Certified Programming with Dependent Types: src/CpdtTactics.v comparison

comparison src/CpdtTactics.v @ 389:76e1dfcb86eb

Comment CpdtTactics.v

author	Adam Chlipala <adam@chlipala.net>
date	Thu, 12 Apr 2012 18:41:32 -0400
parents	d1276004eec9
children	539ed97750bb

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+:76e1dfcb86eb
-(* Copyright (c) 2008-2011, Adam Chlipala
+(* Copyright (c) 2008-2012, Adam Chlipala
 *
 * This work is licensed under a
 * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
 * Unported License.
 * The license text is available at:
 Require Omega.
 Set Implicit Arguments.
+(** A version of [injection] that does some standard simplifications afterward: clear the hypothesis in question, bring the new facts above the double line, and attempt substitution for known variables. *)
 Ltac inject H := injection H; clear H; intros; try subst.
+(** Try calling tactic function [f] on all hypotheses, keeping the first application that doesn't fail. *)
 Ltac appHyps f :=
 match goal with
 | [ H : _ |- _ ] => f H
 end.
+(** Succeed iff [x] is in the list [ls], represented with left-associated nested tuples. *)
 Ltac inList x ls :=
 match ls with
 | x => idtac
 | (_, x) => idtac
 | (?LS, _) => inList x LS
 end.
+(** Try calling tactic function [f] on every element of tupled list [ls], keeping the first call not to fail. *)
 Ltac app f ls :=
 match ls with
 | (?LS, ?X) => f X || app f LS || fail 1
 | _ => f ls
 end.
+(** Run [f] on every element of [ls], not just the first that doesn't fail. *)
 Ltac all f ls :=
 match ls with
 | (?LS, ?X) => f X; all f LS
 | (_, _) => fail 1
 | _ => f ls
 end.
+(** Workhorse tactic to simplify hypotheses for a variety of proofs.
+* Argument [invOne] is a tuple-list of predicates for which we always do inversion automatically. *)
 Ltac simplHyp invOne :=
+(** Helper function to do inversion on certain hypotheses, where [H] is the hypothesis and [F] its head symbol *)
 let invert H F :=
-inList F invOne; (inversion H; fail)
+(** We only proceed for those predicates in [invOne]. *)
-|| (inversion H; [idtac]; clear H; try subst) in
+inList F invOne;
+(** This case covers an inversion that succeeds immediately, meaning no constructors of [F] applied. *)
+(inversion H; fail)
+(** Otherwise, we only proceed if inversion eliminates all but one constructor case. *)
+|| (inversion H; [idtac]; clear H; try subst) in
 match goal with
+(** Eliminate all existential hypotheses. *)
 | [ H : ex _ |- _ ] => destruct H
+(** Find opportunities to take advantage of injectivity of data constructors, for several different arities. *)
 | [ H : ?F ?X = ?F ?Y |- ?G ] =>
+(** This first branch of the [||] fails the whole attempt iff the arguments of the constructor applications are already easy to prove equal. *)
 (assert (X = Y); [ assumption | fail 1 ])
+(** If we pass that filter, then we use injection on [H] and do some simplification as in [inject].
+* The odd-looking check of the goal form is to avoid cases where [injection] gives a more complex result because of dependent typing, which we aren't equipped to handle here. *)
 || (injection H;
 match goal with
 | [ |- X = Y -> G ] =>
 try clear H; intros; try subst
 end)
 match goal with
 | [ |- U = V -> X = Y -> G ] =>
 try clear H; intros; try subst
 end)
+(** Consider some different arities of a predicate [F] in a hypothesis that we might want to invert. *)
 | [ H : ?F _ |- _ ] => invert H F
 | [ H : ?F _ _ |- _ ] => invert H F
 | [ H : ?F _ _ _ |- _ ] => invert H F
 | [ H : ?F _ _ _ _ |- _ ] => invert H F
 | [ H : ?F _ _ _ _ _ |- _ ] => invert H F
+(** Use an (axiom-dependent!) inversion principle for dependent pairs, from the standard library. *)
 | [ H : existT _ ?T _ = existT _ ?T _ |- _ ] => generalize (inj_pair2 _ _ _ _ _ H); clear H
+(** If we're not ready to use that principle yet, try the standard inversion, which often enables the previous rule. *)
 | [ H : existT _ _ _ = existT _ _ _ |- _ ] => inversion H; clear H
+(** Similar logic to the cases for constructor injectivity above, but specialized to [Some], since the above cases won't deal with polymorphic constructors. *)
 | [ H : Some _ = Some _ |- _ ] => injection H; clear H
 end.
