:: GROEB_3 semantic presentation
theorem Th1: :: GROEB_3:1
for b
1 being
set for b
2, b
3 being
bag of b
1 holds
(b2 + b3) / b
3 = b
2
theorem Th2: :: GROEB_3:2
theorem Th3: :: GROEB_3:3
for b
1 being
Ordinalfor b
2 being
TermOrder of b
1for b
3, b
4, b
5 being
bag of b
1 holds
( b
3 <= b
4,b
2 & b
4 < b
5,b
2 implies b
3 < b
5,b
2 )
theorem Th4: :: GROEB_3:4
theorem Th5: :: GROEB_3:5
theorem Th6: :: GROEB_3:6
theorem Th7: :: GROEB_3:7
theorem Th8: :: GROEB_3:8
theorem Th9: :: GROEB_3:9
theorem Th10: :: GROEB_3:10
Lemma11:
for b1 being Ordinal
for b2 being connected TermOrder of b1
for b3 being add-associative right_zeroed right_complementable associative commutative right_unital distributive left_unital Field-like non trivial doubleLoopStr
for b4 being Polynomial of b1,b3
for b5 being set
for b6 being Subset of (Polynom-Ring b1,b3) holds
( PolyRedRel b6,b2 reduces b4,b5 implies b5 is Polynomial of b1,b3 )
theorem Th11: :: GROEB_3:11
theorem Th12: :: GROEB_3:12
theorem Th13: :: GROEB_3:13
theorem Th14: :: GROEB_3:14
theorem Th15: :: GROEB_3:15
:: deftheorem Def1 defines | GROEB_3:def 1 :
Lemma17:
for b1 being set
for b2 being non empty ZeroStr
for b3 being Series of b1,b2
for b4 being Subset of (Bags b1) holds Support (b3 | b4) c= Support b3
theorem Th16: :: GROEB_3:16
theorem Th17: :: GROEB_3:17
theorem Th18: :: GROEB_3:18
definition
let c
1 be
Ordinal;
let c
2 be
connected TermOrder of c
1;
let c
3 be non
empty add-associative right_zeroed right_complementable LoopStr ;
let c
4 be
Polynomial of c
1,c
3;
let c
5 be
Nat;
assume E19:
c
5 <= card (Support c4)
;
func Upper_Support c
4,c
2,c
5 -> finite Subset of
(Bags a1) means :
Def2:
:: GROEB_3:def 2
( a
6 c= Support a
4 &
card a
6 = a
5 & ( for b
1, b
2 being
bag of a
1 holds
( b
1 in a
6 & b
2 in Support a
4 & b
1 <= b
2,a
2 implies b
2 in a
6 ) ) );
existence
ex b1 being finite Subset of (Bags c1) st
( b1 c= Support c4 & card b1 = c5 & ( for b2, b3 being bag of c1 holds
( b2 in b1 & b3 in Support c4 & b2 <= b3,c2 implies b3 in b1 ) ) )
uniqueness
for b1, b2 being finite Subset of (Bags c1) holds
( b1 c= Support c4 & card b1 = c5 & ( for b3, b4 being bag of c1 holds
( b3 in b1 & b4 in Support c4 & b3 <= b4,c2 implies b4 in b1 ) ) & b2 c= Support c4 & card b2 = c5 & ( for b3, b4 being bag of c1 holds
( b3 in b2 & b4 in Support c4 & b3 <= b4,c2 implies b4 in b2 ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines Upper_Support GROEB_3:def 2 :
:: deftheorem Def3 defines Lower_Support GROEB_3:def 3 :
Lemma20:
for b1 being Ordinal
for b2 being connected TermOrder of b1
for b3 being non empty add-associative right_zeroed right_complementable LoopStr
for b4 being Polynomial of b1,b3
for b5 being Nat holds
( b5 <= card (Support b4) implies Lower_Support b4,b2,b5 c= Support b4 )
by XBOOLE_1:36;
theorem Th19: :: GROEB_3:19
theorem Th20: :: GROEB_3:20
theorem Th21: :: GROEB_3:21
theorem Th22: :: GROEB_3:22
theorem Th23: :: GROEB_3:23
theorem Th24: :: GROEB_3:24
for b
1 being
Ordinalfor b
2 being
connected TermOrder of b
1for b
3 being non
empty add-associative right_zeroed right_complementable LoopStr for b
4 being
Polynomial of b
1,b
3for b
5 being
Nat holds
( b
5 <= card (Support b4) implies (
Lower_Support b
4,b
2,b
5 c= Support b
4 &
card (Lower_Support b4,b2,b5) = (card (Support b4)) - b
5 & ( for b
6, b
7 being
bag of b
1 holds
( b
6 in Lower_Support b
4,b
2,b
5 & b
7 in Support b
4 & b
7 <= b
6,b
2 implies b
7 in Lower_Support b
4,b
2,b
5 ) ) ) )
definition
let c
