Article GROEB_2, MML version 4.99.1005
:: GROEB_2:th 2
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT
st b4 <= b3
holds (b2 | b3) | b4 = b2 | b4;
:: GROEB_2:th 3
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being FinSequence of the carrier of b1
for b3 being Element of NAT
st for b4 being Element of NAT
st b4 in dom b2 & b3 < b4
holds b2 . b4 = 0. b1
holds Sum b2 = Sum (b2 | b3);
:: GROEB_2:th 4
theorem
for b1 being non empty Abelian add-associative right_zeroed addLoopStr
for b2 being FinSequence of the carrier of b1
for b3, b4 being Element of NAT holds
Sum Swap(b2,b3,b4) = Sum b2;
:: GROEB_2:funcnot 1 => GROEB_2:func 1
definition
let a1 be set;
let a2, a3 be natural-valued finite-support ManySortedSet of a1;
assume a3 divides a2;
func A2 / A3 -> natural-valued finite-support ManySortedSet of a1 means
a3 + it = a2;
end;
:: GROEB_2:def 1
theorem
for b1 being set
for b2, b3 being natural-valued finite-support ManySortedSet of b1
st b3 divides b2
for b4 being natural-valued finite-support ManySortedSet of b1 holds
b4 = b2 / b3
iff
b3 + b4 = b2;
:: GROEB_2:funcnot 2 => GROEB_2:func 2
definition
let a1 be set;
let a2, a3 be natural-valued finite-support ManySortedSet of a1;
func lcm(A2,A3) -> natural-valued finite-support ManySortedSet of a1 means
for b1 being set holds
it . b1 = max(a2 . b1,a3 . b1);
commutativity;
:: for a1 being set
:: for a2, a3 being natural-valued finite-support ManySortedSet of a1 holds
:: lcm(a2,a3) = lcm(a3,a2);
idempotence;
:: for a1 being set
:: for a2 being natural-valued finite-support ManySortedSet of a1 holds
:: lcm(a2,a2) = a2;
end;
:: GROEB_2:def 2
theorem
for b1 being set
for b2, b3, b4 being natural-valued finite-support ManySortedSet of b1 holds
b4 = lcm(b2,b3)
iff
for b5 being set holds
b4 . b5 = max(b2 . b5,b3 . b5);
:: GROEB_2:funcnot 3 => GROEB_2:func 2
notation
let a1 be set;
let a2, a3 be natural-valued finite-support ManySortedSet of a1;
synonym a2 lcm a3 for lcm(a2,a3);
end;
:: GROEB_2:prednot 1 => GROEB_2:pred 1
definition
let a1 be set;
let a2, a3 be natural-valued finite-support ManySortedSet of a1;
pred A2,A3 are_disjoint means
for b1 being set
st a2 . b1 <> 0
holds a3 . b1 = 0;
end;
:: GROEB_2:dfs 3
definiens
let a1 be set;
let a2, a3 be natural-valued finite-support ManySortedSet of a1;
To prove
a2,a3 are_disjoint
it is sufficient to prove
thus for b1 being set
st a2 . b1 <> 0
holds a3 . b1 = 0;
:: GROEB_2:def 3
theorem
for b1 being set
for b2, b3 being natural-valued finite-support ManySortedSet of b1 holds
b2,b3 are_disjoint
iff
for b4 being set
st b2 . b4 <> 0
holds b3 . b4 = 0;
:: GROEB_2:prednot 2 => not GROEB_2:pred 1
notation
let a1 be set;
let a2, a3 be natural-valued finite-support ManySortedSet of a1;
antonym a2,a3 are_non_disjoint for a2,a3 are_disjoint;
end;
:: GROEB_2:th 7
theorem
for b1 being set
for b2, b3 being natural-valued finite-support ManySortedSet of b1 holds
b2 divides lcm(b2,b3) & b3 divides lcm(b2,b3);
:: GROEB_2:th 8
theorem
for b1 being set
for b2, b3, b4 being natural-valued finite-support ManySortedSet of b1
st b2 divides b4 & b3 divides b4
holds lcm(b2,b3) divides b4;
:: GROEB_2:th 9
theorem
for b1 being set
for b2, b3 being natural-valued finite-support ManySortedSet of b1 holds
b2,b3 are_disjoint
iff
lcm(b2,b3) = b2 + b3;
:: GROEB_2:th 10
theorem
for b1 being set
for b2 being natural-valued finite-support ManySortedSet of b1 holds
b2 / b2 = EmptyBag b1;
:: GROEB_2:th 11
theorem
for b1 being set
for b2, b3 being natural-valued finite-support ManySortedSet of b1 holds
b3 divides b2
iff
lcm(b2,b3) = b2;
:: GROEB_2:th 12
theorem
for b1 being set
for b2, b3, b4 being natural-valued finite-support ManySortedSet of b1
st b2 divides lcm(b3,b4)
holds lcm(b3,b2) divides lcm(b3,b4);
:: GROEB_2:th 13
theorem
for b1 being set
for b2, b3, b4 being natural-valued finite-support ManySortedSet of b1
