Article GROEB_3, MML version 4.99.1005
:: GROEB_3:th 1
theorem
for b1 being set
for b2, b3 being natural-valued finite-support ManySortedSet of b1 holds
(b2 + b3) / b3 = b2;
:: GROEB_3:th 2
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric transitive total admissible Relation of Bags b1,Bags b1
for b3, b4, b5 being natural-valued finite-support ManySortedSet of b1
st b3 <= b4,b2
holds b3 + b5 <= b4 + b5,b2;
:: GROEB_3:th 3
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric transitive total Relation of Bags b1,Bags b1
for b3, b4, b5 being natural-valued finite-support ManySortedSet of b1
st b3 <= b4,b2 & b4 < b5,b2
holds b3 < b5,b2;
:: GROEB_3:th 4
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric transitive total admissible Relation of Bags b1,Bags b1
for b3, b4, b5 being natural-valued finite-support ManySortedSet of b1
st b3 < b4,b2
holds b3 + b5 < b4 + b5,b2;
:: GROEB_3:th 5
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric transitive total admissible Relation of Bags b1,Bags b1
for b3, b4, b5, b6 being natural-valued finite-support ManySortedSet of b1
st b3 < b4,b2 & b5 <= b6,b2
holds b3 + b5 < b4 + b6,b2;
:: GROEB_3:th 6
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric transitive total admissible Relation of Bags b1,Bags b1
for b3, b4, b5, b6 being natural-valued finite-support ManySortedSet of b1
st b3 <= b4,b2 & b5 < b6,b2
holds b3 + b5 < b4 + b6,b2;
:: GROEB_3:th 7
theorem
for b1 being ordinal set
for b2 being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for b3, b4 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b2 holds
term (b3 *' b4) = (term b3) + term b4;
:: GROEB_3:th 8
theorem
for b1 being ordinal set
for b2 being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for b3 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2
for b4 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b2
for b5 being natural-valued finite-support ManySortedSet of b1 holds
b5 in Support b3
iff
(term b4) + b5 in Support (b4 *' b3);
:: GROEB_3:th 9
theorem
for b1 being ordinal set
for b2 being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for b3 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2
for b4 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b2 holds
Support (b4 *' b3) = {(term b4) + b5 where b5 is Element of Bags b1: b5 in Support b3};
:: GROEB_3:th 10
theorem
for b1 being ordinal set
for b2 being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed doubleLoopStr
for b3 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2
for b4 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b2 holds
card Support b3 = card Support (b4 *' b3);
:: GROEB_3:th 11
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 add-associative right_zeroed addLoopStr holds
Red(0_(b1,b3),b2) = 0_(b1,b3);
:: GROEB_3:th 12
theorem
for b1 being ordinal set
for b2 being non trivial right_complementable commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3, b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2
st b3 - b4 = 0_(b1,b2)
holds b3 = b4;
:: GROEB_3:th 13
theorem
for b1 being set
for b2 being non empty right_complementable add-associative right_zeroed addLoopStr holds
- 0_(b1,b2) = 0_(b1,b2);
:: GROEB_3:th 14
theorem
for b1 being 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 holds
(0_(b1,b2)) - b3 = - b3;
:: GROEB_3:th 15
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 add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
b4 - Red(b4,b2) = HM(b4,b2);
:: GROEB_3:funcreg 1
registration
let a1 be ordinal set;
let a2 be non empty right_complementable add-associative right_zeroed addLoopStr;
let a3 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a2;
cluster Support a3 -> finite;
end;
:: GROEB_3:funcnot 1 => GROEB_3: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, a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a2;
redefine func {a3, a4} -> Element of bool the carrier of Polynom-Ring(a1,a2);
commutativity;
:: for a1 being ordinal set
:: for a2 being non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
:: for a3, a4 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a2 holds
:: {a3,a4} = {a4,a3};
end;
:: GROEB_3:funcnot 2 => GROEB_3:func 2
definition
let a1 be set;
let a2 be non empty ZeroStr;
let a3 be Function-like quasi_total Relation of Bags a1,the carrier of a2;
let a4 be Element of bool Bags a1;
func A3 | A4 -> Function-like quasi_total Relation of Bags a1,the carrier of a2 equals
a3 +* (((Support a3) \ a4) --> 0. a2);
end;
:: GROEB_3:def 1
theorem
for b1 being set
for b2 being non empty ZeroStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b4 being Element of bool Bags b1 holds
b3 | b4 = b3 +* (((Support b3) \ b4) --> 0. b2);
:: GROEB_3:funcreg 2
registration
let a1 be ordinal set;
let a2 be non empty ZeroStr;
let a3 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a2;
let a4 be Element of bool Bags a1;
