:: GROEB_1 semantic presentation
theorem Th1: :: GROEB_1:1
theorem Th2: :: GROEB_1:2
for b
1 being
Ordinalfor 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, b
5, b
6 being
Polynomial of b
1,b
3 holds
not ( b
4 reduces_to b
6,b
5,b
2 & ( for b
7 being
Monomial of b
1,b
3 holds
not ( b
6 = b
4 - (b7 *' b5) & not
HT (b7 *' b5),b
2 in Support b
6 &
HT (b7 *' b5),b
2 <= HT b
4,b
2,b
2 ) ) )
Lemma2:
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 )
Lemma3:
for b1 being Ordinal
for b2 being connected admissible TermOrder of b1
for b3 being non empty 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
( b6 = b4 & b7 = b5 implies b6 - b7 = b4 - b5 )
Lemma4:
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 holds
b4 is_irreducible_wrt 0_ b1,b3,b2
theorem Th3: :: GROEB_1:3
theorem Th4: :: GROEB_1:4
theorem Th5: :: GROEB_1:5
theorem Th6: :: GROEB_1:6
theorem Th7: :: GROEB_1:7
theorem Th8: :: GROEB_1:8
theorem Th9: :: GROEB_1:9
theorem Th10: :: GROEB_1:10
theorem Th11: :: GROEB_1:11
:: deftheorem Def1 defines HT GROEB_1:def 1 :
:: deftheorem Def2 defines multiples GROEB_1:def 2 :
theorem Th12: :: GROEB_1:12
theorem Th13: :: GROEB_1:13
theorem Th14: :: GROEB_1:14
Lemma16:
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 )
Lemma17:
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, b5 being Polynomial of b1,b3
for b6 being Subset of (Polynom-Ring b1,b3) holds
not ( PolyRedRel b6,b2 reduces b4,b5 & b5 <> b4 & ( for b7 being Polynomial of b1,b3 holds
not ( b4 reduces_to b7,b6,b2 & PolyRedRel b6,b2 reduces b7,b5 ) ) )
theorem Th15: :: GROEB_1:15
theorem Th16: :: GROEB_1:16
theorem Th17: :: GROEB_1:17
theorem Th18: :: GROEB_1:18
theorem Th19: :: GROEB_1:19
theorem Th20: :: GROEB_1:20
:: deftheorem Def3 defines is_Groebner_basis_wrt GROEB_1:def 3 :
:: deftheorem Def4 defines is_Groebner_basis_of GROEB_1:def 4 :
Lemma24:
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 Subset of (Polynom-Ring b1,b3)
for b5, b6 being set holds
( b5 <> b6 & PolyRedRel b4,b2 reduces b5,b6 implies ( b5 is Polynomial of b1,b3 & b6 is Polynomial of b1,b3 ) )
theorem Th21: :: GROEB_1:21
theorem Th22: :: GROEB_1:22
theorem Th23: :: GROEB_1:23
Lemma25:
for b1 being Ordinal
for b2 being connected TermOrder of b1
for b3 being Abelian add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non trivial doubleLoopStr
for b4 being LeftIdeal of (Polynom-Ring b1,b3)
for b5 being non empty Subset of (Polynom-Ring b1,b3) holds
( b5 c= b4 & ( for b6 being Polynomial of b1,b3 holds
( b6 in b4 implies PolyRedRel b5,b2 reduces b6, 0_ b1,b3 ) ) implies b5 -Ideal = b4 )
theorem Th24: :: GROEB_1:24
theorem Th25: :: GROEB_1:25
theorem Th26: :: GROEB_1:26
theorem Th27: :: GROEB_1:27
theorem Th28: :: GROEB_1:28
theorem Th29: :: GROEB_1:29
theorem Th30: :: GROEB_1:30
theorem Th31: :: GROEB_1:31
theorem Th32: :: GROEB_1:32
theorem Th33: :: GROEB_1:33
Lemma33:
for b1 being Ordinal
for b2, b3, b4 being bag of b1 holds
( b2 divides b3 & b3 divides b4 implies b2 divides b4 )
:: deftheorem Def5 defines DivOrder GROEB_1:def 5 :
theorem Th34: :: GROEB_1:34
theorem Th35: :: GROEB_1:35
Lemma37:
for b1 being non empty add-associative right_zeroed right_complementable unital associative distributive left_zeroed doubleLoopStr
for b2, b3 being non empty Subset of b1 holds
( 0. b1 in b2 & b3 = b2 \ {(0. b1)} implies for b4 being LinearCombination of b2
for b5 being set holds
not ( b5 = Sum b4 & ( for b6 being LinearCombination of b3 holds
not b5 = Sum b6 ) ) )
theorem Th36: :: GROEB_1:36
:: deftheorem Def6 defines is_monic_wrt GROEB_1:def 6 :
:: deftheorem Def7 defines is_reduced_wrt GROEB_1:def 7 :
theorem Th37: :: GROEB_1:37
for b
1 being
Ordinalfor 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
add-closed left-ideal Subset of
(Polynom-Ring b1,b3)for b
5 being
Monomial of b
1,b
3for b
6, b
7 being
Polynomial of b
1,b
3 holds
( b
6 in b
4 & b
7 in b
4 &
HM b
6,b
2 = b
5 &
HM b
7,b
2 = b
5 & ( for b
8 being
Polynomial of b
1,b
3 holds
not ( b
8 in b
4 & b
8 < b
6,b
2 &
HM b
8,b
2 = b
5 ) ) & ( for b
8 being
Polynomial of b
1,b
3 holds
not ( b
8 in b
4 & b
8 < b
7,b
2 &
HM b
8,b
2 = b
5 ) ) implies b
6 = b
7 )
Lemma41:
for b1 being Nat
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 being Polynomial of b1,b3
for b5 being non empty add-closed left-ideal Subset of (Polynom-Ring b1,b3) holds
not ( b4 in b5 & b4 <> 0_ b1,b3 & ( for b6 being Polynomial of b1,b3 holds
not ( b6 in b5 & HM b6,b2 = Monom (1. b3),(HT b4,b2) & b6 <> 0_ b1,b3 ) ) )
theorem Th38: :: GROEB_1:38
theorem Th39: :: GROEB_1:39
theorem Th40: :: GROEB_1:40