:32;
hence x in the_subsets_of_card (2,X) ; ::_thesis: verum
end;
A27: ( { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } in { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } & { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } in { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } ) by TARSKI:def_2;
A28: now__::_thesis:_not__{__{x,y}_where_x,_y_is_Element_of_Vertices_R_:_(_x_<>_y_&_x_in_X_&_y_in_X_&_{x,y}_in_Edges_R_)__}__=__{__{x,y}_where_x,_y_is_Element_of_Vertices_R_:_(_x_<>_y_&_x_in_X_&_y_in_X_&_{x,y}_nin_Edges_R_)__}_
assume A29: { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } ; ::_thesis: contradiction
consider a, b being set such that
A30: a in An and
A31: b in An and
A32: a <> b by A13, A5, NAT_1:59;
reconsider a = a, b = b as Element of Vertices R by A30, A31;
A33: {a,b} nin Edges R by A30, A31, A32, Lstable;
( a in Y \/ An & b in Y \/ An ) by A30, A31, XBOOLE_0:def_3;
then ( a in X & b in X ) by XBOOLE_0:def_3;
then {a,b} in { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } by A33, A32;
then consider aa, bb being Element of Vertices R such that
A34: {a,b} = {aa,bb} and
( aa <> bb & aa in X & bb in X ) and
A35: {aa,bb} in Edges R by A29;
thus contradiction by A35, A30, A31, A32, Lstable, A34; ::_thesis: verum
end;
A36: { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } c= bool (the_subsets_of_card (2,X))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } or x in bool (the_subsets_of_card (2,X)) )
assume x in { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } ; ::_thesis: x in bool (the_subsets_of_card (2,X))
then x c= the_subsets_of_card (2,X) by A17, A22, TARSKI:def_2;
hence x in bool (the_subsets_of_card (2,X)) ; ::_thesis: verum
end;
A37: union { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } = the_subsets_of_card (2,X)
proof
thus union { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } c= the_subsets_of_card (2,X) :: according to XBOOLE_0:def_10 ::_thesis: the_subsets_of_card (2,X) c= union { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } or x in the_subsets_of_card (2,X) )
assume x in union { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } ; ::_thesis: x in the_subsets_of_card (2,X)
then consider Y being set such that
A38: x in Y and
A39: Y in { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } by TARSKI:def_4;
( Y = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } or Y = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } ) by A39, TARSKI:def_2;
hence x in the_subsets_of_card (2,X) by A38, A17, A22; ::_thesis: verum
end;
thus the_subsets_of_card (2,X) c= union { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the_subsets_of_card (2,X) or x in union { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } )
assume x in the_subsets_of_card (2,X) ; ::_thesis: x in union { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } }
then consider xx being Subset of X such that
A40: x = xx and
A41: card xx = 2 ;
consider a, b being set such that
A42: a <> b and
A43: xx = {a,b} by A41, CARD_2:60;
( a in xx & b in xx ) by A43, TARSKI:def_2;
then ( a in X & b in X ) ;
then reconsider a = a, b = b as Element of Vertices R ;
A44: ( a in xx & b in xx ) by A43, TARSKI:def_2;
( {a,b} in Edges R or {a,b} nin Edges R ) ;
then ( {a,b} in { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } or {a,b} in { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } ) by A44, A42;
hence x in union { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } by A40, A43, A27, TARSKI:def_4; ::_thesis: verum
end;
end;
for a being Subset of (the_subsets_of_card (2,X)) st a in { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } holds
( a <> {} & ( for b being Subset of (the_subsets_of_card (2,X)) holds
( not b in { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } or a = b or a misses b ) ) )
proof
let a be Subset of (the_subsets_of_card (2,X)); ::_thesis: ( a in { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } implies ( a <> {} & ( for b being Subset of (the_subsets_of_card (2,X)) holds
( not b in { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } or a = b or a misses b ) ) ) )
assume A45: a in { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } ; ::_thesis: ( a <> {} & ( for b being Subset of (the_subsets_of_card (2,X)) holds
( not b in { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } or a = b or a misses b ) ) )
thus a <> {} ::_thesis: for b being Subset of (the_subsets_of_card (2,X)) holds
( not b in { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } or a = b or a misses b )
proof
percases ( a = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } or a = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } ) by A45, TARSKI:def_2;
supposeA46: a = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } ; ::_thesis: a <> {}
consider aa, bb being set such that
A47: aa in VC and
A48: bb in VC and
A49: aa <> bb by A13, A4, NAT_1:59;
reconsider aa = aa, bb = bb as Element of Vertices R by A47, A48;
{aa,bb} in C by A47, A48, Clique2a;
then A51: {aa,bb} in Edges R by A49, SG4a;
( aa in (Y \/ An) \/ VC & bb in (Y \/ An) \/ VC ) by A47, A48, XBOOLE_0:def_3;
then {aa,bb} in { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } by A49, A51;
hence a <> {} by A46; ::_thesis: verum
end;
supposeA51: a = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } ; ::_thesis: a <> {}
consider aa, bb being set such that
A52: aa in An and
A53: bb in An and
A54: aa <> bb by A13, A5, NAT_1:59;
reconsider aa = aa, bb = bb as Element of Vertices R by A52, A53;
A55a: {aa,bb} nin Edges R by A52, A53, A54, Lstable;
( aa in Y \/ An & bb in Y \/ An ) by A52, A53, XBOOLE_0:def_3;
then ( aa in X & bb in X ) by XBOOLE_0:def_3;
then {aa,bb} in { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } by A54, A55a;
hence a <> {} by A51; ::_thesis: verum
end;
end;
end;
let b be Subset of (the_subsets_of_card (2,X)); ::_thesis: ( not b in { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } or a = b or a misses b )
assume A56: b in { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } ; ::_thesis: ( a = b or a misses b )
assume A57: a <> b ; ::_thesis: a misses b
assume A58: a meets b ; ::_thesis: contradiction
( ( a = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } or a = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } ) & ( b = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } or b = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } ) ) by A45, A56, TARSKI:def_2;
then { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } /\ { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } <> {} by A57, A58, XBOOLE_0:def_7;
then consider x being set such that
A59: x in { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } /\ { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } by XBOOLE_0:def_1;
x in { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } by A59, XBOOLE_0:def_4;
then consider xx, yy being Element of Vertices R such that
A60: {xx,yy} = x and
( xx <> yy & xx in X & yy in X ) and
A61: {xx,yy} in Edges R ;
x in { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } by A59, XBOOLE_0:def_4;
then consider x2, y2 being Element of Vertices R such that
A62: {x2,y2} = x and
( x2 <> y2 & x2 in X & y2 in X ) and
A63: {x2,y2} nin Edges R ;
thus contradiction by A61, A63, A60, A62; ::_thesis: verum
end;
then reconsider P = { { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } , { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } } as a_partition of the_subsets_of_card (2,X) by A37, A36, EQREL_1:def_4;
card P = 2 by A28, CARD_2:57;
then consider S being Subset of X such that
A64: card S >= m and
A65: S is_homogeneous_for P by A16, A14, A15, XXREAL_0:2;
reconsider S = S as finite Subset of (Vertices R) by XBOOLE_1:1;
max ((clique# R),(stability# R)) >= clique# R by XXREAL_0:25;
then m > clique# R by NAT_1:13;
then A66: card S > clique# R by A64, XXREAL_0:2;
max ((clique# R),(stability# R)) >= stability# R by XXREAL_0:25;
then m > stability# R by NAT_1:13;
then A67: card S > stability# R by A64, XXREAL_0:2;
consider p being Element of P such that
A68: the_subsets_of_card (2,S) c= p by A65, RAMSEY_1:def_1;
percases ( p = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } or p = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } ) by TARSKI:def_2;
supposeA69: p = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } ; ::_thesis: contradiction
set H = R SubgraphInducedBy S;
B72: Vertices (R SubgraphInducedBy S) = S by Sub0c;
now__::_thesis:_for_x,_y_being_set_st_x_<>_y_&_x_in_union_(R_SubgraphInducedBy_S)_&_y_in_union_(R_SubgraphInducedBy_S)_holds_
{x,y}_in_Edges_(R_SubgraphInducedBy_S)
let x, y be set ; ::_thesis: ( x <> y & x in union (R SubgraphInducedBy S) & y in union (R SubgraphInducedBy S) implies {x,y} in Edges (R SubgraphInducedBy S) )
assume that
A72: x <> y and
A70: x in union (R SubgraphInducedBy S) and
A71: y in union (R SubgraphInducedBy S) ; ::_thesis: {x,y} in Edges (R SubgraphInducedBy S)
( {x,y} is Subset of S & card {x,y} = 2 ) by B72, A70, A71, A72, CARD_2:57, ZFMISC_1:32;
then {x,y} in the_subsets_of_card (2,S) ;
then {x,y} in { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} in Edges R ) } by A69, A68;
then consider xx, yy being Element of Vertices R such that
A73: {xx,yy} = {x,y} and
( xx <> yy & xx in X & yy in X ) and
A74: {xx,yy} in Edges R ;
{x,y} in R SubgraphInducedBy S by A74, A70, A71, B72, Sub6, A73;
hence {x,y} in Edges (R SubgraphInducedBy S) by A72, SG4a; ::_thesis: verum
end;
then R SubgraphInducedBy S is finite Clique of R by Lclique1;
then order (R SubgraphInducedBy S) <= clique# R by Lcliqueno;
hence contradiction by A66, Sub0c; ::_thesis: verum
end;
supposeA75: p = { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } ; ::_thesis: contradiction
now__::_thesis:_for_x,_y_being_set_st_x_<>_y_&_x_in_S_&_y_in_S_holds_
{x,y}_nin_R
let x, y be set ; ::_thesis: ( x <> y & x in S & y in S implies {x,y} nin R )
assume that
A78: x <> y and
A76: x in S and
A77: y in S ; ::_thesis: {x,y} nin R
( {x,y} is Subset of S & card {x,y} = 2 ) by A76, A77, A78, CARD_2:57, ZFMISC_1:32;
then {x,y} in the_subsets_of_card (2,S) ;
then {x,y} in { {x,y} where x, y is Element of Vertices R : ( x <> y & x in X & y in X & {x,y} nin Edges R ) } by A75, A68;
then consider xx, yy being Element of Vertices R such that
A79: {xx,yy} = {x,y} and
( xx <> yy & xx in X & yy in X ) and
A80: {xx,yy} nin Edges R ;
thus {x,y} nin R by A78, A80, SG4a, A79; ::_thesis: verum
end;
then S is stable by Lstable;
hence contradiction by A67, Lstabno; ::_thesis: verum
end;
end;
end;
end;
end;
end;
theorem CliStaCompl: :: SCMYCIEL:72
for G being SimpleGraph
for C being Clique of G holds Vertices C is StableSet of (Complement G)
proof
let G be SimpleGraph; ::_thesis: for C being Clique of G holds Vertices C is StableSet of (Complement G)
let C be Clique of G; ::_thesis: Vertices C is StableSet of (Complement G)
set CG = Complement G;
A: Vertices G = Vertices (Complement G) by Compl1;
reconsider uC = union C as Subset of (Vertices (Complement G)) by A, ZFMISC_1:77;
now__::_thesis:_for_x,_y_being_set_st_x_<>_y_&_x_in_uC_&_y_in_uC_holds_
{x,y}_nin_Complement_G
let x, y be set ; ::_thesis: ( x <> y & x in uC & y in uC implies {x,y} nin Complement G )
assume that
A3: x <> y and
A1: x in uC and
A2: y in uC ; ::_thesis: {x,y} nin Complement G
{x,y} in C by A1, A2, Clique2a;
then {x,y} in Edges G by A3, SG4a;
hence {x,y} nin Complement G by XBOOLE_0:def_5; ::_thesis: verum
end;
hence union C is StableSet of (Complement G) by Lstable; ::_thesis: verum
end;
theorem CliComplSta: :: SCMYCIEL:73
for G being SimpleGraph
for C being Clique of (Complement G) holds Vertices C is StableSet of G
proof
let G be SimpleGraph; ::_thesis: for C being Clique of (Complement G) holds Vertices C is StableSet of G
let C be Clique of (Complement G); ::_thesis: Vertices C is StableSet of G
Vertices C is StableSet of (Complement (Complement G)) by CliStaCompl;
hence Vertices C is StableSet of G ; ::_thesis: verum
end;
theorem StaCliCompl: :: SCMYCIEL:74
for G being SimpleGraph
for C being StableSet of G holds (Complement G) SubgraphInducedBy C is Clique of (Complement G)
proof
let G be SimpleGraph; ::_thesis: for C being StableSet of G holds (Complement G) SubgraphInducedBy C is Clique of (Complement G)
let C be StableSet of G; ::_thesis: (Complement G) SubgraphInducedBy C is Clique of (Complement G)
set CG = Complement G;
set CGSC = (Complement G) SubgraphInducedBy C;
set uCGSC = union ((Complement G) SubgraphInducedBy C);
now__::_thesis:_for_a,_b_being_set_st_a_<>_b_&_a_in_union_((Complement_G)_SubgraphInducedBy_C)_&_b_in_union_((Complement_G)_SubgraphInducedBy_C)_holds_
{a,b}_in_Edges_((Complement_G)_SubgraphInducedBy_C)
let a, b be set ; ::_thesis: ( a <> b & a in union ((Complement G) SubgraphInducedBy C) & b in union ((Complement G) SubgraphInducedBy C) implies {a,b} in Edges ((Complement G) SubgraphInducedBy C) )
assume that
A4: a <> b and
A2: a in union ((Complement G) SubgraphInducedBy C) and
A3: b in union ((Complement G) SubgraphInducedBy C) ; ::_thesis: {a,b} in Edges ((Complement G) SubgraphInducedBy C)
B1: ( a in C & b in C ) by A2, A3, Sub1;
D1: {a,b} nin Edges G by B1, A4, Lstable;
E1: {a,b} in CompleteSGraph (Vertices G) by B1, CSG1;
{a,b} in Complement G by D1, E1, XBOOLE_0:def_5;
then {a,b} in (Complement G) SubgraphInducedBy C by B1, Sub6;
hence {a,b} in Edges ((Complement G) SubgraphInducedBy C) by A4, SG4a; ::_thesis: verum
end;
hence (Complement G) SubgraphInducedBy C is Clique of (Complement G) by Lclique1; ::_thesis: verum
end;
theorem StaComplCli: :: SCMYCIEL:75
for G being SimpleGraph
for C being StableSet of (Complement G) holds G SubgraphInducedBy C is Clique of G
proof
let G be SimpleGraph; ::_thesis: for C being StableSet of (Complement G) holds G SubgraphInducedBy C is Clique of G
let C be StableSet of (Complement G); ::_thesis: G SubgraphInducedBy C is Clique of G
(Complement (Complement G)) SubgraphInducedBy C is Clique of (Complement (Complement G)) by StaCliCompl;
hence G SubgraphInducedBy C is Clique of G ; ::_thesis: verum
end;
registration
let G be with_finite_clique# SimpleGraph;
cluster Complement G -> with_finite_stability# ;
correctness
coherence
Complement G is with_finite_stability# ;
proof
set CG = Complement G;
consider A being finite Clique of G such that
A: for B being finite Clique of G holds order B <= order A by Lwfcno;
B: Vertices G = Vertices (Complement G) by Compl1;
set C = union A;
reconsider C = union A as finite StableSet of (Complement G) by CliStaCompl;
take C ; :: according to SCMYCIEL:def_23 ::_thesis: for B being finite StableSet of (Complement G) holds card B <= card C
let D be finite StableSet of (Complement G); ::_thesis: card D <= card C
A1: G SubgraphInducedBy D is finite Clique of G by StaComplCli;
order (G SubgraphInducedBy D) <= order A by A1, A;
hence card D <= card C by B, Sub0c; ::_thesis: verum
end;
end;
registration
let G be with_finite_stability# SimpleGraph;
cluster Complement G -> with_finite_clique# ;
correctness
coherence
Complement G is with_finite_clique# ;
proof
set CG = Complement G;
consider A being finite StableSet of G such that
A: for B being finite StableSet of G holds card B <= card A by Lwfstab;
B: Vertices G = Vertices (Complement G) by Compl1;
set C = (Complement G) SubgraphInducedBy A;
reconsider C = (Complement G) SubgraphInducedBy A as finite Clique of (Complement G) by StaCliCompl;
take C ; :: according to SCMYCIEL:def_14 ::_thesis: for D being finite Clique of (Complement G) holds order D <= order C
let D be finite Clique of (Complement G); ::_thesis: order D <= order C
A1: union D is StableSet of G by CliComplSta;
A = union C by B, Sub0c;
hence order D <= order C by A, A1; ::_thesis: verum
end;
end;
theorem cliRstaCR: :: SCMYCIEL:76
for G being with_finite_clique# SimpleGraph holds clique# G = stability# (Complement G)
proof
let G be with_finite_clique# SimpleGraph; ::_thesis: clique# G = stability# (Complement G)
set CG = Complement G;
set sCG = stability# (Complement G);
set cG = clique# G;
consider C being finite Clique of G such that
A: order C = clique# G by Lcliqueno;
B: Vertices G = Vertices (Complement G) by Compl1;
reconsider A = union C as StableSet of (Complement G) by CliStaCompl;
X: card A = clique# G by A;
now__::_thesis:_for_T_being_finite_StableSet_of_(Complement_G)_holds_card_T_<=_clique#_G
let T be finite StableSet of (Complement G); ::_thesis: card T <= clique# G
G SubgraphInducedBy T is Clique of G by StaComplCli;
then order (G SubgraphInducedBy T) <= clique# G by Lcliqueno;
hence card T <= clique# G by B, Sub0c; ::_thesis: verum
end;
hence clique# G = stability# (Complement G) by X, Lstabno; ::_thesis: verum
end;
theorem :: SCMYCIEL:77
for G being with_finite_stability# SimpleGraph holds stability# G = clique# (Complement G)
proof
let G be with_finite_stability# SimpleGraph; ::_thesis: stability# G = clique# (Complement G)
Complement (Complement G) = G ;
hence stability# G = clique# (Complement G) by cliRstaCR; ::_thesis: verum
end;
theorem ClicoComplChr: :: SCMYCIEL:78
for G being SimpleGraph
for C being Clique-partition of (Complement G) holds C is Coloring of G
proof
let G be SimpleGraph; ::_thesis: for C being Clique-partition of (Complement G) holds C is Coloring of G
let C be Clique-partition of (Complement G); ::_thesis: C is Coloring of G
set CG = Complement G;
now__::_thesis:_for_x_being_set_st_x_in_C_holds_
x_is_StableSet_of_G
let x be set ; ::_thesis: ( x in C implies x is StableSet of G )
assume A0: x in C ; ::_thesis: x is StableSet of G
then A1: (Complement G) SubgraphInducedBy x is Clique of (Complement G) by LCliquewise;
union ((Complement G) SubgraphInducedBy x) = x by A0, Sub0c;
hence x is StableSet of G by A1, CliComplSta; ::_thesis: verum
end;
hence C is Coloring of G by Compl1, LStableSetwise; ::_thesis: verum
end;
theorem ClicoChrCompl: :: SCMYCIEL:79
for G being SimpleGraph
for C being Clique-partition of G holds C is Coloring of (Complement G)
proof
let G be SimpleGraph; ::_thesis: for C being Clique-partition of G holds C is Coloring of (Complement G)
let C be Clique-partition of G; ::_thesis: C is Coloring of (Complement G)
Complement (Complement G) = G ;
hence C is Coloring of (Complement G) by ClicoComplChr; ::_thesis: verum
