:: LFUZZY_1 semantic presentation
Lemma39:
for a, b, c, d being real number st a <= b & c <= d holds
min a,c <= min b,d
by XXREAL_0:18;
:: deftheorem Def1 defines is_less_than LFUZZY_1:def 1 :
theorem Th1: :: LFUZZY_1:1
theorem Th2: :: LFUZZY_1:2
theorem Th3: :: LFUZZY_1:3
theorem Th4: :: LFUZZY_1:4
theorem Th5: :: LFUZZY_1:5
theorem Th6: :: LFUZZY_1:6
theorem Th7: :: LFUZZY_1:7
theorem Th8: :: LFUZZY_1:8
:: deftheorem Def2 defines reflexive LFUZZY_1:def 2 :
:: deftheorem Def3 defines reflexive LFUZZY_1:def 3 :
:: deftheorem Def4 defines symmetric LFUZZY_1:def 4 :
:: deftheorem Def5 defines symmetric LFUZZY_1:def 5 :
:: deftheorem Def6 defines transitive LFUZZY_1:def 6 :
Lemma70:
for X, Y, Z being non empty set
for R being RMembership_Func of X,Y
for S being RMembership_Func of Y,Z
for x being Element of X
for z being Element of Z holds
( rng (min R,S,x,z) = { ((R . [x,y]) "/\" (S . [y,z])) where y is Element of Y : verum } & rng (min R,S,x,z) <> {} )
definition
let X be non
empty set ;
let R be
RMembership_Func of
X,
X;
redefine attr a2 is
transitive means :: LFUZZY_1:def 7
for
x,
y,
z being
Element of
X holds
(R . [x,y]) "/\" (R . [y,z]) <<= R . [x,z];
compatibility
( R is transitive iff for x, y, z being Element of X holds (R . [x,y]) "/\" (R . [y,z]) <<= R . [x,z] )
end;
:: deftheorem Def7 defines transitive LFUZZY_1:def 7 :
:: deftheorem Def8 defines antisymmetric LFUZZY_1:def 8 :
:: deftheorem Def9 defines antisymmetric LFUZZY_1:def 9 :
theorem Th9: :: LFUZZY_1:9
theorem Th10: :: LFUZZY_1:10
theorem Th11: :: LFUZZY_1:11
theorem Th12: :: LFUZZY_1:12
theorem Th13: :: LFUZZY_1:13
theorem Th14: :: LFUZZY_1:14
theorem Th15: :: LFUZZY_1:15
theorem Th16: :: LFUZZY_1:16
theorem Th17: :: LFUZZY_1:17
theorem Th18: :: LFUZZY_1:18
theorem Th19: :: LFUZZY_1:19
theorem Th20: :: LFUZZY_1:20
theorem Th21: :: LFUZZY_1:21
definition
let X be non
empty set ;
let R be
RMembership_Func of
X,
X;
let n be
natural number ;
func c3 iter c2 -> RMembership_Func of
a1,
a1 means :
Def10:
:: LFUZZY_1:def 10
ex
F being
Function of
NAT ,
Funcs [:X,X:],
[.0,1.] st
(
it = F . n &
F . 0
= Imf X,
X & ( for
k being
natural number ex
Q being
RMembership_Func of
X,
X st
(
F . k = Q &
F . (k + 1) = Q (#) R ) ) );
existence
ex b1 being RMembership_Func of X,X ex F being Function of NAT , Funcs [:X,X:],[.0,1.] st
( b1 = F . n & F . 0 = Imf X,X & ( for k being natural number ex Q being RMembership_Func of X,X st
( F . k = Q & F . (k + 1) = Q (#) R ) ) )
uniqueness
for b1, b2 being RMembership_Func of X,X st ex F being Function of NAT , Funcs [:X,X:],[.0,1.] st
( b1 = F . n & F . 0 = Imf X,X & ( for k being natural number ex Q being RMembership_Func of X,X st
( F . k = Q & F . (k + 1) = Q (#) R ) ) ) & ex F being Function of NAT , Funcs [:X,X:],[.0,1.] st
( b2 = F . n & F . 0 = Imf X,X & ( for k being natural number ex Q being RMembership_Func of X,X st
( F . k = Q & F . (k + 1) = Q (#) R ) ) ) holds
b1 = b2
end;
:: deftheorem Def10 defines iter LFUZZY_1:def 10 :
theorem Th22: :: LFUZZY_1:22
theorem Th23: :: LFUZZY_1:23
theorem Th24: :: LFUZZY_1:24
theorem Th25: :: LFUZZY_1:25
theorem Th26: :: LFUZZY_1:26
theorem Th27: :: LFUZZY_1:27
theorem Th28: :: LFUZZY_1:28
:: deftheorem Def11 defines TrCl LFUZZY_1:def 11 :
theorem Th29: :: LFUZZY_1:29
theorem Th30: :: LFUZZY_1:30
theorem Th31: :: LFUZZY_1:31
theorem Th32: :: LFUZZY_1:32
Lemma111:
for X being non empty set
for R being RMembership_Func of X,X
for Q being Subset of (FuzzyLattice [:X,X:])
