:: RFINSEQ2 semantic presentation
definition
let c
1 be
FinSequence of
REAL ;
func max_p c
1 -> Nat means :
Def1:
:: RFINSEQ2:def 1
( (
len a
1 = 0 implies a
2 = 0 ) & (
len a
1 > 0 implies ( a
2 in dom a
1 & ( for b
1 being
Natfor b
2, b
3 being
Real holds
( b
1 in dom a
1 & b
2 = a
1 . b
1 & b
3 = a
1 . a
2 implies b
2 <= b
3 ) ) & ( for b
1 being
Nat holds
( b
1 in dom a
1 & a
1 . b
1 = a
1 . a
2 implies a
2 <= b
1 ) ) ) ) );
existence
ex b1 being Nat st
( ( len c1 = 0 implies b1 = 0 ) & ( len c1 > 0 implies ( b1 in dom c1 & ( for b2 being Nat
for b3, b4 being Real holds
( b2 in dom c1 & b3 = c1 . b2 & b4 = c1 . b1 implies b3 <= b4 ) ) & ( for b2 being Nat holds
( b2 in dom c1 & c1 . b2 = c1 . b1 implies b1 <= b2 ) ) ) ) )
uniqueness
for b1, b2 being Nat holds
( ( len c1 = 0 implies b1 = 0 ) & ( len c1 > 0 implies ( b1 in dom c1 & ( for b3 being Nat
for b4, b5 being Real holds
( b3 in dom c1 & b4 = c1 . b3 & b5 = c1 . b1 implies b4 <= b5 ) ) & ( for b3 being Nat holds
( b3 in dom c1 & c1 . b3 = c1 . b1 implies b1 <= b3 ) ) ) ) & ( len c1 = 0 implies b2 = 0 ) & ( len c1 > 0 implies ( b2 in dom c1 & ( for b3 being Nat
for b4, b5 being Real holds
( b3 in dom c1 & b4 = c1 . b3 & b5 = c1 . b2 implies b4 <= b5 ) ) & ( for b3 being Nat holds
( b3 in dom c1 & c1 . b3 = c1 . b2 implies b2 <= b3 ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines max_p RFINSEQ2:def 1 :
for b
1 being
FinSequence of
REAL for b
2 being
Nat holds
( b
2 = max_p b
1 iff ( (
len b
1 = 0 implies b
2 = 0 ) & (
len b
1 > 0 implies ( b
2 in dom b
1 & ( for b
3 being
Natfor b
4, b
5 being
Real holds
( b
3 in dom b
1 & b
4 = b
1 . b
3 & b
5 = b
1 . b
2 implies b
4 <= b
5 ) ) & ( for b
3 being
Nat holds
( b
3 in dom b
1 & b
1 . b
3 = b
1 . b
2 implies b
2 <= b
3 ) ) ) ) ) );
definition
let c
1 be
FinSequence of
REAL ;
func min_p c
1 -> Nat means :
Def2:
:: RFINSEQ2:def 2
( (
len a
1 = 0 implies a
2 = 0 ) & (
len a
1 > 0 implies ( a
2 in dom a
1 & ( for b
1 being
Natfor b
2, b
3 being
Real holds
( b
1 in dom a
1 & b
2 = a
1 . b
1 & b
3 = a
1 . a
2 implies b
2 >= b
3 ) ) & ( for b
1 being
Nat holds
( b
1 in dom a
1 & a
1 . b
1 = a
1 . a
2 implies a
2 <= b
1 ) ) ) ) );
existence
ex b1 being Nat st
( ( len c1 = 0 implies b1 = 0 ) & ( len c1 > 0 implies ( b1 in dom c1 & ( for b2 being Nat
for b3, b4 being Real holds
( b2 in dom c1 & b3 = c1 . b2 & b4 = c1 . b1 implies b3 >= b4 ) ) & ( for b2 being Nat holds
( b2 in dom c1 & c1 . b2 = c1 . b1 implies b1 <= b2 ) ) ) ) )
uniqueness
for b1, b2 being Nat holds
( ( len c1 = 0 implies b1 = 0 ) & ( len c1 > 0 implies ( b1 in dom c1 & ( for b3 being Nat