end;
theorem ChrClicoCompl: :: SCMYCIEL:80
for G being SimpleGraph
for C being Coloring of G holds C is Clique-partition of (Complement G)
proof
let G be SimpleGraph; ::_thesis: for C being Coloring of G holds C is Clique-partition of (Complement G)
let C be Coloring of G; ::_thesis: C is Clique-partition of (Complement G)
set CG = Complement G;
now__::_thesis:_for_x_being_set_st_x_in_C_holds_
(Complement_G)_SubgraphInducedBy_x_is_Clique_of_(Complement_G)
let x be set ; ::_thesis: ( x in C implies (Complement G) SubgraphInducedBy x is Clique of (Complement G) )
assume x in C ; ::_thesis: (Complement G) SubgraphInducedBy x is Clique of (Complement G)
then x is StableSet of G by LStableSetwise;
hence (Complement G) SubgraphInducedBy x is Clique of (Complement G) by StaCliCompl; ::_thesis: verum
end;
hence C is Clique-partition of (Complement G) by Compl1, LCliquewise; ::_thesis: verum
end;
theorem :: SCMYCIEL:81
for G being SimpleGraph
for C being Coloring of (Complement G) holds C is Clique-partition of G
proof
let G be SimpleGraph; ::_thesis: for C being Coloring of (Complement G) holds C is Clique-partition of G
let C be Coloring of (Complement G); ::_thesis: C is Clique-partition of G
Complement (Complement G) = G ;
hence C is Clique-partition of G by ChrClicoCompl; ::_thesis: verum
end;
registration
let G be finitely_colorable SimpleGraph;
cluster Complement G -> with_finite_cliquecover# ;
correctness
coherence
Complement G is with_finite_cliquecover# ;
proof
consider C being Coloring of G such that
A1: C is finite by Lfc;
C is Clique-partition of (Complement G) by ChrClicoCompl;
hence Complement G is with_finite_cliquecover# by A1, Lwfclicov; ::_thesis: verum
end;
end;
registration
let G be with_finite_cliquecover# SimpleGraph;
cluster Complement G -> finitely_colorable ;
correctness
coherence
Complement G is finitely_colorable ;
proof
consider C being Clique-partition of G such that
A1: C is finite by Lwfclicov;
C is Coloring of (Complement G) by ClicoChrCompl;
hence Complement G is finitely_colorable by A1, Lfc; ::_thesis: verum
end;
end;
theorem chrRcovCR: :: SCMYCIEL:82
for G being finitely_colorable SimpleGraph holds chromatic# G = cliquecover# (Complement G)
proof
let G be finitely_colorable SimpleGraph; ::_thesis: chromatic# G = cliquecover# (Complement G)
set CG = Complement G;
set k = cliquecover# (Complement G);
consider C being finite Clique-partition of (Complement G) such that
A1: card C = cliquecover# (Complement G) by Lclicovno;
A2a: C is Coloring of G by ClicoComplChr;
now__::_thesis:_for_C_being_finite_Coloring_of_G_holds_not_cliquecover#_(Complement_G)_>_card_C
let C be finite Coloring of G; ::_thesis: not cliquecover# (Complement G) > card C
assume A3: cliquecover# (Complement G) > card C ; ::_thesis: contradiction
C is Clique-partition of (Complement G) by ChrClicoCompl;
hence contradiction by A3, Lclicovno; ::_thesis: verum
end;
hence chromatic# G = cliquecover# (Complement G) by A2a, A1, Lchro; ::_thesis: verum
end;
theorem :: SCMYCIEL:83
for G being with_finite_cliquecover# SimpleGraph holds cliquecover# G = chromatic# (Complement G)
proof
let G be with_finite_cliquecover# SimpleGraph; ::_thesis: cliquecover# G = chromatic# (Complement G)
Complement (Complement G) = G ;
hence cliquecover# G = chromatic# (Complement G) by chrRcovCR; ::_thesis: verum
end;
begin
definition
let G be SimpleGraph;
func Mycielskian G -> SimpleGraph equals :: SCMYCIEL:def 25
((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ;
correctness
coherence
((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } is SimpleGraph;
proof
set uG = union G;
set C = { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ;
set A = { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ;
set B = { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ;
set M = ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ;
reconsider N = ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } as non empty set ;
B: ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } is subset-closed
proof
let a, b be set ; :: according to CLASSES1:def_1 ::_thesis: ( not a in ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } or not b c= a or b in ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } )
assume that
A1: a in ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } and
B1: b c= a ; ::_thesis: b in ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G }
C1a: {} in {{}} by TARSKI:def_1;
then C1: {} in ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } by MYCIELSK:4;
percases ( a in {{}} or a in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } or a in Edges G or a in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } or a in { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ) by A1, MYCIELSK:4;
suppose a in {{}} ; ::_thesis: b in ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G }
then a = {} by TARSKI:def_1;
hence b in ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } by B1, C1; ::_thesis: verum
end;
suppose a in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ; ::_thesis: b in ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G }
then consider x being Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} such that
A2: a = {x} and
verum ;
thus b in ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } by C1, A2, A1, B1, ZFMISC_1:33; ::_thesis: verum
end;
suppose a in Edges G ; ::_thesis: b in ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G }
then consider x, y being set such that
x <> y and
B2: x in Vertices G and
C2: y in Vertices G and
D2: a = {x,y} by SG4;
E2: ( b = {} or b = {x} or b = {y} or b = {x,y} ) by D2, B1, ZFMISC_1:36;
( x in (union G) \/ [:(union G),{(union G)}:] & y in (union G) \/ [:(union G),{(union G)}:] ) by B2, C2, XBOOLE_0:def_3;
then ( x in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} & y in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} ) by XBOOLE_0:def_3;
then ( {x} in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } & {y} in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) ;
hence b in ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } by E2, C1a, D2, A1, MYCIELSK:4; ::_thesis: verum
end;
suppose a in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ; ::_thesis: b in ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G }
then consider x, y being Element of union G such that
A2: a = {x,[y,(union G)]} and
B2: {x,y} in Edges G ;
C2: x in union G by B2, SG5;
E2: ( b = {} or b = {x} or b = {[y,(union G)]} or b = {x,[y,(union G)]} ) by A2, B1, ZFMISC_1:36;
x in (union G) \/ [:(union G),{(union G)}:] by C2, XBOOLE_0:def_3;
then x in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} by XBOOLE_0:def_3;
then F2: {x} in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ;
( y in union G & union G in {(union G)} ) by B2, SG5, TARSKI:def_1;
then [y,(union G)] in [:(union G),{(union G)}:] by ZFMISC_1:def_2;
then [y,(union G)] in (union G) \/ [:(union G),{(union G)}:] by XBOOLE_0:def_3;
then [y,(union G)] in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} by XBOOLE_0:def_3;
then {[y,(union G)]} in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ;
hence b in ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } by A2, A1, E2, C1a, F2, MYCIELSK:4; ::_thesis: verum
end;
suppose a in { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ; ::_thesis: b in ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G }
then consider x being Element of union G such that
A2: a = {(union G),[x,(union G)]} and
C2: x in Vertices G ;
E2: ( b = {} or b = {(union G)} or b = {[x,(union G)]} or b = {(union G),[x,(union G)]} ) by A2, B1, ZFMISC_1:36;
union G in {(union G)} by TARSKI:def_1;
then union G in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} by XBOOLE_0:def_3;
then F2: {(union G)} in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ;
( x in union G & union G in {(union G)} ) by C2, TARSKI:def_1;
then [x,(union G)] in [:(union G),{(union G)}:] by ZFMISC_1:def_2;
then [x,(union G)] in (union G) \/ [:(union G),{(union G)}:] by XBOOLE_0:def_3;
then [x,(union G)] in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} by XBOOLE_0:def_3;
then {[x,(union G)]} in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ;
hence b in ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } by A2, A1, E2, C1a, F2, MYCIELSK:4; ::_thesis: verum
end;
end;
end;
C: N is 1 -at_most_dimensional
proof
let a be set ; :: according to SCMYCIEL:def_4 ::_thesis: ( a in N implies card a c= 1 + 1 )
assume Aa: a in N ; ::_thesis: card a c= 1 + 1
percases ( a in {{}} or a in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } or a in Edges G or a in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } or a in { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ) by Aa, MYCIELSK:4;
suppose a in {{}} ; ::_thesis: card a c= 1 + 1
then a = {} by TARSKI:def_1;
hence card a c= 1 + 1 ; ::_thesis: verum
end;
suppose a in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ; ::_thesis: card a c= 1 + 1
then consider x being Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} such that
A1: a = {x} and
verum ;
card a = 1 by A1, CARD_1:30;
hence card a c= 1 + 1 by NAT_1:39; ::_thesis: verum
end;
suppose a in Edges G ; ::_thesis: card a c= 1 + 1
hence card a c= 1 + 1 by Lnatmost; ::_thesis: verum
end;
suppose a in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ; ::_thesis: card a c= 1 + 1
then consider x, y being Element of union G such that
A1: a = {x,[y,(union G)]} and
{x,y} in Edges G ;
card {x,[y,(union G)]} <= 2 by CARD_2:50;
hence card a c= 1 + 1 by A1, NAT_1:39; ::_thesis: verum
end;
suppose a in { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ; ::_thesis: card a c= 1 + 1
then consider x being Element of union G such that
A1: a = {(union G),[x,(union G)]} and
x in Vertices G ;
card {(union G),[x,(union G)]} <= 2 by CARD_2:50;
hence card a c= 1 + 1 by A1, NAT_1:39; ::_thesis: verum
end;
end;
end;
thus ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } is SimpleGraph by B, C; ::_thesis: verum
end;
end;
:: deftheorem defines Mycielskian SCMYCIEL:def_25_:_
for G being SimpleGraph holds Mycielskian G = ((({{}} \/ { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ) \/ (Edges G)) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ;
theorem M0: :: SCMYCIEL:84
for G being SimpleGraph holds G c= Mycielskian G
proof
let G be SimpleGraph; ::_thesis: G c= Mycielskian G
set MG = Mycielskian G;
set uG = union G;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in G or x in Mycielskian G )
assume x in G ; ::_thesis: x in Mycielskian G
then x in ({{}} \/ (singletons (union G))) \/ (Edges G) by SG0;
then A: ( x in {{}} \/ (singletons (union G)) or x in Edges G ) by XBOOLE_0:def_3;
percases ( x in {{}} or x in singletons (union G) or x in Edges G ) by A, XBOOLE_0:def_3;
suppose x in {{}} ; ::_thesis: x in Mycielskian G
then x = {} by TARSKI:def_1;
hence x in Mycielskian G by SG1; ::_thesis: verum
end;
suppose x in singletons (union G) ; ::_thesis: x in Mycielskian G
then consider f being Subset of (union G) such that
B: x = f and
C: f is 1 -element ;
consider a being set such that
D: a in union G and
E: f = {a} by C, BSPACEdef9;
a in (union G) \/ [:(union G),{(union G)}:] by D, XBOOLE_0:def_3;
then a in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} by XBOOLE_0:def_3;
then x in { {xx} where xx is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } by B, E;
hence x in Mycielskian G by MYCIELSK:4; ::_thesis: verum
end;
supposeB: x in Edges G ; ::_thesis: x in Mycielskian G
Edges G c= Mycielskian G by MYCIELSK:3;
hence x in Mycielskian G by B; ::_thesis: verum
end;
end;
end;
theorem M0v1: :: SCMYCIEL:85
for G being SimpleGraph
for v being set holds
( v in Vertices (Mycielskian G) iff ( v in union G or ex x being set st
( x in union G & v = [x,(union G)] ) or v = union G ) )
proof
let G be SimpleGraph; ::_thesis: for v being set holds
( v in Vertices (Mycielskian G) iff ( v in union G or ex x being set st
( x in union G & v = [x,(union G)] ) or v = union G ) )
let v be set ; ::_thesis: ( v in Vertices (Mycielskian G) iff ( v in union G or ex x being set st
( x in union G & v = [x,(union G)] ) or v = union G ) )
set uG = union G;
set MG = Mycielskian G;
set uMG = union (Mycielskian G);
hereby ::_thesis: ( ( v in union G or ex x being set st
( x in union G & v = [x,(union G)] ) or v = union G ) implies v in Vertices (Mycielskian G) )
assume v in Vertices (Mycielskian G) ; ::_thesis: S1[]
then consider g being set such that
B: v in g and
C: g in Mycielskian G by TARSKI:def_4;
defpred S1[] means ( v in union G or ex x being set st
( x in union G & v = [x,(union G)] ) or v = union G );
percases ( g in {{}} or g in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } or g in Edges G or g in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } or g in { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ) by C, MYCIELSK:4;
suppose g in {{}} ; ::_thesis: S1[]
hence S1[] by B, TARSKI:def_1; ::_thesis: verum
end;
suppose g in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ; ::_thesis: S1[]
then consider h being Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} such that
A1: g = {h} and
verum ;
B1: ( h in (union G) \/ [:(union G),{(union G)}:] or h in {(union G)} ) by XBOOLE_0:def_3;
C1: v = h by A1, B, TARSKI:def_1;
percases ( h in union G or h in [:(union G),{(union G)}:] or h in {(union G)} ) by B1, XBOOLE_0:def_3;
suppose h in union G ; ::_thesis: S1[]
hence S1[] by A1, B, TARSKI:def_1; ::_thesis: verum
end;
suppose h in [:(union G),{(union G)}:] ; ::_thesis: S1[]
then consider h1, h2 being set such that
A2: h1 in union G and
B2: h2 in {(union G)} and
C2: h = [h1,h2] by ZFMISC_1:def_2;
h2 = union G by B2, TARSKI:def_1;
hence S1[] by C1, C2, A2; ::_thesis: verum
end;
suppose h in {(union G)} ; ::_thesis: S1[]
hence S1[] by C1, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
suppose g in Edges G ; ::_thesis: S1[]
then consider g1, g2 being set such that
g1 <> g2 and
A1: g1 in Vertices G and
B1: g2 in Vertices G and
C1: g = {g1,g2} by SG4;
thus S1[] by A1, B1, B, C1, TARSKI:def_2; ::_thesis: verum
end;
suppose g in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ; ::_thesis: S1[]
then consider g1, g2 being Element of union G such that
A1: g = {g1,[g2,(union G)]} and
B1: {g1,g2} in Edges G ;
C1: ( g1 in union G & g2 in union G ) by B1, SG5;
( v = g1 or v = [g2,(union G)] ) by A1, B, TARSKI:def_2;
hence S1[] by C1; ::_thesis: verum
end;
suppose g in { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ; ::_thesis: S1[]
then consider x being Element of union G such that
A1: g = {(union G),[x,(union G)]} and
B1: x in union G ;
( v = union G or v = [x,(union G)] ) by B, A1, TARSKI:def_2;
hence S1[] by B1; ::_thesis: verum
end;
end;
end;
assume A: ( v in union G or ex x being set st
( x in union G & v = [x,(union G)] ) or v = union G ) ; ::_thesis: v in Vertices (Mycielskian G)
B: for a being set st a in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} holds
a in union (Mycielskian G)
proof
let a be set ; ::_thesis: ( a in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} implies a in union (Mycielskian G) )
assume a in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} ; ::_thesis: a in union (Mycielskian G)
then C2: {a} in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ;
B2a: { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } c= Mycielskian G by MYCIELSK:3;
a in {a} by TARSKI:def_1;
hence a in union (Mycielskian G) by B2a, C2, TARSKI:def_4; ::_thesis: verum
end;
percases ( v in union G or ex x being set st
( x in union G & v = [x,(union G)] ) or v = union G ) by A;
suppose v in union G ; ::_thesis: v in Vertices (Mycielskian G)
then v in (union G) \/ [:(union G),{(union G)}:] by XBOOLE_0:def_3;
then v in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} by XBOOLE_0:def_3;
hence v in Vertices (Mycielskian G) by B; ::_thesis: verum
end;
suppose ex x being set st
( x in union G & v = [x,(union G)] ) ; ::_thesis: v in Vertices (Mycielskian G)
then consider x being set such that
A2: x in union G and
B2: v = [x,(union G)] ;
union G in {(union G)} by TARSKI:def_1;
then v in [:(union G),{(union G)}:] by A2, B2, ZFMISC_1:def_2;
then v in (union G) \/ [:(union G),{(union G)}:] by XBOOLE_0:def_3;
then v in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} by XBOOLE_0:def_3;
hence v in Vertices (Mycielskian G) by B; ::_thesis: verum
end;
suppose v = union G ; ::_thesis: v in Vertices (Mycielskian G)
then v in {(union G)} by TARSKI:def_1;
then v in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} by XBOOLE_0:def_3;
hence v in Vertices (Mycielskian G) by B; ::_thesis: verum
end;
end;
end;
theorem M0v2: :: SCMYCIEL:86
for G being SimpleGraph holds Vertices (Mycielskian G) = ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)}
proof
let G be SimpleGraph; ::_thesis: Vertices (Mycielskian G) = ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)}
set uG = union G;
set MG = Mycielskian G;
set uMG = union (Mycielskian G);
A: union G in {(union G)} by TARSKI:def_1;
thus union (Mycielskian G) c= ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} :: according to XBOOLE_0:def_10 ::_thesis: ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} c= Vertices (Mycielskian G)
proof
let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in union (Mycielskian G) or v in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} )
assume A2: v in union (Mycielskian G) ; ::_thesis: v in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)}
percases ( v in union G or ex x being set st
( x in union G & v = [x,(union G)] ) or v = union G ) by A2, M0v1;
suppose v in union G ; ::_thesis: v in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)}
then v in (union G) \/ ([:(union G),{(union G)}:] \/ {(union G)}) by XBOOLE_0:def_3;
hence v in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} by XBOOLE_1:4; ::_thesis: verum
end;
suppose ex x being set st
( x in union G & v = [x,(union G)] ) ; ::_thesis: v in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)}
then consider x being set such that
B2: x in union G and
C2: v = [x,(union G)] ;
v in [:(union G),{(union G)}:] by A, B2, C2, ZFMISC_1:def_2;
then v in (union G) \/ [:(union G),{(union G)}:] by XBOOLE_0:def_3;
