simproc | The simple microprocessor | Command Line Interface library

I would start easy and ignore the pi / 2 part for now.

You probably want to build a simproc that matches on anything of the form sin x. Then you want to write a conversion that takes that term x (which is assumed to be a sum of several terms) and brings it into the form a + of_int b * p.

A conversion is essentially a function of type cterm → thm which takes a cterm ct and returns a theorem of the form ct ≡ …, i.e. it's a form of deterministic rewriting (a conversion can also fail by throwing a CTERM exception, by convention). There are a lot of combinators for building and using these in Pure/conv.ML.

This is probably a bit fiddly. You essentially have to descend through the term and, for each atom (i.e. anything not of the form _ + _) you have to figure out whether it can be brought into the form of_int … * pi (e.g. again by writing a conversion that does this transformation – to make it easy you can omit this part so that your procedure only works if the terms are already in that form) and then group all the terms of the form of_int … * pi to the right and all the terms not of that form to the left using associativity and commutativity.

I would suggest this:

• Define a function SIN_SIMPROC_ATOM x n = x + of_int n * pi
• Write a conversion sin_atom_conv that rewrites of_int n * pi to SIN_SIMPROC_ATOM 0 n and everything else into SIN_SIMPROC_ATOM x 0
• Write a conversion that descends through +, applies sin_atom_conv to every atom, and then applies some kind of combination rule like SIN_SIMPROC_ATOM x1 n1 + SIN_SIMPROC_ATOM x2 n2 = SIN_SIMPROC_ATOM (x1 + x2) (n1 + n2)
• In the end, you have rewritten your entire form to the form sin (SIN_SIMPROC_ATOM x n), and then you can apply some suitable rule to that.

It's not quite clear to me how to best handle the parity of n. You could rewrite sin (SIN_SIMPROC_ATOM x n) = (-1) ^ nat ¦n¦ * sin x but I'm not sure if that's what the user really wants in most cases. It might make more sense to only do that if you can deduce the parity of n statically (e.g. by using the simplifier) and then directly simplify to sin x or -sin x.

The situation becomes even more complicated if you want to include halves of π. You can of course extend SIN_SIMPROC_ATOM by a second term for halves of π (and one for doubles of π as well to make it more uniform). Or you could ad all of them together so that you just have a single integer n that describes your multiples of π/2, and k multiples of π simply contribute 2k to that term. And then you have to figure out what n mod 4 is – possibly again with the simplifier or with some clever static method.

In the first example, your simproc is not called becaus simprocs (if I remember correctly) are only called after exhausting all ‘normal’ rewriting, and your goal is immediately rewritten to sin (17 * pi / 2 + x) = cos x. Since your simproc does not match that goal anymore, it is not called.

In the second example, your simproc is called (you can verify this by inserting e.g. a val _ = @{print} "foo" into your let block) and it indeed produces a rewrite rule. Unfortunately, this rewrite rule still has the precondition 8 ≡ 2 * real_of_int 4, which simp cannot solve with the very basic simplifier setup you are using, so it fails to apply the rule.

You can find out what's going on there using the simplifier trace:

That feature is enabled by the splitter. The simplifier itself is a big loop that not only performs rewriting, but also afterwards a variety of other things, before it starts rewriting again.

For if, the relevant split rules are if_splits: