:: FUZZY_2 semantic presentation

theorem Th1: :: FUZZY_2:1
for C being non empty set
for x being Element of C
for h being Membership_Func of C holds
( 0 <= h . x & h . x <= 1 )
proof end;

theorem Th2: :: FUZZY_2:2
for C being non empty set
for f, h, g being Membership_Func of C st h c= holds
max f,(min g,h) = min (max f,g),h
proof end;

definition
let C be non empty set ;
let f be Membership_Func of C, g be Membership_Func of C;
func c2 \ c3 -> Membership_Func of a1 equals :: FUZZY_2:def 1
min f,(1_minus g);
correctness
coherence
min f,(1_minus g) is Membership_Func of C
;
;
end;

:: deftheorem Def1 defines \ FUZZY_2:def 1 :
for C being non empty set
for f, g being Membership_Func of C holds f \ g = min f,(1_minus g);

theorem Th3: :: FUZZY_2:3
for C being non empty set
for f, g being Membership_Func of C holds 1_minus (f \ g) = max (1_minus f),g
proof end;

theorem Th4: :: FUZZY_2:4
for C being non empty set
for f, g, h being Membership_Func of C st g c= holds
g \ h c=
proof end;

theorem Th5: :: FUZZY_2:5
for C being non empty set
for f, g, h being Membership_Func of C st g c= holds
h \ f c=
proof end;

theorem Th6: :: FUZZY_2:6
for C being non empty set
for f, g, h, h1 being Membership_Func of C st g c= & h1 c= holds
g \ h c=
proof end;

theorem Th7: :: FUZZY_2:7
for C being non empty set
for f, g being Membership_Func of C holds f c= by FUZZY_1:18;

theorem Th8: :: FUZZY_2:8
for C being non empty set
for f, g being Membership_Func of C holds f \+\ g c= by FUZZY_1:18;

theorem Th9: :: FUZZY_2:9
for C being non empty set
for f being Membership_Func of C holds f \ (EMF C) = f
proof end;

theorem Th10: :: FUZZY_2:10
for C being non empty set
for f being Membership_Func of C holds (EMF C) \ f = EMF C by FUZZY_1:19;

theorem Th11: :: FUZZY_2:11
for C being non empty set
for f, g being Membership_Func of C holds f \ (min f,g) c=
proof end;

theorem Th12: :: FUZZY_2:12
for C being non empty set
for f, g being Membership_Func of C holds f c=
proof end;

theorem Th13: :: FUZZY_2:13
for C being non empty set
for f, g being Membership_Func of C holds max f,g c=
proof end;

theorem Th14: :: FUZZY_2:14
for C being non empty set
for f, g, h being Membership_Func of C holds f \ (g \ h) = max (f \ g),(min f,h)
proof end;

theorem Th15: :: FUZZY_2:15
for C being non empty set
for f, g being Membership_Func of C holds f \ (f \ g) c=
proof end;

theorem Th16: :: FUZZY_2:16
for C being non empty set
for f, g being Membership_Func of C holds (max f,g) \ g c=
proof end;

theorem Th17: :: FUZZY_2:17
for C being non empty set
for f, g, h being Membership_Func of C holds f \ (max g,h) = min (f \ g),(f \ h)
proof end;

theorem Th18: :: FUZZY_2:18
for C being non empty set
for f, g, h being Membership_Func of C holds f \ (min g,h) = max (f \ g),(f \ h)
proof end;

theorem Th19: :: FUZZY_2:19
for C being non empty set
for f, g, h being Membership_Func of C holds (f \ g) \ h = f \ (max g,h)
proof end;

theorem Th20: :: FUZZY_2:20
for C being non empty set
for f, g being Membership_Func of C holds (max f,g) \ (min f,g) c=
proof end;

theorem Th21: :: FUZZY_2:21
for C being non empty set
for f, g, h being Membership_Func of C holds (max f,g) \ h = max (f \ h),(g \ h) by FUZZY_1:10;

theorem Th22: :: FUZZY_2:22
for C being non empty set
for f, g, h being Membership_Func of C st h c= & h c= holds
h c=
proof end;

theorem Th23: :: FUZZY_2:23
for C being non empty set
for f, g, h being Membership_Func of C holds min f,(g \ h) = (min f,g) \ h by FUZZY_1:8;

