:: COHSP_1 semantic presentation
begin
Lm1: for X, Y being non empty set
for f being Function of X,Y
for x being Element of X
for y being set st y in f . x holds
y in union Y
by TARSKI:def_4;
definition
let X be set ;
redefine attr X is binary_complete means :Def1: :: COHSP_1:def 1
for A being set st ( for a, b being set st a in A & b in A holds
a \/ b in X ) holds
union A in X;
compatibility
( X is binary_complete iff for A being set st ( for a, b being set st a in A & b in A holds
a \/ b in X ) holds
union A in X )
proof
thus ( X is binary_complete implies for A being set st ( for a, b being set st a in A & b in A holds
a \/ b in X ) holds
union A in X ) ::_thesis: ( ( for A being set st ( for a, b being set st a in A & b in A holds
a \/ b in X ) holds
union A in X ) implies X is binary_complete )
proof
assume A1: for A being set st A c= X & ( for a, b being set st a in A & b in A holds
a \/ b in X ) holds
union A in X ; :: according to COH_SP:def_1 ::_thesis: for A being set st ( for a, b being set st a in A & b in A holds
a \/ b in X ) holds
union A in X
let A be set ; ::_thesis: ( ( for a, b being set st a in A & b in A holds
a \/ b in X ) implies union A in X )
assume A2: for a, b being set st a in A & b in A holds
a \/ b in X ; ::_thesis: union A in X
A c= X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in X )
assume x in A ; ::_thesis: x in X
then x \/ x in X by A2;
hence x in X ; ::_thesis: verum
end;
hence union A in X by A1, A2; ::_thesis: verum
end;
assume for A being set st ( for a, b being set st a in A & b in A holds
a \/ b in X ) holds
union A in X ; ::_thesis: X is binary_complete
hence for A being set st A c= X & ( for a, b being set st a in A & b in A holds
a \/ b in X ) holds
union A in X ; :: according to COH_SP:def_1 ::_thesis: verum
end;
end;
:: deftheorem Def1 defines binary_complete COHSP_1:def_1_:_
for X being set holds
( X is binary_complete iff for A being set st ( for a, b being set st a in A & b in A holds
a \/ b in X ) holds
union A in X );
registration
cluster non empty finite subset-closed binary_complete for set ;
existence
ex b1 being Coherence_Space st b1 is finite by COH_SP:3;
end;
definition
let X be set ;
func FlatCoh X -> set equals :: COHSP_1:def 2
CohSp (id X);
correctness
coherence
CohSp (id X) is set ;
;
func Sub_of_Fin X -> Subset of X means :Def3: :: COHSP_1:def 3
for x being set holds
( x in it iff ( x in X & x is finite ) );
existence
ex b1 being Subset of X st
for x being set holds
( x in b1 iff ( x in X & x is finite ) )
proof
defpred S1[ set ] means $1 is finite ;
thus ex W being Subset of X st
for x being set holds
( x in W iff ( x in X & S1[x] ) ) from SUBSET_1:sch_1(); ::_thesis: verum
end;
correctness
uniqueness
for b1, b2 being Subset of X st ( for x being set holds
( x in b1 iff ( x in X & x is finite ) ) ) & ( for x being set holds
( x in b2 iff ( x in X & x is finite ) ) ) holds
b1 = b2;
proof
let X1, X2 be Subset of X; ::_thesis: ( ( for x being set holds
( x in X1 iff ( x in X & x is finite ) ) ) & ( for x being set holds
( x in X2 iff ( x in X & x is finite ) ) ) implies X1 = X2 )
assume A1: ( ( for x being set holds
( x in X1 iff ( x in X & x is finite ) ) ) & ( for x being set holds
( x in X2 iff ( x in X & x is finite ) ) ) & not X1 = X2 ) ; ::_thesis: contradiction
then consider x being set such that
A2: ( ( x in X1 & not x in X2 ) or ( x in X2 & not x in X1 ) ) by TARSKI:1;
( x in X2 iff ( not x in X or not x is finite ) ) by A1, A2;
hence contradiction by A1; ::_thesis: verum
end;
end;
:: deftheorem defines FlatCoh COHSP_1:def_2_:_
for X being set holds FlatCoh X = CohSp (id X);
:: deftheorem Def3 defines Sub_of_Fin COHSP_1:def_3_:_
for X being set
for b2 being Subset of X holds
( b2 = Sub_of_Fin X iff for x being set holds
( x in b2 iff ( x in X & x is finite ) ) );
theorem Th1: :: COHSP_1:1
for X, x being set holds
( x in FlatCoh X iff ( x = {} or ex y being set st
( x = {y} & y in X ) ) )
proof
let X, x be set ; ::_thesis: ( x in FlatCoh X iff ( x = {} or ex y being set st
( x = {y} & y in X ) ) )
hereby ::_thesis: ( ( x = {} or ex y being set st
( x = {y} & y in X ) ) implies x in FlatCoh X )
assume A1: x in FlatCoh X ; ::_thesis: ( x <> {} implies ex z being set st
( x = {z} & z in X ) )
assume x <> {} ; ::_thesis: ex z being set st
( x = {z} & z in X )
then reconsider y = x as non empty set ;
set z = the Element of y;
reconsider z = the Element of y as set ;
take z = z; ::_thesis: ( x = {z} & z in X )
thus x = {z} ::_thesis: z in X
proof
hereby :: according to XBOOLE_0:def_10,TARSKI:def_3 ::_thesis: {z} c= x
let c be set ; ::_thesis: ( c in x implies c in {z} )
assume c in x ; ::_thesis: c in {z}
then [z,c] in id X by A1, COH_SP:def_3;
then c = z by RELAT_1:def_10;
hence c in {z} by TARSKI:def_1; ::_thesis: verum
end;
thus {z} c= x by ZFMISC_1:31; ::_thesis: verum
end;
[z,z] in id X by A1, COH_SP:def_3;
hence z in X by RELAT_1:def_10; ::_thesis: verum
end;
A2: now__::_thesis:_(_ex_a_being_set_st_
(_x_=_{a}_&_a_in_X_)_implies_for_y,_z_being_set_st_y_in_x_&_z_in_x_holds_
[y,z]_in_id_X_)
given a being set such that A3: x = {a} and
A4: a in X ; ::_thesis: for y, z being set st y in x & z in x holds
[y,z] in id X
let y, z be set ; ::_thesis: ( y in x & z in x implies [y,z] in id X )
assume ( y in x & z in x ) ; ::_thesis: [y,z] in id X
then ( y = a & z = a ) by A3, TARSKI:def_1;
hence [y,z] in id X by A4, RELAT_1:def_10; ::_thesis: verum
end;
assume ( x = {} or ex y being set st
( x = {y} & y in X ) ) ; ::_thesis: x in FlatCoh X
hence x in FlatCoh X by A2, COH_SP:1, COH_SP:def_3; ::_thesis: verum
end;
theorem Th2: :: COHSP_1:2
for X being set holds union (FlatCoh X) = X
proof
let X be set ; ::_thesis: union (FlatCoh X) = X
hereby :: according to XBOOLE_0:def_10,TARSKI:def_3 ::_thesis: X c= union (FlatCoh X)
let x be set ; ::_thesis: ( x in union (FlatCoh X) implies x in X )
assume x in union (FlatCoh X) ; ::_thesis: x in X
then consider y being set such that
A1: x in y and
A2: y in FlatCoh X by TARSKI:def_4;
ex z being set st
( y = {z} & z in X ) by A1, A2, Th1;
hence x in X by A1, TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in union (FlatCoh X) )
assume x in X ; ::_thesis: x in union (FlatCoh X)
then ( x in {x} & {x} in FlatCoh X ) by Th1, TARSKI:def_1;
hence x in union (FlatCoh X) by TARSKI:def_4; ::_thesis: verum
end;
theorem :: COHSP_1:3
for X being finite subset-closed set holds Sub_of_Fin X = X
proof
let X be finite subset-closed set ; ::_thesis: Sub_of_Fin X = X
thus Sub_of_Fin X c= X ; :: according to XBOOLE_0:def_10 ::_thesis: X c= Sub_of_Fin X
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in Sub_of_Fin X )
assume A1: x in X ; ::_thesis: x in Sub_of_Fin X
bool x c= X
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in bool x or y in X )
assume y in bool x ; ::_thesis: y in X
hence y in X by A1, CLASSES1:def_1; ::_thesis: verum
end;
then x is finite ;
hence x in Sub_of_Fin X by A1, Def3; ::_thesis: verum
end;
registration
cluster{{}} -> subset-closed binary_complete ;
coherence
( {{}} is subset-closed & {{}} is binary_complete ) by COH_SP:3;
let X be set ;
cluster bool X -> subset-closed binary_complete ;
coherence
( bool X is subset-closed & bool X is binary_complete ) by COH_SP:2;
cluster FlatCoh X -> non empty subset-closed binary_complete ;
coherence
( not FlatCoh X is empty & FlatCoh X is subset-closed & FlatCoh X is binary_complete ) ;
end;
registration
let C be non empty subset-closed set ;
cluster Sub_of_Fin C -> non empty subset-closed ;
coherence
( not Sub_of_Fin C is empty & Sub_of_Fin C is subset-closed )
proof
set c = the Element of C;
{} c= the Element of C by XBOOLE_1:2;
then {} in C by CLASSES1:def_1;
hence not Sub_of_Fin C is empty by Def3; ::_thesis: Sub_of_Fin C is subset-closed
let a, b be set ; :: according to CLASSES1:def_1 ::_thesis: ( not a in Sub_of_Fin C or not b c= a or b in Sub_of_Fin C )
assume A1: a in Sub_of_Fin C ; ::_thesis: ( not b c= a or b in Sub_of_Fin C )
then A2: a is finite by Def3;
assume A3: b c= a ; ::_thesis: b in Sub_of_Fin C
then b in C by A1, CLASSES1:def_1;
hence b in Sub_of_Fin C by A2, A3, Def3; ::_thesis: verum
end;
end;
theorem :: COHSP_1:4
Web {{}} = {}
proof
union {{}} = {} by ZFMISC_1:25;
hence Web {{}} = {} ; ::_thesis: verum
end;
scheme :: COHSP_1:sch 1
MinimalElementwrtIncl{ F1() -> set , F2() -> set , P1[ set ] } :
ex a being set st
( a in F2() & P1[a] & ( for b being set st b in F2() & P1[b] & b c= a holds
b = a ) )
provided
A1: ( F1() in F2() & P1[F1()] ) and
A2: F1() is finite
proof
reconsider a = F1() as finite set by A2;
defpred S1[ set ] means ( $1 c= F1() & P1[$1] );
consider X being set such that
A3: for x being set holds
( x in X iff ( x in F2() & S1[x] ) ) from XBOOLE_0:sch_1();
A4: ( bool a is finite & X c= bool a )
proof
thus bool a is finite ; ::_thesis: X c= bool a
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in bool a )
assume x in X ; ::_thesis: x in bool a
then x c= a by A3;
hence x in bool a ; ::_thesis: verum
end;
defpred S2[ set , set ] means $1 c= $2;
A5: for x, y being set st S2[x,y] & S2[y,x] holds
x = y by XBOOLE_0:def_10;
A6: for x, y, z being set st S2[x,y] & S2[y,z] holds
S2[x,z] by XBOOLE_1:1;
reconsider X = X as finite set by A4;
A7: X <> {} by A1, A3;
consider x being set such that
A8: ( x in X & ( for y being set st y in X & y <> x holds
not S2[y,x] ) ) from CARD_2:sch_1(A7, A5, A6);
take x ; ::_thesis: ( x in F2() & P1[x] & ( for b being set st b in F2() & P1[b] & b c= x holds
b = x ) )
thus ( x in F2() & P1[x] ) by A3, A8; ::_thesis: for b being set st b in F2() & P1[b] & b c= x holds
b = x
let b be set ; ::_thesis: ( b in F2() & P1[b] & b c= x implies b = x )
assume that
A9: ( b in F2() & P1[b] ) and
A10: b c= x ; ::_thesis: b = x
x c= a by A3, A8;
then b c= a by A10, XBOOLE_1:1;
then b in X by A3, A9;
hence b = x by A8, A10; ::_thesis: verum
end;
registration
let C be Coherence_Space;
cluster finite for Element of C;
existence
ex b1 being Element of C st b1 is finite
proof
reconsider E = {} as Element of C by COH_SP:1;
take E ; ::_thesis: E is finite
thus E is finite ; ::_thesis: verum
end;
end;
definition
let X be set ;
attrX is c=directed means :: COHSP_1:def 4
for Y being finite Subset of X ex a being set st
( union Y c= a & a in X );
attrX is c=filtered means :: COHSP_1:def 5
for Y being finite Subset of X ex a being set st
( ( for y being set st y in Y holds
a c= y ) & a in X );
end;
:: deftheorem defines c=directed COHSP_1:def_4_:_
for X being set holds
( X is c=directed iff for Y being finite Subset of X ex a being set st
( union Y c= a & a in X ) );
:: deftheorem defines c=filtered COHSP_1:def_5_:_
for X being set holds
( X is c=filtered iff for Y being finite Subset of X ex a being set st
( ( for y being set st y in Y holds
a c= y ) & a in X ) );
registration
cluster c=directed -> non empty for set ;
coherence
for b1 being set st b1 is c=directed holds
not b1 is empty
proof
let X be set ; ::_thesis: ( X is c=directed implies not X is empty )
assume for Y being finite Subset of X ex a being set st
( union Y c= a & a in X ) ; :: according to COHSP_1:def_4 ::_thesis: not X is empty
then ex a being set st
( union ({} X) c= a & a in X ) ;
hence not X is empty ; ::_thesis: verum
end;
cluster c=filtered -> non empty for set ;
coherence
for b1 being set st b1 is c=filtered holds
not b1 is empty
proof
let X be set ; ::_thesis: ( X is c=filtered implies not X is empty )
assume for Y being finite Subset of X ex a being set st
( ( for y being set st y in Y holds
a c= y ) & a in X ) ; :: according to COHSP_1:def_5 ::_thesis: not X is empty
then ex a being set st
( ( for y being set st y in {} X holds
a c= y ) & a in X ) ;
hence not X is empty ; ::_thesis: verum
end;
end;
theorem Th5: :: COHSP_1:5
for X being set st X is c=directed holds
for a, b being set st a in X & b in X holds
ex c being set st
( a \/ b c= c & c in X )
proof
let X be set ; ::_thesis: ( X is c=directed implies for a, b being set st a in X & b in X holds
ex c being set st
( a \/ b c= c & c in X ) )
assume A1: for Y being finite Subset of X ex a being set st
( union Y c= a & a in X ) ; :: according to COHSP_1:def_4 ::_thesis: for a, b being set st a in X & b in X holds
ex c being set st
( a \/ b c= c & c in X )
let a, b be set ; ::_thesis: ( a in X & b in X implies ex c being set st
( a \/ b c= c & c in X ) )
assume ( a in X & b in X ) ; ::_thesis: ex c being set st
( a \/ b c= c & c in X )
then A2: {a,b} is finite Subset of X by ZFMISC_1:32;
union {a,b} = a \/ b by ZFMISC_1:75;
hence ex c being set st
( a \/ b c= c & c in X ) by A1, A2; ::_thesis: verum
end;
theorem Th6: :: COHSP_1:6
for X being non empty set st ( for a, b being set st a in X & b in X holds
ex c being set st
( a \/ b c= c & c in X ) ) holds
X is c=directed
proof
let X be non empty set ; ::_thesis: ( ( for a, b being set st a in X & b in X holds
ex c being set st
( a \/ b c= c & c in X ) ) implies X is c=directed )
assume A1: for a, b being set st a in X & b in X holds
ex c being set st
( a \/ b c= c & c in X ) ; ::_thesis: X is c=directed
set a = the Element of X;
defpred S1[ set ] means ex a being set st
( union $1 c= a & a in X );
let Y be finite Subset of X; :: according to COHSP_1:def_4 ::_thesis: ex a being set st
( union Y c= a & a in X )
A2: now__::_thesis:_for_x,_B_being_set_st_x_in_Y_&_B_c=_Y_&_S1[B]_holds_
S1[B_\/_{x}]
let x, B be set ; ::_thesis: ( x in Y & B c= Y & S1[B] implies S1[B \/ {x}] )
assume that
A3: x in Y and
B c= Y ; ::_thesis: ( S1[B] implies S1[B \/ {x}] )
assume S1[B] ; ::_thesis: S1[B \/ {x}]
then consider a being set such that
A4: union B c= a and
A5: a in X ;
consider c being set such that
A6: ( a \/ x c= c & c in X ) by A1, A3, A5;
thus S1[B \/ {x}] ::_thesis: verum
proof
take c ; ::_thesis: ( union (B \/ {x}) c= c & c in X )
union (B \/ {x}) = (union B) \/ (union {x}) by ZFMISC_1:78
.= (union B) \/ x by ZFMISC_1:25 ;
then union (B \/ {x}) c= a \/ x by A4, XBOOLE_1:9;
hence ( union (B \/ {x}) c= c & c in X ) by A6, XBOOLE_1:1; ::_thesis: verum
end;
end;
union {} c= the Element of X by XBOOLE_1:2, ZFMISC_1:2;
then A7: S1[ {} ] ;
A8: Y is finite ;
thus S1[Y] from FINSET_1:sch_2(A8, A7, A2); ::_thesis: verum
end;
theorem :: COHSP_1:7
for X being set st X is c=filtered holds
for a, b being set st a in X & b in X holds
ex c being set st
( c c= a /\ b & c in X )
proof
let X be set ; ::_thesis: ( X is c=filtered implies for a, b being set st a in X & b in X holds
ex c being set st
( c c= a /\ b & c in X ) )
assume A1: for Y being finite Subset of X ex a being set st
( ( for y being set st y in Y holds
a c= y ) & a in X ) ; :: according to COHSP_1:def_5 ::_thesis: for a, b being set st a in X & b in X holds
ex c being set st
( c c= a /\ b & c in X )
let a, b be set ; ::_thesis: ( a in X & b in X implies ex c being set st
( c c= a /\ b & c in X ) )
assume ( a in X & b in X ) ; ::_thesis: ex c being set st
( c c= a /\ b & c in X )
then {a,b} c= X by ZFMISC_1:32;
then consider c being set such that
A2: for y being set st y in {a,b} holds
c c= y and
A3: c in X by A1;
take c ; ::_thesis: ( c c= a /\ b & c in X )
b in {a,b} by TARSKI:def_2;
then A4: c c= b by A2;
a in {a,b} by TARSKI:def_2;
then c c= a by A2;
hence ( c c= a /\ b & c in X ) by A3, A4, XBOOLE_1:19; ::_thesis: verum
end;
theorem Th8: :: COHSP_1:8
for X being non empty set st ( for a, b being set st a in X & b in X holds
ex c being set st
( c c= a /\ b & c in X ) ) holds
X is c=filtered
proof
let X be non empty set ; ::_thesis: ( ( for a, b being set st a in X & b in X holds
ex c being set st
( c c= a /\ b & c in X ) ) implies X is c=filtered )
assume A1: for a, b being set st a in X & b in X holds
ex c being set st
( c c= a /\ b & c in X ) ; ::_thesis: X is c=filtered
set a = the Element of X;
defpred S1[ set ] means ex a being set st
( ( for y being set st y in $1 holds
a c= y ) & a in X );
let Y be finite Subset of X; :: according to COHSP_1:def_5 ::_thesis: ex a being set st
( ( for y being set st y in Y holds
a c= y ) & a in X )
A2: now__::_thesis:_for_x,_B_being_set_st_x_in_Y_&_B_c=_Y_&_S1[B]_holds_
S1[B_\/_{x}]
let x, B be set ; ::_thesis: ( x in Y & B c= Y & S1[B] implies S1[B \/ {x}] )
assume that
A3: x in Y and
B c= Y ; ::_thesis: ( S1[B] implies S1[B \/ {x}] )
assume S1[B] ; ::_thesis: S1[B \/ {x}]
then consider a being set such that
A4: for y being set st y in B holds
a c= y and
A5: a in X ;
consider c being set such that
A6: c c= a /\ x and
A7: c in X by A1, A3, A5;
A8: ( a /\ x c= a & a /\ x c= x ) by XBOOLE_1:17;
thus S1[B \/ {x}] ::_thesis: verum
proof
take c ; ::_thesis: ( ( for y being set st y in B \/ {x} holds
c c= y ) & c in X )
hereby ::_thesis: c in X
let y be set ; ::_thesis: ( y in B \/ {x} implies c c= y )
assume y in B \/ {x} ; ::_thesis: c c= y
then ( y in B or y in {x} ) by XBOOLE_0:def_3;
then ( ( a c= y & c c= a ) or ( y = x & c c= x ) ) by A4, A6, A8, TARSKI:def_1, XBOOLE_1:1;
hence c c= y by XBOOLE_1:1; ::_thesis: verum
end;
thus c in X by A7; ::_thesis: verum
end;
end;
for y being set st y in {} holds
the Element of X c= y ;
then A9: S1[ {} ] ;
A10: Y is finite ;
thus S1[Y] from FINSET_1:sch_2(A10, A9, A2); ::_thesis: verum
end;
theorem Th9: :: COHSP_1:9
for x being set holds
( {x} is c=directed & {x} is c=filtered )
proof
let x be set ; ::_thesis: ( {x} is c=directed & {x} is c=filtered )
set X = {x};
hereby :: according to COHSP_1:def_4 ::_thesis: {x} is c=filtered
let Y be finite Subset of {x}; ::_thesis: ex x being set st
( union Y c= x & x in {x} )
take x = x; ::_thesis: ( union Y c= x & x in {x} )
union Y c= union {x} by ZFMISC_1:77;
hence ( union Y c= x & x in {x} ) by TARSKI:def_1, ZFMISC_1:25; ::_thesis: verum
end;
let Y be finite Subset of {x}; :: according to COHSP_1:def_5 ::_thesis: ex a being set st
( ( for y being set st y in Y holds
a c= y ) & a in {x} )
take x ; ::_thesis: ( ( for y being set st y in Y holds
x c= y ) & x in {x} )
thus for y being set st y in Y holds
x c= y by TARSKI:def_1; ::_thesis: x in {x}
thus x in {x} by TARSKI:def_1; ::_thesis: verum
end;
Lm2: now__::_thesis:_for_x,_y_being_set_holds_union_{x,y,(x_\/_y)}_=_x_\/_y
let x, y be set ; ::_thesis: union {x,y,(x \/ y)} = x \/ y
thus union {x,y,(x \/ y)} = union ({x,y} \/ {(x \/ y)}) by ENUMSET1:3
.= (union {x,y}) \/ (union {(x \/ y)}) by ZFMISC_1:78
.= (x \/ y) \/ (union {(x \/ y)}) by ZFMISC_1:75
.= (x \/ y) \/ (x \/ y) by ZFMISC_1:25
.= x \/ y ; ::_thesis: verum
end;
theorem :: COHSP_1:10
for x, y being set holds {x,y,(x \/ y)} is c=directed
proof
let x, y be set ; ::_thesis: {x,y,(x \/ y)} is c=directed
set X = {x,y,(x \/ y)};
let A be finite Subset of {x,y,(x \/ y)}; :: according to COHSP_1:def_4 ::_thesis: ex a being set st
( union A c= a & a in {x,y,(x \/ y)} )
take a = x \/ y; ::_thesis: ( union A c= a & a in {x,y,(x \/ y)} )
union {x,y,(x \/ y)} = a by Lm2;
hence union A c= a by ZFMISC_1:77; ::_thesis: a in {x,y,(x \/ y)}
thus a in {x,y,(x \/ y)} by ENUMSET1:def_1; ::_thesis: verum
end;
theorem :: COHSP_1:11
for x, y being set holds {x,y,(x /\ y)} is c=filtered
proof
let x, y be set ; ::_thesis: {x,y,(x /\ y)} is c=filtered
let A be finite Subset of {x,y,(x /\ y)}; :: according to COHSP_1:def_5 ::_thesis: ex a being set st
( ( for y being set st y in A holds
a c= y ) & a in {x,y,(x /\ y)} )
take a = x /\ y; ::_thesis: ( ( for y being set st y in A holds
a c= y ) & a in {x,y,(x /\ y)} )
hereby ::_thesis: a in {x,y,(x /\ y)}
let b be set ; ::_thesis: ( b in A implies a c= b )
assume b in A ; ::_thesis: a c= b
then ( b = x or b = y or b = x /\ y ) by ENUMSET1:def_1;
hence a c= b by XBOOLE_1:17; ::_thesis: verum
end;
thus a in {x,y,(x /\ y)} by ENUMSET1:def_1; ::_thesis: verum
end;
registration
cluster finite c=directed c=filtered for set ;
existence
ex b1 being set st
( b1 is c=directed & b1 is c=filtered & b1 is finite )
proof
take {{}} ; ::_thesis: ( {{}} is c=directed & {{}} is c=filtered & {{}} is finite )
thus ( {{}} is c=directed & {{}} is c=filtered & {{}} is finite ) by Th9; ::_thesis: verum
end;
end;
registration
let C be non empty set ;
cluster finite c=directed c=filtered for Element of bool C;
existence
ex b1 being Subset of C st
( b1 is c=directed & b1 is c=filtered & b1 is finite )
proof
set x = the Element of C;
( { the Element of C} is Subset of C & { the Element of C} is c=directed & { the Element of C} is c=filtered & { the Element of C} is finite ) by Th9, ZFMISC_1:31;
hence ex b1 being Subset of C st
( b1 is c=directed & b1 is c=filtered & b1 is finite ) ; ::_thesis: verum
end;
end;
theorem Th12: :: COHSP_1:12
for X being set holds
( Fin X is c=directed & Fin X is c=filtered )
proof
let X be set ; ::_thesis: ( Fin X is c=directed & Fin X is c=filtered )
now__::_thesis:_for_a,_b_being_set_st_a_in_Fin_X_&_b_in_Fin_X_holds_
ex_c_being_set_st_
(_a_\/_b_c=_c_&_c_in_Fin_X_)
let a, b be set ; ::_thesis: ( a in Fin X & b in Fin X implies ex c being set st
( a \/ b c= c & c in Fin X ) )
assume A1: ( a in Fin X & b in Fin X ) ; ::_thesis: ex c being set st
( a \/ b c= c & c in Fin X )
take c = a \/ b; ::_thesis: ( a \/ b c= c & c in Fin X )
thus a \/ b c= c ; ::_thesis: c in Fin X
( a c= X & b c= X ) by A1, FINSUB_1:def_5;
then a \/ b c= X by XBOOLE_1:8;
hence c in Fin X by A1, FINSUB_1:def_5; ::_thesis: verum
end;
hence Fin X is c=directed by Th6; ::_thesis: Fin X is c=filtered
now__::_thesis:_for_a,_b_being_set_st_a_in_Fin_X_&_b_in_Fin_X_holds_
ex_c_being_set_st_
(_c_c=_a_/\_b_&_c_in_Fin_X_)
reconsider c = {} as set ;
let a, b be set ; ::_thesis: ( a in Fin X & b in Fin X implies ex c being set st
( c c= a /\ b & c in Fin X ) )
assume that
a in Fin X and
b in Fin X ; ::_thesis: ex c being set st
( c c= a /\ b & c in Fin X )
take c = c; ::_thesis: ( c c= a /\ b & c in Fin X )
thus c c= a /\ b by XBOOLE_1:2; ::_thesis: c in Fin X
thus c in Fin X by FINSUB_1:7; ::_thesis: verum
end;
hence Fin X is c=filtered by Th8; ::_thesis: verum
end;
registration
let X be set ;
cluster Fin X -> c=directed c=filtered ;
coherence
( Fin X is c=directed & Fin X is c=filtered ) by Th12;
end;
Lm3: now__::_thesis:_for_C_being_non_empty_subset-closed_set_
for_a_being_Element_of_C_holds_Fin_a_c=_C
let C be non empty subset-closed set ; ::_thesis: for a being Element of C holds Fin a c= C
let a be Element of C; ::_thesis: Fin a c= C
thus Fin a c= C ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Fin a or x in C )
assume x in Fin a ; ::_thesis: x in C
then x c= a by FINSUB_1:def_5;
hence x in C by CLASSES1:def_1; ::_thesis: verum
end;
end;
registration
let C be non empty subset-closed set ;
cluster non empty preBoolean for Element of bool C;
existence
ex b1 being Subset of C st
( b1 is preBoolean & not b1 is empty )
proof
set a = the Element of C;
reconsider b = Fin the Element of C as Subset of C by Lm3;
take b ; ::_thesis: ( b is preBoolean & not b is empty )
thus ( b is preBoolean & not b is empty ) ; ::_thesis: verum
end;
end;
definition
let C be non empty subset-closed set ;
let a be Element of C;
:: original: Fin
redefine func Fin a -> non empty preBoolean Subset of C;
coherence
Fin a is non empty preBoolean Subset of C
proof
Fin a c= C
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Fin a or x in C )
assume x in Fin a ; ::_thesis: x in C
then x c= a by FINSUB_1:def_5;
hence x in C by CLASSES1:def_1; ::_thesis: verum
end;
hence Fin a is non empty preBoolean Subset of C ; ::_thesis: verum
end;
end;
theorem Th13: :: COHSP_1:13
for X being non empty set
for Y being set st X is c=directed & Y c= union X & Y is finite holds
ex Z being set st
( Z in X & Y c= Z )
proof
let X be non empty set ; ::_thesis: for Y being set st X is c=directed & Y c= union X & Y is finite holds
ex Z being set st
( Z in X & Y c= Z )
let Y be set ; ::_thesis: ( X is c=directed & Y c= union X & Y is finite implies ex Z being set st
( Z in X & Y c= Z ) )
set x = the Element of X;
defpred S1[ Element of NAT ] means for Y being set st Y c= union X & Y is finite & card Y = $1 holds
ex Z being set st
( Z in X & Y c= Z );
assume A1: X is c=directed ; ::_thesis: ( not Y c= union X or not Y is finite or ex Z being set st
( Z in X & Y c= Z ) )
A2: now__::_thesis:_for_n_being_Element_of_NAT_st_S1[n]_holds_
S1[n_+_1]
let n be Element of NAT ; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A3: S1[n] ; ::_thesis: S1[n + 1]
thus S1[n + 1] ::_thesis: verum
proof
let Y be set ; ::_thesis: ( Y c= union X & Y is finite & card Y = n + 1 implies ex Z being set st
( Z in X & Y c= Z ) )
assume that
A4: Y c= union X and
A5: Y is finite and
A6: card Y = n + 1 ; ::_thesis: ex Z being set st
( Z in X & Y c= Z )
reconsider Y9 = Y as non empty set by A6;
set y = the Element of Y9;
A7: Y \ { the Element of Y9} c= union X by A4, XBOOLE_1:1;
the Element of Y9 in Y ;
then consider Z9 being set such that
A8: the Element of Y9 in Z9 and
A9: Z9 in X by A4, TARSKI:def_4;
A10: (n + 1) - 1 = n by XCMPLX_1:26;
( { the Element of Y9} c= Y & card { the Element of Y9} = 1 ) by CARD_1:30, ZFMISC_1:31;
then card (Y \ { the Element of Y9}) = n by A5, A6, A10, CARD_2:44;
then consider Z being set such that
A11: Z in X and
A12: Y \ { the Element of Y9} c= Z by A3, A5, A7;
consider V being set such that
A13: Z \/ Z9 c= V and
A14: V in X by A1, A11, A9, Th5;
take V ; ::_thesis: ( V in X & Y c= V )
thus V in X by A14; ::_thesis: Y c= V
thus Y c= V ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Y or x in V )
A15: ( x in { the Element of Y9} or not x in { the Element of Y9} ) ;
assume x in Y ; ::_thesis: x in V
then ( x = the Element of Y9 or x in Y \ { the Element of Y9} ) by A15, TARSKI:def_1, XBOOLE_0:def_5;
then x in Z \/ Z9 by A12, A8, XBOOLE_0:def_3;
hence x in V by A13; ::_thesis: verum
end;
end;
end;
A16: S1[ 0 ]
proof
let Y be set ; ::_thesis: ( Y c= union X & Y is finite & card Y = 0 implies ex Z being set st
( Z in X & Y c= Z ) )
assume that
Y c= union X and
Y is finite and
A17: card Y = 0 ; ::_thesis: ex Z being set st
( Z in X & Y c= Z )
Y = {} by A17;
then Y c= the Element of X by XBOOLE_1:2;
hence ex Z being set st
( Z in X & Y c= Z ) ; ::_thesis: verum
end;
A18: for n being Element of NAT holds S1[n] from NAT_1:sch_1(A16, A2);
assume that
A19: Y c= union X and
A20: Y is finite ; ::_thesis: ex Z being set st
( Z in X & Y c= Z )
reconsider Y9 = Y as finite set by A20;
card Y = card Y9 ;
hence ex Z being set st
( Z in X & Y c= Z ) by A18, A19; ::_thesis: verum
end;
notation
let X be set ;
synonym multiplicative X for cap-closed ;
end;
definition
let X be set ;
attrX is d.union-closed means :Def6: :: COHSP_1:def 6
for A being Subset of X st A is c=directed holds
union A in X;
end;
:: deftheorem Def6 defines d.union-closed COHSP_1:def_6_:_
for X being set holds
( X is d.union-closed iff for A being Subset of X st A is c=directed holds
union A in X );
registration
cluster subset-closed -> multiplicative for set ;
coherence
for b1 being set st b1 is subset-closed holds
b1 is multiplicative
proof
let C be set ; ::_thesis: ( C is subset-closed implies C is multiplicative )
assume A1: C is subset-closed ; ::_thesis: C is multiplicative
let x, y be set ; :: according to FINSUB_1:def_2 ::_thesis: ( not x in C or not y in C or x /\ y in C )
x /\ y c= x by XBOOLE_1:17;
hence ( not x in C or not y in C or x /\ y in C ) by A1, CLASSES1:def_1; ::_thesis: verum
end;
end;
theorem Th14: :: COHSP_1:14
for C being Coherence_Space
for A being c=directed Subset of C holds union A in C
proof
let C be Coherence_Space; ::_thesis: for A being c=directed Subset of C holds union A in C
let A be c=directed Subset of C; ::_thesis: union A in C
now__::_thesis:_for_a,_b_being_set_st_a_in_A_&_b_in_A_holds_
a_\/_b_in_C
let a, b be set ; ::_thesis: ( a in A & b in A implies a \/ b in C )
assume ( a in A & b in A ) ; ::_thesis: a \/ b in C
then ex c being set st
( a \/ b c= c & c in A ) by Th5;
hence a \/ b in C by CLASSES1:def_1; ::_thesis: verum
end;
hence union A in C by Def1; ::_thesis: verum
end;
registration
cluster non empty subset-closed binary_complete -> d.union-closed for set ;
coherence
for b1 being Coherence_Space holds b1 is d.union-closed
proof
let C be Coherence_Space; ::_thesis: C is d.union-closed
let A be Subset of C; :: according to COHSP_1:def_6 ::_thesis: ( A is c=directed implies union A in C )
thus ( A is c=directed implies union A in C ) by Th14; ::_thesis: verum
end;
end;
registration
cluster non empty multiplicative subset-closed binary_complete d.union-closed for set ;
existence
ex b1 being Coherence_Space st
( b1 is multiplicative & b1 is d.union-closed )
proof
set C = the Coherence_Space;
take the Coherence_Space ; ::_thesis: ( the Coherence_Space is multiplicative & the Coherence_Space is d.union-closed )
thus ( the Coherence_Space is multiplicative & the Coherence_Space is d.union-closed ) ; ::_thesis: verum
end;
end;
definition
let C be non empty d.union-closed set ;
let A be c=directed Subset of C;
:: original: union
redefine func union A -> Element of C;
coherence
union A is Element of C by Def6;
end;
definition
let X, Y be set ;
predX includes_lattice_of Y means :: COHSP_1:def 7
for a, b being set st a in Y & b in Y holds
( a /\ b in X & a \/ b in X );
end;
:: deftheorem defines includes_lattice_of COHSP_1:def_7_:_
for X, Y being set holds
( X includes_lattice_of Y iff for a, b being set st a in Y & b in Y holds
( a /\ b in X & a \/ b in X ) );
theorem :: COHSP_1:15
for X being non empty set st X includes_lattice_of X holds
( X is c=directed & X is c=filtered )
proof
let X be non empty set ; ::_thesis: ( X includes_lattice_of X implies ( X is c=directed & X is c=filtered ) )
assume A1: for a, b being set st a in X & b in X holds
( a /\ b in X & a \/ b in X ) ; :: according to COHSP_1:def_7 ::_thesis: ( X is c=directed & X is c=filtered )
for a, b being set st a in X & b in X holds
ex c being set st
( a \/ b c= c & c in X ) by A1;
hence X is c=directed by Th6; ::_thesis: X is c=filtered
for a, b being set st a in X & b in X holds
ex c being set st
( c c= a /\ b & c in X ) by A1;
hence X is c=filtered by Th8; ::_thesis: verum
end;
definition
let X, x, y be set ;
predX includes_lattice_of x,y means :: COHSP_1:def 8
X includes_lattice_of {x,y};
end;
:: deftheorem defines includes_lattice_of COHSP_1:def_8_:_
for X, x, y being set holds
( X includes_lattice_of x,y iff X includes_lattice_of {x,y} );
