:: FINSEQ_1 semantic presentation
:: deftheorem Def1 defines Seg FINSEQ_1:def 1 :
theorem Th1: :: FINSEQ_1:1
canceled;
theorem Th2: :: FINSEQ_1:2
canceled;
theorem Th3: :: FINSEQ_1:3
theorem Th4: :: FINSEQ_1:4
theorem Th5: :: FINSEQ_1:5
theorem Th6: :: FINSEQ_1:6
theorem Th7: :: FINSEQ_1:7
theorem Th8: :: FINSEQ_1:8
theorem Th9: :: FINSEQ_1:9
theorem Th10: :: FINSEQ_1:10
theorem Th11: :: FINSEQ_1:11
theorem Th12: :: FINSEQ_1:12
:: deftheorem Def2 defines FinSequence-like FINSEQ_1:def 2 :
defpred S1[ set , set ] means ex k being Element of NAT st
( a1 = k & a2 = k + 1 );
Lemma60:
for n being Element of NAT holds Seg n,n are_equipotent
Lemma61:
for n being Element of NAT holds Card (Seg n) = Card n
:: deftheorem Def3 defines len FINSEQ_1:def 3 :
theorem Th13: :: FINSEQ_1:13
canceled;
theorem Th14: :: FINSEQ_1:14
theorem Th15: :: FINSEQ_1:15
theorem Th16: :: FINSEQ_1:16
theorem Th17: :: FINSEQ_1:17
theorem Th18: :: FINSEQ_1:18
theorem Th19: :: FINSEQ_1:19
theorem Th20: :: FINSEQ_1:20
theorem Th21: :: FINSEQ_1:21
:: deftheorem Def4 defines FinSequence FINSEQ_1:def 4 :
Lemma73:
for D being set
for f being FinSequence of D holds f is PartFunc of NAT ,D
theorem Th22: :: FINSEQ_1:22
canceled;
theorem Th23: :: FINSEQ_1:23
theorem Th24: :: FINSEQ_1:24
Lemma75:
for q being FinSequence holds
( q = {} iff len q = 0 )
theorem Th25: :: FINSEQ_1:25
theorem Th26: :: FINSEQ_1:26
theorem Th27: :: FINSEQ_1:27
theorem Th28: :: FINSEQ_1:28
canceled;
theorem Th29: :: FINSEQ_1:29
:: deftheorem Def5 defines <* FINSEQ_1:def 5 :
:: deftheorem Def6 defines <*> FINSEQ_1:def 6 :
theorem Th30: :: FINSEQ_1:30
canceled;
theorem Th31: :: FINSEQ_1:31
canceled;
theorem Th32: :: FINSEQ_1:32
:: deftheorem Def7 defines ^ FINSEQ_1:def 7 :
theorem Th33: :: FINSEQ_1:33
theorem Th34: :: FINSEQ_1:34
canceled;
theorem Th35: :: FINSEQ_1:35
theorem Th36: :: FINSEQ_1:36
theorem Th37: :: FINSEQ_1:37
theorem Th38: :: FINSEQ_1:38
theorem Th39: :: FINSEQ_1:39
theorem Th40: :: FINSEQ_1:40
theorem Th41: :: FINSEQ_1:41
theorem Th42: :: FINSEQ_1:42
theorem Th43: :: FINSEQ_1:43
theorem Th44: :: FINSEQ_1:44
theorem Th45: :: FINSEQ_1:45
theorem Th46: :: FINSEQ_1:46
theorem Th47: :: FINSEQ_1:47
theorem Th48: :: FINSEQ_1:48
Lemma103:
for x, y being set holds {[x,y]} is Function
Lemma104:
for x, y, x1, y1 being set st [x,y] in {[x1,y1]} holds
( x = x1 & y = y1 )
:: deftheorem Def8 defines <* FINSEQ_1:def 8 :
theorem Th49: :: FINSEQ_1:49
canceled;
theorem Th50: :: FINSEQ_1:50
:: deftheorem Def9 defines <* FINSEQ_1:def 9 :
:: deftheorem Def10 defines <* FINSEQ_1:def 10 :
theorem Th51: :: FINSEQ_1:51
canceled;
theorem Th52: :: FINSEQ_1:52
theorem Th53: :: FINSEQ_1:53
canceled;
theorem Th54: :: FINSEQ_1:54
canceled;
theorem Th55: :: FINSEQ_1:55
theorem Th56: :: FINSEQ_1:56
theorem Th57: :: FINSEQ_1:57
theorem Th58: :: FINSEQ_1:58
theorem Th59: :: FINSEQ_1:59
theorem Th60: :: FINSEQ_1:60
theorem Th61: :: FINSEQ_1:61
theorem Th62: :: FINSEQ_1:62
theorem Th63: :: FINSEQ_1:63
theorem Th64: :: FINSEQ_1:64
:: deftheorem Def11 defines * FINSEQ_1:def 11 :
theorem Th65: :: FINSEQ_1:65
theorem Th66: :: FINSEQ_1:66
:: deftheorem Def12 defines FinSubsequence-like FINSEQ_1:def 12 :
theorem Th67: :: FINSEQ_1:67
canceled;
theorem Th68: :: FINSEQ_1:68
theorem Th69: :: FINSEQ_1:69
definition
let X be
set ;
given k being
natural number such that E31:
X c= Seg k
;
func Sgm c1 -> FinSequence of
NAT means :
Def13:
:: FINSEQ_1:def 13
(
