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Apply a function to some but not all arguments

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Here is an example of "non-partial" application of addition in Language:Haskell: 

> 41 + 1
42

In this example, we actually increment 41. Let us thus define the increment function as a partial application of addition so that we can model the same computation as follows: 

> let inc = (+) 1
> inc 41
42

It is important that "+" was surrounded by "(...)" because "+" is an infix operator and we need to use in an prefix manner when aiming at partial application. The notation "(+)" does indeed produce a prefix operator.

In Haskell, sections for infix operators correspond to a special form of partial application. A section applies an infix operator to one of its two operands by using parenthesization in the following way: 

> let inc = (+1)
> inc 41
42

In this example, we have applied "+" to its second operand. For what it matters (and because addition is commutative), we could also define the increment function in terms of a section for the first operand: 

> let inc = (1+)
> inc 41
42

Discussion

In languages with type-level functions such as parametrized type synonyms or data types, e.g., Language:Haskell, partial application makes sense at the type level as well.

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