Article POLYRED, MML version 4.99.1005
:: POLYRED:exreg 1
registration
let a1 be ordinal set;
let a2 be non trivial ZeroStr;
cluster Relation-like Function-like quasi_total finite-Support non-zero monomial-like Relation of Bags a1,the carrier of a2;
end;
:: POLYRED:exreg 2
registration
cluster non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr;
end;
:: POLYRED:funcreg 1
registration
let a1 be ordinal set;
let a2 be non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed doubleLoopStr;
let a3, a4 be Function-like quasi_total finite-Support non-zero Relation of Bags a1,the carrier of a2;
cluster a3 *' a4 -> Function-like quasi_total non-zero;
end;
:: POLYRED:th 1
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3, b4 being Function-like quasi_total Relation of Bags b1,the carrier of b2 holds
- (b3 + b4) = (- b3) + - b4;
:: POLYRED:th 2
theorem
for b1 being set
for b2 being non empty left_zeroed addLoopStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2 holds
(0_(b1,b2)) + b3 = b3;
:: POLYRED:th 3
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
(- b3) + b3 = 0_(b1,b2) & b3 + - b3 = 0_(b1,b2);
:: POLYRED:th 4
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
b3 - 0_(b1,b2) = b3;
:: POLYRED:th 5
theorem
for b1 being ordinal set
for b2 being non empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2 holds
(0_(b1,b2)) *' b3 = 0_(b1,b2);
:: POLYRED:th 6
theorem
for b1 being ordinal set
for b2 being non trivial right_complementable associative 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 holds
- (b3 *' b4) = (- b3) *' b4 & - (b3 *' b4) = b3 *' - b4;
:: POLYRED:th 7
theorem
for b1 being ordinal set
for b2 being non empty right_complementable 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 Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b2
for b5 being natural-valued finite-support ManySortedSet of b1 holds
(b4 *' b3) . ((term b4) + b5) = (b4 . term b4) * (b3 . b5);
:: POLYRED:th 8
theorem
for b1 being set
for b2 being non empty left_add-cancelable left-distributive right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2 holds
(0. b2) * b3 = 0_(b1,b2);
:: POLYRED:th 9
theorem
for b1 being set
for b2 being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b4 being Element of the carrier of b2 holds
- (b4 * b3) = (- b4) * b3 & - (b4 * b3) = b4 * - b3;
:: POLYRED:th 10
theorem
for b1 being set
for b2 being non empty left-distributive doubleLoopStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b4, b5 being Element of the carrier of b2 holds
(b4 * b3) + (b5 * b3) = (b4 + b5) * b3;
:: POLYRED:th 11
theorem
for b1 being set
for b2 being non empty associative multLoopStr_0
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b4, b5 being Element of the carrier of b2 holds
(b4 * b5) * b3 = b4 * (b5 * b3);
:: POLYRED:th 12
theorem
for b1 being ordinal set
for b2 being non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed 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) = b3 *' (b5 * b4);
:: POLYRED:funcnot 1 => POLYRED:func 1
definition
let a1 be ordinal set;
let a2 be natural-valued finite-support ManySortedSet of a1;
let a3 be non empty ZeroStr;
let a4 be Function-like quasi_total Relation of Bags a1,the carrier of a3;
func A2 *' A4 -> Function-like quasi_total Relation of Bags a1,the carrier of a3 means
for b1 being natural-valued finite-support ManySortedSet of a1
st a2 divides b1
holds it . b1 = a4 . (b1 -' a2) &
(for b2 being natural-valued finite-support ManySortedSet of a1
st not a2 divides b2
holds it . b2 = 0. a3);
end;
:: POLYRED:def 1
theorem
for b1 being ordinal set
for b2 being natural-valued finite-support ManySortedSet of b1
for b3 being non empty ZeroStr
for b4, b5 being Function-like quasi_total Relation of Bags b1,the carrier of b3 holds
b5 = b2 *' b4
iff
for b6 being natural-valued finite-support ManySortedSet of b1
st b2 divides b6
holds b5 . b6 = b4 . (b6 -' b2) &
(for b7 being natural-valued finite-support ManySortedSet of b1
st not b2 divides b7
holds b5 . b7 = 0. b3);
:: POLYRED:funcreg 2
registration
let a1 be ordinal set;
let a2 be natural-valued finite-support ManySortedSet of a1;
let a3 be non empty ZeroStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
cluster a2 *' a4 -> Function-like quasi_total finite-Support;
end;
:: POLYRED:th 13
theorem
for b1 being ordinal set
for b2, b3 being natural-valued finite-support ManySortedSet of b1
for b4 being non empty ZeroStr
for b5 being Function-like quasi_total Relation of Bags b1,the carrier of b4 holds
(b2 *' b5) . (b3 + b2) = b5 . b3;
:: POLYRED:th 14
theorem
for b1 being ordinal set
for b2 being non empty ZeroStr
