Article TERMORD, MML version 4.99.1005
:: TERMORD:exreg 1
registration
cluster non trivial addLoopStr;
end;
:: TERMORD:exreg 2
registration
cluster non empty non trivial right_complementable add-associative right_zeroed addLoopStr;
end;
:: TERMORD:attrnot 1 => TERMORD:attr 1
definition
let a1 be set;
let a2 be natural-valued finite-support ManySortedSet of a1;
attr a2 is non-zero means
a2 <> EmptyBag a1;
end;
:: TERMORD:dfs 1
definiens
let a1 be set;
let a2 be natural-valued finite-support ManySortedSet of a1;
To prove
a2 is non-zero
it is sufficient to prove
thus a2 <> EmptyBag a1;
:: TERMORD:def 1
theorem
for b1 being set
for b2 being natural-valued finite-support ManySortedSet of b1 holds
b2 is non-zero(b1)
iff
b2 <> EmptyBag b1;
:: TERMORD:th 1
theorem
for b1 being set
for b2, b3 being natural-valued finite-support ManySortedSet of b1 holds
b2 divides b3
iff
ex b4 being natural-valued finite-support ManySortedSet of b1 st
b3 = b2 + b4;
:: TERMORD:th 2
theorem
for b1 being ordinal set
for b2 being non empty right_complementable well-unital 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);
:: TERMORD:funcreg 1
registration
let a1 be ordinal set;
let a2 be non empty right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
let a3, a4 be Function-like quasi_total monomial-like Relation of Bags a1,the carrier of a2;
cluster a3 *' a4 -> Function-like quasi_total monomial-like;
end;
:: TERMORD:funcreg 2
registration
let a1 be ordinal set;
let a2 be non empty right_complementable distributive add-associative right_zeroed doubleLoopStr;
let a3, a4 be Function-like quasi_total Constant Relation of Bags a1,the carrier of a2;
cluster a3 *' a4 -> Function-like quasi_total Constant;
end;
:: TERMORD:th 3
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 natural-valued finite-support ManySortedSet of b1
for b5, b6 being non-zero Element of the carrier of b2 holds
Monom(b5 * b6,b3 + b4) = (Monom(b5,b3)) *' Monom(b6,b4);
:: TERMORD:th 4
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 Element of the carrier of b2 holds
(b3 * b4) |(b1,b2) = (b3 |(b1,b2)) *' (b4 |(b1,b2));
:: TERMORD:exreg 3
registration
let a1 be ordinal set;
cluster Relation-like reflexive antisymmetric connected transitive total admissible Relation of Bags a1,Bags a1;
end;
:: TERMORD:condreg 1
registration
let a1 be natural set;
cluster reflexive antisymmetric transitive total admissible -> well_founded (Relation of Bags a1,Bags a1);
end;
:: TERMORD:prednot 1 => TERMORD:pred 1
definition
let a1 be ordinal set;
let a2 be reflexive antisymmetric transitive total Relation of Bags a1,Bags a1;
let a3, a4 be natural-valued finite-support ManySortedSet of a1;
pred A3 <= A4,A2 means
[a3,a4] in a2;
end;
:: TERMORD:dfs 2
definiens
let a1 be ordinal set;
let a2 be reflexive antisymmetric transitive total Relation of Bags a1,Bags a1;
let a3, a4 be natural-valued finite-support ManySortedSet of a1;
To prove
a3 <= a4,a2
it is sufficient to prove
thus [a3,a4] in a2;
:: TERMORD:def 2
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric transitive total Relation of Bags b1,Bags b1
for b3, b4 being natural-valued finite-support ManySortedSet of b1 holds
b3 <= b4,b2
iff
[b3,b4] in b2;
:: TERMORD:prednot 2 => TERMORD:pred 2
definition
let a1 be ordinal set;
let a2 be reflexive antisymmetric transitive total Relation of Bags a1,Bags a1;
let a3, a4 be natural-valued finite-support ManySortedSet of a1;
pred A3 < A4,A2 means
a3 <= a4,a2 & a3 <> a4;
end;
:: TERMORD:dfs 3
definiens
let a1 be ordinal set;
let a2 be reflexive antisymmetric transitive total Relation of Bags a1,Bags a1;
let a3, a4 be natural-valued finite-support ManySortedSet of a1;
To prove
a3 < a4,a2
it is sufficient to prove
thus a3 <= a4,a2 & a3 <> a4;
:: TERMORD:def 3
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric transitive total Relation of Bags b1,Bags b1
for b3, b4 being natural-valued finite-support ManySortedSet of b1 holds
b3 < b4,b2
iff
b3 <= b4,b2 & b3 <> b4;
:: TERMORD:funcnot 1 => TERMORD:func 1
definition
let a1 be ordinal set;
let a2 be reflexive antisymmetric transitive total Relation of Bags a1,Bags a1;
let a3, a4 be natural-valued finite-support ManySortedSet of a1;
func min(A3,A4,A2) -> natural-valued finite-support ManySortedSet of a1 equals
a3
if a3 <= a4,a2
otherwise a4;
end;
:: TERMORD:def 4
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric transitive total Relation of Bags b1,Bags b1
for b3, b4 being natural-valued finite-support ManySortedSet of b1 holds
(b3 <= b4,b2 implies min(b3,b4,b2) = b3) & (b3 <= b4,b2 or min(b3,b4,b2) = b4);
:: TERMORD:funcnot 2 => TERMORD:func 2
definition
let a1 be ordinal set;
let a2 be reflexive antisymmetric transitive total Relation of Bags a1,Bags a1;
let a3, a4 be natural-valued finite-support ManySortedSet of a1;
