:: GROEB_2 semantic presentation
theorem Th1: :: GROEB_2:1
canceled;
theorem Th2: :: GROEB_2:2
theorem Th3: :: GROEB_2:3
theorem Th4: :: GROEB_2:4
:: deftheorem Def1 defines / GROEB_2:def 1 :
for b
1 being
set for b
2, b
3 being
bag of b
1 holds
( b
3 divides b
2 implies for b
4 being
bag of b
1 holds
( b
4 = b
2 / b
3 iff b
3 + b
4 = b
2 ) );
definition
let c
1 be
set ;
let c
2, c
3 be
bag of c
1;
func lcm c
2,c
3 -> bag of a
1 means :
Def2:
:: GROEB_2:def 2
for b
1 being
set holds a
4 . b
1 = max (a2 . b1),
(a3 . b1);
existence
ex b1 being bag of c1 st
for b2 being set holds b1 . b2 = max (c2 . b2),(c3 . b2)
uniqueness
for b1, b2 being bag of c1 holds
( ( for b3 being set holds b1 . b3 = max (c2 . b3),(c3 . b3) ) & ( for b3 being set holds b2 . b3 = max (c2 . b3),(c3 . b3) ) implies b1 = b2 )
commutativity
for b1, b2, b3 being bag of c1 holds
( ( for b4 being set holds b1 . b4 = max (b2 . b4),(b3 . b4) ) implies for b4 being set holds b1 . b4 = max (b3 . b4),(b2 . b4) )
;
idempotence
for b1 being bag of c1
for b2 being set holds b1 . b2 = max (b1 . b2),(b1 . b2)
;
end;
:: deftheorem Def2 defines lcm GROEB_2:def 2 :
for b
1 being
set for b
2, b
3, b
4 being
bag of b
1 holds
( b
4 = lcm b
2,b
3 iff for b
5 being
set holds b
4 . b
5 = max (b2 . b5),
(b3 . b5) );
:: deftheorem Def3 defines are_disjoint GROEB_2:def 3 :
for b
1 being
set for b
2, b
3 being
bag of b
1 holds
( b
2,b
3 are_disjoint iff for b
4 being
set holds
( b
2 . b
4 = 0 or b
3 . b
4 = 0 ) );
theorem Th5: :: GROEB_2:5
canceled;
theorem Th6: :: GROEB_2:6
canceled;
theorem Th7: :: GROEB_2:7
theorem Th8: :: GROEB_2:8
theorem Th9: :: GROEB_2:9
theorem Th10: :: GROEB_2:10
theorem Th11: :: GROEB_2:11
for b
1 being
set for b
2, b
3 being
bag of b
1 holds
( b
3 divides b
2 iff
lcm b
2,b
3 = b
2 )
theorem Th12: :: GROEB_2:12
theorem Th13: :: GROEB_2:13
theorem Th14: :: GROEB_2:14
theorem Th15: :: GROEB_2:15
theorem Th16: :: GROEB_2:16
theorem Th17: :: GROEB_2:17
theorem Th18: :: GROEB_2:18
theorem Th19: :: GROEB_2:19
theorem Th20: :: GROEB_2:20
Lemma14:
for b1 being Ordinal
for b2 being connected TermOrder of b1
for b3 being add-associative right_zeroed right_complementable unital associative commutative distributive 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 Th21: :: GROEB_2:21
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 unital associative commutative distributive 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 being
Polynomial of b
1,b
3 holds
( b
5 <> b
6 & b
5 in b
4 & b
6 in b
4 implies for b
7, b
8 being
Monomial of b
1,b
3 holds
(
HM (b7 *' b5),b
2 = HM (b8 *' b6),b
2 implies
PolyRedRel b
4,b
2 reduces (b7 *' b5) - (b8 *' b6),
0_ b
1,b
3 ) ) ) implies b
4 is_Groebner_basis_wrt b
2 )
definition
let c
1 be
Ordinal;
let c
2 be
connected TermOrder of c
1;
let c
3 be
add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
trivial doubleLoopStr ;
let c
4, c
5 be
Polynomial of c
1,c
3;
func S-Poly c
4,c
5,c
2 -> Polynomial of a
1,a
3 equals :: GROEB_2:def 4
((HC a5,a2) * (((lcm (HT a4,a2),(HT a5,a2)) / (HT a4,a2)) *' a4)) - ((HC a4,a2) * (((lcm (HT a4,a2),(HT a5,a2)) / (HT a5,a2)) *' a5));
coherence
