:: Monotone Real Sequences. Subsequences
:: by Jaros{\l}aw Kotowicz
::
:: Received November 23, 1989
:: Copyright (c) 1990 Association of Mizar Users

begin

Lm1: for n, m being Element of NAT st n < m holds
ex k being Element of NAT st m = (n + 1) + k

proof end;

Lm2: for seq being Real_Sequence holds
( ( ( for n being Element of NAT holds seq . n < seq . (n + 1) ) implies for n, k being Element of NAT holds seq . n < seq . ((n + 1) + k) ) & ( ( for n, k being Element of NAT holds seq . n < seq . ((n + 1) + k) ) implies for n, m being Element of NAT st n < m holds
seq . n < seq . m ) & ( ( for n, m being Element of NAT st n < m holds
seq . n < seq . m ) implies for n being Element of NAT holds seq . n < seq . (n + 1) ) )

proof end;

Lm3: for seq being Real_Sequence holds
( ( ( for n being Element of NAT holds seq . (n + 1) < seq . n ) implies for n, k being Element of NAT holds seq . ((n + 1) + k) < seq . n ) & ( ( for n, k being Element of NAT holds seq . ((n + 1) + k) < seq . n ) implies for n, m being Element of NAT st n < m holds
seq . m < seq . n ) & ( ( for n, m being Element of NAT st n < m holds
seq . m < seq . n ) implies for n being Element of NAT holds seq . (n + 1) < seq . n ) )

proof end;

Lm4: for seq being Real_Sequence holds
( ( ( for n being Element of NAT holds seq . n <= seq . (n + 1) ) implies for n, k being Element of NAT holds seq . n <= seq . (n + k) ) & ( ( for n, k being Element of NAT holds seq . n <= seq . (n + k) ) implies for n, m being Element of NAT st n <= m holds
seq . n <= seq . m ) & ( ( for n, m being Element of NAT st n <= m holds
seq . n <= seq . m ) implies for n being Element of NAT holds seq . n <= seq . (n + 1) ) )

proof end;

Lm5: for seq being Real_Sequence holds
( ( ( for n being Element of NAT holds seq . (n + 1) <= seq . n ) implies for n, k being Element of NAT holds seq . (n + k) <= seq . n ) & ( ( for n, k being Element of NAT holds seq . (n + k) <= seq . n ) implies for n, m being Element of NAT st n <= m holds
seq . m <= seq . n ) & ( ( for n, m being Element of NAT st n <= m holds
seq . m <= seq . n ) implies for n being Element of NAT holds seq . (n + 1) <= seq . n ) )

proof end;

definition

let f be PartFunc of NAT,REAL;
redefine attr f is increasing means :Def1: :: SEQM_3:def 1
for m, n being Element of NAT st m in dom f & n in dom f & m < n holds
f . m < f . n;
compatibility
( f is increasing iff for m, n being Element of NAT st m in dom f & n in dom f & m < n holds
f . m < f . n )

proof end;

redefine attr f is decreasing means :Def2: :: SEQM_3:def 2
for m, n being Element of NAT st m in dom f & n in dom f & m < n holds
f . m > f . n;
compatibility
( f is decreasing iff for m, n being Element of NAT st m in dom f & n in dom f & m < n holds
f . m > f . n )

proof end;

redefine attr f is non-decreasing means :Def3: :: SEQM_3:def 3
for m, n being Element of NAT st m in dom f & n in dom f & m <= n holds
f . m <= f . n;
compatibility
( f is non-decreasing iff for m, n being Element of NAT st m in dom f & n in dom f & m <= n holds
f . m <= f . n )

proof end;

redefine attr f is non-increasing means :Def4: :: SEQM_3:def 4
for m, n being Element of NAT st m in dom f & n in dom f & m <= n holds
f . m >= f . n;
compatibility
( f is non-increasing iff for m, n being Element of NAT st m in dom f & n in dom f & m <= n holds
f . m >= f . n )

proof end;

end;