hence v in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose v = union G ; ::_thesis: v in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)}
then v in {(union G)} by TARSKI:def_1;
hence v in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
thus ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} c= union (Mycielskian G) ::_thesis: verum
proof
let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} or v in union (Mycielskian G) )
assume v in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} ; ::_thesis: v in union (Mycielskian G)
then A2: ( v in (union G) \/ [:(union G),{(union G)}:] or v in {(union G)} ) by XBOOLE_0:def_3;
percases ( v in union G or v in [:(union G),{(union G)}:] or v in {(union G)} ) by A2, XBOOLE_0:def_3;
suppose v in union G ; ::_thesis: v in union (Mycielskian G)
hence v in union (Mycielskian G) by M0v1; ::_thesis: verum
end;
suppose v in [:(union G),{(union G)}:] ; ::_thesis: v in union (Mycielskian G)
then consider x, y being set such that
A3: x in union G and
B3: y in {(union G)} and
C3: v = [x,y] by ZFMISC_1:def_2;
y = union G by B3, TARSKI:def_1;
hence v in union (Mycielskian G) by A3, C3, M0v1; ::_thesis: verum
end;
suppose v in {(union G)} ; ::_thesis: v in union (Mycielskian G)
then v = union G by TARSKI:def_1;
hence v in union (Mycielskian G) by M0v1; ::_thesis: verum
end;
end;
end;
end;
theorem M00: :: SCMYCIEL:87
for G being SimpleGraph holds union G in union (Mycielskian G)
proof
let G be SimpleGraph; ::_thesis: union G in union (Mycielskian G)
union G in {(union G)} by TARSKI:def_1;
then union G in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} by XBOOLE_0:def_3;
hence union G in union (Mycielskian G) by M0v2; ::_thesis: verum
end;
theorem MGvoid: :: SCMYCIEL:88
for G being void SimpleGraph holds Mycielskian G = {{},{(union G)}}
proof
let G be void SimpleGraph; ::_thesis: Mycielskian G = {{},{(union G)}}
set uG = union G;
A: { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } = {}
proof
assume not { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } = {} ; ::_thesis: contradiction
then consider e being set such that
A1: e in { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } by XBOOLE_0:def_1;
consider x being Element of union G such that
e = {(union G),[x,(union G)]} and
B1: x in Vertices G by A1;
thus contradiction by B1; ::_thesis: verum
end;
B: { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } = {}
proof
assume not { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } = {} ; ::_thesis: contradiction
then consider e being set such that
A1: e in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } by XBOOLE_0:def_1;
consider x, y being Element of union G such that
e = {x,[y,(union G)]} and
B1: {x,y} in Edges G by A1;
thus contradiction by B1, SG5; ::_thesis: verum
end;
C: Edges G = {}
proof
assume not Edges G = {} ; ::_thesis: contradiction
then consider e being set such that
A1: e in Edges G by XBOOLE_0:def_1;
consider x, y being set such that
x <> y and
B1: x in Vertices G and
( y in Vertices G & e = {x,y} ) by A1, SG4;
thus contradiction by B1; ::_thesis: verum
end;
D: { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } = {{(union G)}}
proof
thus { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } c= {{(union G)}} :: according to XBOOLE_0:def_10 ::_thesis: {{(union G)}} c= { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum }
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } or a in {{(union G)}} )
assume a in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ; ::_thesis: a in {{(union G)}}
then consider x being Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} such that
A1: a = {x} and
verum ;
x = union G by TARSKI:def_1;
hence a in {{(union G)}} by A1, TARSKI:def_1; ::_thesis: verum
end;
thus {{(union G)}} c= { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ::_thesis: verum
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in {{(union G)}} or a in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } )
assume a in {{(union G)}} ; ::_thesis: a in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum }
then A1: a = {(union G)} by TARSKI:def_1;
union G in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} by TARSKI:def_1;
hence a in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } by A1; ::_thesis: verum
end;
end;
thus Mycielskian G = {{},{(union G)}} by A, B, C, D, ENUMSET1:1; ::_thesis: verum
end;
registration
let G be finite SimpleGraph;
cluster Mycielskian G -> finite ;
correctness
coherence
Mycielskian G is finite ;
proof
set uG = union G;
set MG = Mycielskian G;
set C = { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ;
set A = { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ;
set B = { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ;
percases ( G is void or not G is void ) ;
suppose G is void ; ::_thesis: Mycielskian G is finite
then Mycielskian G = {{},{(union G)}} by MGvoid;
hence Mycielskian G is finite ; ::_thesis: verum
end;
suppose not G is void ; ::_thesis: Mycielskian G is finite
then reconsider uGf = union G as non empty set by VoidGV;
Ba: uGf is finite ;
deffunc H1( set ) -> set = {(union G),[G,(union G)]};
Bb: { H1(x) where x is Element of uGf : x in uGf } is finite from FRAENKEL:sch_21(Ba);
Aa: union G is finite ;
deffunc H2( set , set ) -> set = {G,[c2,(union G)]};
set AA = { H2(x,y) where x, y is Element of uGf : ( x in union G & y in union G ) } ;
Ab: { H2(x,y) where x, y is Element of uGf : ( x in union G & y in union G ) } is finite from FRAENKEL:sch_22(Aa, Aa);
Ac: { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } c= { H2(x,y) where x, y is Element of uGf : ( x in union G & y in union G ) }
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } or a in { H2(x,y) where x, y is Element of uGf : ( x in union G & y in union G ) } )
assume a in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ; ::_thesis: a in { H2(x,y) where x, y is Element of uGf : ( x in union G & y in union G ) }
then consider x, y being Element of union G such that
A1: a = {x,[y,(union G)]} and
{x,y} in Edges G ;
thus a in { H2(x,y) where x, y is Element of uGf : ( x in union G & y in union G ) } by A1; ::_thesis: verum
end;
defpred S1[ set ] means verum;
deffunc H3( set ) -> set = {G};
{ H3(x) where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : S1[x] } is finite from PRE_CIRC:sch_1();
hence Mycielskian G is finite by Bb, Ab, Ac; ::_thesis: verum
end;
end;
end;
end;
theorem M0order: :: SCMYCIEL:89
for G being finite SimpleGraph holds order (Mycielskian G) = (2 * (order G)) + 1
proof
let G be finite SimpleGraph; ::_thesis: order (Mycielskian G) = (2 * (order G)) + 1
set uG = union G;
set MG = Mycielskian G;
B: card [:(union G),{(union G)}:] = order G by CARD_2:6;
C: union G misses [:(union G),{(union G)}:]
proof
assume union G meets [:(union G),{(union G)}:] ; ::_thesis: contradiction
then consider a being set such that
A0: a in union G and
B0: a in [:(union G),{(union G)}:] by XBOOLE_0:3;
consider x, y being set such that
x in union G and
B1: y in {(union G)} and
C1: a = [x,y] by B0, ZFMISC_1:def_2;
y = union G by B1, TARSKI:def_1;
hence contradiction by C1, A0, Aux1; ::_thesis: verum
end;
D: now__::_thesis:_not_union_G_in_(union_G)_\/_[:(union_G),{(union_G)}:]
assume union G in (union G) \/ [:(union G),{(union G)}:] ; ::_thesis: contradiction
then ( union G in union G or union G in [:(union G),{(union G)}:] ) by XBOOLE_0:def_3;
then consider x, y being set such that
x in union G and
B1: y in {(union G)} and
C1: union G = [x,y] by ZFMISC_1:def_2;
y = union G by B1, TARSKI:def_1;
hence contradiction by C1, Aux2; ::_thesis: verum
end;
thus order (Mycielskian G) = card (((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)}) by M0v2
.= (card ((union G) \/ [:(union G),{(union G)}:])) + 1 by D, CARD_2:41
.= ((card (union G)) + (card [:(union G),{(union G)}:])) + 1 by C, CARD_2:40
.= (2 * (order G)) + 1 by B ; ::_thesis: verum
end;
theorem M0e0: :: SCMYCIEL:90
for G being SimpleGraph
for e being set holds
( e in Edges (Mycielskian G) iff ( e in Edges G or ex x, y being Element of union G st
( e = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( e = {(union G),[y,(union G)]} & y in union G ) ) )
proof
let G be SimpleGraph; ::_thesis: for e being set holds
( e in Edges (Mycielskian G) iff ( e in Edges G or ex x, y being Element of union G st
( e = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( e = {(union G),[y,(union G)]} & y in union G ) ) )
let e be set ; ::_thesis: ( e in Edges (Mycielskian G) iff ( e in Edges G or ex x, y being Element of union G st
( e = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( e = {(union G),[y,(union G)]} & y in union G ) ) )
set uG = union G;
set MG = Mycielskian G;
set C = { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ;
set A = { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ;
set B = { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ;
hereby ::_thesis: ( ( e in Edges G or ex x, y being Element of union G st
( e = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( e = {(union G),[y,(union G)]} & y in union G ) ) implies e in Edges (Mycielskian G) )
assume A0: e in Edges (Mycielskian G) ; ::_thesis: ( e in Edges G or ex x, y being Element of union G st
( e = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( e = {(union G),[y,(union G)]} & y in union G ) )
then consider x, y being set such that
A1: x <> y and
x in Vertices (Mycielskian G) and
y in Vertices (Mycielskian G) and
D1: e = {x,y} by SG4;
percases ( e in {{}} or e in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } or e in Edges G or e in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } or e in { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ) by A0, MYCIELSK:4;
suppose e in {{}} ; ::_thesis: ( e in Edges G or ex x, y being Element of union G st
( e = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( e = {(union G),[y,(union G)]} & y in union G ) )
hence ( e in Edges G or ex x, y being Element of union G st
( e = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( e = {(union G),[y,(union G)]} & y in union G ) ) by D1, TARSKI:def_1; ::_thesis: verum
end;
suppose e in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ; ::_thesis: ( e in Edges G or ex x, y being Element of union G st
( e = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( e = {(union G),[y,(union G)]} & y in union G ) )
then consider a being Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} such that
A2: e = {a} and
verum ;
thus ( e in Edges G or ex x, y being Element of union G st
( e = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( e = {(union G),[y,(union G)]} & y in union G ) ) by A2, D1, A1, ZFMISC_1:5; ::_thesis: verum
end;
suppose ( e in Edges G or e in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } or e in { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ) ; ::_thesis: ( e in Edges G or ex x, y being Element of union G st
( e = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( e = {(union G),[y,(union G)]} & y in union G ) )
hence ( e in Edges G or ex x, y being Element of union G st
( e = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( e = {(union G),[y,(union G)]} & y in union G ) ) ; ::_thesis: verum
end;
end;
end;
assume B: ( e in Edges G or ex x, y being Element of union G st
( e = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( e = {(union G),[y,(union G)]} & y in union G ) ) ; ::_thesis: e in Edges (Mycielskian G)
percases ( e in Edges G or ex x, y being Element of union G st
( e = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( e = {(union G),[y,(union G)]} & y in union G ) ) by B;
supposeS1: e in Edges G ; ::_thesis: e in Edges (Mycielskian G)
A2: card e = 2 by S1, LEdges;
e in Mycielskian G by S1, MYCIELSK:4;
hence e in Edges (Mycielskian G) by A2, LEdges; ::_thesis: verum
end;
suppose ex x, y being Element of union G st
( e = {x,[y,(union G)]} & {x,y} in Edges G ) ; ::_thesis: e in Edges (Mycielskian G)
then consider x, y being Element of union G such that
A2: e = {x,[y,(union G)]} and
B2: {x,y} in Edges G ;
C2: e in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } by A2, B2;
D2: e in Mycielskian G by C2, MYCIELSK:4;
y in union G by B2, SG5;
then x <> [y,(union G)] by Aux1;
then card e = 2 by A2, CARD_2:57;
hence e in Edges (Mycielskian G) by D2, LEdges; ::_thesis: verum
end;
suppose ex y being Element of union G st
( e = {(union G),[y,(union G)]} & y in union G ) ; ::_thesis: e in Edges (Mycielskian G)
then consider y being Element of union G such that
A2: e = {(union G),[y,(union G)]} and
B2: y in union G ;
C2: e in { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } by A2, B2;
D2: e in Mycielskian G by C2, MYCIELSK:4;
card e = 2 by Aux2, A2, CARD_2:57;
hence e in Edges (Mycielskian G) by D2, LEdges; ::_thesis: verum
end;
end;
end;
theorem M0e: :: SCMYCIEL:91
for G being SimpleGraph holds Edges (Mycielskian G) = ((Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[y,(union G)]} where y is Element of union G : y in union G }
proof
let G be SimpleGraph; ::_thesis: Edges (Mycielskian G) = ((Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[y,(union G)]} where y is Element of union G : y in union G }
set uG = union G;
set A = { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ;
set B = { {(union G),[y,(union G)]} where y is Element of union G : y in union G } ;
thus Edges (Mycielskian G) c= ((Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[y,(union G)]} where y is Element of union G : y in union G } :: according to XBOOLE_0:def_10 ::_thesis: ((Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[y,(union G)]} where y is Element of union G : y in union G } c= Edges (Mycielskian G)
proof
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in Edges (Mycielskian G) or e in ((Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[y,(union G)]} where y is Element of union G : y in union G } )
assume A: e in Edges (Mycielskian G) ; ::_thesis: e in ((Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[y,(union G)]} where y is Element of union G : y in union G }
percases ( e in Edges G or ex x, y being Element of union G st
( e = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( e = {(union G),[y,(union G)]} & y in union G ) ) by A, M0e0;
suppose e in Edges G ; ::_thesis: e in ((Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[y,(union G)]} where y is Element of union G : y in union G }
then e in (Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } by XBOOLE_0:def_3;
hence e in ((Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[y,(union G)]} where y is Element of union G : y in union G } by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose ex x, y being Element of union G st
( e = {x,[y,(union G)]} & {x,y} in Edges G ) ; ::_thesis: e in ((Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[y,(union G)]} where y is Element of union G : y in union G }
then e in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ;
then e in (Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } by XBOOLE_0:def_3;
hence e in ((Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[y,(union G)]} where y is Element of union G : y in union G } by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose ex y being Element of union G st
( e = {(union G),[y,(union G)]} & y in union G ) ; ::_thesis: e in ((Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[y,(union G)]} where y is Element of union G : y in union G }
then e in { {(union G),[y,(union G)]} where y is Element of union G : y in union G } ;
hence e in ((Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[y,(union G)]} where y is Element of union G : y in union G } by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
thus ((Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[y,(union G)]} where y is Element of union G : y in union G } c= Edges (Mycielskian G) ::_thesis: verum
proof
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in ((Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[y,(union G)]} where y is Element of union G : y in union G } or e in Edges (Mycielskian G) )
assume e in ((Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[y,(union G)]} where y is Element of union G : y in union G } ; ::_thesis: e in Edges (Mycielskian G)
then A: ( e in (Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } or e in { {(union G),[y,(union G)]} where y is Element of union G : y in union G } ) by XBOOLE_0:def_3;
percases ( e in Edges G or e in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } or e in { {(union G),[y,(union G)]} where y is Element of union G : y in union G } ) by A, XBOOLE_0:def_3;
suppose e in Edges G ; ::_thesis: e in Edges (Mycielskian G)
hence e in Edges (Mycielskian G) by M0e0; ::_thesis: verum
end;
suppose e in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ; ::_thesis: e in Edges (Mycielskian G)
then consider x, y being Element of union G such that
B: ( e = {x,[y,(union G)]} & {x,y} in Edges G ) ;
thus e in Edges (Mycielskian G) by B, M0e0; ::_thesis: verum
end;
suppose e in { {(union G),[y,(union G)]} where y is Element of union G : y in union G } ; ::_thesis: e in Edges (Mycielskian G)
then consider y being Element of union G such that
B: ( e = {(union G),[y,(union G)]} & y in union G ) ;
thus e in Edges (Mycielskian G) by B, M0e0; ::_thesis: verum
end;
end;
end;
end;
theorem M0size: :: SCMYCIEL:92
for G being finite SimpleGraph holds size (Mycielskian G) = (3 * (size G)) + (order G)
proof
let G be finite SimpleGraph; ::_thesis: size (Mycielskian G) = (3 * (size G)) + (order G)
set uG = union G;
set MG = Mycielskian G;
set A = { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ;
set B = { {(union G),[y,(union G)]} where y is Element of union G : y in union G } ;
percases ( G is void or not G is void ) ;
supposeS1: G is void ; ::_thesis: size (Mycielskian G) = (3 * (size G)) + (order G)
then M1: Mycielskian G = {{},{(union G)}} by MGvoid;
B1: size G = 0 by S1, VoidGE, CARD_1:27;
size (Mycielskian G) = 0
proof
assume not size (Mycielskian G) = 0 ; ::_thesis: contradiction
then Edges (Mycielskian G) <> {} ;
then consider e being set such that
A2: e in Edges (Mycielskian G) by XBOOLE_0:def_1;
consider x, y being set such that
B2: x <> y and