st lcm(b3,b2) divides lcm(b3,b4)
holds lcm(b2,b4) divides lcm(b3,b4);
:: GROEB_2:th 14
theorem
for b1 being set
for b2, b3, b4 being natural-valued finite-support ManySortedSet of b1
st lcm(b2,b4) divides lcm(b3,b4)
holds b2 divides lcm(b3,b4);
:: GROEB_2:th 15
theorem
for b1 being Element of NAT
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non empty Element of bool Bags b1 holds
ex b4 being natural-valued finite-support ManySortedSet of b1 st
b4 in b3 &
(for b5 being natural-valued finite-support ManySortedSet of b1
st b5 in b3
holds b4 <= b5,b2);
:: GROEB_2:funcreg 1
registration
let a1 be non trivial right_complementable add-associative right_zeroed addLoopStr;
let a2 be non-zero Element of the carrier of a1;
cluster - a2 -> non-zero;
end;
:: GROEB_2:funcreg 2
registration
let a1 be set;
let a2 be non empty add-cancelable distributive right_zeroed left_zeroed doubleLoopStr;
let a3 be Function-like quasi_total monomial-like Relation of Bags a1,the carrier of a2;
let a4 be Element of the carrier of a2;
cluster a4 * a3 -> Function-like quasi_total monomial-like;
end;
:: GROEB_2:funcreg 3
registration
let a1 be ordinal set;
let a2 be non trivial add-cancelable distributive right_zeroed domRing-like left_zeroed doubleLoopStr;
let a3 be Function-like quasi_total finite-Support non-zero Relation of Bags a1,the carrier of a2;
let a4 be non-zero Element of the carrier of a2;
cluster a4 * a3 -> Function-like quasi_total non-zero;
end;
:: GROEB_2:th 16
theorem
for b1 being ordinal set
for b2 being non empty right-distributive right_zeroed doubleLoopStr
for b3, b4 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b5 being natural-valued finite-support ManySortedSet of b1 holds
b5 *' (b3 + b4) = (b5 *' b3) + (b5 *' b4);
:: GROEB_2:th 17
theorem
for b1 being ordinal set
for b2 being non empty right_complementable add-associative right_zeroed addLoopStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b4 being natural-valued finite-support ManySortedSet of b1 holds
b4 *' - b3 = - (b4 *' b3);
:: GROEB_2:th 18
theorem
for b1 being ordinal set
for b2 being non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b4 being natural-valued finite-support ManySortedSet of b1
for b5 being Element of the carrier of b2 holds
b4 *' (b5 * b3) = b5 * (b4 *' b3);
:: GROEB_2:th 19
theorem
for b1 being ordinal set
for b2 being non empty right-distributive doubleLoopStr
for b3, b4 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b5 being Element of the carrier of b2 holds
b5 * (b3 + b4) = (b5 * b3) + (b5 * b4);
:: GROEB_2:th 20
theorem
for b1 being set
for b2 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b3 being Element of the carrier of b2 holds
- (b3 |(b1,b2)) = (- b3) |(b1,b2);
:: GROEB_2:th 21
theorem
for b1 being Element of NAT
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being Element of bool the carrier of Polynom-Ring(b1,b3)
st not 0_(b1,b3) in b4 &
(for b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b5 <> b6 & b5 in b4 & b6 in b4
for b7, b8 being Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b3
st HM(b7 *' b5,b2) = HM(b8 *' b6,b2)
holds PolyRedRel(b4,b2) reduces (b7 *' b5) - (b8 *' b6),0_(b1,b3))
holds b4 is_Groebner_basis_wrt b2;
:: GROEB_2:funcnot 4 => GROEB_2:func 3
definition
let a1 be ordinal set;
let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
let a3 be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr;
let a4, a5 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
func S-Poly(A4,A5,A2) -> Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 equals
((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));
end;
:: GROEB_2:def 4
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
S-Poly(b4,b5,b2) = ((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));
:: GROEB_2:th 22