theorem Th16: :: COHSP_1:16
for X, x, y being set holds
( X includes_lattice_of x,y iff ( x in X & y in X & x /\ y in X & x \/ y in X ) )
proof
let X, x, y be set ; ::_thesis: ( X includes_lattice_of x,y iff ( x in X & y in X & x /\ y in X & x \/ y in X ) )
thus ( X includes_lattice_of x,y implies ( x in X & y in X & x /\ y in X & x \/ y in X ) ) ::_thesis: ( x in X & y in X & x /\ y in X & x \/ y in X implies X includes_lattice_of x,y )
proof
A1: ( x \/ x = x & y \/ y = y ) ;
A2: ( x in {x,y} & y in {x,y} ) by TARSKI:def_2;
assume for a, b being set st a in {x,y} & b in {x,y} holds
( a /\ b in X & a \/ b in X ) ; :: according to COHSP_1:def_7,COHSP_1:def_8 ::_thesis: ( x in X & y in X & x /\ y in X & x \/ y in X )
hence ( x in X & y in X & x /\ y in X & x \/ y in X ) by A2, A1; ::_thesis: verum
end;
assume A3: ( x in X & y in X & x /\ y in X & x \/ y in X ) ; ::_thesis: X includes_lattice_of x,y
let a, b be set ; :: according to COHSP_1:def_7,COHSP_1:def_8 ::_thesis: ( a in {x,y} & b in {x,y} implies ( a /\ b in X & a \/ b in X ) )
assume that
A4: a in {x,y} and
A5: b in {x,y} ; ::_thesis: ( a /\ b in X & a \/ b in X )
A6: ( b = x or b = y ) by A5, TARSKI:def_2;
( a = x or a = y ) by A4, TARSKI:def_2;
hence ( a /\ b in X & a \/ b in X ) by A3, A6; ::_thesis: verum
end;
begin
definition
let f be Function;
attrf is union-distributive means :Def9: :: COHSP_1:def 9
for A being Subset of (dom f) st union A in dom f holds
f . (union A) = union (f .: A);
attrf is d.union-distributive means :Def10: :: COHSP_1:def 10
for A being Subset of (dom f) st A is c=directed & union A in dom f holds
f . (union A) = union (f .: A);
end;
:: deftheorem Def9 defines union-distributive COHSP_1:def_9_:_
for f being Function holds
( f is union-distributive iff for A being Subset of (dom f) st union A in dom f holds
f . (union A) = union (f .: A) );
:: deftheorem Def10 defines d.union-distributive COHSP_1:def_10_:_
for f being Function holds
( f is d.union-distributive iff for A being Subset of (dom f) st A is c=directed & union A in dom f holds
f . (union A) = union (f .: A) );
definition
let f be Function;
attrf is c=-monotone means :Def11: :: COHSP_1:def 11
for a, b being set st a in dom f & b in dom f & a c= b holds
f . a c= f . b;
attrf is cap-distributive means :Def12: :: COHSP_1:def 12
for a, b being set st dom f includes_lattice_of a,b holds
f . (a /\ b) = (f . a) /\ (f . b);
end;
:: deftheorem Def11 defines c=-monotone COHSP_1:def_11_:_
for f being Function holds
( f is c=-monotone iff for a, b being set st a in dom f & b in dom f & a c= b holds
f . a c= f . b );
:: deftheorem Def12 defines cap-distributive COHSP_1:def_12_:_
for f being Function holds
( f is cap-distributive iff for a, b being set st dom f includes_lattice_of a,b holds
f . (a /\ b) = (f . a) /\ (f . b) );
registration
cluster Relation-like Function-like d.union-distributive -> c=-monotone for set ;
coherence
for b1 being Function st b1 is d.union-distributive holds
b1 is c=-monotone
proof
let f be Function; ::_thesis: ( f is d.union-distributive implies f is c=-monotone )
assume A1: for A being Subset of (dom f) st A is c=directed & union A in dom f holds
f . (union A) = union (f .: A) ; :: according to COHSP_1:def_10 ::_thesis: f is c=-monotone
let a, b be set ; :: according to COHSP_1:def_11 ::_thesis: ( a in dom f & b in dom f & a c= b implies f . a c= f . b )
assume that
A2: a in dom f and
A3: b in dom f and
A4: a c= b ; ::_thesis: f . a c= f . b
reconsider A = {a,b} as Subset of (dom f) by A2, A3, ZFMISC_1:32;
A5: A is c=directed
proof
let Y be finite Subset of A; :: according to COHSP_1:def_4 ::_thesis: ex a being set st
( union Y c= a & a in A )
take b ; ::_thesis: ( union Y c= b & b in A )
union Y c= union A by ZFMISC_1:77;
then union Y c= a \/ b by ZFMISC_1:75;
hence ( union Y c= b & b in A ) by A4, TARSKI:def_2, XBOOLE_1:12; ::_thesis: verum
end;
a in A by TARSKI:def_2;
then A6: f . a in f .: A by FUNCT_1:def_6;
union A = a \/ b by ZFMISC_1:75
.= b by A4, XBOOLE_1:12 ;
then union (f .: A) = f . b by A1, A3, A5;
hence f . a c= f . b by A6, ZFMISC_1:74; ::_thesis: verum
end;
cluster Relation-like Function-like union-distributive -> d.union-distributive for set ;
coherence
for b1 being Function st b1 is union-distributive holds
b1 is d.union-distributive
proof
let f be Function; ::_thesis: ( f is union-distributive implies f is d.union-distributive )
assume A7: for A being Subset of (dom f) st union A in dom f holds
f . (union A) = union (f .: A) ; :: according to COHSP_1:def_9 ::_thesis: f is d.union-distributive
let A be Subset of (dom f); :: according to COHSP_1:def_10 ::_thesis: ( A is c=directed & union A in dom f implies f . (union A) = union (f .: A) )
thus ( A is c=directed & union A in dom f implies f . (union A) = union (f .: A) ) by A7; ::_thesis: verum
end;
end;
theorem :: COHSP_1:17
for f being Function st f is union-distributive holds
for x, y being set st x in dom f & y in dom f & x \/ y in dom f holds
f . (x \/ y) = (f . x) \/ (f . y)
proof
let f be Function; ::_thesis: ( f is union-distributive implies for x, y being set st x in dom f & y in dom f & x \/ y in dom f holds
f . (x \/ y) = (f . x) \/ (f . y) )
assume A1: f is union-distributive ; ::_thesis: for x, y being set st x in dom f & y in dom f & x \/ y in dom f holds
f . (x \/ y) = (f . x) \/ (f . y)
let x, y be set ; ::_thesis: ( x in dom f & y in dom f & x \/ y in dom f implies f . (x \/ y) = (f . x) \/ (f . y) )
set X = {x,y};
assume that
A2: ( x in dom f & y in dom f ) and
A3: x \/ y in dom f ; ::_thesis: f . (x \/ y) = (f . x) \/ (f . y)
A4: union {x,y} = x \/ y by ZFMISC_1:75;
{x,y} c= dom f by A2, ZFMISC_1:32;
hence f . (x \/ y) = union (f .: {x,y}) by A1, A3, A4, Def9
.= union {(f . x),(f . y)} by A2, FUNCT_1:60
.= (f . x) \/ (f . y) by ZFMISC_1:75 ;
::_thesis: verum
end;
theorem Th18: :: COHSP_1:18
for f being Function st f is union-distributive holds
f . {} = {}
proof
let f be Function; ::_thesis: ( f is union-distributive implies f . {} = {} )
assume A1: for A being Subset of (dom f) st union A in dom f holds
f . (union A) = union (f .: A) ; :: according to COHSP_1:def_9 ::_thesis: f . {} = {}
A2: ( {} c= dom f & f .: {} = {} ) by XBOOLE_1:2;
( not {} in dom f implies f . {} = {} ) by FUNCT_1:def_2;
hence f . {} = {} by A1, A2, ZFMISC_1:2; ::_thesis: verum
end;
registration
let C1, C2 be Coherence_Space;
cluster Relation-like C1 -defined C2 -valued Function-like V34(C1,C2) union-distributive cap-distributive for Element of bool [:C1,C2:];
existence
ex b1 being Function of C1,C2 st
( b1 is union-distributive & b1 is cap-distributive )
proof
reconsider a = {} as Element of C2 by COH_SP:1;
take f = C1 --> a; ::_thesis: ( f is union-distributive & f is cap-distributive )
A1: dom f = C1 by FUNCOP_1:13;
thus f is union-distributive ::_thesis: f is cap-distributive
proof
let A be Subset of (dom f); :: according to COHSP_1:def_9 ::_thesis: ( union A in dom f implies f . (union A) = union (f .: A) )
assume union A in dom f ; ::_thesis: f . (union A) = union (f .: A)
then A2: f . (union A) = {} by A1, FUNCOP_1:7;
f .: A c= {{}}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in f .: A or x in {{}} )
assume x in f .: A ; ::_thesis: x in {{}}
then ex y being set st
( y in dom f & y in A & x = f . y ) by FUNCT_1:def_6;
then x = {} by A1, FUNCOP_1:7;
hence x in {{}} by TARSKI:def_1; ::_thesis: verum
end;
then A3: union (f .: A) c= union {{}} by ZFMISC_1:77;
union {{}} = {} by ZFMISC_1:25;
hence f . (union A) = union (f .: A) by A2, A3; ::_thesis: verum
end;
let a, b be set ; :: according to COHSP_1:def_12 ::_thesis: ( dom f includes_lattice_of a,b implies f . (a /\ b) = (f . a) /\ (f . b) )
assume A4: dom f includes_lattice_of a,b ; ::_thesis: f . (a /\ b) = (f . a) /\ (f . b)
then a in dom f by Th16;
then A5: f . a = {} by A1, FUNCOP_1:7;
a /\ b in dom f by A4, Th16;
hence f . (a /\ b) = (f . a) /\ (f . b) by A1, A5, FUNCOP_1:7; ::_thesis: verum
end;
end;
registration
let C be Coherence_Space;
cluster Relation-like C -defined Function-like V30(C) union-distributive cap-distributive for set ;
existence
ex b1 being ManySortedSet of C st
( b1 is union-distributive & b1 is cap-distributive )
proof
set f = the union-distributive cap-distributive Function of C,C;
dom the union-distributive cap-distributive Function of C,C = C by FUNCT_2:52;
then reconsider f = the union-distributive cap-distributive Function of C,C as ManySortedSet of C by PARTFUN1:def_2;
take f ; ::_thesis: ( f is union-distributive & f is cap-distributive )
thus ( f is union-distributive & f is cap-distributive ) ; ::_thesis: verum
end;
end;
definition
let f be Function;
attrf is U-continuous means :Def13: :: COHSP_1:def 13
( dom f is d.union-closed & f is d.union-distributive );
end;
:: deftheorem Def13 defines U-continuous COHSP_1:def_13_:_
for f being Function holds
( f is U-continuous iff ( dom f is d.union-closed & f is d.union-distributive ) );
definition
let f be Function;
attrf is U-stable means :Def14: :: COHSP_1:def 14
( dom f is multiplicative & f is U-continuous & f is cap-distributive );
end;
:: deftheorem Def14 defines U-stable COHSP_1:def_14_:_
for f being Function holds
( f is U-stable iff ( dom f is multiplicative & f is U-continuous & f is cap-distributive ) );
definition
let f be Function;
attrf is U-linear means :Def15: :: COHSP_1:def 15
( f is U-stable & f is union-distributive );
end;
:: deftheorem Def15 defines U-linear COHSP_1:def_15_:_
for f being Function holds
( f is U-linear iff ( f is U-stable & f is union-distributive ) );
registration
cluster Relation-like Function-like U-continuous -> d.union-distributive for set ;
coherence
for b1 being Function st b1 is U-continuous holds
b1 is d.union-distributive by Def13;
cluster Relation-like Function-like U-stable -> cap-distributive U-continuous for set ;
coherence
for b1 being Function st b1 is U-stable holds
( b1 is cap-distributive & b1 is U-continuous ) by Def14;
cluster Relation-like Function-like U-linear -> union-distributive U-stable for set ;
coherence
for b1 being Function st b1 is U-linear holds
( b1 is union-distributive & b1 is U-stable ) by Def15;
end;
registration
let X be d.union-closed set ;
cluster Relation-like X -defined Function-like V30(X) d.union-distributive -> U-continuous for set ;
coherence
for b1 being ManySortedSet of X st b1 is d.union-distributive holds
b1 is U-continuous
proof
let f be ManySortedSet of X; ::_thesis: ( f is d.union-distributive implies f is U-continuous )
dom f = X by PARTFUN1:def_2;
hence ( f is d.union-distributive implies f is U-continuous ) by Def13; ::_thesis: verum
end;
end;
registration
let X be multiplicative set ;
cluster Relation-like X -defined Function-like V30(X) cap-distributive U-continuous -> U-stable for set ;
coherence
for b1 being ManySortedSet of X st b1 is U-continuous & b1 is cap-distributive holds
b1 is U-stable
proof
let f be ManySortedSet of X; ::_thesis: ( f is U-continuous & f is cap-distributive implies f is U-stable )
dom f = X by PARTFUN1:def_2;
hence ( f is U-continuous & f is cap-distributive implies f is U-stable ) by Def14; ::_thesis: verum
end;
end;
registration
cluster Relation-like Function-like union-distributive U-stable -> U-linear for set ;
coherence
for b1 being Function st b1 is U-stable & b1 is union-distributive holds
b1 is U-linear by Def15;
end;
registration
cluster Relation-like Function-like U-linear for set ;
existence
ex b1 being Function st b1 is U-linear
proof
set C = the Coherence_Space;
set f = the union-distributive cap-distributive ManySortedSet of the Coherence_Space;
take the union-distributive cap-distributive ManySortedSet of the Coherence_Space ; ::_thesis: the union-distributive cap-distributive ManySortedSet of the Coherence_Space is U-linear
thus the union-distributive cap-distributive ManySortedSet of the Coherence_Space is U-linear ; ::_thesis: verum
end;
let C be Coherence_Space;
cluster Relation-like C -defined Function-like V30(C) U-linear for set ;
existence
ex b1 being ManySortedSet of C st b1 is U-linear
proof
set f = the union-distributive cap-distributive ManySortedSet of C;
take the union-distributive cap-distributive ManySortedSet of C ; ::_thesis: the union-distributive cap-distributive ManySortedSet of C is U-linear
thus the union-distributive cap-distributive ManySortedSet of C is U-linear ; ::_thesis: verum
end;
let B be Coherence_Space;
cluster Relation-like B -defined C -valued Function-like V34(B,C) U-linear for Element of bool [:B,C:];
existence
ex b1 being Function of B,C st b1 is U-linear
proof
set f = the union-distributive cap-distributive Function of B,C;
take the union-distributive cap-distributive Function of B,C ; ::_thesis: the union-distributive cap-distributive Function of B,C is U-linear
dom the union-distributive cap-distributive Function of B,C = B by FUNCT_2:def_1;
then reconsider f = the union-distributive cap-distributive Function of B,C as union-distributive cap-distributive ManySortedSet of B by PARTFUN1:def_2;
f is U-linear ;
hence the union-distributive cap-distributive Function of B,C is U-linear ; ::_thesis: verum
end;
end;
registration
let f be U-continuous Function;
cluster proj1 f -> d.union-closed ;
coherence
dom f is d.union-closed by Def13;
end;
registration
let f be U-stable Function;
cluster proj1 f -> multiplicative ;
coherence
dom f is multiplicative by Def14;
end;
theorem Th19: :: COHSP_1:19
for X being set holds union (Fin X) = X
proof
let X be set ; ::_thesis: union (Fin X) = X
union (Fin X) c= union (bool X) by FINSUB_1:13, ZFMISC_1:77;
hence union (Fin X) c= X by ZFMISC_1:81; :: according to XBOOLE_0:def_10 ::_thesis: X c= union (Fin X)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in union (Fin X) )
assume x in X ; ::_thesis: x in union (Fin X)
then {x} c= X by ZFMISC_1:31;
then A1: {x} in Fin X by FINSUB_1:def_5;
x in {x} by TARSKI:def_1;
hence x in union (Fin X) by A1, TARSKI:def_4; ::_thesis: verum
end;
theorem Th20: :: COHSP_1:20
for f being U-continuous Function st dom f is subset-closed holds
for a being set st a in dom f holds
f . a = union (f .: (Fin a))
proof
let f be U-continuous Function; ::_thesis: ( dom f is subset-closed implies for a being set st a in dom f holds
f . a = union (f .: (Fin a)) )
assume A1: dom f is subset-closed ; ::_thesis: for a being set st a in dom f holds
f . a = union (f .: (Fin a))
let a be set ; ::_thesis: ( a in dom f implies f . a = union (f .: (Fin a)) )
assume A2: a in dom f ; ::_thesis: f . a = union (f .: (Fin a))
then reconsider C = dom f as non empty subset-closed d.union-closed set by A1;
reconsider a = a as Element of C by A2;
f . a = f . (union (Fin a)) by Th19;
hence f . a = union (f .: (Fin a)) by Def10; ::_thesis: verum
end;
theorem Th21: :: COHSP_1:21
for f being Function st dom f is subset-closed holds
( f is U-continuous iff ( dom f is d.union-closed & f is c=-monotone & ( for a, y being set st a in dom f & y in f . a holds
ex b being set st
( b is finite & b c= a & y in f . b ) ) ) )
proof
let f be Function; ::_thesis: ( dom f is subset-closed implies ( f is U-continuous iff ( dom f is d.union-closed & f is c=-monotone & ( for a, y being set st a in dom f & y in f . a holds
ex b being set st
( b is finite & b c= a & y in f . b ) ) ) ) )
assume A1: dom f is subset-closed ; ::_thesis: ( f is U-continuous iff ( dom f is d.union-closed & f is c=-monotone & ( for a, y being set st a in dom f & y in f . a holds
ex b being set st
( b is finite & b c= a & y in f . b ) ) ) )
hereby ::_thesis: ( dom f is d.union-closed & f is c=-monotone & ( for a, y being set st a in dom f & y in f . a holds
ex b being set st
( b is finite & b c= a & y in f . b ) ) implies f is U-continuous )
assume A2: f is U-continuous ; ::_thesis: ( dom f is d.union-closed & f is c=-monotone & ( for a, y being set st a in dom f & y in f . a holds
ex b being set st
( b is finite & b c= a & y in f . b ) ) )
hence ( dom f is d.union-closed & f is c=-monotone ) ; ::_thesis: for a, y being set st a in dom f & y in f . a holds
ex b being set st
( b is finite & b c= a & y in f . b )
reconsider C = dom f as subset-closed d.union-closed set by A1, A2;
let a, y be set ; ::_thesis: ( a in dom f & y in f . a implies ex b being set st
( b is finite & b c= a & y in f . b ) )
assume that
A3: a in dom f and
A4: y in f . a ; ::_thesis: ex b being set st
( b is finite & b c= a & y in f . b )
reconsider A = { b where b is Subset of a : b is finite } as set ;
A5: A is c=directed
proof
let Y be finite Subset of A; :: according to COHSP_1:def_4 ::_thesis: ex a being set st
( union Y c= a & a in A )
take union Y ; ::_thesis: ( union Y c= union Y & union Y in A )
now__::_thesis:_for_x_being_set_st_x_in_Y_holds_
x_c=_a
let x be set ; ::_thesis: ( x in Y implies x c= a )
assume x in Y ; ::_thesis: x c= a
then x in A ;
then ex c being Subset of a st
( x = c & c is finite ) ;
hence x c= a ; ::_thesis: verum
end;
then A6: union Y c= a by ZFMISC_1:76;
now__::_thesis:_for_b_being_set_st_b_in_Y_holds_
b_is_finite
let b be set ; ::_thesis: ( b in Y implies b is finite )
assume b in Y ; ::_thesis: b is finite
then b in A ;
then ex c being Subset of a st
( b = c & c is finite ) ;
hence b is finite ; ::_thesis: verum
end;
then union Y is finite by FINSET_1:7;
hence ( union Y c= union Y & union Y in A ) by A6; ::_thesis: verum
end;
A7: union A = a
proof
thus union A c= a :: according to XBOOLE_0:def_10 ::_thesis: a c= union A
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union A or x in a )
assume x in union A ; ::_thesis: x in a
then consider b being set such that
A8: x in b and
A9: b in A by TARSKI:def_4;
ex c being Subset of a st
( b = c & c is finite ) by A9;
hence x in a by A8; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in a or x in union A )
assume x in a ; ::_thesis: x in union A
then {x} c= a by ZFMISC_1:31;
then ( x in {x} & {x} in A ) by TARSKI:def_1;
hence x in union A by TARSKI:def_4; ::_thesis: verum
end;
A10: A c= C
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in C )
assume x in A ; ::_thesis: x in C
then ex b being Subset of a st
( x = b & b is finite ) ;
hence x in C by A3, CLASSES1:def_1; ::_thesis: verum
end;
then union A in C by A5, Def6;
then f . (union A) = union (f .: A) by A2, A5, A10, Def10;
then consider B being set such that
A11: y in B and
A12: B in f .: A by A4, A7, TARSKI:def_4;
consider b being set such that
b in dom f and
A13: b in A and
A14: B = f . b by A12, FUNCT_1:def_6;
take b = b; ::_thesis: ( b is finite & b c= a & y in f . b )
ex c being Subset of a st
( b = c & c is finite ) by A13;
hence ( b is finite & b c= a & y in f . b ) by A11, A14; ::_thesis: verum
end;
assume dom f is d.union-closed ; ::_thesis: ( not f is c=-monotone or ex a, y being set st
( a in dom f & y in f . a & ( for b being set holds
( not b is finite or not b c= a or not y in f . b ) ) ) or f is U-continuous )
then reconsider C = dom f as d.union-closed set ;
assume that
A15: for a, b being set st a in dom f & b in dom f & a c= b holds
f . a c= f . b and
A16: for a, y being set st a in dom f & y in f . a holds
ex b being set st
( b is finite & b c= a & y in f . b ) ; :: according to COHSP_1:def_11 ::_thesis: f is U-continuous
C is d.union-closed ;
hence dom f is d.union-closed ; :: according to COHSP_1:def_13 ::_thesis: f is d.union-distributive
thus f is d.union-distributive ::_thesis: verum
proof
let A be Subset of (dom f); :: according to COHSP_1:def_10 ::_thesis: ( A is c=directed & union A in dom f implies f . (union A) = union (f .: A) )
assume that
A17: A is c=directed and
A18: union A in dom f ; ::_thesis: f . (union A) = union (f .: A)
reconsider A9 = A as Subset of C ;
thus f . (union A) c= union (f .: A) :: according to XBOOLE_0:def_10 ::_thesis: union (f .: A) c= f . (union A)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in f . (union A) or x in union (f .: A) )
assume x in f . (union A) ; ::_thesis: x in union (f .: A)
then consider b being set such that
A19: ( b is finite & b c= union A9 ) and
A20: x in f . b by A16, A18;
consider c being set such that
A21: c in A and
A22: b c= c by A17, A19, Th13;
b in C by A1, A21, A22, CLASSES1:def_1;
then A23: f . b c= f . c by A15, A21, A22;
f . c in f .: A by A21, FUNCT_1:def_6;
hence x in union (f .: A) by A20, A23, TARSKI:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union (f .: A) or x in f . (union A) )
assume x in union (f .: A) ; ::_thesis: x in f . (union A)
then consider B being set such that
A24: x in B and
A25: B in f .: A by TARSKI:def_4;
ex b being set st
( b in dom f & b in A & B = f . b ) by A25, FUNCT_1:def_6;
then B c= f . (union A9) by A15, A18, ZFMISC_1:74;
hence x in f . (union A) by A24; ::_thesis: verum
end;
end;
theorem Th22: :: COHSP_1:22
for f being Function st dom f is subset-closed & dom f is d.union-closed holds
( f is U-stable iff ( f is c=-monotone & ( for a, y being set st a in dom f & y in f . a holds
ex b being set st
( b is finite & b c= a & y in f . b & ( for c being set st c c= a & y in f . c holds
b c= c ) ) ) ) )
proof
let f be Function; ::_thesis: ( dom f is subset-closed & dom f is d.union-closed implies ( f is U-stable iff ( f is c=-monotone & ( for a, y being set st a in dom f & y in f . a holds
ex b being set st
( b is finite & b c= a & y in f . b & ( for c being set st c c= a & y in f . c holds
b c= c ) ) ) ) ) )
assume A1: ( dom f is subset-closed & dom f is d.union-closed ) ; ::_thesis: ( f is U-stable iff ( f is c=-monotone & ( for a, y being set st a in dom f & y in f . a holds
ex b being set st
( b is finite & b c= a & y in f . b & ( for c being set st c c= a & y in f . c holds
b c= c ) ) ) ) )
reconsider C = dom f as subset-closed d.union-closed set by A1;
hereby ::_thesis: ( f is c=-monotone & ( for a, y being set st a in dom f & y in f . a holds
ex b being set st
( b is finite & b c= a & y in f . b & ( for c being set st c c= a & y in f . c holds
b c= c ) ) ) implies f is U-stable )
assume f is U-stable ; ::_thesis: ( f is c=-monotone & ( for a, y being set st a in dom f & y in f . a holds
ex c being set st
( c is finite & c c= a & y in f . c & ( for d being set st d c= a & y in f . d holds
c c= d ) ) ) )
then reconsider f9 = f as U-stable Function ;
dom f9 is multiplicative ;
hence f is c=-monotone ; ::_thesis: for a, y being set st a in dom f & y in f . a holds
ex c being set st
( c is finite & c c= a & y in f . c & ( for d being set st d c= a & y in f . d holds
c c= d ) )
defpred S1[ set , set ] means $1 c= $2;
let a, y be set ; ::_thesis: ( a in dom f & y in f . a implies ex c being set st
( c is finite & c c= a & y in f . c & ( for d being set st d c= a & y in f . d holds
c c= d ) ) )
set C = dom f9;
assume that
A2: a in dom f and
A3: y in f . a ; ::_thesis: ex c being set st
( c is finite & c c= a & y in f . c & ( for d being set st d c= a & y in f . d holds
c c= d ) )
consider b being set such that
A4: b is finite and
A5: b c= a and
A6: y in f9 . b by A1, A2, A3, Th21;
b c= b ;
then b in { c where c is Subset of b : y in f . c } by A6;
then reconsider A = { c where c is Subset of b : y in f . c } as non empty set ;
A7: ( bool b is finite & A c= bool b )
proof
thus bool b is finite by A4; ::_thesis: A c= bool b
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in bool b )
assume x in A ; ::_thesis: x in bool b
then ex c being Subset of b st
( x = c & y in f . c ) ;
hence x in bool b ; ::_thesis: verum
end;
A8: for x, y, z being set st S1[x,y] & S1[y,z] holds
S1[x,z] by XBOOLE_1:1;
A9: for x, y being set st S1[x,y] & S1[y,x] holds
x = y by XBOOLE_0:def_10;
reconsider A = A as non empty finite set by A7;
A10: A <> {} ;
consider c being set such that
A11: ( c in A & ( for y being set st y in A & y <> c holds
not S1[y,c] ) ) from CARD_2:sch_1(A10, A9, A8);
ex d being Subset of b st
( c = d & y in f . d ) by A11;
then reconsider c9 = c as Subset of b ;
reconsider c9 = c9 as finite Subset of b by A4;
take c = c; ::_thesis: ( c is finite & c c= a & y in f . c & ( for d being set st d c= a & y in f . d holds
c c= d ) )
A12: ex d being Subset of b st
( c = d & y in f . d ) by A11;
hence A13: ( c is finite & c c= a & y in f . c ) by A4, A5, XBOOLE_1:1; ::_thesis: for d being set st d c= a & y in f . d holds
c c= d
then A14: c9 in dom f9 by A1, A2, CLASSES1:def_1;
let d be set ; ::_thesis: ( d c= a & y in f . d implies c c= d )
assume that
A15: d c= a and
A16: y in f . d ; ::_thesis: c c= d
A17: d in dom f9 by A1, A2, A15, CLASSES1:def_1;
c \/ d c= a by A13, A15, XBOOLE_1:8;
then A18: c \/ d in dom f by A1, A2, CLASSES1:def_1;
A19: c /\ d c= c9 by XBOOLE_1:17;
then c /\ d in dom f by A1, A14, CLASSES1:def_1;
then dom f includes_lattice_of c,d by A14, A17, A18, Th16;
then f . (c /\ d) = (f . c) /\ (f . d) by A14, Def12;
then A20: y in f . (c /\ d) by A12, A16, XBOOLE_0:def_4;
c /\ d is finite Subset of b by A19, XBOOLE_1:1;
then ( c /\ d c= d & c /\ d in A ) by A20, XBOOLE_1:17;
hence c c= d by A11, XBOOLE_1:17; ::_thesis: verum
end;
assume that
A21: f is c=-monotone and
A22: for a, y being set st a in dom f & y in f . a holds
ex b being set st
( b is finite & b c= a & y in f . b & ( for c being set st c c= a & y in f . c holds
b c= c ) ) ; ::_thesis: f is U-stable
C is subset-closed set ;
hence dom f is multiplicative ; :: according to COHSP_1:def_14 ::_thesis: ( f is U-continuous & f is cap-distributive )
now__::_thesis:_for_a,_y_being_set_st_a_in_dom_f_&_y_in_f_._a_holds_
ex_b_being_set_st_
(_b_is_finite_&_b_c=_a_&_y_in_f_._b_)
let a, y be set ; ::_thesis: ( a in dom f & y in f . a implies ex b being set st
( b is finite & b c= a & y in f . b ) )
assume ( a in dom f & y in f . a ) ; ::_thesis: ex b being set st
( b is finite & b c= a & y in f . b )
then ex b being set st
( b is finite & b c= a & y in f . b & ( for c being set st c c= a & y in f . c holds
b c= c ) ) by A22;
hence ex b being set st
( b is finite & b c= a & y in f . b ) ; ::_thesis: verum
end;
hence f is U-continuous by A1, A21, Th21; ::_thesis: f is cap-distributive
thus f is cap-distributive ::_thesis: verum
proof
let a, b be set ; :: according to COHSP_1:def_12 ::_thesis: ( dom f includes_lattice_of a,b implies f . (a /\ b) = (f . a) /\ (f . b) )
A23: a /\ b c= b by XBOOLE_1:17;
assume A24: dom f includes_lattice_of a,b ; ::_thesis: f . (a /\ b) = (f . a) /\ (f . b)
then A25: a /\ b in dom f by Th16;
b in dom f by A24, Th16;
then A26: f . (a /\ b) c= f . b by A21, A25, A23, Def11;
A27: a in dom f by A24, Th16;
a /\ b c= a by XBOOLE_1:17;
then f . (a /\ b) c= f . a by A21, A27, A25, Def11;
hence f . (a /\ b) c= (f . a) /\ (f . b) by A26, XBOOLE_1:19; :: according to XBOOLE_0:def_10 ::_thesis: (f . a) /\ (f . b) c= f . (a /\ b)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (f . a) /\ (f . b) or x in f . (a /\ b) )
assume A28: x in (f . a) /\ (f . b) ; ::_thesis: x in f . (a /\ b)
then A29: x in f . a by XBOOLE_0:def_4;
A30: a \/ b in dom f by A24, Th16;
a c= a \/ b by XBOOLE_1:7;
then f . a c= f . (a \/ b) by A21, A27, A30, Def11;
then consider c being set such that
c is finite and
c c= a \/ b and
A31: x in f . c and
A32: for d being set st d c= a \/ b & x in f . d holds
c c= d by A22, A30, A29;
A33: c c= a by A29, A32, XBOOLE_1:7;
x in f . b by A28, XBOOLE_0:def_4;
then c c= b by A32, XBOOLE_1:7;
then A34: c c= a /\ b by A33, XBOOLE_1:19;
C = dom f ;
then c in dom f by A27, A33, CLASSES1:def_1;
then f . c c= f . (a /\ b) by A21, A25, A34, Def11;
hence x in f . (a /\ b) by A31; ::_thesis: verum
end;
end;
theorem Th23: :: COHSP_1:23
for f being Function st dom f is subset-closed & dom f is d.union-closed holds
( f is U-linear iff ( f is c=-monotone & ( for a, y being set st a in dom f & y in f . a holds
ex x being set st
( x in a & y in f . {x} & ( for b being set st b c= a & y in f . b holds
x in b ) ) ) ) )
proof
let f be Function; ::_thesis: ( dom f is subset-closed & dom f is d.union-closed implies ( f is U-linear iff ( f is c=-monotone & ( for a, y being set st a in dom f & y in f . a holds
ex x being set st
( x in a & y in f . {x} & ( for b being set st b c= a & y in f . b holds
x in b ) ) ) ) ) )
assume A1: ( dom f is subset-closed & dom f is d.union-closed ) ; ::_thesis: ( f is U-linear iff ( f is c=-monotone & ( for a, y being set st a in dom f & y in f . a holds
ex x being set st
( x in a & y in f . {x} & ( for b being set st b c= a & y in f . b holds
x in b ) ) ) ) )
then reconsider C = dom f as subset-closed d.union-closed set ;
hereby ::_thesis: ( f is c=-monotone & ( for a, y being set st a in dom f & y in f . a holds
ex x being set st
( x in a & y in f . {x} & ( for b being set st b c= a & y in f . b holds
x in b ) ) ) implies f is U-linear )
A2: {} is Subset of (dom f) by XBOOLE_1:2;
assume A3: f is U-linear ; ::_thesis: ( f is c=-monotone & ( for a, y being set st a in dom f & y in f . a holds
ex x being set st
( x in a & y in f . {x} & ( for c being set st c c= a & y in f . c holds
x in c ) ) ) )
hence f is c=-monotone ; ::_thesis: for a, y being set st a in dom f & y in f . a holds
ex x being set st
( x in a & y in f . {x} & ( for c being set st c c= a & y in f . c holds
x in c ) )
let a, y be set ; ::_thesis: ( a in dom f & y in f . a implies ex x being set st
( x in a & y in f . {x} & ( for c being set st c c= a & y in f . c holds
x in c ) ) )
assume that
A4: a in dom f and
A5: y in f . a ; ::_thesis: ex x being set st
( x in a & y in f . {x} & ( for c being set st c c= a & y in f . c holds
x in c ) )
consider b being set such that
b is finite and
A6: b c= a and
A7: y in f . b and
A8: for c being set st c c= a & y in f . c holds
b c= c by A1, A3, A4, A5, Th22;
A9: dom f = C ;
{} c= a by XBOOLE_1:2;
then {} in dom f by A4, A9, CLASSES1:def_1;
then f . {} = union (f .: {}) by A3, A2, Def9, ZFMISC_1:2
.= {} by ZFMISC_1:2 ;
then reconsider b = b as non empty set by A7;
reconsider A = { {x} where x is Element of b : verum } as set ;
A10: b in dom f by A4, A6, A9, CLASSES1:def_1;
A11: A c= dom f
proof
let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in A or X in dom f )
assume X in A ; ::_thesis: X in dom f
then ex x being Element of b st X = {x} ;
then X c= b by ZFMISC_1:31;
hence X in dom f by A9, A10, CLASSES1:def_1; ::_thesis: verum
end;
now__::_thesis:_for_X_being_set_st_X_in_A_holds_