( x in Vertices (Mycielskian G) & y in Vertices (Mycielskian G) ) and
C2: e = {x,y} by A2, SG4;
( e = {} or e = {(union G)} ) by M1, A2, TARSKI:def_2;
hence contradiction by C2, B2, ZFMISC_1:5; ::_thesis: verum
end;
hence size (Mycielskian G) = (3 * (size G)) + (order G) by S1, B1; ::_thesis: verum
end;
suppose not G is void ; ::_thesis: size (Mycielskian G) = (3 * (size G)) + (order G)
then reconsider uGf = union G as non empty set by VoidGV;
Ba: uGf is finite ;
deffunc H1( set ) -> set = {(union G),[$1,(union G)]};
{ H1(x) where x is Element of uGf : x in uGf } is finite from FRAENKEL:sch_21(Ba);
then reconsider B = { {(union G),[y,(union G)]} where y is Element of union G : y in union G } as finite set ;
Aa: union G is finite ;
deffunc H2( set , set ) -> set = {$1,[$2,(union G)]};
set AA = { H2(x,y) where x, y is Element of uGf : ( x in union G & y in union G ) } ;
Ab: { H2(x,y) where x, y is Element of uGf : ( x in union G & y in union G ) } is finite from FRAENKEL:sch_22(Aa, Aa);
{ {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } c= { H2(x,y) where x, y is Element of uGf : ( x in union G & y in union G ) }
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } or a in { H2(x,y) where x, y is Element of uGf : ( x in union G & y in union G ) } )
assume a in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ; ::_thesis: a in { H2(x,y) where x, y is Element of uGf : ( x in union G & y in union G ) }
then consider x, y being Element of union G such that
A1: a = {x,[y,(union G)]} and
{x,y} in Edges G ;
thus a in { H2(x,y) where x, y is Element of uGf : ( x in union G & y in union G ) } by A1; ::_thesis: verum
end;
then reconsider A = { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } as finite set by Ab;
B: card B = order G by McopyV;
C: card A = 2 * (size G) by MnewE;
D: now__::_thesis:_not_B_meets_(Edges_G)_\/_A
assume B meets (Edges G) \/ A ; ::_thesis: contradiction
then consider a being set such that
A1: a in B and
B1: a in (Edges G) \/ A by XBOOLE_0:3;
consider y being Element of union G such that
C1: a = {(union G),[y,(union G)]} and
y in union G by A1;
percases ( a in Edges G or a in A ) by B1, XBOOLE_0:def_3;
suppose a in Edges G ; ::_thesis: contradiction
then consider xa, ya being set such that
xa <> ya and
B2: xa in Vertices G and
ya in Vertices G and
D2: a = {xa,ya} by SG4;
percases ( xa = union G or xa = [y,(union G)] ) by C1, D2, ZFMISC_1:6;
suppose xa = union G ; ::_thesis: contradiction
hence contradiction by B2; ::_thesis: verum
end;
suppose xa = [y,(union G)] ; ::_thesis: contradiction
hence contradiction by B2, Aux1; ::_thesis: verum
end;
end;
end;
suppose a in A ; ::_thesis: contradiction
then consider xa, ya being Element of union G such that
A2: a = {xa,[ya,(union G)]} and
B2: {xa,ya} in Edges G ;
C2: xa in union G by B2, SG5;
percases ( xa = union G or xa = [y,(union G)] ) by C1, A2, ZFMISC_1:6;
suppose xa = union G ; ::_thesis: contradiction
hence contradiction by C2; ::_thesis: verum
end;
suppose xa = [y,(union G)] ; ::_thesis: contradiction
hence contradiction by C2, Aux1; ::_thesis: verum
end;
end;
end;
end;
end;
E: now__::_thesis:_not_A_meets_Edges_G
assume A meets Edges G ; ::_thesis: contradiction
then consider a being set such that
A1: a in A and
B1: a in Edges G by XBOOLE_0:3;
consider xa, ya being Element of union G such that
A2: a = {xa,[ya,(union G)]} and
{xa,ya} in Edges G by A1;
consider xe, ye being set such that
xe <> ye and
B2a: xe in Vertices G and
C2a: ye in Vertices G and
D2a: a = {xe,ye} by B1, SG4;
percases ( xe = [ya,(union G)] or ye = [ya,(union G)] ) by A2, D2a, ZFMISC_1:6;
suppose xe = [ya,(union G)] ; ::_thesis: contradiction
hence contradiction by B2a, Aux1; ::_thesis: verum
end;
suppose ye = [ya,(union G)] ; ::_thesis: contradiction
hence contradiction by C2a, Aux1; ::_thesis: verum
end;
end;
end;
thus size (Mycielskian G) = card (((Edges G) \/ A) \/ B) by M0e
.= (card ((Edges G) \/ A)) + (order G) by B, D, CARD_2:40
.= ((card (Edges G)) + (2 * (size G))) + (order G) by C, E, CARD_2:40
.= (3 * (size G)) + (order G) ; ::_thesis: verum
end;
end;
end;
theorem M0e1: :: SCMYCIEL:93
for G being SimpleGraph
for s, t being set holds
( not {s,t} in Edges (Mycielskian G) or {s,t} in Edges G or ( ( s in union G or s = union G ) & ex y being set st
( y in union G & t = [y,(union G)] ) ) or ( ( t in union G or t = union G ) & ex y being set st
( y in union G & s = [y,(union G)] ) ) )
proof
let G be SimpleGraph; ::_thesis: for s, t being set holds
( not {s,t} in Edges (Mycielskian G) or {s,t} in Edges G or ( ( s in union G or s = union G ) & ex y being set st
( y in union G & t = [y,(union G)] ) ) or ( ( t in union G or t = union G ) & ex y being set st
( y in union G & s = [y,(union G)] ) ) )
let s, t be set ; ::_thesis: ( not {s,t} in Edges (Mycielskian G) or {s,t} in Edges G or ( ( s in union G or s = union G ) & ex y being set st
( y in union G & t = [y,(union G)] ) ) or ( ( t in union G or t = union G ) & ex y being set st
( y in union G & s = [y,(union G)] ) ) )
assume A: {s,t} in Edges (Mycielskian G) ; ::_thesis: ( {s,t} in Edges G or ( ( s in union G or s = union G ) & ex y being set st
( y in union G & t = [y,(union G)] ) ) or ( ( t in union G or t = union G ) & ex y being set st
( y in union G & s = [y,(union G)] ) ) )
set uG = union G;
percases ( {s,t} in Edges G or ex x, y being Element of union G st
( {s,t} = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( {s,t} = {(union G),[y,(union G)]} & y in union G ) ) by A, M0e0;
suppose {s,t} in Edges G ; ::_thesis: ( {s,t} in Edges G or ( ( s in union G or s = union G ) & ex y being set st
( y in union G & t = [y,(union G)] ) ) or ( ( t in union G or t = union G ) & ex y being set st
( y in union G & s = [y,(union G)] ) ) )
hence ( {s,t} in Edges G or ( ( s in union G or s = union G ) & ex y being set st
( y in union G & t = [y,(union G)] ) ) or ( ( t in union G or t = union G ) & ex y being set st
( y in union G & s = [y,(union G)] ) ) ) ; ::_thesis: verum
end;
suppose ex x, y being Element of union G st
( {s,t} = {x,[y,(union G)]} & {x,y} in Edges G ) ; ::_thesis: ( {s,t} in Edges G or ( ( s in union G or s = union G ) & ex y being set st
( y in union G & t = [y,(union G)] ) ) or ( ( t in union G or t = union G ) & ex y being set st
( y in union G & s = [y,(union G)] ) ) )
then consider x, y being Element of union G such that
A1: {s,t} = {x,[y,(union G)]} and
B1: {x,y} in Edges G ;
C1: ( x in union G & y in union G ) by B1, SG5;
( ( s = x & t = [y,(union G)] ) or ( t = x & s = [y,(union G)] ) ) by A1, ZFMISC_1:6;
hence ( {s,t} in Edges G or ( ( s in union G or s = union G ) & ex y being set st
( y in union G & t = [y,(union G)] ) ) or ( ( t in union G or t = union G ) & ex y being set st
( y in union G & s = [y,(union G)] ) ) ) by C1; ::_thesis: verum
end;
suppose ex y being Element of union G st
( {s,t} = {(union G),[y,(union G)]} & y in union G ) ; ::_thesis: ( {s,t} in Edges G or ( ( s in union G or s = union G ) & ex y being set st
( y in union G & t = [y,(union G)] ) ) or ( ( t in union G or t = union G ) & ex y being set st
( y in union G & s = [y,(union G)] ) ) )
then consider y being Element of union G such that
A1: {s,t} = {(union G),[y,(union G)]} and
B1: y in union G ;
( ( s = union G & t = [y,(union G)] ) or ( t = union G & s = [y,(union G)] ) ) by A1, ZFMISC_1:6;
hence ( {s,t} in Edges G or ( ( s in union G or s = union G ) & ex y being set st
( y in union G & t = [y,(union G)] ) ) or ( ( t in union G or t = union G ) & ex y being set st
( y in union G & s = [y,(union G)] ) ) ) by B1; ::_thesis: verum
end;
end;
end;
theorem M0e2: :: SCMYCIEL:94
for G being SimpleGraph
for u being set st {(union G),u} in Edges (Mycielskian G) holds
ex x being set st
( x in union G & u = [x,(union G)] )
proof
let G be SimpleGraph; ::_thesis: for u being set st {(union G),u} in Edges (Mycielskian G) holds
ex x being set st
( x in union G & u = [x,(union G)] )
let u be set ; ::_thesis: ( {(union G),u} in Edges (Mycielskian G) implies ex x being set st
( x in union G & u = [x,(union G)] ) )
assume A: {(union G),u} in Edges (Mycielskian G) ; ::_thesis: ex x being set st
( x in union G & u = [x,(union G)] )
set uG = union G;
percases ( {(union G),u} in Edges G or ( ( union G in union G or union G = union G ) & ex y being set st
( y in union G & u = [y,(union G)] ) ) or ( ( u in union G or u = union G ) & ex y being set st
( y in union G & union G = [y,(union G)] ) ) ) by A, M0e1;
suppose {(union G),u} in Edges G ; ::_thesis: ex x being set st
( x in union G & u = [x,(union G)] )
then union G in union G by SG5;
hence ex x being set st
( x in union G & u = [x,(union G)] ) ; ::_thesis: verum
end;
suppose ( ( union G in union G or union G = union G ) & ex y being set st
( y in union G & u = [y,(union G)] ) ) ; ::_thesis: ex x being set st
( x in union G & u = [x,(union G)] )
hence ex x being set st
( x in union G & u = [x,(union G)] ) ; ::_thesis: verum
end;
suppose ( ( u in union G or u = union G ) & ex y being set st
( y in union G & union G = [y,(union G)] ) ) ; ::_thesis: ex x being set st
( x in union G & u = [x,(union G)] )
then consider y being set such that
y in union G and
A1: union G = [y,(union G)] ;
thus ex x being set st
( x in union G & u = [x,(union G)] ) by A1, Aux2; ::_thesis: verum
end;
end;
end;
theorem M0e2aa: :: SCMYCIEL:95
for G being SimpleGraph
for u being set st u in Vertices G holds
{[u,(union G)]} in Mycielskian G
proof
let G be SimpleGraph; ::_thesis: for u being set st u in Vertices G holds
{[u,(union G)]} in Mycielskian G
let u be set ; ::_thesis: ( u in Vertices G implies {[u,(union G)]} in Mycielskian G )
assume A: u in Vertices G ; ::_thesis: {[u,(union G)]} in Mycielskian G
{[u,(union G)],(union G)} in Edges (Mycielskian G) by A, M0e0;
then [u,(union G)] in Vertices (Mycielskian G) by SG5;
hence {[u,(union G)]} in Mycielskian G by Vertices0; ::_thesis: verum
end;
theorem M0e2a: :: SCMYCIEL:96
for G being SimpleGraph
for u being set st u in Vertices G holds
{[u,(union G)],(union G)} in Mycielskian G
proof
let G be SimpleGraph; ::_thesis: for u being set st u in Vertices G holds
{[u,(union G)],(union G)} in Mycielskian G
let u be set ; ::_thesis: ( u in Vertices G implies {[u,(union G)],(union G)} in Mycielskian G )
assume A: u in Vertices G ; ::_thesis: {[u,(union G)],(union G)} in Mycielskian G
{[u,(union G)],(union G)} in Edges (Mycielskian G) by A, M0e0;
hence {[u,(union G)],(union G)} in Mycielskian G ; ::_thesis: verum
end;
theorem M0e3: :: SCMYCIEL:97
for G being SimpleGraph
for x, y being set holds not {[x,(union G)],[y,(union G)]} in Edges (Mycielskian G)
proof
let G be SimpleGraph; ::_thesis: for x, y being set holds not {[x,(union G)],[y,(union G)]} in Edges (Mycielskian G)
let x, y be set ; ::_thesis: not {[x,(union G)],[y,(union G)]} in Edges (Mycielskian G)
assume A: {[x,(union G)],[y,(union G)]} in Edges (Mycielskian G) ; ::_thesis: contradiction
Ab: union G in {x,(union G)} by TARSKI:def_2;
Ac: {x,(union G)} in {{x},{x,(union G)}} by TARSKI:def_2;
B: not [x,(union G)] in union G by Ab, Ac, XREGULAR:7;
C: not [x,(union G)] = union G by Ab, TARSKI:def_2;
Ab1: union G in {y,(union G)} by TARSKI:def_2;
Ac1: {y,(union G)} in {{y},{y,(union G)}} by TARSKI:def_2;
B1: not [y,(union G)] in union G by Ab1, Ac1, XREGULAR:7;
C1: not [y,(union G)] = union G by Ab1, TARSKI:def_2;
{[x,(union G)],[y,(union G)]} in Edges G by A, B, C, B1, C1, M0e1;
hence contradiction by B, SG5; ::_thesis: verum
end;
theorem M0e3a: :: SCMYCIEL:98
for G being SimpleGraph
for x, y being set st x <> y holds
not {[x,(union G)],[y,(union G)]} in Mycielskian G
proof
let G be SimpleGraph; ::_thesis: for x, y being set st x <> y holds
not {[x,(union G)],[y,(union G)]} in Mycielskian G
let x, y be set ; ::_thesis: ( x <> y implies not {[x,(union G)],[y,(union G)]} in Mycielskian G )
assume that
A: x <> y and
B: {[x,(union G)],[y,(union G)]} in Mycielskian G ; ::_thesis: contradiction
[x,(union G)] <> [y,(union G)] by A, XTUPLE_0:1;
then card {[x,(union G)],[y,(union G)]} = 2 by CARD_2:57;
then {[x,(union G)],[y,(union G)]} in Edges (Mycielskian G) by B, LEdges;
hence contradiction by M0e3; ::_thesis: verum
end;
theorem M0e4: :: SCMYCIEL:99
for G being SimpleGraph
for x, y being set st {[x,(union G)],y} in Edges (Mycielskian G) holds
( x <> y & x in union G & ( y in union G or y = union G ) )
proof
let G be SimpleGraph; ::_thesis: for x, y being set st {[x,(union G)],y} in Edges (Mycielskian G) holds
( x <> y & x in union G & ( y in union G or y = union G ) )
let x, y be set ; ::_thesis: ( {[x,(union G)],y} in Edges (Mycielskian G) implies ( x <> y & x in union G & ( y in union G or y = union G ) ) )
assume A: {[x,(union G)],y} in Edges (Mycielskian G) ; ::_thesis: ( x <> y & x in union G & ( y in union G or y = union G ) )
set uG = union G;
set e = {[x,(union G)],y};
percases ( {[x,(union G)],y} in Edges G or ex x, y being Element of union G st
( {[x,(union G)],y} = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( {[x,(union G)],y} = {(union G),[y,(union G)]} & y in union G ) ) by A, M0e0;
suppose {[x,(union G)],y} in Edges G ; ::_thesis: ( x <> y & x in union G & ( y in union G or y = union G ) )
then [x,(union G)] in union G by SG5;
hence ( x <> y & x in union G & ( y in union G or y = union G ) ) by Aux1; ::_thesis: verum
end;
suppose ex x, y being Element of union G st
( {[x,(union G)],y} = {x,[y,(union G)]} & {x,y} in Edges G ) ; ::_thesis: ( x <> y & x in union G & ( y in union G or y = union G ) )
then consider xa, ya being Element of union G such that
A1: {[x,(union G)],y} = {xa,[ya,(union G)]} and
B1: {xa,ya} in Edges G ;
consider xx, yy being set such that
C1: xx <> yy and
D1: ( xx in Vertices G & yy in Vertices G ) and
E1: {xa,ya} = {xx,yy} by B1, SG4;
F1: ( ( xa = xx & ya = yy ) or ( xa = yy & ya = xx ) ) by E1, ZFMISC_1:6;
percases ( ( xa = [x,(union G)] & y = [ya,(union G)] ) or ( xa = y & [ya,(union G)] = [x,(union G)] ) ) by A1, ZFMISC_1:6;
suppose ( xa = [x,(union G)] & y = [ya,(union G)] ) ; ::_thesis: ( x <> y & x in union G & ( y in union G or y = union G ) )
hence ( x <> y & x in union G & ( y in union G or y = union G ) ) by D1, Aux1; ::_thesis: verum
end;
suppose ( xa = y & [ya,(union G)] = [x,(union G)] ) ; ::_thesis: ( x <> y & x in union G & ( y in union G or y = union G ) )
hence ( x <> y & x in union G & ( y in union G or y = union G ) ) by C1, D1, F1, XTUPLE_0:1; ::_thesis: verum
end;
end;
end;
suppose ex y being Element of union G st
( {[x,(union G)],y} = {(union G),[y,(union G)]} & y in union G ) ; ::_thesis: ( x <> y & x in union G & ( y in union G or y = union G ) )
then consider yy being Element of union G such that
A1: {[x,(union G)],y} = {(union G),[yy,(union G)]} and
B1: yy in union G ;
C1: ( ( union G = [x,(union G)] & y = [yy,(union G)] ) or ( union G = y & [x,(union G)] = [yy,(union G)] ) ) by A1, ZFMISC_1:6;
x = yy by C1, Aux2, XTUPLE_0:1;
hence ( x <> y & x in union G & ( y in union G or y = union G ) ) by C1, B1; ::_thesis: verum
end;
end;
end;
theorem M0e4a: :: SCMYCIEL:100
for G being SimpleGraph
for x, y being set st {[x,(union G)],y} in Mycielskian G holds
x <> y
proof
let G be SimpleGraph; ::_thesis: for x, y being set st {[x,(union G)],y} in Mycielskian G holds
x <> y
let x, y be set ; ::_thesis: ( {[x,(union G)],y} in Mycielskian G implies x <> y )
set MG = Mycielskian G;
set uG = union G;
assume A: {[x,(union G)],y} in Mycielskian G ; ::_thesis: x <> y
assume B: x = y ; ::_thesis: contradiction
then [x,(union G)] <> y by Aux3;
then {[x,(union G)],y} in Edges (Mycielskian G) by A, SG4a;
hence contradiction by B, M0e4; ::_thesis: verum
end;
theorem M0e4b: :: SCMYCIEL:101
for G being SimpleGraph
for x, y being set st y in union G & {[x,(union G)],y} in Mycielskian G holds
{x,y} in G
proof
let G be SimpleGraph; ::_thesis: for x, y being set st y in union G & {[x,(union G)],y} in Mycielskian G holds
{x,y} in G
set MG = Mycielskian G;
set uG = union G;
set A = { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ;
set B = { {(union G),[y,(union G)]} where y is Element of union G : y in union G } ;
let x, y be set ; ::_thesis: ( y in union G & {[x,(union G)],y} in Mycielskian G implies {x,y} in G )
assume A0: y in union G ; ::_thesis: ( not {[x,(union G)],y} in Mycielskian G or {x,y} in G )
assume {[x,(union G)],y} in Mycielskian G ; ::_thesis: {x,y} in G
then {[x,(union G)],y} in ({{}} \/ (singletons (Vertices (Mycielskian G)))) \/ (Edges (Mycielskian G)) by SG0;
then A: ( {[x,(union G)],y} in {{}} \/ (singletons (Vertices (Mycielskian G))) or {[x,(union G)],y} in Edges (Mycielskian G) ) by XBOOLE_0:def_3;
percases ( {[x,(union G)],y} in {{}} or {[x,(union G)],y} in singletons (Vertices (Mycielskian G)) or {[x,(union G)],y} in Edges (Mycielskian G) ) by A, XBOOLE_0:def_3;
suppose {[x,(union G)],y} in {{}} ; ::_thesis: {x,y} in G
hence {x,y} in G by TARSKI:def_1; ::_thesis: verum
end;
suppose {[x,(union G)],y} in singletons (Vertices (Mycielskian G)) ; ::_thesis: {x,y} in G
then consider f being Subset of (Vertices (Mycielskian G)) such that
A1: f = {[x,(union G)],y} and
B1: f is 1 -element ;
consider s being set such that
s in Vertices (Mycielskian G) and
D1: f = {s} by B1, BSPACEdef9;
E1: card f = 1 by D1, CARD_1:30;
y = [x,(union G)] by E1, A1, CARD_2:57;
hence {x,y} in G by A0, Aux1; ::_thesis: verum
end;
suppose {[x,(union G)],y} in Edges (Mycielskian G) ; ::_thesis: {x,y} in G
then {[x,(union G)],y} in ((Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ) \/ { {(union G),[y,(union G)]} where y is Element of union G : y in union G } by M0e;
then A2: ( {[x,(union G)],y} in (Edges G) \/ { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } or {[x,(union G)],y} in { {(union G),[y,(union G)]} where y is Element of union G : y in union G } ) by XBOOLE_0:def_3;
percases ( {[x,(union G)],y} in Edges G or {[x,(union G)],y} in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } or {[x,(union G)],y} in { {(union G),[y,(union G)]} where y is Element of union G : y in union G } ) by A2, XBOOLE_0:def_3;
suppose {[x,(union G)],y} in Edges G ; ::_thesis: {x,y} in G
then [x,(union G)] in union G by SG5;
hence {x,y} in G by Aux1; ::_thesis: verum
end;
suppose {[x,(union G)],y} in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ; ::_thesis: {x,y} in G
then consider xx, yy being Element of union G such that