for x, z being Element of X holds { ((R . x,y) "/\" ((@ ("\/" Q,(FuzzyLattice [:X,X:]))) . y,z)) where y is Element of X : verum } = { ((R . [x,y]) "/\" ("\/" (pi Q,[y,z]),(RealPoset [.0,1.]))) where y is Element of X : verum }
theorem Th33: :: LFUZZY_1:33
Lemma113:
for X being non empty set
for R being RMembership_Func of X,X
for Q being Subset of (FuzzyLattice [:X,X:])
for x, z being Element of X holds { ((R . [x,y]) "/\" ("\/" (pi Q,[y,z]),(RealPoset [.0,1.]))) where y is Element of X : verum } = { ("\/" { ((R . [x,y']) "/\" b) where b is Element of (RealPoset [.0,1.]) : b in pi Q,[y',z] } ,(RealPoset [.0,1.])) where y' is Element of X : verum }
Lemma114:
for X being non empty set
for R being RMembership_Func of X,X
for Q being Subset of (FuzzyLattice [:X,X:])
for x, z being Element of X holds { ("\/" { ((R . [x,y]) "/\" b) where b is Element of (RealPoset [.0,1.]) : b in pi Q,[y,z] } ,(RealPoset [.0,1.])) where y is Element of X : verum } = { ("\/" { ((R . [x,y']) "/\" (r . [y',z])) where r is Element of (FuzzyLattice [:X,X:]) : r in Q } ,(RealPoset [.0,1.])) where y' is Element of X : verum }
Lemma118:
for X, Y, Z being non empty set
for R being RMembership_Func of X,Y
for S being RMembership_Func of Y,Z
for x being Element of X
for z being Element of Z holds
( rng (min R,S,x,z) = { ((R . [x,y]) "/\" (S . [y,z])) where y is Element of Y : verum } & rng (min R,S,x,z) <> {} )
Lemma119:
for X, Y, Z being non empty set
for R being RMembership_Func of X,Y
for S being RMembership_Func of Y,Z
for x being Element of X
for z being Element of Z holds (R (#) S) . [x,z] = "\/" { ((R . [x,y]) "/\" (S . [y,z])) where y is Element of Y : verum } ,(RealPoset [.0,1.])
Lemma120:
for X being non empty set
for R being RMembership_Func of X,X
for Q being Subset of (FuzzyLattice [:X,X:])
for x, z being Element of X holds { ("\/" { ((R . [x,y]) "/\" (r . [y,z])) where y is Element of X : verum } ,(RealPoset [.0,1.])) where r is Element of (FuzzyLattice [:X,X:]) : r in Q } = { ("\/" { ((R . [x,y]) "/\" ((@ r') . [y,z])) where y is Element of X : verum } ,(RealPoset [.0,1.])) where r' is Element of (FuzzyLattice [:X,X:]) : r' in Q }
Lemma122:
for X being non empty set
for R being RMembership_Func of X,X
for Q being Subset of (FuzzyLattice [:X,X:])
for x, z being Element of X holds { ("\/" { ((R . [x,y]) "/\" ((@ r) . [y,z])) where y is Element of X : verum } ,(RealPoset [.0,1.])) where r is Element of (FuzzyLattice [:X,X:]) : r in Q } = { ((R (#) (@ r)) . [x,z]) where r is Element of (FuzzyLattice [:X,X:]) : r in Q }
Lemma123:
for X being non empty set
for R being RMembership_Func of X,X
for Q being Subset of (FuzzyLattice [:X,X:])
for x, z being Element of X holds { ((R (#) (@ r)) . [x,z]) where r is Element of (FuzzyLattice [:X,X:]) : r in Q } = pi { (R (#) (@ r)) where r is Element of (FuzzyLattice [:X,X:]) : r in Q } ,[x,z]
Lemma124:
for X being non empty set
for R being RMembership_Func of X,X
for Q being Subset of (FuzzyLattice [:X,X:])
for x, z being Element of X holds "\/" { ("\/" { ((R . [x,y]) "/\" (r . [y,z])) where r is Element of (FuzzyLattice [:X,X:]) : r in Q } ,(RealPoset [.0,1.])) where y is Element of X : verum } ,(RealPoset [.0,1.]) = "\/" { ("\/" { ((R . [x,y]) "/\" (r' . [y,z])) where y is Element of X : verum } ,(RealPoset [.0,1.])) where r' is Element of (FuzzyLattice [:X,X:]) : r' in Q } ,(RealPoset [.0,1.])
theorem Th34: :: LFUZZY_1:34
theorem Th35: :: LFUZZY_1:35
theorem Th36: :: LFUZZY_1:36
theorem Th37: :: LFUZZY_1:37
theorem Th38: :: LFUZZY_1:38
theorem Th39: :: LFUZZY_1:39
theorem Th40: :: LFUZZY_1:40