for b3 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2
for b4 being natural-valued finite-support ManySortedSet of b1 holds
Support (b4 *' b3) c= {b4 + b5 where b5 is Element of Bags b1: b5 in Support b3};
:: POLYRED:th 15
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 ZeroStr
for b4 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
for b5 being natural-valued finite-support ManySortedSet of b1 holds
HT(b5 *' b4,b2) = b5 + HT(b4,b2);
:: POLYRED:th 16
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 ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5, b6 being natural-valued finite-support ManySortedSet of b1
st b6 in Support (b5 *' b4)
holds b6 <= b5 + HT(b4,b2),b2;
:: POLYRED:th 17
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 ZeroStr
for b4 being Function-like quasi_total Relation of Bags b1,the carrier of b3 holds
(EmptyBag b1) *' b4 = b4;
:: POLYRED:th 18
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 ZeroStr
for b4 being Function-like quasi_total Relation of Bags b1,the carrier of b3
for b5, b6 being natural-valued finite-support ManySortedSet of b1 holds
(b5 + b6) *' b4 = b5 *' (b6 *' b4);
:: POLYRED:th 19
theorem
for b1 being ordinal set
for b2 being non trivial right_complementable 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 Element of the carrier of b2 holds
Support (b4 * b3) c= Support b3;
:: POLYRED:th 20
theorem
for b1 being ordinal set
for b2 being non trivial domRing-like doubleLoopStr
for b3 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2
for b4 being non-zero Element of the carrier of b2 holds
Support b3 c= Support (b4 * b3);
:: POLYRED: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 trivial right_complementable 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 non-zero Element of the carrier of b3 holds
HT(b5 * b4,b2) = HT(b4,b2);
:: POLYRED:th 22
theorem
for b1 being ordinal set
for b2 being non trivial right_complementable distributive add-associative right_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
b5 * (b4 *' b3) = (Monom(b5,b4)) *' b3;
:: POLYRED:th 23
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 non-zero Relation of Bags b1,the carrier of b3
for b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b6 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b3
st HT(b4,b2) in Support b5
holds HT(b6 *' b4,b2) in Support (b6 *' b5);
:: POLYRED:funcreg 3
registration
let a1 be ordinal set;
let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
cluster RelStr(#Bags a1,a2#) -> strict connected;
end;
:: POLYRED:funcreg 4
registration
let a1 be natural set;
let a2 be reflexive antisymmetric transitive total admissible Relation of Bags a1,Bags a1;
cluster RelStr(#Bags a1,a2#) -> strict well_founded;
end;
:: POLYRED:prednot 1 => POLYRED:pred 1
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 ZeroStr;
let a4, a5 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
pred A4 <= A5,A2 means
[Support a4,Support a5] in FinOrd RelStr(#Bags a1,a2#);
end;
:: POLYRED:dfs 2
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 ZeroStr;
let a4, a5 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
To prove
a4 <= a5,a2
it is sufficient to prove
thus [Support a4,Support a5] in FinOrd RelStr(#Bags a1,a2#);
:: POLYRED: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 ZeroStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
b4 <= b5,b2
iff
[Support b4,Support b5] in FinOrd RelStr(#Bags b1,b2#);
:: POLYRED:prednot 2 => POLYRED:pred 2
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 ZeroStr;
let a4, a5 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
pred A4 < A5,A2 means
a4 <= a5,a2 & Support a4 <> Support a5;
end;
:: POLYRED:dfs 3
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 ZeroStr;
let a4, a5 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
To prove
a4 < a5,a2
it is sufficient to prove
thus a4 <= a5,a2 & Support a4 <> Support a5;
:: POLYRED: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 ZeroStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
b4 < b5,b2
iff
b4 <= b5,b2 & Support b4 <> Support b5;
:: POLYRED:funcnot 2 => POLYRED:func 2
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 ZeroStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
func Support(A4,A2) -> Element of Fin the carrier of RelStr(#Bags a1,a2#) equals
Support a4;
end;
:: POLYRED: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 ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
Support(b4,b2) = Support b4;
:: POLYRED: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 ZeroStr
for b4 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 holds
