How do you selectively simplify arguments to each time a function is called, without evaluting the function itself?
I'm using Coq 8.5pl1. To make a contrived but illustrative example, (* fix so simpl will automatically unfold. *) Definition double := fix f n := 2*n. Theorem contrived n : double (2 + n) = 2 + double (1 + n). Now, I only want to simplify the arguments to double, and not any part outside of it. (For example, because the rest has already carefully been put into the correct form.) simpl. S (S (n + S (S (n + 0)))) = S (S (S (n + S (n + 0)))) This converted the outside (2 + ...) to (S (S ...)) as well as unfolding double. I can match one of them by doing: match goal with | |- (double ?A) = _ => simpl A end. double (S (S n)) = 2 + double (1 + n) How do I say that I want to simplify all of them? That is, I want to get double (S (S n)) = 2 + double (S n) without having to put a separate pattern for each call to double. I can simplify except for double itself with remember double as x; simpl; subst x. double (S (S n)) = S (S (double (S n)))
Opaque/Transparent approach Combining the results of this answer and this one, we get the following solution: Opaque double. simpl (double _). Transparent double. We use the pattern capability of simpl to narrow down its "action domain", and Opaque/Transparent to forbid (allow resp.) unfolding of double. Custom tactic approach We can also define a custom tactic for simplifying arguments: (* simplifies the first argument of a function *) Ltac simpl_arg_of function := repeat multimatch goal with | |- context [function ?A] => let A' := eval cbn -[function] in A in change A with A' end. That let A' := ... construction is needed to protect nested functions from simplification. Here is a simple test: Fact test n : 82 + double (2 + n) = double (1 + double (1 + 20)) + double (1 * n). Proof. simpl_arg_of double. The above results in 82 + double (S (S n)) = double (S (double 21)) + double (n + 0) Had we used something like context [function ?A] => simpl A in the definition of simpl_arg_of, we would've gotten 82 + double (S (S n)) = double 43 + double (n + 0) instead. Arguments directive approach As suggested by #eponier in comments, we can take advantage of yet another form of simpl -- simpl <qualid>, which the manual (sect. 8.7.4) defines as: This applies simpl only to the applicative subterms whose head occurrence is the unfoldable constant qualid (the constant can be referred to by its notation using string if such a notation exists). The Opaque/Transparent approach doesn't work with it, however we can block unfolding of double using the Arguments directive: Arguments double : simpl never. simpl double.
You may find the ssreflect pattern selection language and rewrite mechanism useful here. For instance, you can have a finer grained control with patterns + the simpl operator /=: From mathcomp Require Import ssreflect. Definition double := fix f n := 2*n. Theorem contrived n : double (2 + n) = 2 + double (1 + n). rewrite ![_+n]/=. Will display: n : nat ============================ double (S (S n)) = 2 + double (S n) You can also use anonymous rewrite rules: rewrite (_ : double (2+n) = 2 + double (1+n)) //. I would personally factor the rewrite in smaller lemmas: Lemma doubleE n : double n = n + n. Proof. by elim: n => //= n ihn; rewrite -!plus_n_Sm -plus_n_O. Qed. Lemma doubleS n : double (1 + n) = 2 + double n. Proof. by rewrite !doubleE /= -plus_n_Sm. Qed. Theorem contrived n : double (1+n) = 2 + double n. rewrite doubleS.
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