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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.

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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 $x$: $$(\lambda z. (y(\lambda x.xz)))$$

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Thanks. The process would be the same whether we do it in normal or applicative order, right? –  John Roberts Apr 17 '13 at 0:25
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Yes, that shouldn't matter. –  Abel Apr 17 '13 at 0:26
    
Could you check this one out too? math.stackexchange.com/questions/363890/… –  John Roberts Apr 17 '13 at 0:39

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