((HC c5,c2) * (((lcm (HT c4,c2),(HT c5,c2)) / (HT c4,c2)) *' c4)) - ((HC c4,c2) * (((lcm (HT c4,c2),(HT c5,c2)) / (HT c5,c2)) *' c5)) is Polynomial of c1,c3
;
end;
:: deftheorem Def4 defines S-Poly GROEB_2:def 4 :
for b
1 being
Ordinalfor b
2 being
connected TermOrder of b
1for b
3 being
add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
trivial doubleLoopStr for b
4, b
5 being
Polynomial of b
1,b
3 holds
S-Poly b
4,b
5,b
2 = ((HC b5,b2) * (((lcm (HT b4,b2),(HT b5,b2)) / (HT b4,b2)) *' b4)) - ((HC b4,b2) * (((lcm (HT b4,b2),(HT b5,b2)) / (HT b5,b2)) *' b5));
Lemma16:
for b1 being non empty add-associative right_zeroed right_complementable unital associative distributive left_zeroed add-cancelable doubleLoopStr
for b2 being Subset of b1
for b3 being Element of b1 holds
( b3 in b2 implies b3 in b2 -Ideal )
Lemma17:
for b1 being Ordinal
for b2 being connected TermOrder of b1
for b3 being add-associative right_zeroed right_complementable unital distributive non trivial doubleLoopStr
for b4, b5 being Polynomial of b1,b3
for b6, b7 being Element of (Polynom-Ring b1,b3) holds
( b6 = b4 & b7 = b5 implies b6 - b7 = b4 - b5 )
theorem Th22: :: GROEB_2:22
theorem Th23: :: GROEB_2:23
theorem Th24: :: GROEB_2:24
theorem Th25: :: GROEB_2:25
theorem Th26: :: GROEB_2:26
for b
1 being
Ordinalfor b
2 being
connected admissible TermOrder of b
1for b
3 being
add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
trivial doubleLoopStr for b
4, b
5 being
Polynomial of b
1,b
3 holds
(
S-Poly b
4,b
5,b
2 = 0_ b
1,b
3 or
HT (S-Poly b4,b5,b2),b
2 < lcm (HT b4,b2),
(HT b5,b2),b
2 )
theorem Th27: :: GROEB_2:27
theorem Th28: :: GROEB_2:28
theorem Th29: :: GROEB_2:29
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 unital associative commutative distributive Field-like non
degenerated doubleLoopStr for b
4 being
Subset of
(Polynom-Ring b1,b3) holds
( ( 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 & 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 for b
5, b
6 being
Polynomial of b
1,b
3 holds
( b
5 in b
4 & b
6 in b
4 implies
PolyRedRel b
4,b
2 reduces S-Poly b
5,b
6,b
2,
0_ b
1,b
3 ) )
theorem Th30: :: GROEB_2:30
definition
let c
1 be
Ordinal;
let c
2 be
connected TermOrder of c
1;
let c
3 be
add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
trivial doubleLoopStr ;
let c
4 be
Subset of
(Polynom-Ring c1,c3);
func S-Poly c
4,c
2 -> Subset of
(Polynom-Ring a1,a3) equals :: GROEB_2:def 5
{ (S-Poly b1,b2,a2) where B is Polynomial of a1,a3, B is Polynomial of a1,a3 : ( b1 in a4 & b2 in a4 ) } ;
coherence
{ (S-Poly b1,b2,c2) where B is Polynomial of c1,c3, B is Polynomial of c1,c3 : ( b1 in c4 & b2 in c4 ) } is Subset of (Polynom-Ring c1,c3)
end;
:: deftheorem Def5 defines S-Poly GROEB_2:def 5 :
theorem Th31: :: GROEB_2:31
theorem Th32: :: GROEB_2:32
theorem Th33: :: GROEB_2:33
theorem Th34: :: GROEB_2:34
:: deftheorem Def6 defines is_MonomialRepresentation_of GROEB_2:def 6 :
theorem Th35: :: GROEB_2:35
theorem Th36: :: GROEB_2:36
Lemma23:
for b1 being Ordinal
for b2 being connected TermOrder of b1
for b3 being non empty add-associative right_zeroed right_complementable unital distributive non trivial doubleLoopStr
for b4 being Polynomial of b1,b3
for b5 being non empty Subset of (Polynom-Ring b1,b3)
for b6 being LeftLinearCombination of b5 holds