rng it = X & ( for
l,
m,
k1,
k2 being
natural number st 1
<= l &
l < m &
m <= len it &
k1 = it . l &
k2 = it . m holds
k1 < k2 ) );
existence
ex b1 being FinSequence of NAT st
( rng b1 = X & ( for l, m, k1, k2 being natural number st 1 <= l & l < m & m <= len b1 & k1 = b1 . l & k2 = b1 . m holds
k1 < k2 ) )
uniqueness
for b1, b2 being FinSequence of NAT st rng b1 = X & ( for l, m, k1, k2 being natural number st 1 <= l & l < m & m <= len b1 & k1 = b1 . l & k2 = b1 . m holds
k1 < k2 ) & rng b2 = X & ( for l, m, k1, k2 being natural number st 1 <= l & l < m & m <= len b2 & k1 = b2 . l & k2 = b2 . m holds
k1 < k2 ) holds
b1 = b2
end;
:: deftheorem Def13 defines Sgm FINSEQ_1:def 13 :
theorem Th70: :: FINSEQ_1:70
canceled;
theorem Th71: :: FINSEQ_1:71
:: deftheorem Def14 defines Seq FINSEQ_1:def 14 :
theorem Th72: :: FINSEQ_1:72
theorem Th73: :: FINSEQ_1:73
theorem Th74: :: FINSEQ_1:74
theorem Th75: :: FINSEQ_1:75
theorem Th76: :: FINSEQ_1:76
theorem Th77: :: FINSEQ_1:77
theorem Th78: :: FINSEQ_1:78
:: deftheorem Def15 defines | FINSEQ_1:def 15 :
theorem Th79: :: FINSEQ_1:79
theorem Th80: :: FINSEQ_1:80
theorem Th81: :: FINSEQ_1:81
theorem Th82: :: FINSEQ_1:82
definition
let R be
Relation;
func c1 [*] -> Relation means :: FINSEQ_1:def 16
for
x,
y being
set holds
(
[x,y] in it iff (
x in field R &
y in field R & ex
p being
FinSequence st
(
len p >= 1 &
p . 1
= x &
p . (len p) = y & ( for
i being
Element of
NAT st
i >= 1 &
i < len p holds
[(p . i),(p . (i + 1))] in R ) ) ) );
existence
ex b1 being Relation st
for x, y being set holds
( [x,y] in b1 iff ( x in field R & y in field R & ex p being FinSequence st
( len p >= 1 & p . 1 = x & p . (len p) = y & ( for i being Element of NAT st i >= 1 & i < len p holds
[(p . i),(p . (i + 1))] in R ) ) ) )
uniqueness
for b1, b2 being Relation st ( for x, y being set holds
( [x,y] in b1 iff ( x in field R & y in field R & ex p being FinSequence st
( len p >= 1 & p . 1 = x & p . (len p) = y & ( for i being Element of NAT st i >= 1 & i < len p holds
[(p . i),(p . (i + 1))] in R ) ) ) ) ) & ( for x, y being set holds
( [x,y] in b2 iff ( x in field R & y in field R & ex p being FinSequence st
( len p >= 1 & p . 1 = x & p . (len p) = y & ( for i being Element of NAT st i >= 1 & i < len p holds
[(p . i),(p . (i + 1))] in R ) ) ) ) ) holds
b1 = b2
end;
:: deftheorem Def16 defines [*] FINSEQ_1:def 16 :
theorem Th83: :: FINSEQ_1:83
for
D1,
D2 being
set st
D1 c= D2 holds
D1 * c= D2 *
theorem Th84: :: FINSEQ_1:84
theorem Th85: :: FINSEQ_1:85
theorem Th86: :: FINSEQ_1:86
Lemma211:
( 1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3 )
;
Lemma212:
( 1 in Seg 4 & 2 in Seg 4 & 3 in Seg 4 & 4 in Seg 4 )
;
Lemma213:
( 1 in Seg 5 & 2 in Seg 5 & 3 in Seg 5 & 4 in Seg 5 & 5 in Seg 5 )
;
Lemma214:
( 1 in Seg 6 & 2 in Seg 6 & 3 in Seg 6 & 4 in Seg 6 & 5 in Seg 6 & 6 in Seg 6 )
;
Lemma215:
( 1 in Seg 7 & 2 in Seg 7 & 3 in Seg 7 & 4 in Seg 7 & 5 in Seg 7 & 6 in Seg 7 & 7 in Seg 7 )
;
Lemma216:
( 1 in Seg 8 & 2 in Seg 8 & 3 in Seg 8 & 4 in Seg 8 & 5 in Seg 8 & 6 in Seg 8 & 7 in Seg 8 & 8 in Seg 8 )
;
theorem Th87: :: FINSEQ_1:87
theorem Th88: :: FINSEQ_1:88
theorem Th89: :: FINSEQ_1:89
theorem Th90: :: FINSEQ_1:90
theorem Th91: :: FINSEQ_1:91
theorem Th92: :: FINSEQ_1:92
theorem Th93: :: FINSEQ_1:93
theorem Th94: :: FINSEQ_1:94
theorem Th95: :: FINSEQ_1:95
theorem Th96: :: FINSEQ_1:96
Lemma236:
for R being Relation st dom R <> {} holds
R <> {}
by RELAT_1:60;