:: RFINSEQ semantic presentation

definition

let F, G be Relation;
pred F,G are_fiberwise_equipotent means :Def1: :: RFINSEQ:def 1
for x being set holds Card (F " {x}) = Card (G " {x});
reflexivity ;
symmetry ;

end;

:: deftheorem Def1 defines are_fiberwise_equipotent RFINSEQ:def 1 :

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

proof end;

theorem Th1: :: RFINSEQ:1

for F, G being Function st F,G are_fiberwise_equipotent holds
rng F = rng G

proof end;

theorem :: RFINSEQ:2

for F, G, H being Function st F,G are_fiberwise_equipotent & F,H are_fiberwise_equipotent holds
G,H are_fiberwise_equipotent

proof end;

theorem Th3: :: RFINSEQ:3

for F, G being Function holds
( F,G are_fiberwise_equipotent iff ex H being Function st
( dom H = dom F & rng H = dom G & H is one-to-one & F = G * H ) )

proof end;

theorem Th4: :: RFINSEQ:4

for F, G being Function holds
( F,G are_fiberwise_equipotent iff for X being set holds Card (F " X) = Card (G " X) )

proof end;

theorem Th5: :: RFINSEQ:5

for D being non empty set
for F, G being Function st rng F c= D & rng G c= D holds
( F,G are_fiberwise_equipotent iff for d being Element of D holds Card (F " {d}) = Card (G " {d}) )

proof end;

theorem Th6: :: RFINSEQ:6

for F, G being Function st dom F = dom G holds
( F,G are_fiberwise_equipotent iff ex P being Permutation of dom F st F = G * P )

proof end;

theorem :: RFINSEQ:7

for F, G being Function st F,G are_fiberwise_equipotent holds
Card (dom F) = Card (dom G)

proof end;

theorem :: RFINSEQ:8

canceled;

theorem :: RFINSEQ:9

for F, G being finite Function holds
( F,G are_fiberwise_equipotent iff for X being set holds card (F " X) = card (G " X) ) by Th4;

theorem :: RFINSEQ:10

for F, G being finite Function st F,G are_fiberwise_equipotent holds
card (dom F) = card (dom G)

proof end;

theorem :: RFINSEQ:11

for D being non empty set
for F, G being finite Function st rng F c= D & rng G c= D holds
( F,G are_fiberwise_equipotent iff for d being Element of D holds card (F " {d}) = card (G " {d}) ) by Th5;

theorem :: RFINSEQ:12

canceled;

theorem :: RFINSEQ:13

for f, g being FinSequence holds
( f,g are_fiberwise_equipotent iff for X being set holds card (f " X) = card (g " X) ) by Th4;

theorem Th14: :: RFINSEQ:14

for f, g, h being FinSequence holds
( f,g are_fiberwise_equipotent iff f ^ h,g ^ h are_fiberwise_equipotent )

proof end;

theorem :: RFINSEQ:15

for f, g being FinSequence holds f ^ g,g ^ f are_fiberwise_equipotent

proof end;

theorem Th16: :: RFINSEQ:16

for f, g being FinSequence st f,g are_fiberwise_equipotent holds
( len f = len g & dom f = dom g )

proof end;

theorem :: RFINSEQ:17

for f, g being FinSequence holds
( f,g are_fiberwise_equipotent iff ex P being Permutation of dom g st f = g * P )

proof end;

registration

let F be Function;
let X be finite set ;
cluster F | X -> Function-like finite ;
coherence

proof end;

end;

defpred S₁[ Nat] means for X being finite set
for F being Function st card (dom (F | X)) = $1 holds
ex a being FinSequence st F | X,a are_fiberwise_equipotent ;

Lm2: S₁[0]

proof end;

Lm3: for n being Nat st S₁[n] holds
S₁[n + 1]

proof end;

theorem :: RFINSEQ:18

for F being Function
for X being finite set ex f being FinSequence st F | X,f are_fiberwise_equipotent

proof end;

definition

let D be set ;
let f be FinSequence of D;
let n be Nat;
func f /^ n -> FinSequence of D means :Def2: :: RFINSEQ:def 2
( len it = (len f) - n & ( for m being Nat st m in dom it holds
it . m = f . (m + n) ) ) if n <= len f
otherwise it = <*> D;
existence

proof end;

uniqueness

proof end;

consistency ;

end;

:: deftheorem Def2 defines /^ RFINSEQ:def 2 :

