:: Introduction to Theory of Rearrangment
:: by Yuji Sakai and Jaros{\l}aw Kotowicz
::
:: Received May 22, 1993
:: Copyright (c) 1993-2011 Association of Mizar Users

begin

definition

let D be non empty set ;
let E be real-membered set ;
let F be PartFunc of D,E;
let r be Real;
:: original: (#)
redefine func r (#) F -> Element of PFuncs (D,REAL);
coherence
r (#) F is Element of PFuncs (D,REAL)

proof end;

end;

Lm1: for f being Function
for x being set st not x in rng f holds
f " {x} = {}

proof end;

definition

let IT be FinSequence;
attr IT is terms've_same_card_as_number means :Def1: :: REARRAN1:def 1
for n being Nat st 1 <= n & n <= len IT holds
for B being finite set st B = IT . n holds
card B = n;
attr IT is ascending means :Def2: :: REARRAN1:def 2
for n being Nat st 1 <= n & n <= (len IT) - 1 holds
IT . n c= IT . (n + 1);

end;

:: deftheorem Def1 defines terms've_same_card_as_number REARRAN1:def 1 :
for IT being FinSequence holds
( IT is terms've_same_card_as_number iff for n being Nat st 1 <= n & n <= len IT holds
for B being finite set st B = IT . n holds
card B = n );

:: deftheorem Def2 defines ascending REARRAN1:def 2 :
for IT being FinSequence holds
( IT is ascending iff for n being Nat st 1 <= n & n <= (len IT) - 1 holds
IT . n c= IT . (n + 1) );

Lm2: 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

proof end;

Lm3: 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

proof end;

Lm4: 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 )

proof end;

definition

let X be set ;
let IT be FinSequence of X;
attr IT is lenght_equal_card_of_set means :Def3: :: REARRAN1:def 3
ex B being finite set st
( B = union X & len IT = card B );

end;

:: deftheorem Def3 defines lenght_equal_card_of_set REARRAN1:def 3 :
for X being set
for IT being FinSequence of X holds
( IT is lenght_equal_card_of_set iff ex B being finite set st
( B = union X & len IT = card B ) );

registration

let D be non empty finite set ;
cluster Relation-like NAT -defined bool D -valued Function-like finite FinSequence-like FinSubsequence-like terms've_same_card_as_number ascending lenght_equal_card_of_set FinSequence of bool D;
existence
ex b₁ being FinSequence of bool D st
( b₁ is terms've_same_card_as_number & b₁ is ascending & b₁ is lenght_equal_card_of_set )

proof end;

end;

definition

let D be non empty finite set ;
mode RearrangmentGen of D is terms've_same_card_as_number ascending lenght_equal_card_of_set FinSequence of bool D;

end;

theorem Th1: :: REARRAN1:1

for D being non empty finite set
for a being FinSequence of bool D holds
( a is lenght_equal_card_of_set iff len a = card D )

proof end;

theorem Th2: :: REARRAN1:2

for a being FinSequence holds
( a is ascending iff for n, m being Element of NAT st n <= m & n in dom a & m in dom a holds
a . n c= a . m )

proof end;

theorem Th3: :: REARRAN1:3

for D being non empty finite set
for a being terms've_same_card_as_number lenght_equal_card_of_set FinSequence of bool D holds a . (len a) = D

proof end;

theorem Th4: :: REARRAN1:4

for D being non empty finite set
for a being lenght_equal_card_of_set FinSequence of bool D holds len a <> 0

proof end;

theorem Th5: :: REARRAN1:5

for D being non empty finite set
for a being terms've_same_card_as_number ascending FinSequence of bool D
for n, m being Element of NAT st n in dom a & m in dom a & n <> m holds
a . n <> a . m

proof end;

theorem :: REARRAN1:6

for D being non empty finite set
for a being terms've_same_card_as_number ascending FinSequence of bool D
for n being Element of NAT st 1 <= n & n <= (len a) - 1 holds
a . n <> a . (n + 1)

proof end;