cluster a3 | a4 -> Function-like quasi_total finite-Support;
end;
:: GROEB_3:th 16
theorem
for b1 being set
for b2 being non empty ZeroStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b4 being Element of bool Bags b1 holds
Support (b3 | b4) = (Support b3) /\ b4 &
(for b5 being natural-valued finite-support ManySortedSet of b1
st b5 in Support (b3 | b4)
holds (b3 | b4) . b5 = b3 . b5);
:: GROEB_3:th 17
theorem
for b1 being set
for b2 being non empty ZeroStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b4 being Element of bool Bags b1 holds
Support (b3 | b4) c= Support b3;
:: GROEB_3:th 18
theorem
for b1 being set
for b2 being non empty ZeroStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2 holds
b3 | Support b3 = b3 & b3 | {} Bags b1 = 0_(b1,b2);
:: GROEB_3:funcnot 3 => GROEB_3: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 empty right_complementable add-associative right_zeroed addLoopStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be Element of NAT;
assume a5 <= card Support a4;
func Upper_Support(A4,A2,A5) -> finite Element of bool Bags a1 means
it c= Support a4 &
card it = a5 &
(for b1, b2 being natural-valued finite-support ManySortedSet of a1
st b1 in it & b2 in Support a4 & b1 <= b2,a2
holds b2 in it);
end;
:: GROEB_3:def 2
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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st b5 <= card Support b4
for b6 being finite Element of bool Bags b1 holds
b6 = Upper_Support(b4,b2,b5)
iff
b6 c= Support b4 &
card b6 = b5 &
(for b7, b8 being natural-valued finite-support ManySortedSet of b1
st b7 in b6 & b8 in Support b4 & b7 <= b8,b2
holds b8 in b6);
:: GROEB_3:funcnot 4 => GROEB_3: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 empty right_complementable add-associative right_zeroed addLoopStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be Element of NAT;
func Lower_Support(A4,A2,A5) -> finite Element of bool Bags a1 equals
(Support a4) \ Upper_Support(a4,a2,a5);
end;
:: GROEB_3:def 3
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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT holds
Lower_Support(b4,b2,b5) = (Support b4) \ Upper_Support(b4,b2,b5);
:: GROEB_3:th 19
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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st b5 <= card Support b4
holds (Upper_Support(b4,b2,b5)) \/ Lower_Support(b4,b2,b5) = Support b4 &
(Upper_Support(b4,b2,b5)) /\ Lower_Support(b4,b2,b5) = {};
:: GROEB_3:th 20
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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st b5 <= card Support b4
for b6, b7 being natural-valued finite-support ManySortedSet of b1
st b6 in Upper_Support(b4,b2,b5) & b7 in Lower_Support(b4,b2,b5)
holds b7 < b6,b2;
:: GROEB_3:th 21
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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
Upper_Support(b4,b2,0) = {} & Lower_Support(b4,b2,0) = Support b4;
:: GROEB_3: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 empty right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
Upper_Support(b4,b2,card Support b4) = Support b4 &
Lower_Support(b4,b2,card Support b4) = {};
:: GROEB_3:th 23
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 add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st 1 <= b5 & b5 <= card Support b4
holds HT(b4,b2) in Upper_Support(b4,b2,b5);
:: GROEB_3: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 empty right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st b5 <= card Support b4
holds Lower_Support(b4,b2,b5) c= Support b4 &
card Lower_Support(b4,b2,b5) = (card Support b4) - b5 &
(for b6, b7 being natural-valued finite-support ManySortedSet of b1
st b6 in Lower_Support(b4,b2,b5) & b7 in Support b4 & b7 <= b6,b2
holds b7 in Lower_Support(b4,b2,b5));
:: GROEB_3:funcnot 5 => GROEB_3:func 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 right_complementable add-associative right_zeroed addLoopStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be Element of NAT;
func Up(A4,A2,A5) -> Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 equals
a4 | Upper_Support(a4,a2,a5);
end;
:: GROEB_3: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 empty right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT holds
Up(b4,b2,b5) = b4 | Upper_Support(b4,b2,b5);
:: GROEB_3:funcnot 6 => GROEB_3:func 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 right_complementable add-associative right_zeroed addLoopStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be Element of NAT;
func Low(A4,A2,A5) -> Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 equals
a4 | Lower_Support(a4,a2,a5);
end;
:: GROEB_3: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 empty right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT holds
Low(b4,b2,b5) = b4 | Lower_Support(b4,b2,b5);