:: deftheorem Def1 defines increasing SEQM_3:def 1 :
for f being PartFunc of NAT,REAL holds
( f is increasing iff for m, n being Element of NAT st m in dom f & n in dom f & m < n holds
f . m < f . n );

:: deftheorem Def2 defines decreasing SEQM_3:def 2 :
for f being PartFunc of NAT,REAL holds
( f is decreasing iff for m, n being Element of NAT st m in dom f & n in dom f & m < n holds
f . m > f . n );

:: deftheorem Def3 defines non-decreasing SEQM_3:def 3 :
for f being PartFunc of NAT,REAL holds
( f is non-decreasing iff for m, n being Element of NAT st m in dom f & n in dom f & m <= n holds
f . m <= f . n );

:: deftheorem Def4 defines non-increasing SEQM_3:def 4 :
for f being PartFunc of NAT,REAL holds
( f is non-increasing iff for m, n being Element of NAT st m in dom f & n in dom f & m <= n holds
f . m >= f . n );

Lm6: for f being Real_Sequence holds
( f is increasing iff for n being Element of NAT holds f . n < f . (n + 1) )

proof end;

Lm7: for f being Real_Sequence holds
( f is decreasing iff for n being Element of NAT holds f . n > f . (n + 1) )

proof end;

Lm8: for f being Real_Sequence holds
( f is non-decreasing iff for n being Element of NAT holds f . n <= f . (n + 1) )

proof end;

Lm9: for f being Real_Sequence holds
( f is non-increasing iff for n being Element of NAT holds f . n >= f . (n + 1) )

proof end;

definition

canceled;
canceled;
let seq be Real_Sequence;
attr seq is monotone means :Def7: :: SEQM_3:def 7
( seq is non-decreasing or seq is non-increasing );

end;

:: deftheorem SEQM_3:def 5 :
canceled;

:: deftheorem SEQM_3:def 6 :
canceled;

:: deftheorem Def7 defines monotone SEQM_3:def 7 :
for seq being Real_Sequence holds
( seq is monotone iff ( seq is non-decreasing or seq is non-increasing ) );

theorem :: SEQM_3:1

canceled;

theorem :: SEQM_3:2

canceled;

theorem :: SEQM_3:3

canceled;

theorem :: SEQM_3:4

canceled;

theorem :: SEQM_3:5

canceled;

theorem :: SEQM_3:6

canceled;

theorem Th7: :: SEQM_3:7

for seq being Real_Sequence holds
( seq is increasing iff for n, m being Element of NAT st n < m holds
seq . n < seq . m )

proof end;

theorem :: SEQM_3:8

for seq being Real_Sequence holds
( seq is increasing iff for n, k being Element of NAT holds seq . n < seq . ((n + 1) + k) )

proof end;

theorem Th9: :: SEQM_3:9

for seq being Real_Sequence holds
( seq is decreasing iff for n, k being Element of NAT holds seq . ((n + 1) + k) < seq . n )

proof end;

theorem Th10: :: SEQM_3:10

for seq being Real_Sequence holds
( seq is decreasing iff for n, m being Element of NAT st n < m holds
seq . m < seq . n )

proof end;

theorem Th11: :: SEQM_3:11

for seq being Real_Sequence holds
( seq is non-decreasing iff for n, k being Element of NAT holds seq . n <= seq . (n + k) )

proof end;

theorem Th12: :: SEQM_3:12

for seq being Real_Sequence holds
( seq is non-decreasing iff for n, m being Element of NAT st n <= m holds
seq . n <= seq . m )

proof end;

theorem Th13: :: SEQM_3:13

for seq being Real_Sequence holds
( seq is non-increasing iff for n, k being Element of NAT holds seq . (n + k) <= seq . n )

proof end;

theorem Th14: :: SEQM_3:14

for seq being Real_Sequence holds
( seq is non-increasing iff for n, m being Element of NAT st n <= m holds
seq . m <= seq . n )

proof end;

theorem :: SEQM_3:15

canceled;

theorem :: SEQM_3:16

canceled;

theorem :: SEQM_3:17

canceled;

