I'm trying to define some notation so that Mathematica code would be more functional, similar to Haskell (just for fun): currying, lambdas, infix operator to function conversion, etc.. And I have some questions about it:
- Is it possible to make all Mathematica
h_[x1_,x2_,...]
functions to work ash[x1][x2][..]
? - Can I distinguish inside
Notation
box between<+1>
and<1+>
, how do I check for + there? - How to define right-associate apply operator with highest precedence (
$
)?
This is what I have so far:
<< Notation`;
lapply[x_, y_] := x[y]
rapply[x_, y_] := x[y]
InfixNotation[ParsedBoxWrapper["\\"], lapply]
InfixNotation[ParsedBoxWrapper["$"], rapply]
x_ \ y_ \ z_ := x[y][z]
x_ $ y_ $ z_ := x[y[z]]
Notation[ParsedBoxWrapper[
RowBox[{
RowBox[{"λ", " ", "x__"}], "->",
"y_"}]] ⟺ ParsedBoxWrapper[
RowBox[{"Function", "[",
RowBox[{
RowBox[{"{", "x__", "}"}], ",", "y_"}], "]"}]]]
Notation[ParsedBoxWrapper[
RowBox[{"〈",
RowBox[{"op_", " ", "x_"}],
"〉"}]] ⟺
ParsedBoxWrapper[
RowBox[{
RowBox[{"#", "op_", " ", "x_"}], "&"}]]]
AddInputAlias["f" -> ParsedBoxWrapper[
RowBox[{"〈",
RowBox[{"\[Placeholder]", "\[Placeholder]"}],
"〉"}]]]
Notation[ParsedBoxWrapper[
RowBox[{"{",
RowBox[{
RowBox[{"x_", " ", ".."}], " ", "y_"}],
"}"}]] ⟺ ParsedBoxWrapper[
RowBox[{"Range", "[",
RowBox[{"x_", ",", "y_"}], "]"}]]]
filter[f_][x_List] := Select[x, f]
map[f_][x__List] := Map[f, x]
filter\PrimeQ $ map\〈-1〉 $ map\ \
(λ x -> 2^x)\ {1 .. 100}
EDIT: Also did some kinda lazy lists, soon it will be haskell inside Mathematica :)
SetAttributes[list, HoldAll]
list[h_, l_][x_] := list[h, l[x]]
list[x_] := list[x, list]
map[f_][list] := list
map[f_][list[x_, xs_]] := list[f[x], map[f][xs]]
take[0][_] := list
take[_][list] := list
take[n_Integer][list[x_, xs_]] := list[x, take[n - 1][xs]]
range[n_Integer] := range[1, n]
range[m_, n_] := list[m, range[m + 1, n]]
range[n_, n_] := list[n, list]
show[list] := "[]"
show[list[x_, l_]] := ToString[x] <> "," <> show[l]
show $ (take[10] $ map\ (λ x -> x^2) $ range[10000])
Answer
That's how I finally defined haskell operators:
rapply[x_] := x
rapply[x_, y__] := x[rapply[y]]
InfixNotation[ParsedBoxWrapper["|"], rapply]
lapply[x_] := x
lapply[x__, y_] := lapply[x][y]
InfixNotation[ParsedBoxWrapper["∘"], lapply]
InfixNotation[ParsedBoxWrapper["·"], Composition]
Now $\circ$, $\dot{}{}$ and | act exactly like haskell's space, . and $ respectfully. Also if we have only single left application then @
is still helpful and it can be hidden with escape characters :@:
.
And beautiful code like $\bf{show\cdot take\ 10\cdot map\ (\lambda\ x\to x{}^{\wedge}2)\cdot range | \infty }$ is possible. There are invisible @
's between take and 10, map and ($\lambda\ x\to x{}^{\wedge}2)$. In haskell the same would look like $\bf{show . take\ 10.map(\backslash x->x{}^{\wedge}2)$[1..]}$.
With double left application, $\circ$ is necessary: $\bf{map\circ(\lambda\ x\to x+1)\circ \{1,2,3\}}$
UPDATE: I made this TextCell
hack to make partial infix operators:
infix[f_String] := Block[{x,y},Head[ToExpression["x" <> f <> "y"]]]
〈TextCell[s_][x_]〉 := infix[s][#, x] &
〈x_[TextCell[s_]]〉 := infix[s][x, #] &
We again can use invisible @
for application. Aliases can be made with TextCell
on the left and on the right within AngleBrackets
to enter them conveniently. Now stuff like <~Mod~10>, <2^>, <^3> also works.
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