:: ALGSEQ_1 semantic presentation

theorem :: ALGSEQ_1:1
canceled;

theorem :: ALGSEQ_1:2
canceled;

theorem :: ALGSEQ_1:3
canceled;

theorem :: ALGSEQ_1:4
canceled;

theorem :: ALGSEQ_1:5
canceled;

theorem :: ALGSEQ_1:6
canceled;

theorem :: ALGSEQ_1:7
canceled;

theorem :: ALGSEQ_1:8
canceled;

theorem :: ALGSEQ_1:9
canceled;

theorem :: ALGSEQ_1:10
for k, n being Element of NAT holds
( k in Segm n iff k < n ) by CARD_1:98;

theorem :: ALGSEQ_1:11
( Segm 0 = {} & Segm 1 = {0} & Segm 2 = {0,1} ) by CARD_1:87, CARD_1:88;

theorem :: ALGSEQ_1:12
for n being Element of NAT holds n in Segm (n + 1) by CARD_1:99;

theorem :: ALGSEQ_1:13
for n, m being Element of NAT holds
( n <= m iff Segm n c= Segm m ) by CARD_1:100;

theorem :: ALGSEQ_1:14
for n, m being Element of NAT st Segm n = Segm m holds
n = m ;

theorem :: ALGSEQ_1:15
for k, n being Element of NAT st k <= n holds
( Segm k = (Segm k) /\ (Segm n) & Segm k = (Segm n) /\ (Segm k) ) by CARD_1:101;

theorem :: ALGSEQ_1:16
for k, n being Element of NAT st ( Segm k = (Segm k) /\ (Segm n) or Segm k = (Segm n) /\ (Segm k) ) holds
k <= n by CARD_1:101;

theorem :: ALGSEQ_1:17
canceled;

definition
let R be non empty ZeroStr ;
canceled;
let F be sequence of R;
attr F is finite-Support means :Def2: :: ALGSEQ_1:def 2
ex n being Element of NAT st
for i being Element of NAT st i >= n holds
F . i = 0. R;
end;

:: deftheorem ALGSEQ_1:def 1 :
canceled;

:: deftheorem Def2 defines finite-Support ALGSEQ_1:def 2 :
for R being non empty ZeroStr
for F being sequence of R holds
( F is finite-Support iff ex n being Element of NAT st
for i being Element of NAT st i >= n holds
F . i = 0. R );

registration
let R be non empty ZeroStr ;
cluster finite-Support M4( NAT ,the carrier of R);
existence
ex b1 being sequence of R st b1 is finite-Support
proof end;
end;

definition
let R be non empty ZeroStr ;
mode AlgSequence of R is finite-Support sequence of R;
end;

definition
let R be non empty ZeroStr ;
let p be AlgSequence of R;
let k be Element of NAT ;
pred k is_at_least_length_of p means :Def3: :: ALGSEQ_1:def 3
for i being Element of NAT st i >= k holds
p . i = 0. R;
end;

:: deftheorem Def3 defines is_at_least_length_of ALGSEQ_1:def 3 :
for R being non empty ZeroStr
for p being AlgSequence of R
for k being Element of NAT holds
( k is_at_least_length_of p iff for i being Element of NAT st i >= k holds
p . i = 0. R );

Lm1: for R being non empty ZeroStr
for p being AlgSequence of R ex m being Element of NAT st m is_at_least_length_of p
proof end;

Lm2: for R being non empty ZeroStr
for p being AlgSequence of R ex k being Element of NAT st
( k is_at_least_length_of p & ( for n being Element of NAT st n is_at_least_length_of p holds
k <= n ) )
proof end;

Lm3: for k, l being Element of NAT
for R being non empty ZeroStr
for p being AlgSequence of R st k is_at_least_length_of p & ( for m being Element of NAT st m is_at_least_length_of p holds
k <= m ) & l is_at_least_length_of p & ( for m being Element of NAT st m is_at_least_length_of p holds
l <= m ) holds
k = l
proof end;

definition
let R be non empty ZeroStr ;
let p be AlgSequence of R;
func len p -> Element of NAT means :Def4: :: ALGSEQ_1:def 4
( it is_at_least_length_of p & ( for m being Element of NAT st m is_at_least_length_of p holds
it <= m ) );
existence
ex b1 being Element of NAT st
( b1 is_at_least_length_of p & ( for m being Element of NAT st m is_at_least_length_of p holds
b1 <= m ) )
by Lm2;
uniqueness
for b1, b2 being Element of NAT st b1 is_at_least_length_of p & ( for m being Element of NAT st m is_at_least_length_of p holds
b1 <= m ) & b2 is_at_least_length_of p & ( for m being Element of NAT st m is_at_least_length_of p holds
b2 <= m ) holds
b1 = b2
by Lm3;
end;

