Lambda calculus definition - Wikipedia - Recent changes [en]

If no variable names are [[#Free and Bound Variable Sets|free]] in the actual parameter and [[#Free and Bound Variable Sets|bound]] in the body, β-reduction may be performed on the lambda abstraction without canonical renaming.

If no variable names are [[#Free and Bound Variable Sets|free]] in the actual parameter and [[#Free and Bound Variable Sets|bound]] in the body, β-reduction may be performed on the lambda abstraction without canonical renaming.

: <math>(\forall z: z \not \in FV(y) \lor z \not \in BV(b)) \to \operatorname{beta-redex}[(\lambda x.b) \ y] = b[x:=y] </math>

: <math>FV(y) \cap BV(b) = \{\} \to \operatorname{beta-redex}[(\lambda x.b) \ y] = b[x:=y] </math>

Alpha renaming may be used on <math>b</math> to rename names that are free in <math>y</math> but bound in <math>b</math>, to meet the pre-condition for this transformation.

Alpha renaming may be used on <math>b</math> to rename names that are free in <math>y</math> but bound in <math>b</math>, to meet the pre-condition for this transformation.