theorem Th24: :: FUZZY_2:24
for C being non empty set
for f, g, h being Membership_Func of C holds (min f,g) \ (min f,h) c=
proof end;

theorem Th25: :: FUZZY_2:25
for C being non empty set
for f, g being Membership_Func of C holds (max f,g) \ (min f,g) c=
proof end;

theorem Th26: :: FUZZY_2:26
for C being non empty set
for f, g being Membership_Func of C holds 1_minus (f \+\ g) c=
proof end;

theorem Th27: :: FUZZY_2:27
for C being non empty set
for f, g, h being Membership_Func of C holds (f \+\ g) \ h = max (f \ (max g,h)),(g \ (max f,h))
proof end;

theorem Th28: :: FUZZY_2:28
for C being non empty set
for f, g, h being Membership_Func of C holds f \ (g \+\ h) c=
proof end;

theorem Th29: :: FUZZY_2:29
for C being non empty set
for f, g being Membership_Func of C st g c= holds
g c=
proof end;

theorem Th30: :: FUZZY_2:30
for C being non empty set
for f, g being Membership_Func of C holds max f,g c=
proof end;

theorem Th31: :: FUZZY_2:31
for C being non empty set
for f, g being Membership_Func of C st f \ g = EMF C holds
g c=
proof end;

theorem Th32: :: FUZZY_2:32
for C being non empty set
for f, g being Membership_Func of C st min f,g = EMF C holds
f \ g = f
proof end;

definition
let C be non empty set ;
let h be Membership_Func of C, g be Membership_Func of C;
func c2 * c3 -> Membership_Func of a1 means :Def2: :: FUZZY_2:def 2
for c being Element of C holds it . c = (h . c) * (g . c);
existence
ex b1 being Membership_Func of C st
for c being Element of C holds b1 . c = (h . c) * (g . c)
proof end;
uniqueness
for b1, b2 being Membership_Func of C st ( for c being Element of C holds b1 . c = (h . c) * (g . c) ) & ( for c being Element of C holds b2 . c = (h . c) * (g . c) ) holds
b1 = b2
proof end;
commutativity
for b1, h, g being Membership_Func of C st ( for c being Element of C holds b1 . c = (h . c) * (g . c) ) holds
for c being Element of C holds b1 . c = (g . c) * (h . c)
;
end;

:: deftheorem Def2 defines * FUZZY_2:def 2 :
for C being non empty set
for h, g, b4 being Membership_Func of C holds
( b4 = h * g iff for c being Element of C holds b4 . c = (h . c) * (g . c) );

definition
let C be non empty set ;
let h be Membership_Func of C, g be Membership_Func of C;
func c2 ++ c3 -> Membership_Func of a1 means :Def3: :: FUZZY_2:def 3
for c being Element of C holds it . c = ((h . c) + (g . c)) - ((h . c) * (g . c));
existence
ex b1 being Membership_Func of C st
for c being Element of C holds b1 . c = ((h . c) + (g . c)) - ((h . c) * (g . c))
proof end;
uniqueness
for b1, b2 being Membership_Func of C st ( for c being Element of C holds b1 . c = ((h . c) + (g . c)) - ((h . c) * (g . c)) ) & ( for c being Element of C holds b2 . c = ((h . c) + (g . c)) - ((h . c) * (g . c)) ) holds
b1 = b2
proof end;
commutativity
for b1, h, g being Membership_Func of C st ( for c being Element of C holds b1 . c = ((h . c) + (g . c)) - ((h . c) * (g . c)) ) holds
for c being Element of C holds b1 . c = ((g . c) + (h . c)) - ((g . c) * (h . c))
;
end;