X_c=_b
let X be set ; ::_thesis: ( X in A implies X c= b )
assume X in A ; ::_thesis: X c= b
then ex x being Element of b st X = {x} ;
hence X c= b by ZFMISC_1:31; ::_thesis: verum
end;
then union A c= b by ZFMISC_1:76;
then A12: union A in dom f by A9, A10, CLASSES1:def_1;
reconsider A = A as Subset of (dom f) by A11;
b c= union A
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in b or x in union A )
assume x in b ; ::_thesis: x in union A
then {x} in A ;
then {x} c= union A by ZFMISC_1:74;
hence x in union A by ZFMISC_1:31; ::_thesis: verum
end;
then A13: f . b c= f . (union A) by A3, A10, A12, Def11;
f . (union A) = union (f .: A) by A3, A12, Def9;
then consider Y being set such that
A14: y in Y and
A15: Y in f .: A by A7, A13, TARSKI:def_4;
consider X being set such that
X in dom f and
A16: X in A and
A17: Y = f . X by A15, FUNCT_1:def_6;
consider x being Element of b such that
A18: X = {x} by A16;
reconsider x = x as set ;
take x = x; ::_thesis: ( x in a & y in f . {x} & ( for c being set st c c= a & y in f . c holds
x in c ) )
x in b ;
hence ( x in a & y in f . {x} ) by A6, A14, A17, A18; ::_thesis: for c being set st c c= a & y in f . c holds
x in c
let c be set ; ::_thesis: ( c c= a & y in f . c implies x in c )
assume ( c c= a & y in f . c ) ; ::_thesis: x in c
then ( x in b & b c= c ) by A8;
hence x in c ; ::_thesis: verum
end;
assume that
A19: f is c=-monotone and
A20: for a, y being set st a in dom f & y in f . a holds
ex x being set st
( x in a & y in f . {x} & ( for b being set st b c= a & y in f . b holds
x in b ) ) ; ::_thesis: f is U-linear
now__::_thesis:_for_a,_y_being_set_st_a_in_dom_f_&_y_in_f_._a_holds_
ex_b_being_set_st_
(_b_is_finite_&_b_c=_a_&_y_in_f_._b_&_(_for_c_being_set_st_c_c=_a_&_y_in_f_._c_holds_
b_c=_c_)_)
let a, y be set ; ::_thesis: ( a in dom f & y in f . a implies ex b being set st
( b is finite & b c= a & y in f . b & ( for c being set st c c= a & y in f . c holds
b c= c ) ) )
assume ( a in dom f & y in f . a ) ; ::_thesis: ex b being set st
( b is finite & b c= a & y in f . b & ( for c being set st c c= a & y in f . c holds
b c= c ) )
then consider x being set such that
A21: ( x in a & y in f . {x} ) and
A22: for b being set st b c= a & y in f . b holds
x in b by A20;
reconsider b = {x} as set ;
take b = b; ::_thesis: ( b is finite & b c= a & y in f . b & ( for c being set st c c= a & y in f . c holds
b c= c ) )
thus ( b is finite & b c= a & y in f . b ) by A21, ZFMISC_1:31; ::_thesis: for c being set st c c= a & y in f . c holds
b c= c
let c be set ; ::_thesis: ( c c= a & y in f . c implies b c= c )
assume ( c c= a & y in f . c ) ; ::_thesis: b c= c
then x in c by A22;
hence b c= c by ZFMISC_1:31; ::_thesis: verum
end;
hence f is U-stable by A1, A19, Th22; :: according to COHSP_1:def_15 ::_thesis: f is union-distributive
thus f is union-distributive ::_thesis: verum
proof
let A be Subset of (dom f); :: according to COHSP_1:def_9 ::_thesis: ( union A in dom f implies f . (union A) = union (f .: A) )
assume A23: union A in dom f ; ::_thesis: f . (union A) = union (f .: A)
thus f . (union A) c= union (f .: A) :: according to XBOOLE_0:def_10 ::_thesis: union (f .: A) c= f . (union A)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in f . (union A) or y in union (f .: A) )
assume y in f . (union A) ; ::_thesis: y in union (f .: A)
then consider x being set such that
A24: x in union A and
A25: y in f . {x} and
for b being set st b c= union A & y in f . b holds
x in b by A20, A23;
consider a being set such that
A26: x in a and
A27: a in A by A24, TARSKI:def_4;
A28: {x} c= a by A26, ZFMISC_1:31;
then {x} in C by A27, CLASSES1:def_1;
then A29: f . {x} c= f . a by A19, A27, A28, Def11;
f . a in f .: A by A27, FUNCT_1:def_6;
hence y in union (f .: A) by A25, A29, TARSKI:def_4; ::_thesis: verum
end;
now__::_thesis:_for_X_being_set_st_X_in_f_.:_A_holds_
X_c=_f_._(union_A)
let X be set ; ::_thesis: ( X in f .: A implies X c= f . (union A) )
assume X in f .: A ; ::_thesis: X c= f . (union A)
then consider a being set such that
A30: a in dom f and
A31: a in A and
A32: X = f . a by FUNCT_1:def_6;
a c= union A by A31, ZFMISC_1:74;
hence X c= f . (union A) by A19, A23, A30, A32, Def11; ::_thesis: verum
end;
hence union (f .: A) c= f . (union A) by ZFMISC_1:76; ::_thesis: verum
end;
end;
begin
definition
let f be Function;
func graph f -> set means :Def16: :: COHSP_1:def 16
for x being set holds
( x in it iff ex y being finite set ex z being set st
( x = [y,z] & y in dom f & z in f . y ) );
existence
ex b1 being set st
for x being set holds
( x in b1 iff ex y being finite set ex z being set st
( x = [y,z] & y in dom f & z in f . y ) )
proof
defpred S1[ set ] means ex y being finite set ex z being set st
( $1 = [y,z] & y in dom f & z in f . y );
consider X being set such that
A1: for x being set holds
( x in X iff ( x in [:(dom f),(union (rng f)):] & S1[x] ) ) from XBOOLE_0:sch_1();
take X ; ::_thesis: for x being set holds
( x in X iff ex y being finite set ex z being set st
( x = [y,z] & y in dom f & z in f . y ) )
let x be set ; ::_thesis: ( x in X iff ex y being finite set ex z being set st
( x = [y,z] & y in dom f & z in f . y ) )
now__::_thesis:_(_ex_y_being_finite_set_ex_z_being_set_st_
(_x_=_[y,z]_&_y_in_dom_f_&_z_in_f_._y_)_implies_x_in_[:(dom_f),(union_(rng_f)):]_)
given y being finite set , z being set such that A2: x = [y,z] and
A3: y in dom f and
A4: z in f . y ; ::_thesis: x in [:(dom f),(union (rng f)):]
f . y in rng f by A3, FUNCT_1:def_3;
then z in union (rng f) by A4, TARSKI:def_4;
hence x in [:(dom f),(union (rng f)):] by A2, A3, ZFMISC_1:87; ::_thesis: verum
end;
hence ( x in X iff ex y being finite set ex z being set st
( x = [y,z] & y in dom f & z in f . y ) ) by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff ex y being finite set ex z being set st
( x = [y,z] & y in dom f & z in f . y ) ) ) & ( for x being set holds
( x in b2 iff ex y being finite set ex z being set st
( x = [y,z] & y in dom f & z in f . y ) ) ) holds
b1 = b2
proof
let X1, X2 be set ; ::_thesis: ( ( for x being set holds
( x in X1 iff ex y being finite set ex z being set st
( x = [y,z] & y in dom f & z in f . y ) ) ) & ( for x being set holds
( x in X2 iff ex y being finite set ex z being set st
( x = [y,z] & y in dom f & z in f . y ) ) ) implies X1 = X2 )
assume A5: ( ( for x being set holds
( x in X1 iff ex y being finite set ex z being set st
( x = [y,z] & y in dom f & z in f . y ) ) ) & ( for x being set holds
( x in X2 iff ex y being finite set ex z being set st
( x = [y,z] & y in dom f & z in f . y ) ) ) & not X1 = X2 ) ; ::_thesis: contradiction
then consider x being set such that
A6: ( ( x in X1 & not x in X2 ) or ( x in X2 & not x in X1 ) ) by TARSKI:1;
( x in X2 iff for y being finite set
for z being set holds
( not x = [y,z] or not y in dom f or not z in f . y ) ) by A5, A6;
hence contradiction by A5; ::_thesis: verum
end;
end;
:: deftheorem Def16 defines graph COHSP_1:def_16_:_
for f being Function
for b2 being set holds
( b2 = graph f iff for x being set holds
( x in b2 iff ex y being finite set ex z being set st
( x = [y,z] & y in dom f & z in f . y ) ) );
definition
let C1, C2 be non empty set ;
let f be Function of C1,C2;
:: original: graph
redefine func graph f -> Subset of [:C1,(union C2):];
coherence
graph f is Subset of [:C1,(union C2):]
proof
graph f c= [:C1,(union C2):]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in graph f or x in [:C1,(union C2):] )
assume x in graph f ; ::_thesis: x in [:C1,(union C2):]
then consider y being finite set , z being set such that
A1: x = [y,z] and
A2: y in dom f and
A3: z in f . y by Def16;
( rng f c= C2 & f . y in rng f ) by A2, FUNCT_1:def_3, RELAT_1:def_19;
then ( dom f = C1 & z in union C2 ) by A3, FUNCT_2:def_1, TARSKI:def_4;
hence x in [:C1,(union C2):] by A1, A2, ZFMISC_1:87; ::_thesis: verum
end;
hence graph f is Subset of [:C1,(union C2):] ; ::_thesis: verum
end;
end;
registration
let f be Function;
cluster graph f -> Relation-like ;
coherence
graph f is Relation-like
proof
let x be set ; :: according to RELAT_1:def_1 ::_thesis: ( not x in graph f or ex b1, b2 being set st x = [b1,b2] )
assume x in graph f ; ::_thesis: ex b1, b2 being set st x = [b1,b2]
then ex y being finite set ex z being set st
( x = [y,z] & y in dom f & z in f . y ) by Def16;
hence ex b1, b2 being set st x = [b1,b2] ; ::_thesis: verum
end;
end;
theorem Th24: :: COHSP_1:24
for f being Function
for x, y being set holds
( [x,y] in graph f iff ( x is finite & x in dom f & y in f . x ) )
proof
let f be Function; ::_thesis: for x, y being set holds
( [x,y] in graph f iff ( x is finite & x in dom f & y in f . x ) )
let x, y be set ; ::_thesis: ( [x,y] in graph f iff ( x is finite & x in dom f & y in f . x ) )
now__::_thesis:_(_ex_y9_being_finite_set_ex_z_being_set_st_
(_[x,y]_=_[y9,z]_&_y9_in_dom_f_&_z_in_f_._y9_)_implies_(_x_is_finite_&_x_in_dom_f_&_y_in_f_._x_)_)
given y9 being finite set , z being set such that A1: [x,y] = [y9,z] and
A2: ( y9 in dom f & z in f . y9 ) ; ::_thesis: ( x is finite & x in dom f & y in f . x )
x = y9 by A1, XTUPLE_0:1;
hence ( x is finite & x in dom f & y in f . x ) by A1, A2, XTUPLE_0:1; ::_thesis: verum
end;
hence ( [x,y] in graph f iff ( x is finite & x in dom f & y in f . x ) ) by Def16; ::_thesis: verum
end;
theorem Th25: :: COHSP_1:25
for f being c=-monotone Function
for a, b being set st b in dom f & a c= b & b is finite holds
for y being set st [a,y] in graph f holds
[b,y] in graph f
proof
let f be c=-monotone Function; ::_thesis: for a, b being set st b in dom f & a c= b & b is finite holds
for y being set st [a,y] in graph f holds
[b,y] in graph f
let a, b be set ; ::_thesis: ( b in dom f & a c= b & b is finite implies for y being set st [a,y] in graph f holds
[b,y] in graph f )
assume that
A1: b in dom f and
A2: a c= b and
A3: b is finite ; ::_thesis: for y being set st [a,y] in graph f holds
[b,y] in graph f
let y be set ; ::_thesis: ( [a,y] in graph f implies [b,y] in graph f )
assume A4: [a,y] in graph f ; ::_thesis: [b,y] in graph f
then a in dom f by Th24;
then A5: f . a c= f . b by A1, A2, Def11;
y in f . a by A4, Th24;
hence [b,y] in graph f by A1, A3, A5, Th24; ::_thesis: verum
end;
theorem Th26: :: COHSP_1:26
for C1, C2 being Coherence_Space
for f being Function of C1,C2
for a being Element of C1
for y1, y2 being set st [a,y1] in graph f & [a,y2] in graph f holds
{y1,y2} in C2
proof
let C1, C2 be Coherence_Space; ::_thesis: for f being Function of C1,C2
for a being Element of C1
for y1, y2 being set st [a,y1] in graph f & [a,y2] in graph f holds
{y1,y2} in C2
let f be Function of C1,C2; ::_thesis: for a being Element of C1
for y1, y2 being set st [a,y1] in graph f & [a,y2] in graph f holds
{y1,y2} in C2
let a be Element of C1; ::_thesis: for y1, y2 being set st [a,y1] in graph f & [a,y2] in graph f holds
{y1,y2} in C2
let y1, y2 be set ; ::_thesis: ( [a,y1] in graph f & [a,y2] in graph f implies {y1,y2} in C2 )
assume ( [a,y1] in graph f & [a,y2] in graph f ) ; ::_thesis: {y1,y2} in C2
then ( y1 in f . a & y2 in f . a ) by Th24;
then {y1,y2} c= f . a by ZFMISC_1:32;
hence {y1,y2} in C2 by CLASSES1:def_1; ::_thesis: verum
end;
theorem :: COHSP_1:27
for C1, C2 being Coherence_Space
for f being c=-monotone Function of C1,C2
for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being set st [a,y1] in graph f & [b,y2] in graph f holds
{y1,y2} in C2
proof
let C1, C2 be Coherence_Space; ::_thesis: for f being c=-monotone Function of C1,C2
for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being set st [a,y1] in graph f & [b,y2] in graph f holds
{y1,y2} in C2
let f be c=-monotone Function of C1,C2; ::_thesis: for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being set st [a,y1] in graph f & [b,y2] in graph f holds
{y1,y2} in C2
let a, b be Element of C1; ::_thesis: ( a \/ b in C1 implies for y1, y2 being set st [a,y1] in graph f & [b,y2] in graph f holds
{y1,y2} in C2 )
assume A1: a \/ b in C1 ; ::_thesis: for y1, y2 being set st [a,y1] in graph f & [b,y2] in graph f holds
{y1,y2} in C2
let y1, y2 be set ; ::_thesis: ( [a,y1] in graph f & [b,y2] in graph f implies {y1,y2} in C2 )
assume A2: ( [a,y1] in graph f & [b,y2] in graph f ) ; ::_thesis: {y1,y2} in C2
then ( a is finite & b is finite ) by Th24;
then reconsider c = a \/ b as finite Element of C1 by A1;
dom f = C1 by FUNCT_2:def_1;
then ( [c,y1] in graph f & [c,y2] in graph f ) by A2, Th25, XBOOLE_1:7;
hence {y1,y2} in C2 by Th26; ::_thesis: verum
end;
theorem Th28: :: COHSP_1:28
for C1, C2 being Coherence_Space
for f, g being U-continuous Function of C1,C2 st graph f = graph g holds
f = g
proof
let C1, C2 be Coherence_Space; ::_thesis: for f, g being U-continuous Function of C1,C2 st graph f = graph g holds
f = g
let f, g be U-continuous Function of C1,C2; ::_thesis: ( graph f = graph g implies f = g )
A1: dom f = C1 by FUNCT_2:def_1;
A2: dom g = C1 by FUNCT_2:def_1;
A3: now__::_thesis:_for_x_being_finite_Element_of_C1
for_f,_g_being_U-continuous_Function_of_C1,C2_st_graph_f_=_graph_g_holds_
f_._x_c=_g_._x
let x be finite Element of C1; ::_thesis: for f, g being U-continuous Function of C1,C2 st graph f = graph g holds
f . x c= g . x
let f, g be U-continuous Function of C1,C2; ::_thesis: ( graph f = graph g implies f . x c= g . x )
A4: dom f = C1 by FUNCT_2:def_1;
assume A5: graph f = graph g ; ::_thesis: f . x c= g . x
thus f . x c= g . x ::_thesis: verum
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in f . x or z in g . x )
assume z in f . x ; ::_thesis: z in g . x
then [x,z] in graph f by A4, Th24;
hence z in g . x by A5, Th24; ::_thesis: verum
end;
end;
A6: now__::_thesis:_for_a_being_Element_of_C1
for_f,_g_being_U-continuous_Function_of_C1,C2_st_graph_f_=_graph_g_holds_
f_.:_(Fin_a)_c=_g_.:_(Fin_a)
let a be Element of C1; ::_thesis: for f, g being U-continuous Function of C1,C2 st graph f = graph g holds
f .: (Fin a) c= g .: (Fin a)
let f, g be U-continuous Function of C1,C2; ::_thesis: ( graph f = graph g implies f .: (Fin a) c= g .: (Fin a) )
A7: dom g = C1 by FUNCT_2:def_1;
assume A8: graph f = graph g ; ::_thesis: f .: (Fin a) c= g .: (Fin a)
thus f .: (Fin a) c= g .: (Fin a) ::_thesis: verum
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in f .: (Fin a) or y in g .: (Fin a) )
assume y in f .: (Fin a) ; ::_thesis: y in g .: (Fin a)
then consider x being set such that
x in dom f and
A9: x in Fin a and
A10: y = f . x by FUNCT_1:def_6;
( f . x c= g . x & g . x c= f . x ) by A3, A8, A9;
then f . x = g . x by XBOOLE_0:def_10;
hence y in g .: (Fin a) by A7, A9, A10, FUNCT_1:def_6; ::_thesis: verum
end;
end;
assume A11: graph f = graph g ; ::_thesis: f = g
now__::_thesis:_for_a_being_Element_of_C1_holds_f_._a_=_g_._a
let a be Element of C1; ::_thesis: f . a = g . a
( f .: (Fin a) c= g .: (Fin a) & g .: (Fin a) c= f .: (Fin a) ) by A11, A6;
then A12: f .: (Fin a) = g .: (Fin a) by XBOOLE_0:def_10;
thus f . a = union (f .: (Fin a)) by A1, Th20
.= g . a by A2, A12, Th20 ; ::_thesis: verum
end;
hence f = g by FUNCT_2:63; ::_thesis: verum
end;
Lm4: for C1, C2 being Coherence_Space
for X being Subset of [:C1,(union C2):] st ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being finite Element of C1 st a c= b holds
for y being set st [a,y] in X holds
[b,y] in X ) & ( for a being finite Element of C1
for y1, y2 being set st [a,y1] in X & [a,y2] in X holds
{y1,y2} in C2 ) holds
ex f being U-continuous Function of C1,C2 st
( X = graph f & ( for a being Element of C1 holds f . a = X .: (Fin a) ) )
proof
let C1, C2 be Coherence_Space; ::_thesis: for X being Subset of [:C1,(union C2):] st ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being finite Element of C1 st a c= b holds
for y being set st [a,y] in X holds
[b,y] in X ) & ( for a being finite Element of C1
for y1, y2 being set st [a,y1] in X & [a,y2] in X holds
{y1,y2} in C2 ) holds
ex f being U-continuous Function of C1,C2 st
( X = graph f & ( for a being Element of C1 holds f . a = X .: (Fin a) ) )
let X be Subset of [:C1,(union C2):]; ::_thesis: ( ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being finite Element of C1 st a c= b holds
for y being set st [a,y] in X holds
[b,y] in X ) & ( for a being finite Element of C1
for y1, y2 being set st [a,y1] in X & [a,y2] in X holds
{y1,y2} in C2 ) implies ex f being U-continuous Function of C1,C2 st
( X = graph f & ( for a being Element of C1 holds f . a = X .: (Fin a) ) ) )
assume that
A1: for x being set st x in X holds
x `1 is finite and
A2: for a, b being finite Element of C1 st a c= b holds
for y being set st [a,y] in X holds
[b,y] in X and
A3: for a being finite Element of C1
for y1, y2 being set st [a,y1] in X & [a,y2] in X holds
{y1,y2} in C2 ; ::_thesis: ex f being U-continuous Function of C1,C2 st
( X = graph f & ( for a being Element of C1 holds f . a = X .: (Fin a) ) )
deffunc H1( set ) -> set = X .: (Fin $1);
consider f being Function such that
A4: ( dom f = C1 & ( for a being set st a in C1 holds
f . a = H1(a) ) ) from FUNCT_1:sch_3();
A5: rng f c= C2
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng f or x in C2 )
assume x in rng f ; ::_thesis: x in C2
then consider a being set such that
A6: a in dom f and
A7: x = f . a by FUNCT_1:def_3;
reconsider a = a as Element of C1 by A4, A6;
A8: x = X .: (Fin a) by A4, A7;
now__::_thesis:_for_z,_y_being_set_st_z_in_x_&_y_in_x_holds_
{z,y}_in_C2
let z, y be set ; ::_thesis: ( z in x & y in x implies {z,y} in C2 )
assume z in x ; ::_thesis: ( y in x implies {z,y} in C2 )
then consider z1 being set such that
A9: [z1,z] in X and
A10: z1 in Fin a by A8, RELAT_1:def_13;
assume y in x ; ::_thesis: {z,y} in C2
then consider y1 being set such that
A11: [y1,y] in X and
A12: y1 in Fin a by A8, RELAT_1:def_13;
reconsider z1 = z1, y1 = y1 as finite Element of C1 by A10, A12;
z1 \/ y1 in Fin a by A10, A12, FINSUB_1:1;
then reconsider b = z1 \/ y1 as finite Element of C1 ;
A13: [b,y] in X by A2, A11, XBOOLE_1:7;
[b,z] in X by A2, A9, XBOOLE_1:7;
hence {z,y} in C2 by A3, A13; ::_thesis: verum
end;
hence x in C2 by COH_SP:6; ::_thesis: verum
end;
A14: now__::_thesis:_for_a,_y_being_set_st_a_in_dom_f_&_y_in_f_._a_holds_
ex_x_being_set_st_
(_x_is_finite_&_x_c=_a_&_y_in_f_._x_)
let a, y be set ; ::_thesis: ( a in dom f & y in f . a implies ex x being set st
( x is finite & x c= a & y in f . x ) )
assume that
A15: a in dom f and
A16: y in f . a ; ::_thesis: ex x being set st
( x is finite & x c= a & y in f . x )
y in X .: (Fin a) by A4, A15, A16;
then consider x being set such that
A17: [x,y] in X and
A18: x in Fin a by RELAT_1:def_13;
x c= a by A18, FINSUB_1:def_5;
then x in C1 by A4, A15, CLASSES1:def_1;
then A19: f . x = X .: (Fin x) by A4;
take x = x; ::_thesis: ( x is finite & x c= a & y in f . x )
x in Fin x by A18, FINSUB_1:def_5;
hence ( x is finite & x c= a & y in f . x ) by A17, A18, A19, FINSUB_1:def_5, RELAT_1:def_13; ::_thesis: verum
end;
f is c=-monotone
proof
let a, b be set ; :: according to COHSP_1:def_11 ::_thesis: ( a in dom f & b in dom f & a c= b implies f . a c= f . b )
assume that
A20: ( a in dom f & b in dom f ) and
A21: a c= b ; ::_thesis: f . a c= f . b
reconsider a = a, b = b as Element of C1 by A4, A20;
Fin a c= Fin b by A21, FINSUB_1:10;
then A22: X .: (Fin a) c= X .: (Fin b) by RELAT_1:123;
f . a = X .: (Fin a) by A4;
hence f . a c= f . b by A4, A22; ::_thesis: verum
end;
then reconsider f = f as U-continuous Function of C1,C2 by A4, A5, A14, Th21, FUNCT_2:def_1, RELSET_1:4;
take f ; ::_thesis: ( X = graph f & ( for a being Element of C1 holds f . a = X .: (Fin a) ) )
thus X = graph f ::_thesis: for a being Element of C1 holds f . a = X .: (Fin a)
proof
let a, b be set ; :: according to RELAT_1:def_2 ::_thesis: ( ( not [a,b] in X or [a,b] in graph f ) & ( not [a,b] in graph f or [a,b] in X ) )
hereby ::_thesis: ( not [a,b] in graph f or [a,b] in X )
assume A23: [a,b] in X ; ::_thesis: [a,b] in graph f
[a,b] `1 = a ;
then reconsider a9 = a as finite Element of C1 by A1, A23, ZFMISC_1:87;
a9 in Fin a by FINSUB_1:def_5;
then A24: b in X .: (Fin a) by A23, RELAT_1:def_13;
f . a9 = X .: (Fin a) by A4;
hence [a,b] in graph f by A4, A24, Th24; ::_thesis: verum
end;
assume A25: [a,b] in graph f ; ::_thesis: [a,b] in X
then reconsider a = a as finite Element of C1 by A4, Th24;
A26: f . a = X .: (Fin a) by A4;
b in f . a by A25, Th24;
then consider x being set such that
A27: [x,b] in X and
A28: x in Fin a by A26, RELAT_1:def_13;
x c= a by A28, FINSUB_1:def_5;
hence [a,b] in X by A2, A27, A28; ::_thesis: verum
end;
thus for a being Element of C1 holds f . a = X .: (Fin a) by A4; ::_thesis: verum
end;
theorem :: COHSP_1:29
for C1, C2 being Coherence_Space
for X being Subset of [:C1,(union C2):] st ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being finite Element of C1 st a c= b holds
for y being set st [a,y] in X holds
[b,y] in X ) & ( for a being finite Element of C1
for y1, y2 being set st [a,y1] in X & [a,y2] in X holds
{y1,y2} in C2 ) holds
ex f being U-continuous Function of C1,C2 st X = graph f
proof
let C1, C2 be Coherence_Space; ::_thesis: for X being Subset of [:C1,(union C2):] st ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being finite Element of C1 st a c= b holds
for y being set st [a,y] in X holds
[b,y] in X ) & ( for a being finite Element of C1
for y1, y2 being set st [a,y1] in X & [a,y2] in X holds
{y1,y2} in C2 ) holds
ex f being U-continuous Function of C1,C2 st X = graph f
let X be Subset of [:C1,(union C2):]; ::_thesis: ( ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being finite Element of C1 st a c= b holds
for y being set st [a,y] in X holds
[b,y] in X ) & ( for a being finite Element of C1
for y1, y2 being set st [a,y1] in X & [a,y2] in X holds
{y1,y2} in C2 ) implies ex f being U-continuous Function of C1,C2 st X = graph f )
assume A1: ( ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being finite Element of C1 st a c= b holds
for y being set st [a,y] in X holds
[b,y] in X ) & ( for a being finite Element of C1
for y1, y2 being set st [a,y1] in X & [a,y2] in X holds
{y1,y2} in C2 ) & ( for f being U-continuous Function of C1,C2 holds not X = graph f ) ) ; ::_thesis: contradiction
then ex f being U-continuous Function of C1,C2 st
( X = graph f & ( for a being Element of C1 holds f . a = X .: (Fin a) ) ) by Lm4;
hence contradiction by A1; ::_thesis: verum
end;
theorem :: COHSP_1:30
for C1, C2 being Coherence_Space
for f being U-continuous Function of C1,C2
for a being Element of C1 holds f . a = (graph f) .: (Fin a)
proof
let C1, C2 be Coherence_Space; ::_thesis: for f being U-continuous Function of C1,C2
for a being Element of C1 holds f . a = (graph f) .: (Fin a)
let f be U-continuous Function of C1,C2; ::_thesis: for a being Element of C1 holds f . a = (graph f) .: (Fin a)
let a be Element of C1; ::_thesis: f . a = (graph f) .: (Fin a)
set X = graph f;
A1: now__::_thesis:_for_x_being_set_st_x_in_graph_f_holds_
x_`1_is_finite
let x be set ; ::_thesis: ( x in graph f implies x `1 is finite )
assume x in graph f ; ::_thesis: x `1 is finite
then ex y being finite set ex z being set st
( x = [y,z] & y in dom f & z in f . y ) by Def16;
hence x `1 is finite by MCART_1:7; ::_thesis: verum
end;
dom f = C1 by FUNCT_2:def_1;
then A2: for a, b being finite Element of C1 st a c= b holds
for y being set st [a,y] in graph f holds
[b,y] in graph f by Th25;
for a being finite Element of C1
for y1, y2 being set st [a,y1] in graph f & [a,y2] in graph f holds
{y1,y2} in C2 by Th26;
then consider g being U-continuous Function of C1,C2 such that
A3: graph f = graph g and
A4: for a being Element of C1 holds g . a = (graph f) .: (Fin a) by A1, A2, Lm4;
g . a = (graph f) .: (Fin a) by A4;
hence f . a = (graph f) .: (Fin a) by A3, Th28; ::_thesis: verum
end;
begin
definition
let f be Function;
func Trace f -> set means :Def17: :: COHSP_1:def 17
for x being set holds
( x in it iff ex a, y being set st
( x = [a,y] & a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) ) );
existence
ex b1 being set st
for x being set holds
( x in b1 iff ex a, y being set st
( x = [a,y] & a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) ) )
proof
defpred S1[ set ] means ex a, y being set st
( $1 = [a,y] & a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) );
consider T being set such that
A1: for x being set holds
( x in T iff ( x in [:(dom f),(Union f):] & S1[x] ) ) from XBOOLE_0:sch_1();
take T ; ::_thesis: for x being set holds
( x in T iff ex a, y being set st
( x = [a,y] & a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) ) )
let x be set ; ::_thesis: ( x in T iff ex a, y being set st
( x = [a,y] & a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) ) )
now__::_thesis:_(_ex_a,_y_being_set_st_
(_x_=_[a,y]_&_a_in_dom_f_&_y_in_f_._a_&_(_for_b_being_set_st_b_in_dom_f_&_b_c=_a_&_y_in_f_._b_holds_
a_=_b_)_)_implies_x_in_[:(dom_f),(Union_f):]_)
given a, y being set such that A2: x = [a,y] and
A3: a in dom f and
A4: y in f . a and
for b being set st b in dom f & b c= a & y in f . b holds
a = b ; ::_thesis: x in [:(dom f),(Union f):]
y in Union f by A3, A4, CARD_5:2;
hence x in [:(dom f),(Union f):] by A2, A3, ZFMISC_1:87; ::_thesis: verum
end;
hence ( x in T iff ex a, y being set st
( x = [a,y] & a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) ) ) by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff ex a, y being set st
( x = [a,y] & a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) ) ) ) & ( for x being set holds
( x in b2 iff ex a, y being set st
( x = [a,y] & a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) ) ) ) holds
b1 = b2
proof
let T1, T2 be set ; ::_thesis: ( ( for x being set holds
( x in T1 iff ex a, y being set st
( x = [a,y] & a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) ) ) ) & ( for x being set holds
( x in T2 iff ex a, y being set st
( x = [a,y] & a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) ) ) ) implies T1 = T2 )
assume A5: ( ( for x being set holds
( x in T1 iff ex a, y being set st
( x = [a,y] & a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) ) ) ) & ( for x being set holds
( x in T2 iff ex a, y being set st
( x = [a,y] & a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) ) ) ) & not T1 = T2 ) ; ::_thesis: contradiction
then consider x being set such that
A6: ( ( x in T1 & not x in T2 ) or ( x in T2 & not x in T1 ) ) by TARSKI:1;
( x in T2 iff for a, y being set holds
( not x = [a,y] or not a in dom f or not y in f . a or ex b being set st
( b in dom f & b c= a & y in f . b & not a = b ) ) ) by A5, A6;
hence contradiction by A5; ::_thesis: verum
end;
end;
:: deftheorem Def17 defines Trace COHSP_1:def_17_:_
for f being Function
for b2 being set holds
( b2 = Trace f iff for x being set holds
( x in b2 iff ex a, y being set st
( x = [a,y] & a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) ) ) );
theorem Th31: :: COHSP_1:31
for f being Function
for a, y being set holds
( [a,y] in Trace f iff ( a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) ) )
proof
let f be Function; ::_thesis: for a, y being set holds
( [a,y] in Trace f iff ( a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) ) )
let a9, y9 be set ; ::_thesis: ( [a9,y9] in Trace f iff ( a9 in dom f & y9 in f . a9 & ( for b being set st b in dom f & b c= a9 & y9 in f . b holds
a9 = b ) ) )
now__::_thesis:_(_ex_a,_y_being_set_st_
(_[a9,y9]_=_[a,y]_&_a_in_dom_f_&_y_in_f_._a_&_(_for_b_being_set_st_b_in_dom_f_&_b_c=_a_&_y_in_f_._b_holds_
a_=_b_)_)_implies_(_a9_in_dom_f_&_y9_in_f_._a9_&_(_for_b_being_set_st_b_in_dom_f_&_b_c=_a9_&_y9_in_f_._b_holds_
a9_=_b_)_)_)
given a, y being set such that A1: [a9,y9] = [a,y] and
A2: ( a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) ) ; ::_thesis: ( a9 in dom f & y9 in f . a9 & ( for b being set st b in dom f & b c= a9 & y9 in f . b holds
a9 = b ) )
( a9 = a & y9 = y ) by A1, XTUPLE_0:1;
hence ( a9 in dom f & y9 in f . a9 & ( for b being set st b in dom f & b c= a9 & y9 in f . b holds
a9 = b ) ) by A2; ::_thesis: verum
end;
hence ( [a9,y9] in Trace f iff ( a9 in dom f & y9 in f . a9 & ( for b being set st b in dom f & b c= a9 & y9 in f . b holds
a9 = b ) ) ) by Def17; ::_thesis: verum
end;
definition
let C1, C2 be non empty set ;
let f be Function of C1,C2;
:: original: Trace
redefine func Trace f -> Subset of [:C1,(union C2):];
coherence
Trace f is Subset of [:C1,(union C2):]
proof
Trace f c= [:C1,(union C2):]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Trace f or x in [:C1,(union C2):] )
assume x in Trace f ; ::_thesis: x in [:C1,(union C2):]
then consider a, y being set such that
A1: x = [a,y] and
A2: a in dom f and
A3: y in f . a and
for b being set st b in dom f & b c= a & y in f . b holds
a = b by Def17;
( rng f c= C2 & f . a in rng f ) by A2, FUNCT_1:def_3, RELAT_1:def_19;
then ( dom f = C1 & y in union C2 ) by A3, FUNCT_2:def_1, TARSKI:def_4;
hence x in [:C1,(union C2):] by A1, A2, ZFMISC_1:87; ::_thesis: verum
end;
hence Trace f is Subset of [:C1,(union C2):] ; ::_thesis: verum
end;
end;
registration
let f be Function;
cluster Trace f -> Relation-like ;
coherence
Trace f is Relation-like
proof
let x be set ; :: according to RELAT_1:def_1 ::_thesis: ( not x in Trace f or ex b1, b2 being set st x = [b1,b2] )
assume x in Trace f ; ::_thesis: ex b1, b2 being set st x = [b1,b2]
then ex a, y being set st
( x = [a,y] & a in dom f & y in f . a & ( for b being set st b in dom f & b c= a & y in f . b holds
a = b ) ) by Def17;
hence ex b1, b2 being set st x = [b1,b2] ; ::_thesis: verum
end;
end;
theorem :: COHSP_1:32
for f being U-continuous Function st dom f is subset-closed holds
Trace f c= graph f
proof
let f be U-continuous Function; ::_thesis: ( dom f is subset-closed implies Trace f c= graph f )
assume A1: dom f is subset-closed ; ::_thesis: Trace f c= graph f
let x, z be set ; :: according to RELAT_1:def_3 ::_thesis: ( not [x,z] in Trace f or [x,z] in graph f )
assume [x,z] in Trace f ; ::_thesis: [x,z] in graph f
then consider a, y being set such that
A2: [x,z] = [a,y] and
A3: a in dom f and
A4: y in f . a and
A5: for b being set st b in dom f & b c= a & y in f . b holds
a = b by Def17;
consider b being set such that
A6: b is finite and
A7: b c= a and
A8: y in f . b by A1, A3, A4, Th21;