Lm4: for n being Nat
for D being non empty set
for f being FinSequence of D st len f <= n holds
f | n = f

proof end;

theorem Th19: :: RFINSEQ:19

for D being non empty set
for f being FinSequence of D
for n, m being Nat st n in dom f & m in Seg n holds
( (f | n) . m = f . m & m in dom f )

proof end;

theorem Th20: :: RFINSEQ:20

for D being non empty set
for f being FinSequence of D
for n being Nat
for x being set st len f = n + 1 & x = f . (n + 1) holds
f = (f | n) ^ <*x*>

proof end;

theorem Th21: :: RFINSEQ:21

for D being non empty set
for f being FinSequence of D
for n being Nat holds (f | n) ^ (f /^ n) = f

proof end;

theorem :: RFINSEQ:22

for R1, R2 being FinSequence of REAL st R1,R2 are_fiberwise_equipotent holds
Sum R1 = Sum R2

proof end;

definition

let R be FinSequence of REAL ;
func MIM R -> FinSequence of REAL means :Def3: :: RFINSEQ:def 3
( len it = len R & it . (len it) = R . (len R) & ( for n being Nat st 1 <= n & n <= (len it) - 1 holds
it . n = (R . n) - (R . (n + 1)) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def3 defines MIM RFINSEQ:def 3 :

theorem Th23: :: RFINSEQ:23

for R being FinSequence of REAL
for r being Real
for n being Nat st len R = n + 2 & R . (n + 1) = r holds
MIM (R | (n + 1)) = ((MIM R) | n) ^ <*r*>

proof end;

theorem Th24: :: RFINSEQ:24

for R being FinSequence of REAL
for r, s being Real
for n being Nat st len R = n + 2 & R . (n + 1) = r & R . (n + 2) = s holds
MIM R = ((MIM R) | n) ^ <*(r - s),s*>

proof end;

theorem Th25: :: RFINSEQ:25

MIM (<*> REAL ) = <*> REAL

proof end;

theorem Th26: :: RFINSEQ:26

for r being Real holds MIM <*r*> = <*r*>

proof end;

theorem :: RFINSEQ:27

for r, s being Real holds MIM <*r,s*> = <*(r - s),s*>

proof end;

theorem :: RFINSEQ:28

for R being FinSequence of REAL
for n being Nat holds (MIM R) /^ n = MIM (R /^ n)

proof end;

theorem Th29: :: RFINSEQ:29

for R being FinSequence of REAL st len R <> 0 holds
Sum (MIM R) = R . 1

proof end;

theorem :: RFINSEQ:30

for R being FinSequence of REAL
for n being Nat st 1 <= n & n < len R holds
Sum (MIM (R /^ n)) = R . (n + 1)

proof end;

definition

let IT be FinSequence of REAL ;
attr IT is non-increasing means :Def4: :: RFINSEQ:def 4
for n being Nat st n in dom IT & n + 1 in dom IT holds
IT . n >= IT . (n + 1);

end;

:: deftheorem Def4 defines non-increasing RFINSEQ:def 4 :

registration

cluster non-increasing FinSequence of REAL ;
existence

proof end;

end;

theorem Th31: :: RFINSEQ:31

for R being FinSequence of REAL st ( len R = 0 or len R = 1 ) holds
R is non-increasing

proof end;

theorem Th32: :: RFINSEQ:32

for R being FinSequence of REAL holds
( R is non-increasing iff for n, m being Nat st n in dom R & m in dom R & n < m holds
R . n >= R . m )

proof end;

theorem Th33: :: RFINSEQ:33

for R being non-increasing FinSequence of REAL
for n being Nat holds R | n is non-increasing FinSequence of REAL

proof end;

theorem :: RFINSEQ:34

for R being non-increasing FinSequence of REAL
for n being Nat holds R /^ n is non-increasing FinSequence of REAL

proof end;

Lm5: for f, g being non-increasing FinSequence of REAL
for n being Nat st len f = n + 1 & len f = len g & f,g are_fiberwise_equipotent holds
( f . (len f) = g . (len g) & f | n,g | n are_fiberwise_equipotent )

proof end;

defpred S₂[ Nat] means for R being FinSequence of REAL st $1 = len R holds
ex b being non-increasing FinSequence of REAL st R,b are_fiberwise_equipotent ;

Lm6: S₂[0]

proof end;

Lm7: for n being Nat st S₂[n] holds
S₂[n + 1]

proof end;

theorem :: RFINSEQ:35

for R being FinSequence of REAL ex R1 being non-increasing FinSequence of REAL st R,R1 are_fiberwise_equipotent

proof end;

Lm8: for n being Nat
for g1, g2 being non-increasing FinSequence of REAL st n = len g1 & g1,g2 are_fiberwise_equipotent holds
g1 = g2

proof end;

theorem :: RFINSEQ:36

for R1, R2 being non-increasing FinSequence of REAL st R1,R2 are_fiberwise_equipotent holds
R1 = R2

proof end;

theorem :: RFINSEQ:37

for R being FinSequence of REAL
for r, s being Real st r <> 0 holds
R " {(s / r)} = (r * R) " {s}

proof end;

theorem :: RFINSEQ:38

for R being FinSequence of REAL holds (0 * R) " {0} = dom R

proof end;