:: DICKSON semantic presentation
theorem Th1: :: DICKSON:1
theorem Th2: :: DICKSON:2
for b
1 being
Nat holds b
1 c= b
1 + 1
theorem Th3: :: DICKSON:3
:: deftheorem Def1 defines ascending DICKSON:def 1 :
:: deftheorem Def2 defines weakly-ascending DICKSON:def 2 :
theorem Th4: :: DICKSON:4
theorem Th5: :: DICKSON:5
theorem Th6: :: DICKSON:6
canceled;
theorem Th7: :: DICKSON:7
theorem Th8: :: DICKSON:8
:: deftheorem Def3 defines quasi_ordered DICKSON:def 3 :
:: deftheorem Def4 defines EqRel DICKSON:def 4 :
theorem Th9: :: DICKSON:9
definition
let c
1 be
RelStr ;
func <=E c
1 -> Relation of
Class (EqRel a1) means :
Def5:
:: DICKSON:def 5
for b
1, b
2 being
set holds
(
[b1,b2] in a
2 iff ex b
3, b
4 being
Element of a
1 st
( b
1 = Class (EqRel a1),b
3 & b
2 = Class (EqRel a1),b
4 & b
3 <= b
4 ) );
existence
ex b1 being Relation of Class (EqRel c1) st
for b2, b3 being set holds
( [b2,b3] in b1 iff ex b4, b5 being Element of c1 st
( b2 = Class (EqRel c1),b4 & b3 = Class (EqRel c1),b5 & b4 <= b5 ) )
uniqueness
for b1, b2 being Relation of Class (EqRel c1) holds
( ( for b3, b4 being set holds
( [b3,b4] in b1 iff ex b5, b6 being Element of c1 st
( b3 = Class (EqRel c1),b5 & b4 = Class (EqRel c1),b6 & b5 <= b6 ) ) ) & ( for b3, b4 being set holds
( [b3,b4] in b2 iff ex b5, b6 being Element of c1 st
( b3 = Class (EqRel c1),b5 & b4 = Class (EqRel c1),b6 & b5 <= b6 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines <=E DICKSON:def 5 :
theorem Th10: :: DICKSON:10
theorem Th11: :: DICKSON:11
:: deftheorem Def6 defines \~ DICKSON:def 6 :
:: deftheorem Def7 defines \~ DICKSON:def 7 :
theorem Th12: :: DICKSON:12
theorem Th13: :: DICKSON:13
theorem Th14: :: DICKSON:14
theorem Th15: :: DICKSON:15
theorem Th16: :: DICKSON:16
theorem Th17: :: DICKSON:17
theorem Th18: :: DICKSON:18
theorem Th19: :: DICKSON:19
:: deftheorem Def8 defines min-classes DICKSON:def 8 :
theorem Th20: :: DICKSON:20
theorem Th21: :: DICKSON:21
theorem Th22: :: DICKSON:22
theorem Th23: :: DICKSON:23
theorem Th24: :: DICKSON:24
theorem Th25: :: DICKSON:25
:: deftheorem Def9 defines is_Dickson-basis_of DICKSON:def 9 :
theorem Th26: :: DICKSON:26
theorem Th27: :: DICKSON:27
:: deftheorem Def10 defines Dickson DICKSON:def 10 :
theorem Th28: :: DICKSON:28
theorem Th29: :: DICKSON:29
:: deftheorem Def11 defines mindex DICKSON:def 11 :
for b
1 being
Functionfor b
2 being
set holds
(
dom b
1 = NAT & b
2 in rng b
1 implies for b
3 being
Nat holds
( b
3 = b
1 mindex b
2 iff ( b
1 . b
3 = b
2 & ( for b
4 being
Nat holds
( b
1 . b
4 = b
2 implies b
3 <= b
4 ) ) ) ) );
definition
let c
1 be non
empty 1-sorted ;
let c
2 be
sequence of c
1;
let c
3 be
set ;
let c
4 be
Nat;
assume E30:
ex b
1 being
Nat st
( c
4 < b
1 & c
2 . b
1 = c
3 )
;
func c
2 mindex c
3,c
4 -> Nat means :
Def12:
:: DICKSON:def 12
( a
2 . a
5 = a
3 & a
4 < a
5 & ( for b
1 being
Nat holds
( a
4 < b
1 & a
2 . b
1 = a
3 implies a
5 <= b
1 ) ) );
existence
ex b1 being Nat st
( c2 . b1 = c3 & c4 < b1 & ( for b2 being Nat holds
( c4 < b2 & c2 . b2 = c3 implies b1 <= b2 ) ) )
uniqueness
for b1, b2 being Nat holds
( c2 . b1 = c3 & c4 < b1 & ( for b3 being Nat holds
( c4 < b3 & c2 . b3 = c3 implies b1 <= b3 ) ) & c2 . b2 = c3 & c4 < b2 & ( for b3 being Nat holds
( c4 < b3 & c2 . b3 = c3 implies b2 <= b3 ) ) implies b1 = b2 )
end;
:: deftheorem Def12 defines mindex DICKSON:def 12 :
for b
1 being non
empty 1-sorted for b
2 being
sequence of b
1for b
3 being
set for b
4 being
Nat holds
( ex b
5 being
Nat st
( b
4 < b
5 & b
2 . b
5 = b
3 ) implies for b
5 being
Nat holds
( b
5 = b
2 mindex b
3,b
4 iff ( b
2 . b
5 = b
3 & b
4 < b
5 & ( for b
6 being
Nat holds
( b
4 < b
6 & b
2 . b
6 = b
3 implies b
5 <= b
6 ) ) ) ) );
theorem Th30: :: DICKSON:30
theorem Th31: :: DICKSON:31
theorem Th32: :: DICKSON:32
theorem Th33: :: DICKSON:33
theorem Th34: :: DICKSON:34
theorem Th35: :: DICKSON:35
:: deftheorem Def13 defines Dickson-bases DICKSON:def 13 :
theorem Th36: :: DICKSON:36
theorem Th37: :: DICKSON:37
theorem Th38: :: DICKSON:38
theorem Th39: :: DICKSON:39
theorem Th40: :: DICKSON:40
theorem Th41: :: DICKSON:41
theorem Th42: :: DICKSON:42
theorem Th43: :: DICKSON:43
Lemma45:
for b1 being RelStr-yielding ManySortedSet of {} holds
( not product b1 is empty & product b1 is quasi_ordered & product b1 is Dickson & product b1 is antisymmetric )
:: deftheorem Def14 defines NATOrd DICKSON:def 14 :
theorem Th44: :: DICKSON:44
theorem Th45: :: DICKSON:45
theorem Th46: :: DICKSON:46
theorem Th47: :: DICKSON:47
:: deftheorem Def15 defines OrderedNAT DICKSON:def 15 :
theorem Th48: :: DICKSON:48
theorem Th49: :: DICKSON:49
theorem Th50: :: DICKSON:50