A3: {[x,(union G)],y} = {xx,[yy,(union G)]} and
B3: {xx,yy} in Edges G ;
C3: ( xx in union G & yy in union G ) by B3, SG5;
( ( [x,(union G)] = xx & y = [yy,(union G)] ) or ( [x,(union G)] = [yy,(union G)] & y = xx ) ) by A3, ZFMISC_1:6;
then ( x = yy & y = xx ) by XTUPLE_0:1, C3, Aux1;
hence {x,y} in G by B3; ::_thesis: verum
end;
suppose {[x,(union G)],y} in { {(union G),[y,(union G)]} where y is Element of union G : y in union G } ; ::_thesis: {x,y} in G
then consider yy being Element of union G such that
A3: {[x,(union G)],y} = {(union G),[yy,(union G)]} and
yy in union G ;
( ( [x,(union G)] = union G & y = [yy,(union G)] ) or ( [x,(union G)] = [yy,(union G)] & y = union G ) ) by A3, ZFMISC_1:6;
hence {x,y} in G by Aux1, A0; ::_thesis: verum
end;
end;
end;
end;
end;
theorem M0e4c: :: SCMYCIEL:102
for G being SimpleGraph
for x, y being set st {x,y} in Edges G holds
{[x,(union G)],y} in Mycielskian G
proof
let G be SimpleGraph; ::_thesis: for x, y being set st {x,y} in Edges G holds
{[x,(union G)],y} in Mycielskian G
set uG = union G;
let x, y be set ; ::_thesis: ( {x,y} in Edges G implies {[x,(union G)],y} in Mycielskian G )
A0: { {xx,[yy,(union G)]} where xx, yy is Element of union G : {xx,yy} in Edges G } c= Mycielskian G by MYCIELSK:3;
assume A: {x,y} in Edges G ; ::_thesis: {[x,(union G)],y} in Mycielskian G
then ( x in union G & y in union G ) by SG5;
then {[x,(union G)],y} in { {xx,[yy,(union G)]} where xx, yy is Element of union G : {xx,yy} in Edges G } by A;
hence {[x,(union G)],y} in Mycielskian G by A0; ::_thesis: verum
end;
theorem M1: :: SCMYCIEL:103
for G being SimpleGraph
for x, y being set st x in Vertices G & y in Vertices G & {x,y} in Mycielskian G holds
{x,y} in G
proof
let G be SimpleGraph; ::_thesis: for x, y being set st x in Vertices G & y in Vertices G & {x,y} in Mycielskian G holds
{x,y} in G
let s, t be set ; ::_thesis: ( s in Vertices G & t in Vertices G & {s,t} in Mycielskian G implies {s,t} in G )
assume that
A: s in Vertices G and
B: t in Vertices G and
C: {s,t} in Mycielskian G ; ::_thesis: {s,t} in G
percases ( s = t or s <> t ) ;
suppose s = t ; ::_thesis: {s,t} in G
then {s,t} = {s} by ENUMSET1:29;
hence {s,t} in G by A, Vertices0; ::_thesis: verum
end;
suppose s <> t ; ::_thesis: {s,t} in G
then card {s,t} = 2 by CARD_2:57;
then A1: {s,t} in Edges (Mycielskian G) by C, LEdges;
percases ( {s,t} in Edges G or ( ( s in union G or s = union G ) & ex y being set st
( y in union G & t = [y,(union G)] ) ) or ( ( t in union G or t = union G ) & ex y being set st
( y in union G & s = [y,(union G)] ) ) ) by A1, M0e1;
suppose {s,t} in Edges G ; ::_thesis: {s,t} in G
hence {s,t} in G ; ::_thesis: verum
end;
suppose ( ( s in union G or s = union G ) & ex y being set st
( y in union G & t = [y,(union G)] ) ) ; ::_thesis: {s,t} in G
then consider y being set such that
y in union G and
B1: t = [y,(union G)] ;
thus {s,t} in G by B1, B, Aux1; ::_thesis: verum
end;
suppose ( ( t in union G or t = union G ) & ex y being set st
( y in union G & s = [y,(union G)] ) ) ; ::_thesis: {s,t} in G
then consider y being set such that
y in union G and
B1: s = [y,(union G)] ;
thus {s,t} in G by B1, A, Aux1; ::_thesis: verum
end;
end;
end;
end;
end;
theorem GsubMG: :: SCMYCIEL:104
for G being SimpleGraph holds G = (Mycielskian G) SubgraphInducedBy (Vertices G)
proof
let G be SimpleGraph; ::_thesis: G = (Mycielskian G) SubgraphInducedBy (Vertices G)
set L = Vertices G;
set MG = Mycielskian G;
thus G c= (Mycielskian G) SubgraphInducedBy (Vertices G) by M0, Sub0b; :: according to XBOOLE_0:def_10 ::_thesis: (Mycielskian G) SubgraphInducedBy (Vertices G) c= G
thus (Mycielskian G) SubgraphInducedBy (Vertices G) c= G ::_thesis: verum
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (Mycielskian G) SubgraphInducedBy (Vertices G) or a in G )
assume Aa: a in (Mycielskian G) SubgraphInducedBy (Vertices G) ; ::_thesis: a in G
set m = a;
C1a: a in bool (Vertices G) by Aa, XBOOLE_0:def_4;
percases ( a in {{}} or a in { {x} where x is Element of ((Vertices G) \/ [:(Vertices G),{(Vertices G)}:]) \/ {(Vertices G)} : verum } or a in Edges G or a in { {x,[y,(Vertices G)]} where x, y is Element of Vertices G : {x,y} in Edges G } or a in { {(Vertices G),[x,(Vertices G)]} where x is Element of Vertices G : x in Vertices G } ) by Aa, MYCIELSK:4;
suppose a in {{}} ; ::_thesis: a in G
then a = {} by TARSKI:def_1;
hence a in G by SG1; ::_thesis: verum
end;
suppose a in { {x} where x is Element of ((Vertices G) \/ [:(Vertices G),{(Vertices G)}:]) \/ {(Vertices G)} : verum } ; ::_thesis: a in G
then consider x being Element of ((Vertices G) \/ [:(Vertices G),{(Vertices G)}:]) \/ {(Vertices G)} such that
A2: a = {x} and
verum ;
x in a by A2, TARSKI:def_1;
hence a in G by C1a, A2, Vertices0; ::_thesis: verum
end;
suppose a in Edges G ; ::_thesis: a in G
hence a in G ; ::_thesis: verum
end;
suppose a in { {x,[y,(Vertices G)]} where x, y is Element of Vertices G : {x,y} in Edges G } ; ::_thesis: a in G
then consider x, y being Element of Vertices G such that
A2: a = {x,[y,(Vertices G)]} and
{x,y} in Edges G ;
[y,(Vertices G)] in a by A2, TARSKI:def_2;
hence a in G by C1a, Aux1; ::_thesis: verum
end;
suppose a in { {(Vertices G),[x,(Vertices G)]} where x is Element of Vertices G : x in Vertices G } ; ::_thesis: a in G
then consider x being Element of Vertices G such that
A2: a = {(Vertices G),[x,(Vertices G)]} and
x in Vertices G ;
Vertices G in a by A2, TARSKI:def_2;
then Vertices G in Vertices G by C1a;
hence a in G ; ::_thesis: verum
end;
end;
end;
end;
theorem MClique3: :: SCMYCIEL:105
for G being SimpleGraph
for C being finite Clique of (Mycielskian G) st 3 <= order C holds
for v being Vertex of C holds v <> union G
proof
let G be SimpleGraph; ::_thesis: for C being finite Clique of (Mycielskian G) st 3 <= order C holds
for v being Vertex of C holds v <> union G
let C be finite Clique of (Mycielskian G); ::_thesis: ( 3 <= order C implies for v being Vertex of C holds v <> union G )
assume A: 3 <= order C ; ::_thesis: for v being Vertex of C holds v <> union G
set MG = Mycielskian G;
let v be Vertex of C; ::_thesis: v <> union G
assume B: v = union G ; ::_thesis: contradiction
3 c= order C by A, NAT_1:39;
then consider v1, v2 being set such that
D: v1 in Vertices C and
E: v2 in Vertices C and
F: v1 <> v and
G: v2 <> v and
H: v1 <> v2 by card3;
Ia: {v,v1} in C by D, Clique2a;
Ja: {v,v2} in C by E, Clique2a;
I1: {v,v1} in Edges (Mycielskian G) by Ia, F, SG4a;
J1: {v,v2} in Edges (Mycielskian G) by G, Ja, SG4a;
consider x1 being set such that
x1 in union G and
Kb: v1 = [x1,(union G)] by B, I1, M0e2;
consider x2 being set such that
x2 in union G and
Lb: v2 = [x2,(union G)] by B, J1, M0e2;
{v1,v2} in C by D, E, Clique2a;
hence contradiction by Kb, Lb, H, M0e3a; ::_thesis: verum
end;
theorem MClique0: :: SCMYCIEL:106
for G being with_finite_clique# SimpleGraph st clique# G = 0 holds
for D being finite Clique of (Mycielskian G) holds order D <= 1
proof
let G be with_finite_clique# SimpleGraph; ::_thesis: ( clique# G = 0 implies for D being finite Clique of (Mycielskian G) holds order D <= 1 )
assume A: clique# G = 0 ; ::_thesis: for D being finite Clique of (Mycielskian G) holds order D <= 1
set uG = union G;
B: Vertices G = {} by A, Cno0;
C: G is void by B, VoidGV;
D: union (Mycielskian G) = union {{},{(union G)}} by C, MGvoid
.= {} \/ {(union G)} by ZFMISC_1:75
.= {(union G)} ;
let D be finite Clique of (Mycielskian G); ::_thesis: order D <= 1
Vertices D c= {(union G)} by D, ZFMISC_1:77;
then card (Vertices D) c= card {(union G)} by CARD_1:11;
then card (Vertices D) <= card {(union G)} by NAT_1:39;
hence order D <= 1 by CARD_1:30; ::_thesis: verum
end;
theorem :: SCMYCIEL:107
for G being SimpleGraph
for x being set st Vertices G = {x} holds
Mycielskian G = {{},{x},{[x,(union G)]},{(union G)},{[x,(union G)],(union G)}}
proof
let G be SimpleGraph; ::_thesis: for x being set st Vertices G = {x} holds
Mycielskian G = {{},{x},{[x,(union G)]},{(union G)},{[x,(union G)],(union G)}}
let a be set ; ::_thesis: ( Vertices G = {a} implies Mycielskian G = {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}} )
assume Aa: Vertices G = {a} ; ::_thesis: Mycielskian G = {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}}
A: card (Vertices G) = 1 by Aa, CARD_1:30;
B: a in Vertices G by Aa, TARSKI:def_1;
set uG = union G;
set MG = Mycielskian G;
set A = { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ;
set B = { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ;
set C = { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ;
consider aa being set such that
Ca: union G = {aa} by A, CARD_2:42;
C: a = aa by Ca, B, TARSKI:def_1;
D: [:(union G),{(union G)}:] = {[a,(union G)]} by Ca, C, ZFMISC_1:29;
E0: G is edgeless by A, GsingleE;
thus Mycielskian G c= {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}} :: according to XBOOLE_0:def_10 ::_thesis: {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}} c= Mycielskian G
proof
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in Mycielskian G or e in {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}} )
assume A1: e in Mycielskian G ; ::_thesis: e in {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}}
percases ( e in {{}} or e in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } or e in Edges G or e in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } or e in { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ) by A1, MYCIELSK:4;
suppose e in {{}} ; ::_thesis: e in {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}}
then e = {} by TARSKI:def_1;
hence e in {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}} by ENUMSET1:def_3; ::_thesis: verum
end;
suppose e in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ; ::_thesis: e in {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}}
then consider x being Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} such that
B1: e = {x} and
verum ;
( x in (union G) \/ [:(union G),{(union G)}:] or x in {(union G)} ) by XBOOLE_0:def_3;
then ( x in union G or x in [:(union G),{(union G)}:] or x in {(union G)} ) by XBOOLE_0:def_3;
then ( x = a or x = [a,(union G)] or x = union G ) by Ca, C, D, TARSKI:def_1;
hence e in {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}} by B1, ENUMSET1:def_3; ::_thesis: verum
end;
suppose e in Edges G ; ::_thesis: e in {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}}
hence e in {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}} by E0, Ledgeless; ::_thesis: verum
end;
suppose e in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ; ::_thesis: e in {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}}
then consider x, y being Element of union G such that
e = {x,[y,(union G)]} and
B1: {x,y} in Edges G ;
thus e in {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}} by B1, E0, Ledgeless; ::_thesis: verum
end;
suppose e in { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ; ::_thesis: e in {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}}
then consider x being Element of union G such that
A1: e = {(union G),[x,(union G)]} and
x in Vertices G ;
x = a by Ca, C, TARSKI:def_1;
hence e in {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}} by A1, ENUMSET1:def_3; ::_thesis: verum
end;
end;
end;
thus {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}} c= Mycielskian G ::_thesis: verum
proof
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}} or e in Mycielskian G )
assume A1: e in {{},{a},{[a,(union G)]},{(union G)},{[a,(union G)],(union G)}} ; ::_thesis: e in Mycielskian G
percases ( e = {} or e = {a} or e = {[a,(union G)]} or e = {(union G)} or e = {[a,(union G)],(union G)} ) by A1, ENUMSET1:def_3;
suppose e = {} ; ::_thesis: e in Mycielskian G
hence e in Mycielskian G by SG1; ::_thesis: verum
end;
supposeS1: e = {a} ; ::_thesis: e in Mycielskian G
a in (union G) \/ [:(union G),{(union G)}:] by B, XBOOLE_0:def_3;
then a in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} by XBOOLE_0:def_3;
then e in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } by S1;
hence e in Mycielskian G by MYCIELSK:4; ::_thesis: verum
end;
supposeS1: e = {[a,(union G)]} ; ::_thesis: e in Mycielskian G
[a,(union G)] in [:(union G),{(union G)}:] by D, TARSKI:def_1;
then [a,(union G)] in (union G) \/ [:(union G),{(union G)}:] by XBOOLE_0:def_3;
then [a,(union G)] in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} by XBOOLE_0:def_3;
then e in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } by S1;
hence e in Mycielskian G by MYCIELSK:4; ::_thesis: verum
end;
supposeS1: e = {(union G)} ; ::_thesis: e in Mycielskian G
union G in {(union G)} by TARSKI:def_1;
then union G in ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} by XBOOLE_0:def_3;
then e in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } by S1;
hence e in Mycielskian G by MYCIELSK:4; ::_thesis: verum
end;
suppose e = {[a,(union G)],(union G)} ; ::_thesis: e in Mycielskian G
then e in { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } by B;
hence e in Mycielskian G by MYCIELSK:4; ::_thesis: verum
end;
end;
end;
end;
theorem MClique1: :: SCMYCIEL:108
for G being with_finite_clique# SimpleGraph st clique# G = 1 holds
for D being finite Clique of (Mycielskian G) holds order D <= 2
proof
let G be with_finite_clique# SimpleGraph; ::_thesis: ( clique# G = 1 implies for D being finite Clique of (Mycielskian G) holds order D <= 2 )
assume A: clique# G = 1 ; ::_thesis: for D being finite Clique of (Mycielskian G) holds order D <= 2
set uG = union G;
set MG = Mycielskian G;
set uMG = union (Mycielskian G);
let D be finite Clique of (Mycielskian G); ::_thesis: order D <= 2
set uD = union D;
assume H0: order D > 2 ; ::_thesis: contradiction
then H: order D >= 2 + 1 by NAT_1:13;
not union D is empty by H0;
then consider v being set such that
A0: v in union D by XBOOLE_0:def_1;
C: v <> union G by A0, H, MClique3;
3 c= order D by H, NAT_1:39;
then consider v1, v2 being set such that
B1: v1 in union D and
v2 in union D and
B3: v1 <> v and
v2 <> v and
v1 <> v2 by card3;
C1: v1 <> union G by B1, H, MClique3;
set e = {v,v1};
{v,v1} in D by A0, B1, Clique2a;
then F0: {v,v1} in Edges (Mycielskian G) by B3, SG4a;
percases ( {v,v1} in Edges G or ex x, y being Element of union G st
( {v,v1} = {x,[y,(union G)]} & {x,y} in Edges G ) or ex y being Element of union G st
( {v,v1} = {(union G),[y,(union G)]} & y in union G ) ) by F0, M0e0;
suppose {v,v1} in Edges G ; ::_thesis: contradiction
hence contradiction by A, Cno2; ::_thesis: verum
end;
suppose ex x, y being Element of union G st
( {v,v1} = {x,[y,(union G)]} & {x,y} in Edges G ) ; ::_thesis: contradiction
then consider x, y being Element of union G such that
{v,v1} = {x,[y,(union G)]} and
H1: {x,y} in Edges G ;
thus contradiction by A, H1, Cno2; ::_thesis: verum
end;
suppose ex y being Element of union G st
( {v,v1} = {(union G),[y,(union G)]} & y in union G ) ; ::_thesis: contradiction
then consider y being Element of union G such that
H1: {v,v1} = {(union G),[y,(union G)]} and
y in union G ;
thus contradiction by C, C1, H1, ZFMISC_1:6; ::_thesis: verum
end;
end;
end;
theorem MClique2: :: SCMYCIEL:109
for G being with_finite_clique# SimpleGraph st 2 <= clique# G holds
for D being finite Clique of (Mycielskian G) holds order D <= clique# G
proof
let G be with_finite_clique# SimpleGraph; ::_thesis: ( 2 <= clique# G implies for D being finite Clique of (Mycielskian G) holds order D <= clique# G )
assume A: 2 <= clique# G ; ::_thesis: for D being finite Clique of (Mycielskian G) holds order D <= clique# G
let D be finite Clique of (Mycielskian G); ::_thesis: order D <= clique# G
assume Ba: order D > clique# G ; ::_thesis: contradiction
set MG = Mycielskian G;
set uG = union G;
Da1: Vertices D c= Vertices (Mycielskian G) by ZFMISC_1:77;
Da2: Edges D c= Edges (Mycielskian G) by SG6e;
2 < order D by Ba, A, XXREAL_0:2;
then Fz: 2 + 1 <= order D by NAT_1:13;
percases ( D c= G or not D c= G ) ;
suppose D c= G ; ::_thesis: contradiction
hence contradiction by Ba, Lcliqueno; ::_thesis: verum
end;
suppose not D c= G ; ::_thesis: contradiction
then consider e being set such that
B2: e in D and
C2: not e in G by TARSKI:def_3;
now__::_thesis:_not_Vertices_D_c=_Vertices_G
assume A3: Vertices D c= Vertices G ; ::_thesis: contradiction
B3za: e <> {} by C2, SG1;
now__::_thesis:_e_in_Edges_D
assume not e in Edges D ; ::_thesis: contradiction
then consider y being set such that
A4: e = {y} and
B4: y in Vertices D by B3za, B2, SG2;
thus contradiction by C2, A4, B4, A3, Vertices0; ::_thesis: verum
end;
then consider x, y being set such that
x <> y and
D3a: x in Vertices D and
D3b: y in Vertices D and
D3: e = {x,y} by SG4;
thus contradiction by B2, A3, D3, C2, M1, D3a, D3b; ::_thesis: verum
end;
then consider v being set such that
A1: v in Vertices D and
B1: not v in Vertices G by TARSKI:def_3;
3 c= order D by Fz, NAT_1:39;
then consider v1, v2 being set such that
C1a: v1 in Vertices D and
C1b: v2 in Vertices D and
C1c: v1 <> v and
C1d: v2 <> v and
C1e: v1 <> v2 by card3;
{v,v1} in D by A1, C1a, Clique2a;
then E1a: {v,v1} in Edges D by C1c, SG4a;
{v,v2} in D by A1, C1b, Clique2a;
then E1b: {v,v2} in Edges D by C1d, SG4a;
{v1,v2} in D by C1a, C1b, Clique2a;
then E1c: {v1,v2} in Edges D by C1e, SG4a;
percases ( v = union G or ex x being set st
( x in union G & v = [x,(union G)] ) ) by A1, Da1, B1, M0v1;
supposeS2: v = union G ; ::_thesis: contradiction
consider x being set such that
x in union G and
F1a: v1 = [x,(union G)] by S2, E1a, Da2, M0e2;
consider y being set such that
y in union G and
F1b: v2 = [y,(union G)] by S2, E1b, Da2, M0e2;
thus contradiction by E1c, Da2, F1a, F1b, M0e3; ::_thesis: verum
end;
suppose ex x being set st
( x in union G & v = [x,(union G)] ) ; ::_thesis: contradiction
then consider x being set such that
S2a: x in union G and
S2b: v = [x,(union G)] ;
set E = D SubgraphInducedBy (union G);
reconsider F = G SubgraphInducedBy ({x} \/ (union (D SubgraphInducedBy (union G)))) as finite SimpleGraph ;
Z2b: Vertices F = {x} \/ (Vertices (D SubgraphInducedBy (union G)))
proof
thus Vertices F c= {x} \/ (Vertices (D SubgraphInducedBy (union G))) :: according to XBOOLE_0:def_10 ::_thesis: {x} \/ (Vertices (D SubgraphInducedBy (union G))) c= Vertices F
