:: Zero Based Finite Sequences
:: by Tetsuya Tsunetou , Grzegorz Bancerek and Yatsuka Nakamura
::
:: Received September 28, 2001
:: Copyright (c) 2001 Association of Mizar Users

theorem Th1: :: AFINSQ_1:1

for n being Element of NAT holds n in n + 1

proof end;

theorem Th2: :: AFINSQ_1:2

for k, n being Element of NAT st k <= n holds
k = k /\ n

proof end;

theorem :: AFINSQ_1:3

for k, n being Element of NAT st k = k /\ n holds
k <= n by Th2;

theorem Th4: :: AFINSQ_1:4

for n being Nat holds n \/ {n} = n + 1

proof end;

theorem Th5: :: AFINSQ_1:5

for n being Element of NAT holds Seg n c= n + 1

proof end;

theorem :: AFINSQ_1:6

for n being Element of NAT holds n + 1 = {0 } \/ (Seg n)

proof end;

theorem Th7: :: AFINSQ_1:7

for r being Function holds
( ( r is finite & r is T-Sequence-like ) iff ex n being Element of NAT st dom r = n )

proof end;

definition

mode XFinSequence is finite T-Sequence;

end;

registration

let p be XFinSequence;
cluster dom p -> natural ;
coherence
dom p is natural

proof end;

end;

notation

let p be XFinSequence;
synonym len p for card p;

end;

registration

let p be XFinSequence;
identify len p with dom p;
compatibility
len p = dom p

proof end;

end;

definition

let p be XFinSequence;
:: original: len
redefine func len p -> Element of NAT ;
coherence
len p is Element of NAT

proof end;

end;

definition

canceled;
let p be XFinSequence;
:: original: dom
redefine func dom p -> Subset of NAT ;
coherence
dom p is Subset of NAT

proof end;

end;

:: deftheorem AFINSQ_1:def 1 :

canceled;

theorem :: AFINSQ_1:8

for f being Function st ex k being Element of NAT st dom f c= k holds
ex p being XFinSequence st f c= p

proof end;

scheme :: AFINSQ_1:sch 1
XSeqEx{ F₁() -> Nat, P₁[ set , set ] } :

ex p being XFinSequence st
( dom p = F₁() & ( for k being Element of NAT st k in F₁() holds
P₁[k,p . k] ) )

provided

A2: for k being Element of NAT st k in F₁() holds
ex x being set st P₁[k,x]

proof end;

scheme :: AFINSQ_1:sch 2
XSeqLambda{ F₁() -> Nat, F₂( set ) -> set } :

ex p being XFinSequence st
( len p = F₁() & ( for k being Element of NAT st k in F₁() holds
p . k = F₂(k) ) )

proof end;

theorem :: AFINSQ_1:9

for z being set
for p being XFinSequence st z in p holds
ex k being Element of NAT st
( k in dom p & z = [k,(p . k)] )

proof end;

theorem Th10: :: AFINSQ_1:10

for p, q being XFinSequence st dom p = dom q & ( for k being Element of NAT st k in dom p holds
p . k = q . k ) holds
p = q

proof end;

theorem :: AFINSQ_1:11

for p, q being XFinSequence st len p = len q & ( for k being Element of NAT st k < len p holds
p . k = q . k ) holds
p = q

proof end;

theorem Th12: :: AFINSQ_1:12

for n being Element of NAT
for p being XFinSequence holds p | n is XFinSequence

proof end;

theorem :: AFINSQ_1:13

for f being Function
for p being XFinSequence st rng p c= dom f holds
f * p is XFinSequence

proof end;

theorem :: AFINSQ_1:14

for k being Element of NAT
for p being XFinSequence st k < len p holds
dom (p | k) = k

proof end;

registration

let D be set ;
cluster finite set ;
existence
ex b₁ being T-Sequence of st b₁ is finite

proof end;

end;

definition

let D be set ;
mode XFinSequence of D is finite T-Sequence of ;

end;

theorem Th15: :: AFINSQ_1:15

for D being set
for f being XFinSequence of holds f is PartFunc of NAT ,D

proof end;

registration

cluster empty -> T-Sequence-like set ;
coherence
for b₁ being Function st b₁ is empty holds
b₁ is T-Sequence-like ;

end;

registration

let D be set ;
cluster finite Element of bool [:NAT ,D:];
existence
ex b₁ being PartFunc of NAT ,D st b₁ is finite