func max(A3,A4,A2) -> natural-valued finite-support ManySortedSet of a1 equals
a3
if a4 <= a3,a2
otherwise a4;
end;
:: TERMORD:def 5
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric transitive total Relation of Bags b1,Bags b1
for b3, b4 being natural-valued finite-support ManySortedSet of b1 holds
(b4 <= b3,b2 implies max(b3,b4,b2) = b3) & (b4 <= b3,b2 or max(b3,b4,b2) = b4);
:: TERMORD:th 5
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3, b4 being natural-valued finite-support ManySortedSet of b1 holds
b3 <= b4,b2
iff
not b4 < b3,b2;
:: TERMORD:th 6
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric transitive total Relation of Bags b1,Bags b1
for b3 being natural-valued finite-support ManySortedSet of b1 holds
b3 <= b3,b2;
:: TERMORD:th 7
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric transitive total Relation of Bags b1,Bags b1
for b3, b4 being natural-valued finite-support ManySortedSet of b1
st b3 <= b4,b2 & b4 <= b3,b2
holds b3 = b4;
:: TERMORD:th 8
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;
:: TERMORD:th 9
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric transitive total admissible Relation of Bags b1,Bags b1
for b3 being natural-valued finite-support ManySortedSet of b1 holds
EmptyBag b1 <= b3,b2;
:: TERMORD:th 10
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric transitive total admissible Relation of Bags b1,Bags b1
for b3, b4 being natural-valued finite-support ManySortedSet of b1
st b3 divides b4
holds b3 <= b4,b2;
:: TERMORD:th 11
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric transitive total Relation of Bags b1,Bags b1
for b3, b4 being natural-valued finite-support ManySortedSet of b1
st not min(b3,b4,b2) = b3
holds min(b3,b4,b2) = b4;
:: TERMORD:th 12
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric transitive total Relation of Bags b1,Bags b1
for b3, b4 being natural-valued finite-support ManySortedSet of b1
st not max(b3,b4,b2) = b3
holds max(b3,b4,b2) = b4;
:: TERMORD:th 13
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3, b4 being natural-valued finite-support ManySortedSet of b1 holds
min(b3,b4,b2) <= b3,b2 & min(b3,b4,b2) <= b4,b2;
:: TERMORD:th 14
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3, b4 being natural-valued finite-support ManySortedSet of b1 holds
b3 <= max(b3,b4,b2),b2 & b4 <= max(b3,b4,b2),b2;
:: TERMORD:th 15
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3, b4 being natural-valued finite-support ManySortedSet of b1 holds
min(b3,b4,b2) = min(b4,b3,b2) & max(b3,b4,b2) = max(b4,b3,b2);
:: TERMORD:th 16
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3, b4 being natural-valued finite-support ManySortedSet of b1 holds
min(b3,b4,b2) = b3
iff
max(b3,b4,b2) = b4;
:: TERMORD:funcnot 3 => TERMORD: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 ZeroStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
func HT(A4,A2) -> Element of Bags a1 means
((Support a4 = {} implies not it = EmptyBag a1)) implies it in Support a4 &
(for b1 being natural-valued finite-support ManySortedSet of a1
st b1 in Support a4
holds b1 <= it,a2);
end;
:: TERMORD: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 empty ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being Element of Bags b1 holds
b5 = HT(b4,b2)
iff
(Support b4 = {} & b5 = EmptyBag b1 or b5 in Support b4 &
(for b6 being natural-valued finite-support ManySortedSet of b1
st b6 in Support b4
holds b6 <= b5,b2));
:: TERMORD:funcnot 4 => TERMORD: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 ZeroStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
func HC(A4,A2) -> Element of the carrier of a3 equals
a4 . HT(a4,a2);
end;
:: TERMORD:def 7
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
HC(b4,b2) = b4 . HT(b4,b2);
:: TERMORD:funcnot 5 => TERMORD: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 ZeroStr;
let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
func HM(A4,A2) -> Function-like quasi_total monomial-like Relation of Bags a1,the carrier of a3 equals
Monom(HC(a4,a2),HT(a4,a2));
end;
:: TERMORD:def 8
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
HM(b4,b2) = Monom(HC(b4,b2),HT(b4,b2));
:: TERMORD: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 ZeroStr;
let a4 be Function-like quasi_total finite-Support non-zero Relation of Bags a1,the carrier of a3;
cluster HM(a4,a2) -> Function-like quasi_total non-zero monomial-like;
end;
:: TERMORD: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 ZeroStr;
let a4 be Function-like quasi_total finite-Support non-zero Relation of Bags a1,the carrier of a3;
cluster HC(a4,a2) -> non-zero;
end;
:: TERMORD: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 finite-Support Relation of Bags b1,the carrier of b3 holds