1 be
Ordinal;
let c
2 be
connected TermOrder of c
1;
let c
3 be non
empty add-associative right_zeroed right_complementable LoopStr ;
let c
4 be
Polynomial of c
1,c
3;
let c
5 be
Nat;
func Up c
4,c
2,c
5 -> Polynomial of a
1,a
3 equals :: GROEB_3:def 4
a
4 | (Upper_Support a4,a2,a5);
coherence
c4 | (Upper_Support c4,c2,c5) is Polynomial of c1,c3
;
func Low c
4,c
2,c
5 -> Polynomial of a
1,a
3 equals :: GROEB_3:def 5
a
4 | (Lower_Support a4,a2,a5);
coherence
c4 | (Lower_Support c4,c2,c5) is Polynomial of c1,c3
;
end;
:: deftheorem Def4 defines Up GROEB_3:def 4 :
:: deftheorem Def5 defines Low GROEB_3:def 5 :
Lemma26:
for b1 being Ordinal
for b2 being connected TermOrder of b1
for b3 being non empty add-associative right_zeroed right_complementable LoopStr
for b4 being Polynomial of b1,b3
for b5 being Nat holds
( b5 <= card (Support b4) implies ( Support (b4 | (Upper_Support b4,b2,b5)) = Upper_Support b4,b2,b5 & Support (b4 | (Lower_Support b4,b2,b5)) = Lower_Support b4,b2,b5 ) )
theorem Th25: :: GROEB_3:25
theorem Th26: :: GROEB_3:26
theorem Th27: :: GROEB_3:27
theorem Th28: :: GROEB_3:28
theorem Th29: :: GROEB_3:29
theorem Th30: :: GROEB_3:30
theorem Th31: :: GROEB_3:31
theorem Th32: :: GROEB_3:32
theorem Th33: :: GROEB_3:33
theorem Th34: :: GROEB_3:34
theorem Th35: :: GROEB_3:35
theorem Th36: :: GROEB_3:36
registration
let c
1 be
Ordinal;
let c
2 be
connected TermOrder of c
1;
let c
3 be
Abelian add-associative right_zeroed right_complementable non
trivial doubleLoopStr ;
let c
4 be
non-zero Polynomial of c
1,c
3;
cluster Up a
4,a
2,1
-> non-zero monomial-like ;
coherence
( Up c4,c2,1 is non-zero & Up c4,c2,1 is monomial-like )
cluster Low a
4,a
2,
(card (Support a4)) -> monomial-like ;
coherence
Low c4,c2,(card (Support c4)) is monomial-like
end;
theorem Th37: :: GROEB_3:37
theorem Th38: :: GROEB_3:38
theorem Th39: :: GROEB_3:39
theorem Th40: :: GROEB_3:40
theorem Th41: :: GROEB_3:41
theorem Th42: :: GROEB_3:42
theorem Th43: :: GROEB_3:43
theorem Th44: :: GROEB_3:44
Lemma47:
for b1 being Ordinal
for b2 being connected TermOrder of b1
for b3 being add-associative right_zeroed right_complementable associative commutative right_unital distributive left_unital Field-like non trivial doubleLoopStr
for b4 being Subset of (Polynom-Ring b1,b3)
for b5 being RedSequence of PolyRedRel b4,b2
for b6 being Nat holds
( 1 <= b6 & b6 <= len b5 & len b5 > 1 implies b5 . b6 is Polynomial of b1,b3 )
theorem Th45: :: GROEB_3:45
Lemma49:
for b1 being Ordinal
for b2 being connected admissible TermOrder of b1
for b3 being add-associative right_zeroed right_complementable associative commutative right_unital distributive left_unital Field-like non trivial doubleLoopStr
for b4, b5 being Polynomial of b1,b3 holds
( HT b4,b2, HT b5,b2 are_disjoint implies for b6, b7 being bag of b1 holds
not ( b6 in Support b4 & b7 in Support b5 & not ( b6 = HT b4,b2 & b7 = HT b5,b2 ) & (HT b4,b2) + b7 = (HT b5,b2) + b6 ) )
theorem Th46: :: GROEB_3:46
theorem Th47: :: GROEB_3:47
theorem Th48: :: GROEB_3:48
theorem Th49: :: GROEB_3:49
theorem Th50: :: GROEB_3:50
theorem Th51: :: GROEB_3:51
theorem Th52: :: GROEB_3:52
for b
1 being
Ordinalfor b
2 being
connected admissible TermOrder of b
1for b
3 being
add-associative right_zeroed right_complementable associative commutative right_unital distributive left_unital Field-like non
trivial doubleLoopStr for b
4, b
5 being
Polynomial of b
1,b
3 holds
(
HT b
4,b
2,
HT b
5,b
2 are_disjoint implies for b
6, b
7 being