Lm5: 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

proof end;

theorem Th7: :: REARRAN1:7

for n being Element of NAT
for D being non empty finite set
for a being terms've_same_card_as_number FinSequence of bool D st n in dom a holds
a . n <> {}

proof end;

theorem Th8: :: REARRAN1:8

for n being Element of NAT
for D being non empty finite set
for a being terms've_same_card_as_number FinSequence of bool D st 1 <= n & n <= (len a) - 1 holds
(a . (n + 1)) \ (a . n) <> {}

proof end;

theorem Th9: :: REARRAN1:9

for D being non empty finite set
for a being terms've_same_card_as_number lenght_equal_card_of_set FinSequence of bool D ex d being Element of D st a . 1 = {d}

proof end;

theorem Th10: :: REARRAN1:10

for n being Element of NAT
for D being non empty finite set
for a being terms've_same_card_as_number ascending FinSequence of bool D st 1 <= n & n <= (len a) - 1 holds
ex d being Element of D st
( (a . (n + 1)) \ (a . n) = {d} & a . (n + 1) = (a . n) \/ {d} & (a . (n + 1)) \ {d} = a . n )

proof end;

definition

let D be non empty finite set ;
let A be RearrangmentGen of D;
func Co_Gen A -> RearrangmentGen of D means :Def4: :: REARRAN1:def 4
for m being Nat st 1 <= m & m <= (len it) - 1 holds
it . m = D \ (A . ((len A) - m));
existence
ex b₁ being RearrangmentGen of D st
for m being Nat st 1 <= m & m <= (len b₁) - 1 holds
b₁ . m = D \ (A . ((len A) - m))

proof end;

uniqueness
for b₁, b₂ being RearrangmentGen of D st ( for m being Nat st 1 <= m & m <= (len b₁) - 1 holds
b₁ . m = D \ (A . ((len A) - m)) ) & ( for m being Nat st 1 <= m & m <= (len b₂) - 1 holds
b₂ . m = D \ (A . ((len A) - m)) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def4 defines Co_Gen REARRAN1:def 4 :
for D being non empty finite set
for A, b₃ being RearrangmentGen of D holds
( b₃ = Co_Gen A iff for m being Nat st 1 <= m & m <= (len b₃) - 1 holds
b₃ . m = D \ (A . ((len A) - m)) );

theorem :: REARRAN1:11

for D being non empty finite set
for A being RearrangmentGen of D holds Co_Gen (Co_Gen A) = A

proof end;

theorem Th12: :: REARRAN1:12

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
len (MIM (FinS (F,D))) = len (CHI (A,C))

proof end;

definition

let D, C be non empty finite set ;
let A be RearrangmentGen of C;
let F be PartFunc of D,REAL;
func Rland (F,A) -> PartFunc of C,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 (F,A) -> PartFunc of C,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 defines Rland REARRAN1:def 5 :
for D, C being non empty finite set
for A being RearrangmentGen of C
for F being PartFunc of D,REAL holds Rland (F,A) = Sum ((MIM (FinS (F,D))) (#) (CHI (A,C)));

:: deftheorem defines Rlor REARRAN1:def 6 :
for D, C being non empty finite set
for A being RearrangmentGen of C
for F being PartFunc of D,REAL holds Rlor (F,A) = Sum ((MIM (FinS (F,D))) (#) (CHI ((Co_Gen A),C)));

theorem Th13: :: REARRAN1:13

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
dom (Rland (F,A)) = C

proof end;

theorem Th14: :: REARRAN1:14

for C, D being non empty finite set
for c being Element of C
for F being PartFunc of D,REAL
for A being RearrangmentGen of C st F is total & card C = card D holds
( ( c in A . 1 implies ((MIM (FinS (F,D))) (#) (CHI (A,C))) # c = MIM (FinS (F,D)) ) & ( for n being Element of NAT st 1 <= n & n < len A & c in (A . (n + 1)) \ (A . n) holds
((MIM (FinS (F,D))) (#) (CHI (A,C))) # c = (n |-> 0) ^ (MIM ((FinS (F,D)) /^ n)) ) )

proof end;

