:: CHAIN_1 semantic presentation
theorem Th1: :: CHAIN_1:1
for b
1, b
2 being
real number holds
not ( b
1 < b
2 & ( for b
3 being
Real holds
not ( b
1 < b
3 & b
3 < b
2 ) ) )
theorem Th2: :: CHAIN_1:2
:: deftheorem Def1 defines zero CHAIN_1:def 1 :
:: deftheorem Def2 defines zero CHAIN_1:def 2 :
for b
1 being
Nat holds
( b
1 is
zero iff not b
1 >= 1 );
:: deftheorem Def3 defines trivial CHAIN_1:def 3 :
for b
1 being
set holds
( b
1 is
trivial iff for b
2, b
3 being
set holds
( b
2 in b
1 & b
3 in b
1 implies b
2 = b
3 ) );
theorem Th3: :: CHAIN_1:3
canceled;
theorem Th4: :: CHAIN_1:4
theorem Th5: :: CHAIN_1:5
theorem Th6: :: CHAIN_1:6
for b
1 being
set holds
(
Card b
1 = 2 iff ex b
2, b
3 being
set st
( b
2 in b
1 & b
3 in b
1 & b
2 <> b
3 & ( for b
4 being
set holds
not ( b
4 in b
1 & not b
4 = b
2 & not b
4 = b
3 ) ) ) )
theorem Th7: :: CHAIN_1:7
for b
1, b
2 being
Nat holds
( ( b
1 is
even iff b
2 is
even ) iff b
1 + b
2 is
even )
theorem Th8: :: CHAIN_1:8
theorem Th9: :: CHAIN_1:9
:: deftheorem Def4 defines REAL CHAIN_1:def 4 :
:: deftheorem Def5 defines Grating CHAIN_1:def 5 :
theorem Th10: :: CHAIN_1:10
theorem Th11: :: CHAIN_1:11
theorem Th12: :: CHAIN_1:12
theorem Th13: :: CHAIN_1:13
theorem Th14: :: CHAIN_1:14
theorem Th15: :: CHAIN_1:15
theorem Th16: :: CHAIN_1:16
:: deftheorem Def6 defines Gap CHAIN_1:def 6 :
theorem Th17: :: CHAIN_1:17
for b
1 being
finite non
trivial Subset of
REAL for b
2, b
3 being
Real holds
(
[b2,b3] is
Gap of b
1 iff ( b
2 in b
1 & b
3 in b
1 & ( ( b
2 < b
3 & ( for b
4 being
Real holds
not ( b
4 in b
1 & b
2 < b
4 & b
4 < b
3 ) ) ) or ( b
3 < b
2 & ( for b
4 being
Real holds
( b
4 in b
1 implies ( not b
2 < b
4 & not b
4 < b
3 ) ) ) ) ) ) )
theorem Th18: :: CHAIN_1:18
deffunc H1( set ) -> set = a1;
theorem Th19: :: CHAIN_1:19
theorem Th20: :: CHAIN_1:20
theorem Th21: :: CHAIN_1:21
theorem Th22: :: CHAIN_1:22
theorem Th23: :: CHAIN_1:23
:: deftheorem Def7 defines cell CHAIN_1:def 7 :
theorem Th24: :: CHAIN_1:24
theorem Th25: :: CHAIN_1:25
theorem Th26: :: CHAIN_1:26
theorem Th27: :: CHAIN_1:27
theorem Th28: :: CHAIN_1:28
theorem Th29: :: CHAIN_1:29
theorem Th30: :: CHAIN_1:30
theorem Th31: :: CHAIN_1:31
theorem Th32: :: CHAIN_1:32
:: deftheorem Def8 defines cells CHAIN_1:def 8 :
theorem Th33: :: CHAIN_1:33
theorem Th34: :: CHAIN_1:34
theorem Th35: :: CHAIN_1:35
theorem Th36: :: CHAIN_1:36
theorem Th37: :: CHAIN_1:37
theorem Th38: :: CHAIN_1:38
theorem Th39: :: CHAIN_1:39
