custom notation - Going full functional (Haskell style)

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 as h[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.