theorem :: SEQM_3:18

canceled;

theorem :: SEQM_3:19

for seq being Real_Sequence st seq is increasing holds
for n being Element of NAT st 0 < n holds
seq . 0 < seq . n by Th7;

theorem :: SEQM_3:20

for seq being Real_Sequence st seq is decreasing holds
for n being Element of NAT st 0 < n holds
seq . n < seq . 0 by Th10;

theorem Th21: :: SEQM_3:21

for seq being Real_Sequence st seq is non-decreasing holds
for n being Element of NAT holds seq . 0 <= seq . n by Th12;

theorem Th22: :: SEQM_3:22

for seq being Real_Sequence st seq is non-increasing holds
for n being Element of NAT holds seq . n <= seq . 0 by Th14;

registration

cluster Function-like constant -> non-decreasing non-increasing Element of K21(K22(NAT,REAL));
coherence
for b₁ being PartFunc of NAT,REAL st b₁ is constant holds
( b₁ is non-decreasing & b₁ is non-increasing )

proof end;

cluster Function-like non-decreasing non-increasing -> constant Element of K21(K22(NAT,REAL));
coherence
for b₁ being PartFunc of NAT,REAL st b₁ is non-decreasing & b₁ is non-increasing holds
b₁ is constant

proof end;

end;

theorem :: SEQM_3:23

canceled;

theorem :: SEQM_3:24

canceled;

theorem :: SEQM_3:25

canceled;

theorem :: SEQM_3:26

canceled;

theorem :: SEQM_3:27

canceled;

Lm10: ( incl NAT is increasing & incl NAT is natural-valued )

proof end;

Lm11: ( id NAT is increasing & id NAT is natural-valued )
;

registration

cluster Relation-like NAT -defined REAL -valued Function-like non empty total V15( NAT , REAL ) complex-valued ext-real-valued real-valued natural-valued increasing Element of K21(K22(NAT,REAL));
existence
ex b₁ being Real_Sequence st
( b₁ is increasing & b₁ is natural-valued ) by Lm10;

end;

registration

cluster Relation-like NAT -defined NAT -valued Function-like non empty total V15( NAT , NAT ) complex-valued ext-real-valued real-valued rational-valued integer-valued natural-valued increasing Element of K21(K22(NAT,NAT));
existence
ex b₁ being sequence of NAT st b₁ is increasing by Lm11;

end;

Lm12: for f being sequence of NAT holds
( f is increasing iff for n being Element of NAT holds f . n < f . (n + 1) )

proof end;

theorem :: SEQM_3:28

canceled;

theorem :: SEQM_3:29

for seq being Real_Sequence holds
( seq is sequence of NAT iff for n being Element of NAT holds seq . n is Element of NAT )

proof end;

theorem :: SEQM_3:30

canceled;

theorem :: SEQM_3:31

canceled;

registration

let Nseq be increasing sequence of NAT;
let k be Element of NAT ;
cluster Nseq ^\ k -> natural-valued increasing ;
coherence
( Nseq ^\ k is increasing & Nseq ^\ k is natural-valued )

proof end;

end;

definition

canceled;
canceled;
canceled;
let f be Real_Sequence;
redefine attr f is increasing means :: SEQM_3:def 11
for n being Element of NAT holds f . n < f . (n + 1);
compatibility
( f is increasing iff for n being Element of NAT holds f . n < f . (n + 1) ) by Lm6;
redefine attr f is decreasing means :: SEQM_3:def 12
for n being Element of NAT holds f . n > f . (n + 1);
compatibility
( f is decreasing iff for n being Element of NAT holds f . n > f . (n + 1) ) by Lm7;
redefine attr f is non-decreasing means :: SEQM_3:def 13
for n being Element of NAT holds f . n <= f . (n + 1);
compatibility
( f is non-decreasing iff for n being Element of NAT holds f . n <= f . (n + 1) ) by Lm8;
redefine attr f is non-increasing means :: SEQM_3:def 14
for n being Element of NAT holds f . n >= f . (n + 1);
compatibility
( f is non-increasing iff for n being Element of NAT holds f . n >= f . (n + 1) ) by Lm9;

end;

