# Curry

Curry is being phased out in favor of CurryApplied and OperatorApplied, which were introduced experimentally in Version 12.1.

Curry[f,n]

represents an operator form of the function f of n arguments so that Curry[f,n][x1][xn] is equivalent to f[x1,,xn].

Curry[f]

represents an operator form of the function f of two arguments so that Curry[f][y][x] is equivalent to f[x,y].

Curry[f,{i1,,in}]

represents an operator form of the function f of n arguments so that Curry[f,{i1,,in}][x1][xn] is equivalent to f[xi1,,xin].

Curry[f,k{i1,,in}]

represents a k-arguments operator form of the function f of n arguments so that Curry[f,k{i1,,in}][x1][xk] is equivalent to f[xi1,,xin], with kMax[{i1,,in}].

# Details • Curry[f] is equivalent to Curry[f,{2,1}].
• Curry[f,n] is equivalent to Curry[f,{1,2,,n}].
• Curry[f,{i1,,in}] is equivalent to Curry[f,Max[{i1,,in}]->{i1,,in}].
• Curry[f,{i1,,in,opts}][x1][xk] is equivalent to f[xi1,,xin,opts] for a sequence opts of options.
• The ip curried argument of Curry[f,{i1,,in}] will be the p argument of f.
• Curry[f,arity][x1,][y1,][z1,] is equivalent to Curry[f,arity][x1,,y1,,z1,], so that the structure of brackets is not relevant, only the number of arguments.

# Examples

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## Basic Examples(3)

Curry the second argument of a function of two arguments:

Curry a function of three arguments, keeping their order:

This is a curried form of Integrate that curries two integration variables:

Apply it to a function of variables and :

That is equivalent to:

## Scope(6)

Curry the second argument of a function:

Apply the operator:

Curry a function of 3 arguments, keeping their order:

Curry a function of 3 arguments, applying a permutation before they are passed to the function:

These are curried operators taking 4 arguments, but only 2 of them are passed to the function f:

Use arguments of the curried operator with any bracketing structure:

Curry Level with default option values:

Pass options to Level:

## Applications(5)

Curry Composition of 3 functions:

Feed the 3 functions sequentially:

Apply the composition to an expression:

Specify how many arguments are functions to be composed:

Use Curry to construct the opposite order of a given ordering function:

Build an array of subscripted variables:

Build the K and S combinators using Curry:

The combinations SKK and SKS are equivalent to the identity:

Build the B and C combinators using S and K:

## Properties & Relations(6)

Curry[f] is equivalent to Curry[f,{2,1}]:

For a function of zero arguments, Curry[f,0] returns f[]:

If additional arguments are provided, the empty pair of brackets is still inserted:

Curry Curry itself:

Compare to Construct:

For positive n, Curry[Construct,n][f] is equivalent to Curry[f,n-1]:

The relation also holds for n=1:

Compose two Curry operators with a permutation and its inverse:

The result is equivalent to using Curry without reordering the arguments:

Take two permutation lists of the same length:

Compose the corresponding Curry operators:

Alternatively, use Curry with their permutation product, in the same order: