The `x + 1` that is evaluated to give the value that `y` gets bound to uses the (more local) binding of `x` to `5`, not the (previous, less local) binding of `x` to `0`. By the way, the parentheses in that displayed expression were just to focus your attention. It would have parsed and meant the same without them.
-Now we can allow ourselves to introduce λ-expressions in the following way. If a λ-expression is applied to an argument, as in: `(`λ `x.` φ`) M`, for any (simple or complex) expressions φ and `M`, this means the same as: `let x be M in` φ. That is, the argument to the λ-expression provides (when evaluated) a value for the variable `x` to be bound to, and then the result of the whole thing is whatever φ evaluates to, under that binding to `x`.
+Now we can allow ourselves to introduce λ-expressions in the following way. If a λ-expression is applied to an argument, as in: `(`λ `x.` φ`) M`, for any (simple or complex) expressions φ and `M`, this means the same as: `let x be M in` φ. That is, the argument `M` to the λ-expression provides (when evaluated) a value for the variable `x` to be bound to, and then the result of the whole thing is whatever φ evaluates to, under that binding to `x`.
If we restricted ourselves to only that usage of λ-expressions, that is when they were applied to all the arguments they're expecting, then we wouldn't have moved very far from the decidable fragment of arithmetic we began with.
For the time being, these are the only patterns we'll allow. But since the definition of patterns is recursive, this permits very complex patterns. What would this evaluate to:
let
- ([x, y], [z:zs, w]) match ([[], 'true], [[10, 20, 30], 'false])
- in (z, y)
+ ([xs, ys], [z:zs, ws]) match ([[], [1]], [[10, 20, 30], [0]])
+ in z & ys
Also, we will permit complex patterns in λ-expressions, too. So you can write:
`factorial match` λ `n. case n of 0 then 1; _ then n * factorial (n - 1) end`
`in factorial`
-But there's a problem here. What value does `factorial` have when evaluating the expression `factorial (n - 1)`?
+But there's a problem here. What value does `factorial` have when evaluating the subexpression `factorial (n - 1)`?
As we said in class, the natural precedent for this with non-function variables would go something like this:
`length match` λ `xs. case xs of [] then 0; _:ys then 1 + length ys end`
`in length`
-This function accept a sequence `xs`, and if its empty returns `0`, else it says that its length is `1` plus whatever is the length of its remainder when you take away the first element. In programming circles, this remainder is commonly called the sequence's "tail" (and the first element is its "head").
+This function accept a sequence `xs`, and if it's empty returns `0`, else it says that its length is `1` plus whatever is the length of its remainder when you take away the first element. In programming circles, this remainder is commonly called the sequence's "tail" (and the first element is its "head").
Thus if we evaluated `length [10, 20, 30]`, that would give the same result as `1 + length [20, 30]`, which would give the same result as `1 + (1 + length [30])`, which would give the same result as `1 + (1 + (1 + length []))`. But `length []` is `0`, so our original expression evaluates to `1 + (1 + (1 + 0))`, or `3`.