b in dom f by A1, A3, A7, CLASSES1:def_1;
then a = b by A5, A7, A8;
hence [x,z] in graph f by A2, A3, A4, A6, Th24; ::_thesis: verum
end;
theorem Th33: :: COHSP_1:33
for f being U-continuous Function st dom f is subset-closed holds
for a, y being set st [a,y] in Trace f holds
a is finite
proof
let f be U-continuous Function; ::_thesis: ( dom f is subset-closed implies for a, y being set st [a,y] in Trace f holds
a is finite )
assume A1: dom f is subset-closed ; ::_thesis: for a, y being set st [a,y] in Trace f holds
a is finite
let a, y be set ; ::_thesis: ( [a,y] in Trace f implies a is finite )
assume A2: [a,y] in Trace f ; ::_thesis: a is finite
then A3: a in dom f by Th31;
y in f . a by A2, Th31;
then consider b being set such that
A4: b is finite and
A5: b c= a and
A6: y in f . b by A1, A3, Th21;
b in dom f by A1, A3, A5, CLASSES1:def_1;
hence a is finite by A2, A4, A5, A6, Th31; ::_thesis: verum
end;
theorem Th34: :: COHSP_1:34
for C1, C2 being Coherence_Space
for f being c=-monotone Function of C1,C2
for a1, a2 being set st a1 \/ a2 in C1 holds
for y1, y2 being set st [a1,y1] in Trace f & [a2,y2] in Trace f holds
{y1,y2} in C2
proof
let C1, C2 be Coherence_Space; ::_thesis: for f being c=-monotone Function of C1,C2
for a1, a2 being set st a1 \/ a2 in C1 holds
for y1, y2 being set st [a1,y1] in Trace f & [a2,y2] in Trace f holds
{y1,y2} in C2
let f be c=-monotone Function of C1,C2; ::_thesis: for a1, a2 being set st a1 \/ a2 in C1 holds
for y1, y2 being set st [a1,y1] in Trace f & [a2,y2] in Trace f holds
{y1,y2} in C2
A1: dom f = C1 by FUNCT_2:def_1;
let a1, a2 be set ; ::_thesis: ( a1 \/ a2 in C1 implies for y1, y2 being set st [a1,y1] in Trace f & [a2,y2] in Trace f holds
{y1,y2} in C2 )
set a = a1 \/ a2;
assume a1 \/ a2 in C1 ; ::_thesis: for y1, y2 being set st [a1,y1] in Trace f & [a2,y2] in Trace f holds
{y1,y2} in C2
then reconsider a = a1 \/ a2 as Element of C1 ;
A2: a2 c= a by XBOOLE_1:7;
then a2 in C1 by CLASSES1:def_1;
then A3: f . a2 c= f . a by A1, A2, Def11;
let y1, y2 be set ; ::_thesis: ( [a1,y1] in Trace f & [a2,y2] in Trace f implies {y1,y2} in C2 )
assume ( [a1,y1] in Trace f & [a2,y2] in Trace f ) ; ::_thesis: {y1,y2} in C2
then A4: ( y1 in f . a1 & y2 in f . a2 ) by Th31;
A5: a1 c= a by XBOOLE_1:7;
then a1 in C1 by CLASSES1:def_1;
then f . a1 c= f . a by A1, A5, Def11;
then {y1,y2} c= f . a by A3, A4, ZFMISC_1:32;
hence {y1,y2} in C2 by CLASSES1:def_1; ::_thesis: verum
end;
theorem Th35: :: COHSP_1:35
for C1, C2 being Coherence_Space
for f being cap-distributive Function of C1,C2
for a1, a2 being set st a1 \/ a2 in C1 holds
for y being set st [a1,y] in Trace f & [a2,y] in Trace f holds
a1 = a2
proof
let C1, C2 be Coherence_Space; ::_thesis: for f being cap-distributive Function of C1,C2
for a1, a2 being set st a1 \/ a2 in C1 holds
for y being set st [a1,y] in Trace f & [a2,y] in Trace f holds
a1 = a2
let f be cap-distributive Function of C1,C2; ::_thesis: for a1, a2 being set st a1 \/ a2 in C1 holds
for y being set st [a1,y] in Trace f & [a2,y] in Trace f holds
a1 = a2
A1: dom f = C1 by FUNCT_2:def_1;
let a1, a2 be set ; ::_thesis: ( a1 \/ a2 in C1 implies for y being set st [a1,y] in Trace f & [a2,y] in Trace f holds
a1 = a2 )
set a = a1 \/ a2;
assume A2: a1 \/ a2 in C1 ; ::_thesis: for y being set st [a1,y] in Trace f & [a2,y] in Trace f holds
a1 = a2
a2 c= a1 \/ a2 by XBOOLE_1:7;
then A3: a2 in C1 by A2, CLASSES1:def_1;
a1 c= a1 \/ a2 by XBOOLE_1:7;
then A4: a1 in C1 by A2, CLASSES1:def_1;
then reconsider b = a1 /\ a2 as Element of C1 by A3, FINSUB_1:def_2;
b in C1 ;
then A5: C1 includes_lattice_of a1,a2 by A2, A4, A3, Th16;
let y be set ; ::_thesis: ( [a1,y] in Trace f & [a2,y] in Trace f implies a1 = a2 )
assume that
A6: [a1,y] in Trace f and
A7: [a2,y] in Trace f ; ::_thesis: a1 = a2
( y in f . a1 & y in f . a2 ) by A6, A7, Th31;
then y in (f . a1) /\ (f . a2) by XBOOLE_0:def_4;
then A8: y in f . b by A1, A5, Def12;
b c= a1 by XBOOLE_1:17;
then ( b c= a2 & b = a1 ) by A1, A6, A8, Th31, XBOOLE_1:17;
hence a1 = a2 by A1, A7, A8, Th31; ::_thesis: verum
end;
theorem Th36: :: COHSP_1:36
for C1, C2 being Coherence_Space
for f, g being U-stable Function of C1,C2 st Trace f c= Trace g holds
for a being Element of C1 holds f . a c= g . a
proof
let C1, C2 be Coherence_Space; ::_thesis: for f, g being U-stable Function of C1,C2 st Trace f c= Trace g holds
for a being Element of C1 holds f . a c= g . a
let f, g be U-stable Function of C1,C2; ::_thesis: ( Trace f c= Trace g implies for a being Element of C1 holds f . a c= g . a )
assume A1: Trace f c= Trace g ; ::_thesis: for a being Element of C1 holds f . a c= g . a
let x be Element of C1; ::_thesis: f . x c= g . x
A2: dom f = C1 by FUNCT_2:def_1;
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in f . x or z in g . x )
assume z in f . x ; ::_thesis: z in g . x
then consider b being set such that
b is finite and
A3: b c= x and
A4: z in f . b and
A5: for c being set st c c= x & z in f . c holds
b c= c by A2, Th22;
reconsider b = b as Element of C1 by A3, CLASSES1:def_1;
now__::_thesis:_for_c_being_set_st_c_in_dom_f_&_c_c=_b_&_z_in_f_._c_holds_
b_=_c
let c be set ; ::_thesis: ( c in dom f & c c= b & z in f . c implies b = c )
assume that
c in dom f and
A6: c c= b and
A7: z in f . c ; ::_thesis: b = c
c c= x by A3, A6, XBOOLE_1:1;
then b c= c by A5, A7;
hence b = c by A6, XBOOLE_0:def_10; ::_thesis: verum
end;
then [b,z] in Trace f by A2, A4, Th31;
then A8: z in g . b by A1, Th31;
dom g = C1 by FUNCT_2:def_1;
then g . b c= g . x by A3, Def11;
hence z in g . x by A8; ::_thesis: verum
end;
theorem Th37: :: COHSP_1:37
for C1, C2 being Coherence_Space
for f, g being U-stable Function of C1,C2 st Trace f = Trace g holds
f = g
proof
let C1, C2 be Coherence_Space; ::_thesis: for f, g being U-stable Function of C1,C2 st Trace f = Trace g holds
f = g
let f, g be U-stable Function of C1,C2; ::_thesis: ( Trace f = Trace g implies f = g )
A1: dom f = C1 by FUNCT_2:def_1;
A2: dom g = C1 by FUNCT_2:def_1;
A3: now__::_thesis:_for_a_being_Element_of_C1
for_f,_g_being_U-stable_Function_of_C1,C2_st_Trace_f_=_Trace_g_holds_
f_.:_(Fin_a)_c=_g_.:_(Fin_a)
let a be Element of C1; ::_thesis: for f, g being U-stable Function of C1,C2 st Trace f = Trace g holds
f .: (Fin a) c= g .: (Fin a)
let f, g be U-stable Function of C1,C2; ::_thesis: ( Trace f = Trace g implies f .: (Fin a) c= g .: (Fin a) )
A4: dom g = C1 by FUNCT_2:def_1;
assume A5: Trace f = Trace g ; ::_thesis: f .: (Fin a) c= g .: (Fin a)
thus f .: (Fin a) c= g .: (Fin a) ::_thesis: verum
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in f .: (Fin a) or y in g .: (Fin a) )
assume y in f .: (Fin a) ; ::_thesis: y in g .: (Fin a)
then consider x being set such that
x in dom f and
A6: x in Fin a and
A7: y = f . x by FUNCT_1:def_6;
( f . x c= g . x & g . x c= f . x ) by A5, A6, Th36;
then f . x = g . x by XBOOLE_0:def_10;
hence y in g .: (Fin a) by A4, A6, A7, FUNCT_1:def_6; ::_thesis: verum
end;
end;
assume A8: Trace f = Trace g ; ::_thesis: f = g
now__::_thesis:_for_a_being_Element_of_C1_holds_f_._a_=_g_._a
let a be Element of C1; ::_thesis: f . a = g . a
( f .: (Fin a) c= g .: (Fin a) & g .: (Fin a) c= f .: (Fin a) ) by A8, A3;
then A9: f .: (Fin a) = g .: (Fin a) by XBOOLE_0:def_10;
thus f . a = union (f .: (Fin a)) by A1, Th20
.= g . a by A2, A9, Th20 ; ::_thesis: verum
end;
hence f = g by FUNCT_2:63; ::_thesis: verum
end;
Lm5: for C1, C2 being Coherence_Space
for X being Subset of [:C1,(union C2):] st ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ) holds
ex f being U-stable Function of C1,C2 st
( X = Trace f & ( for a being Element of C1 holds f . a = X .: (Fin a) ) )
proof
let C1, C2 be Coherence_Space; ::_thesis: for X being Subset of [:C1,(union C2):] st ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ) holds
ex f being U-stable Function of C1,C2 st
( X = Trace f & ( for a being Element of C1 holds f . a = X .: (Fin a) ) )
let X be Subset of [:C1,(union C2):]; ::_thesis: ( ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ) implies ex f being U-stable Function of C1,C2 st
( X = Trace f & ( for a being Element of C1 holds f . a = X .: (Fin a) ) ) )
assume that
A1: for x being set st x in X holds
x `1 is finite and
A2: for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 and
A3: for a, b being Element of C1 st a \/ b in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ; ::_thesis: ex f being U-stable Function of C1,C2 st
( X = Trace f & ( for a being Element of C1 holds f . a = X .: (Fin a) ) )
deffunc H1( set ) -> set = X .: (Fin $1);
consider f being Function such that
A4: ( dom f = C1 & ( for a being set st a in C1 holds
f . a = H1(a) ) ) from FUNCT_1:sch_3();
A5: now__::_thesis:_for_a,_y_being_set_st_a_in_dom_f_&_y_in_f_._a_holds_
ex_x_being_set_st_
(_x_is_finite_&_x_c=_a_&_y_in_f_._x_&_(_for_c_being_set_st_c_c=_a_&_y_in_f_._c_holds_
x_c=_c_)_)
let a, y be set ; ::_thesis: ( a in dom f & y in f . a implies ex x being set st
( x is finite & x c= a & y in f . x & ( for c being set st c c= a & y in f . c holds
x c= c ) ) )
assume that
A6: a in dom f and
A7: y in f . a ; ::_thesis: ex x being set st
( x is finite & x c= a & y in f . x & ( for c being set st c c= a & y in f . c holds
x c= c ) )
reconsider a9 = a as Element of C1 by A4, A6;
y in X .: (Fin a) by A4, A6, A7;
then consider x being set such that
A8: [x,y] in X and
A9: x in Fin a by RELAT_1:def_13;
x c= a by A9, FINSUB_1:def_5;
then x in C1 by A4, A6, CLASSES1:def_1;
then A10: f . x = X .: (Fin x) by A4;
take x = x; ::_thesis: ( x is finite & x c= a & y in f . x & ( for c being set st c c= a & y in f . c holds
x c= c ) )
x in Fin x by A9, FINSUB_1:def_5;
hence ( x is finite & x c= a & y in f . x ) by A8, A9, A10, FINSUB_1:def_5, RELAT_1:def_13; ::_thesis: for c being set st c c= a & y in f . c holds
x c= c
let c be set ; ::_thesis: ( c c= a & y in f . c implies x c= c )
assume that
A11: c c= a and
A12: y in f . c ; ::_thesis: x c= c
c c= a9 by A11;
then c in dom f by A4, CLASSES1:def_1;
then y in X .: (Fin c) by A4, A12;
then consider z being set such that
A13: [z,y] in X and
A14: z in Fin c by RELAT_1:def_13;
A15: Fin c c= Fin a by A11, FINSUB_1:10;
then A16: z in Fin a9 by A14;
x \/ z in Fin a9 by A9, A14, A15, FINSUB_1:1;
then x = z by A3, A8, A9, A13, A16;
hence x c= c by A14, FINSUB_1:def_5; ::_thesis: verum
end;
A17: rng f c= C2
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng f or x in C2 )
assume x in rng f ; ::_thesis: x in C2
then consider a being set such that
A18: a in dom f and
A19: x = f . a by FUNCT_1:def_3;
reconsider a = a as Element of C1 by A4, A18;
A20: x = X .: (Fin a) by A4, A19;
now__::_thesis:_for_z,_y_being_set_st_z_in_x_&_y_in_x_holds_
{z,y}_in_C2
let z, y be set ; ::_thesis: ( z in x & y in x implies {z,y} in C2 )
assume z in x ; ::_thesis: ( y in x implies {z,y} in C2 )
then consider z1 being set such that
A21: [z1,z] in X and
A22: z1 in Fin a by A20, RELAT_1:def_13;
assume y in x ; ::_thesis: {z,y} in C2
then consider y1 being set such that
A23: [y1,y] in X and
A24: y1 in Fin a by A20, RELAT_1:def_13;
z1 \/ y1 in Fin a by A22, A24, FINSUB_1:1;
hence {z,y} in C2 by A2, A21, A22, A23, A24; ::_thesis: verum
end;
hence x in C2 by COH_SP:6; ::_thesis: verum
end;
f is c=-monotone
proof
let a, b be set ; :: according to COHSP_1:def_11 ::_thesis: ( a in dom f & b in dom f & a c= b implies f . a c= f . b )
assume that
A25: ( a in dom f & b in dom f ) and
A26: a c= b ; ::_thesis: f . a c= f . b
reconsider a = a, b = b as Element of C1 by A4, A25;
Fin a c= Fin b by A26, FINSUB_1:10;
then A27: X .: (Fin a) c= X .: (Fin b) by RELAT_1:123;
f . a = X .: (Fin a) by A4;
hence f . a c= f . b by A4, A27; ::_thesis: verum
end;
then reconsider f = f as U-stable Function of C1,C2 by A4, A17, A5, Th22, FUNCT_2:def_1, RELSET_1:4;
take f ; ::_thesis: ( X = Trace f & ( for a being Element of C1 holds f . a = X .: (Fin a) ) )
thus X = Trace f ::_thesis: for a being Element of C1 holds f . a = X .: (Fin a)
proof
let a, b be set ; :: according to RELAT_1:def_2 ::_thesis: ( ( not [a,b] in X or [a,b] in Trace f ) & ( not [a,b] in Trace f or [a,b] in X ) )
hereby ::_thesis: ( not [a,b] in Trace f or [a,b] in X )
assume A28: [a,b] in X ; ::_thesis: [a,b] in Trace f
[a,b] `1 = a ;
then reconsider a9 = a as finite Element of C1 by A1, A28, ZFMISC_1:87;
a9 in Fin a by FINSUB_1:def_5;
then A29: b in X .: (Fin a) by A28, RELAT_1:def_13;
A30: now__::_thesis:_for_c_being_set_st_c_in_dom_f_&_c_c=_a9_&_b_in_f_._c_holds_
a9_=_c
let c be set ; ::_thesis: ( c in dom f & c c= a9 & b in f . c implies a9 = c )
assume that
A31: c in dom f and
A32: c c= a9 and
A33: b in f . c ; ::_thesis: a9 = c
reconsider c9 = c as Element of C1 by A4, A31;
b in X .: (Fin c9) by A4, A33;
then consider x being set such that
A34: [x,b] in X and
A35: x in Fin c9 by RELAT_1:def_13;
A36: x c= c by A35, FINSUB_1:def_5;
then x \/ a9 = a9 by A32, XBOOLE_1:1, XBOOLE_1:12;
then x = a by A3, A28, A34, A35;
hence a9 = c by A32, A36, XBOOLE_0:def_10; ::_thesis: verum
end;
f . a9 = X .: (Fin a) by A4;
hence [a,b] in Trace f by A4, A29, A30, Th31; ::_thesis: verum
end;
assume A37: [a,b] in Trace f ; ::_thesis: [a,b] in X
then ( a in dom f & b in f . a ) by Th31;
then b in X .: (Fin a) by A4;
then consider x being set such that
A38: [x,b] in X and
A39: x in Fin a by RELAT_1:def_13;
reconsider a = a as Element of C1 by A4, A37, Th31;
x in Fin a by A39;
then reconsider x = x as finite Element of C1 ;
x in Fin x by FINSUB_1:def_5;
then b in X .: (Fin x) by A38, RELAT_1:def_13;
then A40: b in f . x by A4;
x c= a by A39, FINSUB_1:def_5;
hence [a,b] in X by A4, A37, A38, A40, Th31; ::_thesis: verum
end;
thus for a being Element of C1 holds f . a = X .: (Fin a) by A4; ::_thesis: verum
end;
theorem Th38: :: COHSP_1:38
for C1, C2 being Coherence_Space
for X being Subset of [:C1,(union C2):] st ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ) holds
ex f being U-stable Function of C1,C2 st X = Trace f
proof
let C1, C2 be Coherence_Space; ::_thesis: for X being Subset of [:C1,(union C2):] st ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ) holds
ex f being U-stable Function of C1,C2 st X = Trace f
let X be Subset of [:C1,(union C2):]; ::_thesis: ( ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ) implies ex f being U-stable Function of C1,C2 st X = Trace f )
assume A1: ( ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ) & ( for f being U-stable Function of C1,C2 holds not X = Trace f ) ) ; ::_thesis: contradiction
then ex f being U-stable Function of C1,C2 st
( X = Trace f & ( for a being Element of C1 holds f . a = X .: (Fin a) ) ) by Lm5;
hence contradiction by A1; ::_thesis: verum
end;
theorem :: COHSP_1:39
for C1, C2 being Coherence_Space
for f being U-stable Function of C1,C2
for a being Element of C1 holds f . a = (Trace f) .: (Fin a)
proof
let C1, C2 be Coherence_Space; ::_thesis: for f being U-stable Function of C1,C2
for a being Element of C1 holds f . a = (Trace f) .: (Fin a)
let f be U-stable Function of C1,C2; ::_thesis: for a being Element of C1 holds f . a = (Trace f) .: (Fin a)
let a be Element of C1; ::_thesis: f . a = (Trace f) .: (Fin a)
set X = Trace f;
A1: dom f = C1 by FUNCT_2:def_1;
A2: now__::_thesis:_for_x_being_set_st_x_in_Trace_f_holds_
x_`1_is_finite
let x be set ; ::_thesis: ( x in Trace f implies x `1 is finite )
assume A3: x in Trace f ; ::_thesis: x `1 is finite
then consider a, y being set such that
A4: x = [a,y] and
a in dom f and
y in f . a and
for b being set st b in dom f & b c= a & y in f . b holds
a = b by Def17;
a is finite by A1, A3, A4, Th33;
hence x `1 is finite by A4, MCART_1:7; ::_thesis: verum
end;
( ( for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being set st [a,y1] in Trace f & [b,y2] in Trace f holds
{y1,y2} in C2 ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y being set st [a,y] in Trace f & [b,y] in Trace f holds
a = b ) ) by Th34, Th35;
then consider g being U-stable Function of C1,C2 such that
A5: Trace f = Trace g and
A6: for a being Element of C1 holds g . a = (Trace f) .: (Fin a) by A2, Lm5;
g . a = (Trace f) .: (Fin a) by A6;
hence f . a = (Trace f) .: (Fin a) by A5, Th37; ::_thesis: verum
end;
theorem Th40: :: COHSP_1:40
for C1, C2 being Coherence_Space
for f being U-stable Function of C1,C2
for a being Element of C1
for y being set holds
( y in f . a iff ex b being Element of C1 st
( [b,y] in Trace f & b c= a ) )
proof
let C1, C2 be Coherence_Space; ::_thesis: for f being U-stable Function of C1,C2
for a being Element of C1
for y being set holds
( y in f . a iff ex b being Element of C1 st
( [b,y] in Trace f & b c= a ) )
let f be U-stable Function of C1,C2; ::_thesis: for a being Element of C1
for y being set holds
( y in f . a iff ex b being Element of C1 st
( [b,y] in Trace f & b c= a ) )
let a be Element of C1; ::_thesis: for y being set holds
( y in f . a iff ex b being Element of C1 st
( [b,y] in Trace f & b c= a ) )
let y be set ; ::_thesis: ( y in f . a iff ex b being Element of C1 st
( [b,y] in Trace f & b c= a ) )
A1: dom f = C1 by FUNCT_2:def_1;
hereby ::_thesis: ( ex b being Element of C1 st
( [b,y] in Trace f & b c= a ) implies y in f . a )
assume y in f . a ; ::_thesis: ex b being Element of C1 st
( [b,y] in Trace f & b c= a )
then consider b being set such that
b is finite and
A2: b c= a and
A3: y in f . b and
A4: for c being set st c c= a & y in f . c holds
b c= c by A1, Th22;
reconsider b = b as Element of C1 by A2, CLASSES1:def_1;
take b = b; ::_thesis: ( [b,y] in Trace f & b c= a )
now__::_thesis:_for_c_being_set_st_c_in_dom_f_&_c_c=_b_&_y_in_f_._c_holds_
b_=_c
let c be set ; ::_thesis: ( c in dom f & c c= b & y in f . c implies b = c )
assume that
c in dom f and
A5: c c= b and
A6: y in f . c ; ::_thesis: b = c
c c= a by A2, A5, XBOOLE_1:1;
then b c= c by A4, A6;
hence b = c by A5, XBOOLE_0:def_10; ::_thesis: verum
end;
hence [b,y] in Trace f by A1, A3, Th31; ::_thesis: b c= a
thus b c= a by A2; ::_thesis: verum
end;
given b being Element of C1 such that A7: [b,y] in Trace f and
A8: b c= a ; ::_thesis: y in f . a
A9: y in f . b by A7, Th31;
f . b c= f . a by A1, A8, Def11;
hence y in f . a by A9; ::_thesis: verum
end;
theorem :: COHSP_1:41
for C1, C2 being Coherence_Space ex f being U-stable Function of C1,C2 st Trace f = {}
proof
let C1, C2 be Coherence_Space; ::_thesis: ex f being U-stable Function of C1,C2 st Trace f = {}
reconsider X = {} as Subset of [:C1,(union C2):] by XBOOLE_1:2;
A1: for a, b being Element of C1 st a \/ b in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ;
( ( for x being set st x in X holds
x `1 is finite ) & ( for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) ) ;
hence ex f being U-stable Function of C1,C2 st Trace f = {} by A1, Th38; ::_thesis: verum
end;
theorem Th42: :: COHSP_1:42
for C1, C2 being Coherence_Space
for a being finite Element of C1
for y being set st y in union C2 holds
ex f being U-stable Function of C1,C2 st Trace f = {[a,y]}
proof
let C1, C2 be Coherence_Space; ::_thesis: for a being finite Element of C1
for y being set st y in union C2 holds
ex f being U-stable Function of C1,C2 st Trace f = {[a,y]}
let a be finite Element of C1; ::_thesis: for y being set st y in union C2 holds
ex f being U-stable Function of C1,C2 st Trace f = {[a,y]}
let y be set ; ::_thesis: ( y in union C2 implies ex f being U-stable Function of C1,C2 st Trace f = {[a,y]} )
assume A1: y in union C2 ; ::_thesis: ex f being U-stable Function of C1,C2 st Trace f = {[a,y]}
then [a,y] in [:C1,(union C2):] by ZFMISC_1:87;
then reconsider X = {[a,y]} as Subset of [:C1,(union C2):] by ZFMISC_1:31;
A2: now__::_thesis:_for_a1,_b_being_Element_of_C1_st_a1_\/_b_in_C1_holds_
for_y1,_y2_being_set_st_[a1,y1]_in_X_&_[b,y2]_in_X_holds_
{y1,y2}_in_C2
let a1, b be Element of C1; ::_thesis: ( a1 \/ b in C1 implies for y1, y2 being set st [a1,y1] in X & [b,y2] in X holds
{y1,y2} in C2 )
assume a1 \/ b in C1 ; ::_thesis: for y1, y2 being set st [a1,y1] in X & [b,y2] in X holds
{y1,y2} in C2
let y1, y2 be set ; ::_thesis: ( [a1,y1] in X & [b,y2] in X implies {y1,y2} in C2 )
assume that
A3: [a1,y1] in X and
A4: [b,y2] in X ; ::_thesis: {y1,y2} in C2
[b,y2] = [a,y] by A4, TARSKI:def_1;
then A5: y2 = y by XTUPLE_0:1;
[a1,y1] = [a,y] by A3, TARSKI:def_1;
then y1 = y by XTUPLE_0:1;
then {y1,y2} = {y} by A5, ENUMSET1:29;
hence {y1,y2} in C2 by A1, COH_SP:4; ::_thesis: verum
end;
A6: now__::_thesis:_for_a1,_b_being_Element_of_C1_st_a1_\/_b_in_C1_holds_
for_y1_being_set_st_[a1,y1]_in_X_&_[b,y1]_in_X_holds_
a1_=_b
let a1, b be Element of C1; ::_thesis: ( a1 \/ b in C1 implies for y1 being set st [a1,y1] in X & [b,y1] in X holds
a1 = b )
assume a1 \/ b in C1 ; ::_thesis: for y1 being set st [a1,y1] in X & [b,y1] in X holds
a1 = b
let y1 be set ; ::_thesis: ( [a1,y1] in X & [b,y1] in X implies a1 = b )
assume ( [a1,y1] in X & [b,y1] in X ) ; ::_thesis: a1 = b
then ( [a1,y1] = [a,y] & [b,y1] = [a,y] ) by TARSKI:def_1;
hence a1 = b by XTUPLE_0:1; ::_thesis: verum
end;
now__::_thesis:_for_x_being_set_st_x_in_X_holds_
x_`1_is_finite
let x be set ; ::_thesis: ( x in X implies x `1 is finite )
assume x in X ; ::_thesis: x `1 is finite
then x = [a,y] by TARSKI:def_1;
hence x `1 is finite by MCART_1:7; ::_thesis: verum
end;
hence ex f being U-stable Function of C1,C2 st Trace f = {[a,y]} by A2, A6, Th38; ::_thesis: verum
end;
theorem :: COHSP_1:43
for C1, C2 being Coherence_Space
for a being Element of C1
for y being set
for f being U-stable Function of C1,C2 st Trace f = {[a,y]} holds
for b being Element of C1 holds
( ( a c= b implies f . b = {y} ) & ( not a c= b implies f . b = {} ) )
proof
let C1, C2 be Coherence_Space; ::_thesis: for a being Element of C1
for y being set
for f being U-stable Function of C1,C2 st Trace f = {[a,y]} holds
for b being Element of C1 holds
( ( a c= b implies f . b = {y} ) & ( not a c= b implies f . b = {} ) )
let a be Element of C1; ::_thesis: for y being set
for f being U-stable Function of C1,C2 st Trace f = {[a,y]} holds
for b being Element of C1 holds
( ( a c= b implies f . b = {y} ) & ( not a c= b implies f . b = {} ) )
let y be set ; ::_thesis: for f being U-stable Function of C1,C2 st Trace f = {[a,y]} holds
for b being Element of C1 holds
( ( a c= b implies f . b = {y} ) & ( not a c= b implies f . b = {} ) )
let f be U-stable Function of C1,C2; ::_thesis: ( Trace f = {[a,y]} implies for b being Element of C1 holds
( ( a c= b implies f . b = {y} ) & ( not a c= b implies f . b = {} ) ) )
assume A1: Trace f = {[a,y]} ; ::_thesis: for b being Element of C1 holds
( ( a c= b implies f . b = {y} ) & ( not a c= b implies f . b = {} ) )
let b be Element of C1; ::_thesis: ( ( a c= b implies f . b = {y} ) & ( not a c= b implies f . b = {} ) )
A2: [a,y] in Trace f by A1, TARSKI:def_1;
hereby ::_thesis: ( not a c= b implies f . b = {} )
A3: f . b c= {y}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in f . b or x in {y} )
assume x in f . b ; ::_thesis: x in {y}
then consider c being Element of C1 such that
A4: [c,x] in Trace f and
c c= b by Th40;
[c,x] = [a,y] by A1, A4, TARSKI:def_1;
then x = y by XTUPLE_0:1;
hence x in {y} by TARSKI:def_1; ::_thesis: verum
end;
assume a c= b ; ::_thesis: f . b = {y}
then y in f . b by A2, Th40;
then {y} c= f . b by ZFMISC_1:31;
hence f . b = {y} by A3, XBOOLE_0:def_10; ::_thesis: verum
end;
assume that
A5: not a c= b and
A6: f . b <> {} ; ::_thesis: contradiction
reconsider B = f . b as non empty set by A6;
set z = the Element of B;
consider c being Element of C1 such that
A7: [c, the Element of B] in Trace f and
A8: c c= b by Th40;
[c, the Element of B] = [a,y] by A1, A7, TARSKI:def_1;
hence contradiction by A5, A8, XTUPLE_0:1; ::_thesis: verum
end;
theorem Th44: :: COHSP_1:44
for C1, C2 being Coherence_Space
for f being U-stable Function of C1,C2
for X being Subset of (Trace f) ex g being U-stable Function of C1,C2 st Trace g = X
proof
let C1, C2 be Coherence_Space; ::_thesis: for f being U-stable Function of C1,C2
for X being Subset of (Trace f) ex g being U-stable Function of C1,C2 st Trace g = X
let f be U-stable Function of C1,C2; ::_thesis: for X being Subset of (Trace f) ex g being U-stable Function of C1,C2 st Trace g = X
let X be Subset of (Trace f); ::_thesis: ex g being U-stable Function of C1,C2 st Trace g = X
A1: for a, b being Element of C1 st a \/ b in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b by Th35;
A2: now__::_thesis:_for_x_being_set_st_x_in_X_holds_
x_`1_is_finite
let x be set ; ::_thesis: ( x in X implies x `1 is finite )
assume A3: x in X ; ::_thesis: x `1 is finite
then consider a, y being set such that
A4: x = [a,y] and
a in dom f and
y in f . a and
for b being set st b in dom f & b c= a & y in f . b holds
a = b by Def17;
dom f = C1 by FUNCT_2:def_1;
then a is finite by A3, A4, Th33;
hence x `1 is finite by A4, MCART_1:7; ::_thesis: verum
end;
( X is Subset of [:C1,(union C2):] & ( for a, b being Element of C1 st a \/ b in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) ) by Th34, XBOOLE_1:1;
hence ex g being U-stable Function of C1,C2 st Trace g = X by A2, A1, Th38; ::_thesis: verum
end;
theorem Th45: :: COHSP_1:45
for C1, C2 being Coherence_Space
for A being set st ( for x, y being set st x in A & y in A holds
ex f being U-stable Function of C1,C2 st x \/ y = Trace f ) holds
ex f being U-stable Function of C1,C2 st union A = Trace f
proof
let C1, C2 be Coherence_Space; ::_thesis: for A being set st ( for x, y being set st x in A & y in A holds
ex f being U-stable Function of C1,C2 st x \/ y = Trace f ) holds
ex f being U-stable Function of C1,C2 st union A = Trace f
let A be set ; ::_thesis: ( ( for x, y being set st x in A & y in A holds
ex f being U-stable Function of C1,C2 st x \/ y = Trace f ) implies ex f being U-stable Function of C1,C2 st union A = Trace f )
assume A1: for x, y being set st x in A & y in A holds
ex f being U-stable Function of C1,C2 st x \/ y = Trace f ; ::_thesis: ex f being U-stable Function of C1,C2 st union A = Trace f
set X = union A;
A2: now__::_thesis:_for_a,_b_being_Element_of_C1_st_a_\/_b_in_C1_holds_
for_y1,_y2_being_set_st_[a,y1]_in_union_A_&_[b,y2]_in_union_A_holds_
{y1,y2}_in_C2
let a, b be Element of C1; ::_thesis: ( a \/ b in C1 implies for y1, y2 being set st [a,y1] in union A & [b,y2] in union A holds
{y1,y2} in C2 )
assume A3: a \/ b in C1 ; ::_thesis: for y1, y2 being set st [a,y1] in union A & [b,y2] in union A holds
{y1,y2} in C2
let y1, y2 be set ; ::_thesis: ( [a,y1] in union A & [b,y2] in union A implies {y1,y2} in C2 )
assume [a,y1] in union A ; ::_thesis: ( [b,y2] in union A implies {y1,y2} in C2 )
then consider x1 being set such that
A4: [a,y1] in x1 and
A5: x1 in A by TARSKI:def_4;
assume [b,y2] in union A ; ::_thesis: {y1,y2} in C2
then consider x2 being set such that
A6: [b,y2] in x2 and
A7: x2 in A by TARSKI:def_4;
A8: ( x1 c= x1 \/ x2 & x2 c= x1 \/ x2 ) by XBOOLE_1:7;
ex f being U-stable Function of C1,C2 st x1 \/ x2 = Trace f by A1, A5, A7;
hence {y1,y2} in C2 by A3, A4, A6, A8, Th34; ::_thesis: verum
end;
A9: now__::_thesis:_for_a,_b_being_Element_of_C1_st_a_\/_b_in_C1_holds_
for_y_being_set_st_[a,y]_in_union_A_&_[b,y]_in_union_A_holds_
a_=_b
let a, b be Element of C1; ::_thesis: ( a \/ b in C1 implies for y being set st [a,y] in union A & [b,y] in union A holds
a = b )
assume A10: a \/ b in C1 ; ::_thesis: for y being set st [a,y] in union A & [b,y] in union A holds
a = b
let y be set ; ::_thesis: ( [a,y] in union A & [b,y] in union A implies a = b )
assume [a,y] in union A ; ::_thesis: ( [b,y] in union A implies a = b )
then consider x1 being set such that
A11: [a,y] in x1 and
A12: x1 in A by TARSKI:def_4;
assume [b,y] in union A ; ::_thesis: a = b
then consider x2 being set such that
A13: [b,y] in x2 and
A14: x2 in A by TARSKI:def_4;
A15: ( x1 c= x1 \/ x2 & x2 c= x1 \/ x2 ) by XBOOLE_1:7;
ex f being U-stable Function of C1,C2 st x1 \/ x2 = Trace f by A1, A12, A14;
hence a = b by A10, A11, A13, A15, Th35; ::_thesis: verum
end;
A16: now__::_thesis:_for_x_being_set_st_x_in_union_A_holds_
x_`1_is_finite
let x be set ; ::_thesis: ( x in union A implies x `1 is finite )
assume x in union A ; ::_thesis: x `1 is finite
then consider y being set such that
A17: x in y and
A18: y in A by TARSKI:def_4;
y \/ y = y ;
then consider f being U-stable Function of C1,C2 such that
A19: y = Trace f by A1, A18;
consider a, y being set such that
A20: x = [a,y] and
a in dom f and
y in f . a and
for b being set st b in dom f & b c= a & y in f . b holds
a = b by A17, A19, Def17;
dom f = C1 by FUNCT_2:def_1;
then a is finite by A17, A19, A20, Th33;
hence x `1 is finite by A20, MCART_1:7; ::_thesis: verum
end;
union A c= [:C1,(union C2):]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union A or x in [:C1,(union C2):] )
assume x in union A ; ::_thesis: x in [:C1,(union C2):]
then consider y being set such that
A21: x in y and
A22: y in A by TARSKI:def_4;
y \/ y = y ;
then ex f being U-stable Function of C1,C2 st y = Trace f by A1, A22;
hence x in [:C1,(union C2):] by A21; ::_thesis: verum
end;
hence ex f being U-stable Function of C1,C2 st union A = Trace f by A16, A2, A9, Th38; ::_thesis: verum
end;
definition
let C1, C2 be Coherence_Space;
func StabCoh (C1,C2) -> set means :Def18: :: COHSP_1:def 18
for x being set holds
( x in it iff ex f being U-stable Function of C1,C2 st x = Trace f );
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff ex f being U-stable Function of C1,C2 st x = Trace f ) ) & ( for x being set holds
( x in b2 iff ex f being U-stable Function of C1,C2 st x = Trace f ) ) holds
b1 = b2
proof
let X1, X2 be set ; ::_thesis: ( ( for x being set holds
( x in X1 iff ex f being U-stable Function of C1,C2 st x = Trace f ) ) & ( for x being set holds
( x in X2 iff ex f being U-stable Function of C1,C2 st x = Trace f ) ) implies X1 = X2 )
assume A1: ( ( for x being set holds
( x in X1 iff ex f being U-stable Function of C1,C2 st x = Trace f ) ) & ( for x being set holds
( x in X2 iff ex f being U-stable Function of C1,C2 st x = Trace f ) ) & not X1 = X2 ) ; ::_thesis: contradiction
then consider x being set such that
A2: ( ( x in X1 & not x in X2 ) or ( x in X2 & not x in X1 ) ) by TARSKI:1;
( x in X2 iff for f being U-stable Function of C1,C2 holds not x = Trace f ) by A1, A2;