:: GROEB_3: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 empty right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st b5 <= card Support b4
holds Support Up(b4,b2,b5) = Upper_Support(b4,b2,b5) & Support Low(b4,b2,b5) = Lower_Support(b4,b2,b5);
:: GROEB_3:th 26
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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st b5 <= card Support b4
holds Support Up(b4,b2,b5) c= Support b4 & Support Low(b4,b2,b5) c= Support b4;
:: GROEB_3: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 add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st 1 <= b5 & b5 <= card Support b4
holds Support Low(b4,b2,b5) c= Support Red(b4,b2);
:: GROEB_3:th 28
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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st b5 <= card Support b4
for b6 being natural-valued finite-support ManySortedSet of b1
st b6 in Support b4
holds (b6 in Support Up(b4,b2,b5) or b6 in Support Low(b4,b2,b5)) &
not b6 in (Support Up(b4,b2,b5)) /\ Support Low(b4,b2,b5);
:: GROEB_3:th 29
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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st b5 <= card Support b4
for b6, b7 being natural-valued finite-support ManySortedSet of b1
st b6 in Support Low(b4,b2,b5) & b7 in Support Up(b4,b2,b5)
holds b6 < b7,b2;
:: GROEB_3:th 30
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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st 1 <= b5 & b5 <= card Support b4
holds HT(b4,b2) in Support Up(b4,b2,b5);
:: GROEB_3:th 31
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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st b5 <= card Support b4
for b6 being natural-valued finite-support ManySortedSet of b1
st b6 in Support Low(b4,b2,b5)
holds (Low(b4,b2,b5)) . b6 = b4 . b6 & (Up(b4,b2,b5)) . b6 = 0. b3;
:: GROEB_3:th 32
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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st b5 <= card Support b4
for b6 being natural-valued finite-support ManySortedSet of b1
st b6 in Support Up(b4,b2,b5)
holds (Up(b4,b2,b5)) . b6 = b4 . b6 & (Low(b4,b2,b5)) . b6 = 0. b3;
:: GROEB_3:th 33
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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st b5 <= card Support b4
holds (Up(b4,b2,b5)) + Low(b4,b2,b5) = b4;
:: GROEB_3:th 34
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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
Up(b4,b2,0) = 0_(b1,b3) & Low(b4,b2,0) = b4;
:: GROEB_3:th 35
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 right_complementable Abelian add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
Up(b4,b2,card Support b4) = b4 &
Low(b4,b2,card Support b4) = 0_(b1,b3);
:: GROEB_3:th 36
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 Abelian add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 holds
Up(b4,b2,1) = HM(b4,b2) & Low(b4,b2,1) = Red(b4,b2);
:: GROEB_3:funcreg 3
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 add-associative right_zeroed addLoopStr;
let a4 be Function-like quasi_total finite-Support non-zero Relation of Bags a1,the carrier of a3;
cluster Up(a4,a2,0) -> Function-like quasi_total finite-Support monomial-like;
end;
:: GROEB_3: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 Abelian add-associative right_zeroed doubleLoopStr;
let a4 be Function-like quasi_total finite-Support non-zero Relation of Bags a1,the carrier of a3;
cluster Up(a4,a2,1) -> Function-like quasi_total finite-Support non-zero monomial-like;
end;
:: GROEB_3:funcreg 5
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 Abelian add-associative right_zeroed doubleLoopStr;
let a4 be Function-like quasi_total finite-Support non-zero Relation of Bags a1,the carrier of a3;
cluster Low(a4,a2,card Support a4) -> Function-like quasi_total finite-Support monomial-like;
end;
:: GROEB_3: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 trivial right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st b5 = (card Support b4) - 1
holds Low(b4,b2,b5) is Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b3;
:: GROEB_3:th 38
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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st b5 < card Support b4
holds HT(Low(b4,b2,b5 + 1),b2) <= HT(Low(b4,b2,b5),b2),b2;
:: GROEB_3: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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st 0 < b5 & b5 < card Support b4
holds HT(Low(b4,b2,b5),b2) < HT(b4,b2),b2;
:: GROEB_3:th 40
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 well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b3
for b6 being Element of NAT
st b6 <= card Support b4
for b7 being natural-valued finite-support ManySortedSet of b1 holds
(term b5) + b7 in Support Low(b5 *' b4,b2,b6)
iff
b7 in Support Low(b4,b2,b6);
:: GROEB_3:th 41
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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st b5 < card Support b4