:: deftheorem Def4 defines len ALGSEQ_1:def 4 :
for R being non empty ZeroStr
for p being AlgSequence of R
for b3 being Element of NAT holds
( b3 = len p iff ( b3 is_at_least_length_of p & ( for m being Element of NAT st m is_at_least_length_of p holds
b3 <= m ) ) );

theorem :: ALGSEQ_1:18
canceled;

theorem :: ALGSEQ_1:19
canceled;

theorem :: ALGSEQ_1:20
canceled;

theorem :: ALGSEQ_1:21
canceled;

theorem Th22: :: ALGSEQ_1:22
for i being Element of NAT
for R being non empty ZeroStr
for p being AlgSequence of R st i >= len p holds
p . i = 0. R
proof end;

theorem :: ALGSEQ_1:23
canceled;

theorem Th24: :: ALGSEQ_1:24
for k being Element of NAT
for R being non empty ZeroStr
for p being AlgSequence of R st ( for i being Element of NAT st i < k holds
p . i <> 0. R ) holds
len p >= k
proof end;

theorem Th25: :: ALGSEQ_1:25
for k being Element of NAT
for R being non empty ZeroStr
for p being AlgSequence of R st len p = k + 1 holds
p . k <> 0. R
proof end;

definition
let R be non empty ZeroStr ;
let p be AlgSequence of R;
func support p -> Subset of NAT equals :: ALGSEQ_1:def 5
Segm (len p);
coherence
Segm (len p) is Subset of NAT
;
end;

:: deftheorem defines support ALGSEQ_1:def 5 :
for R being non empty ZeroStr
for p being AlgSequence of R holds support p = Segm (len p);

theorem :: ALGSEQ_1:26
canceled;

theorem :: ALGSEQ_1:27
for k being Element of NAT
for R being non empty ZeroStr
for p being AlgSequence of R holds
( k = len p iff Segm k = support p ) ;

scheme :: ALGSEQ_1:sch 1
AlgSeqLambdaF{ F1() -> non empty ZeroStr , F2() -> Element of NAT , F3( Element of NAT ) -> Element of F1() } :
ex p being AlgSequence of F1() st
( len p <= F2() & ( for k being Element of NAT st k < F2() holds
p . k = F3(k) ) )
proof end;

theorem Th28: :: ALGSEQ_1:28
for R being non empty ZeroStr
for p, q being AlgSequence of R 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 :: ALGSEQ_1:29
for R being non empty ZeroStr st the carrier of R <> {(0. R)} holds
for k being Element of NAT ex p being AlgSequence of R st len p = k
proof end;

definition
let R be non empty ZeroStr ;
let x be Element of R;
func <%x%> -> AlgSequence of R means :Def6: :: ALGSEQ_1:def 6
( len it <= 1 & it . 0 = x );
existence
ex b1 being AlgSequence of R st
( len b1 <= 1 & b1 . 0 = x )
proof end;
uniqueness
for b1, b2 being AlgSequence of R st len b1 <= 1 & b1 . 0 = x & len b2 <= 1 & b2 . 0 = x holds
b1 = b2
proof end;
end;

:: deftheorem Def6 defines <% ALGSEQ_1:def 6 :
for R being non empty ZeroStr
for x being Element of R
for b3 being AlgSequence of R holds
( b3 = <%x%> iff ( len b3 <= 1 & b3 . 0 = x ) );

Lm4: for R being non empty ZeroStr
for p being AlgSequence of R st p = <%(0. R)%> holds
len p = 0
proof end;

theorem :: ALGSEQ_1:30
canceled;

theorem Th31: :: ALGSEQ_1:31
for R being non empty ZeroStr
for p being AlgSequence of R holds
( p = <%(0. R)%> iff len p = 0 )
proof end;

theorem :: ALGSEQ_1:32
for R being non empty ZeroStr
for p being AlgSequence of R holds
( p = <%(0. R)%> iff support p = {} ) by Th31;

theorem Th33: :: ALGSEQ_1:33
for i being Element of NAT
for R being non empty ZeroStr holds <%(0. R)%> . i = 0. R
proof end;

theorem :: ALGSEQ_1:34
for R being non empty ZeroStr
for p being AlgSequence of R holds
( p = <%(0. R)%> iff rng p = {(0. R)} )
proof end;