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in Vertices F or a in {x} \/ (Vertices (D SubgraphInducedBy (union G))) )
assume a in Vertices F ; ::_thesis: a in {x} \/ (Vertices (D SubgraphInducedBy (union G)))
then a in (union G) /\ ({x} \/ (union (D SubgraphInducedBy (union G)))) by Sub5;
then C5: a in {x} \/ (union (D SubgraphInducedBy (union G))) by XBOOLE_0:def_4;
percases ( a in {x} or a in union (D SubgraphInducedBy (union G)) ) by C5, XBOOLE_0:def_3;
suppose a in {x} ; ::_thesis: a in {x} \/ (Vertices (D SubgraphInducedBy (union G)))
hence a in {x} \/ (Vertices (D SubgraphInducedBy (union G))) by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose a in union (D SubgraphInducedBy (union G)) ; ::_thesis: a in {x} \/ (Vertices (D SubgraphInducedBy (union G)))
hence a in {x} \/ (Vertices (D SubgraphInducedBy (union G))) by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in {x} \/ (Vertices (D SubgraphInducedBy (union G))) or a in Vertices F )
assume A3: a in {x} \/ (Vertices (D SubgraphInducedBy (union G))) ; ::_thesis: a in Vertices F
percases ( a in {x} or a in Vertices (D SubgraphInducedBy (union G)) ) by A3, XBOOLE_0:def_3;
suppose a in {x} ; ::_thesis: a in Vertices F
then A4: a = x by TARSKI:def_1;
x in {x} by TARSKI:def_1;
then x in {x} \/ (union (D SubgraphInducedBy (union G))) by XBOOLE_0:def_3;
then x in (union G) /\ ({x} \/ (union (D SubgraphInducedBy (union G)))) by S2a, XBOOLE_0:def_4;
hence a in Vertices F by A4, Sub5; ::_thesis: verum
end;
suppose a in Vertices (D SubgraphInducedBy (union G)) ; ::_thesis: a in Vertices F
then a in (union D) /\ (union G) by Sub5;
then a in union G by XBOOLE_0:def_4;
then a in (union G) /\ ({x} \/ (union (D SubgraphInducedBy (union G)))) by A3, XBOOLE_0:def_4;
hence a in Vertices F by Sub5; ::_thesis: verum
end;
end;
end;
Z2d: union (D SubgraphInducedBy (union G)) c= union D by ZFMISC_1:77;
Z2c: now__::_thesis:_not_x_in_union_(D_SubgraphInducedBy_(union_G))
assume x in union (D SubgraphInducedBy (union G)) ; ::_thesis: contradiction
then {[x,(union G)],x} in D by Z2d, A1, S2b, Clique2a;
hence contradiction by M0e4a; ::_thesis: verum
end;
reconsider Fs = F as SimpleGraph-like Subset of G ;
now__::_thesis:_for_a,_b_being_set_st_a_<>_b_&_a_in_union_Fs_&_b_in_union_Fs_holds_
{a,b}_in_Edges_Fs
let a, b be set ; ::_thesis: ( a <> b & a in union Fs & b in union Fs implies {a,b} in Edges Fs )
assume that
A4: a <> b and
B4: a in union Fs and
C4: b in union Fs ; ::_thesis: {a,b} in Edges Fs
D4ba: a in (union G) /\ ({x} \/ (union (D SubgraphInducedBy (union G)))) by B4, Sub5;
then D4b: ( a in union G & a in {x} \/ (union (D SubgraphInducedBy (union G))) ) by XBOOLE_0:def_4;
E4ba: b in (union G) /\ ({x} \/ (union (D SubgraphInducedBy (union G)))) by C4, Sub5;
then E4b: ( b in union G & b in {x} \/ (union (D SubgraphInducedBy (union G))) ) by XBOOLE_0:def_4;
F4a: a in Vertices G by D4ba, XBOOLE_0:def_4;
F4b: b in Vertices G by E4ba, XBOOLE_0:def_4;
x in {x} by TARSKI:def_1;
then H4: x in {x} \/ (union (D SubgraphInducedBy (union G))) by XBOOLE_0:def_3;
{a,b} in Fs
proof
percases ( ( a in {x} & b in {x} ) or ( a in {x} & b in union (D SubgraphInducedBy (union G)) ) or ( b in {x} & a in union (D SubgraphInducedBy (union G)) ) or ( a in union (D SubgraphInducedBy (union G)) & b in union (D SubgraphInducedBy (union G)) ) ) by D4b, E4b, XBOOLE_0:def_3;
suppose ( a in {x} & b in {x} ) ; ::_thesis: {a,b} in Fs
then A5: ( a = x & b = x ) by TARSKI:def_1;
then {a,b} = {x} by ENUMSET1:29;
hence {a,b} in Fs by A5, B4, Vertices0; ::_thesis: verum
end;
supposeS4: ( a in {x} & b in union (D SubgraphInducedBy (union G)) ) ; ::_thesis: {a,b} in Fs
then A5: a = x by TARSKI:def_1;
b in (union D) /\ (union G) by S4, Sub5;
then B5: ( b in union D & b in union G ) by XBOOLE_0:def_4;
then {[x,(union G)],b} in D by A1, S2b, Clique2a;
then {x,b} in G by B5, M0e4b;
hence {a,b} in Fs by H4, A5, E4b, Sub6; ::_thesis: verum
end;
supposeS4: ( b in {x} & a in union (D SubgraphInducedBy (union G)) ) ; ::_thesis: {a,b} in Fs
then A5: b = x by TARSKI:def_1;
a in (union D) /\ (union G) by S4, Sub5;
then B5: ( a in union D & a in union G ) by XBOOLE_0:def_4;
then {[x,(union G)],a} in D by A1, S2b, Clique2a;
then {x,a} in G by B5, M0e4b;
hence {a,b} in Fs by H4, A5, D4b, Sub6; ::_thesis: verum
end;
suppose ( a in union (D SubgraphInducedBy (union G)) & b in union (D SubgraphInducedBy (union G)) ) ; ::_thesis: {a,b} in Fs
then ( a in (union D) /\ (union G) & b in (union D) /\ (union G) ) by Sub5;
then ( a in union D & b in union D ) by XBOOLE_0:def_4;
then {a,b} in D by Clique2a;
then {a,b} in G by F4a, F4b, M1;
hence {a,b} in Fs by D4b, E4b, Sub6; ::_thesis: verum
end;
end;
end;
hence {a,b} in Edges Fs by A4, SG4a; ::_thesis: verum
end;
then Y2a: Fs is clique by Lclique1;
U2: Vertices D c= {v} \/ (Vertices (D SubgraphInducedBy (union G)))
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in Vertices D or a in {v} \/ (Vertices (D SubgraphInducedBy (union G))) )
assume A3: a in Vertices D ; ::_thesis: a in {v} \/ (Vertices (D SubgraphInducedBy (union G)))
percases ( a = v or a <> v ) ;
suppose a = v ; ::_thesis: a in {v} \/ (Vertices (D SubgraphInducedBy (union G)))
then a in {v} by TARSKI:def_1;
hence a in {v} \/ (Vertices (D SubgraphInducedBy (union G))) by XBOOLE_0:def_3; ::_thesis: verum
end;
supposeS3: a <> v ; ::_thesis: a in {v} \/ (Vertices (D SubgraphInducedBy (union G)))
{a,[x,(union G)]} in D by S2b, A1, A3, Clique2a;
then {a,[x,(union G)]} in Edges D by S3, S2b, SG4a;
then ( a in union G or a = union G ) by Da2, M0e4;
then a in Vertices (D SubgraphInducedBy (union G)) by Fz, MClique3, A3, Sub3;
hence a in {v} \/ (Vertices (D SubgraphInducedBy (union G))) by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
U2a: Vertices (D SubgraphInducedBy (union G)) c= Vertices D by ZFMISC_1:77;
Z2a1: {v} \/ (Vertices (D SubgraphInducedBy (union G))) c= Vertices D
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in {v} \/ (Vertices (D SubgraphInducedBy (union G))) or a in Vertices D )
assume a in {v} \/ (Vertices (D SubgraphInducedBy (union G))) ; ::_thesis: a in Vertices D
then ( a in {v} or a in Vertices (D SubgraphInducedBy (union G)) ) by XBOOLE_0:def_3;
hence a in Vertices D by A1, U2a, TARSKI:def_1; ::_thesis: verum
end;
Z2d: not v in Vertices (D SubgraphInducedBy (union G)) by S2b, Sub1, Aux1;
order F = 1 + (card (Vertices (D SubgraphInducedBy (union G)))) by Z2b, Z2c, CARD_2:41
.= card ({v} \/ (Vertices (D SubgraphInducedBy (union G)))) by Z2d, CARD_2:41
.= order D by Z2a1, U2, XBOOLE_0:def_10 ;
hence contradiction by Ba, Y2a, Lcliqueno; ::_thesis: verum
end;
end;
end;
end;
end;
registration
let G be with_finite_clique# SimpleGraph;
cluster Mycielskian G -> with_finite_clique# ;
coherence
Mycielskian G is with_finite_clique#
proof
set MG = Mycielskian G;
set uG = union G;
percases ( clique# G = 0 or clique# G = 1 or clique# G > 1 ) by NAT_1:25;
supposeS1: clique# G = 0 ; ::_thesis: Mycielskian G is with_finite_clique#
then union G = {} by Cno0;
then {} in union (Mycielskian G) by M00;
then consider C being finite Clique of (Mycielskian G) such that
A1: Vertices C = {{}} by CliqueS;
take C ; :: according to SCMYCIEL:def_14 ::_thesis: for D being finite Clique of (Mycielskian G) holds order D <= order C
order C = 1 by A1, CARD_1:30;
hence for D being finite Clique of (Mycielskian G) holds order D <= order C by S1, MClique0; ::_thesis: verum
end;
supposeS1: clique# G = 1 ; ::_thesis: Mycielskian G is with_finite_clique#
then consider C being finite Clique of G such that
A1: order C = 1 by Lcliqueno;
A1b: union C c= union G by ZFMISC_1:77;
Vertices C <> {} by A1;
then consider v being set such that
B1: v in Vertices C by XBOOLE_0:def_1;
C1: [v,(union G)] in union (Mycielskian G) by B1, A1b, M0v1;
D1: union G in union (Mycielskian G) by M00;
E1: {[v,(union G)],(union G)} in Mycielskian G by B1, A1b, M0e2a;
reconsider CC = {{},{[v,(union G)]},{(union G)},{[v,(union G)],(union G)}} as finite Clique of (Mycielskian G) by C1, D1, E1, Cliqueon2;
B1a: CC = CompleteSGraph {[v,(union G)],(union G)} by P2;
B1d: Vertices CC = {[v,(union G)],(union G)} by B1a, CSGLem1;
take CC ; :: according to SCMYCIEL:def_14 ::_thesis: for D being finite Clique of (Mycielskian G) holds order D <= order CC
order CC = 2 by B1d, Aux2, CARD_2:57;
hence for D being finite Clique of (Mycielskian G) holds order D <= order CC by S1, MClique1; ::_thesis: verum
end;
suppose clique# G > 1 ; ::_thesis: Mycielskian G is with_finite_clique#
then A1: clique# G >= 1 + 1 by NAT_1:13;
consider C being finite Clique of G such that
B1: order C = clique# G by Lcliqueno;
G c= Mycielskian G by M0;
then reconsider CC = C as finite Clique of (Mycielskian G) by XBOOLE_1:1;
take CC ; :: according to SCMYCIEL:def_14 ::_thesis: for D being finite Clique of (Mycielskian G) holds order D <= order CC
thus for D being finite Clique of (Mycielskian G) holds order D <= order CC by B1, A1, MClique2; ::_thesis: verum
end;
end;
end;
end;
theorem MClique: :: SCMYCIEL:110
for G being with_finite_clique# SimpleGraph st 2 <= clique# G holds
clique# (Mycielskian G) = clique# G
proof
let G be with_finite_clique# SimpleGraph; ::_thesis: ( 2 <= clique# G implies clique# (Mycielskian G) = clique# G )
assume that
A: 2 <= clique# G and
B: clique# (Mycielskian G) <> clique# G ; ::_thesis: contradiction
set MG = Mycielskian G;
consider D being finite Clique of (Mycielskian G) such that
D: order D = clique# (Mycielskian G) by Lcliqueno;
clique# G <= clique# (Mycielskian G) by M0, CliqueSubno0;
then clique# G < clique# (Mycielskian G) by B, XXREAL_0:1;
hence contradiction by A, D, MClique2; ::_thesis: verum
end;
theorem Mfc1: :: SCMYCIEL:111
for G being finitely_colorable SimpleGraph ex E being Coloring of (Mycielskian G) st card E = 1 + (chromatic# G)
proof
let G be finitely_colorable SimpleGraph; ::_thesis: ex E being Coloring of (Mycielskian G) st card E = 1 + (chromatic# G)
set uG = union G;
set MG = Mycielskian G;
set uMG = union (Mycielskian G);
set cnG = chromatic# G;
consider C being finite Coloring of G such that
A: card C = chromatic# G by Lchro;
defpred S1[ set , set ] means $2 = { [x,(union G)] where x is Element of union G : x in $1 } ;
P: for e being set st e in C holds
ex u being set st S1[e,u] ;
consider r being Function such that
dom r = C and
C: for e being set st e in C holds
S1[e,r . e] from CLASSES1:sch_1(P);
set D = { (d \/ (r . d)) where d is Element of C : d in C } ;
D1: card { (d \/ (r . d)) where d is Element of C : d in C } = card C
proof
percases ( { (d \/ (r . d)) where d is Element of C : d in C } is empty or not { (d \/ (r . d)) where d is Element of C : d in C } is empty ) ;
supposeA7: { (d \/ (r . d)) where d is Element of C : d in C } is empty ; ::_thesis: card { (d \/ (r . d)) where d is Element of C : d in C } = card C
now__::_thesis:_C_is_empty
assume not C is empty ; ::_thesis: contradiction
then consider c being set such that
A8: c in C by XBOOLE_0:def_1;
c \/ (r . c) in { (d \/ (r . d)) where d is Element of C : d in C } by A8;
hence contradiction by A7; ::_thesis: verum
end;
hence card { (d \/ (r . d)) where d is Element of C : d in C } = card C by A7; ::_thesis: verum
end;
supposeA9: not { (d \/ (r . d)) where d is Element of C : d in C } is empty ; ::_thesis: card { (d \/ (r . d)) where d is Element of C : d in C } = card C
defpred S2[ set , set ] means $2 = $1 \/ (r . $1);
A10: for e being set st e in C holds
ex u being set st S2[e,u] ;
consider s being Function such that
A11: dom s = C and
A12: for e being set st e in C holds
S2[e,s . e] from CLASSES1:sch_1(A10);
A13: rng s c= { (d \/ (r . d)) where d is Element of C : d in C }
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng s or y in { (d \/ (r . d)) where d is Element of C : d in C } )
assume y in rng s ; ::_thesis: y in { (d \/ (r . d)) where d is Element of C : d in C }
then consider x being set such that
A14: x in dom s and
A15: y = s . x by FUNCT_1:def_3;
y = x \/ (r . x) by A14, A15, A11, A12;
hence y in { (d \/ (r . d)) where d is Element of C : d in C } by A14, A11; ::_thesis: verum
end;
then reconsider s = s as Function of C, { (d \/ (r . d)) where d is Element of C : d in C } by A11, FUNCT_2:2;
{ (d \/ (r . d)) where d is Element of C : d in C } c= rng s
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (d \/ (r . d)) where d is Element of C : d in C } or x in rng s )
assume x in { (d \/ (r . d)) where d is Element of C : d in C } ; ::_thesis: x in rng s
then consider c being Element of C such that
A16: x = c \/ (r . c) and
A17: c in C ;
x = s . c by A16, A17, A12;
hence x in rng s by A17, A11, FUNCT_1:def_3; ::_thesis: verum
end;
then rng s = { (d \/ (r . d)) where d is Element of C : d in C } by A13, XBOOLE_0:def_10;
then A18: s is onto by FUNCT_2:def_3;
s is one-to-one
proof
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom s or not x2 in dom s or not s . x1 = s . x2 or x1 = x2 )
assume that
A19: x1 in dom s and
A20: x2 in dom s and
A21: s . x1 = s . x2 ; ::_thesis: x1 = x2
A22: s . x1 = x1 \/ (r . x1) by A19, A11, A12;
A23: s . x2 = x2 \/ (r . x2) by A20, A11, A12;
thus x1 c= x2 :: according to XBOOLE_0:def_10 ::_thesis: x2 c= x1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in x1 or x in x2 )
assume A24: x in x1 ; ::_thesis: x in x2
A25: x in s . x1 by A22, A24, XBOOLE_0:def_3;
percases ( x in x2 or x in r . x2 ) by A25, A21, A23, XBOOLE_0:def_3;
suppose x in x2 ; ::_thesis: x in x2
hence x in x2 ; ::_thesis: verum
end;
suppose x in r . x2 ; ::_thesis: x in x2
then x in { [xx,(union G)] where xx is Element of union G : xx in x2 } by C, A11, A20;
then consider xx being Element of union G such that
A26: x = [xx,(union G)] and
xx in x2 ;
thus x in x2 by A26, A19, A24, A11, Aux1; ::_thesis: verum
end;
end;
end;
thus x2 c= x1 ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in x2 or x in x1 )
assume A27: x in x2 ; ::_thesis: x in x1
A28: x in s . x2 by A23, A27, XBOOLE_0:def_3;
percases ( x in x1 or x in r . x1 ) by A28, A21, A22, XBOOLE_0:def_3;
suppose x in x1 ; ::_thesis: x in x1
hence x in x1 ; ::_thesis: verum
end;
suppose x in r . x1 ; ::_thesis: x in x1
then x in { [xx,(union G)] where xx is Element of union G : xx in x1 } by C, A11, A19;
then consider xx being Element of union G such that
A26: x = [xx,(union G)] and
xx in x1 ;
thus x in x1 by A26, A20, A27, A11, Aux1; ::_thesis: verum
end;
end;
end;
end;
hence card { (d \/ (r . d)) where d is Element of C : d in C } = card C by A18, A9, EULER_1:11; ::_thesis: verum
end;
end;
end;
D1a: { (d \/ (r . d)) where d is Element of C : d in C } is finite by D1;
set E = { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}};
E1: union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) = union (Mycielskian G)
proof
thus union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) c= union (Mycielskian G) :: according to XBOOLE_0:def_10 ::_thesis: union (Mycielskian G) c= union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}})
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) or x in union (Mycielskian G) )
assume x in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) ; ::_thesis: x in union (Mycielskian G)
then consider Y being set such that
A2: x in Y and
B2: Y in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} by TARSKI:def_4;
percases ( Y in { (d \/ (r . d)) where d is Element of C : d in C } or Y in {{(union G)}} ) by B2, XBOOLE_0:def_3;
suppose Y in { (d \/ (r . d)) where d is Element of C : d in C } ; ::_thesis: x in union (Mycielskian G)
then consider d being Element of C such that
A3: Y = d \/ (r . d) and
B3: d in C ;
percases ( x in d or x in r . d ) by A3, A2, XBOOLE_0:def_3;
supposeS3: x in d ; ::_thesis: x in union (Mycielskian G)
B4: union G c= union (Mycielskian G) by M0, ZFMISC_1:77;
x in union G by S3;
hence x in union (Mycielskian G) by B4; ::_thesis: verum
end;
suppose x in r . d ; ::_thesis: x in union (Mycielskian G)
then x in { [yy,(union G)] where yy is Element of union G : yy in d } by B3, C;
then consider yy being Element of union G such that
A8: x = [yy,(union G)] and
B8: yy in d ;
{x} in Mycielskian G by A8, B8, M0e2aa;
hence x in union (Mycielskian G) by Vertices0; ::_thesis: verum
end;
end;
end;
suppose Y in {{(union G)}} ; ::_thesis: x in union (Mycielskian G)
then Y = {(union G)} by TARSKI:def_1;
then x = union G by A2, TARSKI:def_1;
hence x in union (Mycielskian G) by M00; ::_thesis: verum
end;
end;
end;
thus union (Mycielskian G) c= union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) ::_thesis: verum
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in union (Mycielskian G) or a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) )
assume a in union (Mycielskian G) ; ::_thesis: a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}})
then consider Y being set such that
A2: a in Y and
B2: Y in Mycielskian G by TARSKI:def_4;
C2: ( a in union G implies a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) )
proof
assume a in union G ; ::_thesis: a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}})
then a in union C by EQREL_1:def_4;
then consider d being set such that
D4: a in d and
E4: d in C by TARSKI:def_4;
F4: a in d \/ (r . d) by D4, XBOOLE_0:def_3;
d \/ (r . d) in { (d \/ (r . d)) where d is Element of C : d in C } by E4;
then d \/ (r . d) in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} by XBOOLE_0:def_3;
hence a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) by F4, TARSKI:def_4; ::_thesis: verum
end;
D2: now__::_thesis:_for_x_being_set_st_a_=_[x,(union_G)]_&_x_in_union_G_holds_
a_in_union_(_{__(d_\/_(r_._d))_where_d_is_Element_of_C_:_d_in_C__}__\/_{{(union_G)}})
let x be set ; ::_thesis: ( a = [x,(union G)] & x in union G implies a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) )
assume A4: a = [x,(union G)] ; ::_thesis: ( x in union G implies a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) )
assume B4: x in union G ; ::_thesis: a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}})
then x in union C by EQREL_1:def_4;
then consider d being set such that
D4: x in d and
E4: d in C by TARSKI:def_4;
d \/ (r . d) in { (d \/ (r . d)) where d is Element of C : d in C } by E4;
then G4: d \/ (r . d) in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} by XBOOLE_0:def_3;
a in { [xx,(union G)] where xx is Element of union G : xx in d } by D4, A4, B4;
