:: LFUZZY_0 semantic presentation
:: deftheorem Def1 defines real LFUZZY_0:def 1 :
:: deftheorem Def2 defines interval LFUZZY_0:def 2 :
theorem Th1: :: LFUZZY_0:1
theorem Th2: :: LFUZZY_0:2
:: deftheorem Def3 defines RealPoset LFUZZY_0:def 3 :
theorem Th3: :: LFUZZY_0:3
theorem Th4: :: LFUZZY_0:4
theorem Th5: :: LFUZZY_0:5
theorem Th6: :: LFUZZY_0:6
theorem Th7: :: LFUZZY_0:7
theorem Th8: :: LFUZZY_0:8
theorem Th9: :: LFUZZY_0:9
theorem Th10: :: LFUZZY_0:10
theorem Th11: :: LFUZZY_0:11
theorem Th12: :: LFUZZY_0:12
theorem Th13: :: LFUZZY_0:13
:: deftheorem Def4 defines FuzzyLattice LFUZZY_0:def 4 :
theorem Th14: :: LFUZZY_0:14
Lemma18:
for b1 being non empty set holds FuzzyLattice b1 is constituted-Functions
theorem Th15: :: LFUZZY_0:15
Lemma20:
for b1 being non empty set
for b2 being Element of (FuzzyLattice b1) holds
b2 is Membership_Func of b1
:: deftheorem Def5 defines @ LFUZZY_0:def 5 :
Lemma21:
for b1 being non empty set
for b2 being Membership_Func of b1 holds
b2 is Element of (FuzzyLattice b1)
:: deftheorem Def6 defines @ LFUZZY_0:def 6 :
theorem Th16: :: LFUZZY_0:16
theorem Th17: :: LFUZZY_0:17
theorem Th18: :: LFUZZY_0:18
theorem Th19: :: LFUZZY_0:19
theorem Th20: :: LFUZZY_0:20
theorem Th21: :: LFUZZY_0:21
scheme :: LFUZZY_0:sch 1
s1{ F
1()
-> complete LATTICE, F
2()
-> non
empty set , F
3()
-> non
empty set , F
4(
set ,
set )
-> Element of F
1(), P
1[
set ], P
2[
set ] } :
"\/" { ("\/" { F4(b1,b2) where B is Element of F3() : P2[b2] } ,F1()) where B is Element of F2() : P1[b1] } ,F
1()
= "\/" { F4(b1,b2) where B is Element of F2(), B is Element of F3() : ( P1[b1] & P2[b2] ) } ,F
1()
scheme :: LFUZZY_0:sch 2
s2{ F
1()
-> complete LATTICE, F
2()
-> non
empty set , F
3()
-> non
empty set , F
4(
set ,
set )
-> Element of F
1(), P
1[
set ], P
2[
set ] } :
"\/" { ("\/" { F4(b2,b1) where B is Element of F2() : P1[b2] } ,F1()) where B is Element of F3() : P2[b1] } ,F
1()
= "\/" { F4(b1,b2) where B is Element of F2(), B is Element of F3() : ( P1[b1] & P2[b2] ) } ,F
1()
scheme :: LFUZZY_0:sch 3
s3{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3(
set ,
set )
-> set , F
4(
set ,
set )
-> set , P
1[
set ,
set ] } :
{ F3(b1,b2) where B is Element of F1(), B is Element of F2() : P1[b1,b2] } = { F4(b1,b2) where B is Element of F1(), B is Element of F2() : P1[b1,b2] }
provided
E25:
for b
1 being
Element of F
1()
for b
2 being
Element of F
2() holds
( P
1[b
1,b
2] implies F
3(b
1,b
2)
= F
4(b
1,b
2) )
scheme :: LFUZZY_0:sch 4
s4{ F
1()
-> non
empty set , F
2()
-> non
empty set , F
3(
set ,
set )
-> set , F
4(
set ,
set )
-> set , P
1[
set ,
set ], P
2[
set ,
set ] } :
{ F3(b1,b2) where B is Element of F1(), B is Element of F2() : P1[b1,b2] } = { F4(b1,b2) where B is Element of F1(), B is Element of F2() : P2[b1,b2] }
provided
E25:
for b
1 being
Element of F
1()
for b
2 being
Element of F
2() holds
( P
1[b
1,b
2] iff P
2[b
1,b
2] )
and
E26:
for b
1 being
Element of F
1()
for b
2 being
Element of F
2() holds
( P
1[b
1,b
2] implies F
3(b
1,b
2)
= F
4(b
1,b
2) )
scheme :: LFUZZY_0:sch 5
s5{ F
1()
-> complete LATTICE, F
2()
-> non
empty set , F
3()
-> non
empty set , F
4(
set ,
set )
-> Element of F
1(), F
5(
set ,
set )
-> Element of F
1(), P
1[
set ], P
2[
set ] } :
"\/" { ("\/" { F4(b1,b2) where B is Element of F3() : P2[b2] } ,F1()) where B is Element of F2() : P1[b1] } ,F
1()
= "\/" { ("\/" { F5(b2,b1) where B is Element of F2() : P1[b2] } ,F1()) where B is Element of F3() : P2[b1] } ,F
1()
provided
E25:
for b
1 being
Element of F
2()
for b
2 being
Element of F
3() holds
( P
1[b
1] & P
2[b
2] implies F
4(b
1,b
2)
= F
5(b
1,b
2) )
Lemma25:
for b1 being Element of (RealPoset [.0,1.]) holds
b1 is Real
Lemma26:
for b1, b2, b3 being non empty set
for b4 being RMembership_Func of b1,b2
for b5 being RMembership_Func of b2,b3
for b6 being Element of b1
for b7 being Element of b3 holds
( rng (min b4,b5,b6,b7) = { ((b4 . [b6,b8]) "/\" (b5 . [b8,b7])) where B is Element of b2 : verum } & rng (min b4,b5,b6,b7) <> {} )
theorem Th22: :: LFUZZY_0:22
Lemma28:
for b1, b2, b3 being non empty set
for b4 being RMembership_Func of b1,b2
for b5 being RMembership_Func of b2,b3
for b6 being Element of b1
for b7 being Element of b3
for b8 being Element of (RealPoset [.0,1.]) holds ((b4 (#) b5) . [b6,b7]) "/\" b8 = "\/" { (((b4 . [b6,b9]) "/\" (b5 . [b9,b7])) "/\" b8) where B is Element of b2 : verum } ,(RealPoset [.0,1.])
Lemma29:
for b1, b2, b3 being non empty set
for b4 being RMembership_Func of b1,b2
for b5 being RMembership_Func of b2,b3
for b6 being Element of b1
for b7 being Element of b3
for b8 being Element of (RealPoset [.0,1.]) holds b8 "/\" ((b4 (#) b5) . [b6,b7]) = "\/" { ((b8 "/\" (b4 . [b6,b9])) "/\" (b5 . [b9,b7])) where B is Element of b2 : verum } ,(RealPoset [.0,1.])
theorem Th23: :: LFUZZY_0:23