PosetMax Support(b4,b2) = HT(b4,b2);
:: POLYRED: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 addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
b4 <= b4,b2;
:: POLYRED: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 addLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
b4 <= b5,b2 & b5 <= b4,b2
iff
Support b4 = Support b5;
:: POLYRED: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 empty addLoopStr
for b4, b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b4 <= b5,b2 & b5 <= b6,b2
holds b4 <= b6,b2;
:: POLYRED: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 addLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st not b4 <= b5,b2
holds b5 <= b4,b2;
:: POLYRED: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 addLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
b4 <= b5,b2
iff
not b5 < b4,b2;
:: POLYRED: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 ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
0_(b1,b3) <= b4,b2;
:: POLYRED:th 31
theorem
for b1 being natural 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 doubleLoopStr
for b4 being non empty Element of bool the carrier of Polynom-Ring(b1,b3) holds
ex b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 st
b5 in b4 &
(for b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b6 in b4
holds b5 <= b6,b2);
:: POLYRED:th 32
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, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
b4 < b5,b2
iff
((b4 = 0_(b1,b3) implies b5 = 0_(b1,b3)) & not HT(b4,b2) < HT(b5,b2),b2 implies HT(b4,b2) = HT(b5,b2) & Red(b4,b2) < Red(b5,b2),b2);
:: POLYRED:th 33
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 non-zero Relation of Bags b1,the carrier of b3 holds
Red(b4,b2) < HM(b4,b2),b2;
:: POLYRED: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 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 holds
HM(b4,b2) <= b4,b2;
:: POLYRED:th 35
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 non-zero Relation of Bags b1,the carrier of b3 holds
Red(b4,b2) < b4,b2;
:: POLYRED:prednot 3 => POLYRED: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 trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr;
let a4, a5, a6 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a7 be natural-valued finite-support ManySortedSet of a1;
pred A4 reduces_to A6,A5,A7,A2 means
a4 <> 0_(a1,a3) &
a5 <> 0_(a1,a3) &
a7 in Support a4 &
(ex b1 being natural-valued finite-support ManySortedSet of a1 st
b1 + HT(a5,a2) = a7 &
a6 = a4 - (((a4 . a7) / HC(a5,a2)) * (b1 *' a5)));
end;
:: POLYRED:dfs 5
definiens
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, a6 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a7 be natural-valued finite-support ManySortedSet of a1;
To prove
a4 reduces_to a6,a5,a7,a2
it is sufficient to prove
thus a4 <> 0_(a1,a3) &
a5 <> 0_(a1,a3) &
a7 in Support a4 &
(ex b1 being natural-valued finite-support ManySortedSet of a1 st
b1 + HT(a5,a2) = a7 &
a6 = a4 - (((a4 . a7) / HC(a5,a2)) * (b1 *' a5)));
:: POLYRED: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, b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b7 being natural-valued finite-support ManySortedSet of b1 holds
b4 reduces_to b6,b5,b7,b2
iff
b4 <> 0_(b1,b3) &
b5 <> 0_(b1,b3) &
b7 in Support b4 &
(ex b8 being natural-valued finite-support ManySortedSet of b1 st
b8 + HT(b5,b2) = b7 &
b6 = b4 - (((b4 . b7) / HC(b5,b2)) * (b8 *' b5)));
:: POLYRED:prednot 4 => POLYRED: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 trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr;
let a4, a5, a6 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
pred A4 reduces_to A6,A5,A2 means
ex b1 being natural-valued finite-support ManySortedSet of a1 st
a4 reduces_to a6,a5,b1,a2;
end;
:: POLYRED:dfs 6
definiens
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, a6 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
To prove
a4 reduces_to a6,a5,a2
it is sufficient to prove
thus ex b1 being natural-valued finite-support ManySortedSet of a1 st
a4 reduces_to a6,a5,b1,a2;
:: POLYRED:def 6
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, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
b4 reduces_to b6,b5,b2
iff
ex b7 being natural-valued finite-support ManySortedSet of b1 st
b4 reduces_to b6,b5,b7,b2;
:: POLYRED:prednot 5 => POLYRED: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 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;
let a6 be Element of bool the carrier of Polynom-Ring(a1,a3);
pred A4 reduces_to A5,A6,A2 means
ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