:: deftheorem Def3 defines ++ FUZZY_2:def 3 :
for C being non empty set
for h, g, b4 being Membership_Func of C holds
( b4 = h ++ g iff for c being Element of C holds b4 . c = ((h . c) + (g . c)) - ((h . c) * (g . c)) );

theorem Th33: :: FUZZY_2:33
for C being non empty set
for f being Membership_Func of C holds
( f c= & f ++ f c= )
proof end;

theorem Th34: :: FUZZY_2:34
for C being non empty set
for f, g, h being Membership_Func of C holds (f * g) * h = f * (g * h)
proof end;

theorem Th35: :: FUZZY_2:35
for C being non empty set
for f, g, h being Membership_Func of C holds (f ++ g) ++ h = f ++ (g ++ h)
proof end;

theorem Th36: :: FUZZY_2:36
for C being non empty set
for f, g being Membership_Func of C holds
( f c= & f ++ (f * g) c= )
proof end;

theorem Th37: :: FUZZY_2:37
for C being non empty set
for f, g, h being Membership_Func of C holds (f * g) ++ (f * h) c=
proof end;

theorem Th38: :: FUZZY_2:38
for C being non empty set
for f, g, h being Membership_Func of C holds f ++ (g * h) c=
proof end;

theorem Th39: :: FUZZY_2:39
for C being non empty set
for f, g being Membership_Func of C holds 1_minus (f * g) = (1_minus f) ++ (1_minus g)
proof end;

theorem Th40: :: FUZZY_2:40
for C being non empty set
for f, g being Membership_Func of C holds 1_minus (f ++ g) = (1_minus f) * (1_minus g)
proof end;

theorem Th41: :: FUZZY_2:41
for C being non empty set
for f, g being Membership_Func of C holds f ++ g = 1_minus ((1_minus f) * (1_minus g))
proof end;

theorem Th42: :: FUZZY_2:42
for C being non empty set
for f being Membership_Func of C holds
( f * (EMF C) = EMF C & f * (UMF C) = f )
proof end;

theorem Th43: :: FUZZY_2:43
for C being non empty set
for f being Membership_Func of C holds
( f ++ (EMF C) = f & f ++ (UMF C) = UMF C )
proof end;

theorem Th44: :: FUZZY_2:44
for C being non empty set
for f, g being Membership_Func of C holds min f,g c=
proof end;

theorem Th45: :: FUZZY_2:45
for C being non empty set
for f, g being Membership_Func of C holds f ++ g c=
proof end;

Lemma55: for a, b, c being Real st 0 <= c holds
c * (min a,b) = min (c * a),(c * b)
proof end;

Lemma58: for a, b, c being Real st 0 <= c holds
c * (max a,b) = max (c * a),(c * b)
proof end;

theorem Th46: :: FUZZY_2:46
for a, b, c being Real st 0 <= c holds
( c * (max a,b) = max (c * a),(c * b) & c * (min a,b) = min (c * a),(c * b) ) by , ;

theorem Th47: :: FUZZY_2:47
for a, b, c being Real holds
( c + (max a,b) = max (c + a),(c + b) & c + (min a,b) = min (c + a),(c + b) )
proof end;

theorem Th48: :: FUZZY_2:48
for C being non empty set
for f, g, h being Membership_Func of C holds f * (max g,h) = max (f * g),(f * h)
proof end;

theorem Th49: :: FUZZY_2:49
for C being non empty set
for f, g, h being Membership_Func of C holds f * (min g,h) = min (f * g),(f * h)
proof end;

theorem Th50: :: FUZZY_2:50
for C being non empty set
for f, g, h being Membership_Func of C holds f ++ (max g,h) = max (f ++ g),(f ++ h)
proof end;

theorem Th51: :: FUZZY_2:51
for C being non empty set
for f, g, h being Membership_Func of C holds f ++ (min g,h) = min (f ++ g),(f ++ h)
proof end;

theorem Th52: :: FUZZY_2:52
for C being non empty set
for f, g, h being Membership_Func of C holds max f,(g * h) c=
proof end;

theorem Th53: :: FUZZY_2:53
for C being non empty set
for f, g, h being Membership_Func of C holds min f,(g * h) c=
proof end;

theorem Th54: :: FUZZY_2:54
for C being non empty set
for c being Element of C
for f, g being Membership_Func of C holds (f ++ g) . c = 1 - ((1 - (f . c)) * (1 - (g . c)))
proof end;

theorem Th55: :: FUZZY_2:55
for C being non empty set
for f, g, h being Membership_Func of C holds (max f,g) ++ (max f,h) c=
proof end;

theorem Th56: :: FUZZY_2:56
for C being non empty set
for f, g, h being Membership_Func of C holds (min f,g) ++ (min f,h) c=
proof end;