hence contradiction by A1; ::_thesis: verum
end;
existence
ex b1 being set st
for x being set holds
( x in b1 iff ex f being U-stable Function of C1,C2 st x = Trace f )
proof
defpred S1[ set ] means ex f being U-stable Function of C1,C2 st $1 = Trace f;
consider X being set such that
A3: for x being set holds
( x in X iff ( x in bool [:C1,(union C2):] & S1[x] ) ) from XBOOLE_0:sch_1();
take X ; ::_thesis: for x being set holds
( x in X iff ex f being U-stable Function of C1,C2 st x = Trace f )
let x be set ; ::_thesis: ( x in X iff ex f being U-stable Function of C1,C2 st x = Trace f )
thus ( x in X iff ex f being U-stable Function of C1,C2 st x = Trace f ) by A3; ::_thesis: verum
end;
end;
:: deftheorem Def18 defines StabCoh COHSP_1:def_18_:_
for C1, C2 being Coherence_Space
for b3 being set holds
( b3 = StabCoh (C1,C2) iff for x being set holds
( x in b3 iff ex f being U-stable Function of C1,C2 st x = Trace f ) );
registration
let C1, C2 be Coherence_Space;
cluster StabCoh (C1,C2) -> non empty subset-closed binary_complete ;
coherence
( not StabCoh (C1,C2) is empty & StabCoh (C1,C2) is subset-closed & StabCoh (C1,C2) is binary_complete )
proof
set C = StabCoh (C1,C2);
set f = the U-stable Function of C1,C2;
Trace the U-stable Function of C1,C2 in StabCoh (C1,C2) by Def18;
hence not StabCoh (C1,C2) is empty ; ::_thesis: ( StabCoh (C1,C2) is subset-closed & StabCoh (C1,C2) is binary_complete )
thus StabCoh (C1,C2) is subset-closed ::_thesis: StabCoh (C1,C2) is binary_complete
proof
let a, b be set ; :: according to CLASSES1:def_1 ::_thesis: ( not a in StabCoh (C1,C2) or not b c= a or b in StabCoh (C1,C2) )
assume a in StabCoh (C1,C2) ; ::_thesis: ( not b c= a or b in StabCoh (C1,C2) )
then A1: ex f being U-stable Function of C1,C2 st a = Trace f by Def18;
assume b c= a ; ::_thesis: b in StabCoh (C1,C2)
then ex g being U-stable Function of C1,C2 st Trace g = b by A1, Th44;
hence b in StabCoh (C1,C2) by Def18; ::_thesis: verum
end;
let A be set ; :: according to COHSP_1:def_1 ::_thesis: ( ( for a, b being set st a in A & b in A holds
a \/ b in StabCoh (C1,C2) ) implies union A in StabCoh (C1,C2) )
assume A2: for a, b being set st a in A & b in A holds
a \/ b in StabCoh (C1,C2) ; ::_thesis: union A in StabCoh (C1,C2)
now__::_thesis:_for_x,_y_being_set_st_x_in_A_&_y_in_A_holds_
ex_f_being_U-stable_Function_of_C1,C2_st_x_\/_y_=_Trace_f
let x, y be set ; ::_thesis: ( x in A & y in A implies ex f being U-stable Function of C1,C2 st x \/ y = Trace f )
assume ( x in A & y in A ) ; ::_thesis: ex f being U-stable Function of C1,C2 st x \/ y = Trace f
then x \/ y in StabCoh (C1,C2) by A2;
hence ex f being U-stable Function of C1,C2 st x \/ y = Trace f by Def18; ::_thesis: verum
end;
then ex f being U-stable Function of C1,C2 st union A = Trace f by Th45;
hence union A in StabCoh (C1,C2) by Def18; ::_thesis: verum
end;
end;
theorem Th46: :: COHSP_1:46
for C1, C2 being Coherence_Space
for f being U-stable Function of C1,C2 holds Trace f c= [:(Sub_of_Fin C1),(union C2):]
proof
let C1, C2 be Coherence_Space; ::_thesis: for f being U-stable Function of C1,C2 holds Trace f c= [:(Sub_of_Fin C1),(union C2):]
let f be U-stable Function of C1,C2; ::_thesis: Trace f c= [:(Sub_of_Fin C1),(union C2):]
let x1, x2 be set ; :: according to RELAT_1:def_3 ::_thesis: ( not [x1,x2] in Trace f or [x1,x2] in [:(Sub_of_Fin C1),(union C2):] )
assume A1: [x1,x2] in Trace f ; ::_thesis: [x1,x2] in [:(Sub_of_Fin C1),(union C2):]
then consider a, y being set such that
A2: [x1,x2] = [a,y] and
A3: a in dom f and
A4: y in f . a and
for b being set st b in dom f & b c= a & y in f . b holds
a = b by Def17;
A5: dom f = C1 by FUNCT_2:def_1;
then a is finite by A1, A2, Th33;
then A6: a in Sub_of_Fin C1 by A3, A5, Def3;
y in union C2 by A3, A4, A5, Lm1;
hence [x1,x2] in [:(Sub_of_Fin C1),(union C2):] by A2, A6, ZFMISC_1:87; ::_thesis: verum
end;
theorem :: COHSP_1:47
for C1, C2 being Coherence_Space holds union (StabCoh (C1,C2)) = [:(Sub_of_Fin C1),(union C2):]
proof
let C1, C2 be Coherence_Space; ::_thesis: union (StabCoh (C1,C2)) = [:(Sub_of_Fin C1),(union C2):]
thus union (StabCoh (C1,C2)) c= [:(Sub_of_Fin C1),(union C2):] :: according to XBOOLE_0:def_10 ::_thesis: [:(Sub_of_Fin C1),(union C2):] c= union (StabCoh (C1,C2))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union (StabCoh (C1,C2)) or x in [:(Sub_of_Fin C1),(union C2):] )
assume x in union (StabCoh (C1,C2)) ; ::_thesis: x in [:(Sub_of_Fin C1),(union C2):]
then consider a being set such that
A1: x in a and
A2: a in StabCoh (C1,C2) by TARSKI:def_4;
ex f being U-stable Function of C1,C2 st a = Trace f by A2, Def18;
then a c= [:(Sub_of_Fin C1),(union C2):] by Th46;
hence x in [:(Sub_of_Fin C1),(union C2):] by A1; ::_thesis: verum
end;
let x, y be set ; :: according to RELAT_1:def_3 ::_thesis: ( not [x,y] in [:(Sub_of_Fin C1),(union C2):] or [x,y] in union (StabCoh (C1,C2)) )
assume A3: [x,y] in [:(Sub_of_Fin C1),(union C2):] ; ::_thesis: [x,y] in union (StabCoh (C1,C2))
then A4: y in union C2 by ZFMISC_1:87;
A5: x in Sub_of_Fin C1 by A3, ZFMISC_1:87;
then x is finite by Def3;
then ex f being U-stable Function of C1,C2 st Trace f = {[x,y]} by A5, A4, Th42;
then ( [x,y] in {[x,y]} & {[x,y]} in StabCoh (C1,C2) ) by Def18, TARSKI:def_1;
hence [x,y] in union (StabCoh (C1,C2)) by TARSKI:def_4; ::_thesis: verum
end;
theorem Th48: :: COHSP_1:48
for C1, C2 being Coherence_Space
for a, b being finite Element of C1
for y1, y2 being set holds
( [[a,y1],[b,y2]] in Web (StabCoh (C1,C2)) iff ( ( not a \/ b in C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies a = b ) ) ) )
proof
let C1, C2 be Coherence_Space; ::_thesis: for a, b being finite Element of C1
for y1, y2 being set holds
( [[a,y1],[b,y2]] in Web (StabCoh (C1,C2)) iff ( ( not a \/ b in C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies a = b ) ) ) )
let a, b be finite Element of C1; ::_thesis: for y1, y2 being set holds
( [[a,y1],[b,y2]] in Web (StabCoh (C1,C2)) iff ( ( not a \/ b in C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies a = b ) ) ) )
let y1, y2 be set ; ::_thesis: ( [[a,y1],[b,y2]] in Web (StabCoh (C1,C2)) iff ( ( not a \/ b in C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies a = b ) ) ) )
hereby ::_thesis: ( ( ( not a \/ b in C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies a = b ) ) ) implies [[a,y1],[b,y2]] in Web (StabCoh (C1,C2)) )
assume [[a,y1],[b,y2]] in Web (StabCoh (C1,C2)) ; ::_thesis: ( ( a \/ b in C1 or not y1 in union C2 or not y2 in union C2 ) implies ( [y1,y2] in Web C2 & ( y1 = y2 implies a = b ) ) )
then {[a,y1],[b,y2]} in StabCoh (C1,C2) by COH_SP:5;
then A1: ex f being U-stable Function of C1,C2 st {[a,y1],[b,y2]} = Trace f by Def18;
A2: ( [a,y1] in {[a,y1],[b,y2]} & [b,y2] in {[a,y1],[b,y2]} ) by TARSKI:def_2;
assume A3: ( a \/ b in C1 or not y1 in union C2 or not y2 in union C2 ) ; ::_thesis: ( [y1,y2] in Web C2 & ( y1 = y2 implies a = b ) )
then {y1,y2} in C2 by A1, A2, Th34, ZFMISC_1:87;
hence [y1,y2] in Web C2 by COH_SP:5; ::_thesis: ( y1 = y2 implies a = b )
thus ( y1 = y2 implies a = b ) by A1, A2, A3, Th35, ZFMISC_1:87; ::_thesis: verum
end;
assume A4: ( ( not a \/ b in C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies a = b ) ) ) ; ::_thesis: [[a,y1],[b,y2]] in Web (StabCoh (C1,C2))
then A5: y2 in union C2 by ZFMISC_1:87;
then A6: [b,y2] in [:C1,(union C2):] by ZFMISC_1:87;
A7: y1 in union C2 by A4, ZFMISC_1:87;
then [a,y1] in [:C1,(union C2):] by ZFMISC_1:87;
then reconsider X = {[a,y1],[b,y2]} as Subset of [:C1,(union C2):] by A6, ZFMISC_1:32;
A8: now__::_thesis:_for_a1,_b1_being_Element_of_C1_st_a1_\/_b1_in_C1_holds_
for_z1,_z2_being_set_st_[a1,z1]_in_X_&_[b1,z2]_in_X_holds_
{z1,z2}_in_C2
let a1, b1 be Element of C1; ::_thesis: ( a1 \/ b1 in C1 implies for z1, z2 being set st [a1,z1] in X & [b1,z2] in X holds
{z1,z2} in C2 )
assume A9: a1 \/ b1 in C1 ; ::_thesis: for z1, z2 being set st [a1,z1] in X & [b1,z2] in X holds
{z1,z2} in C2
let z1, z2 be set ; ::_thesis: ( [a1,z1] in X & [b1,z2] in X implies {z1,z2} in C2 )
assume that
A10: [a1,z1] in X and
A11: [b1,z2] in X ; ::_thesis: {z1,z2} in C2
( [b1,z2] = [a,y1] or [b1,z2] = [b,y2] ) by A11, TARSKI:def_2;
then A12: ( ( z2 = y1 & b1 = a ) or ( b1 = b & z2 = y2 ) ) by XTUPLE_0:1;
( [a1,z1] = [a,y1] or [a1,z1] = [b,y2] ) by A10, TARSKI:def_2;
then ( ( z1 = y1 & a1 = a ) or ( a1 = b & z1 = y2 ) ) by XTUPLE_0:1;
then ( {z1,z2} = {y1} or {z1,z2} in C2 or {z1,z2} = {y2} ) by A4, A9, A12, COH_SP:5, ENUMSET1:29;
hence {z1,z2} in C2 by A7, A5, COH_SP:4; ::_thesis: verum
end;
A13: now__::_thesis:_for_a1,_b1_being_Element_of_C1_st_a1_\/_b1_in_C1_holds_
for_y_being_set_st_[a1,y]_in_X_&_[b1,y]_in_X_holds_
a1_=_b1
let a1, b1 be Element of C1; ::_thesis: ( a1 \/ b1 in C1 implies for y being set st [a1,y] in X & [b1,y] in X holds
a1 = b1 )
assume A14: a1 \/ b1 in C1 ; ::_thesis: for y being set st [a1,y] in X & [b1,y] in X holds
a1 = b1
let y be set ; ::_thesis: ( [a1,y] in X & [b1,y] in X implies a1 = b1 )
assume that
A15: [a1,y] in X and
A16: [b1,y] in X ; ::_thesis: a1 = b1
( [a1,y] = [a,y1] or [a1,y] = [b,y2] ) by A15, TARSKI:def_2;
then A17: ( ( a1 = a & y = y1 ) or ( a1 = b & y = y2 ) ) by XTUPLE_0:1;
( [b1,y] = [a,y1] or [b1,y] = [b,y2] ) by A16, TARSKI:def_2;
hence a1 = b1 by A4, A14, A17, XTUPLE_0:1; ::_thesis: verum
end;
now__::_thesis:_for_x_being_set_st_x_in_X_holds_
x_`1_is_finite
let x be set ; ::_thesis: ( x in X implies x `1 is finite )
assume x in X ; ::_thesis: x `1 is finite
then ( x = [a,y1] or x = [b,y2] ) by TARSKI:def_2;
hence x `1 is finite by MCART_1:7; ::_thesis: verum
end;
then ex f being U-stable Function of C1,C2 st X = Trace f by A8, A13, Th38;
then X in StabCoh (C1,C2) by Def18;
hence [[a,y1],[b,y2]] in Web (StabCoh (C1,C2)) by COH_SP:5; ::_thesis: verum
end;
begin
theorem Th49: :: COHSP_1:49
for C1, C2 being Coherence_Space
for f being U-stable Function of C1,C2 holds
( f is U-linear iff for a, y being set st [a,y] in Trace f holds
ex x being set st a = {x} )
proof
let C1, C2 be Coherence_Space; ::_thesis: for f being U-stable Function of C1,C2 holds
( f is U-linear iff for a, y being set st [a,y] in Trace f holds
ex x being set st a = {x} )
let f be U-stable Function of C1,C2; ::_thesis: ( f is U-linear iff for a, y being set st [a,y] in Trace f holds
ex x being set st a = {x} )
A1: dom f = C1 by FUNCT_2:def_1;
hereby ::_thesis: ( ( for a, y being set st [a,y] in Trace f holds
ex x being set st a = {x} ) implies f is U-linear )
assume A2: f is U-linear ; ::_thesis: for a, y being set st [a,y] in Trace f holds
ex x being set st a = {x}
let a, y be set ; ::_thesis: ( [a,y] in Trace f implies ex x being set st a = {x} )
assume A3: [a,y] in Trace f ; ::_thesis: ex x being set st a = {x}
then A4: a in dom f by Th31;
y in f . a by A3, Th31;
then consider x being set such that
A5: x in a and
A6: y in f . {x} and
for b being set st b c= a & y in f . b holds
x in b by A1, A2, A4, Th23;
A7: {x} c= a by A5, ZFMISC_1:31;
take x = x; ::_thesis: a = {x}
A8: {x,x} = {x} by ENUMSET1:29;
{x,x} in C1 by A1, A4, A5, COH_SP:6;
hence a = {x} by A1, A3, A6, A7, A8, Th31; ::_thesis: verum
end;
assume A9: for a, y being set st [a,y] in Trace f holds
ex x being set st a = {x} ; ::_thesis: f is U-linear
now__::_thesis:_for_a,_y_being_set_st_a_in_dom_f_&_y_in_f_._a_holds_
ex_x_being_set_st_
(_x_in_a_&_y_in_f_._{x}_&_(_for_c_being_set_st_c_c=_a_&_y_in_f_._c_holds_
x_in_c_)_)
let a, y be set ; ::_thesis: ( a in dom f & y in f . a implies ex x being set st
( x in a & y in f . {x} & ( for c being set st c c= a & y in f . c holds
x in c ) ) )
assume that
A10: a in dom f and
A11: y in f . a ; ::_thesis: ex x being set st
( x in a & y in f . {x} & ( for c being set st c c= a & y in f . c holds
x in c ) )
consider b being set such that
b is finite and
A12: b c= a and
A13: y in f . b and
A14: for c being set st c c= a & y in f . c holds
b c= c by A1, A10, A11, Th22;
now__::_thesis:_(_b_in_dom_f_&_(_for_c_being_set_st_c_in_dom_f_&_c_c=_b_&_y_in_f_._c_holds_
b_=_c_)_)
thus b in dom f by A1, A10, A12, CLASSES1:def_1; ::_thesis: for c being set st c in dom f & c c= b & y in f . c holds
b = c
let c be set ; ::_thesis: ( c in dom f & c c= b & y in f . c implies b = c )
assume that
c in dom f and
A15: c c= b and
A16: y in f . c ; ::_thesis: b = c
c c= a by A12, A15, XBOOLE_1:1;
then b c= c by A14, A16;
hence b = c by A15, XBOOLE_0:def_10; ::_thesis: verum
end;
then [b,y] in Trace f by A13, Th31;
then consider x being set such that
A17: b = {x} by A9;
take x = x; ::_thesis: ( x in a & y in f . {x} & ( for c being set st c c= a & y in f . c holds
x in c ) )
x in b by A17, TARSKI:def_1;
hence ( x in a & y in f . {x} ) by A12, A13, A17; ::_thesis: for c being set st c c= a & y in f . c holds
x in c
let c be set ; ::_thesis: ( c c= a & y in f . c implies x in c )
assume ( c c= a & y in f . c ) ; ::_thesis: x in c
then b c= c by A14;
hence x in c by A17, ZFMISC_1:31; ::_thesis: verum
end;
hence f is U-linear by A1, Th23; ::_thesis: verum
end;
definition
let f be Function;
func LinTrace f -> set means :Def19: :: COHSP_1:def 19
for x being set holds
( x in it iff ex y, z being set st
( x = [y,z] & [{y},z] in Trace f ) );
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff ex y, z being set st
( x = [y,z] & [{y},z] in Trace f ) ) ) & ( for x being set holds
( x in b2 iff ex y, z being set st
( x = [y,z] & [{y},z] in Trace f ) ) ) holds
b1 = b2
proof
let X1, X2 be set ; ::_thesis: ( ( for x being set holds
( x in X1 iff ex y, z being set st
( x = [y,z] & [{y},z] in Trace f ) ) ) & ( for x being set holds
( x in X2 iff ex y, z being set st
( x = [y,z] & [{y},z] in Trace f ) ) ) implies X1 = X2 )
assume A1: ( ( for x being set holds
( x in X1 iff ex y, z being set st
( x = [y,z] & [{y},z] in Trace f ) ) ) & ( for x being set holds
( x in X2 iff ex y, z being set st
( x = [y,z] & [{y},z] in Trace f ) ) ) & not X1 = X2 ) ; ::_thesis: contradiction
then consider x being set such that
A2: ( ( x in X1 & not x in X2 ) or ( x in X2 & not x in X1 ) ) by TARSKI:1;
( x in X2 iff for y, z being set holds
( not x = [y,z] or not [{y},z] in Trace f ) ) by A1, A2;
hence contradiction by A1; ::_thesis: verum
end;
existence
ex b1 being set st
for x being set holds
( x in b1 iff ex y, z being set st
( x = [y,z] & [{y},z] in Trace f ) )
proof
defpred S1[ set ] means ex y, z being set st
( $1 = [y,z] & [{y},z] in Trace f );
set C1 = dom f;
set C2 = rng f;
consider X being set such that
A3: for x being set holds
( x in X iff ( x in [:(union (dom f)),(union (rng f)):] & S1[x] ) ) from XBOOLE_0:sch_1();
take X ; ::_thesis: for x being set holds
( x in X iff ex y, z being set st
( x = [y,z] & [{y},z] in Trace f ) )
let x be set ; ::_thesis: ( x in X iff ex y, z being set st
( x = [y,z] & [{y},z] in Trace f ) )
now__::_thesis:_(_ex_y,_z_being_set_st_
(_x_=_[y,z]_&_[{y},z]_in_Trace_f_)_implies_x_in_[:(union_(dom_f)),(union_(rng_f)):]_)
given y, z being set such that A4: x = [y,z] and
A5: [{y},z] in Trace f ; ::_thesis: x in [:(union (dom f)),(union (rng f)):]
A6: {y} in dom f by A5, Th31;
then A7: f . {y} in rng f by FUNCT_1:def_3;
z in f . {y} by A5, Th31;
then A8: z in union (rng f) by A7, TARSKI:def_4;
y in {y} by TARSKI:def_1;
then y in union (dom f) by A6, TARSKI:def_4;
hence x in [:(union (dom f)),(union (rng f)):] by A4, A8, ZFMISC_1:87; ::_thesis: verum
end;
hence ( x in X iff ex y, z being set st
( x = [y,z] & [{y},z] in Trace f ) ) by A3; ::_thesis: verum
end;
end;
:: deftheorem Def19 defines LinTrace COHSP_1:def_19_:_
for f being Function
for b2 being set holds
( b2 = LinTrace f iff for x being set holds
( x in b2 iff ex y, z being set st
( x = [y,z] & [{y},z] in Trace f ) ) );
theorem Th50: :: COHSP_1:50
for f being Function
for x, y being set holds
( [x,y] in LinTrace f iff [{x},y] in Trace f )
proof
let f be Function; ::_thesis: for x, y being set holds
( [x,y] in LinTrace f iff [{x},y] in Trace f )
let x, y be set ; ::_thesis: ( [x,y] in LinTrace f iff [{x},y] in Trace f )
now__::_thesis:_(_ex_v,_z_being_set_st_
(_[x,y]_=_[v,z]_&_[{v},z]_in_Trace_f_)_implies_[{x},y]_in_Trace_f_)
given v, z being set such that A1: [x,y] = [v,z] and
A2: [{v},z] in Trace f ; ::_thesis: [{x},y] in Trace f
x = v by A1, XTUPLE_0:1;
hence [{x},y] in Trace f by A1, A2, XTUPLE_0:1; ::_thesis: verum
end;
hence ( [x,y] in LinTrace f iff [{x},y] in Trace f ) by Def19; ::_thesis: verum
end;
theorem Th51: :: COHSP_1:51
for f being Function st f . {} = {} holds
for x, y being set st {x} in dom f & y in f . {x} holds
[x,y] in LinTrace f
proof
let f be Function; ::_thesis: ( f . {} = {} implies for x, y being set st {x} in dom f & y in f . {x} holds
[x,y] in LinTrace f )
assume A1: f . {} = {} ; ::_thesis: for x, y being set st {x} in dom f & y in f . {x} holds
[x,y] in LinTrace f
let x, y be set ; ::_thesis: ( {x} in dom f & y in f . {x} implies [x,y] in LinTrace f )
set a = {x};
( [x,y] in LinTrace f iff [{x},y] in Trace f ) by Th50;
then ( [x,y] in LinTrace f iff ( {x} in dom f & y in f . {x} & ( for b being set st b in dom f & b c= {x} & y in f . b holds
{x} = b ) ) ) by Th31;
hence ( {x} in dom f & y in f . {x} implies [x,y] in LinTrace f ) by A1, ZFMISC_1:33; ::_thesis: verum
end;
theorem Th52: :: COHSP_1:52
for f being Function
for x, y being set st [x,y] in LinTrace f holds
( {x} in dom f & y in f . {x} )
proof
let f be Function; ::_thesis: for x, y being set st [x,y] in LinTrace f holds
( {x} in dom f & y in f . {x} )
let x, y be set ; ::_thesis: ( [x,y] in LinTrace f implies ( {x} in dom f & y in f . {x} ) )
assume [x,y] in LinTrace f ; ::_thesis: ( {x} in dom f & y in f . {x} )
then [{x},y] in Trace f by Th50;
hence ( {x} in dom f & y in f . {x} ) by Th31; ::_thesis: verum
end;
definition
let C1, C2 be non empty set ;
let f be Function of C1,C2;
:: original: LinTrace
redefine func LinTrace f -> Subset of [:(union C1),(union C2):];
coherence
LinTrace f is Subset of [:(union C1),(union C2):]
proof
LinTrace f c= [:(union C1),(union C2):]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LinTrace f or x in [:(union C1),(union C2):] )
assume x in LinTrace f ; ::_thesis: x in [:(union C1),(union C2):]
then consider y, z being set such that
A1: x = [y,z] and
A2: [{y},z] in Trace f by Def19;
A3: y in {y} by TARSKI:def_1;
dom f = C1 by FUNCT_2:def_1;
then {y} in C1 by A2, Th31;
then A4: y in union C1 by A3, TARSKI:def_4;
z in union C2 by A2, ZFMISC_1:87;
hence x in [:(union C1),(union C2):] by A1, A4, ZFMISC_1:87; ::_thesis: verum
end;
hence LinTrace f is Subset of [:(union C1),(union C2):] ; ::_thesis: verum
end;
end;
registration
let f be Function;
cluster LinTrace f -> Relation-like ;
coherence
LinTrace f is Relation-like
proof
let x be set ; :: according to RELAT_1:def_1 ::_thesis: ( not x in LinTrace f or ex b1, b2 being set st x = [b1,b2] )
assume x in LinTrace f ; ::_thesis: ex b1, b2 being set st x = [b1,b2]
then ex y, z being set st
( x = [y,z] & [{y},z] in Trace f ) by Def19;
hence ex b1, b2 being set st x = [b1,b2] ; ::_thesis: verum
end;
end;
definition
let C1, C2 be Coherence_Space;
func LinCoh (C1,C2) -> set means :Def20: :: COHSP_1:def 20
for x being set holds
( x in it iff ex f being U-linear Function of C1,C2 st x = LinTrace f );
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff ex f being U-linear Function of C1,C2 st x = LinTrace f ) ) & ( for x being set holds
( x in b2 iff ex f being U-linear Function of C1,C2 st x = LinTrace f ) ) holds
b1 = b2
proof
let X1, X2 be set ; ::_thesis: ( ( for x being set holds
( x in X1 iff ex f being U-linear Function of C1,C2 st x = LinTrace f ) ) & ( for x being set holds
( x in X2 iff ex f being U-linear Function of C1,C2 st x = LinTrace f ) ) implies X1 = X2 )
assume A1: ( ( for x being set holds
( x in X1 iff ex f being U-linear Function of C1,C2 st x = LinTrace f ) ) & ( for x being set holds
( x in X2 iff ex f being U-linear Function of C1,C2 st x = LinTrace f ) ) & not X1 = X2 ) ; ::_thesis: contradiction
then consider x being set such that
A2: ( ( x in X1 & not x in X2 ) or ( x in X2 & not x in X1 ) ) by TARSKI:1;
( x in X2 iff for f being U-linear Function of C1,C2 holds not x = LinTrace f ) by A1, A2;
hence contradiction by A1; ::_thesis: verum
end;
existence
ex b1 being set st
for x being set holds
( x in b1 iff ex f being U-linear Function of C1,C2 st x = LinTrace f )
proof
defpred S1[ set ] means ex f being U-linear Function of C1,C2 st $1 = LinTrace f;
consider X being set such that
A3: for x being set holds
( x in X iff ( x in bool [:(union C1),(union C2):] & S1[x] ) ) from XBOOLE_0:sch_1();
take X ; ::_thesis: for x being set holds
( x in X iff ex f being U-linear Function of C1,C2 st x = LinTrace f )
let x be set ; ::_thesis: ( x in X iff ex f being U-linear Function of C1,C2 st x = LinTrace f )
thus ( x in X iff ex f being U-linear Function of C1,C2 st x = LinTrace f ) by A3; ::_thesis: verum
end;
end;
:: deftheorem Def20 defines LinCoh COHSP_1:def_20_:_
for C1, C2 being Coherence_Space
for b3 being set holds
( b3 = LinCoh (C1,C2) iff for x being set holds
( x in b3 iff ex f being U-linear Function of C1,C2 st x = LinTrace f ) );
theorem Th53: :: COHSP_1:53
for C1, C2 being Coherence_Space
for f being c=-monotone Function of C1,C2
for x1, x2 being set st {x1,x2} in C1 holds
for y1, y2 being set st [x1,y1] in LinTrace f & [x2,y2] in LinTrace f holds
{y1,y2} in C2
proof
let C1, C2 be Coherence_Space; ::_thesis: for f being c=-monotone Function of C1,C2
for x1, x2 being set st {x1,x2} in C1 holds
for y1, y2 being set st [x1,y1] in LinTrace f & [x2,y2] in LinTrace f holds
{y1,y2} in C2
let f be c=-monotone Function of C1,C2; ::_thesis: for x1, x2 being set st {x1,x2} in C1 holds
for y1, y2 being set st [x1,y1] in LinTrace f & [x2,y2] in LinTrace f holds
{y1,y2} in C2
A1: dom f = C1 by FUNCT_2:def_1;
let a1, a2 be set ; ::_thesis: ( {a1,a2} in C1 implies for y1, y2 being set st [a1,y1] in LinTrace f & [a2,y2] in LinTrace f holds
{y1,y2} in C2 )
assume {a1,a2} in C1 ; ::_thesis: for y1, y2 being set st [a1,y1] in LinTrace f & [a2,y2] in LinTrace f holds
{y1,y2} in C2
then reconsider a = {a1,a2} as Element of C1 ;
A2: {a2} c= a by ZFMISC_1:7;
then {a2} in C1 by CLASSES1:def_1;
then A3: f . {a2} c= f . a by A1, A2, Def11;
let y1, y2 be set ; ::_thesis: ( [a1,y1] in LinTrace f & [a2,y2] in LinTrace f implies {y1,y2} in C2 )
assume ( [a1,y1] in LinTrace f & [a2,y2] in LinTrace f ) ; ::_thesis: {y1,y2} in C2
then A4: ( y1 in f . {a1} & y2 in f . {a2} ) by Th52;
A5: {a1} c= a by ZFMISC_1:7;
then {a1} in C1 by CLASSES1:def_1;
then f . {a1} c= f . a by A1, A5, Def11;
then {y1,y2} c= f . a by A3, A4, ZFMISC_1:32;
hence {y1,y2} in C2 by CLASSES1:def_1; ::_thesis: verum
end;
theorem Th54: :: COHSP_1:54
for C1, C2 being Coherence_Space
for f being cap-distributive Function of C1,C2
for x1, x2 being set st {x1,x2} in C1 holds
for y being set st [x1,y] in LinTrace f & [x2,y] in LinTrace f holds
x1 = x2
proof
let C1, C2 be Coherence_Space; ::_thesis: for f being cap-distributive Function of C1,C2
for x1, x2 being set st {x1,x2} in C1 holds
for y being set st [x1,y] in LinTrace f & [x2,y] in LinTrace f holds
x1 = x2
let f be cap-distributive Function of C1,C2; ::_thesis: for x1, x2 being set st {x1,x2} in C1 holds
for y being set st [x1,y] in LinTrace f & [x2,y] in LinTrace f holds
x1 = x2
let a1, a2 be set ; ::_thesis: ( {a1,a2} in C1 implies for y being set st [a1,y] in LinTrace f & [a2,y] in LinTrace f holds
a1 = a2 )
set a = {a1,a2};
assume A1: {a1,a2} in C1 ; ::_thesis: for y being set st [a1,y] in LinTrace f & [a2,y] in LinTrace f holds
a1 = a2
let y be set ; ::_thesis: ( [a1,y] in LinTrace f & [a2,y] in LinTrace f implies a1 = a2 )
A2: {a1,a2} = {a1} \/ {a2} by ENUMSET1:1;
assume ( [a1,y] in LinTrace f & [a2,y] in LinTrace f ) ; ::_thesis: a1 = a2
then ( [{a1},y] in Trace f & [{a2},y] in Trace f ) by Th50;
then {a1} = {a2} by A1, A2, Th35;
hence a1 = a2 by ZFMISC_1:3; ::_thesis: verum
end;
theorem Th55: :: COHSP_1:55
for C1, C2 being Coherence_Space
for f, g being U-linear Function of C1,C2 st LinTrace f = LinTrace g holds
f = g
proof
let C1, C2 be Coherence_Space; ::_thesis: for f, g being U-linear Function of C1,C2 st LinTrace f = LinTrace g holds
f = g
let f, g be U-linear Function of C1,C2; ::_thesis: ( LinTrace f = LinTrace g implies f = g )
assume A1: LinTrace f = LinTrace g ; ::_thesis: f = g
Trace f = Trace g
proof
let a, y be set ; :: according to RELAT_1:def_2 ::_thesis: ( ( not [a,y] in Trace f or [a,y] in Trace g ) & ( not [a,y] in Trace g or [a,y] in Trace f ) )
hereby ::_thesis: ( not [a,y] in Trace g or [a,y] in Trace f )
assume A2: [a,y] in Trace f ; ::_thesis: [a,y] in Trace g
then consider x being set such that
A3: a = {x} by Th49;
[x,y] in LinTrace f by A2, A3, Th50;
hence [a,y] in Trace g by A1, A3, Th50; ::_thesis: verum
end;
assume A4: [a,y] in Trace g ; ::_thesis: [a,y] in Trace f
then consider x being set such that
A5: a = {x} by Th49;
[x,y] in LinTrace g by A4, A5, Th50;
hence [a,y] in Trace f by A1, A5, Th50; ::_thesis: verum
end;
hence f = g by Th37; ::_thesis: verum
end;
Lm6: for C1, C2 being Coherence_Space
for X being Subset of [:(union C1),(union C2):] st ( for a, b being set st {a,b} in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being set st {a,b} in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ) holds
ex f being U-linear Function of C1,C2 st
( X = LinTrace f & ( for a being Element of C1 holds f . a = X .: a ) )
proof
let C1, C2 be Coherence_Space; ::_thesis: for X being Subset of [:(union C1),(union C2):] st ( for a, b being set st {a,b} in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being set st {a,b} in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ) holds
ex f being U-linear Function of C1,C2 st
( X = LinTrace f & ( for a being Element of C1 holds f . a = X .: a ) )
let X be Subset of [:(union C1),(union C2):]; ::_thesis: ( ( for a, b being set st {a,b} in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being set st {a,b} in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ) implies ex f being U-linear Function of C1,C2 st
( X = LinTrace f & ( for a being Element of C1 holds f . a = X .: a ) ) )
assume that
A1: for a, b being set st {a,b} in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 and
A2: for a, b being set st {a,b} in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ; ::_thesis: ex f being U-linear Function of C1,C2 st
( X = LinTrace f & ( for a being Element of C1 holds f . a = X .: a ) )
deffunc H1( set ) -> set = X .: $1;
consider f being Function such that
A3: ( dom f = C1 & ( for a being set st a in C1 holds
f . a = H1(a) ) ) from FUNCT_1:sch_3();
A4: now__::_thesis:_for_a,_y_being_set_st_a_in_dom_f_&_y_in_f_._a_holds_
ex_x_being_set_st_
(_x_in_a_&_y_in_f_._{x}_&_(_for_c_being_set_st_c_c=_a_&_y_in_f_._c_holds_
x_in_c_)_)
let a, y be set ; ::_thesis: ( a in dom f & y in f . a implies ex x being set st
( x in a & y in f . {x} & ( for c being set st c c= a & y in f . c holds
x in c ) ) )
assume that
A5: a in dom f and
A6: y in f . a ; ::_thesis: ex x being set st
( x in a & y in f . {x} & ( for c being set st c c= a & y in f . c holds
x in c ) )
reconsider a9 = a as Element of C1 by A3, A5;
y in X .: a by A3, A5, A6;
then consider x being set such that
A7: [x,y] in X and
A8: x in a by RELAT_1:def_13;
take x = x; ::_thesis: ( x in a & y in f . {x} & ( for c being set st c c= a & y in f . c holds
x in c ) )
{x} c= a by A8, ZFMISC_1:31;
then {x} in C1 by A3, A5, CLASSES1:def_1;
then ( x in {x} & f . {x} = X .: {x} ) by A3, TARSKI:def_1;
hence ( x in a & y in f . {x} ) by A7, A8, RELAT_1:def_13; ::_thesis: for c being set st c c= a & y in f . c holds
x in c
let c be set ; ::_thesis: ( c c= a & y in f . c implies x in c )
assume that
A9: c c= a and
A10: y in f . c ; ::_thesis: x in c
c c= a9 by A9;
then c in dom f by A3, CLASSES1:def_1;
then y in X .: c by A3, A10;
then consider z being set such that
A11: [z,y] in X and
A12: z in c by RELAT_1:def_13;
{x,z} c= a9 by A8, A9, A12, ZFMISC_1:32;
then {x,z} in C1 by CLASSES1:def_1;
hence x in c by A2, A7, A11, A12; ::_thesis: verum
end;
A13: rng f c= C2
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng f or x in C2 )
assume x in rng f ; ::_thesis: x in C2
then consider a being set such that
A14: a in dom f and
A15: x = f . a by FUNCT_1:def_3;
reconsider a = a as Element of C1 by A3, A14;
A16: x = X .: a by A3, A15;
now__::_thesis:_for_z,_y_being_set_st_z_in_x_&_y_in_x_holds_
{z,y}_in_C2
let z, y be set ; ::_thesis: ( z in x & y in x implies {z,y} in C2 )
assume z in x ; ::_thesis: ( y in x implies {z,y} in C2 )
then consider z1 being set such that
A17: [z1,z] in X and
A18: z1 in a by A16, RELAT_1:def_13;
assume y in x ; ::_thesis: {z,y} in C2
then consider y1 being set such that
A19: [y1,y] in X and
A20: y1 in a by A16, RELAT_1:def_13;
{z1,y1} in C1 by A18, A20, COH_SP:6;
hence {z,y} in C2 by A1, A17, A19; ::_thesis: verum
end;
hence x in C2 by COH_SP:6; ::_thesis: verum
end;
f is c=-monotone
proof
let a, b be set ; :: according to COHSP_1:def_11 ::_thesis: ( a in dom f & b in dom f & a c= b implies f . a c= f . b )
assume that
A21: ( a in dom f & b in dom f ) and
A22: a c= b ; ::_thesis: f . a c= f . b
reconsider a = a, b = b as Element of C1 by A3, A21;
A23: f . a = X .: a by A3;
X .: a c= X .: b by A22, RELAT_1:123;
hence f . a c= f . b by A3, A23; ::_thesis: verum
end;
then reconsider f = f as U-linear Function of C1,C2 by A3, A13, A4, Th23, FUNCT_2:def_1, RELSET_1:4;
take f ; ::_thesis: ( X = LinTrace f & ( for a being Element of C1 holds f . a = X .: a ) )
thus X = LinTrace f ::_thesis: for a being Element of C1 holds f . a = X .: a
proof
let a, b be set ; :: according to RELAT_1:def_2 ::_thesis: ( ( not [a,b] in X or [a,b] in LinTrace f ) & ( not [a,b] in LinTrace f or [a,b] in X ) )
hereby ::_thesis: ( not [a,b] in LinTrace f or [a,b] in X )
assume A24: [a,b] in X ; ::_thesis: [a,b] in LinTrace f
then a in union C1 by ZFMISC_1:87;
then consider a9 being set such that
A25: a in a9 and
A26: a9 in C1 by TARSKI:def_4;
{a} c= a9 by A25, ZFMISC_1:31;
then reconsider aa = {a} as Element of C1 by A26, CLASSES1:def_1;
A27: ( f . aa = X .: aa & f . {} = {} ) by A3, Th18;
a in {a} by TARSKI:def_1;