holds Support Low(b4,b2,b5 + 1) c= Support Low(b4,b2,b5);
:: GROEB_3:th 42
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 right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st b5 < card Support b4
holds (Support Low(b4,b2,b5)) \ Support Low(b4,b2,b5 + 1) = {HT(Low(b4,b2,b5),b2)};
:: GROEB_3: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 trivial right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of NAT
st b5 < card Support b4
holds Low(b4,b2,b5 + 1) = Red(Low(b4,b2,b5),b2);
:: GROEB_3: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 well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b3
for b6 being Element of NAT
st b6 <= card Support b4
holds Low(b5 *' b4,b2,b6) = b5 *' Low(b4,b2,b6);
:: GROEB_3:th 45
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 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 b5,b6,b2
holds - b4 reduces_to - b5,b6,b2;
:: GROEB_3:th 46
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 Abelian add-associative right_zeroed doubleLoopStr
for b4, b5, b6, b7 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b4 reduces_to b5,{b7},b2 &
(for b8 being natural-valued finite-support ManySortedSet of b1
st b8 in Support b6
holds not HT(b7,b2) divides b8)
holds b4 + b6 reduces_to b5 + b6,{b7},b2;
:: GROEB_3:th 47
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 Abelian 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 holds
b4 *' b5 reduces_to (Red(b4,b2)) *' b5,{b5},b2;
:: GROEB_3:th 48
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 Abelian 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
for b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b6 . HT(b4 *' b5,b2) = 0. b3
holds (b4 *' b5) + b6 reduces_to ((Red(b4,b2)) *' b5) + b6,{b5},b2;
:: GROEB_3:th 49
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 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 PolyRedRel(b4,b2) reduces b5,b6
holds PolyRedRel(b4,b2) reduces - b5,- b6;
:: GROEB_3:th 50
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 Abelian add-associative right_zeroed doubleLoopStr
for b4, b5, b6, b7 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st PolyRedRel({b7},b2) reduces b4,b5 &
(for b8 being natural-valued finite-support ManySortedSet of b1
st b8 in Support b6
holds not HT(b7,b2) divides b8)
holds PolyRedRel({b7},b2) reduces b4 + b6,b5 + b6;
:: GROEB_3:th 51
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 Abelian 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 holds
PolyRedRel({b5},b2) reduces b4 *' b5,0_(b1,b3);
:: GROEB_3:th 52
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 HT(b4,b2),HT(b5,b2) are_disjoint
for b6, b7 being natural-valued finite-support ManySortedSet of b1
st b6 in Support Red(b4,b2) & b7 in Support Red(b5,b2)
holds (HT(b4,b2)) + b7 <> (HT(b5,b2)) + b6;
:: GROEB_3:th 53
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, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st HT(b4,b2),HT(b5,b2) are_disjoint
holds S-Poly(b4,b5,b2) = ((HM(b5,b2)) *' Red(b4,b2)) - ((HM(b4,b2)) *' Red(b5,b2));
:: GROEB_3:th 54
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, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st HT(b4,b2),HT(b5,b2) are_disjoint
holds S-Poly(b4,b5,b2) = ((Red(b4,b2)) *' b5) - ((Red(b5,b2)) *' b4);
:: GROEB_3:th 55
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 Abelian 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(b4,b2),HT(b5,b2) are_disjoint & Red(b4,b2) is non-zero(b1, b3) & Red(b5,b2) is non-zero(b1, b3)
holds PolyRedRel({b4},b2) reduces ((HM(b5,b2)) *' Red(b4,b2)) - ((HM(b4,b2)) *' Red(b5,b2)),b5 *' Red(b4,b2);
:: GROEB_3:th 56
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 Abelian add-associative right_zeroed doubleLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st HT(b4,b2),HT(b5,b2) are_disjoint
holds PolyRedRel({b4,b5},b2) reduces S-Poly(b4,b5,b2),0_(b1,b3);
:: GROEB_3:th 57
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 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b5 in b4 & b6 in b4 & HT(b5,b2),HT(b6,b2) are_non_disjoint
holds PolyRedRel(b4,b2) reduces S-Poly(b5,b6,b2),0_(b1,b3);
:: GROEB_3:th 58
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 degenerated 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)
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 & HT(b5,b2),HT(b6,b2) are_non_disjoint
holds PolyRedRel(b4,b2) reduces S-Poly(b5,b6,b2),0_(b1,b3))
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 & HT(b5,b2),HT(b6,b2) are_non_disjoint & b7 is_a_normal_form_of S-Poly(b5,b6,b2),PolyRedRel(b4,b2)
holds b7 = 0_(b1,b3);
:: GROEB_3:th 59
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, b7 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b5 in b4 & b6 in b4 & HT(b5,b2),HT(b6,b2) are_non_disjoint & b7 is_a_normal_form_of S-Poly(b5,b6,b2),PolyRedRel(b4,b2)
holds b7 = 0_(b1,b3))
holds b4 is_Groebner_basis_wrt b2;