:: STIRL2_1 semantic presentation
theorem Th1: :: STIRL2_1:1
theorem Th2: :: STIRL2_1:2
theorem Th3: :: STIRL2_1:3
theorem Th4: :: STIRL2_1:4
theorem Th5: :: STIRL2_1:5
theorem Th6: :: STIRL2_1:6
theorem Th7: :: STIRL2_1:7
theorem Th8: :: STIRL2_1:8
theorem Th9: :: STIRL2_1:9
theorem Th10: :: STIRL2_1:10
for b
1, b
2 being
Nat holds
( b
1 in b
2 implies ( b
1 <= b
2 - 1 & b
2 - 1 is
Nat ) )
theorem Th11: :: STIRL2_1:11
theorem Th12: :: STIRL2_1:12
theorem Th13: :: STIRL2_1:13
theorem Th14: :: STIRL2_1:14
:: deftheorem Def1 defines "increasing STIRL2_1:def 1 :
theorem Th15: :: STIRL2_1:15
theorem Th16: :: STIRL2_1:16
theorem Th17: :: STIRL2_1:17
for b
1, b
2 being
Natfor b
3 being
Function of b
1,b
2 holds
( b
3 is
onto implies b
1 >= b
2 )
theorem Th18: :: STIRL2_1:18
theorem Th19: :: STIRL2_1:19
theorem Th20: :: STIRL2_1:20
theorem Th21: :: STIRL2_1:21
theorem Th22: :: STIRL2_1:22
for b
1, b
2 being
Nat holds
not ( ( b
1 = 0 implies b
2 = 0 ) & ( b
2 = 0 implies b
1 = 0 ) & ( for b
3 being
Function of b
1,b
2 holds
not b
3 is
"increasing ) )
theorem Th23: :: STIRL2_1:23
for b
1, b
2 being
Nat holds
not ( ( b
1 = 0 implies b
2 = 0 ) & ( b
2 = 0 implies b
1 = 0 ) & b
1 >= b
2 & ( for b
3 being
Function of b
1,b
2 holds
not ( b
3 is
onto & b
3 is
"increasing ) ) )
theorem Th24: :: STIRL2_1:24
theorem Th25: :: STIRL2_1:25
:: deftheorem Def2 defines block STIRL2_1:def 2 :
theorem Th26: :: STIRL2_1:26
theorem Th27: :: STIRL2_1:27
for b
1 being
Nat holds
( b
1 <> 0 implies 0
block b
1 = 0 )
theorem Th28: :: STIRL2_1:28
theorem Th29: :: STIRL2_1:29
for b
1, b
2 being
Nat holds
( b
1 < b
2 implies b
1 block b
2 = 0 )
theorem Th30: :: STIRL2_1:30
for b
1 being
Nat holds
( b
1 block 0
= 1 iff b
1 = 0 )
theorem Th31: :: STIRL2_1:31
for b
1 being
Nat holds
( b
1 <> 0 implies b
1 block 0
= 0 )
theorem Th32: :: STIRL2_1:32
for b
1 being
Nat holds
( b
1 <> 0 implies b
1 block 1
= 1 )
Lemma24:
for b1 being set holds
( Card b1 = 0 iff b1 is empty )
by CARD_2:59, CARD_1:47;
theorem Th33: :: STIRL2_1:33
for b
1, b
2 being
Nat holds
( not ( ( ( 1
<= b
1 & b
1 <= b
2 ) or b
1 = b
2 ) & not b
2 block b
1 > 0 ) & not ( b
2 block b
1 > 0 & not ( 1
<= b
1 & b
1 <= b
2 ) & not b
1 = b
2 ) )
scheme :: STIRL2_1:sch 3
s3{ F
1()
-> set , F
2()
-> set , F
3()
-> set , F
4()
-> set , F
5(
set )
-> set , P
1[
set ,
set ,
set ] } :
Card { b1 where B is Function of F1(),F2() : P1[b1,F1(),F2()] } = Card { b1 where B is Function of F3(),F4() : ( P1[b1,F3(),F4()] & rng (b1 | F1()) c= F2() & ( for b1 being set holds
( b2 in F3() \ F1() implies b1 . b2 = F5(b2) ) ) ) }
provided
E25:
for b
1 being
set holds
( b
1 in F
3()
\ F
1() implies F
5(b
1)
in F
4() )
and
E26:
( F
1()
c= F
3() & F
2()
c= F
4() )
and
E27:
( F
2() is
empty implies F
1() is
empty )
and
E28:
for b
1 being
Function of F
3(),F
4() holds
( ( for b
2 being
set holds
( b
2 in F
3()
\ F
1() implies F
5(b
2)
= b
1 . b
2 ) ) implies ( P
1[b
1,F
3(),F
4()] iff P
1[b
1 | F
1(),F
1(),F
2()] ) )
scheme :: STIRL2_1:sch 4
s4{ F
1()
-> set , F
2()
-> set , F
3()
-> set , F
4()
-> set , P
1[
set ,
set ,
set ] } :
provided
E25:
( F
2() is
empty implies F
1() is
empty )
and
E26:
not F
3()
in F
1()
and
E27:
for b
1 being
Function of F
1()
\/ {F3()},F
2()
