:: 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 b1 being Nat st
( a1 = b1 & a2 = b1 + 1 );
Lemma9:
for b1 being Nat holds Seg b1,b1 are_equipotent
Lemma10:
for b1 being Nat holds Card (Seg b1) = Card b1
:: 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 :
Lemma17:
for b1 being set
for b2 being FinSequence of b1 holds
b2 is PartFunc of NAT ,b1
theorem Th22: :: FINSEQ_1:22
canceled;
theorem Th23: :: FINSEQ_1:23
theorem Th24: :: FINSEQ_1:24
Lemma19:
for b1 being FinSequence holds
( b1 = {} iff len b1 = 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
for b
1, b
2, b
3 being
FinSequence holds
( ( b
1 ^ b
2 = b
3 ^ b
2 or b
2 ^ b
1 = b
2 ^ b
3 ) implies b
1 = b
3 )
theorem Th47: :: FINSEQ_1:47
theorem Th48: :: FINSEQ_1:48
Lemma35:
for b1, b2 being set holds
{[b1,b2]} is Function
Lemma36:
for b1, b2, b3, b4 being set holds
( [b1,b2] in {[b3,b4]} implies ( b1 = b3 & b2 = b4 ) )
:: 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 :
registration
let c
1, c
2 be
set ;
cluster <*a1,a2*> -> Relation-like Function-like ;
coherence
( <*c1,c2*> is Function-like & <*c1,c2*> is Relation-like )
;
let c
3 be
set ;
cluster <*a1,a2,a3*> -> Relation-like Function-like ;
coherence
( <*c1,c2,c3*> is Function-like & <*c1,c2,c3*> is Relation-like )
;
end;
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 :
for b
1, b
2 being
set holds
( b
2 = b
1 * iff for b
3 being
set holds
( b
3 in b
2 iff b
3 is
FinSequence of b
1 ) );
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 c
1 be
set ;
given c
2 being
natural number such that E49:
c
1 c= Seg c
2
;
func Sgm c
1 -> FinSequence of
NAT means :
Def13:
:: FINSEQ_1:def 13
(
rng a
2 = a
1 & ( for b
1, b
2, b
3, b
4 being
natural number holds
not ( 1
<= b
1 & b
1 < b
2 & b
2 <= len a
2 & b
3 = a
2 . b
1 & b
4 = a
2 . b
2 & not b
3 < b
4 ) ) );
existence
ex b1 being FinSequence of NAT st
( rng b1 = c1 & ( for b2, b3, b4, b5 being natural number holds
not ( 1 <= b2 & b2 < b3 & b3 <= len b1 & b4 = b1 . b2 & b5 = b1 . b3 & not b4 < b5 ) ) )
uniqueness
for b1, b2 being FinSequence of NAT holds
( rng b1 = c1 & ( for b3, b4, b5, b6 being natural number holds
not ( 1 <= b3 & b3 < b4 & b4 <= len b1 & b5 = b1 . b3 & b6 = b1 . b4 & not b5 < b6 ) ) & rng b2 = c1 & ( for b3, b4, b5, b6 being natural number holds
not ( 1 <= b3 & b3 < b4 & b4 <= len b2 & b5 = b2 . b3 & b6 = b2 . b4 & not b5 < b6 ) ) implies 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 c
1 be
Relation;
func c
1 [*] -> Relation means :: FINSEQ_1:def 16
for b
1, b
2 being
set holds
(
[b1,b2] in a
2 iff ( b
1 in field a
1 & b
2 in field a
1 & ex b
3 being
FinSequence st
(
len b
3 >= 1 & b
3 . 1
= b
1 & b
3 . (len b3) = b
2 & ( for b
4 being
Nat holds
( b
4 >= 1 & b
4 < len b
3 implies
[(b3 . b4),(b3 . (b4 + 1))] in a
1 ) ) ) ) );
existence
ex b1 being Relation st
for b2, b3 being set holds
( [b2,b3] in b1 iff ( b2 in field c1 & b3 in field c1 & ex b4 being FinSequence st
( len b4 >= 1 & b4 . 1 = b2 & b4 . (len b4) = b3 & ( for b5 being Nat holds
( b5 >= 1 & b5 < len b4 implies [(b4 . b5),(b4 . (b5 + 1))] in c1 ) ) ) ) )
uniqueness
for b1, b2 being Relation holds
( ( for b3, b4 being set holds
( [b3,b4] in b1 iff ( b3 in field c1 & b4 in field c1 & ex b5 being FinSequence st
( len b5 >= 1 & b5 . 1 = b3 & b5 . (len b5) = b4 & ( for b6 being Nat holds
( b6 >= 1 & b6 < len b5 implies [(b5 . b6),(b5 . (b6 + 1))] in c1 ) ) ) ) ) ) & ( for b3, b4 being set holds
( [b3,b4] in b2 iff ( b3 in field c1 & b4 in field c1 & ex b5 being FinSequence st
( len b5 >= 1 & b5 . 1 = b3 & b5 . (len b5) = b4 & ( for b6 being Nat holds
( b6 >= 1 & b6 < len b5 implies [(b5 . b6),(b5 . (b6 + 1))] in c1 ) ) ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def16 defines [*] FINSEQ_1:def 16 :
theorem Th83: :: FINSEQ_1:83
for b
1, b
2 being
set holds
( b
1 c= b
2 implies b
1 * c= b
2 * )
theorem Th84: :: FINSEQ_1:84
theorem Th85: :: FINSEQ_1:85
theorem Th86: :: FINSEQ_1:86
Lemma51:
( 1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3 )
;
Lemma52:
( 1 in Seg 4 & 2 in Seg 4 & 3 in Seg 4 & 4 in Seg 4 )
;
Lemma53:
( 1 in Seg 5 & 2 in Seg 5 & 3 in Seg 5 & 4 in Seg 5 & 5 in Seg 5 )
;
Lemma54:
( 1 in Seg 6 & 2 in Seg 6 & 3 in Seg 6 & 4 in Seg 6 & 5 in Seg 6 & 6 in Seg 6 )
;
Lemma55:
( 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 )
;
Lemma56:
( 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
Lemma63:
for b1 being Relation holds
not ( dom b1 <> {} & not b1 <> {} )
by RELAT_1:60;