HC(b4,b2) = 0. b3
iff
b4 = 0_(b1,b3);
:: TERMORD: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 trivial ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
(HM(b4,b2)) . HT(b4,b2) = b4 . HT(b4,b2);
:: TERMORD: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 trivial ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being natural-valued finite-support ManySortedSet of b1
st b5 <> HT(b4,b2)
holds (HM(b4,b2)) . b5 = 0. b3;
:: TERMORD: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 trivial ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
Support HM(b4,b2) c= Support b4;
:: TERMORD: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 ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
st Support HM(b4,b2) <> {}
holds Support HM(b4,b2) = {HT(b4,b2)};
:: TERMORD:th 22
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non trivial ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
term HM(b4,b2) = HT(b4,b2) & coefficient HM(b4,b2) = HC(b4,b2);
:: TERMORD: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 empty ZeroStr
for b4 being Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b3 holds
HT(b4,b2) = term b4 & HC(b4,b2) = coefficient b4 & HM(b4,b2) = b4;
:: TERMORD: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 ZeroStr
for b4 being Function-like quasi_total Constant Relation of Bags b1,the carrier of b3 holds
HT(b4,b2) = EmptyBag b1 & HC(b4,b2) = b4 . EmptyBag b1;
:: TERMORD: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 ZeroStr
for b4 being Element of the carrier of b3 holds
HT(b4 |(b1,b3),b2) = EmptyBag b1 & HC(b4 |(b1,b3),b2) = b4;
:: TERMORD: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 trivial ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
HT(HM(b4,b2),b2) = HT(b4,b2);
:: TERMORD: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 ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
HC(HM(b4,b2),b2) = HC(b4,b2);
:: TERMORD: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 ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
HM(HM(b4,b2),b2) = HM(b4,b2);
:: TERMORD:th 29
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 left_zeroed doubleLoopStr
for b4, b5 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 holds
(HT(b4,b2)) + HT(b5,b2) in Support (b4 *' b5);
:: TERMORD:th 30
theorem
for b1 being ordinal set
for b2 being non empty right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b3, b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2 holds
Support (b3 *' b4) c= {b5 + b6 where b5 is Element of Bags b1, b6 is Element of Bags b1: b5 in Support b3 & b6 in Support b4};
:: TERMORD:th 31
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, b5 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 holds
HT(b4 *' b5,b2) = (HT(b4,b2)) + HT(b5,b2);
:: TERMORD: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 well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for b4, b5 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 holds
HC(b4 *' b5,b2) = (HC(b4,b2)) * HC(b5,b2);
:: TERMORD: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 well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for b4, b5 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 holds
HM(b4 *' b5,b2) = (HM(b4,b2)) *' HM(b5,b2);
:: TERMORD:th 34
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_zeroed addLoopStr
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;
:: TERMORD:funcnot 6 => TERMORD: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;
func Red(A4,A2) -> Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 equals
a4 - HM(a4,a2);
end;
:: TERMORD:def 9
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty 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
Red(b4,b2) = b4 - HM(b4,b2);
:: TERMORD: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 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
Support Red(b4,b2) c= Support b4;
:: TERMORD: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 add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
Support Red(b4,b2) = (Support b4) \ {HT(b4,b2)};
:: TERMORD: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 holds
Support ((HM(b4,b2)) + Red(b4,b2)) = Support b4;
:: TERMORD:th 38
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non 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)) + Red(b4,b2) = b4;
:: TERMORD: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 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
(Red(b4,b2)) . HT(b4,b2) = 0. b3;
:: TERMORD:th 40
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non 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 natural-valued finite-support ManySortedSet of b1
st b5 in Support b4 & b5 <> HT(b4,b2)
holds (Red(b4,b2)) . b5 = b4 . b5;