:: HILBASIS semantic presentation
theorem Th1: :: HILBASIS:1
theorem Th2: :: HILBASIS:2
theorem Th3: :: HILBASIS:3
theorem Th4: :: HILBASIS:4
for b
1, b
2 being
set for b
3, b
4 being
bag of b
2for b
5, b
6 being
bag of b
1 holds
( b
5 = b
3 | b
1 & b
6 = b
4 | b
1 & b
3 divides b
4 implies b
5 divides b
6 )
theorem Th5: :: HILBASIS:5
for b
1, b
2 being
set for b
3, b
4 being
bag of b
2for b
5, b
6 being
bag of b
1 holds
( b
5 = b
3 | b
1 & b
6 = b
4 | b
1 implies (
(b3 -' b4) | b
1 = b
5 -' b
6 &
(b3 + b4) | b
1 = b
5 + b
6 ) )
:: deftheorem Def1 defines bag_extend HILBASIS:def 1 :
theorem Th6: :: HILBASIS:6
theorem Th7: :: HILBASIS:7
:: deftheorem Def2 defines UnitBag HILBASIS:def 2 :
theorem Th8: :: HILBASIS:8
theorem Th9: :: HILBASIS:9
theorem Th10: :: HILBASIS:10
theorem Th11: :: HILBASIS:11
:: deftheorem Def3 defines 1_1 HILBASIS:def 3 :
theorem Th12: :: HILBASIS:12
theorem Th13: :: HILBASIS:13
theorem Th14: :: HILBASIS:14
theorem Th15: :: HILBASIS:15
theorem Th16: :: HILBASIS:16
:: deftheorem Def4 defines minlen HILBASIS:def 4 :
theorem Th17: :: HILBASIS:17
:: deftheorem Def5 defines monomial HILBASIS:def 5 :
theorem Th18: :: HILBASIS:18
theorem Th19: :: HILBASIS:19
theorem Th20: :: HILBASIS:20
theorem Th21: :: HILBASIS:21
theorem Th22: :: HILBASIS:22
theorem Th23: :: HILBASIS:23
theorem Th24: :: HILBASIS:24
theorem Th25: :: HILBASIS:25
theorem Th26: :: HILBASIS:26
theorem Th27: :: HILBASIS:27
theorem Th28: :: HILBASIS:28
theorem Th29: :: HILBASIS:29
definition
let c
1 be non
empty Abelian add-associative right_zeroed right_complementable unital associative commutative distributive non
trivial doubleLoopStr ;
let c
2 be
Nat;
func upm c
2,c
1 -> Function of
(Polynom-Ring (Polynom-Ring a2,a1)),
(Polynom-Ring (a2 + 1),a1) means :
Def6:
:: HILBASIS:def 6
for b
1 being
Polynomial of
(Polynom-Ring a2,a1)for b
2 being
Polynomial of a
2,a
1for b
3 being
Polynomial of
(a2 + 1),a
1for b
4 being
bag of a
2 + 1 holds
( b
3 = a
3 . b
1 & b
2 = b
1 . (b4 . a2) implies b
3 . b
4 = b
2 . (b4 | a2) );
existence
ex b1 being Function of (Polynom-Ring (Polynom-Ring c2,c1)),(Polynom-Ring (c2 + 1),c1) st
for b2 being Polynomial of (Polynom-Ring c2,c1)
for b3 being Polynomial of c2,c1
for b4 being Polynomial of (c2 + 1),c1
for b5 being bag of c2 + 1 holds
( b4 = b1 . b2 & b3 = b2 . (b5 . c2) implies b4 . b5 = b3 . (b5 | c2) )
uniqueness
for b1, b2 being Function of (Polynom-Ring (Polynom-Ring c2,c1)),(Polynom-Ring (c2 + 1),c1) holds
( ( for b3 being Polynomial of (Polynom-Ring c2,c1)
for b4 being Polynomial of c2,c1
for b5 being Polynomial of (c2 + 1),c1
for b6 being bag of c2 + 1 holds
( b5 = b1 . b3 & b4 = b3 . (b6 . c2) implies b5 . b6 = b4 . (b6 | c2) ) ) & ( for b3 being Polynomial of (Polynom-Ring c2,c1)
for b4 being Polynomial of c2,c1
for b5 being Polynomial of (c2 + 1),c1
for b6 being bag of c2 + 1 holds
( b5 = b2 . b3 & b4 = b3 . (b6 . c2) implies b5 . b6 = b4 . (b6 | c2) ) ) implies b1 = b2 )
end;
:: deftheorem Def6 defines upm HILBASIS:def 6 :
definition
let c
1 be non
empty Abelian add-associative right_zeroed right_complementable unital associative commutative distributive non
trivial doubleLoopStr ;
let c
2 be
Nat;
func mpu c
2,c
1 -> Function of
(Polynom-Ring (a2 + 1),a1),
(Polynom-Ring (Polynom-Ring a2,a1)) means :
Def7:
:: HILBASIS:def 7
for b
1 being
Polynomial of
(a2 + 1),a
1for b
2 being
Polynomial of a
2,a
1for b
3 being
Polynomial of
(Polynom-Ring a2,a1)for b
4 being
Natfor b
5 being
bag of a
2 holds
( b
3 = a
3 . b
1 & b
2 = b
3 . b
4 implies b
2 . b
5 = b
1 . (b5 bag_extend b4) );
existence
ex b1 being Function of (Polynom-Ring (c2 + 1),c1),(Polynom-Ring (Polynom-Ring c2,c1)) st
for b2 being Polynomial of (c2 + 1),c1
for b3 being Polynomial of c2,c1
for b4 being Polynomial of (Polynom-Ring c2,c1)
for b5 being Nat
for b6 being bag of c2 holds
( b4 = b1 . b2 & b3 = b4 . b5 implies b3 . b6 = b2 . (b6 bag_extend b5) )
uniqueness
for b1, b2 being Function of (Polynom-Ring (c2 + 1),c1),(Polynom-Ring (Polynom-Ring c2,c1)) holds
( ( for b3 being Polynomial of (c2 + 1),c1
for b4 being Polynomial of c2,c1
for b5 being Polynomial of (Polynom-Ring c2,c1)
for b6 being Nat
for b7 being bag of c2 holds
( b5 = b1 . b3 & b4 = b5 . b6 implies b4 . b7 = b3 . (b7 bag_extend b6) ) ) & ( for b3 being Polynomial of (c2 + 1),c1
for b4 being Polynomial of c2,c1
for b5 being Polynomial of (Polynom-Ring c2,c1)
for b6 being Nat
for b7 being bag of c2 holds
( b5 = b2 . b3 & b4 = b5 . b6 implies b4 . b7 = b3 . (b7 bag_extend b6) ) ) implies b1 = b2 )
end;
:: deftheorem Def7 defines mpu HILBASIS:def 7 :
theorem Th30: :: HILBASIS:30
theorem Th31: :: HILBASIS:31
Lemma36:
for b1 being non empty Abelian add-associative right_zeroed right_complementable unital associative commutative distributive Noetherian doubleLoopStr holds Polynom-Ring b1 is Noetherian
;
theorem Th32: :: HILBASIS:32
canceled;
theorem Th33: :: HILBASIS:33
theorem Th34: :: HILBASIS:34
theorem Th35: :: HILBASIS:35
theorem Th36: :: HILBASIS:36