Article GROEB_1, MML version 4.99.1005
:: GROEB_1:funcnot 1 => GROEB_1:func 1
definition
let a1 be ordinal set;
let a2 be 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;
redefine func {a3} -> Element of bool the carrier of Polynom-Ring(a1,a2);
end;
:: GROEB_1:th 1
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b4 reduces_to b6,b5,b2
holds ex b7 being Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b3 st
b6 = b4 - (b7 *' b5);
:: GROEB_1:th 2
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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, b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b4 reduces_to b6,b5,b2
holds ex b7 being Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b3 st
b6 = b4 - (b7 *' b5) & not HT(b7 *' b5,b2) in Support b6 & HT(b7 *' b5,b2) <= HT(b4,b2),b2;
:: GROEB_1:th 3
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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
for b6, b7 being Element of bool the carrier of Polynom-Ring(b1,b3)
st b6 c= b7 & b4 reduces_to b5,b6,b2
holds b4 reduces_to b5,b7,b2;
:: GROEB_1:th 4
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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 Element of bool the carrier of Polynom-Ring(b1,b3)
st b4 c= b5
holds PolyRedRel(b4,b2) c= PolyRedRel(b5,b2);
:: GROEB_1:th 5
theorem
for b1 being ordinal set
for b2 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2 holds
Support - b3 = Support b3;
:: GROEB_1:th 6
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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 holds
HT(- b4,b2) = HT(b4,b2);
:: GROEB_1:th 7
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive admissible 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 holds
HT(b4 - b5,b2) <= max(HT(b4,b2),HT(b5,b2),b2),b2;
:: GROEB_1:th 8
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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 Support b5 c= Support b4
holds b5 <= b4,b2;
:: GROEB_1:th 9
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive admissible Relation of Bags b1,Bags b1
for b3 being non empty non degenerated 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 b4 is_reducible_wrt b5,b2
holds HT(b5,b2) <= HT(b4,b2),b2;
:: GROEB_1:th 10
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive admissible 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 Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
PolyRedRel({b4},b2) is locally-confluent;
:: GROEB_1:th 11
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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 ex b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 st
b5 in b4 & b4 -Ideal = {b5} -Ideal
holds PolyRedRel(b4,b2) is locally-confluent;
:: GROEB_1:funcnot 2 => GROEB_1:func 2
definition
let a1 be ordinal set;
let a2 be total reflexive antisymmetric connected transitive 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 Element of bool the carrier of Polynom-Ring(a1,a3);
func HT(A4,A2) -> Element of bool Bags a1 equals
{HT(b1,a2) where b1 is Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3: b1 in a4 & b1 <> 0_(a1,a3)};
end;
:: GROEB_1:def 1
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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 Element of bool the carrier of Polynom-Ring(b1,b3) holds
HT(b4,b2) = {HT(b5,b2) where b5 is Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3: b5 in b4 & b5 <> 0_(b1,b3)};
:: GROEB_1:funcnot 3 => GROEB_1:func 3
definition
let a1 be ordinal set;
let a2 be Element of bool Bags a1;
func multiples A2 -> Element of bool Bags a1 equals
{b1 where b1 is Element of Bags a1: ex b2 being natural-valued finite-support ManySortedSet of a1 st
b2 in a2 & b2 divides b1};
end;
:: GROEB_1:def 2
theorem
for b1 being ordinal set
for b2 being Element of bool Bags b1 holds
multiples b2 = {b3 where b3 is Element of Bags b1: ex b4 being natural-valued finite-support ManySortedSet of b1 st
b4 in b2 & b4 divides b3};
:: GROEB_1:th 12
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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 PolyRedRel(b4,b2) is locally-confluent
holds PolyRedRel(b4,b2) is confluent;
:: GROEB_1:th 13
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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)
st PolyRedRel(b4,b2) is confluent
holds PolyRedRel(b4,b2) is with_UN_property;
:: GROEB_1:th 14
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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 PolyRedRel(b4,b2) is with_UN_property
holds PolyRedRel(b4,b2) is with_Church-Rosser_property;
