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These types are identical, but they implement completely different views of the same data. The definitions of addViaRect and mulViaPolar can also be written using pattern matching in argument position: let add2 (Rect(ar,ai)) (Rect(br,bi)) = Complex.mkRect(ar+br, ai+bi) let mul2 (Polar(r1,th1)) (Polar(r2,th2)) = Complex.mkPolar(r1*r2, th1+th2)
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Note The Microsoft.FSharp.Math.Complex type defines addition and multiplication operators
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accessible via the overloaded operators + and *. The redefinitions in this section are included primarily to show how you can use different active patterns to access the same type in different ways.
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Matching on .NET Object Types
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One of the useful things about active patterns is that they let you use pattern matching with existing .NET object types. For example, the .NET object type System.Type is a runtime representation of types in .NET and F#. Here are the members found on this type: type System.Type with member IsGenericType : bool member GetGenericTypeDefinition : unit -> Type member GetGenericArguments : unit -> Type[] member HasElementType : bool member GetElementType : unit -> Type member IsByRef : bool member IsPointer : bool member IsGenericParameter : bool member GenericParameterPosition : int This type looks very much like one you d like to pattern match against. There are clearly three or four distinct cases here, and pattern matching will help you isolate them. You can define an active pattern to achieve this, as shown in Listing 9-4. Listing 9-4. Defining an Active Pattern for Matching on System.Type Values let (|Named|Array|Ptr|Param|) (typ : System.Type) = if typ.IsGenericType then Named(typ.GetGenericTypeDefinition(),typ.GetGenericArguments()) elif typ.IsGenericParameter then Param(typ.GenericParameterPosition) elif not typ.HasElementType then Named(typ, [| |]) elif typ.IsArray then Array(typ.GetElementType(),typ.GetArrayRank()) elif typ.IsByRef then Ptr(true,typ.GetElementType()) elif typ.IsPointer then Ptr(false,typ.GetElementType()) else failwith "MSDN says this can't happen"
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This then lets you use pattern matching against a value of this type: open System let rec formatType typ = match typ with | Named (con, [| |]) -> sprintf "%s" con.Name | Named (con, args) -> sprintf "%s<%s>" con.Name (formatTypes args) | Array (arg, rank) -> sprintf "Array(%d,%s)" rank (formatType arg) | Ptr(true,arg) -> sprintf "%s&" (formatType arg) | Ptr(false,arg) -> sprintf "%s*" (formatType arg) | Param(pos) -> sprintf "!%d" pos and formatTypes typs = String.Join(",", Array.map formatType typs) or collect the free generic type variables: let rec freeVarsAcc typ acc = match typ with | Array (arg, rank) -> freeVarsAcc arg acc | Ptr (_,arg) -> freeVarsAcc arg acc | Param _ -> (typ :: acc) | Named (con, args) -> Array.fold_right freeVarsAcc args acc let freeVars typ = freeVarsAcc typ []
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Defining Partial and Parameterized Active Patterns
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Active patterns can also be partial. You can recognize a partial pattern by a name such as (|MulThree|_|) and by the fact that it returns a value of type 'a option for some 'a. For example: let (|MulThree|_|) inp = if inp % 3 = 0 then Some(inp/3) else None let (|MulSeven|_|) inp = if inp % 7 = 0 then Some(inp/7) else None Finally, active patterns can also be parameterized. You can recognize a parameterized active pattern by the fact that it takes several arguments. For example: let (|MulN|_|) n inp = if inp % n = 0 then Some(inp/n) else None The F# quotation API Microsoft.FSharp.Quotations uses both parameterized and partial patterns extensively.
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Hiding Abstract Syntax Implementations with Active Patterns
Earlier in this chapter you saw the following type that defines an optimized representation of propositional logic terms using a unique stamp for each syntactically unique term: type Prop = Prop of int and internal PropRepr = | AndRepr of Prop * Prop | OrRepr of Prop * Prop | NotRepr of Prop
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| VarRepr of string | TrueRepr However, what happens if you want to pattern match against values of type Prop Even if you exposed the representation, then all you would get is an integer, which you would have to look up in an internal table. You can define an active pattern for restoring matching on that data structure, as shown in Listing 9-5. Listing 9-5. Extending Listing 9-3 with an Active Pattern for the Optimized Representation module PropOps = ... let (|And|Or|Not|Var|True|) prop = match table.FromUnique(prop) with | AndRepr(x,y) -> And(x,y) | OrRepr(x,y) -> Or(x,y) | NotRepr(x) -> Not(x) | VarRepr(v) -> Var(v) | TrueRepr -> True This code defines an active pattern in the auxiliary module PropOps that lets you pattern match against Prop values, despite that they are using optimized unique-integer references under the hood. For example, you can define a pretty-printer for Prop terms as follows, despite that they are using optimized representations: open PropOps let rec showProp prec prop = let parenIfPrec lim s = if prec < lim then "(" match prop with | Or(p1,p2) -> parenIfPrec 4 (showProp 4 p1 + | And(p1,p2) -> parenIfPrec 3 (showProp 3 p1 + | Not(p) -> parenIfPrec 2 ("not "+showProp | Var(v) -> v | True -> "T"
+ s + ")" else s " || " + showProp 4 p2) " && " + showProp 3 p2) 1 p)
Likewise, you can define functions to place the representation in various normal forms. For example, the following function computes negation normal form (NNF), where all instances of NOT nodes have been pushed to the leaves of the representation: let rec nnf sign prop = match prop with | And(p1,p2) -> if sign then And(nnf sign p1, nnf sign p2) else Or(nnf sign p1, nnf sign p2) | Or(p1,p2) -> if sign then Or(nnf sign p1, nnf sign p2) else And(nnf sign p1, nnf sign p2) | Not(p) -> nnf (not sign) p | Var(_) | True -> if sign then prop else Not(prop) let NNF prop = nnf true prop
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