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being Element of bool the carrier of Polynom-Ring(b1,b3)
for b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b5 in b4 & b6 in b4
holds S-Poly(b5,b6,b2) in b4 -Ideal;
:: GROEB_2:th 24
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b4, b5 being Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b3 holds
S-Poly(b4,b5,b2) = 0_(b1,b3);
:: GROEB_2:th 25
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
S-Poly(b4,0_(b1,b3),b2) = 0_(b1,b3) & S-Poly(0_(b1,b3),b4,b2) = 0_(b1,b3);
:: GROEB_2:th 26
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st S-Poly(b4,b5,b2) <> 0_(b1,b3)
holds HT(S-Poly(b4,b5,b2),b2) < lcm(HT(b4,b2),HT(b5,b2)),b2;
:: GROEB_2:th 27
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b4, b5 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
st HT(b5,b2) divides HT(b4,b2)
holds (HC(b5,b2)) * b4 top_reduces_to S-Poly(b4,b5,b2),b5,b2;
:: GROEB_2:th 28
theorem
for b1 being Element of NAT
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being Element of bool the carrier of Polynom-Ring(b1,b3)
st b4 is_Groebner_basis_wrt b2
for b5, b6, b7 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b5 in b4 & b6 in b4 & b7 is_a_normal_form_of S-Poly(b5,b6,b2),PolyRedRel(b4,b2)
holds b7 = 0_(b1,b3);
:: GROEB_2:th 29
theorem
for b1 being Element of NAT
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being Element of bool the carrier of Polynom-Ring(b1,b3)
st for b5, b6, b7 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b5 in b4 & b6 in b4 & b7 is_a_normal_form_of S-Poly(b5,b6,b2),PolyRedRel(b4,b2)
holds b7 = 0_(b1,b3)
for b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b5 in b4 & b6 in b4
holds PolyRedRel(b4,b2) reduces S-Poly(b5,b6,b2),0_(b1,b3);
:: GROEB_2:th 30
theorem
for b1 being Element of NAT
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being Element of bool the carrier of Polynom-Ring(b1,b3)
st not 0_(b1,b3) in b4 &
(for b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b5 in b4 & b6 in b4
holds PolyRedRel(b4,b2) reduces S-Poly(b5,b6,b2),0_(b1,b3))
holds b4 is_Groebner_basis_wrt b2;
:: GROEB_2:funcnot 5 => GROEB_2:func 4
definition
let a1 be ordinal set;
let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
let a3 be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr;
let a4 be Element of bool the carrier of Polynom-Ring(a1,a3);
func S-Poly(A4,A2) -> Element of bool the carrier of Polynom-Ring(a1,a3) equals
{S-Poly(b1,b2,a2) where b1 is Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3, b2 is Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3: b1 in a4 & b2 in a4};
end;
:: GROEB_2:def 5
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Element of bool the carrier of Polynom-Ring(b1,b3) holds
S-Poly(b4,b2) = {S-Poly(b5,b6,b2) where b5 is Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3, b6 is Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3: b5 in b4 & b6 in b4};
:: GROEB_2:funcreg 4
registration
let a1 be ordinal set;
let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
let a3 be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr;
let a4 be finite Element of bool the carrier of Polynom-Ring(a1,a3);
cluster S-Poly(a4,a2) -> finite;
end;
:: GROEB_2:th 31
theorem
for b1 being Element of NAT
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being Element of bool the carrier of Polynom-Ring(b1,b3)
st not 0_(b1,b3) in b4 &
(for b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b5 in b4
holds b5 is Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b3)
holds b4 is_Groebner_basis_wrt b2;
:: GROEB_2:th 32
theorem
for b1 being non empty multLoopStr
for b2 being non empty Element of bool the carrier of b1
for b3 being LeftLinearCombination of b2
for b4 being Element of NAT holds
b3 | b4 is LeftLinearCombination of b2;