:: GROEB_1:th 15
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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)
st PolyRedRel(b4,b2) is with_Church-Rosser_property
for b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b5 in b4 -Ideal
holds PolyRedRel(b4,b2) reduces b5,0_(b1,b3);
:: GROEB_1:th 16
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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)
st for b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b5 in b4 -Ideal
holds PolyRedRel(b4,b2) reduces b5,0_(b1,b3)
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 is_reducible_wrt b4,b2;
:: GROEB_1:th 17
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
st b5 in b4 -Ideal
holds b5 is_reducible_wrt b4,b2
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 is_top_reducible_wrt b4,b2;
:: GROEB_1:th 18
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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)
st 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 is_top_reducible_wrt b4,b2
for b5 being natural-valued finite-support ManySortedSet of b1
st b5 in HT(b4 -Ideal,b2)
holds ex b6 being natural-valued finite-support ManySortedSet of b1 st
b6 in HT(b4,b2) & b6 divides b5;
:: GROEB_1:th 19
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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)
st for b5 being natural-valued finite-support ManySortedSet of b1
st b5 in HT(b4 -Ideal,b2)
holds ex b6 being natural-valued finite-support ManySortedSet of b1 st
b6 in HT(b4,b2) & b6 divides b5
holds HT(b4 -Ideal,b2) c= multiples HT(b4,b2);
:: GROEB_1:th 20
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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 HT(b4 -Ideal,b2) c= multiples HT(b4,b2)
holds PolyRedRel(b4,b2) is locally-confluent;
:: GROEB_1:prednot 1 => GROEB_1:pred 1
definition
let a1 be ordinal set;
let a2 be total reflexive antisymmetric connected transitive 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);
pred A4 is_Groebner_basis_wrt A2 means
PolyRedRel(a4,a2) is locally-confluent;
end;
:: GROEB_1:dfs 3
definiens
let a1 be ordinal set;
let a2 be total reflexive antisymmetric connected transitive 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);
To prove
a4 is_Groebner_basis_wrt a2
it is sufficient to prove
thus PolyRedRel(a4,a2) is locally-confluent;
:: GROEB_1:def 3
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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
b4 is_Groebner_basis_wrt b2
iff
PolyRedRel(b4,b2) is locally-confluent;
:: GROEB_1:prednot 2 => GROEB_1:pred 2
definition
let a1 be ordinal set;
let a2 be total reflexive antisymmetric connected transitive 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 Element of bool the carrier of Polynom-Ring(a1,a3);
pred A4 is_Groebner_basis_of A5,A2 means
a4 -Ideal = a5 & PolyRedRel(a4,a2) is locally-confluent;
end;
:: GROEB_1:dfs 4
definiens
let a1 be ordinal set;
let a2 be total reflexive antisymmetric connected transitive 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 Element of bool the carrier of Polynom-Ring(a1,a3);
To prove
a4 is_Groebner_basis_of a5,a2
it is sufficient to prove
thus a4 -Ideal = a5 & PolyRedRel(a4,a2) is locally-confluent;
:: GROEB_1:def 4
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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 Element of bool the carrier of Polynom-Ring(b1,b3) holds
b4 is_Groebner_basis_of b5,b2
iff
b4 -Ideal = b5 & PolyRedRel(b4,b2) is locally-confluent;
:: GROEB_1:th 21
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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, b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
st b4 is_Groebner_basis_of b5,b2
holds PolyRedRel(b4,b2) is Completion of PolyRedRel(b5,b2);
:: GROEB_1:th 22
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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, b5 being Element of the carrier of Polynom-Ring(b1,b3)
for b6 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
st b6 is_Groebner_basis_wrt b2
holds b4,b5 are_congruent_mod b6 -Ideal
iff
nf(b4,PolyRedRel(b6,b2)) = nf(b5,PolyRedRel(b6,b2));
:: GROEB_1:th 23
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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 b5 is_Groebner_basis_wrt b2
holds b4 in b5 -Ideal
iff
PolyRedRel(b5,b2) reduces b4,0_(b1,b3);
:: GROEB_1:th 24
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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)
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
st b5 is_Groebner_basis_of b4,b2