then a in r . d by E4, C;
then a in d \/ (r . d) by XBOOLE_0:def_3;
hence a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) by G4, TARSKI:def_4; ::_thesis: verum
end;
percases ( Y in {{}} or Y in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } or Y in Edges G or Y in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } or Y in { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ) by B2, MYCIELSK:4;
suppose Y in {{}} ; ::_thesis: a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}})
hence a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) by A2, TARSKI:def_1; ::_thesis: verum
end;
suppose Y in { {x} where x is Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} : verum } ; ::_thesis: a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}})
then consider x being Element of ((union G) \/ [:(union G),{(union G)}:]) \/ {(union G)} such that
A3: Y = {x} and
verum ;
C3: a = x by A3, A2, TARSKI:def_1;
D3: ( a in (union G) \/ [:(union G),{(union G)}:] or a in {(union G)} ) by C3, XBOOLE_0:def_3;
percases ( a in union G or a in [:(union G),{(union G)}:] or a in {(union G)} ) by D3, XBOOLE_0:def_3;
suppose a in union G ; ::_thesis: a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}})
hence a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) by C2; ::_thesis: verum
end;
suppose a in [:(union G),{(union G)}:] ; ::_thesis: a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}})
then consider x, y being set such that
A4: x in union G and
B4: y in {(union G)} and
C4: a = [x,y] by ZFMISC_1:def_2;
y = union G by B4, TARSKI:def_1;
hence a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) by A4, C4, D2; ::_thesis: verum
end;
supposeS4: a in {(union G)} ; ::_thesis: a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}})
{(union G)} in {{(union G)}} by TARSKI:def_1;
then {(union G)} in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} by XBOOLE_0:def_3;
hence a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) by S4, TARSKI:def_4; ::_thesis: verum
end;
end;
end;
suppose Y in Edges G ; ::_thesis: a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}})
then consider p, r being set such that
p <> r and
B3: p in Vertices G and
C3: r in Vertices G and
D3: Y = {p,r} by SG4;
thus a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) by C2, B3, C3, D3, A2, TARSKI:def_2; ::_thesis: verum
end;
suppose Y in { {x,[y,(union G)]} where x, y is Element of union G : {x,y} in Edges G } ; ::_thesis: a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}})
then consider x, y being Element of union G such that
A3: Y = {x,[y,(union G)]} and
B3: {x,y} in Edges G ;
C3: ( a = x or a = [y,(union G)] ) by A2, A3, TARSKI:def_2;
x in union G by B3, SG5;
hence a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) by C3, C2, D2; ::_thesis: verum
end;
suppose Y in { {(union G),[x,(union G)]} where x is Element of union G : x in Vertices G } ; ::_thesis: a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}})
then consider x being Element of union G such that
A3: Y = {(union G),[x,(union G)]} and
B3: x in Vertices G ;
percases ( a = union G or a = [x,(union G)] ) by A2, A3, TARSKI:def_2;
suppose a = union G ; ::_thesis: a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}})
then A4: a in {(union G)} by TARSKI:def_1;
{(union G)} in {{(union G)}} by TARSKI:def_1;
then {(union G)} in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} by XBOOLE_0:def_3;
hence a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) by A4, TARSKI:def_4; ::_thesis: verum
end;
supposeA4: a = [x,(union G)] ; ::_thesis: a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}})
thus a in union ( { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}}) by A4, D2, B3; ::_thesis: verum
end;
end;
end;
end;
end;
end;
F1: now__::_thesis:_for_A_being_Subset_of_(union_(Mycielskian_G))_st_A_in__{__(d_\/_(r_._d))_where_d_is_Element_of_C_:_d_in_C__}__\/_{{(union_G)}}_holds_
(_A_<>_{}_&_(_for_B_being_Subset_of_(union_(Mycielskian_G))_holds_
(_not_B_in__{__(d_\/_(r_._d))_where_d_is_Element_of_C_:_d_in_C__}__\/_{{(union_G)}}_or_A_=_B_or_A_misses_B_)_)_)
let A be Subset of (union (Mycielskian G)); ::_thesis: ( A in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} implies ( b1 <> {} & ( for B being Subset of (union (Mycielskian G)) holds
( not b2 in { (b3 \/ (r . b3)) where d is Element of C : b3 in C } \/ {{(union G)}} or B = b2 or B misses b2 ) ) ) )
assume A2: A in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} ; ::_thesis: ( b1 <> {} & ( for B being Subset of (union (Mycielskian G)) holds
( not b2 in { (b3 \/ (r . b3)) where d is Element of C : b3 in C } \/ {{(union G)}} or B = b2 or B misses b2 ) ) )
percases ( A in { (d \/ (r . d)) where d is Element of C : d in C } or A in {{(union G)}} ) by A2, XBOOLE_0:def_3;
suppose A in { (d \/ (r . d)) where d is Element of C : d in C } ; ::_thesis: ( b1 <> {} & ( for B being Subset of (union (Mycielskian G)) holds
( not b2 in { (b3 \/ (r . b3)) where d is Element of C : b3 in C } \/ {{(union G)}} or B = b2 or B misses b2 ) ) )
then consider d being Element of C such that
A3: A = d \/ (r . d) and
B3: d in C ;
thus A <> {} by A3, B3; ::_thesis: for B being Subset of (union (Mycielskian G)) holds
( not B in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} or A = B or A misses B )
thus for B being Subset of (union (Mycielskian G)) holds
( not B in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} or A = B or A misses B ) ::_thesis: verum
proof
let B be Subset of (union (Mycielskian G)); ::_thesis: ( not B in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} or A = B or A misses B )
assume A4: B in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} ; ::_thesis: ( A = B or A misses B )
percases ( B in { (d \/ (r . d)) where d is Element of C : d in C } or B in {{(union G)}} ) by A4, XBOOLE_0:def_3;
suppose B in { (d \/ (r . d)) where d is Element of C : d in C } ; ::_thesis: ( A = B or A misses B )
then consider e being Element of C such that
A5: B = e \/ (r . e) and
B5: e in C ;
now__::_thesis:_(_A_meets_B_implies_A_=_B_)
assume A meets B ; ::_thesis: A = B
then consider x being set such that
A6: x in A and
B6: x in B by XBOOLE_0:3;
percases ( ( x in d & x in e ) or ( x in d & x in r . e ) or ( x in r . d & x in e ) or ( x in r . d & x in r . e ) ) by A6, B6, A5, A3, XBOOLE_0:def_3;
suppose ( x in d & x in e ) ; ::_thesis: A = B
then d = e by EQREL_1:70;
hence A = B by A5, A3; ::_thesis: verum
end;
supposeA7: ( x in d & x in r . e ) ; ::_thesis: A = B
x in { [yy,(union G)] where yy is Element of union G : yy in e } by A7, C;
then consider yy being Element of union G such that
A8: x = [yy,(union G)] and
yy in e ;
thus A = B by A8, Aux1, A7; ::_thesis: verum
end;
supposeA7: ( x in r . d & x in e ) ; ::_thesis: A = B
x in { [yy,(union G)] where yy is Element of union G : yy in d } by A7, C;
then consider yy being Element of union G such that
A8: x = [yy,(union G)] and
yy in d ;
thus A = B by A8, Aux1, A7; ::_thesis: verum
end;
supposeA7: ( x in r . d & x in r . e ) ; ::_thesis: A = B
x in { [yy,(union G)] where yy is Element of union G : yy in d } by A7, B3, C;
then consider yy being Element of union G such that
A8: x = [yy,(union G)] and
B8: yy in d ;
x in { [zz,(union G)] where zz is Element of union G : zz in e } by A7, B5, C;
then consider zz being Element of union G such that
C8: x = [zz,(union G)] and
D8: zz in e ;
yy = zz by A8, C8, XTUPLE_0:1;
then d = e by B8, D8, EQREL_1:70;
hence A = B by A5, A3; ::_thesis: verum
end;
end;
end;
hence ( A = B or A misses B ) ; ::_thesis: verum
end;
suppose B in {{(union G)}} ; ::_thesis: ( A = B or A misses B )
then B5: B = {(union G)} by TARSKI:def_1;
now__::_thesis:_not_A_meets_B
assume A meets B ; ::_thesis: contradiction
then consider x being set such that
A4: x in A and
B4: x in B by XBOOLE_0:3;
C4: x = union G by B4, B5, TARSKI:def_1;
percases ( union G in d or union G in r . d ) by C4, A4, A3, XBOOLE_0:def_3;
suppose union G in d ; ::_thesis: contradiction
then union G in union G ;
hence contradiction ; ::_thesis: verum
end;
suppose union G in r . d ; ::_thesis: contradiction
then union G in { [yy,(union G)] where yy is Element of union G : yy in d } by B3, C;
then consider yy being Element of union G such that
A4: union G = [yy,(union G)] and
yy in d ;
thus contradiction by A4, Aux2; ::_thesis: verum
end;
end;
end;
hence ( A = B or A misses B ) ; ::_thesis: verum
end;
end;
end;
end;
supposeA2a: A in {{(union G)}} ; ::_thesis: ( b1 <> {} & ( for B being Subset of (union (Mycielskian G)) holds
( not b2 in { (b3 \/ (r . b3)) where d is Element of C : b3 in C } \/ {{(union G)}} or B = b2 or B misses b2 ) ) )
then A2: A = {(union G)} by TARSKI:def_1;
thus A <> {} by A2a; ::_thesis: for B being Subset of (union (Mycielskian G)) holds
( not B in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} or A = B or A misses B )
thus for B being Subset of (union (Mycielskian G)) holds
( not B in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} or A = B or A misses B ) ::_thesis: verum
proof
let B be Subset of (union (Mycielskian G)); ::_thesis: ( not B in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} or A = B or A misses B )
assume B2: B in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} ; ::_thesis: ( A = B or A misses B )
percases ( B in { (d \/ (r . d)) where d is Element of C : d in C } or B in {{(union G)}} ) by B2, XBOOLE_0:def_3;
suppose B in { (d \/ (r . d)) where d is Element of C : d in C } ; ::_thesis: ( A = B or A misses B )
then consider d being Element of C such that
A3: B = d \/ (r . d) and
B3: d in C ;
now__::_thesis:_not_A_meets_B
assume A meets B ; ::_thesis: contradiction
then consider x being set such that
A4: x in A and
B4: x in B by XBOOLE_0:3;
C4: x = union G by A4, A2, TARSKI:def_1;
percases ( union G in d or union G in r . d ) by C4, B4, A3, XBOOLE_0:def_3;
suppose union G in d ; ::_thesis: contradiction
then union G in union G ;
hence contradiction ; ::_thesis: verum
end;
suppose union G in r . d ; ::_thesis: contradiction
then union G in { [yy,(union G)] where yy is Element of union G : yy in d } by B3, C;
then consider yy being Element of union G such that
A4: union G = [yy,(union G)] and
yy in d ;
thus contradiction by A4, Aux2; ::_thesis: verum
end;
end;
end;
hence ( A = B or A misses B ) ; ::_thesis: verum
end;
suppose B in {{(union G)}} ; ::_thesis: ( A = B or A misses B )
hence ( A = B or A misses B ) by A2, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
end;
end;
end;
G1: { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} c= bool (union (Mycielskian G))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} or x in bool (union (Mycielskian G)) )
assume A2: x in { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} ; ::_thesis: x in bool (union (Mycielskian G))
percases ( x in { (d \/ (r . d)) where d is Element of C : d in C } or x in {{(union G)}} ) by A2, XBOOLE_0:def_3;
suppose x in { (d \/ (r . d)) where d is Element of C : d in C } ; ::_thesis: x in bool (union (Mycielskian G))
then consider d being Element of C such that
A3: x = d \/ (r . d) and
B3: d in C ;
E3: union G c= union (Mycielskian G) by M0, ZFMISC_1:77;
C3: d c= union (Mycielskian G) by E3, XBOOLE_1:1;
r . d c= union (Mycielskian G)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in r . d or y in union (Mycielskian G) )
assume y in r . d ; ::_thesis: y in union (Mycielskian G)
then y in { [yy,(union G)] where yy is Element of union G : yy in d } by B3, C;
then consider yy being Element of union G such that
A4: y = [yy,(union G)] and
B4: yy in d ;
{y} in Mycielskian G by A4, B4, M0e2aa;
hence y in union (Mycielskian G) by Vertices0; ::_thesis: verum
end;
then x c= union (Mycielskian G) by A3, C3, XBOOLE_1:8;
hence x in bool (union (Mycielskian G)) ; ::_thesis: verum
end;
suppose x in {{(union G)}} ; ::_thesis: x in bool (union (Mycielskian G))
then A3: x = {(union G)} by TARSKI:def_1;
union G in union (Mycielskian G) by M00;
then x c= union (Mycielskian G) by A3, ZFMISC_1:31;
hence x in bool (union (Mycielskian G)) ; ::_thesis: verum
end;
end;
end;
reconsider E = { (d \/ (r . d)) where d is Element of C : d in C } \/ {{(union G)}} as a_partition of union (Mycielskian G) by E1, F1, G1, EQREL_1:def_4;
E is StableSet-wise
proof
let e be set ; :: according to SCMYCIEL:def_20 ::_thesis: ( e in E implies e is StableSet of (Mycielskian G) )
assume A1: e in E ; ::_thesis: e is StableSet of (Mycielskian G)
reconsider e1 = e as Subset of (union (Mycielskian G)) by A1;
e1 is stable
proof
let x, y be set ; :: according to SCMYCIEL:def_19 ::_thesis: ( x <> y & x in e1 & y in e1 implies {x,y} nin Mycielskian G )
assume that
A2: x <> y and
B2: x in e1 and
C2: y in e1 ; ::_thesis: {x,y} nin Mycielskian G
percases ( e in { (d \/ (r . d)) where d is Element of C : d in C } or e in {{(union G)}} ) by A1, XBOOLE_0:def_3;
suppose e in { (d \/ (r . d)) where d is Element of C : d in C } ; ::_thesis: {x,y} nin Mycielskian G
then consider d being Element of C such that
A3: e = d \/ (r . d) and
B3: d in C ;
C3: S1[d,r . d] by C, B3;
D3: d is stable by B3, LStableSetwise;
percases ( ( x in d & y in d ) or ( x in d & y in r . d ) or ( x in r . d & y in d ) or ( x in r . d & y in r . d ) ) by A3, B2, C2, XBOOLE_0:def_3;
supposeS3: ( x in d & y in d ) ; ::_thesis: {x,y} nin Mycielskian G
then {x,y} nin G by D3, A2, Lstable;
hence {x,y} nin Mycielskian G by S3, M1; ::_thesis: verum
end;
supposethat S3a: x in d and
S3b: y in r . d ; ::_thesis: {x,y} nin Mycielskian G
consider x1 being Element of union G such that
A4: y = [x1,(union G)] and
B4: x1 in d by S3b, C3;
percases ( x1 = x or x1 <> x ) ;
suppose x1 = x ; ::_thesis: {x,y} nin Mycielskian G
hence {x,y} nin Mycielskian G by A4, M0e4a; ::_thesis: verum
end;
suppose x1 <> x ; ::_thesis: {x,y} nin Mycielskian G
then {x1,x} nin G by S3a, B4, D3, Lstable;
hence {x,y} nin Mycielskian G by S3a, A4, M0e4b; ::_thesis: verum
end;
end;
end;
supposethat S3a: x in r . d and
S3b: y in d ; ::_thesis: {x,y} nin Mycielskian G
consider x1 being Element of union G such that
A4: x = [x1,(union G)] and
B4: x1 in d by S3a, C3;
percases ( x1 = y or x1 <> y ) ;
suppose x1 = y ; ::_thesis: {x,y} nin Mycielskian G
hence {x,y} nin Mycielskian G by A4, M0e4a; ::_thesis: verum
end;
suppose x1 <> y ; ::_thesis: {x,y} nin Mycielskian G
then {x1,y} nin G by S3b, B4, D3, Lstable;
hence {x,y} nin Mycielskian G by A4, S3b, M0e4b; ::_thesis: verum
end;
end;
end;
supposethat S3a: x in r . d and
S3b: y in r . d ; ::_thesis: {x,y} nin Mycielskian G
consider x1 being Element of union G such that
A4: x = [x1,(union G)] and
x1 in d by S3a, C3;
consider y1 being Element of union G such that
C4: y = [y1,(union G)] and
y1 in d by S3b, C3;
thus {x,y} nin Mycielskian G by A4, C4, A2, M0e3a; ::_thesis: verum
end;
end;
end;
suppose e in {{(union G)}} ; ::_thesis: {x,y} nin Mycielskian G
then e = {(union G)} by TARSKI:def_1;
then ( x = union G & y = union G ) by B2, C2, TARSKI:def_1;
hence {x,y} nin Mycielskian G by A2; ::_thesis: verum
end;
end;
end;
hence e is StableSet of (Mycielskian G) ; ::_thesis: verum
end;
then reconsider E = E as Coloring of (Mycielskian G) ;
take E ; ::_thesis: card E = 1 + (chromatic# G)
now__::_thesis:_not_{(union_G)}_in__{__(d_\/_(r_._d))_where_d_is_Element_of_C_:_d_in_C__}_
assume {(union G)} in { (d \/ (r . d)) where d is Element of C : d in C } ; ::_thesis: contradiction
then consider d being Element of C such that
A2: {(union G)} = d \/ (r . d) and
A2a: d in C ;
B2: union G in d \/ (r . d) by A2, TARSKI:def_1;
percases ( union G in d or union G in r . d ) by B2, XBOOLE_0:def_3;
suppose union G in d ; ::_thesis: contradiction
then union G in union G ;
hence contradiction ; ::_thesis: verum
end;
suppose union G in r . d ; ::_thesis: contradiction
then union G in { [x,(union G)] where x is Element of union G : x in d } by A2a, C;
then consider x being Element of union G such that
A3: union G = [x,(union G)] and
x in d ;
thus contradiction by A3, Aux2; ::_thesis: verum
end;
end;
end;
hence card E = 1 + (chromatic# G) by D1, D1a, A, CARD_2:41; ::_thesis: verum
end;
registration
let G be finitely_colorable SimpleGraph;
cluster Mycielskian G -> finitely_colorable ;
coherence
Mycielskian G is finitely_colorable
proof
consider E being Coloring of (Mycielskian G) such that
A: card E = 1 + (chromatic# G) by Mfc1;
E is finite by A;
hence Mycielskian G is finitely_colorable by Lfc; ::_thesis: verum
end;
end;
theorem Mcn1: :: SCMYCIEL:112
for G being finitely_colorable SimpleGraph holds chromatic# (Mycielskian G) = 1 + (chromatic# G)
proof
let G be finitely_colorable SimpleGraph; ::_thesis: chromatic# (Mycielskian G) = 1 + (chromatic# G)
set uG = union G;
set MG = Mycielskian G;
set uMG = union (Mycielskian G);
set cnG = chromatic# G;
set cnMG = chromatic# (Mycielskian G);
consider D being Coloring of (Mycielskian G) such that
A: card D = 1 + (chromatic# G) by Mfc1;
D is finite by A;
then Z: chromatic# (Mycielskian G) <= 1 + (chromatic# G) by A, Lchro;
now__::_thesis:_not_1_+_(chromatic#_G)_>_chromatic#_(Mycielskian_G)
assume A1: 1 + (chromatic# G) > chromatic# (Mycielskian G) ; ::_thesis: contradiction
B1: chromatic# G >= chromatic# (Mycielskian G) by A1, NAT_1:13;
C1: chromatic# G <= chromatic# (Mycielskian G) by M0, Subchro;
D1: chromatic# G = chromatic# (Mycielskian G) by B1, C1, XXREAL_0:1;
consider E being finite Coloring of (Mycielskian G) such that
E1: card E = chromatic# (Mycielskian G) by Lchro;
E1a: union E = union (Mycielskian G) by EQREL_1:def_4;
EE: G = (Mycielskian G) SubgraphInducedBy (union G) by GsubMG;
reconsider S = union G as Subset of (Vertices (Mycielskian G)) by M0, ZFMISC_1:77;
reconsider C = E | S as finite Coloring of G by EE, Tsr0;
F1: card C >= chromatic# G by Lchro;
G1: card C <= chromatic# G by D1, E1, MYCIELSK:8;
H1: card C = chromatic# G by F1, G1, XXREAL_0:1;
H1a: union G in union (Mycielskian G) by M00;
then consider EuG being set such that
I1: union G in EuG and
J1: EuG in E by E1a, TARSKI:def_4;
reconsider EuG = EuG as Subset of (Vertices (Mycielskian G)) by J1;
reconsider uG = union G as Element of Vertices (Mycielskian G) by I1, J1;
set se = EuG /\ S;
K1: EuG meets S by J1, D1, E1, H1, MYCIELSK:9;
EuG /\ S in C by J1, K1;
then consider sev being Element of Vertices G such that
