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copied!<p>The first step is to turn your top-level set of equations for <code>foldl</code> into a lambda expression which uses case analysis, like so:</p> <pre><code>val rec foldl = fn f => fn b => fn lst => case lst of [] => b | (h::t) => foldl f (f(h,b)) t </code></pre> <p>Now you can use the same logic as before. Taking as an example the function <code>fn (x, y) => x * y</code>, we can see that</p> <pre><code>val prod = foldl (fn (x, y) => x * y) </code></pre> <p>is equivalent to</p> <pre><code>val prod = (fn f => fn b => fn lst => case lst of [] => b | (h::t) => foldl f (f(h,b)) t) (fn (x, y) => x * y) </code></pre> <p>which <a href="http://en.wikipedia.org/wiki/Beta_reduction#Reduction" rel="nofollow">beta-reduces</a> to</p> <pre><code>val prod = fn b => fn lst => case lst of [] => b | (h::t) => foldl (fn (x, y) => x * y) ((fn (x, y) => x * y)(h,b)) t </code></pre> <p>which beta-reduces to</p> <pre><code>val prod = fn b => fn lst => case lst of [] => b | (h::t) => foldl (fn (x, y) => x * y) (h * b) t </code></pre> <p>Now since we know from our first definition that <code>prod</code> is equivalent to <code>foldl (fn (x, y) => x * y)</code>, we can substitute it into its own definition:</p> <pre><code>val rec prod = fn b => fn lst => case lst of [] => b | (h::t) => prod (h * b) t </code></pre> <p>We can then mentally convert this back to a function defined by equations if we like:</p> <pre><code>fun prod b [] = b | prod b (h::t) = prod (h * b) t </code></pre> <p>That's about what you'd expect, right?</p>

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