for b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b6 in b4
holds PolyRedRel(b5,b2) reduces b6,0_(b1,b3);
:: GROEB_1:th 25
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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 Element of bool the carrier of Polynom-Ring(b1,b3)
st for b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b6 in b5
holds PolyRedRel(b4,b2) reduces b6,0_(b1,b3)
for b6 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
st b6 in b5
holds b6 is_reducible_wrt b4,b2;
:: GROEB_1:th 26
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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 add-closed left-ideal Element of bool the carrier of Polynom-Ring(b1,b3)
for b5 being Element of bool the carrier of Polynom-Ring(b1,b3)
st b5 c= b4 &
(for b6 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
st b6 in b4
holds b6 is_reducible_wrt b5,b2)
for b6 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
st b6 in b4
holds b6 is_top_reducible_wrt b5,b2;
:: GROEB_1:th 27
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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 Element of bool the carrier of Polynom-Ring(b1,b3)
st for b6 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
st b6 in b5
holds b6 is_top_reducible_wrt b4,b2
for b6 being natural-valued finite-support ManySortedSet of b1
st b6 in HT(b5,b2)
holds ex b7 being natural-valued finite-support ManySortedSet of b1 st
b7 in HT(b4,b2) & b7 divides b6;
:: GROEB_1:th 28
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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 Element of bool the carrier of Polynom-Ring(b1,b3)
st for b6 being natural-valued finite-support ManySortedSet of b1
st b6 in HT(b5,b2)
holds ex b7 being natural-valued finite-support ManySortedSet of b1 st
b7 in HT(b4,b2) & b7 divides b6
holds HT(b5,b2) c= multiples HT(b4,b2);
:: GROEB_1:th 29
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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 add-closed left-ideal Element of bool the carrier of Polynom-Ring(b1,b3)
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
st b5 c= b4 & HT(b4,b2) c= multiples HT(b5,b2)
holds b5 is_Groebner_basis_of b4,b2;
:: GROEB_1:th 30
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive admissible 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 holds
{0_(b1,b3)} is_Groebner_basis_of {0_(b1,b3)},b2;
:: GROEB_1:th 31
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive admissible 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 Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
{b4} is_Groebner_basis_of {b4} -Ideal,b2;
:: GROEB_1:th 32
theorem
for b1 being total reflexive antisymmetric connected transitive admissible Relation of Bags {},Bags {}
for b2 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3 being non empty add-closed left-ideal Element of bool the carrier of Polynom-Ring({},b2)
for b4 being non empty Element of bool the carrier of Polynom-Ring({},b2)
st b4 c= b3 & b4 <> {0_({},b2)}
holds b4 is_Groebner_basis_of b3,b1;
:: GROEB_1:th 33
theorem
for b1 being ordinal non empty set
for b2 being total reflexive antisymmetric connected transitive admissible Relation of Bags b1,Bags b1
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr holds
ex b4 being Element of bool the carrier of Polynom-Ring(b1,b3) st
not b4 is_Groebner_basis_wrt b2;
:: GROEB_1:funcnot 4 => GROEB_1:func 4
definition
let a1 be ordinal set;
func DivOrder A1 -> total reflexive antisymmetric transitive Relation of Bags a1,Bags a1 means
for b1, b2 being natural-valued finite-support ManySortedSet of a1 holds
[b1,b2] in it
iff
b1 divides b2;
end;
:: GROEB_1:def 5
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric transitive Relation of Bags b1,Bags b1 holds
b2 = DivOrder b1
iff
for b3, b4 being natural-valued finite-support ManySortedSet of b1 holds
[b3,b4] in b2
iff
b3 divides b4;
:: GROEB_1:funcreg 1
registration
let a1 be Element of NAT;
cluster RelStr(#Bags a1,DivOrder a1#) -> strict Dickson;
end;
:: GROEB_1:th 34
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of RelStr(#Bags b1,DivOrder b1#) holds
ex b3 being finite Element of bool Bags b1 st
b3 is_Dickson-basis_of b2,RelStr(#Bags b1,DivOrder b1#);
:: GROEB_1:th 35
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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 add-closed left-ideal Element of bool the carrier of Polynom-Ring(b1,b3) holds
ex b5 being finite Element of bool the carrier of Polynom-Ring(b1,b3) st
b5 is_Groebner_basis_of b4,b2;
:: GROEB_1:th 36