+(** Find some hypothesis to rewrite with, ensuring that [auto] proves all of the extra subgoals added by [rewrite]. *)
 Ltac rewriteHyp :=
 match goal with
-| [ H : _ |- _ ] => rewrite H; auto; [idtac]
+| [ H : _ |- _ ] => rewrite H by solve [ auto ]
 end.
+(** Combine [autorewrite] with automatic hypothesis rewrites. *)
 Ltac rewriterP := repeat (rewriteHyp; autorewrite with core in *).
 Ltac rewriter := autorewrite with core in *; rewriterP.
+(** This one is just so darned useful, let's add it as a hint here. *)
 Hint Rewrite app_ass.
+(** Devious marker predicate to use for encoding state within proof goals *)
 Definition done (T : Type) (x : T) := True.
+(** Try a new instantiation of a universally quantified fact, proved by [e].
+* [trace] is an accumulator recording which instantiations we choose. *)
 Ltac inster e trace :=
+(** Does [e] have any quantifiers left? *)
 match type of e with
 | forall x : _, _ =>
+(** Yes, so let's pick the first context variable of the right type. *)
 match goal with
 | [ H : _ |- _ ] =>
 inster (e H) (trace, H)
 | _ => fail 2
 end
 | _ =>
+(** No more quantifiers, so now we check if the trace we computed was already used. *)
 match trace with
 | (_, _) =>
+(** We only reach this case if the trace is nonempty, ensuring that [inster] fails if no progress can be made. *)
 match goal with
-| [ H : done (trace, _) |- _ ] => fail 1
+| [ H : done (trace, _) |- _ ] =>
+(** Uh oh, found a record of this trace in the context!  Abort to backtrack to try another trace. *)
+fail 1
 | _ =>
+(** What is the type of the proof [e] now? *)
 let T := type of e in
 match type of T with
 | Prop =>
+(** [e] should be thought of as a proof, so let's add it to the context, and also add a new marker hypothesis recording our choice of trace. *)
 generalize e; intro;
-assert (done (trace, tt)); [constructor | idtac]
+assert (done (trace, tt)) by constructor
 | _ =>
+(** [e] is something beside a proof.  Better make sure no element of our current trace was generated by a previous call to [inster], or we might get stuck in an infinite loop!  (We store previous [inster] terms in second positions of tuples used as arguments to [done] in hypotheses.  Proofs instantiated by [inster] merely use [tt] in such positions.) *)
 all ltac:(fun X =>
 match goal with
 | [ H : done (_, X) |- _ ] => fail 1
 | _ => idtac
 end) trace;
+(** Pick a new name for our new instantiation. *)
 let i := fresh "i" in (pose (i := e);
-assert (done (trace, i)); [constructor | idtac])
+assert (done (trace, i)) by constructor)
 end
 end
 end
 end.
+(** After a round of application with the above, we will have a lot of junk [done] markers to clean up; hence this tactic. *)
 Ltac un_done :=
 repeat match goal with
 | [ H : done _ |- _ ] => clear H
 end.
 Require Import JMeq.
