Lambda Calculus Expression Evaluation

Question

I am looking at the following lambda calculus expression: (λx.(λy.(x(λx.xy))))y. Could somebody help me to evaluate it? I am guessing that the first step would be to pass the outermost y into the outer most function λx, but I am unsure where to go from there.

Answer 1

First we have to rename the bound $y$ to $z$ to avoid capture: $$(\lambda x.(\lambda z. (x(\lambda x.xz))))y$$ Then we may substitute $y$ for any bound occurence of $x$: $$(\lambda z. (y(\lambda x.xz)))$$