:: FINSEQ_7 semantic presentation
theorem Th1: :: FINSEQ_7:1
theorem Th2: :: FINSEQ_7:2
theorem Th3: :: FINSEQ_7:3
theorem Th4: :: FINSEQ_7:4
Lemma2:
for b1, b2 being Nat holds (b1 -' b2) -' 1 = (b1 -' 1) -' b2
:: deftheorem Def1 defines Replace FINSEQ_7:def 1 :
theorem Th5: :: FINSEQ_7:5
canceled;
theorem Th6: :: FINSEQ_7:6
canceled;
theorem Th7: :: FINSEQ_7:7
theorem Th8: :: FINSEQ_7:8
theorem Th9: :: FINSEQ_7:9
Lemma5:
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being Element of b1
for b4 being Nat holds
( 1 <= b4 & b4 <= len b2 implies (Replace b2,b4,b3) . b4 = b3 )
theorem Th10: :: FINSEQ_7:10
theorem Th11: :: FINSEQ_7:11
theorem Th12: :: FINSEQ_7:12
theorem Th13: :: FINSEQ_7:13
theorem Th14: :: FINSEQ_7:14
theorem Th15: :: FINSEQ_7:15
theorem Th16: :: FINSEQ_7:16
theorem Th17: :: FINSEQ_7:17
theorem Th18: :: FINSEQ_7:18
theorem Th19: :: FINSEQ_7:19
definition
let c
1 be non
empty set ;
let c
2 be
FinSequence of c
1;
let c
3, c
4 be
Nat;
func Swap c
2,c
3,c
4 -> FinSequence of a
1 equals :
Def2:
:: FINSEQ_7:def 2
Replace (Replace a2,a3,(a2 /. a4)),a
4,
(a2 /. a3) if ( 1
<= a
3 & a
3 <= len a
2 & 1
<= a
4 & a
4 <= len a
2 )
otherwise a
2;
correctness
coherence
( ( 1 <= c3 & c3 <= len c2 & 1 <= c4 & c4 <= len c2 implies Replace (Replace c2,c3,(c2 /. c4)),c4,(c2 /. c3) is FinSequence of c1 ) & ( not ( 1 <= c3 & c3 <= len c2 & 1 <= c4 & c4 <= len c2 ) implies c2 is FinSequence of c1 ) );
consistency
for b1 being FinSequence of c1 holds
verum;
;
end;
:: deftheorem Def2 defines Swap FINSEQ_7:def 2 :
theorem Th20: :: FINSEQ_7:20
Lemma15:
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Nat holds
( 1 <= b3 & b3 <= len b2 & 1 <= b4 & b4 <= len b2 implies ( (Swap b2,b3,b4) . b3 = b2 . b4 & (Swap b2,b3,b4) . b4 = b2 . b3 ) )
theorem Th21: :: FINSEQ_7:21
theorem Th22: :: FINSEQ_7:22
theorem Th23: :: FINSEQ_7:23
theorem Th24: :: FINSEQ_7:24
theorem Th25: :: FINSEQ_7:25
theorem Th26: :: FINSEQ_7:26
theorem Th27: :: FINSEQ_7:27
theorem Th28: :: FINSEQ_7:28
theorem Th29: :: FINSEQ_7:29
theorem Th30: :: FINSEQ_7:30
theorem Th31: :: FINSEQ_7:31
Lemma19:
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4, b5 being Nat holds
( b3 <> b4 & b5 <> b4 implies (Swap b2,b3,b5) . b4 = b2 . b4 )
theorem Th32: :: FINSEQ_7:32
theorem Th33: :: FINSEQ_7:33
theorem Th34: :: FINSEQ_7:34
theorem Th35: :: FINSEQ_7:35
for b
1 being non
empty set for b
2 being
FinSequence of b
1for b
3 being
Element of b
1for b
4, b
5, b
6 being
Nat holds
( b
4 <> b
5 & b
6 <> b
5 & 1
<= b
4 & b
4 <= len b
2 & 1
<= b
6 & b
6 <= len b
2 & 1
<= b
5 & b
5 <= len b
2 implies
Swap (Replace b2,b5,b3),b
4,b
6 = Replace (Swap b2,b4,b6),b
5,b
3 )
theorem Th36: :: FINSEQ_7:36
for b
1 being non
empty set for b
2 being
FinSequence of b
1for b
3, b
4, b
5 being
Nat holds
( b
3 <> b
4 & b
5 <> b
4 & 1
<= b
3 & b
3 <= len b
2 & 1
<= b
5 & b
5 <= len b
2 & 1
<= b
4 & b
4 <= len b
2 implies
Swap (Swap b2,b3,b5),b
5,b
4 = Swap (Swap b2,b3,b4),b
3,b
5 )
theorem Th37: :: FINSEQ_7:37
for b
1 being non
empty set for b
2 being
FinSequence of b
1for b
3, b
4, b
5, b
6 being
Nat holds
( b
3 <> b
4 & b
5 <> b
4 & b
6 <> b
3 & b
6 <> b
5 & 1
<= b
3 & b
3 <= len b
2 & 1
<= b
5 & b
5 <= len b
2 & 1
<= b
4 & b
4 <= len b
2 & 1
<= b
6 & b
6 <= len b
2 implies
Swap (Swap b2,b3,b5),b
4,b
6 = Swap (Swap b2,b4,b6),b
3,b
5 )