b1 in a6 & a4 reduces_to a5,b1,a2;
end;
:: POLYRED: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 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;
let a6 be Element of bool the carrier of Polynom-Ring(a1,a3);
To prove
a4 reduces_to a5,a6,a2
it is sufficient to prove
thus ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
b1 in a6 & a4 reduces_to a5,b1,a2;
:: POLYRED: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 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 being Element of bool the carrier of Polynom-Ring(b1,b3) holds
b4 reduces_to b5,b6,b2
iff
ex b7 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 st
b7 in b6 & b4 reduces_to b5,b7,b2;
:: POLYRED:prednot 6 => POLYRED: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 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;
pred A4 is_reducible_wrt A5,A2 means
ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
a4 reduces_to b1,a5,a2;
end;
:: POLYRED: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 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;
To prove
a4 is_reducible_wrt a5,a2
it is sufficient to prove
thus ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
a4 reduces_to b1,a5,a2;
:: POLYRED: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 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
b4 is_reducible_wrt b5,b2
iff
ex b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 st
b4 reduces_to b6,b5,b2;
:: POLYRED:prednot 7 => not POLYRED:pred 6
notation
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;
antonym a4 is_irreducible_wrt a5,a2 for a4 is_reducible_wrt a5,a2;
end;
:: POLYRED:prednot 8 => not POLYRED:pred 6
notation
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;
antonym a4 is_in_normalform_wrt a5,a2 for a4 is_reducible_wrt a5,a2;
end;
:: POLYRED:prednot 9 => POLYRED:pred 7
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 Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be Element of bool the carrier of Polynom-Ring(a1,a3);
pred A4 is_reducible_wrt A5,A2 means
ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
a4 reduces_to b1,a5,a2;
end;
:: POLYRED: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 trivial right_complementable almost_left_invertible associative commutative 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 Element of bool the carrier of Polynom-Ring(a1,a3);
To prove
a4 is_reducible_wrt a5,a2
it is sufficient to prove
thus ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
a4 reduces_to b1,a5,a2;
:: POLYRED: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 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
for b5 being Element of bool the carrier of Polynom-Ring(b1,b3) holds
b4 is_reducible_wrt b5,b2
iff
ex b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 st
b4 reduces_to b6,b5,b2;
:: POLYRED:prednot 10 => not POLYRED:pred 7
notation
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 Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be Element of bool the carrier of Polynom-Ring(a1,a3);
antonym a4 is_irreducible_wrt a5,a2 for a4 is_reducible_wrt a5,a2;
end;
:: POLYRED:prednot 11 => not POLYRED:pred 7
notation
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 Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be Element of bool the carrier of Polynom-Ring(a1,a3);
antonym a4 is_in_normalform_wrt a5,a2 for a4 is_reducible_wrt a5,a2;
end;
:: POLYRED:prednot 12 => POLYRED:pred 8
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, a6 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
pred A4 top_reduces_to A6,A5,A2 means
a4 reduces_to a6,a5,HT(a4,a2),a2;
end;
:: POLYRED: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 trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr;
let a4, a5, a6 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
To prove
a4 top_reduces_to a6,a5,a2
it is sufficient to prove
thus a4 reduces_to a6,a5,HT(a4,a2),a2;
:: POLYRED: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 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 holds
b4 top_reduces_to b6,b5,b2
iff
b4 reduces_to b6,b5,HT(b4,b2),b2;
:: POLYRED:prednot 13 => POLYRED:pred 9
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;
pred A4 is_top_reducible_wrt A5,A2 means
ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
a4 top_reduces_to b1,a5,a2;
end;
:: POLYRED:dfs 11
definiens
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;
To prove
a4 is_top_reducible_wrt a5,a2
it is sufficient to prove
thus ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
a4 top_reduces_to b1,a5,a2;
