:: CONVEX1 semantic presentation
:: deftheorem Def1 defines * CONVEX1:def 1 :
:: deftheorem Def2 defines convex CONVEX1:def 2 :
theorem Th1: :: CONVEX1:1
theorem Th2: :: CONVEX1:2
theorem Th3: :: CONVEX1:3
theorem Th4: :: CONVEX1:4
theorem Th5: :: CONVEX1:5
theorem Th6: :: CONVEX1:6
Lemma64:
for V being non empty RealLinearSpace-like RLSStruct
for M being Subset of V holds 1 * M = M
Lemma65:
for V being RealLinearSpace
for M being non empty Subset of V holds 0 * M = {(0. V)}
Lemma66:
for V being non empty add-associative LoopStr
for M1, M2, M3 being Subset of V holds (M1 + M2) + M3 = M1 + (M2 + M3)
Lemma72:
for V being non empty RealLinearSpace-like RLSStruct
for M being Subset of V
for r1, r2 being Real holds r1 * (r2 * M) = (r1 * r2) * M
Lemma75:
for V being non empty RealLinearSpace-like RLSStruct
for M1, M2 being Subset of V
for r being Real holds r * (M1 + M2) = (r * M1) + (r * M2)
theorem Th7: :: CONVEX1:7
theorem Th8: :: CONVEX1:8
theorem Th9: :: CONVEX1:9
theorem Th10: :: CONVEX1:10
theorem Th11: :: CONVEX1:11
theorem Th12: :: CONVEX1:12
Lemma84:
for V being non empty RLSStruct
for M, N being Subset of V
for r being Real st M c= N holds
r * M c= r * N
Lemma85:
for V being non empty RLSStruct
for M being empty Subset of V
for r being Real holds r * M = {}
Lemma86:
for V being non empty LoopStr
for M being empty Subset of V
for N being Subset of V holds M + N = {}
Lemma87:
for V being non empty right_zeroed LoopStr
for M being Subset of V holds M + {(0. V)} = M
Lemma88:
for V being RealLinearSpace
for M being Subset of V
for r1, r2 being Real st r1 >= 0 & r2 >= 0 & M is convex holds
(r1 * M) + (r2 * M) c= (r1 + r2) * M
theorem Th13: :: CONVEX1:13
theorem Th14: :: CONVEX1:14
theorem Th15: :: CONVEX1:15
theorem Th16: :: CONVEX1:16
theorem Th17: :: CONVEX1:17
theorem Th18: :: CONVEX1:18
theorem Th19: :: CONVEX1:19
theorem Th20: :: CONVEX1:20
:: deftheorem Def3 defines convex CONVEX1:def 3 :
theorem Th21: :: CONVEX1:21
theorem Th22: :: CONVEX1:22
theorem Th23: :: CONVEX1:23
Lemma97:
for V being RealLinearSpace
for v being VECTOR of V
for L being Linear_Combination of V st L is convex & Carrier L = {v} holds
( L . v = 1 & Sum L = (L . v) * v )
Lemma98:
for V being RealLinearSpace
for v1, v2 being VECTOR of V
for L being Linear_Combination of V st L is convex & Carrier L = {v1,v2} & v1 <> v2 holds
( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 & Sum L = ((L . v1) * v1) + ((L . v2) * v2) )
Lemma99:
for p being FinSequence
for x, y, z being set st p is one-to-one & rng p = {x,y,z} & x <> y & y <> z & z <> x & not p = <*x,y,z*> & not p = <*x,z,y*> & not p = <*y,x,z*> & not p = <*y,z,x*> & not p = <*z,x,y*> holds
p = <*z,y,x*>
Lemma101:
for V being RealLinearSpace
for v1, v2, v3 being VECTOR of V
for L being Linear_Combination of {v1,v2,v3} st v1 <> v2 & v2 <> v3 & v3 <> v1 holds
Sum L = (((L . v1) * v1) + ((L . v2) * v2)) + ((L . v3) * v3)
Lemma105:
for V being RealLinearSpace
for v1, v2, v3 being VECTOR of V
for L being Linear_Combination of V st L is convex & Carrier L = {v1,v2,v3} & v1 <> v2 & v2 <> v3 & v3 <> v1 holds
( ((L . v1) + (L . v2)) + (L . v3) = 1 & L . v1 >= 0 & L . v2 >= 0 & L . v3 >= 0 & Sum L = (((L . v1) * v1) + ((L . v2) * v2)) + ((L . v3) * v3) )
theorem Th24: :: CONVEX1:24
theorem Th25: :: CONVEX1:25
theorem Th26: :: CONVEX1:26
theorem Th27: :: CONVEX1:27
theorem Th28: :: CONVEX1:28
theorem Th29: :: CONVEX1:29
:: deftheorem Def4 defines Convex-Family CONVEX1:def 4 :
:: deftheorem Def5 defines conv CONVEX1:def 5 :
theorem Th30: :: CONVEX1:30
theorem Th31: :: CONVEX1:31
for
p being
FinSequence for
x,
y,
z being
set st
p is
one-to-one &
rng p = {x,y,z} &
x <> y &
y <> z &
z <> x & not
p = <*x,y,z*> & not
p = <*x,z,y*> & not
p = <*y,x,z*> & not
p = <*y,z,x*> & not
p = <*z,x,y*> holds
p = <*z,y,x*> by ;
theorem Th32: :: CONVEX1:32
theorem Th33: :: CONVEX1:33
theorem Th34: :: CONVEX1:34
theorem Th35: :: CONVEX1:35
theorem Th36: :: CONVEX1:36
theorem Th37: :: CONVEX1:37
theorem Th38: :: CONVEX1:38
theorem Th39: :: CONVEX1:39
theorem Th40: :: CONVEX1:40