:: deftheorem SEQM_3:def 8 :
canceled;

:: deftheorem SEQM_3:def 9 :
canceled;

:: deftheorem SEQM_3:def 10 :
canceled;

:: deftheorem defines increasing SEQM_3:def 11 :
for f being Real_Sequence holds
( f is increasing iff for n being Element of NAT holds f . n < f . (n + 1) );

:: deftheorem defines decreasing SEQM_3:def 12 :
for f being Real_Sequence holds
( f is decreasing iff for n being Element of NAT holds f . n > f . (n + 1) );

:: deftheorem defines non-decreasing SEQM_3:def 13 :
for f being Real_Sequence holds
( f is non-decreasing iff for n being Element of NAT holds f . n <= f . (n + 1) );

:: deftheorem defines non-increasing SEQM_3:def 14 :
for f being Real_Sequence holds
( f is non-increasing iff for n being Element of NAT holds f . n >= f . (n + 1) );

theorem :: SEQM_3:32

canceled;

theorem :: SEQM_3:33

for Nseq being increasing sequence of NAT
for n being Element of NAT holds n <= Nseq . n

proof end;

theorem :: SEQM_3:34

canceled;

theorem :: SEQM_3:35

canceled;

theorem :: SEQM_3:36

canceled;

theorem Th37: :: SEQM_3:37

for k being Element of NAT
for seq, seq1 being Real_Sequence holds (seq + seq1) ^\ k = (seq ^\ k) + (seq1 ^\ k)

proof end;

theorem Th38: :: SEQM_3:38

for k being Element of NAT
for seq being Real_Sequence holds (- seq) ^\ k = - (seq ^\ k)

proof end;

theorem :: SEQM_3:39

for k being Element of NAT
for seq, seq1 being Real_Sequence holds (seq - seq1) ^\ k = (seq ^\ k) - (seq1 ^\ k)

proof end;

theorem :: SEQM_3:40

canceled;

theorem Th41: :: SEQM_3:41

for k being Element of NAT
for seq being Real_Sequence holds (seq ") ^\ k = (seq ^\ k) "

proof end;

theorem Th42: :: SEQM_3:42

for k being Element of NAT
for seq, seq1 being Real_Sequence holds (seq (#) seq1) ^\ k = (seq ^\ k) (#) (seq1 ^\ k)

proof end;

theorem :: SEQM_3:43

for k being Element of NAT
for seq, seq1 being Real_Sequence holds (seq /" seq1) ^\ k = (seq ^\ k) /" (seq1 ^\ k)

proof end;

theorem :: SEQM_3:44

for k being Element of NAT
for r being real number
for seq being Real_Sequence holds (r (#) seq) ^\ k = r (#) (seq ^\ k)

proof end;

theorem :: SEQM_3:45

canceled;

theorem :: SEQM_3:46

canceled;

theorem :: SEQM_3:47

canceled;

theorem :: SEQM_3:48

canceled;

theorem :: SEQM_3:49

for seq, seq1 being Real_Sequence st seq is increasing & seq1 is subsequence of seq holds
seq1 is increasing

proof end;

theorem :: SEQM_3:50

for seq, seq1 being Real_Sequence st seq is decreasing & seq1 is subsequence of seq holds
seq1 is decreasing

proof end;

theorem Th51: :: SEQM_3:51

for seq, seq1 being Real_Sequence st seq is non-decreasing & seq1 is subsequence of seq holds
seq1 is non-decreasing

proof end;

theorem Th52: :: SEQM_3:52

for seq, seq1 being Real_Sequence st seq is non-increasing & seq1 is subsequence of seq holds
seq1 is non-increasing

proof end;

theorem :: SEQM_3:53

for seq, seq1 being Real_Sequence st seq is monotone & seq1 is subsequence of seq holds
seq1 is monotone

proof end;

theorem :: SEQM_3:54

canceled;