for b4, b5 being Real holds
( b3 in dom c1 & b4 = c1 . b3 & b5 = c1 . b1 implies b4 >= b5 ) ) & ( for b3 being Nat holds
( b3 in dom c1 & c1 . b3 = c1 . b1 implies b1 <= b3 ) ) ) ) & ( len c1 = 0 implies b2 = 0 ) & ( len c1 > 0 implies ( b2 in dom c1 & ( for b3 being Nat
for b4, b5 being Real holds
( b3 in dom c1 & b4 = c1 . b3 & b5 = c1 . b2 implies b4 >= b5 ) ) & ( for b3 being Nat holds
( b3 in dom c1 & c1 . b3 = c1 . b2 implies b2 <= b3 ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines min_p RFINSEQ2:def 2 :
for b
1 being
FinSequence of
REAL for b
2 being
Nat holds
( b
2 = min_p b
1 iff ( (
len b
1 = 0 implies b
2 = 0 ) & (
len b
1 > 0 implies ( b
2 in dom b
1 & ( for b
3 being
Natfor b
4, b
5 being
Real holds
( b
3 in dom b
1 & b
4 = b
1 . b
3 & b
5 = b
1 . b
2 implies b
4 >= b
5 ) ) & ( for b
3 being
Nat holds
( b
3 in dom b
1 & b
1 . b
3 = b
1 . b
2 implies b
2 <= b
3 ) ) ) ) ) );
:: deftheorem Def3 defines max RFINSEQ2:def 3 :
:: deftheorem Def4 defines min RFINSEQ2:def 4 :
theorem Th1: :: RFINSEQ2:1
theorem Th2: :: RFINSEQ2:2
theorem Th3: :: RFINSEQ2:3
theorem Th4: :: RFINSEQ2:4
theorem Th5: :: RFINSEQ2:5
theorem Th6: :: RFINSEQ2:6
theorem Th7: :: RFINSEQ2:7
theorem Th8: :: RFINSEQ2:8
theorem Th9: :: RFINSEQ2:9
theorem Th10: :: RFINSEQ2:10
theorem Th11: :: RFINSEQ2:11
theorem Th12: :: RFINSEQ2:12
theorem Th13: :: RFINSEQ2:13
Lemma5:
for b1, b2 being FinSequence of REAL holds
( b1,b2 are_fiberwise_equipotent implies max b1 <= max b2 )
theorem Th14: :: RFINSEQ2:14
Lemma7:
for b1, b2 being FinSequence of REAL holds
( b1,b2 are_fiberwise_equipotent implies min b1 >= min b2 )
theorem Th15: :: RFINSEQ2:15
:: deftheorem Def5 defines sort_d RFINSEQ2:def 5 :
theorem Th16: :: RFINSEQ2:16
theorem Th17: :: RFINSEQ2:17
Lemma12:
for b1, b2 being non-decreasing FinSequence of REAL
for b3 being Nat holds
( len b1 = b3 + 1 & len b1 = len b2 & b1,b2 are_fiberwise_equipotent implies ( b1 . (len b1) = b2 . (len b2) & b1 | b3,b2 | b3 are_fiberwise_equipotent ) )
theorem Th18: :: RFINSEQ2:18
Lemma14:
for b1 being Nat
for b2, b3 being non-decreasing FinSequence of REAL holds
( b1 = len b2 & b2,b3 are_fiberwise_equipotent implies b2 = b3 )
theorem Th19: :: RFINSEQ2:19
:: deftheorem Def6 defines sort_a RFINSEQ2:def 6 :
theorem Th20: :: RFINSEQ2:20
theorem Th21: :: RFINSEQ2:21
theorem Th22: :: RFINSEQ2:22
theorem Th23: :: RFINSEQ2:23
theorem Th24: :: RFINSEQ2:24
theorem Th25: :: RFINSEQ2:25
theorem Th26: :: RFINSEQ2:26
theorem Th27: :: RFINSEQ2:27
theorem Th28: :: RFINSEQ2:28
theorem Th29: :: RFINSEQ2:29
theorem Th30: :: RFINSEQ2:30
theorem Th31: :: RFINSEQ2:31
theorem Th32: :: RFINSEQ2:32