diff options
Diffstat (limited to 'src/algebra/indexedp.spad.pamphlet')
| -rw-r--r-- | src/algebra/indexedp.spad.pamphlet | 114 |
1 files changed, 42 insertions, 72 deletions
diff --git a/src/algebra/indexedp.spad.pamphlet b/src/algebra/indexedp.spad.pamphlet index dab47a6a..bbb78cd0 100644 --- a/src/algebra/indexedp.spad.pamphlet +++ b/src/algebra/indexedp.spad.pamphlet @@ -26,7 +26,7 @@ ++ respect to an ordered indexing set. IndexedDirectProductCategory(A:BasicType,S:OrderedType): Category == - BasicType with + Join(BasicType,ConvertibleFrom List IndexedProductTerm(A,S)) with if A has SetCategory and S has SetCategory then SetCategory map: (A -> A, %) -> % ++ map(f,z) returns the new element created by applying the @@ -47,7 +47,7 @@ IndexedDirectProductCategory(A:BasicType,S:OrderedType): Category == ++ reductum(z) returns a new element created by removing the ++ leading coefficient/support pair from the element z. ++ Error: if z has no support. - terms: % -> List Pair(S,A) + terms: % -> List IndexedProductTerm(A,S) ++ \spad{terms x} returns the list of terms in \spad{x}. ++ Each term is a pair of a support (the first component) ++ and the corresponding value (the second component). @@ -94,46 +94,25 @@ IndexedDirectProductObject(A,S): Public == Private where A: BasicType S: OrderedType Public == IndexedDirectProductCategory(A,S) - Private == add - Term == Pair(S,A) - Rep == List Term - -- Return the index of a term - termIndex(t: Term): S == first t - -- Return the value of a term - termValue(t: Term): A == second t - - x = y == - x' := rep x - y' := rep y - while not null x' and not null y' repeat - termIndex first x' ~= termIndex first y' => return false - termValue first x' ~= termValue first y' => return false - x' := rest x' - y' := rest y' - null x' and null y' - + Private == List IndexedProductTerm(A,S) add if A has CoercibleTo OutputForm and S has CoercibleTo OutputForm then coerce(x:%):OutputForm == - bracket [rarrow(termIndex(t)::OutputForm, termValue(t)::OutputForm) + bracket [rarrow(index(t)::OutputForm, coefficient(t)::OutputForm) for t in rep x] - -- sample():% == [[sample()$S,sample()$A]$Term]$Rep - - monomial(r,s) == per [[s,r]] - map(f,x) == per [[termIndex tm,f termValue tm] for tm in rep x] - - reductum x == - per rest rep x + monomial(r,s) == per [term(s,r)] + map(f,x) == per [term(index tm,f coefficient tm) for tm in rep x] + reductum x == per rest rep x leadingCoefficient x == null rep x => error "Can't take leadingCoefficient of empty product element" - termValue first rep x + coefficient first rep x leadingSupport x == null rep x => error "Can't take leadingCoefficient of empty product element" - termIndex first rep x - + index first rep x terms x == rep x + convert l == per l @ \section{domain IDPAM IndexedDirectProductAbelianMonoid} @@ -145,18 +124,14 @@ IndexedDirectProductObject(A,S): Public == Private where IndexedDirectProductAbelianMonoid(A:AbelianMonoid,S:OrderedType): Join(AbelianMonoid,IndexedDirectProductCategory(A,S)) == IndexedDirectProductObject(A,S) add - --representations - Term == Pair(S,A) + Term == IndexedProductTerm(A,S) import Term - import List Term - termIndex(t: Term): S == first t - termValue(t: Term): A == second t r: A n: NonNegativeInteger f: A -> A s: S - 0 == nil$List(Term) pretend % + 0 == convert nil$List(Term) zero? x == null terms x import %tail: List Term -> List Term from Foreign Builtin @@ -178,17 +153,17 @@ IndexedDirectProductAbelianMonoid(A:AbelianMonoid,S:OrderedType): res: List Term := nil while not empty? x' and not empty? y' repeat