# CurryApplied

CurryApplied[f,n]

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

CurryApplied[n]

represents an operator form of CurryApplied that can be applied to a function to represent an operator form with n arguments.

CurryApplied[f,{i1,,in}]

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

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

represents an operator form that takes k arguments.

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

# Examples

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## Basic Examples(3)

Curry a function of two arguments:

Use the operator form of CurryApplied to do the same thing:

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(7)

Curry the first argument of a function of two arguments:

Apply the function to any date object:

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 function with any bracketing structure:

Curry Level with default option values:

Pass options to Level:

## Applications(4)

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:

Build an array of subscripted variables:

Build the K and S combinators using CurryApplied:

The combinations SKK and SKS are equivalent to the identity:

Build the B and C combinators using S and K:

## Properties & Relations(7)

CurryApplied[f,arity] represents the same operator as OperatorApplied[f,arity]:

CurryApplied[n][f] is equivalent to CurryApplied[f,n]:

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

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

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

Curry CurryApplied itself:

Compare to Construct:

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

The relation also holds for n=1:

Compose two CurryApplied operators with a permutation and its inverse:

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

Take two permutation lists of the same length:

Compose the corresponding CurryApplied operators:

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