:: GROEB_2:th 33
theorem
for b1 being non empty multLoopStr
for b2 being non empty Element of bool the carrier of b1
for b3 being LeftLinearCombination of b2
for b4 being Element of NAT holds
b3 /^ b4 is LeftLinearCombination of b2;
:: GROEB_2:th 34
theorem
for b1 being non empty multLoopStr
for b2, b3 being non empty Element of bool the carrier of b1
for b4 being LeftLinearCombination of b2
st b2 c= b3
holds b4 is LeftLinearCombination of b3;
:: GROEB_2:funcnot 6 => GROEB_2:func 5
definition
let a1 be ordinal set;
let a2 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
let a3 be non empty Element of bool the carrier of Polynom-Ring(a1,a2);
let a4, a5 be LeftLinearCombination of a3;
redefine func a4 ^ a5 -> LeftLinearCombination of a3;
end;
:: GROEB_2:prednot 3 => GROEB_2:pred 2
definition
let a1 be ordinal set;
let a2 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
let a3 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a2;
let a4 be non empty Element of bool the carrier of Polynom-Ring(a1,a2);
let a5 be LeftLinearCombination of a4;
pred A5 is_MonomialRepresentation_of A3 means
Sum a5 = a3 &
(for b1 being Element of NAT
st b1 in dom a5
holds ex b2 being Function-like quasi_total monomial-like Relation of Bags a1,the carrier of a2 st
ex b3 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a2 st
b3 in a4 & a5 /. b1 = b2 *' b3);
end;
:: GROEB_2:dfs 6
definiens
let a1 be ordinal set;
let a2 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
let a3 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a2;
let a4 be non empty Element of bool the carrier of Polynom-Ring(a1,a2);
let a5 be LeftLinearCombination of a4;
To prove
a5 is_MonomialRepresentation_of a3
it is sufficient to prove
thus Sum a5 = a3 &
(for b1 being Element of NAT
st b1 in dom a5
holds ex b2 being Function-like quasi_total monomial-like Relation of Bags a1,the carrier of a2 st
ex b3 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a2 st
b3 in a4 & a5 /. b1 = b2 *' b3);
:: GROEB_2:def 6
theorem
for b1 being ordinal set
for b2 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2
for b4 being non empty Element of bool the carrier of Polynom-Ring(b1,b2)
for b5 being LeftLinearCombination of b4 holds
b5 is_MonomialRepresentation_of b3
iff
Sum b5 = b3 &
(for b6 being Element of NAT
st b6 in dom b5
holds ex b7 being Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b2 st
ex b8 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2 st
b8 in b4 & b5 /. b6 = b7 *' b8);
:: GROEB_2:th 35
theorem
for b1 being ordinal set
for b2 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2
for b4 being non empty Element of bool the carrier of Polynom-Ring(b1,b2)
for b5 being LeftLinearCombination of b4
st b5 is_MonomialRepresentation_of b3
holds Support b3 c= union {Support (b6 *' b7) where b6 is Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b2, b7 is Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2: ex b8 being Element of NAT st
b8 in dom b5 & b5 /. b8 = b6 *' b7};
:: GROEB_2:th 36
theorem
for b1 being ordinal set
for b2 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b3, b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b2)
for b6, b7 being LeftLinearCombination of b5
st b6 is_MonomialRepresentation_of b3 & b7 is_MonomialRepresentation_of b4
holds b6 ^ b7 is_MonomialRepresentation_of b3 + b4;
:: GROEB_2:prednot 4 => GROEB_2:pred 3
definition
let a1 be ordinal set;
let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
let a3 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be non empty Element of bool the carrier of Polynom-Ring(a1,a3);
let a6 be LeftLinearCombination of a5;
let a7 be natural-valued finite-support ManySortedSet of a1;
pred A6 is_Standard_Representation_of A4,A5,A7,A2 means
Sum a6 = a4 &
(for b1 being Element of NAT
st b1 in dom a6