M1: sev in EuG /\ S and
N1: for d being Element of C st d <> EuG /\ S holds
ex w being Element of Vertices G st
( w in Adjacent sev & w in d ) by F1, G1, XXREAL_0:1, AdjCol;
N1a: not uG is empty by XBOOLE_1:65, K1;
then {[sev,uG]} in Mycielskian G by M0e2aa;
then reconsider csev = [sev,uG] as Element of Vertices (Mycielskian G) by Vertices0;
csev in Vertices (Mycielskian G) by H1a;
then csev in union E by EQREL_1:def_4;
then consider Ecse being set such that
O1: csev in Ecse and
P1: Ecse in E by TARSKI:def_4;
reconsider Ecse = Ecse as Subset of (Vertices (Mycielskian G)) by P1;
Q1: now__::_thesis:_not_EuG_<>_Ecse
assume A2: EuG <> Ecse ; ::_thesis: contradiction
set sf = Ecse /\ S;
B2: Ecse meets S by P1, D1, E1, H1, MYCIELSK:9;
C2: Ecse /\ S in C by B2, P1;
now__::_thesis:_not_EuG_/\_S_=_Ecse_/\_S
assume EuG /\ S = Ecse /\ S ; ::_thesis: contradiction
then ( sev in EuG & sev in Ecse ) by M1, XBOOLE_0:def_4;
then EuG meets Ecse by XBOOLE_0:3;
hence contradiction by A2, P1, J1, EQREL_1:def_4; ::_thesis: verum
end;
then consider w being Element of Vertices G such that
D2: w in Adjacent sev and
E2: w in Ecse /\ S by C2, N1;
F2: w in Ecse by E2, XBOOLE_0:def_4;
G2: Ecse is stable by P1, LStableSetwise;
H2: csev <> w by N1a, Aux1;
{sev,w} in Edges G by D2, Ladj;
then {csev,w} in Mycielskian G by M0e4c;
hence contradiction by G2, H2, F2, O1, Lstable; ::_thesis: verum
end;
R1a: {csev,uG} in Edges (Mycielskian G) by N1a, M0e0;
S1: csev <> uG by Aux2;
EuG is stable by J1, LStableSetwise;
hence contradiction by S1, R1a, Q1, O1, I1, Lstable; ::_thesis: verum
end;
hence chromatic# (Mycielskian G) = 1 + (chromatic# G) by Z, XXREAL_0:1; ::_thesis: verum
end;
definition
let G be SimpleGraph;
func MycielskianSeq G -> ManySortedSet of NAT means :LMycielskianSeq: :: SCMYCIEL:def 26
ex myc being Function st
( it = myc & myc . 0 = G & ( for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G ) );
existence
ex b1 being ManySortedSet of NAT ex myc being Function st
( b1 = myc & myc . 0 = G & ( for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G ) )
proof
defpred S1[ Nat, set , set ] means ( ( $2 is SimpleGraph implies ex G being SimpleGraph st
( $2 = G & $3 = Mycielskian G ) ) & ( $2 is not SimpleGraph implies $3 = $2 ) );
P: for n being Element of NAT
for x being set ex y being set st S1[n,x,y]
proof
let n be Element of NAT ; ::_thesis: for x being set ex y being set st S1[n,x,y]
let x be set ; ::_thesis: ex y being set st S1[n,x,y]
percases ( x is SimpleGraph or not x is SimpleGraph ) ;
suppose x is SimpleGraph ; ::_thesis: ex y being set st S1[n,x,y]
then reconsider G = x as SimpleGraph ;
Mycielskian G = Mycielskian G ;
hence ex y being set st S1[n,x,y] ; ::_thesis: verum
end;
suppose x is not SimpleGraph ; ::_thesis: ex y being set st S1[n,x,y]
hence ex y being set st S1[n,x,y] ; ::_thesis: verum
end;
end;
end;
consider f being Function such that
A: dom f = NAT and
B: f . 0 = G and
C: for n being Element of NAT holds S1[n,f . n,f . (n + 1)] from RECDEF_1:sch_1(P);
reconsider f = f as NAT -defined Function by A, RELAT_1:def_18;
reconsider f = f as ManySortedSet of NAT by A, PARTFUN1:def_2;
take f ; ::_thesis: ex myc being Function st
( f = myc & myc . 0 = G & ( for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G ) )
take myc = f; ::_thesis: ( f = myc & myc . 0 = G & ( for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G ) )
thus f = myc ; ::_thesis: ( myc . 0 = G & ( for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G ) )
thus myc . 0 = G by B; ::_thesis: for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G
let k be Nat; ::_thesis: for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G
let G be SimpleGraph; ::_thesis: ( G = myc . k implies myc . (k + 1) = Mycielskian G )
assume Z: G = myc . k ; ::_thesis: myc . (k + 1) = Mycielskian G
k in NAT by ORDINAL1:def_12;
then ex G being SimpleGraph st
( f . k = G & f . (k + 1) = Mycielskian G ) by C, Z;
hence myc . (k + 1) = Mycielskian G by Z; ::_thesis: verum
end;
uniqueness
for b1, b2 being ManySortedSet of NAT st ex myc being Function st
( b1 = myc & myc . 0 = G & ( for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G ) ) & ex myc being Function st
( b2 = myc & myc . 0 = G & ( for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G ) ) holds
b1 = b2
proof
let it1, it2 be ManySortedSet of NAT ; ::_thesis: ( ex myc being Function st
( it1 = myc & myc . 0 = G & ( for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G ) ) & ex myc being Function st
( it2 = myc & myc . 0 = G & ( for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G ) ) implies it1 = it2 )
given myc1 being Function such that A1: it1 = myc1 and
B1: myc1 . 0 = G and
C1: for k being Nat
for G being SimpleGraph st G = myc1 . k holds
myc1 . (k + 1) = Mycielskian G ; ::_thesis: ( for myc being Function holds
( not it2 = myc or not myc . 0 = G or ex k being Nat ex G being SimpleGraph st
( G = myc . k & not myc . (k + 1) = Mycielskian G ) ) or it1 = it2 )
given myc2 being Function such that A2: it2 = myc2 and
B2: myc2 . 0 = G and
C2: for k being Nat
for G being SimpleGraph st G = myc2 . k holds
myc2 . (k + 1) = Mycielskian G ; ::_thesis: it1 = it2
defpred S1[ Nat] means ( myc1 . $1 is SimpleGraph & myc1 . $1 = myc2 . $1 );
P0: S1[ 0 ] by B1, B2;
P1: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume IH: S1[k] ; ::_thesis: S1[k + 1]
reconsider H = myc1 . k as SimpleGraph by IH;
myc1 . (k + 1) = Mycielskian H by C1;
hence myc1 . (k + 1) is SimpleGraph ; ::_thesis: myc1 . (k + 1) = myc2 . (k + 1)
thus myc1 . (k + 1) = Mycielskian H by C1
.= myc2 . (k + 1) by IH, C2 ; ::_thesis: verum
end;
D: for k being Nat holds S1[k] from NAT_1:sch_2(P0, P1);
for i being set st i in NAT holds
myc1 . i = myc2 . i by D;
hence it1 = it2 by A1, A2, PBOOLE:3; ::_thesis: verum
end;
end;
:: deftheorem LMycielskianSeq defines MycielskianSeq SCMYCIEL:def_26_:_
for G being SimpleGraph
for b2 being ManySortedSet of NAT holds
( b2 = MycielskianSeq G iff ex myc being Function st
( b2 = myc & myc . 0 = G & ( for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G ) ) );
theorem MSeq0: :: SCMYCIEL:113
for G being SimpleGraph holds (MycielskianSeq G) . 0 = G
proof
let G be SimpleGraph; ::_thesis: (MycielskianSeq G) . 0 = G
consider myc being Function such that
A: MycielskianSeq G = myc and
B: myc . 0 = G and
for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G by LMycielskianSeq;
thus (MycielskianSeq G) . 0 = G by A, B; ::_thesis: verum
end;
theorem MGS0: :: SCMYCIEL:114
for G being SimpleGraph
for n being Nat holds (MycielskianSeq G) . n is SimpleGraph
proof
let G be SimpleGraph; ::_thesis: for n being Nat holds (MycielskianSeq G) . n is SimpleGraph
let n be Nat; ::_thesis: (MycielskianSeq G) . n is SimpleGraph
set MG = MycielskianSeq G;
defpred S1[ Nat] means (MycielskianSeq G) . $1 is SimpleGraph;
consider myc being Function such that
A: MycielskianSeq G = myc and
B: myc . 0 = G and
C: for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G by LMycielskianSeq;
P0: S1[ 0 ] by A, B;
P1: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume S1[k] ; ::_thesis: S1[k + 1]
then reconsider H = (MycielskianSeq G) . k as SimpleGraph ;
(MycielskianSeq G) . (k + 1) = Mycielskian H by A, C;
hence S1[k + 1] ; ::_thesis: verum
end;
for k being Nat holds S1[k] from NAT_1:sch_2(P0, P1);
hence (MycielskianSeq G) . n is SimpleGraph ; ::_thesis: verum
end;
registration
let G be SimpleGraph;
let n be Nat;
cluster(MycielskianSeq G) . n -> SimpleGraph-like ;
coherence
(MycielskianSeq G) . n is SimpleGraph-like by MGS0;
end;
theorem MSeq1: :: SCMYCIEL:115
for G, H being SimpleGraph
for n being Nat holds (MycielskianSeq G) . (n + 1) = Mycielskian ((MycielskianSeq G) . n)
proof
let G, H be SimpleGraph; ::_thesis: for n being Nat holds (MycielskianSeq G) . (n + 1) = Mycielskian ((MycielskianSeq G) . n)
let n be Nat; ::_thesis: (MycielskianSeq G) . (n + 1) = Mycielskian ((MycielskianSeq G) . n)
set H = (MycielskianSeq G) . n;
consider myc being Function such that
A: MycielskianSeq G = myc and
myc . 0 = G and
C: for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G by LMycielskianSeq;
thus (MycielskianSeq G) . (n + 1) = Mycielskian ((MycielskianSeq G) . n) by A, C; ::_thesis: verum
end;
registration
let G be with_finite_clique# SimpleGraph;
let n be Nat;
cluster(MycielskianSeq G) . n -> with_finite_clique# ;
coherence
(MycielskianSeq G) . n is with_finite_clique#
proof
set MG = MycielskianSeq G;
defpred S1[ Nat] means (MycielskianSeq G) . G is with_finite_clique# ;
consider myc being Function such that
A: MycielskianSeq G = myc and
B: myc . 0 = G and
C: for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G by LMycielskianSeq;
P0: S1[ 0 ] by A, B;
P1: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume S1[k] ; ::_thesis: S1[k + 1]
then reconsider H = (MycielskianSeq G) . k as with_finite_clique# SimpleGraph ;
(MycielskianSeq G) . (k + 1) = Mycielskian H by A, C;
hence S1[k + 1] ; ::_thesis: verum
end;
for k being Nat holds S1[k] from NAT_1:sch_2(P0, P1);
hence (MycielskianSeq G) . n is with_finite_clique# ; ::_thesis: verum
end;
end;
registration
let G be finitely_colorable SimpleGraph;
let n be Nat;
cluster(MycielskianSeq G) . n -> finitely_colorable ;
coherence
(MycielskianSeq G) . n is finitely_colorable
proof
set MG = MycielskianSeq G;
defpred S1[ Nat] means (MycielskianSeq G) . G is finitely_colorable ;
consider myc being Function such that
A: MycielskianSeq G = myc and
B: myc . 0 = G and
C: for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G by LMycielskianSeq;
P0: S1[ 0 ] by A, B;
P1: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume S1[k] ; ::_thesis: S1[k + 1]
then reconsider H = (MycielskianSeq G) . k as finitely_colorable SimpleGraph ;
(MycielskianSeq G) . (k + 1) = Mycielskian H by A, C;
hence S1[k + 1] ; ::_thesis: verum
end;
for k being Nat holds S1[k] from NAT_1:sch_2(P0, P1);
hence (MycielskianSeq G) . n is finitely_colorable ; ::_thesis: verum
end;
end;
registration
let G be finite SimpleGraph;
let n be Nat;
cluster(MycielskianSeq G) . n -> finite ;
coherence
(MycielskianSeq G) . n is finite
proof
defpred S1[ Nat] means (MycielskianSeq G) . G is finite ;
P0: S1[ 0 ] by MSeq0;
P1: now__::_thesis:_for_k_being_Nat_st_S1[k]_holds_
S1[k_+_1]
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A: S1[k] ; ::_thesis: S1[k + 1]
set H = (MycielskianSeq G) . k;
(MycielskianSeq G) . (k + 1) = Mycielskian ((MycielskianSeq G) . k) by MSeq1;
hence S1[k + 1] by A; ::_thesis: verum
end;
for k being Nat holds S1[k] from NAT_1:sch_2(P0, P1);
hence (MycielskianSeq G) . n is finite ; ::_thesis: verum
end;
end;
theorem MSnorder: :: SCMYCIEL:116
for G being finite SimpleGraph
for n being Nat holds order ((MycielskianSeq G) . n) = (((2 |^ n) * (order G)) + (2 |^ n)) - 1
proof
let G be finite SimpleGraph; ::_thesis: for n being Nat holds order ((MycielskianSeq G) . n) = (((2 |^ n) * (order G)) + (2 |^ n)) - 1
let n be Nat; ::_thesis: order ((MycielskianSeq G) . n) = (((2 |^ n) * (order G)) + (2 |^ n)) - 1
set g = order G;
set MG = MycielskianSeq G;
defpred S1[ Nat] means order ((MycielskianSeq G) . $1) = (((2 |^ $1) * (order G)) + (2 |^ $1)) - 1;
P0: S1[ 0 ]
proof
thus order ((MycielskianSeq G) . 0) = ((order G) + 1) - 1 by MSeq0
.= ((1 * (order G)) + (2 |^ 0)) - 1 by NEWTON:4
.= (((2 |^ 0) * (order G)) + (2 |^ 0)) - 1 by NEWTON:4 ; ::_thesis: verum
end;
P1: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A1: S1[n] ; ::_thesis: S1[n + 1]
thus order ((MycielskianSeq G) . (n + 1)) = order (Mycielskian ((MycielskianSeq G) . n)) by MSeq1
.= (2 * ((((2 |^ n) * (order G)) + (2 |^ n)) - 1)) + 1 by A1, M0order
.= ((((2 * (2 |^ n)) * (order G)) + (2 * (2 |^ n))) - (2 * 1)) + 1
.= ((((2 |^ (n + 1)) * (order G)) + (2 * (2 |^ n))) - (2 * 1)) + 1 by NEWTON:6
.= ((((2 |^ (n + 1)) * (order G)) + (2 |^ (n + 1))) - 2) + 1 by NEWTON:6
.= (((2 |^ (n + 1)) * (order G)) + (2 |^ (n + 1))) - 1 ; ::_thesis: verum
end;
for n being Nat holds S1[n] from NAT_1:sch_2(P0, P1);
hence order ((MycielskianSeq G) . n) = (((2 |^ n) * (order G)) + (2 |^ n)) - 1 ; ::_thesis: verum
end;
theorem :: SCMYCIEL:117
for G being finite SimpleGraph
for n being Nat holds size ((MycielskianSeq G) . n) = (((3 |^ n) * (size G)) + (((3 |^ n) - (2 |^ n)) * (order G))) + ((n + 1) block 3)
proof
let G be finite SimpleGraph; ::_thesis: for n being Nat holds size ((MycielskianSeq G) . n) = (((3 |^ n) * (size G)) + (((3 |^ n) - (2 |^ n)) * (order G))) + ((n + 1) block 3)
let n be Nat; ::_thesis: size ((MycielskianSeq G) . n) = (((3 |^ n) * (size G)) + (((3 |^ n) - (2 |^ n)) * (order G))) + ((n + 1) block 3)
set g = order G;
set s = size G;
set MG = MycielskianSeq G;
defpred S1[ Nat] means size ((MycielskianSeq G) . $1) = (((3 |^ $1) * (size G)) + (((3 |^ $1) - (2 |^ $1)) * (order G))) + (($1 + 1) block 3);
P0: S1[ 0 ]
proof
thus size ((MycielskianSeq G) . 0) = ((1 * (size G)) + (0 * (order G))) + 0 by MSeq0
.= (((3 |^ 0) * (size G)) + ((1 - 1) * (order G))) + 0 by NEWTON:4
.= (((3 |^ 0) * (size G)) + (((3 |^ 0) - 1) * (order G))) + 0 by NEWTON:4
.= (((3 |^ 0) * (size G)) + (((3 |^ 0) - (2 |^ 0)) * (order G))) + 0 by NEWTON:4
.= (((3 |^ 0) * (size G)) + (((3 |^ 0) - (2 |^ 0)) * (order G))) + ((0 + 1) block 3) by STIRL2_1:29 ; ::_thesis: verum
end;
P1: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A1: S1[n] ; ::_thesis: S1[n + 1]
C1: n + 1 >= 0 + 1 by XREAL_1:6;
B1: (1 / 2) * ((2 |^ (n + 1)) - 2) = (1 / 2) * ((2 * (2 |^ n)) - (2 * 1)) by NEWTON:6
.= (2 |^ n) - 1 ;
thus size ((MycielskianSeq G) . (n + 1)) = size (Mycielskian ((MycielskianSeq G) . n)) by MSeq1
.= (3 * ((((3 |^ n) * (size G)) + (((3 |^ n) - (2 |^ n)) * (order G))) + ((n + 1) block 3))) + (order ((MycielskianSeq G) . n)) by A1, M0size
.= ((((3 * (3 |^ n)) * (size G)) + ((3 * ((3 |^ n) - (2 |^ n))) * (order G))) + (3 * ((n + 1) block 3))) + (order ((MycielskianSeq G) . n))
.= ((((3 |^ (n + 1)) * (size G)) + ((3 * ((3 |^ n) - (2 |^ n))) * (order G))) + (3 * ((n + 1) block 3))) + (order ((MycielskianSeq G) . n)) by NEWTON:6
.= ((((3 |^ (n + 1)) * (size G)) + ((3 * ((3 |^ n) - (2 |^ n))) * (order G))) + (3 * ((n + 1) block 3))) + ((((2 |^ n) * (order G)) + (2 |^ n)) - 1) by MSnorder
.= ((((3 |^ (n + 1)) * (size G)) + ((3 * ((3 |^ n) - (2 |^ n))) * (order G))) + ((2 |^ n) * (order G))) + ((3 * ((n + 1) block 3)) + ((2 |^ n) - 1))
.= ((((3 |^ (n + 1)) * (size G)) + ((3 * ((3 |^ n) - (2 |^ n))) * (order G))) + ((2 |^ n) * (order G))) + (((2 + 1) * ((n + 1) block (2 + 1))) + ((n + 1) block 2)) by B1, C1, STIRL2_1:47
.= (((3 |^ (n + 1)) * (size G)) + ((((3 * (3 |^ n)) * (order G)) - (((2 * (2 |^ n)) * (order G)) + ((2 |^ n) * (order G)))) + ((2 |^ n) * (order G)))) + (((n + 1) + 1) block 3) by STIRL2_1:46
.= (((3 |^ (n + 1)) * (size G)) + ((((3 * (3 |^ n)) * (order G)) - (((2 |^ (n + 1)) * (order G)) + ((2 |^ n) * (order G)))) + ((2 |^ n) * (order G)))) + (((n + 1) + 1) block 3) by NEWTON:6
.= (((3 |^ (n + 1)) * (size G)) + (((((3 * (3 |^ n)) * (order G)) - ((2 |^ (n + 1)) * (order G))) - ((2 |^ n) * (order G))) + ((2 |^ n) * (order G)))) + (((n + 1) + 1) block 3)
.= (((3 |^ (n + 1)) * (size G)) + (((((3 |^ (n + 1)) * (order G)) - ((2 |^ (n + 1)) * (order G))) - ((2 |^ n) * (order G))) + ((2 |^ n) * (order G)))) + (((n + 1) + 1) block 3) by NEWTON:6
.= (((3 |^ (n + 1)) * (size G)) + (((3 |^ (n + 1)) - (2 |^ (n + 1))) * (order G))) + (((n + 1) + 1) block 3) ; ::_thesis: verum
end;
for n being Nat holds S1[n] from NAT_1:sch_2(P0, P1);
hence size ((MycielskianSeq G) . n) = (((3 |^ n) * (size G)) + (((3 |^ n) - (2 |^ n)) * (order G))) + ((n + 1) block 3) ; ::_thesis: verum
end;
theorem MycTh: :: SCMYCIEL:118
for n being Nat holds
( clique# ((MycielskianSeq (CompleteSGraph 2)) . n) = 2 & chromatic# ((MycielskianSeq (CompleteSGraph 2)) . n) = n + 2 )
proof
Aa: card 2 = 2 by CARD_1:50, CARD_2:57;
set P2 = CompleteSGraph 2;
defpred S1[ Nat] means ( clique# ((MycielskianSeq (CompleteSGraph 2)) . $1) = 2 & chromatic# ((MycielskianSeq (CompleteSGraph 2)) . $1) = $1 + 2 );
A: clique# ((MycielskianSeq (CompleteSGraph 2)) . 0) = clique# (CompleteSGraph 2) by MSeq0
.= 2 by Aa, cliqueCSG ;
chromatic# ((MycielskianSeq (CompleteSGraph 2)) . 0) = chromatic# (CompleteSGraph 2) by MSeq0
.= 0 + 2 by Aa, chromaticCSG ;
then P0: S1[ 0 ] by A;
P1: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume IH: S1[n] ; ::_thesis: S1[n + 1]
thus clique# ((MycielskianSeq (CompleteSGraph 2)) . (n + 1)) = clique# (Mycielskian ((MycielskianSeq (CompleteSGraph 2)) . n)) by MSeq1
.= 2 by MClique, IH ; ::_thesis: chromatic# ((MycielskianSeq (CompleteSGraph 2)) . (n + 1)) = (n + 1) + 2
thus chromatic# ((MycielskianSeq (CompleteSGraph 2)) . (n + 1)) = chromatic# (Mycielskian ((MycielskianSeq (CompleteSGraph 2)) . n)) by MSeq1
.= 1 + (n + 2) by IH, Mcn1
.= (n + 1) + 2 ; ::_thesis: verum
end;
for n being Nat holds S1[n] from NAT_1:sch_2(P0, P1);
hence for n being Nat holds
( clique# ((MycielskianSeq (CompleteSGraph 2)) . n) = 2 & chromatic# ((MycielskianSeq (CompleteSGraph 2)) . n) = n + 2 ) ; ::_thesis: verum
end;
theorem :: SCMYCIEL:119
for n being Nat ex G being finite SimpleGraph st
( clique# G = 2 & chromatic# G > n )
proof
let n be Nat; ::_thesis: ex G being finite SimpleGraph st
( clique# G = 2 & chromatic# G > n )
set P2 = CompleteSGraph 2;
reconsider G = (MycielskianSeq (CompleteSGraph 2)) . n as finite SimpleGraph ;
take G ; ::_thesis: ( clique# G = 2 & chromatic# G > n )
( (n + 1) + 1 > n + 1 & n + 1 > n ) by NAT_1:13;
then n + 2 > n by XXREAL_0:2;
hence ( clique# G = 2 & chromatic# G > n ) by MycTh; ::_thesis: verum
end;
theorem :: SCMYCIEL:120
for n being Nat ex G being finite SimpleGraph st
( stability# G = 2 & cliquecover# G > n )
proof
let n be Nat; ::_thesis: ex G being finite SimpleGraph st
( stability# G = 2 & cliquecover# G > n )
set G = (MycielskianSeq (CompleteSGraph 2)) . n;
( (n + 1) + 1 > n + 1 & n + 1 > n ) by NAT_1:13;
then n + 2 > n by XXREAL_0:2;
then A1: ( clique# ((MycielskianSeq (CompleteSGraph 2)) . n) = 2 & chromatic# ((MycielskianSeq (CompleteSGraph 2)) . n) > n ) by MycTh;
take S = Complement ((MycielskianSeq (CompleteSGraph 2)) . n); ::_thesis: ( stability# S = 2 & cliquecover# S > n )
thus stability# S = 2 by A1, cliRstaCR; ::_thesis: cliquecover# S > n
thus cliquecover# S > n by A1, chrRcovCR; ::_thesis: verum
end;