bag of b
1 holds
not ( b
6 in Support (Red b4,b2) & b
7 in Support (Red b5,b2) &
(HT b4,b2) + b
7 = (HT b5,b2) + b
6 ) )
theorem Th53: :: GROEB_3:53
for b
1 being
Ordinalfor b
2 being
connected TermOrder of b
1for b
3 being
Abelian add-associative right_zeroed right_complementable associative commutative right_unital distributive left_unital Field-like non
trivial doubleLoopStr for b
4, b
5 being
Polynomial of b
1,b
3 holds
(
HT b
4,b
2,
HT b
5,b
2 are_disjoint implies
S-Poly b
4,b
5,b
2 = ((HM b5,b2) *' (Red b4,b2)) - ((HM b4,b2) *' (Red b5,b2)) )
theorem Th54: :: GROEB_3:54
theorem Th55: :: GROEB_3:55
for b
1 being
Ordinalfor b
2 being
connected admissible TermOrder of b
1for b
3 being
Abelian add-associative right_zeroed right_complementable associative commutative right_unital distributive left_unital Field-like non
trivial doubleLoopStr for b
4, b
5 being
non-zero Polynomial of b
1,b
3 holds
(
HT b
4,b
2,
HT b
5,b
2 are_disjoint &
Red b
4,b
2 is
non-zero &
Red b
5,b
2 is
non-zero implies
PolyRedRel {b4},b
2 reduces ((HM b5,b2) *' (Red b4,b2)) - ((HM b4,b2) *' (Red b5,b2)),b
5 *' (Red b4,b2) )
theorem Th56: :: GROEB_3:56
for b
1 being
Ordinalfor b
2 being
connected admissible TermOrder of b
1for b
3 being
Abelian add-associative right_zeroed right_complementable associative commutative right_unital distributive left_unital Field-like non
trivial doubleLoopStr for b
4, b
5 being
Polynomial of b
1,b
3 holds
(
HT b
4,b
2,
HT b
5,b
2 are_disjoint implies
PolyRedRel {b4,b5},b
2 reduces S-Poly b
4,b
5,b
2,
0_ b
1,b
3 )
theorem Th57: :: GROEB_3:57
for b
1 being
Natfor b
2 being
connected admissible TermOrder of b
1for b
3 being non
empty Abelian add-associative right_zeroed right_complementable associative commutative right_unital distributive left_unital Field-like non
degenerated doubleLoopStr for b
4 being
Subset of
(Polynom-Ring b1,b3) holds
( b
4 is_Groebner_basis_wrt b
2 implies for b
5, b
6 being
Polynomial of b
1,b
3 holds
( b
5 in b
4 & b
6 in b
4 & not
HT b
5,b
2,
HT b
6,b
2 are_disjoint implies
PolyRedRel b
4,b
2 reduces S-Poly b
5,b
6,b
2,
0_ b
1,b
3 ) )
theorem Th58: :: GROEB_3:58
for b
1 being
Natfor b
2 being
connected admissible TermOrder of b
1for b
3 being
Abelian add-associative right_zeroed right_complementable associative commutative right_unital distributive left_unital Field-like non
degenerated non
trivial doubleLoopStr for b
4 being
Subset of
(Polynom-Ring b1,b3) holds
( not
0_ b
1,b
3 in b
4 & ( for b
5, b
6 being
Polynomial of b
1,b
3 holds
( b
5 in b
4 & b
6 in b
4 & not
HT b
5,b
2,
HT b
6,b
2 are_disjoint implies
PolyRedRel b
4,b
2 reduces S-Poly b
5,b
6,b
2,
0_ b
1,b
3 ) ) implies for b
5, b
6, b
7 being
Polynomial of b
1,b
3 holds
( b
5 in b
4 & b
6 in b
4 & not
HT b
5,b
2,
HT b
6,b
2 are_disjoint & b
7 is_a_normal_form_of S-Poly b
5,b
6,b
2,
PolyRedRel b
4,b
2 implies b
7 = 0_ b
1,b
3 ) )
theorem Th59: :: GROEB_3:59
for b
1 being
Natfor b
2 being
connected admissible TermOrder of b
1for b
3 being non
empty Abelian add-associative right_zeroed right_complementable associative commutative right_unital distributive left_unital Field-like non
degenerated doubleLoopStr for b
4 being
Subset of
(Polynom-Ring b1,b3) holds
( not
0_ b
1,b
3 in b
4 & ( for b
5, b
6, b
7 being
Polynomial of b
1,b
3 holds
( b
5 in b
4 & b
6 in b
4 & not
HT b
5,b
2,
HT b
6,b
2 are_disjoint & b
7 is_a_normal_form_of S-Poly b
5,b
6,b
2,
PolyRedRel b
4,b
2 implies b
7 = 0_ b
1,b
3 ) ) implies b
4 is_Groebner_basis_wrt b
2 )