\/ {F4()} holds
( b
1 . F
3()
= F
4() implies ( P
1[b
1,F
1()
\/ {F3()},F
2()
\/ {F4()}] iff P
1[b
1 | F
1(),F
1(),F
2()] ) )
theorem Th34: :: STIRL2_1:34
theorem Th35: :: STIRL2_1:35
for b
1, b
2 being
Natfor b
3 being
Function of b
1 + 1,b
2 holds
not ( b
2 <> 0 & b
3 " {(b3 . b1)} <> {b1} & ( for b
4 being
Nat holds
not ( b
4 in b
3 " {(b3 . b1)} & b
4 <> b
1 ) ) )
theorem Th36: :: STIRL2_1:36
theorem Th37: :: STIRL2_1:37
theorem Th38: :: STIRL2_1:38
theorem Th39: :: STIRL2_1:39
theorem Th40: :: STIRL2_1:40
theorem Th41: :: STIRL2_1:41
Lemma33:
for b1 being Nat holds b1 + 1 = b1 \/ {b1}
by AFINSQ_1:4;
Lemma34:
for b1, b2 being Nat holds
( b1 < b2 implies b2 \/ {b1} = b2 )
theorem Th42: :: STIRL2_1:42
theorem Th43: :: STIRL2_1:43
:: deftheorem Def3 defines "**" STIRL2_1:def 3 :
theorem Th44: :: STIRL2_1:44
theorem Th45: :: STIRL2_1:45
theorem Th46: :: STIRL2_1:46
theorem Th47: :: STIRL2_1:47
theorem Th48: :: STIRL2_1:48
theorem Th49: :: STIRL2_1:49
:: deftheorem Def4 defines Sum STIRL2_1:def 4 :
theorem Th50: :: STIRL2_1:50
theorem Th51: :: STIRL2_1:51
theorem Th52: :: STIRL2_1:52
theorem Th53: :: STIRL2_1:53
theorem Th54: :: STIRL2_1:54
theorem Th55: :: STIRL2_1:55
theorem Th56: :: STIRL2_1:56
theorem Th57: :: STIRL2_1:57
for b
1 being
Nat holds
( b
1 >= 1 implies b
1 block 2
= (1 / 2) * ((2 |^ b1) - 2) )
theorem Th58: :: STIRL2_1:58
for b
1 being
Nat holds
( b
1 >= 2 implies b
1 block 3
= (1 / 6) * (((3 |^ b1) - (3 * (2 |^ b1))) + 3) )
Lemma49:
for b1 being Nat holds b1 |^ 3 = (b1 * b1) * b1
theorem Th59: :: STIRL2_1:59
for b
1 being
Nat holds
( b
1 >= 3 implies b
1 block 4
= (1 / 24) * ((((4 |^ b1) - (4 * (3 |^ b1))) + (6 * (2 |^ b1))) - 4) )
theorem Th60: :: STIRL2_1:60
theorem Th61: :: STIRL2_1:61
theorem Th62: :: STIRL2_1:62
theorem Th63: :: STIRL2_1:63
theorem Th64: :: STIRL2_1:64
theorem Th65: :: STIRL2_1:65
theorem Th66: :: STIRL2_1:66
for b
1 being
Subset of
NAT holds
not ( b
1 is
finite & ( for b
2 being
Nat holds
ex b
3 being
Nat st
( b
3 in b
1 & not b
3 <= b
2 ) ) )
theorem Th67: :: STIRL2_1:67
for b
1, b
2, b
3, b
4 being
set holds
( ( b
2 is
empty implies b
1 is
empty ) & not b
3 in b
1 implies for b
5 being
Function of b
1,b
2 holds
ex b
6 being
Function of b
1 \/ {b3},b
2 \/ {b4} st
( b
6 | b
1 = b
5 & b
6 . b
3 = b
4 ) )
theorem Th68: :: STIRL2_1:68
Lemma58:
for b1, b2 being set holds
( b2 in b1 implies (b1 \ {b2}) \/ {b2} = b1 )
by ZFMISC_1:140;
theorem Th69: :: STIRL2_1:69
Lemma60:
for b1 being finite set
for b2 being set holds
not ( b2 in b1 & not card (b1 \ {b2}) < card b1 )
theorem Th70: :: STIRL2_1:70
Lemma62:
for b1 being Nat
for b2 being finite Subset of NAT
for b3 being Function of b2, card b2 holds
not ( b1 in b2 & b3 is bijective & ( for b4, b5 being Nat holds
not ( b4 in dom b3 & b5 in dom b3 & b4 < b5 & not b3 . b4 < b3 . b5 ) ) & ( for b4 being Permutation of b2 holds
ex b5 being Nat st
( b5 in b2 & not ( ( b5 < b1 implies b4 . b5 = (b3 " ) . ((b3 . b5) + 1) ) & ( b5 = b1 implies b4 . b5 = (b3 " ) . 0 ) & ( b5 > b1 implies b4 . b5 = b5 ) ) ) ) )
theorem Th71: :: STIRL2_1:71
:: deftheorem Def5 defines "increasing STIRL2_1:def 5 :
theorem Th72: :: STIRL2_1:72
theorem Th73: :: STIRL2_1:73
theorem Th74: :: STIRL2_1:74
theorem Th75: :: STIRL2_1:75