theorem :: SEQM_3:55

canceled;

theorem Th56: :: SEQM_3:56

for seq, seq1 being Real_Sequence st seq is bounded_above & seq1 is subsequence of seq holds
seq1 is bounded_above

proof end;

theorem Th57: :: SEQM_3:57

for seq, seq1 being Real_Sequence st seq is bounded_below & seq1 is subsequence of seq holds
seq1 is bounded_below

proof end;

theorem :: SEQM_3:58

for seq, seq1 being Real_Sequence st seq is bounded & seq1 is subsequence of seq holds
seq1 is bounded

proof end;

theorem :: SEQM_3:59

for r being real number
for seq being Real_Sequence holds
( ( seq is increasing & 0 < r implies r (#) seq is increasing ) & ( 0 = r implies r (#) seq is constant ) & ( seq is increasing & r < 0 implies r (#) seq is decreasing ) )

proof end;

theorem :: SEQM_3:60

for r being real number
for seq being Real_Sequence holds
( ( seq is decreasing & 0 < r implies r (#) seq is decreasing ) & ( seq is decreasing & r < 0 implies r (#) seq is increasing ) )

proof end;

theorem :: SEQM_3:61

for r being real number
for seq being Real_Sequence holds
( ( seq is non-decreasing & 0 <= r implies r (#) seq is non-decreasing ) & ( seq is non-decreasing & r <= 0 implies r (#) seq is non-increasing ) )

proof end;

theorem :: SEQM_3:62

for r being real number
for seq being Real_Sequence holds
( ( seq is non-increasing & 0 <= r implies r (#) seq is non-increasing ) & ( seq is non-increasing & r <= 0 implies r (#) seq is non-decreasing ) )

proof end;

theorem Th63: :: SEQM_3:63

for seq, seq1 being Real_Sequence holds
( ( seq is increasing & seq1 is increasing implies seq + seq1 is increasing ) & ( seq is decreasing & seq1 is decreasing implies seq + seq1 is decreasing ) & ( seq is non-decreasing & seq1 is non-decreasing implies seq + seq1 is non-decreasing ) & ( seq is non-increasing & seq1 is non-increasing implies seq + seq1 is non-increasing ) )

proof end;

theorem :: SEQM_3:64

for seq, seq1 being Real_Sequence holds
( ( seq is increasing & seq1 is constant implies seq + seq1 is increasing ) & ( seq is decreasing & seq1 is constant implies seq + seq1 is decreasing ) & ( seq is non-decreasing & seq1 is constant implies seq + seq1 is non-decreasing ) & ( seq is non-increasing & seq1 is constant implies seq + seq1 is non-increasing ) )

proof end;

theorem :: SEQM_3:65

canceled;

theorem :: SEQM_3:66

canceled;

theorem :: SEQM_3:67

canceled;

theorem Th68: :: SEQM_3:68

for r being real number
for seq being Real_Sequence holds
( ( seq is bounded_above & 0 < r implies r (#) seq is bounded_above ) & ( 0 = r implies r (#) seq is bounded ) & ( seq is bounded_above & r < 0 implies r (#) seq is bounded_below ) )

proof end;

theorem :: SEQM_3:69

canceled;

theorem :: SEQM_3:70

for seq being Real_Sequence holds
( ( seq is bounded implies for r being real number holds r (#) seq is bounded ) & ( seq is bounded implies - seq is bounded ) & ( seq is bounded implies abs seq is bounded ) & ( abs seq is bounded implies seq is bounded ) )

proof end;

theorem :: SEQM_3:71

canceled;

theorem :: SEQM_3:72

canceled;

theorem :: SEQM_3:73

canceled;

theorem :: SEQM_3:74

canceled;

theorem :: SEQM_3:75

canceled;

theorem :: SEQM_3:76

canceled;

theorem :: SEQM_3:77

for seq, seq1 being Real_Sequence st seq is bounded_above & seq1 is non-increasing holds
seq + seq1 is bounded_above

proof end;