then b in X .: aa by A24, RELAT_1:def_13;
hence [a,b] in LinTrace f by A3, A27, Th51; ::_thesis: verum
end;
assume A28: [a,b] in LinTrace f ; ::_thesis: [a,b] in X
then b in f . {a} by Th52;
then b in X .: {a} by A3, A28, Th52;
then ex x being set st
( [x,b] in X & x in {a} ) by RELAT_1:def_13;
hence [a,b] in X by TARSKI:def_1; ::_thesis: verum
end;
thus for a being Element of C1 holds f . a = X .: a by A3; ::_thesis: verum
end;
theorem Th56: :: COHSP_1:56
for C1, C2 being Coherence_Space
for X being Subset of [:(union C1),(union C2):] st ( for a, b being set st {a,b} in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being set st {a,b} in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ) holds
ex f being U-linear Function of C1,C2 st X = LinTrace f
proof
let C1, C2 be Coherence_Space; ::_thesis: for X being Subset of [:(union C1),(union C2):] st ( for a, b being set st {a,b} in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being set st {a,b} in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ) holds
ex f being U-linear Function of C1,C2 st X = LinTrace f
let X be Subset of [:(union C1),(union C2):]; ::_thesis: ( ( for a, b being set st {a,b} in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being set st {a,b} in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ) implies ex f being U-linear Function of C1,C2 st X = LinTrace f )
assume A1: ( ( for a, b being set st {a,b} in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being set st {a,b} in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ) & ( for f being U-linear Function of C1,C2 holds not X = LinTrace f ) ) ; ::_thesis: contradiction
then ex f being U-linear Function of C1,C2 st
( X = LinTrace f & ( for a being Element of C1 holds f . a = X .: a ) ) by Lm6;
hence contradiction by A1; ::_thesis: verum
end;
theorem :: COHSP_1:57
for C1, C2 being Coherence_Space
for f being U-linear Function of C1,C2
for a being Element of C1 holds f . a = (LinTrace f) .: a
proof
let C1, C2 be Coherence_Space; ::_thesis: for f being U-linear Function of C1,C2
for a being Element of C1 holds f . a = (LinTrace f) .: a
let f be U-linear Function of C1,C2; ::_thesis: for a being Element of C1 holds f . a = (LinTrace f) .: a
let a be Element of C1; ::_thesis: f . a = (LinTrace f) .: a
set X = LinTrace f;
( ( for a, b being set st {a,b} in C1 holds
for y1, y2 being set st [a,y1] in LinTrace f & [b,y2] in LinTrace f holds
{y1,y2} in C2 ) & ( for a, b being set st {a,b} in C1 holds
for y being set st [a,y] in LinTrace f & [b,y] in LinTrace f holds
a = b ) ) by Th53, Th54;
then consider g being U-linear Function of C1,C2 such that
A1: LinTrace f = LinTrace g and
A2: for a being Element of C1 holds g . a = (LinTrace f) .: a by Lm6;
g . a = (LinTrace f) .: a by A2;
hence f . a = (LinTrace f) .: a by A1, Th55; ::_thesis: verum
end;
theorem :: COHSP_1:58
for C1, C2 being Coherence_Space ex f being U-linear Function of C1,C2 st LinTrace f = {}
proof
let C1, C2 be Coherence_Space; ::_thesis: ex f being U-linear Function of C1,C2 st LinTrace f = {}
reconsider X = {} as Subset of [:(union C1),(union C2):] by XBOOLE_1:2;
( ( for a, b being set st {a,b} in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) & ( for a, b being set st {a,b} in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b ) ) ;
hence ex f being U-linear Function of C1,C2 st LinTrace f = {} by Th56; ::_thesis: verum
end;
theorem Th59: :: COHSP_1:59
for C1, C2 being Coherence_Space
for x, y being set st x in union C1 & y in union C2 holds
ex f being U-linear Function of C1,C2 st LinTrace f = {[x,y]}
proof
let C1, C2 be Coherence_Space; ::_thesis: for x, y being set st x in union C1 & y in union C2 holds
ex f being U-linear Function of C1,C2 st LinTrace f = {[x,y]}
let a, y be set ; ::_thesis: ( a in union C1 & y in union C2 implies ex f being U-linear Function of C1,C2 st LinTrace f = {[a,y]} )
assume that
A1: a in union C1 and
A2: y in union C2 ; ::_thesis: ex f being U-linear Function of C1,C2 st LinTrace f = {[a,y]}
[a,y] in [:(union C1),(union C2):] by A1, A2, ZFMISC_1:87;
then reconsider X = {[a,y]} as Subset of [:(union C1),(union C2):] by ZFMISC_1:31;
A3: now__::_thesis:_for_a1,_b_being_set_st_{a1,b}_in_C1_holds_
for_y1,_y2_being_set_st_[a1,y1]_in_X_&_[b,y2]_in_X_holds_
{y1,y2}_in_C2
let a1, b be set ; ::_thesis: ( {a1,b} in C1 implies for y1, y2 being set st [a1,y1] in X & [b,y2] in X holds
{y1,y2} in C2 )
assume {a1,b} in C1 ; ::_thesis: for y1, y2 being set st [a1,y1] in X & [b,y2] in X holds
{y1,y2} in C2
let y1, y2 be set ; ::_thesis: ( [a1,y1] in X & [b,y2] in X implies {y1,y2} in C2 )
assume that
A4: [a1,y1] in X and
A5: [b,y2] in X ; ::_thesis: {y1,y2} in C2
[b,y2] = [a,y] by A5, TARSKI:def_1;
then A6: y2 = y by XTUPLE_0:1;
[a1,y1] = [a,y] by A4, TARSKI:def_1;
then y1 = y by XTUPLE_0:1;
then {y1,y2} = {y} by A6, ENUMSET1:29;
hence {y1,y2} in C2 by A2, COH_SP:4; ::_thesis: verum
end;
now__::_thesis:_for_a1,_b_being_set_st_{a1,b}_in_C1_holds_
for_y1_being_set_st_[a1,y1]_in_X_&_[b,y1]_in_X_holds_
a1_=_b
let a1, b be set ; ::_thesis: ( {a1,b} in C1 implies for y1 being set st [a1,y1] in X & [b,y1] in X holds
a1 = b )
assume {a1,b} in C1 ; ::_thesis: for y1 being set st [a1,y1] in X & [b,y1] in X holds
a1 = b
let y1 be set ; ::_thesis: ( [a1,y1] in X & [b,y1] in X implies a1 = b )
assume ( [a1,y1] in X & [b,y1] in X ) ; ::_thesis: a1 = b
then ( [a1,y1] = [a,y] & [b,y1] = [a,y] ) by TARSKI:def_1;
hence a1 = b by XTUPLE_0:1; ::_thesis: verum
end;
hence ex f being U-linear Function of C1,C2 st LinTrace f = {[a,y]} by A3, Th56; ::_thesis: verum
end;
theorem :: COHSP_1:60
for C1, C2 being Coherence_Space
for x, y being set st x in union C1 holds
for f being U-linear Function of C1,C2 st LinTrace f = {[x,y]} holds
for a being Element of C1 holds
( ( x in a implies f . a = {y} ) & ( not x in a implies f . a = {} ) )
proof
let C1, C2 be Coherence_Space; ::_thesis: for x, y being set st x in union C1 holds
for f being U-linear Function of C1,C2 st LinTrace f = {[x,y]} holds
for a being Element of C1 holds
( ( x in a implies f . a = {y} ) & ( not x in a implies f . a = {} ) )
let a, y be set ; ::_thesis: ( a in union C1 implies for f being U-linear Function of C1,C2 st LinTrace f = {[a,y]} holds
for a being Element of C1 holds
( ( a in a implies f . a = {y} ) & ( not a in a implies f . a = {} ) ) )
assume a in union C1 ; ::_thesis: for f being U-linear Function of C1,C2 st LinTrace f = {[a,y]} holds
for a being Element of C1 holds
( ( a in a implies f . a = {y} ) & ( not a in a implies f . a = {} ) )
then reconsider a9 = {a} as Element of C1 by COH_SP:4;
let f be U-linear Function of C1,C2; ::_thesis: ( LinTrace f = {[a,y]} implies for a being Element of C1 holds
( ( a in a implies f . a = {y} ) & ( not a in a implies f . a = {} ) ) )
assume A1: LinTrace f = {[a,y]} ; ::_thesis: for a being Element of C1 holds
( ( a in a implies f . a = {y} ) & ( not a in a implies f . a = {} ) )
let b be Element of C1; ::_thesis: ( ( a in b implies f . b = {y} ) & ( not a in b implies f . b = {} ) )
[a,y] in LinTrace f by A1, TARSKI:def_1;
then A2: y in f . {a} by Th52;
hereby ::_thesis: ( not a in b implies f . b = {} )
A3: f . b c= {y}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in f . b or x in {y} )
assume x in f . b ; ::_thesis: x in {y}
then consider c being Element of C1 such that
A4: [c,x] in Trace f and
c c= b by Th40;
consider d being set such that
A5: c = {d} by A4, Th49;
[d,x] in LinTrace f by A4, A5, Th50;
then [d,x] = [a,y] by A1, TARSKI:def_1;
then x = y by XTUPLE_0:1;
hence x in {y} by TARSKI:def_1; ::_thesis: verum
end;
assume a in b ; ::_thesis: f . b = {y}
then A6: a9 c= b by ZFMISC_1:31;
dom f = C1 by FUNCT_2:def_1;
then f . a9 c= f . b by A6, Def11;
then {y} c= f . b by A2, ZFMISC_1:31;
hence f . b = {y} by A3, XBOOLE_0:def_10; ::_thesis: verum
end;
assume that
A7: not a in b and
A8: f . b <> {} ; ::_thesis: contradiction
reconsider B = f . b as non empty set by A8;
set z = the Element of B;
consider c being Element of C1 such that
A9: [c, the Element of B] in Trace f and
A10: c c= b by Th40;
consider d being set such that
A11: c = {d} by A9, Th49;
d in c by A11, TARSKI:def_1;
then A12: d in b by A10;
[d, the Element of B] in LinTrace f by A9, A11, Th50;
then [d, the Element of B] = [a,y] by A1, TARSKI:def_1;
hence contradiction by A7, A12, XTUPLE_0:1; ::_thesis: verum
end;
theorem Th61: :: COHSP_1:61
for C1, C2 being Coherence_Space
for f being U-linear Function of C1,C2
for X being Subset of (LinTrace f) ex g being U-linear Function of C1,C2 st LinTrace g = X
proof
let C1, C2 be Coherence_Space; ::_thesis: for f being U-linear Function of C1,C2
for X being Subset of (LinTrace f) ex g being U-linear Function of C1,C2 st LinTrace g = X
let f be U-linear Function of C1,C2; ::_thesis: for X being Subset of (LinTrace f) ex g being U-linear Function of C1,C2 st LinTrace g = X
let X be Subset of (LinTrace f); ::_thesis: ex g being U-linear Function of C1,C2 st LinTrace g = X
A1: for a, b being set st {a,b} in C1 holds
for y being set st [a,y] in X & [b,y] in X holds
a = b by Th54;
( X is Subset of [:(union C1),(union C2):] & ( for a, b being set st {a,b} in C1 holds
for y1, y2 being set st [a,y1] in X & [b,y2] in X holds
{y1,y2} in C2 ) ) by Th53, XBOOLE_1:1;
hence ex g being U-linear Function of C1,C2 st LinTrace g = X by A1, Th56; ::_thesis: verum
end;
theorem Th62: :: COHSP_1:62
for C1, C2 being Coherence_Space
for A being set st ( for x, y being set st x in A & y in A holds
ex f being U-linear Function of C1,C2 st x \/ y = LinTrace f ) holds
ex f being U-linear Function of C1,C2 st union A = LinTrace f
proof
let C1, C2 be Coherence_Space; ::_thesis: for A being set st ( for x, y being set st x in A & y in A holds
ex f being U-linear Function of C1,C2 st x \/ y = LinTrace f ) holds
ex f being U-linear Function of C1,C2 st union A = LinTrace f
let A be set ; ::_thesis: ( ( for x, y being set st x in A & y in A holds
ex f being U-linear Function of C1,C2 st x \/ y = LinTrace f ) implies ex f being U-linear Function of C1,C2 st union A = LinTrace f )
assume A1: for x, y being set st x in A & y in A holds
ex f being U-linear Function of C1,C2 st x \/ y = LinTrace f ; ::_thesis: ex f being U-linear Function of C1,C2 st union A = LinTrace f
set X = union A;
A2: now__::_thesis:_for_a,_b_being_set_st_{a,b}_in_C1_holds_
for_y1,_y2_being_set_st_[a,y1]_in_union_A_&_[b,y2]_in_union_A_holds_
{y1,y2}_in_C2
let a, b be set ; ::_thesis: ( {a,b} in C1 implies for y1, y2 being set st [a,y1] in union A & [b,y2] in union A holds
{y1,y2} in C2 )
assume A3: {a,b} in C1 ; ::_thesis: for y1, y2 being set st [a,y1] in union A & [b,y2] in union A holds
{y1,y2} in C2
let y1, y2 be set ; ::_thesis: ( [a,y1] in union A & [b,y2] in union A implies {y1,y2} in C2 )
assume [a,y1] in union A ; ::_thesis: ( [b,y2] in union A implies {y1,y2} in C2 )
then consider x1 being set such that
A4: [a,y1] in x1 and
A5: x1 in A by TARSKI:def_4;
assume [b,y2] in union A ; ::_thesis: {y1,y2} in C2
then consider x2 being set such that
A6: [b,y2] in x2 and
A7: x2 in A by TARSKI:def_4;
A8: ( x1 c= x1 \/ x2 & x2 c= x1 \/ x2 ) by XBOOLE_1:7;
ex f being U-linear Function of C1,C2 st x1 \/ x2 = LinTrace f by A1, A5, A7;
hence {y1,y2} in C2 by A3, A4, A6, A8, Th53; ::_thesis: verum
end;
A9: now__::_thesis:_for_a,_b_being_set_st_{a,b}_in_C1_holds_
for_y_being_set_st_[a,y]_in_union_A_&_[b,y]_in_union_A_holds_
a_=_b
let a, b be set ; ::_thesis: ( {a,b} in C1 implies for y being set st [a,y] in union A & [b,y] in union A holds
a = b )
assume A10: {a,b} in C1 ; ::_thesis: for y being set st [a,y] in union A & [b,y] in union A holds
a = b
let y be set ; ::_thesis: ( [a,y] in union A & [b,y] in union A implies a = b )
assume [a,y] in union A ; ::_thesis: ( [b,y] in union A implies a = b )
then consider x1 being set such that
A11: [a,y] in x1 and
A12: x1 in A by TARSKI:def_4;
assume [b,y] in union A ; ::_thesis: a = b
then consider x2 being set such that
A13: [b,y] in x2 and
A14: x2 in A by TARSKI:def_4;
A15: ( x1 c= x1 \/ x2 & x2 c= x1 \/ x2 ) by XBOOLE_1:7;
ex f being U-linear Function of C1,C2 st x1 \/ x2 = LinTrace f by A1, A12, A14;
hence a = b by A10, A11, A13, A15, Th54; ::_thesis: verum
end;
union A c= [:(union C1),(union C2):]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union A or x in [:(union C1),(union C2):] )
assume x in union A ; ::_thesis: x in [:(union C1),(union C2):]
then consider y being set such that
A16: x in y and
A17: y in A by TARSKI:def_4;
y \/ y = y ;
then ex f being U-linear Function of C1,C2 st y = LinTrace f by A1, A17;
hence x in [:(union C1),(union C2):] by A16; ::_thesis: verum
end;
hence ex f being U-linear Function of C1,C2 st union A = LinTrace f by A2, A9, Th56; ::_thesis: verum
end;
registration
let C1, C2 be Coherence_Space;
cluster LinCoh (C1,C2) -> non empty subset-closed binary_complete ;
coherence
( not LinCoh (C1,C2) is empty & LinCoh (C1,C2) is subset-closed & LinCoh (C1,C2) is binary_complete )
proof
set C = LinCoh (C1,C2);
set f = the U-linear Function of C1,C2;
LinTrace the U-linear Function of C1,C2 in LinCoh (C1,C2) by Def20;
hence not LinCoh (C1,C2) is empty ; ::_thesis: ( LinCoh (C1,C2) is subset-closed & LinCoh (C1,C2) is binary_complete )
thus LinCoh (C1,C2) is subset-closed ::_thesis: LinCoh (C1,C2) is binary_complete
proof
let a, b be set ; :: according to CLASSES1:def_1 ::_thesis: ( not a in LinCoh (C1,C2) or not b c= a or b in LinCoh (C1,C2) )
assume a in LinCoh (C1,C2) ; ::_thesis: ( not b c= a or b in LinCoh (C1,C2) )
then A1: ex f being U-linear Function of C1,C2 st a = LinTrace f by Def20;
assume b c= a ; ::_thesis: b in LinCoh (C1,C2)
then ex g being U-linear Function of C1,C2 st LinTrace g = b by A1, Th61;
hence b in LinCoh (C1,C2) by Def20; ::_thesis: verum
end;
let A be set ; :: according to COHSP_1:def_1 ::_thesis: ( ( for a, b being set st a in A & b in A holds
a \/ b in LinCoh (C1,C2) ) implies union A in LinCoh (C1,C2) )
assume A2: for a, b being set st a in A & b in A holds
a \/ b in LinCoh (C1,C2) ; ::_thesis: union A in LinCoh (C1,C2)
now__::_thesis:_for_x,_y_being_set_st_x_in_A_&_y_in_A_holds_
ex_f_being_U-linear_Function_of_C1,C2_st_x_\/_y_=_LinTrace_f
let x, y be set ; ::_thesis: ( x in A & y in A implies ex f being U-linear Function of C1,C2 st x \/ y = LinTrace f )
assume ( x in A & y in A ) ; ::_thesis: ex f being U-linear Function of C1,C2 st x \/ y = LinTrace f
then x \/ y in LinCoh (C1,C2) by A2;
hence ex f being U-linear Function of C1,C2 st x \/ y = LinTrace f by Def20; ::_thesis: verum
end;
then ex f being U-linear Function of C1,C2 st union A = LinTrace f by Th62;
hence union A in LinCoh (C1,C2) by Def20; ::_thesis: verum
end;
end;
theorem :: COHSP_1:63
for C1, C2 being Coherence_Space holds union (LinCoh (C1,C2)) = [:(union C1),(union C2):]
proof
let C1, C2 be Coherence_Space; ::_thesis: union (LinCoh (C1,C2)) = [:(union C1),(union C2):]
thus union (LinCoh (C1,C2)) c= [:(union C1),(union C2):] :: according to XBOOLE_0:def_10 ::_thesis: [:(union C1),(union C2):] c= union (LinCoh (C1,C2))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union (LinCoh (C1,C2)) or x in [:(union C1),(union C2):] )
assume x in union (LinCoh (C1,C2)) ; ::_thesis: x in [:(union C1),(union C2):]
then consider a being set such that
A1: x in a and
A2: a in LinCoh (C1,C2) by TARSKI:def_4;
ex f being U-linear Function of C1,C2 st a = LinTrace f by A2, Def20;
hence x in [:(union C1),(union C2):] by A1; ::_thesis: verum
end;
let x, y be set ; :: according to RELAT_1:def_3 ::_thesis: ( not [x,y] in [:(union C1),(union C2):] or [x,y] in union (LinCoh (C1,C2)) )
assume A3: [x,y] in [:(union C1),(union C2):] ; ::_thesis: [x,y] in union (LinCoh (C1,C2))
then A4: y in union C2 by ZFMISC_1:87;
x in union C1 by A3, ZFMISC_1:87;
then ex f being U-linear Function of C1,C2 st LinTrace f = {[x,y]} by A4, Th59;
then ( [x,y] in {[x,y]} & {[x,y]} in LinCoh (C1,C2) ) by Def20, TARSKI:def_1;
hence [x,y] in union (LinCoh (C1,C2)) by TARSKI:def_4; ::_thesis: verum
end;
theorem :: COHSP_1:64
for C1, C2 being Coherence_Space
for x1, x2, y1, y2 being set holds
( [[x1,y1],[x2,y2]] in Web (LinCoh (C1,C2)) iff ( x1 in union C1 & x2 in union C1 & ( ( not [x1,x2] in Web C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies x1 = x2 ) ) ) ) )
proof
let C1, C2 be Coherence_Space; ::_thesis: for x1, x2, y1, y2 being set holds
( [[x1,y1],[x2,y2]] in Web (LinCoh (C1,C2)) iff ( x1 in union C1 & x2 in union C1 & ( ( not [x1,x2] in Web C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies x1 = x2 ) ) ) ) )
let x1, x2, y1, y2 be set ; ::_thesis: ( [[x1,y1],[x2,y2]] in Web (LinCoh (C1,C2)) iff ( x1 in union C1 & x2 in union C1 & ( ( not [x1,x2] in Web C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies x1 = x2 ) ) ) ) )
hereby ::_thesis: ( x1 in union C1 & x2 in union C1 & ( ( not [x1,x2] in Web C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies x1 = x2 ) ) ) implies [[x1,y1],[x2,y2]] in Web (LinCoh (C1,C2)) )
assume [[x1,y1],[x2,y2]] in Web (LinCoh (C1,C2)) ; ::_thesis: ( x1 in union C1 & x2 in union C1 & ( ( not [x1,x2] in Web C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies x1 = x2 ) ) ) )
then {[x1,y1],[x2,y2]} in LinCoh (C1,C2) by COH_SP:5;
then consider f being U-linear Function of C1,C2 such that
A1: {[x1,y1],[x2,y2]} = LinTrace f by Def20;
[x1,y1] in LinTrace f by A1, TARSKI:def_2;
then A2: [{x1},y1] in Trace f by Th50;
then A3: {x1} in dom f by Th31;
[x2,y2] in LinTrace f by A1, TARSKI:def_2;
then A4: [{x2},y2] in Trace f by Th50;
then A5: {x2} in dom f by Th31;
A6: ( x1 in {x1} & x2 in {x2} ) by TARSKI:def_1;
A7: Trace f in StabCoh (C1,C2) by Def18;
A8: dom f = C1 by FUNCT_2:def_1;
{[{x1},y1],[{x2},y2]} c= Trace f by A2, A4, ZFMISC_1:32;
then {[{x1},y1],[{x2},y2]} in StabCoh (C1,C2) by A7, CLASSES1:def_1;
then [[{x1},y1],[{x2},y2]] in Web (StabCoh (C1,C2)) by COH_SP:5;
then ( ( not {x1} \/ {x2} in C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies {x1} = {x2} ) ) ) by A3, A5, A8, Th48;
then ( ( not {x1,x2} in C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies x1 = x2 ) ) ) by ENUMSET1:1, ZFMISC_1:3;
hence ( x1 in union C1 & x2 in union C1 & ( ( not [x1,x2] in Web C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies x1 = x2 ) ) ) ) by A3, A5, A8, A6, COH_SP:5, TARSKI:def_4; ::_thesis: verum
end;
assume ( x1 in union C1 & x2 in union C1 ) ; ::_thesis: ( ( not ( not [x1,x2] in Web C1 & y1 in union C2 & y2 in union C2 ) & not ( [y1,y2] in Web C2 & ( y1 = y2 implies x1 = x2 ) ) ) or [[x1,y1],[x2,y2]] in Web (LinCoh (C1,C2)) )
then reconsider a = {x1}, b = {x2} as Element of C1 by COH_SP:4;
assume ( ( not [x1,x2] in Web C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies x1 = x2 ) ) ) ; ::_thesis: [[x1,y1],[x2,y2]] in Web (LinCoh (C1,C2))
then ( ( not {x1,x2} in C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies a = b ) ) ) by COH_SP:5;
then ( ( not a \/ b in C1 & y1 in union C2 & y2 in union C2 ) or ( [y1,y2] in Web C2 & ( y1 = y2 implies a = b ) ) ) by ENUMSET1:1;
then [[a,y1],[b,y2]] in Web (StabCoh (C1,C2)) by Th48;
then {[a,y1],[b,y2]} in StabCoh (C1,C2) by COH_SP:5;
then consider f being U-stable Function of C1,C2 such that
A9: {[a,y1],[b,y2]} = Trace f by Def18;
now__::_thesis:_for_a1,_y_being_set_st_[a1,y]_in_Trace_f_holds_
ex_x_being_set_st_a1_=_{x}
let a1, y be set ; ::_thesis: ( [a1,y] in Trace f implies ex x being set st a1 = {x} )
assume [a1,y] in Trace f ; ::_thesis: ex x being set st a1 = {x}
then ( [a1,y] = [a,y1] or [a1,y] = [b,y2] ) by A9, TARSKI:def_2;
then ( a1 = {x1} or a1 = {x2} ) by XTUPLE_0:1;
hence ex x being set st a1 = {x} ; ::_thesis: verum
end;
then f is U-linear by Th49;
then A10: LinTrace f in LinCoh (C1,C2) by Def20;
{[x1,y1],[x2,y2]} c= LinTrace f
proof
let x, y be set ; :: according to RELAT_1:def_3 ::_thesis: ( not [x,y] in {[x1,y1],[x2,y2]} or [x,y] in LinTrace f )
assume [x,y] in {[x1,y1],[x2,y2]} ; ::_thesis: [x,y] in LinTrace f
then ( ( [x,y] = [x1,y1] & [a,y1] in Trace f ) or ( [x,y] = [x2,y2] & [b,y2] in Trace f ) ) by A9, TARSKI:def_2;
hence [x,y] in LinTrace f by Th50; ::_thesis: verum
end;
then {[x1,y1],[x2,y2]} in LinCoh (C1,C2) by A10, CLASSES1:def_1;
hence [[x1,y1],[x2,y2]] in Web (LinCoh (C1,C2)) by COH_SP:5; ::_thesis: verum
end;
begin
definition
let C be Coherence_Space;
func 'not' C -> set equals :: COHSP_1:def 21
{ a where a is Subset of (union C) : for b being Element of C ex x being set st a /\ b c= {x} } ;
correctness
coherence
{ a where a is Subset of (union C) : for b being Element of C ex x being set st a /\ b c= {x} } is set ;
;
end;
:: deftheorem defines 'not' COHSP_1:def_21_:_
for C being Coherence_Space holds 'not' C = { a where a is Subset of (union C) : for b being Element of C ex x being set st a /\ b c= {x} } ;
theorem Th65: :: COHSP_1:65
for C being Coherence_Space
for x being set holds
( x in 'not' C iff ( x c= union C & ( for a being Element of C ex z being set st x /\ a c= {z} ) ) )
proof
let C be Coherence_Space; ::_thesis: for x being set holds
( x in 'not' C iff ( x c= union C & ( for a being Element of C ex z being set st x /\ a c= {z} ) ) )
let x be set ; ::_thesis: ( x in 'not' C iff ( x c= union C & ( for a being Element of C ex z being set st x /\ a c= {z} ) ) )
( x in 'not' C iff ex X being Subset of (union C) st
( x = X & ( for a being Element of C ex z being set st X /\ a c= {z} ) ) ) ;
hence ( x in 'not' C iff ( x c= union C & ( for a being Element of C ex z being set st x /\ a c= {z} ) ) ) ; ::_thesis: verum
end;
registration
let C be Coherence_Space;
cluster 'not' C -> non empty subset-closed binary_complete ;
coherence
( not 'not' C is empty & 'not' C is subset-closed & 'not' C is binary_complete )
proof
reconsider a = {} as Subset of (union C) by XBOOLE_1:2;
now__::_thesis:_for_b_being_Element_of_C_ex_x_being_set_st_a_/\_b_c=_{x}
let b be Element of C; ::_thesis: ex x being set st a /\ b c= {x}
take x = {} ; ::_thesis: a /\ b c= {x}
thus a /\ b c= {x} by XBOOLE_1:2; ::_thesis: verum
end;
then a in 'not' C ;
hence not 'not' C is empty ; ::_thesis: ( 'not' C is subset-closed & 'not' C is binary_complete )
hereby :: according to CLASSES1:def_1 ::_thesis: 'not' C is binary_complete
let a, b be set ; ::_thesis: ( a in 'not' C & b c= a implies b in 'not' C )
assume a in 'not' C ; ::_thesis: ( b c= a implies b in 'not' C )
then consider aa being Subset of (union C) such that
A1: a = aa and
A2: for b being Element of C ex x being set st aa /\ b c= {x} ;
assume A3: b c= a ; ::_thesis: b in 'not' C
then reconsider bb = b as Subset of (union C) by A1, XBOOLE_1:1;
now__::_thesis:_for_c_being_Element_of_C_ex_x_being_set_st_bb_/\_c_c=_{x}
let c be Element of C; ::_thesis: ex x being set st bb /\ c c= {x}
consider x being set such that
A4: aa /\ c c= {x} by A2;
take x = x; ::_thesis: bb /\ c c= {x}
b /\ c c= a /\ c by A3, XBOOLE_1:26;
hence bb /\ c c= {x} by A1, A4, XBOOLE_1:1; ::_thesis: verum
end;
hence b in 'not' C ; ::_thesis: verum
end;
let A be set ; :: according to COHSP_1:def_1 ::_thesis: ( ( for a, b being set st a in A & b in A holds
a \/ b in 'not' C ) implies union A in 'not' C )
assume A5: for a, b being set st a in A & b in A holds
a \/ b in 'not' C ; ::_thesis: union A in 'not' C
A c= bool (union C)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in bool (union C) )
assume x in A ; ::_thesis: x in bool (union C)
then x \/ x in 'not' C by A5;
then ex a being Subset of (union C) st
( x = a & ( for b being Element of C ex x being set st a /\ b c= {x} ) ) ;
hence x in bool (union C) ; ::_thesis: verum
end;
then union A c= union (bool (union C)) by ZFMISC_1:77;
then reconsider a = union A as Subset of (union C) by ZFMISC_1:81;
now__::_thesis:_for_c_being_Element_of_C_ex_x_being_set_st_a_/\_c_c=_{x}
let c be Element of C; ::_thesis: ex x being set st a /\ x c= {b2}
percases ( a /\ c = {} or a /\ c <> {} ) ;
supposeA6: a /\ c = {} ; ::_thesis: ex x being set st a /\ x c= {b2}
take x = {} ; ::_thesis: a /\ c c= {x}
thus a /\ c c= {x} by A6, XBOOLE_1:2; ::_thesis: verum
end;
suppose a /\ c <> {} ; ::_thesis: ex y being set st a /\ y c= {b2}
then reconsider X = a /\ c as non empty set ;
set x = the Element of X;
reconsider y = the Element of X as set ;
take y = y; ::_thesis: a /\ c c= {y}
thus a /\ c c= {y} ::_thesis: verum
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in a /\ c or z in {y} )
assume A7: z in a /\ c ; ::_thesis: z in {y}
then A8: z in c by XBOOLE_0:def_4;
the Element of X in a by XBOOLE_0:def_4;
then consider w being set such that
A9: the Element of X in w and
A10: w in A by TARSKI:def_4;
z in a by A7, XBOOLE_0:def_4;
then consider v being set such that
A11: z in v and
A12: v in A by TARSKI:def_4;
w \/ v in 'not' C by A5, A12, A10;
then consider aa being Subset of (union C) such that
A13: w \/ v = aa and
A14: for b being Element of C ex x being set st aa /\ b c= {x} ;
consider t being set such that
A15: aa /\ c c= {t} by A14;
( the Element of X in c & the Element of X in aa ) by A9, A13, XBOOLE_0:def_3, XBOOLE_0:def_4;
then A16: the Element of X in aa /\ c by XBOOLE_0:def_4;
z in aa by A11, A13, XBOOLE_0:def_3;
then z in aa /\ c by A8, XBOOLE_0:def_4;
then z in {t} by A15;
hence z in {y} by A15, A16, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
end;
hence union A in 'not' C ; ::_thesis: verum
end;
end;
theorem Th66: :: COHSP_1:66
for C being Coherence_Space holds union ('not' C) = union C
proof
let C be Coherence_Space; ::_thesis: union ('not' C) = union C
hereby :: according to XBOOLE_0:def_10,TARSKI:def_3 ::_thesis: union C c= union ('not' C)
let x be set ; ::_thesis: ( x in union ('not' C) implies x in union C )
assume x in union ('not' C) ; ::_thesis: x in union C
then consider a being set such that
A1: x in a and
A2: a in 'not' C by TARSKI:def_4;
a c= union C by A2, Th65;
hence x in union C by A1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union C or x in union ('not' C) )
assume x in union C ; ::_thesis: x in union ('not' C)
then A3: {x} c= union C by ZFMISC_1:31;
for a being Element of C ex z being set st {x} /\ a c= {z} by XBOOLE_1:17;
then ( x in {x} & {x} in 'not' C ) by A3, ZFMISC_1:31;
hence x in union ('not' C) by TARSKI:def_4; ::_thesis: verum
end;
theorem Th67: :: COHSP_1:67
for C being Coherence_Space
for x, y being set st x <> y & {x,y} in C holds
not {x,y} in 'not' C
proof
let C be Coherence_Space; ::_thesis: for x, y being set st x <> y & {x,y} in C holds
not {x,y} in 'not' C
let x, y be set ; ::_thesis: ( x <> y & {x,y} in C implies not {x,y} in 'not' C )
assume that
A1: x <> y and
A2: ( {x,y} in C & {x,y} in 'not' C ) ; ::_thesis: contradiction
consider z being set such that
A3: {x,y} /\ {x,y} c= {z} by A2, Th65;
x = z by A3, ZFMISC_1:20;
hence contradiction by A1, A3, ZFMISC_1:20; ::_thesis: verum
end;
theorem Th68: :: COHSP_1:68
for C being Coherence_Space
for x, y being set st {x,y} c= union C & not {x,y} in C holds
{x,y} in 'not' C
proof
let C be Coherence_Space; ::_thesis: for x, y being set st {x,y} c= union C & not {x,y} in C holds
{x,y} in 'not' C
let x, y be set ; ::_thesis: ( {x,y} c= union C & not {x,y} in C implies {x,y} in 'not' C )
assume that
A1: {x,y} c= union C and
A2: not {x,y} in C ; ::_thesis: {x,y} in 'not' C
now__::_thesis:_for_a_being_Element_of_C_ex_z_being_set_st_{x,y}_/\_a_c=_{z}
let a be Element of C; ::_thesis: ex z being set st {x,y} /\ a c= {z}
( x in a or not x in a ) ;
then consider z being set such that
A3: ( ( x in a & z = x ) or ( not x in a & z = y ) ) ;
take z = z; ::_thesis: {x,y} /\ a c= {z}
thus {x,y} /\ a c= {z} ::_thesis: verum
proof
let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in {x,y} /\ a or v in {z} )
assume A4: v in {x,y} /\ a ; ::_thesis: v in {z}
then A5: v in {x,y} by XBOOLE_0:def_4;
A6: v in a by A4, XBOOLE_0:def_4;
percases ( v = x or v = y ) by A5, TARSKI:def_2;
suppose v = x ; ::_thesis: v in {z}
hence v in {z} by A3, A4, TARSKI:def_1, XBOOLE_0:def_4; ::_thesis: verum
end;
supposeA7: v = y ; ::_thesis: v in {z}
then ( x in a implies {x,y} c= a ) by A6, ZFMISC_1:32;
hence v in {z} by A2, A3, A7, CLASSES1:def_1, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
end;
hence {x,y} in 'not' C by A1; ::_thesis: verum
end;
theorem :: COHSP_1:69
for C being Coherence_Space
for x, y being set holds
( [x,y] in Web ('not' C) iff ( x in union C & y in union C & ( x = y or not [x,y] in Web C ) ) )
proof
let C be Coherence_Space; ::_thesis: for x, y being set holds
( [x,y] in Web ('not' C) iff ( x in union C & y in union C & ( x = y or not [x,y] in Web C ) ) )
let x, y be set ; ::_thesis: ( [x,y] in Web ('not' C) iff ( x in union C & y in union C & ( x = y or not [x,y] in Web C ) ) )
A1: ( {x,y} c= union C & not {x,y} in C implies {x,y} in 'not' C ) by Th68;
A2: union ('not' C) = union C by Th66;
( x <> y & {x,y} in C implies not {x,y} in 'not' C ) by Th67;
hence ( [x,y] in Web ('not' C) implies ( x in union C & y in union C & ( x = y or not [x,y] in Web C ) ) ) by A2, COH_SP:5, ZFMISC_1:87; ::_thesis: ( x in union C & y in union C & ( x = y or not [x,y] in Web C ) implies [x,y] in Web ('not' C) )
assume that
A3: x in union C and
A4: y in union C and
A5: ( x = y or not [x,y] in Web C ) ; ::_thesis: [x,y] in Web ('not' C)
( ( x = y & {x} in 'not' C & {x} = {x,y} ) or not {x,y} in C ) by A2, A3, A5, COH_SP:4, COH_SP:5, ENUMSET1:29;
hence [x,y] in Web ('not' C) by A1, A3, A4, COH_SP:5, ZFMISC_1:32; ::_thesis: verum
end;
Lm7: for C being Coherence_Space holds 'not' ('not' C) c= C
proof
let C be Coherence_Space; ::_thesis: 'not' ('not' C) c= C
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in 'not' ('not' C) or x in C )
assume x in 'not' ('not' C) ; ::_thesis: x in C
then consider a being Subset of (union ('not' C)) such that
A1: x = a and
A2: for b being Element of 'not' C ex x being set st a /\ b c= {x} ;
A3: union ('not' C) = union C by Th66;
now__::_thesis:_for_x,_y_being_set_st_x_in_a_&_y_in_a_holds_
{x,y}_in_C
let x, y be set ; ::_thesis: ( x in a & y in a implies {x,y} in C )
assume that
A4: x in a and
A5: y in a and
A6: not {x,y} in C ; ::_thesis: contradiction
{x,y} c= union C by A3, A4, A5, ZFMISC_1:32;
then {x,y} in 'not' C by A6, Th68;
then consider z being set such that
A7: a /\ {x,y} c= {z} by A2;