proof end;

end;

theorem :: AFINSQ_1:16

for k being Element of NAT
for D being set
for p being XFinSequence of holds p | k is XFinSequence of by Th12;

theorem :: AFINSQ_1:17

for k being Element of NAT
for D being non empty set ex p being XFinSequence of st len p = k

proof end;

registration

cluster empty set ;
existence
ex b₁ being XFinSequence st b₁ is empty by ORDINAL1:45;

end;

theorem :: AFINSQ_1:18

for p being XFinSequence holds
( len p = 0 iff p = {} ) ;

theorem Th19: :: AFINSQ_1:19

for D being set holds {} is XFinSequence of

proof end;

registration

let D be set ;
cluster empty set ;
existence
ex b₁ being XFinSequence of st b₁ is empty

proof end;

end;

definition

let x be set ;
func <%x%> -> set equals :: AFINSQ_1:def 2
0 .--> x;
coherence
0 .--> x is set ;

end;

:: deftheorem defines <% AFINSQ_1:def 2 :

for x being set holds <%x%> = 0 .--> x;

registration

let x be set ;
cluster <%x%> -> non empty ;
coherence
not <%x%> is empty ;

end;

definition

let D be set ;
func <%> D -> XFinSequence of equals :: AFINSQ_1:def 3
{} ;
coherence
{} is XFinSequence of by Th19;

end;

:: deftheorem defines <%> AFINSQ_1:def 3 :

for D being set holds <%> D = {} ;

registration

let D be set ;
cluster <%> D -> empty ;
coherence
<%> D is empty ;

end;

definition

let p, q be XFinSequence;
redefine func p ^ q means :Def4: :: AFINSQ_1:def 4
( dom it = (len p) + (len q) & ( for k being Element of NAT st k in dom p holds
it . k = p . k ) & ( for k being Element of NAT st k in dom q holds
it . ((len p) + k) = q . k ) );
compatibility
for b₁ being set holds
( b₁ = p ^ q iff ( dom b₁ = (len p) + (len q) & ( for k being Element of NAT st k in dom p holds
b₁ . k = p . k ) & ( for k being Element of NAT st k in dom q holds
b₁ . ((len p) + k) = q . k ) ) )

proof end;

end;

:: deftheorem Def4 defines ^ AFINSQ_1:def 4 :

for p, q being XFinSequence
for b₃ being set holds
( b₃ = p ^ q iff ( dom b₃ = (len p) + (len q) & ( for k being Element of NAT st k in dom p holds
b₃ . k = p . k ) & ( for k being Element of NAT st k in dom q holds
b₃ . ((len p) + k) = q . k ) ) );

registration

let p, q be XFinSequence;
cluster p ^ q -> finite ;
coherence
p ^ q is finite

proof end;

end;

theorem :: AFINSQ_1:20

for p, q being XFinSequence holds len (p ^ q) = (len p) + (len q) by Def4;

theorem Th21: :: AFINSQ_1:21

for k being Element of NAT
for p, q being XFinSequence st len p <= k & k < (len p) + (len q) holds
(p ^ q) . k = q . (k - (len p))

proof end;

theorem :: AFINSQ_1:22

for k being Element of NAT
for p, q being XFinSequence st len p <= k & k < len (p ^ q) holds
(p ^ q) . k = q . (k - (len p))

proof end;

theorem Th23: :: AFINSQ_1:23

for k being Element of NAT
for p, q being XFinSequence holds
( not k in dom (p ^ q) or k in dom p or ex n being Element of NAT st
( n in dom q & k = (len p) + n ) )

proof end;

theorem Th24: :: AFINSQ_1:24

for p, q being T-Sequence holds dom p c= dom (p ^ q)

proof end;

theorem Th25: :: AFINSQ_1:25

for x being set
for q, p being XFinSequence st x in dom q holds
ex k being Element of NAT st
( k = x & (len p) + k in dom (p ^ q) )

proof end;

theorem Th26: :: AFINSQ_1:26

for k being Element of NAT
for q, p being XFinSequence st k in dom q holds
(len p) + k in dom (p ^ q)

proof end;

theorem Th27: :: AFINSQ_1:27

for p, q being XFinSequence holds rng p c= rng (p ^ q)

proof end;

theorem Th28: :: AFINSQ_1:28

for q, p being XFinSequence holds rng q c= rng (p ^ q)