theorem Th40: :: CHAIN_1:40
theorem Th41: :: CHAIN_1:41
theorem Th42: :: CHAIN_1:42
theorem Th43: :: CHAIN_1:43
theorem Th44: :: CHAIN_1:44
theorem Th45: :: CHAIN_1:45
theorem Th46: :: CHAIN_1:46
for b
1, b
2 being
Natfor b
3 being non
zero Natfor b
4, b
5, b
6, b
7 being
Element of
REAL b
3for b
8 being
Grating of b
3 holds
( b
1 <= b
3 & b
2 <= b
3 &
cell b
4,b
5 in cells b
1,b
8 &
cell b
6,b
7 in cells b
2,b
8 &
cell b
4,b
5 c= cell b
6,b
7 implies for b
9 being
Element of
Seg b
3 holds
not ( not ( b
4 . b
9 = b
6 . b
9 & b
5 . b
9 = b
7 . b
9 ) & not ( b
4 . b
9 = b
6 . b
9 & b
5 . b
9 = b
6 . b
9 ) & not ( b
4 . b
9 = b
7 . b
9 & b
5 . b
9 = b
7 . b
9 ) & not ( b
4 . b
9 <= b
5 . b
9 & b
7 . b
9 < b
6 . b
9 & b
7 . b
9 <= b
4 . b
9 & b
5 . b
9 <= b
6 . b
9 ) ) )
theorem Th47: :: CHAIN_1:47
for b
1, b
2 being
Natfor b
3 being non
zero Natfor b
4, b
5, b
6, b
7 being
Element of
REAL b
3for b
8 being
Grating of b
3 holds
not ( b
1 < b
2 & b
2 <= b
3 &
cell b
4,b
5 in cells b
1,b
8 &
cell b
6,b
7 in cells b
2,b
8 &
cell b
4,b
5 c= cell b
6,b
7 & ( for b
9 being
Element of
Seg b
3 holds
( not ( b
4 . b
9 = b
6 . b
9 & b
5 . b
9 = b
6 . b
9 ) & not ( b
4 . b
9 = b
7 . b
9 & b
5 . b
9 = b
7 . b
9 ) ) ) )
theorem Th48: :: CHAIN_1:48
for b
1 being non
zero Natfor b
2, b
3, b
4, b
5 being
Element of
REAL b
1for b
6 being
Grating of b
1for b
7, b
8 being
Subset of
(Seg b1) holds
(
cell b
2,b
3 c= cell b
4,b
5 & ( for b
9 being
Element of
Seg b
1 holds
( ( b
9 in b
7 & b
2 . b
9 < b
3 . b
9 &
[(b2 . b9),(b3 . b9)] is
Gap of b
6 . b
9 ) or ( not b
9 in b
7 & b
2 . b
9 = b
3 . b
9 & b
2 . b
9 in b
6 . b
9 ) ) ) & ( for b
9 being
Element of
Seg b
1 holds
( ( b
9 in b
8 & b
4 . b
9 < b
5 . b
9 &
[(b4 . b9),(b5 . b9)] is
Gap of b
6 . b
9 ) or ( not b
9 in b
8 & b
4 . b
9 = b
5 . b
9 & b
4 . b
9 in b
6 . b
9 ) ) ) implies ( b
7 c= b
8 & ( for b
9 being
Element of
Seg b
1 holds
( not ( not b
9 in b
7 & b
9 in b
8 ) implies ( b
2 . b
9 = b
4 . b
9 & b
3 . b
9 = b
5 . b
9 ) ) ) & ( for b
9 being
Element of
Seg b
1 holds
not ( not b
9 in b
7 & b
9 in b
8 & not ( b
2 . b
9 = b
4 . b
9 & b
3 . b
9 = b
4 . b
9 ) & not ( b
2 . b
9 = b
5 . b
9 & b
3 . b
9 = b
5 . b
9 ) ) ) ) )
:: deftheorem Def9 defines 0_ CHAIN_1:def 9 :
:: deftheorem Def10 defines Omega CHAIN_1:def 10 :
definition
let c
1 be non
zero Nat;
let c
2 be
Grating of c
1;
func infinite-cell c
2 -> Cell of a
1,a
2 means :
Def11:
:: CHAIN_1:def 11
ex b
1, b
2 being
Element of
