:: REARRAN1 semantic presentation
Lemma1:
for b1 being Function
for b2 being set holds
( not b2 in rng b1 implies b1 " {b2} = {} )
:: deftheorem Def1 defines terms've_same_card_as_number REARRAN1:def 1 :
:: deftheorem Def2 defines ascending REARRAN1:def 2 :
Lemma4:
for b1 being non empty finite set
for b2 being FinSequence of bool b1
for b3 being Nat holds
( 1 <= b3 & b3 <= len b2 implies b2 . b3 is finite )
Lemma5:
for b1 being non empty finite set
for b2 being FinSequence of bool b1 holds
( len b2 = card b1 & b2 is terms've_same_card_as_number implies for b3 being finite set holds
( b3 = b2 . (len b2) implies b3 = b1 ) )
Lemma6:
for b1 being non empty finite set holds
ex b2 being FinSequence of bool b1 st
( len b2 = card b1 & b2 is ascending & b2 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
Lemma14:
for b1 being Nat
for b2 being non empty finite set
for b3 being FinSequence of bool b2 holds
( b1 in dom b3 implies b3 . b1 c= b2 )
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 c
1, c
2 be non
empty finite set ;
let c
3 be
RearrangmentGen of c
2;
let c
4 be
PartFunc of c
1,
REAL ;
func Rland c
4,c
3 -> PartFunc of a
2,
REAL equals :: REARRAN1:def 5
Sum ((MIM (FinS a4,a1)) (#) (CHI a3,a2));
correctness
coherence
Sum ((MIM (FinS c4,c1)) (#) (CHI c3,c2)) is PartFunc of c2, REAL ;
;
func Rlor c
4,c
3 -> PartFunc of a
2,
REAL equals :: REARRAN1:def 6
Sum ((MIM (FinS a4,a1)) (#) (CHI (Co_Gen a3),a2));
correctness
coherence
Sum ((MIM (FinS c4,c1)) (#) (CHI (Co_Gen c3),c2)) is PartFunc of c2, 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 b
1, b
2 being non
empty finite set for b
3 being
PartFunc of b
1,
REAL for b
4 being
RearrangmentGen of b
2 holds
( b
3 is
total &
card b
2 = card b
1 implies (
Rlor b
3,b
4,
Rland b
3,b
4 are_fiberwise_equipotent &
FinS (Rlor b3,b4),b
2 = FinS (Rland b3,b4),b
2 &
Sum (Rlor b3,b4),b
2 = Sum (Rland b3,b4),b
2 ) )
theorem Th29: :: REARRAN1:29
for b
1 being
Realfor b
2, b
3 being non
empty finite set for b
4 being
PartFunc of b
2,
REAL for b
5 being
RearrangmentGen of b
3 holds
( b
4 is
total &
card b
3 = card b
2 implies (
max+ ((Rland b4,b5) - b1),
max+ (b4 - b1) are_fiberwise_equipotent &
FinS (max+ ((Rland b4,b5) - b1)),b
3 = FinS (max+ (b4 - b1)),b
2 &
Sum (max+ ((Rland b4,b5) - b1)),b
3 = Sum (max+ (b4 - b1)),b
2 ) )
theorem Th30: :: REARRAN1:30
for b
1 being
Realfor b
2, b
3 being non
empty finite set for b
4 being
PartFunc of b
2,
REAL for b
5 being
RearrangmentGen of b
3 holds
( b
4 is
total &
card b
3 = card b
2 implies (
max- ((Rland b4,b5) - b1),
max- (b4 - b1) are_fiberwise_equipotent &
FinS (max- ((Rland b4,b5) - b1)),b
3 = FinS (max- (b4 - b1)),b
2 &
Sum (max- ((Rland b4,b5) - b1)),b
3 = Sum (max- (b4 - b1)),b
2 ) )
theorem Th31: :: REARRAN1:31
theorem Th32: :: REARRAN1:32
theorem Th33: :: REARRAN1:33
theorem Th34: :: REARRAN1:34
for b
1 being
Realfor b
2, b
3 being non
empty finite set for b
4 being
PartFunc of b
2,
REAL for b
5 being
RearrangmentGen of b
3 holds
( b
4 is
total &
card b
3 = card b
2 implies (
max+ ((Rlor b4,b5) - b1),
max+ (b4 - b1) are_fiberwise_equipotent &
FinS (max+ ((Rlor b4,b5) - b1)),b
3 = FinS (max+ (b4 - b1)),b
2 &
Sum (max+ ((Rlor b4,b5) - b1)),b
3 = Sum (max+ (b4 - b1)),b
2 ) )
theorem Th35: :: REARRAN1:35
for b
1 being
Realfor b
2, b
3 being non
empty finite set for b
4 being
PartFunc of b
2,
REAL for b
5 being
RearrangmentGen of b
3 holds
( b
4 is
total &
card b
3 = card b
2 implies (
max- ((Rlor b4,b5) - b1),
max- (b4 - b1) are_fiberwise_equipotent &
FinS (max- ((Rlor b4,b5) - b1)),b
3 = FinS (max- (b4 - b1)),b
2 &
Sum (max- ((Rlor b4,b5) - b1)),b
3 = Sum (max- (b4 - b1)),b
2 ) )
theorem Th36: :: REARRAN1:36
theorem Th37: :: REARRAN1:37
theorem Th38: :: REARRAN1:38
theorem Th39: :: REARRAN1:39