proof end;

theorem Th29: :: AFINSQ_1:29

for p, q being XFinSequence holds rng (p ^ q) = (rng p) \/ (rng q)

proof end;

theorem Th30: :: AFINSQ_1:30

for p, q, r being XFinSequence holds (p ^ q) ^ r = p ^ (q ^ r)

proof end;

theorem :: AFINSQ_1:31

for p, r, q being XFinSequence st ( p ^ r = q ^ r or r ^ p = r ^ q ) holds
p = q

proof end;

theorem Th32: :: AFINSQ_1:32

for p being XFinSequence holds
( p ^ {} = p & {} ^ p = p )

proof end;

theorem :: AFINSQ_1:33

for p, q being XFinSequence st p ^ q = {} holds
( p = {} & q = {} )

proof end;

definition

let D be set ;
let p, q be XFinSequence of ;
:: original: ^
redefine func p ^ q -> T-Sequence of ;
coherence
p ^ q is T-Sequence of

proof end;

end;

Lm1: for x, y, x1, y1 being set st [x,y] in {[x1,y1]} holds
( x = x1 & y = y1 )

proof end;

definition

let x be set ;
:: original: <%
redefine func <%x%> -> Function means :Def5: :: AFINSQ_1:def 5
( dom it = 1 & it . 0 = x );
coherence
<%x%> is Function ;
compatibility
for b₁ being Function holds
( b₁ = <%x%> iff ( dom b₁ = 1 & b₁ . 0 = x ) )

proof end;

end;

:: deftheorem Def5 defines <% AFINSQ_1:def 5 :

for x being set
for b₂ being Function holds
( b₂ = <%x%> iff ( dom b₂ = 1 & b₂ . 0 = x ) );

registration

let x be set ;
cluster <%x%> -> Relation-like Function-like ;
coherence
( <%x%> is Function-like & <%x%> is Relation-like ) ;

end;

registration

let x be set ;
cluster <%x%> -> T-Sequence-like finite ;
coherence
( <%x%> is finite & <%x%> is T-Sequence-like )

proof end;

end;

theorem :: AFINSQ_1:34

for p, q being XFinSequence
for D being set st p ^ q is XFinSequence of holds
( p is XFinSequence of & q is XFinSequence of )

proof end;

definition

let x, y be set ;
func <%x,y%> -> set equals :: AFINSQ_1:def 6
<%x%> ^ <%y%>;
correctness
coherence
<%x%> ^ <%y%> is set ;
;
let z be set ;
func <%x,y,z%> -> set equals :: AFINSQ_1:def 7
(<%x%> ^ <%y%>) ^ <%z%>;
correctness
coherence
(<%x%> ^ <%y%>) ^ <%z%> is set ;
;

end;

:: deftheorem defines <% AFINSQ_1:def 6 :

for x, y being set holds <%x,y%> = <%x%> ^ <%y%>;

:: deftheorem defines <% AFINSQ_1:def 7 :

for x, y, z being set holds <%x,y,z%> = (<%x%> ^ <%y%>) ^ <%z%>;

registration

let x, y be set ;
cluster <%x,y%> -> Relation-like Function-like ;
coherence
( <%x,y%> is Function-like & <%x,y%> is Relation-like ) ;
let z be set ;
cluster <%x,y,z%> -> Relation-like Function-like ;
coherence
( <%x,y,z%> is Function-like & <%x,y,z%> is Relation-like ) ;

end;

registration

let x, y be set ;
cluster <%x,y%> -> T-Sequence-like finite ;
coherence
( <%x,y%> is finite & <%x,y%> is T-Sequence-like ) ;
let z be set ;
cluster <%x,y,z%> -> T-Sequence-like finite ;
coherence
( <%x,y,z%> is finite & <%x,y,z%> is T-Sequence-like ) ;

end;

theorem :: AFINSQ_1:35

for x being set holds <%x%> = {[0 ,x]} by FUNCT_4:87;

theorem Th36: :: AFINSQ_1:36

for x being set
for p being XFinSequence holds
( p = <%x%> iff ( dom p = 1 & rng p = {x} ) )

proof end;

theorem :: AFINSQ_1:37

canceled;

theorem :: AFINSQ_1:38

for x being set
for p being XFinSequence holds
( p = <%x%> iff ( len p = 1 & p . 0 = x ) ) by Def5;