REAL a
1 st
( a
3 = cell b
1,b
2 & ( for b
3 being
Element of
Seg a
1 holds
( b
2 . b
3 < b
1 . b
3 &
[(b1 . b3),(b2 . b3)] is
Gap of a
2 . b
3 ) ) );
existence
ex b1 being Cell of c1,c2ex b2, b3 being Element of REAL c1 st
( b1 = cell b2,b3 & ( for b4 being Element of Seg c1 holds
( b3 . b4 < b2 . b4 & [(b2 . b4),(b3 . b4)] is Gap of c2 . b4 ) ) )
uniqueness
for b1, b2 being Cell of c1,c2 holds
( ex b3, b4 being Element of REAL c1 st
( b1 = cell b3,b4 & ( for b5 being Element of Seg c1 holds
( b4 . b5 < b3 . b5 & [(b3 . b5),(b4 . b5)] is Gap of c2 . b5 ) ) ) & ex b3, b4 being Element of REAL c1 st
( b2 = cell b3,b4 & ( for b5 being Element of Seg c1 holds
( b4 . b5 < b3 . b5 & [(b3 . b5),(b4 . b5)] is Gap of c2 . b5 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def11 defines infinite-cell CHAIN_1:def 11 :
theorem Th49: :: CHAIN_1:49
theorem Th50: :: CHAIN_1:50
scheme :: CHAIN_1:sch 4
s4{ F
1()
-> non
zero Nat, F
2()
-> Grating of F
1(), F
3()
-> Nat, F
4()
-> Chain of F
3(),F
2(), P
1[
set ] } :
provided
E52:
P
1[
0_ F
3(),F
2()]
and
E53:
for b
1 being
Cell of F
3(),F
2() holds
( b
1 in F
4() implies P
1[
{b1}] )
and
E54:
for b
1, b
2 being
Chain of F
3(),F
2() holds
( b
1 c= F
4() & b
2 c= F
4() & P
1[b
1] & P
1[b
2] implies P
1[b
1 + b
2] )
:: deftheorem Def12 defines star CHAIN_1:def 12 :
theorem Th51: :: CHAIN_1:51
:: deftheorem Def13 defines del CHAIN_1:def 13 :
:: deftheorem Def14 defines bounds CHAIN_1:def 14 :
theorem Th52: :: CHAIN_1:52
theorem Th53: :: CHAIN_1:53
for b
1 being
Natfor b
2 being non
zero Natfor b
3 being
Grating of b
2 holds
( b
1 + 1
> b
2 implies for b
4 being
Chain of
(b1 + 1),b
3 holds
del b
4 = 0_ b
1,b
3 )
theorem Th54: :: CHAIN_1:54
for b
1 being
Natfor b
2 being non
zero Natfor b
3 being
Grating of b
2 holds
( b
1 + 1
<= b
2 implies for b
4 being
Cell of b
1,b
3for b
5 being
Cell of
(b1 + 1),b
3 holds
( b
4 in del {b5} iff b
4 c= b
5 ) )
theorem Th55: :: CHAIN_1:55
theorem Th56: :: CHAIN_1:56
theorem Th57: :: CHAIN_1:57
theorem Th58: :: CHAIN_1:58
theorem Th59: :: CHAIN_1:59
theorem Th60: :: CHAIN_1:60
theorem Th61: :: CHAIN_1:61
theorem Th62: :: CHAIN_1:62
theorem Th63: :: CHAIN_1:63
theorem Th64: :: CHAIN_1:64
theorem Th65: :: CHAIN_1:65
:: deftheorem Def15 defines Cycle CHAIN_1:def 15 :
for b
1 being non
zero Natfor b
2 being
Grating of b
1for b
3 being
Natfor b
4 being
Chain of b
3,b
2 holds
( b
4 is
Cycle of b
3,b
2 iff not ( not ( b
3 = 0 &
card b
4 is
even ) & ( for b