y in {x,y} by TARSKI:def_2;
then y in a /\ {x,y} by A5, XBOOLE_0:def_4;
then A8: y = z by A7, TARSKI:def_1;
x in {x,y} by TARSKI:def_2;
then x in a /\ {x,y} by A4, XBOOLE_0:def_4;
then x = z by A7, TARSKI:def_1;
then {x,y} = {x} by A8, ENUMSET1:29;
hence contradiction by A3, A4, A6, COH_SP:4; ::_thesis: verum
end;
hence x in C by A1, COH_SP:6; ::_thesis: verum
end;
theorem Th70: :: COHSP_1:70
for C being Coherence_Space holds 'not' ('not' C) = C
proof
let C be Coherence_Space; ::_thesis: 'not' ('not' C) = C
thus 'not' ('not' C) c= C by Lm7; :: according to XBOOLE_0:def_10 ::_thesis: C c= 'not' ('not' C)
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in C or a in 'not' ('not' C) )
assume A1: a in C ; ::_thesis: a in 'not' ('not' C)
A2: ( union C = union ('not' C) & union ('not' C) = union ('not' ('not' C)) ) by Th66;
now__::_thesis:_for_x,_y_being_set_st_x_in_a_&_y_in_a_holds_
{x,y}_in_'not'_('not'_C)
let x, y be set ; ::_thesis: ( x in a & y in a implies {x,y} in 'not' ('not' C) )
assume that
A3: x in a and
A4: y in a ; ::_thesis: {x,y} in 'not' ('not' C)
A5: x in union C by A1, A3, TARSKI:def_4;
{x,y} c= a by A3, A4, ZFMISC_1:32;
then {x,y} in C by A1, CLASSES1:def_1;
then A6: ( x <> y implies not {x,y} in 'not' C ) by Th67;
y in union C by A1, A4, TARSKI:def_4;
then A7: {x,y} c= union C by A5, ZFMISC_1:32;
{x,x} = {x} by ENUMSET1:29;
hence {x,y} in 'not' ('not' C) by A2, A5, A7, A6, Th68, COH_SP:4; ::_thesis: verum
end;
hence a in 'not' ('not' C) by COH_SP:6; ::_thesis: verum
end;
theorem :: COHSP_1:71
'not' {{}} = {{}}
proof
union ('not' {{}}) = union {{}} by Th66
.= {} by ZFMISC_1:25 ;
hence 'not' {{}} c= {{}} by ZFMISC_1:1, ZFMISC_1:82; :: according to XBOOLE_0:def_10 ::_thesis: {{}} c= 'not' {{}}
{} in 'not' {{}} by COH_SP:1;
hence {{}} c= 'not' {{}} by ZFMISC_1:31; ::_thesis: verum
end;
theorem :: COHSP_1:72
for X being set holds
( 'not' (FlatCoh X) = bool X & 'not' (bool X) = FlatCoh X )
proof
let X be set ; ::_thesis: ( 'not' (FlatCoh X) = bool X & 'not' (bool X) = FlatCoh X )
thus 'not' (FlatCoh X) = bool X ::_thesis: 'not' (bool X) = FlatCoh X
proof
hereby :: according to XBOOLE_0:def_10,TARSKI:def_3 ::_thesis: bool X c= 'not' (FlatCoh X)
let x be set ; ::_thesis: ( x in 'not' (FlatCoh X) implies x in bool X )
assume x in 'not' (FlatCoh X) ; ::_thesis: x in bool X
then x c= union (FlatCoh X) by Th65;
then x c= X by Th2;
hence x in bool X ; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in bool X or x in 'not' (FlatCoh X) )
A1: now__::_thesis:_for_a_being_Element_of_FlatCoh_X_ex_z_being_set_st_x_/\_a_c=_{z}
let a be Element of FlatCoh X; ::_thesis: ex z being set st x /\ z c= {b2}
percases ( a = {} or ex z being set st
( a = {z} & z in X ) ) by Th1;
suppose a = {} ; ::_thesis: ex z being set st x /\ z c= {b2}
then x /\ a c= {0} by XBOOLE_1:2;
hence ex z being set st x /\ a c= {z} ; ::_thesis: verum
end;
suppose ex z being set st
( a = {z} & z in X ) ; ::_thesis: ex z being set st x /\ z c= {b2}
then consider z being set such that
A2: a = {z} and
z in X ;
take z = z; ::_thesis: x /\ a c= {z}
thus x /\ a c= {z} by A2, XBOOLE_1:17; ::_thesis: verum
end;
end;
end;
assume x in bool X ; ::_thesis: x in 'not' (FlatCoh X)
then x c= X ;
then x c= union (FlatCoh X) by Th2;
hence x in 'not' (FlatCoh X) by A1; ::_thesis: verum
end;
hence 'not' (bool X) = FlatCoh X by Th70; ::_thesis: verum
end;
begin
definition
let x, y be set ;
funcx U+ y -> set equals :: COHSP_1:def 22
Union (disjoin <*x,y*>);
correctness
coherence
Union (disjoin <*x,y*>) is set ;
;
end;
:: deftheorem defines U+ COHSP_1:def_22_:_
for x, y being set holds x U+ y = Union (disjoin <*x,y*>);
theorem Th73: :: COHSP_1:73
for x, y being set holds x U+ y = [:x,{1}:] \/ [:y,{2}:]
proof
let x, y be set ; ::_thesis: x U+ y = [:x,{1}:] \/ [:y,{2}:]
len <*x,y*> = 2 by FINSEQ_1:44;
then A1: dom <*x,y*> = {1,2} by FINSEQ_1:2, FINSEQ_1:def_3;
now__::_thesis:_for_z_being_set_holds_
(_z_in_x_U+_y_iff_z_in_[:x,{1}:]_\/_[:y,{2}:]_)
let z be set ; ::_thesis: ( z in x U+ y iff z in [:x,{1}:] \/ [:y,{2}:] )
A2: ( z `2 in {1,2} iff ( z `2 = 1 or z `2 = 2 ) ) by TARSKI:def_2;
A3: ( z in [:x,{1}:] iff ( z `1 in x & z `2 = 1 & z = [(z `1),(z `2)] ) ) by MCART_1:13, MCART_1:21, ZFMISC_1:106;
A4: ( z in [:y,{2}:] iff ( z `1 in y & z `2 = 2 & z = [(z `1),(z `2)] ) ) by MCART_1:13, MCART_1:21, ZFMISC_1:106;
( z in x U+ y iff ( z `2 in {1,2} & z `1 in <*x,y*> . (z `2) & z = [(z `1),(z `2)] ) ) by A1, CARD_3:22;
hence ( z in x U+ y iff z in [:x,{1}:] \/ [:y,{2}:] ) by A2, A3, A4, FINSEQ_1:44, XBOOLE_0:def_3; ::_thesis: verum
end;
hence x U+ y = [:x,{1}:] \/ [:y,{2}:] by TARSKI:1; ::_thesis: verum
end;
theorem Th74: :: COHSP_1:74
for x being set holds
( x U+ {} = [:x,{1}:] & {} U+ x = [:x,{2}:] )
proof
let x be set ; ::_thesis: ( x U+ {} = [:x,{1}:] & {} U+ x = [:x,{2}:] )
thus x U+ {} = [:x,{1}:] \/ [:{},{2}:] by Th73
.= [:x,{1}:] \/ {} by ZFMISC_1:90
.= [:x,{1}:] ; ::_thesis: {} U+ x = [:x,{2}:]
thus {} U+ x = [:{},{1}:] \/ [:x,{2}:] by Th73
.= {} \/ [:x,{2}:] by ZFMISC_1:90
.= [:x,{2}:] ; ::_thesis: verum
end;
theorem Th75: :: COHSP_1:75
for x, y, z being set st z in x U+ y holds
( z = [(z `1),(z `2)] & ( ( z `2 = 1 & z `1 in x ) or ( z `2 = 2 & z `1 in y ) ) )
proof
let x, y, z be set ; ::_thesis: ( z in x U+ y implies ( z = [(z `1),(z `2)] & ( ( z `2 = 1 & z `1 in x ) or ( z `2 = 2 & z `1 in y ) ) ) )
assume z in x U+ y ; ::_thesis: ( z = [(z `1),(z `2)] & ( ( z `2 = 1 & z `1 in x ) or ( z `2 = 2 & z `1 in y ) ) )
then z in [:x,{1}:] \/ [:y,{2}:] by Th73;
then ( z in [:x,{1}:] or z in [:y,{2}:] ) by XBOOLE_0:def_3;
hence ( z = [(z `1),(z `2)] & ( ( z `2 = 1 & z `1 in x ) or ( z `2 = 2 & z `1 in y ) ) ) by MCART_1:13, MCART_1:21; ::_thesis: verum
end;
theorem Th76: :: COHSP_1:76
for x, y, z being set holds
( [z,1] in x U+ y iff z in x )
proof
let x, y, z be set ; ::_thesis: ( [z,1] in x U+ y iff z in x )
x U+ y = [:x,{1}:] \/ [:y,{2}:] by Th73;
then ( [z,1] in x U+ y iff ( [z,1] in [:x,{1}:] or ( [z,1] in [:y,{2}:] & 1 <> 2 ) ) ) by XBOOLE_0:def_3;
hence ( [z,1] in x U+ y iff z in x ) by ZFMISC_1:106; ::_thesis: verum
end;
theorem Th77: :: COHSP_1:77
for x, y, z being set holds
( [z,2] in x U+ y iff z in y )
proof
let x, y, z be set ; ::_thesis: ( [z,2] in x U+ y iff z in y )
x U+ y = [:x,{1}:] \/ [:y,{2}:] by Th73;
then ( [z,2] in x U+ y iff ( ( [z,2] in [:x,{1}:] & 1 <> 2 ) or [z,2] in [:y,{2}:] ) ) by XBOOLE_0:def_3;
hence ( [z,2] in x U+ y iff z in y ) by ZFMISC_1:106; ::_thesis: verum
end;
theorem Th78: :: COHSP_1:78
for x1, y1, x2, y2 being set holds
( x1 U+ y1 c= x2 U+ y2 iff ( x1 c= x2 & y1 c= y2 ) )
proof
let x1, y1, x2, y2 be set ; ::_thesis: ( x1 U+ y1 c= x2 U+ y2 iff ( x1 c= x2 & y1 c= y2 ) )
hereby ::_thesis: ( x1 c= x2 & y1 c= y2 implies x1 U+ y1 c= x2 U+ y2 )
assume A1: x1 U+ y1 c= x2 U+ y2 ; ::_thesis: ( x1 c= x2 & y1 c= y2 )
thus x1 c= x2 ::_thesis: y1 c= y2
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in x1 or a in x2 )
assume a in x1 ; ::_thesis: a in x2
then [a,1] in x1 U+ y1 by Th76;
hence a in x2 by A1, Th76; ::_thesis: verum
end;
thus y1 c= y2 ::_thesis: verum
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in y1 or a in y2 )
assume a in y1 ; ::_thesis: a in y2
then [a,2] in x1 U+ y1 by Th77;
hence a in y2 by A1, Th77; ::_thesis: verum
end;
end;
assume A2: ( x1 c= x2 & y1 c= y2 ) ; ::_thesis: x1 U+ y1 c= x2 U+ y2
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in x1 U+ y1 or a in x2 U+ y2 )
assume A3: a in x1 U+ y1 ; ::_thesis: a in x2 U+ y2
then A4: ( ( a `2 = 1 & a `1 in x1 ) or ( a `2 = 2 & a `1 in y1 ) ) by Th75;
a = [(a `1),(a `2)] by A3, Th75;
hence a in x2 U+ y2 by A2, A4, Th76, Th77; ::_thesis: verum
end;
theorem Th79: :: COHSP_1:79
for x, y, z being set st z c= x U+ y holds
ex x1, y1 being set st
( z = x1 U+ y1 & x1 c= x & y1 c= y )
proof
let x, y, z be set ; ::_thesis: ( z c= x U+ y implies ex x1, y1 being set st
( z = x1 U+ y1 & x1 c= x & y1 c= y ) )
assume A1: z c= x U+ y ; ::_thesis: ex x1, y1 being set st
( z = x1 U+ y1 & x1 c= x & y1 c= y )
take x1 = proj1 (z /\ [:x,{1}:]); ::_thesis: ex y1 being set st
( z = x1 U+ y1 & x1 c= x & y1 c= y )
take y1 = proj1 (z /\ [:y,{2}:]); ::_thesis: ( z = x1 U+ y1 & x1 c= x & y1 c= y )
A2: x U+ y = [:x,{1}:] \/ [:y,{2}:] by Th73;
thus z = x1 U+ y1 ::_thesis: ( x1 c= x & y1 c= y )
proof
hereby :: according to XBOOLE_0:def_10,TARSKI:def_3 ::_thesis: x1 U+ y1 c= z
let a be set ; ::_thesis: ( a in z implies a in x1 U+ y1 )
assume A3: a in z ; ::_thesis: a in x1 U+ y1
then A4: a = [(a `1),(a `2)] by A1, Th75;
( a in [:x,{1}:] or a in [:y,{2}:] ) by A1, A2, A3, XBOOLE_0:def_3;
then ( ( a in z /\ [:x,{1}:] & a `2 = 1 ) or ( a in z /\ [:y,{2}:] & a `2 = 2 ) ) by A3, A4, XBOOLE_0:def_4, ZFMISC_1:106;
then ( ( a `1 in x1 & a `2 = 1 ) or ( a `1 in y1 & a `2 = 2 ) ) by A4, XTUPLE_0:def_12;
hence a in x1 U+ y1 by A4, Th76, Th77; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in x1 U+ y1 or a in z )
assume A5: a in x1 U+ y1 ; ::_thesis: a in z
then A6: a = [(a `1),(a `2)] by Th75;
percases ( ( a `2 = 1 & a `1 in x1 ) or ( a `2 = 2 & a `1 in y1 ) ) by A5, Th75;
supposeA7: ( a `2 = 1 & a `1 in x1 ) ; ::_thesis: a in z
then consider b being set such that
A8: [(a `1),b] in z /\ [:x,{1}:] by XTUPLE_0:def_12;
( [(a `1),b] in z & [(a `1),b] in [:x,{1}:] ) by A8, XBOOLE_0:def_4;
hence a in z by A6, A7, ZFMISC_1:106; ::_thesis: verum
end;
supposeA9: ( a `2 = 2 & a `1 in y1 ) ; ::_thesis: a in z
then consider b being set such that
A10: [(a `1),b] in z /\ [:y,{2}:] by XTUPLE_0:def_12;
( [(a `1),b] in z & [(a `1),b] in [:y,{2}:] ) by A10, XBOOLE_0:def_4;
hence a in z by A6, A9, ZFMISC_1:106; ::_thesis: verum
end;
end;
end;
z /\ [:y,{2}:] c= [:y,{2}:] by XBOOLE_1:17;
then A11: y1 c= proj1 [:y,{2}:] by XTUPLE_0:8;
z /\ [:x,{1}:] c= [:x,{1}:] by XBOOLE_1:17;
then x1 c= proj1 [:x,{1}:] by XTUPLE_0:8;
hence ( x1 c= x & y1 c= y ) by A11, FUNCT_5:9; ::_thesis: verum
end;
theorem Th80: :: COHSP_1:80
for x1, y1, x2, y2 being set holds
( x1 U+ y1 = x2 U+ y2 iff ( x1 = x2 & y1 = y2 ) )
proof
let x1, y1, x2, y2 be set ; ::_thesis: ( x1 U+ y1 = x2 U+ y2 iff ( x1 = x2 & y1 = y2 ) )
A1: ( x1 U+ y1 c= x2 U+ y2 iff ( x1 c= x2 & y1 c= y2 ) ) by Th78;
( x2 U+ y2 c= x1 U+ y1 iff ( x2 c= x1 & y2 c= y1 ) ) by Th78;
hence ( x1 U+ y1 = x2 U+ y2 iff ( x1 = x2 & y1 = y2 ) ) by A1, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th81: :: COHSP_1:81
for x1, y1, x2, y2 being set holds (x1 U+ y1) \/ (x2 U+ y2) = (x1 \/ x2) U+ (y1 \/ y2)
proof
let x1, y1, x2, y2 be set ; ::_thesis: (x1 U+ y1) \/ (x2 U+ y2) = (x1 \/ x2) U+ (y1 \/ y2)
set X1 = [:x1,{1}:];
set X2 = [:x2,{1}:];
set Y1 = [:y1,{2}:];
set Y2 = [:y2,{2}:];
set X = [:(x1 \/ x2),{1}:];
set Y = [:(y1 \/ y2),{2}:];
A1: [:(x1 \/ x2),{1}:] = [:x1,{1}:] \/ [:x2,{1}:] by ZFMISC_1:97;
A2: ( (x1 \/ x2) U+ (y1 \/ y2) = [:(x1 \/ x2),{1}:] \/ [:(y1 \/ y2),{2}:] & [:(y1 \/ y2),{2}:] = [:y1,{2}:] \/ [:y2,{2}:] ) by Th73, ZFMISC_1:97;
( x1 U+ y1 = [:x1,{1}:] \/ [:y1,{2}:] & x2 U+ y2 = [:x2,{1}:] \/ [:y2,{2}:] ) by Th73;
hence (x1 U+ y1) \/ (x2 U+ y2) = (([:x1,{1}:] \/ [:y1,{2}:]) \/ [:x2,{1}:]) \/ [:y2,{2}:] by XBOOLE_1:4
.= ([:(x1 \/ x2),{1}:] \/ [:y1,{2}:]) \/ [:y2,{2}:] by A1, XBOOLE_1:4
.= (x1 \/ x2) U+ (y1 \/ y2) by A2, XBOOLE_1:4 ;
::_thesis: verum
end;
theorem Th82: :: COHSP_1:82
for x1, y1, x2, y2 being set holds (x1 U+ y1) /\ (x2 U+ y2) = (x1 /\ x2) U+ (y1 /\ y2)
proof
let x1, y1, x2, y2 be set ; ::_thesis: (x1 U+ y1) /\ (x2 U+ y2) = (x1 /\ x2) U+ (y1 /\ y2)
set X1 = [:x1,{1}:];
set X2 = [:x2,{1}:];
set Y1 = [:y1,{2}:];
set Y2 = [:y2,{2}:];
set X = [:(x1 /\ x2),{1}:];
set Y = [:(y1 /\ y2),{2}:];
A1: [:(x1 /\ x2),{1}:] = [:x1,{1}:] /\ [:x2,{1}:] by ZFMISC_1:99;
A2: {1} misses {2} by ZFMISC_1:11;
then [:y1,{2}:] misses [:x2,{1}:] by ZFMISC_1:104;
then A3: ( [:(y1 /\ y2),{2}:] = [:y1,{2}:] /\ [:y2,{2}:] & [:y1,{2}:] /\ [:x2,{1}:] = {} ) by XBOOLE_0:def_7, ZFMISC_1:99;
[:x1,{1}:] misses [:y2,{2}:] by A2, ZFMISC_1:104;
then A4: [:x1,{1}:] /\ [:y2,{2}:] = {} by XBOOLE_0:def_7;
( x1 U+ y1 = [:x1,{1}:] \/ [:y1,{2}:] & x2 U+ y2 = [:x2,{1}:] \/ [:y2,{2}:] ) by Th73;
hence (x1 U+ y1) /\ (x2 U+ y2) = (([:x1,{1}:] \/ [:y1,{2}:]) /\ [:x2,{1}:]) \/ (([:x1,{1}:] \/ [:y1,{2}:]) /\ [:y2,{2}:]) by XBOOLE_1:23
.= ([:(x1 /\ x2),{1}:] \/ ([:y1,{2}:] /\ [:x2,{1}:])) \/ (([:x1,{1}:] \/ [:y1,{2}:]) /\ [:y2,{2}:]) by A1, XBOOLE_1:23
.= [:(x1 /\ x2),{1}:] \/ (([:x1,{1}:] /\ [:y2,{2}:]) \/ [:(y1 /\ y2),{2}:]) by A3, XBOOLE_1:23
.= (x1 /\ x2) U+ (y1 /\ y2) by A4, Th73 ;
::_thesis: verum
end;
definition
let C1, C2 be Coherence_Space;
funcC1 "/\" C2 -> set equals :: COHSP_1:def 23
{ (a U+ b) where a is Element of C1, b is Element of C2 : verum } ;
correctness
coherence
{ (a U+ b) where a is Element of C1, b is Element of C2 : verum } is set ;
;
funcC1 "\/" C2 -> set equals :: COHSP_1:def 24
{ (a U+ {}) where a is Element of C1 : verum } \/ { ({} U+ b) where b is Element of C2 : verum } ;
correctness
coherence
{ (a U+ {}) where a is Element of C1 : verum } \/ { ({} U+ b) where b is Element of C2 : verum } is set ;
;
end;
:: deftheorem defines "/\" COHSP_1:def_23_:_
for C1, C2 being Coherence_Space holds C1 "/\" C2 = { (a U+ b) where a is Element of C1, b is Element of C2 : verum } ;
:: deftheorem defines "\/" COHSP_1:def_24_:_
for C1, C2 being Coherence_Space holds C1 "\/" C2 = { (a U+ {}) where a is Element of C1 : verum } \/ { ({} U+ b) where b is Element of C2 : verum } ;
theorem Th83: :: COHSP_1:83
for C1, C2 being Coherence_Space
for x being set holds
( x in C1 "/\" C2 iff ex a being Element of C1 ex b being Element of C2 st x = a U+ b ) ;
theorem Th84: :: COHSP_1:84
for C1, C2 being Coherence_Space
for x, y being set holds
( x U+ y in C1 "/\" C2 iff ( x in C1 & y in C2 ) )
proof
let C1, C2 be Coherence_Space; ::_thesis: for x, y being set holds
( x U+ y in C1 "/\" C2 iff ( x in C1 & y in C2 ) )
let x, y be set ; ::_thesis: ( x U+ y in C1 "/\" C2 iff ( x in C1 & y in C2 ) )
now__::_thesis:_(_ex_a_being_Element_of_C1_ex_b_being_Element_of_C2_st_x_U+_y_=_a_U+_b_implies_ex_a_being_Element_of_C1_ex_b_being_Element_of_C2_st_
(_x_=_a_&_y_=_b_)_)
given a being Element of C1, b being Element of C2 such that A1: x U+ y = a U+ b ; ::_thesis: ex a being Element of C1 ex b being Element of C2 st
( x = a & y = b )
take a = a; ::_thesis: ex b being Element of C2 st
( x = a & y = b )
take b = b; ::_thesis: ( x = a & y = b )
thus ( x = a & y = b ) by A1, Th80; ::_thesis: verum
end;
hence ( x U+ y in C1 "/\" C2 iff ( x in C1 & y in C2 ) ) ; ::_thesis: verum
end;
theorem Th85: :: COHSP_1:85
for C1, C2 being Coherence_Space holds union (C1 "/\" C2) = (union C1) U+ (union C2)
proof
let C1, C2 be Coherence_Space; ::_thesis: union (C1 "/\" C2) = (union C1) U+ (union C2)
thus union (C1 "/\" C2) c= (union C1) U+ (union C2) :: according to XBOOLE_0:def_10 ::_thesis: (union C1) U+ (union C2) c= union (C1 "/\" C2)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union (C1 "/\" C2) or x in (union C1) U+ (union C2) )
assume x in union (C1 "/\" C2) ; ::_thesis: x in (union C1) U+ (union C2)
then consider a being set such that
A1: x in a and
A2: a in C1 "/\" C2 by TARSKI:def_4;
consider a1 being Element of C1, a2 being Element of C2 such that
A3: a = a1 U+ a2 by A2;
( a1 c= union C1 & a2 c= union C2 ) by ZFMISC_1:74;
then a c= (union C1) U+ (union C2) by A3, Th78;
hence x in (union C1) U+ (union C2) by A1; ::_thesis: verum
end;
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in (union C1) U+ (union C2) or z in union (C1 "/\" C2) )
assume A4: z in (union C1) U+ (union C2) ; ::_thesis: z in union (C1 "/\" C2)
then A5: z = [(z `1),(z `2)] by Th75;
percases ( ( z `2 = 1 & z `1 in union C1 ) or ( z `2 = 2 & z `1 in union C2 ) ) by A4, Th75;
supposeA6: ( z `2 = 1 & z `1 in union C1 ) ; ::_thesis: z in union (C1 "/\" C2)
set b = the Element of C2;
consider a being set such that
A7: z `1 in a and
A8: a in C1 by A6, TARSKI:def_4;
reconsider a = a as Element of C1 by A8;
A9: a U+ the Element of C2 in C1 "/\" C2 ;
z in a U+ the Element of C2 by A5, A6, A7, Th76;
hence z in union (C1 "/\" C2) by A9, TARSKI:def_4; ::_thesis: verum
end;
supposeA10: ( z `2 = 2 & z `1 in union C2 ) ; ::_thesis: z in union (C1 "/\" C2)
set b = the Element of C1;
consider a being set such that
A11: z `1 in a and
A12: a in C2 by A10, TARSKI:def_4;
reconsider a = a as Element of C2 by A12;
A13: the Element of C1 U+ a in C1 "/\" C2 ;
z in the Element of C1 U+ a by A5, A10, A11, Th77;
hence z in union (C1 "/\" C2) by A13, TARSKI:def_4; ::_thesis: verum
end;
end;
end;
theorem Th86: :: COHSP_1:86
for C1, C2 being Coherence_Space
for x, y being set holds
( x U+ y in C1 "\/" C2 iff ( ( x in C1 & y = {} ) or ( x = {} & y in C2 ) ) )
proof
let C1, C2 be Coherence_Space; ::_thesis: for x, y being set holds
( x U+ y in C1 "\/" C2 iff ( ( x in C1 & y = {} ) or ( x = {} & y in C2 ) ) )
let x, y be set ; ::_thesis: ( x U+ y in C1 "\/" C2 iff ( ( x in C1 & y = {} ) or ( x = {} & y in C2 ) ) )
A1: now__::_thesis:_(_ex_a_being_Element_of_C1_st_x_U+_y_=_a_U+_{}_implies_(_x_in_C1_&_y_=_{}_)_)
given a being Element of C1 such that A2: x U+ y = a U+ {} ; ::_thesis: ( x in C1 & y = {} )
x = a by A2, Th80;
hence ( x in C1 & y = {} ) by A2, Th80; ::_thesis: verum
end;
A3: now__::_thesis:_(_ex_a_being_Element_of_C2_st_x_U+_y_=_{}_U+_a_implies_(_y_in_C2_&_x_=_{}_)_)
given a being Element of C2 such that A4: x U+ y = {} U+ a ; ::_thesis: ( y in C2 & x = {} )
y = a by A4, Th80;
hence ( y in C2 & x = {} ) by A4, Th80; ::_thesis: verum
end;
( x U+ y in C1 "\/" C2 iff ( x U+ y in { (a U+ {}) where a is Element of C1 : verum } or x U+ y in { ({} U+ b) where b is Element of C2 : verum } ) ) by XBOOLE_0:def_3;
hence ( x U+ y in C1 "\/" C2 iff ( ( x in C1 & y = {} ) or ( x = {} & y in C2 ) ) ) by A1, A3; ::_thesis: verum
end;
theorem Th87: :: COHSP_1:87
for C1, C2 being Coherence_Space
for x being set st x in C1 "\/" C2 holds
ex a being Element of C1 ex b being Element of C2 st
( x = a U+ b & ( a = {} or b = {} ) )
proof
let C1, C2 be Coherence_Space; ::_thesis: for x being set st x in C1 "\/" C2 holds
ex a being Element of C1 ex b being Element of C2 st
( x = a U+ b & ( a = {} or b = {} ) )
let x be set ; ::_thesis: ( x in C1 "\/" C2 implies ex a being Element of C1 ex b being Element of C2 st
( x = a U+ b & ( a = {} or b = {} ) ) )
assume x in C1 "\/" C2 ; ::_thesis: ex a being Element of C1 ex b being Element of C2 st
( x = a U+ b & ( a = {} or b = {} ) )
then ( x in { (a U+ {}) where a is Element of C1 : verum } or x in { ({} U+ b) where b is Element of C2 : verum } ) by XBOOLE_0:def_3;
then ( ( {} in C2 & ex a being Element of C1 st x = a U+ {} ) or ( {} in C1 & ex b being Element of C2 st x = {} U+ b ) ) by COH_SP:1;
hence ex a being Element of C1 ex b being Element of C2 st
( x = a U+ b & ( a = {} or b = {} ) ) ; ::_thesis: verum
end;
theorem :: COHSP_1:88
for C1, C2 being Coherence_Space holds union (C1 "\/" C2) = (union C1) U+ (union C2)
proof
let C1, C2 be Coherence_Space; ::_thesis: union (C1 "\/" C2) = (union C1) U+ (union C2)
thus union (C1 "\/" C2) c= (union C1) U+ (union C2) :: according to XBOOLE_0:def_10 ::_thesis: (union C1) U+ (union C2) c= union (C1 "\/" C2)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union (C1 "\/" C2) or x in (union C1) U+ (union C2) )
assume x in union (C1 "\/" C2) ; ::_thesis: x in (union C1) U+ (union C2)
then consider a being set such that
A1: x in a and
A2: a in C1 "\/" C2 by TARSKI:def_4;
consider a1 being Element of C1, a2 being Element of C2 such that
A3: a = a1 U+ a2 and
( a1 = {} or a2 = {} ) by A2, Th87;
( a1 c= union C1 & a2 c= union C2 ) by ZFMISC_1:74;
then a c= (union C1) U+ (union C2) by A3, Th78;
hence x in (union C1) U+ (union C2) by A1; ::_thesis: verum
end;
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in (union C1) U+ (union C2) or z in union (C1 "\/" C2) )
assume A4: z in (union C1) U+ (union C2) ; ::_thesis: z in union (C1 "\/" C2)
then A5: z = [(z `1),(z `2)] by Th75;
percases ( ( z `2 = 1 & z `1 in union C1 ) or ( z `2 = 2 & z `1 in union C2 ) ) by A4, Th75;
supposeA6: ( z `2 = 1 & z `1 in union C1 ) ; ::_thesis: z in union (C1 "\/" C2)
reconsider b = {} as Element of C2 by COH_SP:1;
consider a being set such that
A7: z `1 in a and
A8: a in C1 by A6, TARSKI:def_4;
reconsider a = a as Element of C1 by A8;
A9: a U+ b in C1 "\/" C2 by Th86;
z in a U+ b by A5, A6, A7, Th76;
hence z in union (C1 "\/" C2) by A9, TARSKI:def_4; ::_thesis: verum
end;
supposeA10: ( z `2 = 2 & z `1 in union C2 ) ; ::_thesis: z in union (C1 "\/" C2)
reconsider b = {} as Element of C1 by COH_SP:1;
consider a being set such that
A11: z `1 in a and
A12: a in C2 by A10, TARSKI:def_4;
reconsider a = a as Element of C2 by A12;
A13: b U+ a in C1 "\/" C2 by Th86;
z in b U+ a by A5, A10, A11, Th77;
hence z in union (C1 "\/" C2) by A13, TARSKI:def_4; ::_thesis: verum
end;
end;
end;
registration
let C1, C2 be Coherence_Space;
clusterC1 "/\" C2 -> non empty subset-closed binary_complete ;
coherence
( not C1 "/\" C2 is empty & C1 "/\" C2 is subset-closed & C1 "/\" C2 is binary_complete )
proof
set a9 = the Element of C1;
set b9 = the Element of C2;
the Element of C1 U+ the Element of C2 in C1 "/\" C2 ;
hence not C1 "/\" C2 is empty ; ::_thesis: ( C1 "/\" C2 is subset-closed & C1 "/\" C2 is binary_complete )
hereby :: according to CLASSES1:def_1 ::_thesis: C1 "/\" C2 is binary_complete
let a, b be set ; ::_thesis: ( a in C1 "/\" C2 & b c= a implies b in C1 "/\" C2 )
assume a in C1 "/\" C2 ; ::_thesis: ( b c= a implies b in C1 "/\" C2 )
then consider aa being Element of C1, bb being Element of C2 such that
A1: a = aa U+ bb ;
assume b c= a ; ::_thesis: b in C1 "/\" C2
then consider x1, y1 being set such that
A2: b = x1 U+ y1 and
A3: ( x1 c= aa & y1 c= bb ) by A1, Th79;
( x1 in C1 & y1 in C2 ) by A3, CLASSES1:def_1;
hence b in C1 "/\" C2 by A2; ::_thesis: verum
end;
let A be set ; :: according to COHSP_1:def_1 ::_thesis: ( ( for a, b being set st a in A & b in A holds
a \/ b in C1 "/\" C2 ) implies union A in C1 "/\" C2 )
assume A4: for a, b being set st a in A & b in A holds
a \/ b in C1 "/\" C2 ; ::_thesis: union A in C1 "/\" C2
set A2 = { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ;
now__::_thesis:_for_x,_y_being_set_st_x_in__{__b_where_b_is_Element_of_C2_:_ex_a_being_Element_of_C1_st_a_U+_b_in_A__}__&_y_in__{__b_where_b_is_Element_of_C2_:_ex_a_being_Element_of_C1_st_a_U+_b_in_A__}__holds_
x_\/_y_in_C2
let x, y be set ; ::_thesis: ( x in { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } & y in { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } implies x \/ y in C2 )
assume x in { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ; ::_thesis: ( y in { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } implies x \/ y in C2 )
then consider ax being Element of C2 such that
A5: x = ax and
A6: ex b being Element of C1 st b U+ ax in A ;
consider bx being Element of C1 such that
A7: bx U+ ax in A by A6;
assume y in { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ; ::_thesis: x \/ y in C2
then consider ay being Element of C2 such that
A8: y = ay and
A9: ex b being Element of C1 st b U+ ay in A ;
consider by1 being Element of C1 such that
A10: by1 U+ ay in A by A9;
(bx U+ ax) \/ (by1 U+ ay) in C1 "/\" C2 by A4, A7, A10;
then (bx \/ by1) U+ (ax \/ ay) in C1 "/\" C2 by Th81;
hence x \/ y in C2 by A5, A8, Th84; ::_thesis: verum
end;
then A11: union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } in C2 by Def1;
set A1 = { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ;
A12: (union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) U+ (union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ) = union A
proof
hereby :: according to XBOOLE_0:def_10,TARSKI:def_3 ::_thesis: union A c= (union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) U+ (union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } )
let x be set ; ::_thesis: ( x in (union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) U+ (union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ) implies b1 in union A )
assume A13: x in (union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) U+ (union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ) ; ::_thesis: b1 in union A
then A14: x = [(x `1),(x `2)] by Th75;
percases ( ( x `2 = 1 & x `1 in union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) or ( x `2 = 2 & x `1 in union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ) ) by A13, Th75;
supposeA15: ( x `2 = 1 & x `1 in union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) ; ::_thesis: b1 in union A
then consider a being set such that
A16: x `1 in a and
A17: a in { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } by TARSKI:def_4;
consider ax being Element of C1 such that
A18: a = ax and
A19: ex b being Element of C2 st ax U+ b in A by A17;
consider bx being Element of C2 such that
A20: ax U+ bx in A by A19;
x in ax U+ bx by A14, A15, A16, A18, Th76;
hence x in union A by A20, TARSKI:def_4; ::_thesis: verum
end;
supposeA21: ( x `2 = 2 & x `1 in union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ) ; ::_thesis: b1 in union A
then consider a being set such that
A22: x `1 in a and
A23: a in { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } by TARSKI:def_4;
consider bx being Element of C2 such that
A24: a = bx and
A25: ex a being Element of C1 st a U+ bx in A by A23;
consider ax being Element of C1 such that
A26: ax U+ bx in A by A25;
x in ax U+ bx by A14, A21, A22, A24, Th77;
hence x in union A by A26, TARSKI:def_4; ::_thesis: verum
end;
end;
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union A or x in (union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) U+ (union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ) )
assume x in union A ; ::_thesis: x in (union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) U+ (union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } )
then consider a being set such that
A27: x in a and
A28: a in A by TARSKI:def_4;
a \/ a in C1 "/\" C2 by A4, A28;
then consider aa being Element of C1, bb being Element of C2 such that
A29: a = aa U+ bb ;
bb in { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } by A28, A29;
then A30: bb c= union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } by ZFMISC_1:74;
aa in { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } by A28, A29;
then aa c= union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } by ZFMISC_1:74;
then aa U+ bb c= (union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) U+ (union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ) by A30, Th78;
hence x in (union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) U+ (union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ) by A27, A29; ::_thesis: verum
end;
now__::_thesis:_for_x,_y_being_set_st_x_in__{__a_where_a_is_Element_of_C1_:_ex_b_being_Element_of_C2_st_a_U+_b_in_A__}__&_y_in__{__a_where_a_is_Element_of_C1_:_ex_b_being_Element_of_C2_st_a_U+_b_in_A__}__holds_
x_\/_y_in_C1
let x, y be set ; ::_thesis: ( x in { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } & y in { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } implies x \/ y in C1 )
assume x in { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ; ::_thesis: ( y in { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } implies x \/ y in C1 )
then consider ax being Element of C1 such that
A31: x = ax and
A32: ex b being Element of C2 st ax U+ b in A ;
consider bx being Element of C2 such that
A33: ax U+ bx in A by A32;
assume y in { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ; ::_thesis: x \/ y in C1
then consider ay being Element of C1 such that
A34: y = ay and
A35: ex b being Element of C2 st ay U+ b in A ;
consider by1 being Element of C2 such that
A36: ay U+ by1 in A by A35;
(ax U+ bx) \/ (ay U+ by1) in C1 "/\" C2 by A4, A33, A36;
then (ax \/ ay) U+ (bx \/ by1) in C1 "/\" C2 by Th81;
hence x \/ y in C1 by A31, A34, Th84; ::_thesis: verum
end;
then union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } in C1 by Def1;
hence union A in C1 "/\" C2 by A11, A12; ::_thesis: verum
end;
clusterC1 "\/" C2 -> non empty subset-closed binary_complete ;
coherence
( not C1 "\/" C2 is empty & C1 "\/" C2 is subset-closed & C1 "\/" C2 is binary_complete )
proof
set a9 = the Element of C1;
the Element of C1 U+ {} in C1 "\/" C2 by Th86;
hence not C1 "\/" C2 is empty ; ::_thesis: ( C1 "\/" C2 is subset-closed & C1 "\/" C2 is binary_complete )
hereby :: according to CLASSES1:def_1 ::_thesis: C1 "\/" C2 is binary_complete
let a, b be set ; ::_thesis: ( a in C1 "\/" C2 & b c= a implies b in C1 "\/" C2 )
assume a in C1 "\/" C2 ; ::_thesis: ( b c= a implies b in C1 "\/" C2 )
then consider aa being Element of C1, bb being Element of C2 such that
A37: a = aa U+ bb and
A38: ( aa = {} or bb = {} ) by Th87;
assume b c= a ; ::_thesis: b in C1 "\/" C2
then consider x1, y1 being set such that
A39: b = x1 U+ y1 and
A40: ( x1 c= aa & y1 c= bb ) by A37, Th79;
A41: ( x1 in C1 & y1 in C2 ) by A40, CLASSES1:def_1;
( x1 = {} or y1 = {} ) by A38, A40;
hence b in C1 "\/" C2 by A39, A41, Th86; ::_thesis: verum
end;
let A be set ; :: according to COHSP_1:def_1 ::_thesis: ( ( for a, b being set st a in A & b in A holds
a \/ b in C1 "\/" C2 ) implies union A in C1 "\/" C2 )
set A1 = { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ;
set A2 = { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ;
assume A42: for a, b being set st a in A & b in A holds
a \/ b in C1 "\/" C2 ; ::_thesis: union A in C1 "\/" C2
now__::_thesis:_for_x,_y_being_set_st_x_in__{__b_where_b_is_Element_of_C2_:_ex_a_being_Element_of_C1_st_a_U+_b_in_A__}__&_y_in__{__b_where_b_is_Element_of_C2_:_ex_a_being_Element_of_C1_st_a_U+_b_in_A__}__holds_
x_\/_y_in_C2
let x, y be set ; ::_thesis: ( x in { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } & y in { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } implies x \/ y in C2 )
assume x in { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ; ::_thesis: ( y in { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } implies x \/ y in C2 )
then consider ax being Element of C2 such that
A43: x = ax and
A44: ex b being Element of C1 st b U+ ax in A ;
consider bx being Element of C1 such that
A45: bx U+ ax in A by A44;