theorem :: AFINSQ_1:39

for x being set
for p being XFinSequence holds (<%x%> ^ p) . 0 = x

proof end;

theorem :: AFINSQ_1:40

for x being set
for p being XFinSequence holds (p ^ <%x%>) . (len p) = x

proof end;

theorem :: AFINSQ_1:41

for x, y, z being set holds
( <%x,y,z%> = <%x%> ^ <%y,z%> & <%x,y,z%> = <%x,y%> ^ <%z%> ) by Th30;

theorem Th42: :: AFINSQ_1:42

for x, y being set
for p being XFinSequence holds
( p = <%x,y%> iff ( len p = 2 & p . 0 = x & p . 1 = y ) )

proof end;

theorem :: AFINSQ_1:43

for x, y, z being set
for p being XFinSequence holds
( p = <%x,y,z%> iff ( len p = 3 & p . 0 = x & p . 1 = y & p . 2 = z ) )

proof end;

theorem :: AFINSQ_1:44

for p being XFinSequence st p <> {} holds
ex q being XFinSequence ex x being set st p = q ^ <%x%>

proof end;

definition

let D be non empty set ;
let x be Element of D;
:: original: <%
redefine func <%x%> -> XFinSequence of ;
coherence
<%x%> is XFinSequence of

proof end;

end;

scheme :: AFINSQ_1:sch 3
IndXSeq{ P₁[ XFinSequence] } :

for p being XFinSequence holds P₁[p]

provided

A1: P₁[ {} ] and
for p being XFinSequence
for x being set st P₁[p] holds
P₁[p ^ <%x%>]

proof end;

theorem :: AFINSQ_1:45

for p, q, r, s being XFinSequence st p ^ q = r ^ s & len p <= len r holds
ex t being XFinSequence st p ^ t = r

proof end;

definition

let D be set ;
func D ^omega -> set means :Def8: :: AFINSQ_1:def 8
for x being set holds
( x in it iff x is XFinSequence of );
existence
ex b₁ being set st
for x being set holds
( x in b₁ iff x is XFinSequence of )

proof end;

uniqueness
for b₁, b₂ being set st ( for x being set holds
( x in b₁ iff x is XFinSequence of ) ) & ( for x being set holds
( x in b₂ iff x is XFinSequence of ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def8 defines ^omega AFINSQ_1:def 8 :

for D, b₂ being set holds
( b₂ = D ^omega iff for x being set holds
( x in b₂ iff x is XFinSequence of ) );

registration

let D be set ;
cluster D ^omega -> non empty ;
coherence
not D ^omega is empty

proof end;

end;

theorem :: AFINSQ_1:46

for x, D being set holds
( x in D ^omega iff x is XFinSequence of ) by Def8;

theorem :: AFINSQ_1:47

for D being set holds {} in D ^omega

proof end;

scheme :: AFINSQ_1:sch 4
SepXSeq{ F₁() -> non empty set , P₁[ XFinSequence] } :

ex X being set st
for x being set holds
( x in X iff ex p being XFinSequence st
( p in F₁() ^omega & P₁[p] & x = p ) )

proof end;

notation

let p be XFinSequence;
let i, x be set ;
synonym Replace p,i,x for p +* i,x;

end;

registration

let p be XFinSequence;
let i, x be set ;
cluster Replace p,i,x -> T-Sequence-like finite ;
coherence
( p +* i,x is finite & p +* i,x is T-Sequence-like )

proof end;

end;

theorem :: AFINSQ_1:48

for p being XFinSequence
for i being Element of NAT
for x being set holds
( len (Replace p,i,x) = len p & ( i < len p implies (Replace p,i,x) . i = x ) & ( for j being Element of NAT st j <> i holds
(Replace p,i,x) . j = p . j ) )

proof end;

definition

let D be non empty set ;
let p be XFinSequence of ;
let i be Element of NAT ;
let a be Element of D;
:: original: Replace
redefine func Replace p,i,a -> XFinSequence of ;
coherence
Replace p,i,a is XFinSequence of

proof end;

end;

registration

let x, y be set ;
cluster [x,y] -> non natural ;
coherence
not [x,y] is natural

proof end;

end;

registration

cluster -> real-valued set ;
coherence
for b₁ being XFinSequence of holds b₁ is real-valued

proof end;

end;

registration

cluster -> natural-valued set ;
coherence
for b₁ being XFinSequence of holds b₁ is natural-valued

proof end;

end;