holds ex b2 being Function-like quasi_total non-zero monomial-like Relation of Bags a1,the carrier of a3 st
ex b3 being Function-like quasi_total finite-Support non-zero Relation of Bags a1,the carrier of a3 st
b3 in a5 & a6 /. b1 = b2 *' b3 & HT(b2 *' b3,a2) <= a7,a2);
end;
:: GROEB_2:dfs 7
definiens
let a1 be ordinal set;
let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
let a3 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be non empty Element of bool the carrier of Polynom-Ring(a1,a3);
let a6 be LeftLinearCombination of a5;
let a7 be natural-valued finite-support ManySortedSet of a1;
To prove
a6 is_Standard_Representation_of a4,a5,a7,a2
it is sufficient to prove
thus Sum a6 = a4 &
(for b1 being Element of NAT
st b1 in dom a6
holds ex b2 being Function-like quasi_total non-zero monomial-like Relation of Bags a1,the carrier of a3 st
ex b3 being Function-like quasi_total finite-Support non-zero Relation of Bags a1,the carrier of a3 st
b3 in a5 & a6 /. b1 = b2 *' b3 & HT(b2 *' b3,a2) <= a7,a2);
:: GROEB_2:def 7
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b6 being LeftLinearCombination of b5
for b7 being natural-valued finite-support ManySortedSet of b1 holds
b6 is_Standard_Representation_of b4,b5,b7,b2
iff
Sum b6 = b4 &
(for b8 being Element of NAT
st b8 in dom b6
holds ex b9 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b3 st
ex b10 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 st
b10 in b5 & b6 /. b8 = b9 *' b10 & HT(b9 *' b10,b2) <= b7,b2);
:: GROEB_2:prednot 5 => GROEB_2:pred 4
definition
let a1 be ordinal set;
let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
let a3 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be non empty Element of bool the carrier of Polynom-Ring(a1,a3);
let a6 be LeftLinearCombination of a5;
pred A6 is_Standard_Representation_of A4,A5,A2 means
a6 is_Standard_Representation_of a4,a5,HT(a4,a2),a2;
end;
:: GROEB_2:dfs 8
definiens
let a1 be ordinal set;
let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
let a3 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be non empty Element of bool the carrier of Polynom-Ring(a1,a3);
let a6 be LeftLinearCombination of a5;
To prove
a6 is_Standard_Representation_of a4,a5,a2
it is sufficient to prove
thus a6 is_Standard_Representation_of a4,a5,HT(a4,a2),a2;
:: GROEB_2:def 8
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b6 being LeftLinearCombination of b5 holds
b6 is_Standard_Representation_of b4,b5,b2
iff
b6 is_Standard_Representation_of b4,b5,HT(b4,b2),b2;
:: GROEB_2:prednot 6 => GROEB_2:pred 5
definition
let a1 be ordinal set;
let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
let a3 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be non empty Element of bool the carrier of Polynom-Ring(a1,a3);
let a6 be natural-valued finite-support ManySortedSet of a1;
pred A4 has_a_Standard_Representation_of A5,A6,A2 means
ex b1 being LeftLinearCombination of a5 st
b1 is_Standard_Representation_of a4,a5,a6,a2;
end;
:: GROEB_2:dfs 9
definiens
let a1 be ordinal set;
let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
let a3 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be non empty Element of bool the carrier of Polynom-Ring(a1,a3);
let a6 be natural-valued finite-support ManySortedSet of a1;
To prove
a4 has_a_Standard_Representation_of a5,a6,a2
it is sufficient to prove
thus ex b1 being LeftLinearCombination of a5 st
b1 is_Standard_Representation_of a4,a5,a6,a2;
:: GROEB_2:def 9
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b6 being natural-valued finite-support ManySortedSet of b1 holds
b4 has_a_Standard_Representation_of b5,b6,b2
iff
ex b7 being LeftLinearCombination of b5 st
b7 is_Standard_Representation_of b4,b5,b6,b2;
:: GROEB_2:prednot 7 => GROEB_2:pred 6
definition
let a1 be ordinal set;