assume y in { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ; ::_thesis: x \/ y in C2
then consider ay being Element of C2 such that
A46: y = ay and
A47: ex b being Element of C1 st b U+ ay in A ;
consider by1 being Element of C1 such that
A48: by1 U+ ay in A by A47;
(bx U+ ax) \/ (by1 U+ ay) in C1 "\/" C2 by A42, A45, A48;
then (bx \/ by1) U+ (ax \/ ay) in C1 "\/" C2 by Th81;
then ( x \/ y in C2 or x \/ y = {} ) by A43, A46, Th86;
hence x \/ y in C2 by COH_SP:1; ::_thesis: verum
end;
then A49: union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } in C2 by Def1;
A50: (union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) U+ (union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ) = union A
proof
hereby :: according to XBOOLE_0:def_10,TARSKI:def_3 ::_thesis: union A c= (union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) U+ (union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } )
let x be set ; ::_thesis: ( x in (union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) U+ (union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ) implies b1 in union A )
assume A51: x in (union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) U+ (union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ) ; ::_thesis: b1 in union A
then A52: x = [(x `1),(x `2)] by Th75;
percases ( ( x `2 = 1 & x `1 in union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) or ( x `2 = 2 & x `1 in union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ) ) by A51, Th75;
supposeA53: ( x `2 = 1 & x `1 in union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) ; ::_thesis: b1 in union A
then consider a being set such that
A54: x `1 in a and
A55: a in { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } by TARSKI:def_4;
consider ax being Element of C1 such that
A56: a = ax and
A57: ex b being Element of C2 st ax U+ b in A by A55;
consider bx being Element of C2 such that
A58: ax U+ bx in A by A57;
x in ax U+ bx by A52, A53, A54, A56, Th76;
hence x in union A by A58, TARSKI:def_4; ::_thesis: verum
end;
supposeA59: ( x `2 = 2 & x `1 in union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ) ; ::_thesis: b1 in union A
then consider a being set such that
A60: x `1 in a and
A61: a in { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } by TARSKI:def_4;
consider bx being Element of C2 such that
A62: a = bx and
A63: ex a being Element of C1 st a U+ bx in A by A61;
consider ax being Element of C1 such that
A64: ax U+ bx in A by A63;
x in ax U+ bx by A52, A59, A60, A62, Th77;
hence x in union A by A64, TARSKI:def_4; ::_thesis: verum
end;
end;
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union A or x in (union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) U+ (union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ) )
assume x in union A ; ::_thesis: x in (union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) U+ (union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } )
then consider a being set such that
A65: x in a and
A66: a in A by TARSKI:def_4;
a \/ a in C1 "\/" C2 by A42, A66;
then consider aa being Element of C1, bb being Element of C2 such that
A67: a = aa U+ bb and
( aa = {} or bb = {} ) by Th87;
bb in { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } by A66, A67;
then A68: bb c= union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } by ZFMISC_1:74;
aa in { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } by A66, A67;
then aa c= union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } by ZFMISC_1:74;
then aa U+ bb c= (union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) U+ (union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ) by A68, Th78;
hence x in (union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ) U+ (union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } ) by A65, A67; ::_thesis: verum
end;
A69: now__::_thesis:_(_union__{__a_where_a_is_Element_of_C1_:_ex_b_being_Element_of_C2_st_a_U+_b_in_A__}__<>_{}_implies_not_union__{__b_where_b_is_Element_of_C2_:_ex_a_being_Element_of_C1_st_a_U+_b_in_A__}__<>_{}_)
assume union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } <> {} ; ::_thesis: not union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } <> {}
then reconsider AA = union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } as non empty set ;
set aa = the Element of AA;
consider x being set such that
A70: the Element of AA in x and
A71: x in { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } by TARSKI:def_4;
consider ax being Element of C1 such that
A72: x = ax and
A73: ex b being Element of C2 st ax U+ b in A by A71;
consider bx being Element of C2 such that
A74: ax U+ bx in A by A73;
assume union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } <> {} ; ::_thesis: contradiction
then reconsider AA = union { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } as non empty set ;
set bb = the Element of AA;
consider y being set such that
A75: the Element of AA in y and
A76: y in { b where b is Element of C2 : ex a being Element of C1 st a U+ b in A } by TARSKI:def_4;
consider by1 being Element of C2 such that
A77: y = by1 and
A78: ex a being Element of C1 st a U+ by1 in A by A76;
consider ay being Element of C1 such that
A79: ay U+ by1 in A by A78;
(ax U+ bx) \/ (ay U+ by1) in C1 "\/" C2 by A42, A74, A79;
then (ax \/ ay) U+ (bx \/ by1) in C1 "\/" C2 by Th81;
hence contradiction by A70, A72, A75, A77, Th86; ::_thesis: verum
end;
now__::_thesis:_for_x,_y_being_set_st_x_in__{__a_where_a_is_Element_of_C1_:_ex_b_being_Element_of_C2_st_a_U+_b_in_A__}__&_y_in__{__a_where_a_is_Element_of_C1_:_ex_b_being_Element_of_C2_st_a_U+_b_in_A__}__holds_
x_\/_y_in_C1
let x, y be set ; ::_thesis: ( x in { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } & y in { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } implies x \/ y in C1 )
assume x in { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ; ::_thesis: ( y in { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } implies x \/ y in C1 )
then consider ax being Element of C1 such that
A80: x = ax and
A81: ex b being Element of C2 st ax U+ b in A ;
consider bx being Element of C2 such that
A82: ax U+ bx in A by A81;
assume y in { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } ; ::_thesis: x \/ y in C1
then consider ay being Element of C1 such that
A83: y = ay and
A84: ex b being Element of C2 st ay U+ b in A ;
consider by1 being Element of C2 such that
A85: ay U+ by1 in A by A84;
(ax U+ bx) \/ (ay U+ by1) in C1 "\/" C2 by A42, A82, A85;
then (ax \/ ay) U+ (bx \/ by1) in C1 "\/" C2 by Th81;
then ( x \/ y in C1 or x \/ y = {} ) by A80, A83, Th86;
hence x \/ y in C1 by COH_SP:1; ::_thesis: verum
end;
then union { a where a is Element of C1 : ex b being Element of C2 st a U+ b in A } in C1 by Def1;
hence union A in C1 "\/" C2 by A49, A69, A50, Th86; ::_thesis: verum
end;
end;
theorem :: COHSP_1:89
for C1, C2 being Coherence_Space
for x, y being set holds
( [[x,1],[y,1]] in Web (C1 "/\" C2) iff [x,y] in Web C1 )
proof
let C1, C2 be Coherence_Space; ::_thesis: for x, y being set holds
( [[x,1],[y,1]] in Web (C1 "/\" C2) iff [x,y] in Web C1 )
let x, y be set ; ::_thesis: ( [[x,1],[y,1]] in Web (C1 "/\" C2) iff [x,y] in Web C1 )
A1: ( [[x,1],[y,1]] in Web (C1 "/\" C2) iff {[x,1],[y,1]} in C1 "/\" C2 ) by COH_SP:5;
A2: ( [x,y] in Web C1 iff {x,y} in C1 ) by COH_SP:5;
A3: [:{x,y},{1}:] = {[x,1],[y,1]} by ZFMISC_1:30;
( {x,y} U+ {} = [:{x,y},{1}:] & {} in C2 ) by Th74, COH_SP:1;
hence ( [[x,1],[y,1]] in Web (C1 "/\" C2) iff [x,y] in Web C1 ) by A1, A2, A3, Th84; ::_thesis: verum
end;
theorem :: COHSP_1:90
for C1, C2 being Coherence_Space
for x, y being set holds
( [[x,2],[y,2]] in Web (C1 "/\" C2) iff [x,y] in Web C2 )
proof
let C1, C2 be Coherence_Space; ::_thesis: for x, y being set holds
( [[x,2],[y,2]] in Web (C1 "/\" C2) iff [x,y] in Web C2 )
let x, y be set ; ::_thesis: ( [[x,2],[y,2]] in Web (C1 "/\" C2) iff [x,y] in Web C2 )
A1: ( [[x,2],[y,2]] in Web (C1 "/\" C2) iff {[x,2],[y,2]} in C1 "/\" C2 ) by COH_SP:5;
A2: ( [x,y] in Web C2 iff {x,y} in C2 ) by COH_SP:5;
A3: [:{x,y},{2}:] = {[x,2],[y,2]} by ZFMISC_1:30;
( {} U+ {x,y} = [:{x,y},{2}:] & {} in C1 ) by Th74, COH_SP:1;
hence ( [[x,2],[y,2]] in Web (C1 "/\" C2) iff [x,y] in Web C2 ) by A1, A2, A3, Th84; ::_thesis: verum
end;
theorem :: COHSP_1:91
for C1, C2 being Coherence_Space
for x, y being set st x in union C1 & y in union C2 holds
( [[x,1],[y,2]] in Web (C1 "/\" C2) & [[y,2],[x,1]] in Web (C1 "/\" C2) )
proof
let C1, C2 be Coherence_Space; ::_thesis: for x, y being set st x in union C1 & y in union C2 holds
( [[x,1],[y,2]] in Web (C1 "/\" C2) & [[y,2],[x,1]] in Web (C1 "/\" C2) )
let x, y be set ; ::_thesis: ( x in union C1 & y in union C2 implies ( [[x,1],[y,2]] in Web (C1 "/\" C2) & [[y,2],[x,1]] in Web (C1 "/\" C2) ) )
assume ( x in union C1 & y in union C2 ) ; ::_thesis: ( [[x,1],[y,2]] in Web (C1 "/\" C2) & [[y,2],[x,1]] in Web (C1 "/\" C2) )
then ( {x} in C1 & {y} in C2 ) by COH_SP:4;
then {x} U+ {y} in C1 "/\" C2 ;
then [:{x},{1}:] \/ [:{y},{2}:] in C1 "/\" C2 by Th73;
then {[x,1]} \/ [:{y},{2}:] in C1 "/\" C2 by ZFMISC_1:29;
then {[x,1]} \/ {[y,2]} in C1 "/\" C2 by ZFMISC_1:29;
then A1: {[x,1],[y,2]} in C1 "/\" C2 by ENUMSET1:1;
hence [[x,1],[y,2]] in Web (C1 "/\" C2) by COH_SP:5; ::_thesis: [[y,2],[x,1]] in Web (C1 "/\" C2)
thus [[y,2],[x,1]] in Web (C1 "/\" C2) by A1, COH_SP:5; ::_thesis: verum
end;
theorem :: COHSP_1:92
for C1, C2 being Coherence_Space
for x, y being set holds
( [[x,1],[y,1]] in Web (C1 "\/" C2) iff [x,y] in Web C1 )
proof
let C1, C2 be Coherence_Space; ::_thesis: for x, y being set holds
( [[x,1],[y,1]] in Web (C1 "\/" C2) iff [x,y] in Web C1 )
let x, y be set ; ::_thesis: ( [[x,1],[y,1]] in Web (C1 "\/" C2) iff [x,y] in Web C1 )
A1: ( [[x,1],[y,1]] in Web (C1 "\/" C2) iff {[x,1],[y,1]} in C1 "\/" C2 ) by COH_SP:5;
A2: ( [x,y] in Web C1 iff {x,y} in C1 ) by COH_SP:5;
( {x,y} U+ {} = [:{x,y},{1}:] & [:{x,y},{1}:] = {[x,1],[y,1]} ) by Th74, ZFMISC_1:30;
hence ( [[x,1],[y,1]] in Web (C1 "\/" C2) iff [x,y] in Web C1 ) by A1, A2, Th86; ::_thesis: verum
end;
theorem :: COHSP_1:93
for C1, C2 being Coherence_Space
for x, y being set holds
( [[x,2],[y,2]] in Web (C1 "\/" C2) iff [x,y] in Web C2 )
proof
let C1, C2 be Coherence_Space; ::_thesis: for x, y being set holds
( [[x,2],[y,2]] in Web (C1 "\/" C2) iff [x,y] in Web C2 )
let x, y be set ; ::_thesis: ( [[x,2],[y,2]] in Web (C1 "\/" C2) iff [x,y] in Web C2 )
A1: ( [[x,2],[y,2]] in Web (C1 "\/" C2) iff {[x,2],[y,2]} in C1 "\/" C2 ) by COH_SP:5;
A2: ( [x,y] in Web C2 iff {x,y} in C2 ) by COH_SP:5;
( {} U+ {x,y} = [:{x,y},{2}:] & [:{x,y},{2}:] = {[x,2],[y,2]} ) by Th74, ZFMISC_1:30;
hence ( [[x,2],[y,2]] in Web (C1 "\/" C2) iff [x,y] in Web C2 ) by A1, A2, Th86; ::_thesis: verum
end;
theorem :: COHSP_1:94
for C1, C2 being Coherence_Space
for x, y being set holds
( not [[x,1],[y,2]] in Web (C1 "\/" C2) & not [[y,2],[x,1]] in Web (C1 "\/" C2) )
proof
let C1, C2 be Coherence_Space; ::_thesis: for x, y being set holds
( not [[x,1],[y,2]] in Web (C1 "\/" C2) & not [[y,2],[x,1]] in Web (C1 "\/" C2) )
let x, y be set ; ::_thesis: ( not [[x,1],[y,2]] in Web (C1 "\/" C2) & not [[y,2],[x,1]] in Web (C1 "\/" C2) )
A1: {x} U+ {y} = [:{x},{1}:] \/ [:{y},{2}:] by Th73
.= {[x,1]} \/ [:{y},{2}:] by ZFMISC_1:29
.= {[x,1]} \/ {[y,2]} by ZFMISC_1:29
.= {[x,1],[y,2]} by ENUMSET1:1 ;
A2: not {x} U+ {y} in C1 "\/" C2 by Th86;
hence not [[x,1],[y,2]] in Web (C1 "\/" C2) by A1, COH_SP:5; ::_thesis: not [[y,2],[x,1]] in Web (C1 "\/" C2)
thus not [[y,2],[x,1]] in Web (C1 "\/" C2) by A2, A1, COH_SP:5; ::_thesis: verum
end;
theorem :: COHSP_1:95
for C1, C2 being Coherence_Space holds 'not' (C1 "/\" C2) = ('not' C1) "\/" ('not' C2)
proof
let C1, C2 be Coherence_Space; ::_thesis: 'not' (C1 "/\" C2) = ('not' C1) "\/" ('not' C2)
set C = C1 "/\" C2;
thus 'not' (C1 "/\" C2) c= ('not' C1) "\/" ('not' C2) :: according to XBOOLE_0:def_10 ::_thesis: ('not' C1) "\/" ('not' C2) c= 'not' (C1 "/\" C2)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in 'not' (C1 "/\" C2) or x in ('not' C1) "\/" ('not' C2) )
A1: union (C1 "/\" C2) = (union C1) U+ (union C2) by Th85;
assume A2: x in 'not' (C1 "/\" C2) ; ::_thesis: x in ('not' C1) "\/" ('not' C2)
then x c= union (C1 "/\" C2) by Th65;
then consider x1, x2 being set such that
A3: x = x1 U+ x2 and
A4: x1 c= union C1 and
A5: x2 c= union C2 by A1, Th79;
now__::_thesis:_for_a_being_Element_of_C1_ex_z_being_set_st_x1_/\_a_c=_{z}
reconsider b = {} as Element of C2 by COH_SP:1;
let a be Element of C1; ::_thesis: ex z being set st x1 /\ a c= {z}
a U+ b in C1 "/\" C2 ;
then consider z being set such that
A6: x /\ (a U+ b) c= {z} by A2, Th65;
(x1 /\ a) U+ (x2 /\ b) c= {z} by A3, A6, Th82;
then ( [:(x1 /\ a),{1}:] c= [:(x1 /\ a),{1}:] \/ [:(x2 /\ b),{2}:] & [:(x1 /\ a),{1}:] \/ [:(x2 /\ b),{2}:] c= {z} ) by Th73, XBOOLE_1:7;
then A7: [:(x1 /\ a),{1}:] c= {z} by XBOOLE_1:1;
now__::_thesis:_(_(_x1_/\_a_=_{}_implies_x1_/\_a_c=_{0}_)_&_(_x1_/\_a_<>_{}_implies_ex_zz_being_set_st_x1_/\_a_c=_{zz}_)_)
thus ( x1 /\ a = {} implies x1 /\ a c= {0} ) by XBOOLE_1:2; ::_thesis: ( x1 /\ a <> {} implies ex zz being set st x1 /\ a c= {zz} )
assume x1 /\ a <> {} ; ::_thesis: ex zz being set st x1 /\ a c= {zz}
then reconsider A = x1 /\ a as non empty set ;
set z1 = the Element of A;
reconsider zz = the Element of A as set ;
take zz = zz; ::_thesis: x1 /\ a c= {zz}
thus x1 /\ a c= {zz} ::_thesis: verum
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in x1 /\ a or y in {zz} )
A8: 1 in {1} by TARSKI:def_1;
assume y in x1 /\ a ; ::_thesis: y in {zz}
then [y,1] in [:(x1 /\ a),{1}:] by A8, ZFMISC_1:87;
then A9: [y,1] = z by A7, TARSKI:def_1;
[ the Element of A,1] in [:(x1 /\ a),{1}:] by A8, ZFMISC_1:87;
then [ the Element of A,1] = z by A7, TARSKI:def_1;
then y = the Element of A by A9, XTUPLE_0:1;
hence y in {zz} by TARSKI:def_1; ::_thesis: verum
end;
end;
hence ex z being set st x1 /\ a c= {z} ; ::_thesis: verum
end;
then reconsider x1 = x1 as Element of 'not' C1 by A4, Th65;
now__::_thesis:_for_b_being_Element_of_C2_ex_z_being_set_st_x2_/\_b_c=_{z}
reconsider a = {} as Element of C1 by COH_SP:1;
let b be Element of C2; ::_thesis: ex z being set st x2 /\ b c= {z}
a U+ b in C1 "/\" C2 ;
then consider z being set such that
A10: x /\ (a U+ b) c= {z} by A2, Th65;
(x1 /\ a) U+ (x2 /\ b) c= {z} by A3, A10, Th82;
then ( [:(x2 /\ b),{2}:] c= [:(x1 /\ a),{1}:] \/ [:(x2 /\ b),{2}:] & [:(x1 /\ a),{1}:] \/ [:(x2 /\ b),{2}:] c= {z} ) by Th73, XBOOLE_1:7;
then A11: [:(x2 /\ b),{2}:] c= {z} by XBOOLE_1:1;
now__::_thesis:_(_(_x2_/\_b_=_{}_implies_x2_/\_b_c=_{0}_)_&_(_x2_/\_b_<>_{}_implies_ex_zz_being_set_st_x2_/\_b_c=_{zz}_)_)
thus ( x2 /\ b = {} implies x2 /\ b c= {0} ) by XBOOLE_1:2; ::_thesis: ( x2 /\ b <> {} implies ex zz being set st x2 /\ b c= {zz} )
assume x2 /\ b <> {} ; ::_thesis: ex zz being set st x2 /\ b c= {zz}
then reconsider A = x2 /\ b as non empty set ;
set z1 = the Element of A;
reconsider zz = the Element of A as set ;
take zz = zz; ::_thesis: x2 /\ b c= {zz}
thus x2 /\ b c= {zz} ::_thesis: verum
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in x2 /\ b or y in {zz} )
A12: 2 in {2} by TARSKI:def_1;
assume y in x2 /\ b ; ::_thesis: y in {zz}
then [y,2] in [:(x2 /\ b),{2}:] by A12, ZFMISC_1:87;
then A13: [y,2] = z by A11, TARSKI:def_1;
[ the Element of A,2] in [:(x2 /\ b),{2}:] by A12, ZFMISC_1:87;
then [ the Element of A,2] = z by A11, TARSKI:def_1;
then y = the Element of A by A13, XTUPLE_0:1;
hence y in {zz} by TARSKI:def_1; ::_thesis: verum
end;
end;
hence ex z being set st x2 /\ b c= {z} ; ::_thesis: verum
end;
then reconsider x2 = x2 as Element of 'not' C2 by A5, Th65;
now__::_thesis:_(_x1_in_'not'_C1_&_x2_in_'not'_C2_&_(_x1_<>_{}_implies_not_x2_<>_{}_)_)
thus ( x1 in 'not' C1 & x2 in 'not' C2 ) ; ::_thesis: ( x1 <> {} implies not x2 <> {} )
assume ( x1 <> {} & x2 <> {} ) ; ::_thesis: contradiction
then reconsider x1 = x1, x2 = x2 as non empty set ;
set y1 = the Element of x1;
set y2 = the Element of x2;
union ('not' C2) = union C2 by Th66;
then the Element of x2 in union C2 by TARSKI:def_4;
then A14: { the Element of x2} in C2 by COH_SP:4;
union ('not' C1) = union C1 by Th66;
then the Element of x1 in union C1 by TARSKI:def_4;
then { the Element of x1} in C1 by COH_SP:4;
then { the Element of x1} U+ { the Element of x2} in C1 "/\" C2 by A14;
then consider z being set such that
A15: x /\ ({ the Element of x1} U+ { the Element of x2}) c= {z} by A2, Th65;
A16: (x1 /\ { the Element of x1}) U+ (x2 /\ { the Element of x2}) c= {z} by A3, A15, Th82;
the Element of x2 in { the Element of x2} by TARSKI:def_1;
then the Element of x2 in x2 /\ { the Element of x2} by XBOOLE_0:def_4;
then [ the Element of x2,2] in (x1 /\ { the Element of x1}) U+ (x2 /\ { the Element of x2}) by Th77;
then A17: [ the Element of x2,2] = z by A16, TARSKI:def_1;
the Element of x1 in { the Element of x1} by TARSKI:def_1;
then the Element of x1 in x1 /\ { the Element of x1} by XBOOLE_0:def_4;
then [ the Element of x1,1] in (x1 /\ { the Element of x1}) U+ (x2 /\ { the Element of x2}) by Th76;
then [ the Element of x1,1] = z by A16, TARSKI:def_1;
hence contradiction by A17, XTUPLE_0:1; ::_thesis: verum
end;
hence x in ('not' C1) "\/" ('not' C2) by A3, Th86; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ('not' C1) "\/" ('not' C2) or x in 'not' (C1 "/\" C2) )
assume x in ('not' C1) "\/" ('not' C2) ; ::_thesis: x in 'not' (C1 "/\" C2)
then consider x1 being Element of 'not' C1, x2 being Element of 'not' C2 such that
A18: x = x1 U+ x2 and
A19: ( x1 = {} or x2 = {} ) by Th87;
A20: for a being Element of C1 "/\" C2 ex z being set st x /\ a c= {z}
proof
let a be Element of C1 "/\" C2; ::_thesis: ex z being set st x /\ a c= {z}
consider a1 being Element of C1, a2 being Element of C2 such that
A21: a = a1 U+ a2 by Th83;
A22: x /\ a = (x1 /\ a1) U+ (x2 /\ a2) by A18, A21, Th82;
consider z2 being set such that
A23: x2 /\ a2 c= {z2} by Th65;
consider z1 being set such that
A24: x1 /\ a1 c= {z1} by Th65;
( x1 = {} or x1 <> {} ) ;
then consider z being set such that
A25: ( ( z = [z2,2] & x1 = {} ) or ( z = [z1,1] & x1 <> {} ) ) ;
take z ; ::_thesis: x /\ a c= {z}
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in x /\ a or y in {z} )
assume A26: y in x /\ a ; ::_thesis: y in {z}
then A27: y = [(y `1),(y `2)] by A22, Th75;
A28: ( ( y `2 = 1 & y `1 in x1 /\ a1 ) or ( y `2 = 2 & y `1 in x2 /\ a2 ) ) by A22, A26, Th75;
percases ( ( z = [z2,2] & x1 = {} ) or ( z = [z1,1] & x1 <> {} ) ) by A25;
supposeA29: ( z = [z2,2] & x1 = {} ) ; ::_thesis: y in {z}
then y `1 = z2 by A23, A28, TARSKI:def_1;
hence y in {z} by A27, A28, A29, TARSKI:def_1; ::_thesis: verum
end;
supposeA30: ( z = [z1,1] & x1 <> {} ) ; ::_thesis: y in {z}
then y `1 = z1 by A19, A24, A28, TARSKI:def_1;
hence y in {z} by A19, A27, A28, A30, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
x2 c= union ('not' C2) by ZFMISC_1:74;
then A31: x2 c= union C2 by Th66;
x1 c= union ('not' C1) by ZFMISC_1:74;
then x1 c= union C1 by Th66;
then x c= (union C1) U+ (union C2) by A18, A31, Th78;
then x c= union (C1 "/\" C2) by Th85;
hence x in 'not' (C1 "/\" C2) by A20; ::_thesis: verum
end;
definition
let C1, C2 be Coherence_Space;
funcC1 [*] C2 -> set equals :: COHSP_1:def 25
union { (bool [:a,b:]) where a is Element of C1, b is Element of C2 : verum } ;
correctness
coherence
union { (bool [:a,b:]) where a is Element of C1, b is Element of C2 : verum } is set ;
;
end;
:: deftheorem defines [*] COHSP_1:def_25_:_
for C1, C2 being Coherence_Space holds C1 [*] C2 = union { (bool [:a,b:]) where a is Element of C1, b is Element of C2 : verum } ;
theorem Th96: :: COHSP_1:96
for C1, C2 being Coherence_Space
for x being set holds
( x in C1 [*] C2 iff ex a being Element of C1 ex b being Element of C2 st x c= [:a,b:] )
proof
let C1, C2 be Coherence_Space; ::_thesis: for x being set holds
( x in C1 [*] C2 iff ex a being Element of C1 ex b being Element of C2 st x c= [:a,b:] )
let x be set ; ::_thesis: ( x in C1 [*] C2 iff ex a being Element of C1 ex b being Element of C2 st x c= [:a,b:] )
hereby ::_thesis: ( ex a being Element of C1 ex b being Element of C2 st x c= [:a,b:] implies x in C1 [*] C2 )
assume x in C1 [*] C2 ; ::_thesis: ex a being Element of C1 ex b being Element of C2 st x c= [:a,b:]
then consider y being set such that
A1: x in y and
A2: y in { (bool [:a,b:]) where a is Element of C1, b is Element of C2 : verum } by TARSKI:def_4;
consider a being Element of C1, b being Element of C2 such that
A3: y = bool [:a,b:] by A2;
take a = a; ::_thesis: ex b being Element of C2 st x c= [:a,b:]
take b = b; ::_thesis: x c= [:a,b:]
thus x c= [:a,b:] by A1, A3; ::_thesis: verum
end;
given a9 being Element of C1, b9 being Element of C2 such that A4: x c= [:a9,b9:] ; ::_thesis: x in C1 [*] C2
bool [:a9,b9:] in { (bool [:a,b:]) where a is Element of C1, b is Element of C2 : verum } ;
hence x in C1 [*] C2 by A4, TARSKI:def_4; ::_thesis: verum
end;
registration
let C1, C2 be Coherence_Space;
clusterC1 [*] C2 -> non empty ;
coherence
not C1 [*] C2 is empty
proof
set a1 = the Element of C1;
set a2 = the Element of C2;
[: the Element of C1, the Element of C2:] in C1 [*] C2 by Th96;
hence not C1 [*] C2 is empty ; ::_thesis: verum
end;
end;
theorem Th97: :: COHSP_1:97
for C1, C2 being Coherence_Space
for a being Element of C1 [*] C2 holds
( proj1 a in C1 & proj2 a in C2 & a c= [:(proj1 a),(proj2 a):] )
proof
let C1, C2 be Coherence_Space; ::_thesis: for a being Element of C1 [*] C2 holds
( proj1 a in C1 & proj2 a in C2 & a c= [:(proj1 a),(proj2 a):] )
let a be Element of C1 [*] C2; ::_thesis: ( proj1 a in C1 & proj2 a in C2 & a c= [:(proj1 a),(proj2 a):] )
consider a1 being Element of C1, a2 being Element of C2 such that
A1: a c= [:a1,a2:] by Th96;
( proj1 a c= a1 & proj2 a c= a2 ) by A1, FUNCT_5:11;
hence ( proj1 a in C1 & proj2 a in C2 ) by CLASSES1:def_1; ::_thesis: a c= [:(proj1 a),(proj2 a):]
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in a or x in [:(proj1 a),(proj2 a):] )
assume A2: x in a ; ::_thesis: x in [:(proj1 a),(proj2 a):]
then A3: x = [(x `1),(x `2)] by A1, MCART_1:21;
then ( x `1 in proj1 a & x `2 in proj2 a ) by A2, XTUPLE_0:def_12, XTUPLE_0:def_13;
hence x in [:(proj1 a),(proj2 a):] by A3, ZFMISC_1:87; ::_thesis: verum
end;
registration
let C1, C2 be Coherence_Space;
clusterC1 [*] C2 -> subset-closed binary_complete ;
coherence
( C1 [*] C2 is subset-closed & C1 [*] C2 is binary_complete )
proof
thus C1 [*] C2 is subset-closed ::_thesis: C1 [*] C2 is binary_complete
proof
let a, b be set ; :: according to CLASSES1:def_1 ::_thesis: ( not a in C1 [*] C2 or not b c= a or b in C1 [*] C2 )
assume a in C1 [*] C2 ; ::_thesis: ( not b c= a or b in C1 [*] C2 )
then consider a1 being Element of C1, a2 being Element of C2 such that
A1: a c= [:a1,a2:] by Th96;
assume b c= a ; ::_thesis: b in C1 [*] C2
then b c= [:a1,a2:] by A1, XBOOLE_1:1;
hence b in C1 [*] C2 by Th96; ::_thesis: verum
end;
let A be set ; :: according to COHSP_1:def_1 ::_thesis: ( ( for a, b being set st a in A & b in A holds
a \/ b in C1 [*] C2 ) implies union A in C1 [*] C2 )
set A1 = { (proj1 a) where a is Element of C1 [*] C2 : a in A } ;
set A2 = { (proj2 a) where a is Element of C1 [*] C2 : a in A } ;
assume A2: for a, b being set st a in A & b in A holds
a \/ b in C1 [*] C2 ; ::_thesis: union A in C1 [*] C2
now__::_thesis:_for_a1,_b1_being_set_st_a1_in__{__(proj2_a)_where_a_is_Element_of_C1_[*]_C2_:_a_in_A__}__&_b1_in__{__(proj2_a)_where_a_is_Element_of_C1_[*]_C2_:_a_in_A__}__holds_
a1_\/_b1_in_C2
let a1, b1 be set ; ::_thesis: ( a1 in { (proj2 a) where a is Element of C1 [*] C2 : a in A } & b1 in { (proj2 a) where a is Element of C1 [*] C2 : a in A } implies a1 \/ b1 in C2 )
assume a1 in { (proj2 a) where a is Element of C1 [*] C2 : a in A } ; ::_thesis: ( b1 in { (proj2 a) where a is Element of C1 [*] C2 : a in A } implies a1 \/ b1 in C2 )
then consider a being Element of C1 [*] C2 such that
A3: a1 = proj2 a and
A4: a in A ;
assume b1 in { (proj2 a) where a is Element of C1 [*] C2 : a in A } ; ::_thesis: a1 \/ b1 in C2
then consider b being Element of C1 [*] C2 such that
A5: b1 = proj2 b and
A6: b in A ;
a \/ b in C1 [*] C2 by A2, A4, A6;
then proj2 (a \/ b) in C2 by Th97;
hence a1 \/ b1 in C2 by A3, A5, XTUPLE_0:27; ::_thesis: verum
end;
then A7: union { (proj2 a) where a is Element of C1 [*] C2 : a in A } in C2 by Def1;
A8: union A c= [:(union { (proj1 a) where a is Element of C1 [*] C2 : a in A } ),(union { (proj2 a) where a is Element of C1 [*] C2 : a in A } ):]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union A or x in [:(union { (proj1 a) where a is Element of C1 [*] C2 : a in A } ),(union { (proj2 a) where a is Element of C1 [*] C2 : a in A } ):] )
assume x in union A ; ::_thesis: x in [:(union { (proj1 a) where a is Element of C1 [*] C2 : a in A } ),(union { (proj2 a) where a is Element of C1 [*] C2 : a in A } ):]
then consider a being set such that
A9: x in a and
A10: a in A by TARSKI:def_4;
A11: a \/ a in C1 [*] C2 by A2, A10;
then proj2 a in { (proj2 a) where a is Element of C1 [*] C2 : a in A } by A10;
then A12: proj2 a c= union { (proj2 a) where a is Element of C1 [*] C2 : a in A } by ZFMISC_1:74;
a c= [:(proj1 a),(proj2 a):] by A11, Th97;
then A13: x in [:(proj1 a),(proj2 a):] by A9;
proj1 a in { (proj1 a) where a is Element of C1 [*] C2 : a in A } by A10, A11;
then proj1 a c= union { (proj1 a) where a is Element of C1 [*] C2 : a in A } by ZFMISC_1:74;
then [:(proj1 a),(proj2 a):] c= [:(union { (proj1 a) where a is Element of C1 [*] C2 : a in A } ),(union { (proj2 a) where a is Element of C1 [*] C2 : a in A } ):] by A12, ZFMISC_1:96;
hence x in [:(union { (proj1 a) where a is Element of C1 [*] C2 : a in A } ),(union { (proj2 a) where a is Element of C1 [*] C2 : a in A } ):] by A13; ::_thesis: verum
end;
now__::_thesis:_for_a1,_b1_being_set_st_a1_in__{__(proj1_a)_where_a_is_Element_of_C1_[*]_C2_:_a_in_A__}__&_b1_in__{__(proj1_a)_where_a_is_Element_of_C1_[*]_C2_:_a_in_A__}__holds_
a1_\/_b1_in_C1
let a1, b1 be set ; ::_thesis: ( a1 in { (proj1 a) where a is Element of C1 [*] C2 : a in A } & b1 in { (proj1 a) where a is Element of C1 [*] C2 : a in A } implies a1 \/ b1 in C1 )
assume a1 in { (proj1 a) where a is Element of C1 [*] C2 : a in A } ; ::_thesis: ( b1 in { (proj1 a) where a is Element of C1 [*] C2 : a in A } implies a1 \/ b1 in C1 )
then consider a being Element of C1 [*] C2 such that
A14: a1 = proj1 a and
A15: a in A ;
assume b1 in { (proj1 a) where a is Element of C1 [*] C2 : a in A } ; ::_thesis: a1 \/ b1 in C1
then consider b being Element of C1 [*] C2 such that
A16: b1 = proj1 b and
A17: b in A ;
a \/ b in C1 [*] C2 by A2, A15, A17;
then proj1 (a \/ b) in C1 by Th97;
hence a1 \/ b1 in C1 by A14, A16, XTUPLE_0:23; ::_thesis: verum
end;
then union { (proj1 a) where a is Element of C1 [*] C2 : a in A } in C1 by Def1;
hence union A in C1 [*] C2 by A7, A8, Th96; ::_thesis: verum
end;
end;
theorem :: COHSP_1:98
for C1, C2 being Coherence_Space holds union (C1 [*] C2) = [:(union C1),(union C2):]
proof
let C1, C2 be Coherence_Space; ::_thesis: union (C1 [*] C2) = [:(union C1),(union C2):]
thus union (C1 [*] C2) c= [:(union C1),(union C2):] :: according to XBOOLE_0:def_10 ::_thesis: [:(union C1),(union C2):] c= union (C1 [*] C2)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union (C1 [*] C2) or x in [:(union C1),(union C2):] )
assume x in union (C1 [*] C2) ; ::_thesis: x in [:(union C1),(union C2):]
then consider a being set such that
A1: x in a and
A2: a in C1 [*] C2 by TARSKI:def_4;
consider a1 being Element of C1, a2 being Element of C2 such that
A3: a c= [:a1,a2:] by A2, Th96;
( a1 c= union C1 & a2 c= union C2 ) by ZFMISC_1:74;
then [:a1,a2:] c= [:(union C1),(union C2):] by ZFMISC_1:96;
then a c= [:(union C1),(union C2):] by A3, XBOOLE_1:1;
hence x in [:(union C1),(union C2):] by A1; ::_thesis: verum
end;
let x, y be set ; :: according to RELAT_1:def_3 ::_thesis: ( not [x,y] in [:(union C1),(union C2):] or [x,y] in union (C1 [*] C2) )
assume A4: [x,y] in [:(union C1),(union C2):] ; ::_thesis: [x,y] in union (C1 [*] C2)
then x in union C1 by ZFMISC_1:87;
then consider a1 being set such that
A5: x in a1 and
A6: a1 in C1 by TARSKI:def_4;
y in union C2 by A4, ZFMISC_1:87;
then consider a2 being set such that
A7: y in a2 and
A8: a2 in C2 by TARSKI:def_4;
A9: [:a1,a2:] in C1 [*] C2 by A6, A8, Th96;
[x,y] in [:a1,a2:] by A5, A7, ZFMISC_1:87;
hence [x,y] in union (C1 [*] C2) by A9, TARSKI:def_4; ::_thesis: verum
end;
theorem :: COHSP_1:99
for C1, C2 being Coherence_Space
for x1, y1, x2, y2 being set holds
( [[x1,x2],[y1,y2]] in Web (C1 [*] C2) iff ( [x1,y1] in Web C1 & [x2,y2] in Web C2 ) )
proof
let C1, C2 be Coherence_Space; ::_thesis: for x1, y1, x2, y2 being set holds
( [[x1,x2],[y1,y2]] in Web (C1 [*] C2) iff ( [x1,y1] in Web C1 & [x2,y2] in Web C2 ) )
let x1, y1, x2, y2 be set ; ::_thesis: ( [[x1,x2],[y1,y2]] in Web (C1 [*] C2) iff ( [x1,y1] in Web C1 & [x2,y2] in Web C2 ) )
A1: {[x1,x2],[y1,y2]} c= [:{x1,y1},{x2,y2}:]
proof
let x, y be set ; :: according to RELAT_1:def_3 ::_thesis: ( not [x,y] in {[x1,x2],[y1,y2]} or [x,y] in [:{x1,y1},{x2,y2}:] )
assume [x,y] in {[x1,x2],[y1,y2]} ; ::_thesis: [x,y] in [:{x1,y1},{x2,y2}:]
then ( ( [x,y] = [x1,x2] & x1 in {x1,y1} & x2 in {x2,y2} ) or ( [x,y] = [y1,y2] & y1 in {x1,y1} & y2 in {x2,y2} ) ) by TARSKI:def_2;
hence [x,y] in [:{x1,y1},{x2,y2}:] by ZFMISC_1:87; ::_thesis: verum
end;
A2: ( proj1 {[x1,x2],[y1,y2]} = {x1,y1} & proj2 {[x1,x2],[y1,y2]} = {x2,y2} ) by FUNCT_5:13;
hereby ::_thesis: ( [x1,y1] in Web C1 & [x2,y2] in Web C2 implies [[x1,x2],[y1,y2]] in Web (C1 [*] C2) )
assume [[x1,x2],[y1,y2]] in Web (C1 [*] C2) ; ::_thesis: ( [x1,y1] in Web C1 & [x2,y2] in Web C2 )
then {[x1,x2],[y1,y2]} in C1 [*] C2 by COH_SP:5;
then ( {x1,y1} in C1 & {x2,y2} in C2 ) by A2, Th97;
hence ( [x1,y1] in Web C1 & [x2,y2] in Web C2 ) by COH_SP:5; ::_thesis: verum
end;
assume ( [x1,y1] in Web C1 & [x2,y2] in Web C2 ) ; ::_thesis: [[x1,x2],[y1,y2]] in Web (C1 [*] C2)
then ( {x1,y1} in C1 & {x2,y2} in C2 ) by COH_SP:5;
then {[x1,x2],[y1,y2]} in C1 [*] C2 by A1, Th96;
hence [[x1,x2],[y1,y2]] in Web (C1 [*] C2) by COH_SP:5; ::_thesis: verum
end;