( b6 is_MonomialRepresentation_of b4 implies for b7 being bag of b1 holds
( ( for b8 being Nat holds
( b8 in dom b6 implies for b9 being Monomial of b1,b3
for b10 being Polynomial of b1,b3 holds
( b6 . b8 = b9 *' b10 & b10 in b5 implies (b9 *' b10) . b7 = 0. b3 ) ) ) implies b4 . b7 = 0. b3 ) )
:: deftheorem Def7 defines is_Standard_Representation_of GROEB_2:def 7 :
:: deftheorem Def8 defines is_Standard_Representation_of GROEB_2:def 8 :
:: deftheorem Def9 defines has_a_Standard_Representation_of GROEB_2:def 9 :
:: deftheorem Def10 defines has_a_Standard_Representation_of GROEB_2:def 10 :
theorem Th37: :: GROEB_2:37
Lemma28:
for b1 being Ordinal
for b2 being connected admissible TermOrder of b1
for b3 being non empty Abelian add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non degenerated doubleLoopStr
for b4, b5 being Polynomial of b1,b3
for b6, b7 being Element of (Polynom-Ring b1,b3) holds
( b4 = b6 & b5 = b7 implies for b8 being non empty Subset of (Polynom-Ring b1,b3) holds
not ( PolyRedRel b8,b2 reduces b4,b5 & ( for b9 being LeftLinearCombination of b8 holds
not ( b6 = b7 + (Sum b9) & ( for b10 being Nat holds
not ( b10 in dom b9 & ( for b11 being non-zero Monomial of b1,b3
for b12 being non-zero Polynomial of b1,b3 holds
not ( b12 in b8 & b9 . b10 = b11 *' b12 & HT (b11 *' b12),b2 <= HT b4,b2,b2 ) ) ) ) ) ) ) )
theorem Th38: :: GROEB_2:38
for b
1 being
Ordinalfor b
2 being
connected TermOrder of b
1for b
3 being non
empty add-associative right_zeroed right_complementable unital distributive non
trivial doubleLoopStr for b
4, b
5 being
Polynomial of b
1,b
3for b
6 being non
empty Subset of
(Polynom-Ring b1,b3)for b
7, b
8 being
LeftLinearCombination of b
6for b
9 being
bag of b
1 holds
( b
7 is_Standard_Representation_of b
4,b
6,b
9,b
2 & b
8 is_Standard_Representation_of b
5,b
6,b
9,b
2 implies b
7 ^ b
8 is_Standard_Representation_of b
4 + b
5,b
6,b
9,b
2 )
theorem Th39: :: GROEB_2:39
for b
1 being
Ordinalfor b
2 being
connected TermOrder of b
1for b
3 being non
empty add-associative right_zeroed right_complementable unital distributive non
trivial doubleLoopStr for b
4, b
5 being
Polynomial of b
1,b
3for b
6 being non
empty Subset of
(Polynom-Ring b1,b3)for b
7, b
8 being
LeftLinearCombination of b
6for b
9 being
bag of b
1for b
10 being
Nat holds
( b
7 is_Standard_Representation_of b
4,b
6,b
9,b
2 & b
8 = b
7 | b
10 & b
5 = Sum (b7 /^ b10) implies b
8 is_Standard_Representation_of b
4 - b
5,b
6,b
9,b
2 )
theorem Th40: :: GROEB_2:40
for b
1 being
Ordinalfor b
2 being
connected TermOrder of b
1for b
3 being non
empty Abelian add-associative right_zeroed right_complementable unital distributive non
trivial doubleLoopStr for b
4, b
5 being
Polynomial of b
1,b
3for b
6 being non
empty Subset of
(Polynom-Ring b1,b3)for b
7, b
8 being
LeftLinearCombination of b
6for b
9 being
bag of b
1for b
10 being
Nat holds
( b
7 is_Standard_Representation_of b
4,b
6,b
9,b
2 & b
8 = b
7 /^ b
10 & b
5 = Sum (b7 | b10) & b
10 <= len b
7 implies b
8 is_Standard_Representation_of b
4 - b
5,b
6,b
9,b
2 )
theorem Th41: :: GROEB_2:41
theorem Th42: :: GROEB_2:42
theorem Th43: :: GROEB_2:43
theorem Th44: :: GROEB_2:44
theorem Th45: :: GROEB_2:45