fg or Application[f,g]

represents the formal application of f to g.



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Basic Examples  (2)

Represent a combinatory term:

Apply the standard reduction rules of combinatory logic:

Applications  (1)

Prove an identity among combinators:

Properties & Relations  (3)

Convert from a formal application to an application in the Wolfram Language:

Alternatively, replace the operator:

Convert from a Wolfram Language application to a formal application:

Get the reduction rules for some notable combinators:

Apply the rules to combinatory terms:

Possible Issues  (1)

Many interesting combinatory terms do not have a normal form, so they keep changing forever:

This term leads to a cycle of length 2:

Neat Examples  (2)

The reduction rule for the Turing combinator:

Prove that is a fixed-point combinator:

Eliminate variables to convert arbitrary terms into combinator form:

Find the combinator that doubles its argument:

Find the combinator for a function application:

Verify the result by applying the standard combinatory reductions for TemplateBox[{}, CombinatorS] and TemplateBox[{}, CombinatorK]:

Wolfram Research (2020), Application, Wolfram Language function,


Wolfram Research (2020), Application, Wolfram Language function,


Wolfram Language. 2020. "Application." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2020). Application. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_application, author="Wolfram Research", title="{Application}", year="2020", howpublished="\url{}", note=[Accessed: 19-June-2024 ]}


@online{reference.wolfram_2024_application, organization={Wolfram Research}, title={Application}, year={2020}, url={}, note=[Accessed: 19-June-2024 ]}