newcell: List Term := nil - if termIndex x'.first = termIndex y'.first then - r := termValue x'.first + termValue y'.first + if index x'.first = index y'.first then + r := coefficient x'.first + coefficient y'.first if not zero? r then - newcell := cons([termIndex x'.first, r], empty()) + newcell := [term(index x'.first, r)] x' := rest x' y' := rest y' - else if termIndex x'.first > termIndex y'.first then - newcell := cons(x'.first, empty()) + else if index x'.first > index y'.first then + newcell := [x'.first] x' := rest x' else - newcell := cons(y'.first, empty()) + newcell := [y'.first] y' := rest y' if not empty? newcell then if not empty? endcell then @@ -202,32 +177,29 @@ IndexedDirectProductAbelianMonoid(A:AbelianMonoid,S:OrderedType): x' if empty? res then res := end else qsetrest!(endcell, end) - res pretend % + convert res n * x == - n = 0 => 0 - n = 1 => x - [[termIndex u,a] for u in terms x - | not zero?(a:=n * termValue u)] pretend % + zero? n => 0 + one? n => x + convert [term(index u,a) for u in terms x + | not zero?(a:=n * coefficient u)] monomial(r,s) == zero? r => 0 - [[s,r]] pretend % + convert [term(s,r)] map(f,x) == - [[termIndex tm,a] for tm in terms x - | not zero?(a:=f termValue tm)] pretend % + convert [term(index tm,a) for tm in terms x + | not zero?(a:=f coefficient tm)] reductum x == null terms x => 0 - rest(terms x) pretend % + convert rest(terms x) leadingCoefficient x == null terms x => 0 - termValue terms(x).first - - pair2Term(t: Pair(A,S)): Term == - [second t, first t] + coefficient terms(x).first @ \section{domain IDPOAM IndexedDirectProductOrderedAbelianMonoid} @@ -304,16 +276,14 @@ IndexedDirectProductAbelianGroup(A:AbelianGroup,S:OrderedType): Join(AbelianGroup,IndexedDirectProductCategory(A,S)) == IndexedDirectProductAbelianMonoid(A,S) add --representations - Term == Pair(S,A) - termIndex(t: Term): S == first t - termValue(t: Term): A == second t + Term == IndexedProductTerm(A,S) - -x == [[termIndex u,-termValue u] for u in terms x] pretend % + -x == convert [term(index u,-coefficient u) for u in terms x] n:Integer * x:% == - n = 0 => 0 - n = 1 => x - [[termIndex u,a] for u in terms x - | not zero?(a := n * termValue u)] pretend % + zero? n => 0 + one? n => x + convert [term(index u,a) for u in terms x + | not zero?(a := n * coefficient u)] import %tail: List Term -> List Term from Foreign Builtin @@ -330,17 +300,17 @@ IndexedDirectProductAbelianGroup(A:AbelianGroup,S:OrderedType): res: List Term := nil while not empty? x' and not empty? y' repeat newcell: List Term := nil - if termIndex x'.first = termIndex y'.first then - r := termValue x'.first - termValue y'.first + if index x'.first = index y'.first then + r := coefficient x'.first - coefficient y'.first if not zero? r then - newcell := cons([termIndex x'.first, r], empty()) + newcell := [term(index x'.first, r)] x' := rest x' y' := rest y' - else if termIndex x'.first > termIndex y'.first then - newcell := cons(x'.first, empty()) + else if index x'.first > index y'.first then + newcell := [x'.first] x' := rest x' else - newcell := cons([termIndex y'.first,-termValue y'.first], empty()) + newcell := [term(index y'.first,-coefficient y'.first)] y' := rest y' if not empty? newcell then if not empty? endcell then @@ -350,11 +320,11 @@ IndexedDirectProductAbelianGroup(A:AbelianGroup,S:OrderedType): res := newcell; endcell := res end := - empty? x' => terms(-(y' pretend %)) + empty? x' => terms(-convert y') x' if empty? res then res := end else qsetrest!(endcell, end) - res pretend % + convert res @ \section{License} |