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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 add-closed left-ideal Element of bool the carrier of Polynom-Ring(b1,b3)
st b4 <> {0_(b1,b3)}
holds ex b5 being finite Element of bool the carrier of Polynom-Ring(b1,b3) st
b5 is_Groebner_basis_of b4,b2 & not 0_(b1,b3) in b5;
:: GROEB_1:prednot 3 => GROEB_1:pred 3
definition
let a1 be ordinal set;
let a2 be total reflexive antisymmetric connected transitive Relation of Bags a1,Bags a1;
let a3 be non empty multLoopStr_0;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
pred A4 is_monic_wrt A2 means
HC(a4,a2) = 1. a3;
end;
:: GROEB_1:dfs 6
definiens
let a1 be ordinal set;
let a2 be total reflexive antisymmetric connected transitive Relation of Bags a1,Bags a1;
let a3 be non empty multLoopStr_0;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
To prove
a4 is_monic_wrt a2
it is sufficient to prove
thus HC(a4,a2) = 1. a3;
:: GROEB_1:def 6
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive Relation of Bags b1,Bags b1
for b3 being non empty multLoopStr_0
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
b4 is_monic_wrt b2
iff
HC(b4,b2) = 1. b3;
:: GROEB_1:prednot 4 => GROEB_1:pred 4
definition
let a1 be ordinal set;
let a2 be total reflexive antisymmetric connected transitive Relation of Bags a1,Bags a1;
let a3 be non empty 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);
pred A4 is_reduced_wrt A2 means
for b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3
st b1 in a4
holds b1 is_monic_wrt a2 & b1 is_irreducible_wrt a4 \ {b1},a2;
end;
:: GROEB_1:dfs 7
definiens
let a1 be ordinal set;
let a2 be total reflexive antisymmetric connected transitive Relation of Bags a1,Bags a1;
let a3 be non empty 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);
To prove
a4 is_reduced_wrt a2
it is sufficient to prove
thus for b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3
st b1 in a4
holds b1 is_monic_wrt a2 & b1 is_irreducible_wrt a4 \ {b1},a2;
:: GROEB_1:def 7
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive Relation of Bags b1,Bags b1
for b3 being non empty 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
b4 is_reduced_wrt b2
iff
for b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b5 in b4
holds b5 is_monic_wrt b2 & b5 is_irreducible_wrt b4 \ {b5},b2;
:: GROEB_1:prednot 5 => GROEB_1:pred 4
notation
let a1 be ordinal set;
let a2 be total reflexive antisymmetric connected transitive Relation of Bags a1,Bags a1;
let a3 be non empty 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);
synonym a4 is_autoreduced_wrt a2 for a4 is_reduced_wrt a2;
end;
:: GROEB_1:th 37
theorem
for b1 being ordinal set
for b2 being total reflexive antisymmetric connected transitive 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 add-closed left-ideal Element of bool the carrier of Polynom-Ring(b1,b3)
for b5 being Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b3
for b6, b7 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b6 in b4 &
b7 in b4 &
HM(b6,b2) = b5 &
HM(b7,b2) = b5 &
(for b8 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b8 in b4 & b8 < b6,b2
holds HM(b8,b2) <> b5) &
(for b8 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b8 in b4 & b8 < b7,b2
holds HM(b8,b2) <> b5)
holds b6 = b7;
:: GROEB_1:th 38
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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 add-closed left-ideal Element of bool the carrier of Polynom-Ring(b1,b3)
for b5 being Element of bool the carrier of Polynom-Ring(b1,b3)
for b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b7 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
st b6 in b5 & b7 in b5 & b6 <> b7 & HT(b7,b2) divides HT(b6,b2) & b5 is_Groebner_basis_of b4,b2
holds b5 \ {b6} is_Groebner_basis_of b4,b2;
:: GROEB_1:th 39
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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 add-closed left-ideal Element of bool the carrier of Polynom-Ring(b1,b3)
st b4 <> {0_(b1,b3)}
holds ex b5 being finite Element of bool the carrier of Polynom-Ring(b1,b3) st
b5 is_Groebner_basis_of b4,b2 & b5 is_reduced_wrt b2;
:: GROEB_1:th 40
theorem
for b1 being Element of NAT
for b2 being total reflexive antisymmetric connected transitive 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 add-closed left-ideal Element of bool the carrier of Polynom-Ring(b1,b3)
for b5, b6 being non empty finite Element of bool the carrier of Polynom-Ring(b1,b3)
st b5 is_Groebner_basis_of b4,b2 & b5 is_reduced_wrt b2 & b6 is_Groebner_basis_of b4,b2 & b6 is_reduced_wrt b2
holds b5 = b6;