5 being
Nat holds
not ( b
3 = b
5 + 1 & ex b
6 being
Chain of
(b5 + 1),b
2 st
( b
6 = b
4 &
del b
6 = 0_ b
5,b
2 ) ) ) ) );
theorem Th66: :: CHAIN_1:66
theorem Th67: :: CHAIN_1:67
for b
1 being
Natfor b
2 being non
zero Natfor b
3 being
Grating of b
2 holds
( b
1 > b
2 implies for b
4 being
Chain of b
1,b
3 holds
b
4 is
Cycle of b
1,b
3 )
theorem Th68: :: CHAIN_1:68
:: deftheorem Def16 defines del CHAIN_1:def 16 :
theorem Th69: :: CHAIN_1:69
definition
let c
1 be non
zero Nat;
let c
2 be
Grating of c
1;
let c
3 be
Nat;
func Chains c
3,c
2 -> strict AbGroup means :
Def17:
:: CHAIN_1:def 17
( the
carrier of a
4 = bool (cells a3,a2) &
0. a
4 = 0_ a
3,a
2 & ( for b
1, b
2 being
Element of a
4for b
3, b
4 being
Chain of a
3,a
2 holds
( b
1 = b
3 & b
2 = b
4 implies b
1 + b
2 = b
3 + b
4 ) ) );
existence
ex b1 being strict AbGroup st
( the carrier of b1 = bool (cells c3,c2) & 0. b1 = 0_ c3,c2 & ( for b2, b3 being Element of b1
for b4, b5 being Chain of c3,c2 holds
( b2 = b4 & b3 = b5 implies b2 + b3 = b4 + b5 ) ) )
uniqueness
for b1, b2 being strict AbGroup holds
( the carrier of b1 = bool (cells c3,c2) & 0. b1 = 0_ c3,c2 & ( for b3, b4 being Element of b1
for b5, b6 being Chain of c3,c2 holds
( b3 = b5 & b4 = b6 implies b3 + b4 = b5 + b6 ) ) & the carrier of b2 = bool (cells c3,c2) & 0. b2 = 0_ c3,c2 & ( for b3, b4 being Element of b2
for b5, b6 being Chain of c3,c2 holds
( b3 = b5 & b4 = b6 implies b3 + b4 = b5 + b6 ) ) implies b1 = b2 )
end;
:: deftheorem Def17 defines Chains CHAIN_1:def 17 :
theorem Th70: :: CHAIN_1:70
definition
let c
1 be non
zero Nat;
let c
2 be
Grating of c
1;
let c
3 be
Nat;
func del c
3,c
2 -> Homomorphism of
Chains (a3 + 1),a
2,
Chains a
3,a
2 means :: CHAIN_1:def 18
for b
1 being
Element of
(Chains (a3 + 1),a2)for b
2 being
Chain of
(a3 + 1),a
2 holds
( b
1 = b
2 implies a
4 . b
1 = del b
2 );
existence
ex b1 being Homomorphism of Chains (c3 + 1),c2, Chains c3,c2 st
for b2 being Element of (Chains (c3 + 1),c2)
for b3 being Chain of (c3 + 1),c2 holds
( b2 = b3 implies b1 . b2 = del b3 )
uniqueness
for b1, b2 being Homomorphism of Chains (c3 + 1),c2, Chains c3,c2 holds
( ( for b3 being Element of (Chains (c3 + 1),c2)
for b4 being Chain of (c3 + 1),c2 holds
( b3 = b4 implies b1 . b3 = del b4 ) ) & ( for b3 being Element of (Chains (c3 + 1),c2)
for b4 being Chain of (c3 + 1),c2 holds
( b3 = b4 implies b2 . b3 = del b4 ) ) implies b1 = b2 )
end;
:: deftheorem Def18 defines del CHAIN_1:def 18 :