:: POLYRED:def 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 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
b4 is_top_reducible_wrt b5,b2
iff
ex b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 st
b4 top_reduces_to b6,b5,b2;
:: POLYRED:prednot 14 => POLYRED:pred 10
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 Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be Element of bool the carrier of Polynom-Ring(a1,a3);
pred A4 is_top_reducible_wrt A5,A2 means
ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
b1 in a5 & a4 is_top_reducible_wrt b1,a2;
end;
:: POLYRED:dfs 12
definiens
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 Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
let a5 be Element of bool the carrier of Polynom-Ring(a1,a3);
To prove
a4 is_top_reducible_wrt a5,a2
it is sufficient to prove
thus ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
b1 in a5 & a4 is_top_reducible_wrt b1,a2;
:: POLYRED:def 12
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
for b5 being Element of bool the carrier of Polynom-Ring(b1,b3) holds
b4 is_top_reducible_wrt b5,b2
iff
ex b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 st
b6 in b5 & b4 is_top_reducible_wrt b6,b2;
:: POLYRED: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 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
for b5 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 holds
b4 is_reducible_wrt b5,b2
iff
ex b6 being natural-valued finite-support ManySortedSet of b1 st
b6 in Support b4 & HT(b5,b2) divides b6;
:: POLYRED: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 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
0_(b1,b3) is_irreducible_wrt b4,b2;
:: POLYRED: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 non degenerated 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
for b6 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b3
st b4 reduces_to b4 - (b6 *' b5),b5,b2
holds HT(b6 *' b5,b2) in Support b4;
:: POLYRED: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 degenerated 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
for b7 being natural-valued finite-support ManySortedSet of b1
st b4 reduces_to b6,b5,b7,b2
holds not b7 in Support b6;
:: POLYRED: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 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
for b7, b8 being natural-valued finite-support ManySortedSet of b1
st b7 < b8,b2 & b4 reduces_to b6,b5,b7,b2
holds b8 in Support b6
iff
b8 in Support b4;
:: POLYRED: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 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
for b7, b8 being natural-valued finite-support ManySortedSet of b1
st b7 < b8,b2 & b4 reduces_to b6,b5,b7,b2
holds b4 . b8 = b6 . b8;
:: POLYRED: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 non degenerated 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
for b7 being natural-valued finite-support ManySortedSet of b1
st b7 in Support b6
holds b7 <= HT(b4,b2),b2;
:: POLYRED: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, 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 b6 < b4,b2;
:: POLYRED:funcnot 3 => POLYRED: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 be Element of bool the carrier of Polynom-Ring(a1,a3);
func PolyRedRel(A4,A2) -> Relation of (the carrier of Polynom-Ring(a1,a3)) \ {0_(a1,a3)},the carrier of Polynom-Ring(a1,a3) means
for b1, b2 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 holds
[b1,b2] in it
iff
b1 reduces_to b2,a4,a2;
end;
:: POLYRED:def 13
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)
for b5 being Relation of (the carrier of Polynom-Ring(b1,b3)) \ {0_(b1,b3)},the carrier of Polynom-Ring(b1,b3) holds
b5 = PolyRedRel(b4,b2)
iff
for b6, b7 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
[b6,b7] in b5
iff
b6 reduces_to b7,b4,b2;
:: POLYRED: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 empty non degenerated 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
for b6 being Element of bool the carrier of Polynom-Ring(b1,b3)
st PolyRedRel(b6,b2) reduces b4,b5
holds b5 <= b4,b2 &
(b5 = 0_(b1,b3) or HT(b5,b2) <= HT(b4,b2),b2);
:: POLYRED:funcreg 5
registration
let a1 be natural set;
let a2 be reflexive antisymmetric connected transitive total admissible Relation of Bags a1,Bags a1;
let a3 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
let a4 be Element of bool the carrier of Polynom-Ring(a1,a3);
cluster PolyRedRel(a4,a2) -> strongly-normalizing;
end;
:: POLYRED:th 45
theorem
for b1 being natural 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 left_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