let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
let a3 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be non empty Element of bool the carrier of Polynom-Ring(a1,a3);
pred A4 has_a_Standard_Representation_of A5,A2 means
ex b1 being LeftLinearCombination of a5 st
b1 is_Standard_Representation_of a4,a5,a2;
end;
:: GROEB_2:dfs 10
definiens
let a1 be ordinal set;
let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
let a3 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be non empty Element of bool the carrier of Polynom-Ring(a1,a3);
To prove
a4 has_a_Standard_Representation_of a5,a2
it is sufficient to prove
thus ex b1 being LeftLinearCombination of a5 st
b1 is_Standard_Representation_of a4,a5,a2;
:: GROEB_2:def 10
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3) holds
b4 has_a_Standard_Representation_of b5,b2
iff
ex b6 being LeftLinearCombination of b5 st
b6 is_Standard_Representation_of b4,b5,b2;
:: GROEB_2:th 37
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b6 being LeftLinearCombination of b5
for b7 being natural-valued finite-support ManySortedSet of b1
st b6 is_Standard_Representation_of b4,b5,b7,b2
holds b6 is_MonomialRepresentation_of b4;
:: GROEB_2:th 38
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b6 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b7, b8 being LeftLinearCombination of b6
for b9 being natural-valued finite-support ManySortedSet of b1
st b7 is_Standard_Representation_of b4,b6,b9,b2 & b8 is_Standard_Representation_of b5,b6,b9,b2
holds b7 ^ b8 is_Standard_Representation_of b4 + b5,b6,b9,b2;
:: GROEB_2:th 39
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b6 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b7, b8 being LeftLinearCombination of b6
for b9 being natural-valued finite-support ManySortedSet of b1
for b10 being Element of NAT
st b7 is_Standard_Representation_of b4,b6,b9,b2 & b8 = b7 | b10 & b5 = Sum (b7 /^ b10)
holds b8 is_Standard_Representation_of b4 - b5,b6,b9,b2;
:: GROEB_2:th 40
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b6 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b7, b8 being LeftLinearCombination of b6
for b9 being natural-valued finite-support ManySortedSet of b1
for b10 being Element of NAT
st b7 is_Standard_Representation_of b4,b6,b9,b2 & b8 = b7 /^ b10 & b5 = Sum (b7 | b10) & b10 <= len b7
holds b8 is_Standard_Representation_of b4 - b5,b6,b9,b2;
:: GROEB_2:th 41
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b6 being LeftLinearCombination of b5
st b6 is_MonomialRepresentation_of b4
holds ex b7 being Element of NAT st
ex b8 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b3 st
ex b9 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 st
b7 in dom b6 & b9 in b5 & b6 . b7 = b8 *' b9 & HT(b4,b2) <= HT(b8 *' b9,b2),b2;
:: GROEB_2:th 42
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b6 being LeftLinearCombination of b5
st b6 is_Standard_Representation_of b4,b5,b2
holds ex b7 being Element of NAT st
ex b8 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b3 st
ex b9 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 st
b9 in b5 & b7 in dom b6 & b6 /. b7 = b8 *' b9 & HT(b4,b2) = HT(b8 *' b9,b2);
:: GROEB_2:th 43
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
st PolyRedRel(b5,b2) reduces b4,0_(b1,b3)
holds b4 has_a_Standard_Representation_of b5,b2;
:: GROEB_2:th 44
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
st b4 has_a_Standard_Representation_of b5,b2
holds b4 is_top_reducible_wrt b5,b2;
:: GROEB_2:th 45
theorem
for b1 being Element of NAT
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being non empty Element of bool the carrier of Polynom-Ring(b1,b3) holds
b4 is_Groebner_basis_wrt b2
iff
for b5 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
st b5 in b4 -Ideal
holds b5 has_a_Standard_Representation_of b4,b2;