theorem Th15: :: REARRAN1:15

for C, D being non empty finite set
for c being Element of C
for F being PartFunc of D,REAL
for A being RearrangmentGen of C st F is total & card C = card D holds
( ( c in A . 1 implies (Rland (F,A)) . c = (FinS (F,D)) . 1 ) & ( for n being Element of NAT st 1 <= n & n < len A & c in (A . (n + 1)) \ (A . n) holds
(Rland (F,A)) . c = (FinS (F,D)) . (n + 1) ) )

proof end;

theorem Th16: :: REARRAN1:16

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
rng (Rland (F,A)) = rng (FinS (F,D))

proof end;

theorem Th17: :: REARRAN1:17

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
Rland (F,A), FinS (F,D) are_fiberwise_equipotent

proof end;

theorem Th18: :: REARRAN1:18

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
FinS ((Rland (F,A)),C) = FinS (F,D)

proof end;

theorem Th19: :: REARRAN1:19

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
Sum ((Rland (F,A)),C) = Sum (F,D)

proof end;

theorem :: REARRAN1:20

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
( FinS (((Rland (F,A)) - r),C) = FinS ((F - r),D) & Sum (((Rland (F,A)) - r),C) = Sum ((F - r),D) )

proof end;

theorem Th21: :: REARRAN1:21

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
dom (Rlor (F,A)) = C

proof end;

theorem Th22: :: REARRAN1:22

for C, D being non empty finite set
for c being Element of C
for F being PartFunc of D,REAL
for A being RearrangmentGen of C st F is total & card C = card D holds
( ( c in (Co_Gen A) . 1 implies (Rlor (F,A)) . c = (FinS (F,D)) . 1 ) & ( for n being Element of NAT st 1 <= n & n < len (Co_Gen A) & c in ((Co_Gen A) . (n + 1)) \ ((Co_Gen A) . n) holds
(Rlor (F,A)) . c = (FinS (F,D)) . (n + 1) ) )

proof end;

theorem Th23: :: REARRAN1:23

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
rng (Rlor (F,A)) = rng (FinS (F,D))

proof end;

theorem Th24: :: REARRAN1:24

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), FinS (F,D) are_fiberwise_equipotent

proof end;

theorem Th25: :: REARRAN1:25

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
FinS ((Rlor (F,A)),C) = FinS (F,D)

proof end;

theorem Th26: :: REARRAN1:26

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
Sum ((Rlor (F,A)),C) = Sum (F,D)

proof end;

theorem :: REARRAN1:27

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
( FinS (((Rlor (F,A)) - r),C) = FinS ((F - r),D) & Sum (((Rlor (F,A)) - r),C) = Sum ((F - r),D) )

proof end;

theorem :: 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) )

proof end;

theorem :: 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) )

proof end;

theorem :: 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) )

proof end;

theorem Th31: :: REARRAN1:31

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 D = card C holds
( len (FinS ((Rland (F,A)),C)) = card C & 1 <= len (FinS ((Rland (F,A)),C)) )

proof end;

theorem :: REARRAN1:32

for n being Element of NAT
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 D = card C & n in dom A holds
(FinS ((Rland (F,A)),C)) | n = FinS ((Rland (F,A)),(A . n))

proof end;

theorem :: REARRAN1:33

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 D = card C holds
Rland ((F - r),A) = (Rland (F,A)) - r

proof end;

theorem :: 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) )

proof end;

theorem :: 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) )

proof end;

theorem Th36: :: REARRAN1:36

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 D = card C holds
( len (FinS ((Rlor (F,A)),C)) = card C & 1 <= len (FinS ((Rlor (F,A)),C)) )

proof end;

theorem :: REARRAN1:37

for n being Element of NAT
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 D = card C & n in dom A holds
(FinS ((Rlor (F,A)),C)) | n = FinS ((Rlor (F,A)),((Co_Gen A) . n))

proof end;

theorem :: REARRAN1:38

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 D = card C holds
Rlor ((F - r),A) = (Rlor (F,A)) - r

proof end;

theorem :: REARRAN1:39

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 D = card C holds
( Rland (F,A),F are_fiberwise_equipotent & Rlor (F,A),F are_fiberwise_equipotent & rng (Rland (F,A)) = rng F & rng (Rlor (F,A)) = rng F )

proof end;