:: REARRAN1 semantic presentation
Lemma31:
for f being Function
for x being set st not x in rng f holds
f " {x} = {}
:: deftheorem Def1 defines terms've_same_card_as_number REARRAN1:def 1 :
:: deftheorem Def2 defines ascending REARRAN1:def 2 :
Lemma39:
for D being non empty finite set
for A being FinSequence of bool D
for k being Element of NAT st 1 <= k & k <= len A holds
A . k is finite
Lemma41:
for D being non empty finite set
for A being FinSequence of bool D st len A = card D & A is terms've_same_card_as_number holds
for B being finite set st B = A . (len A) holds
B = D
Lemma46:
for D being non empty finite set ex B being FinSequence of bool D st
( len B = card D & B is ascending & B is terms've_same_card_as_number )
:: deftheorem Def3 defines lenght_equal_card_of_set REARRAN1:def 3 :
theorem Th1: :: REARRAN1:1
theorem Th2: :: REARRAN1:2
theorem Th3: :: REARRAN1:3
theorem Th4: :: REARRAN1:4
theorem Th5: :: REARRAN1:5
theorem Th6: :: REARRAN1:6
Lemma80:
for n being Element of NAT
for D being non empty finite set
for a being FinSequence of bool D st n in dom a holds
a . n c= D
theorem Th7: :: REARRAN1:7
theorem Th8: :: REARRAN1:8
theorem Th9: :: REARRAN1:9
theorem Th10: :: REARRAN1:10
:: deftheorem Def4 defines Co_Gen REARRAN1:def 4 :
theorem Th11: :: REARRAN1:11
theorem Th12: :: REARRAN1:12
definition
let D be non
empty finite set ,
C be non
empty finite set ;
let A be
RearrangmentGen of
C;
let F be
PartFunc of
D,
REAL ;
func Rland c4,
c3 -> PartFunc of
a2,
REAL equals :: REARRAN1:def 5
Sum ((MIM (FinS F,D)) (#) (CHI A,C));
correctness
coherence
Sum ((MIM (FinS F,D)) (#) (CHI A,C)) is PartFunc of C, REAL ;
;
func Rlor c4,
c3 -> PartFunc of
a2,
REAL equals :: REARRAN1:def 6
Sum ((MIM (FinS F,D)) (#) (CHI (Co_Gen A),C));
correctness
coherence
Sum ((MIM (FinS F,D)) (#) (CHI (Co_Gen A),C)) is PartFunc of C, REAL ;
;
end;
:: deftheorem Def5 defines Rland REARRAN1:def 5 :
:: deftheorem Def6 defines Rlor REARRAN1:def 6 :
theorem Th13: :: REARRAN1:13
theorem Th14: :: REARRAN1:14
theorem Th15: :: REARRAN1:15
theorem Th16: :: REARRAN1:16
theorem Th17: :: REARRAN1:17
theorem Th18: :: REARRAN1:18
theorem Th19: :: REARRAN1:19
theorem Th20: :: REARRAN1:20
theorem Th21: :: REARRAN1:21
theorem Th22: :: REARRAN1:22
theorem Th23: :: REARRAN1:23
theorem Th24: :: REARRAN1:24
theorem Th25: :: REARRAN1:25
theorem Th26: :: REARRAN1:26
theorem Th27: :: REARRAN1:27
theorem Th28: :: REARRAN1:28
for
D,
C being non
empty finite set for
F being
PartFunc of
D,
REAL for
A being
RearrangmentGen of
C st
F is
total &
card C = card D holds
(
Rlor F,
A,
Rland F,
A are_fiberwise_equipotent &
FinS (Rlor F,A),
C = FinS (Rland F,A),
C &
Sum (Rlor F,A),
C = Sum (Rland F,A),
C )
theorem Th29: :: REARRAN1:29
for
r being
Real for
D,
C being non
empty finite set for
F being
PartFunc of
D,
REAL for
A being
RearrangmentGen of
C st
F is
total &
card C = card D holds
(
max+ ((Rland F,A) - r),
max+ (F - r) are_fiberwise_equipotent &
FinS (max+ ((Rland F,A) - r)),
C = FinS (max+ (F - r)),
D &
Sum (max+ ((Rland F,A) - r)),
C = Sum (max+ (F - r)),
D )
theorem Th30: :: REARRAN1:30
for
r being
Real for
D,
C being non
empty finite set for
F being
PartFunc of
D,
REAL for
A being
RearrangmentGen of
C st
F is
total &
card C = card D holds
(
max- ((Rland F,A) - r),
max- (F - r) are_fiberwise_equipotent &
FinS (max- ((Rland F,A) - r)),
C = FinS (max- (F - r)),
D &
Sum (max- ((Rland F,A) - r)),
C = Sum (max- (F - r)),
D )
theorem Th31: :: REARRAN1:31
theorem Th32: :: REARRAN1:32
theorem Th33: :: REARRAN1:33
theorem Th34: :: REARRAN1:34
for
r being
Real for
D,
C being non
empty finite set for
F being
PartFunc of
D,
REAL for
A being
RearrangmentGen of
C st
F is
total &
card C = card D holds
(
max+ ((Rlor F,A) - r),
max+ (F - r) are_fiberwise_equipotent &
FinS (max+ ((Rlor F,A) - r)),
C = FinS (max+ (F - r)),
D &
Sum (max+ ((Rlor F,A) - r)),
C = Sum (max+ (F - r)),
D )
theorem Th35: :: REARRAN1:35
for
r being
Real for
D,
C being non
empty finite set for
F being
PartFunc of
D,
REAL for
A being
RearrangmentGen of
C st
F is
total &
card C = card D holds
(
max- ((Rlor F,A) - r),
max- (F - r) are_fiberwise_equipotent &
FinS (max- ((Rlor F,A) - r)),
C = FinS (max- (F - r)),
D &
Sum (max- ((Rlor F,A) - r)),
C = Sum (max- (F - r)),
D )
theorem Th36: :: REARRAN1:36
theorem Th37: :: REARRAN1:37
theorem Th38: :: REARRAN1:38
theorem Th39: :: REARRAN1:39