theorem :: SEQM_3:78

for seq, seq1 being Real_Sequence st seq is bounded_below & seq1 is non-decreasing holds
seq + seq1 is bounded_below

proof end;

theorem :: SEQM_3:79

canceled;

theorem :: SEQM_3:80

( incl NAT is increasing & incl NAT is natural-valued ) by Lm10;

registration

cluster -> natural-valued FinSequence of NAT ;
coherence
for b₁ being FinSequence of NAT holds b₁ is natural-valued ;

end;

begin

theorem Th81: :: SEQM_3:81

for r, s being Real holds
( abs (r - s) = 1 iff ( ( r > s & r = s + 1 ) or ( r < s & s = r + 1 ) ) )

proof end;

theorem :: SEQM_3:82

for i, j, n, m being Element of NAT holds
( (abs (i - j)) + (abs (n - m)) = 1 iff ( ( abs (i - j) = 1 & n = m ) or ( abs (n - m) = 1 & i = j ) ) )

proof end;

theorem :: SEQM_3:83

for n being Element of NAT holds
( n > 1 iff ex m being Element of NAT st
( n = m + 1 & m > 0 ) )

proof end;

theorem :: SEQM_3:84

for f being FinSequence
for n, m, k being Element of NAT st len f = m + 1 & n in dom f & k in Seg m & not (Del (f,n)) . k = f . k holds
(Del (f,n)) . k = f . (k + 1)

proof end;

definition

let f be FinSequence;
redefine attr f is constant means :: SEQM_3:def 15
for n, m being Element of NAT st n in dom f & m in dom f holds
f . n = f . m;
compatibility
( f is constant iff for n, m being Element of NAT st n in dom f & m in dom f holds
f . n = f . m )

proof end;

end;

:: deftheorem defines constant SEQM_3:def 15 :
for f being FinSequence holds
( f is constant iff for n, m being Element of NAT st n in dom f & m in dom f holds
f . n = f . m );

registration

cluster -> real-valued FinSequence of REAL ;
coherence
for b₁ being FinSequence of REAL holds b₁ is real-valued ;

end;

registration

cluster Relation-like NAT -defined REAL -valued Function-like non empty complex-valued ext-real-valued real-valued increasing V52() FinSequence-like FinSubsequence-like FinSequence of REAL ;
existence
ex b₁ being FinSequence of REAL st
( not b₁ is empty & b₁ is increasing )

proof end;

end;

registration

cluster Relation-like NAT -defined REAL -valued Function-like constant complex-valued ext-real-valued real-valued V52() FinSequence-like FinSubsequence-like FinSequence of REAL ;
existence
ex b₁ being FinSequence of REAL st b₁ is constant

proof end;

end;

theorem Th85: :: SEQM_3:85

for v being FinSequence of REAL
for n, i, m being Element of NAT st v <> {} & rng v c= Seg n & v . (len v) = n & ( for k being Element of NAT st 1 <= k & k <= (len v) - 1 holds
for r, s being Real st r = v . k & s = v . (k + 1) & not abs (r - s) = 1 holds
r = s ) & i in Seg n & i + 1 in Seg n & m in dom v & v . m = i & ( for k being Element of NAT st k in dom v & v . k = i holds
k <= m ) holds
( m + 1 in dom v & v . (m + 1) = i + 1 )

proof end;

theorem :: SEQM_3:86

for v being FinSequence of REAL
for n being Element of NAT st v <> {} & rng v c= Seg n & v . 1 = 1 & v . (len v) = n & ( for k being Element of NAT st 1 <= k & k <= (len v) - 1 holds
for r, s being Real st r = v . k & s = v . (k + 1) & not abs (r - s) = 1 holds
r = s ) holds
( ( for i being Element of NAT st i in Seg n holds
ex k being Element of NAT st
( k in dom v & v . k = i ) ) & ( for m, k, i being Element of NAT
for r being Real st m in dom v & v . m = i & ( for j being Element of NAT st j in dom v & v . j = i holds
j <= m ) & m < k & k in dom v & r = v . k holds
i < r ) )

proof end;