holds PolyRedRel(b4,b2) reduces b6 *' b5,0_(b1,b3);
:: POLYRED: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 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, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b7 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b3
st b5 reduces_to b6,b4,b2
holds b7 *' b5 reduces_to b7 *' b6,b4,b2;
:: POLYRED: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 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, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b7 being Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b3
st PolyRedRel(b4,b2) reduces b5,b6
holds PolyRedRel(b4,b2) reduces b7 *' b5,b7 *' b6;
:: POLYRED: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 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 Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b6 being Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b3
st PolyRedRel(b4,b2) reduces b5,0_(b1,b3)
holds PolyRedRel(b4,b2) reduces b6 *' b5,0_(b1,b3);
:: POLYRED:th 49
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, b7, b8 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st b5 - b6 = b7 & PolyRedRel(b4,b2) reduces b7,b8
holds ex b9, b10 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 st
b9 - b10 = b8 & PolyRedRel(b4,b2) reduces b5,b9 & PolyRedRel(b4,b2) reduces b6,b10;
:: POLYRED:th 50
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 PolyRedRel(b4,b2) reduces b5 - b6,0_(b1,b3)
holds b5,b6 are_convergent_wrt PolyRedRel(b4,b2);
:: POLYRED:th 51
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 PolyRedRel(b4,b2) reduces b5 - b6,0_(b1,b3)
holds b5,b6 are_convertible_wrt PolyRedRel(b4,b2);
:: POLYRED:prednot 15 => POLYRED:pred 11
definition
let a1 be non empty addLoopStr;
let a2 be Element of bool the carrier of a1;
let a3, a4 be Element of the carrier of a1;
pred A3,A4 are_congruent_mod A2 means
a3 - a4 in a2;
end;
:: POLYRED:dfs 14
definiens
let a1 be non empty addLoopStr;
let a2 be Element of bool the carrier of a1;
let a3, a4 be Element of the carrier of a1;
To prove
a3,a4 are_congruent_mod a2
it is sufficient to prove
thus a3 - a4 in a2;
:: POLYRED:def 14
theorem
for b1 being non empty addLoopStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1 holds
b3,b4 are_congruent_mod b2
iff
b3 - b4 in b2;
:: POLYRED:th 52
theorem
for b1 being non empty right_complementable right-distributive add-associative right_zeroed left_zeroed doubleLoopStr
for b2 being non empty right-ideal Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
b3,b3 are_congruent_mod b2;
:: POLYRED:th 53
theorem
for b1 being non empty right_complementable right-distributive well-unital add-associative right_zeroed doubleLoopStr
for b2 being non empty right-ideal Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1
st b3,b4 are_congruent_mod b2
holds b4,b3 are_congruent_mod b2;
:: POLYRED:th 54
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being non empty add-closed Element of bool the carrier of b1
for b3, b4, b5 being Element of the carrier of b1
st b3,b4 are_congruent_mod b2 & b4,b5 are_congruent_mod b2
holds b3,b5 are_congruent_mod b2;
:: POLYRED:th 55
theorem
for b1 being non trivial right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty add-closed Element of bool the carrier of b1
for b3, b4, b5, b6 being Element of the carrier of b1
st b3,b4 are_congruent_mod b2 & b5,b6 are_congruent_mod b2
holds b3 + b5,b4 + b6 are_congruent_mod b2;
:: POLYRED:th 56
theorem
for b1 being non empty right_complementable commutative distributive add-associative right_zeroed doubleLoopStr
for b2 being non empty add-closed right-ideal Element of bool the carrier of b1
for b3, b4, b5, b6 being Element of the carrier of b1
st b3,b4 are_congruent_mod b2 & b5,b6 are_congruent_mod b2
holds b3 * b5,b4 * b6 are_congruent_mod b2;
:: POLYRED:th 57
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 Element of the carrier of Polynom-Ring(b1,b3)
st b5,b6 are_convertible_wrt PolyRedRel(b4,b2)
holds b5,b6 are_congruent_mod b4 -Ideal;
:: POLYRED:th 58
theorem
for b1 being natural 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 non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b5, b6 being Element of the carrier of Polynom-Ring(b1,b3)
st b5,b6 are_congruent_mod b4 -Ideal
holds b5,b6 are_convertible_wrt PolyRedRel(b4,b2);
:: POLYRED:th 59
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 PolyRedRel(b4,b2) reduces b5,b6
holds b5 - b6 in b4 -Ideal;
:: POLYRED:th 60
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 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st PolyRedRel(b4,b2) reduces b5,0_(b1,b3)
holds b5 in b4 -Ideal;