+(** A more parameterized version of the famous [crush].  Extra arguments are:
+* - A tuple-list of lemmas we try [inster]-ing
+* - A tuple-list of predicates we try inversion for *)
 Ltac crush' lemmas invOne :=
-let sintuition := simpl in *; intuition; try subst; repeat (simplHyp invOne; intuition; try subst); try congruence in
+(** A useful combination of standard automation *)
-let rewriter := autorewrite with core in *;
+let sintuition := simpl in *; intuition; try subst;
-repeat (match goal with
+repeat (simplHyp invOne; intuition; try subst); try congruence in
-| [ H : ?P |- _ ] =>
-match P with
+(** A fancier version of [rewriter] from above, which uses [crush'] to discharge side conditions *)
-| context[JMeq] => fail 1
+let rewriter := autorewrite with core in *;
-| _ => (rewrite H; [])
+repeat (match goal with
-|| (rewrite H; [ | solve [ crush' lemmas invOne ] ])
+| [ H : ?P |- _ ] =>
-|| (rewrite H; [ | solve [ crush' lemmas invOne ] | solve [ crush' lemmas invOne ] ])
+match P with
-end
+| context[JMeq] => fail 1 (** JMeq is too fancy to deal with here. *)
-end; autorewrite with core in *)
+| _ => rewrite H by crush' lemmas invOne
-in (sintuition; rewriter;
+end
-match lemmas with
+end; autorewrite with core in *) in
-| false => idtac
-| _ => repeat ((app ltac:(fun L => inster L L) lemmas || appHyps ltac:(fun L => inster L L));
+(** Now the main sequence of heuristics: *)
-repeat (simplHyp invOne; intuition)); un_done
+(sintuition; rewriter;
-end; sintuition; rewriter; sintuition; try omega; try (elimtype False; omega)).
+match lemmas with
+| false => idtac (** No lemmas?  Nothing to do here *)
+| _ =>
+(** Try a loop of instantiating lemmas... *)
+repeat ((app ltac:(fun L => inster L L) lemmas
+(** ...or instantiating hypotheses... *)
+|| appHyps ltac:(fun L => inster L L));
+(** ...and then simplifying hypotheses. *)
+repeat (simplHyp invOne; intuition)); un_done
+end;
+sintuition; rewriter; sintuition;
+(** End with a last attempt to prove an arithmetic fact with [omega], or prove any sort of fact in a context that is contradictory by reasoning that [omega] can do. *)
+try omega; try (elimtype False; omega)).
+(** [crush] instantiates [crush'] with the simplest possible parameters. *)
 Ltac crush := crush' false fail.
+(** * Wrap Program's [dependent destruction] in a slightly more pleasant form *)
 Require Import Program.Equality.
+(** Run [dependent destruction] on [E] and look for opportunities to simplify the result.
+The weird introduction of [x] helps get around limitations of [dependent destruction], in terms of which sorts of arguments it will accept (e.g., variables bound to hypotheses within Ltac [match]es). *)
 Ltac dep_destruct E :=
 let x := fresh "x" in
 remember E as x; simpl in x; dependent destruction x;
 try match goal with
 | [ H : _ = E |- _ ] => rewrite <- H in *; clear H
 end.
+(** Nuke all hypotheses that we can get away with, without invalidating the goal statement. *)
 Ltac clear_all :=
 repeat match goal with
 | [ H : _ |- _ ] => clear H
 end.
+(** Instantiate a quantifier in a hypothesis [H] with value [v], or, if [v] doesn't have the right type, with a new unification variable.
+* Also prove the lefthand sides of any implications that this exposes, simplifying [H] to leave out those implications. *)
 Ltac guess v H :=
 repeat match type of H with
 | forall x : ?T, _ =>
 match type of T with
 | Prop =>
 (let H' := fresh "H'" in
 assert (H' : T); [
 solve [ eauto 6 ]
-| generalize (H H'); clear H H'; intro H ])
+| specialize (H H'); clear H' ])
 || fail 1
 | _ =>
-(generalize (H v); clear H; intro H)
+specialize (H v)
 || let x := fresh "x" in
 evar (x : T);
-let x' := eval cbv delta [x] in x in
+let x' := eval unfold x in x in
-clear x; generalize (H x'); clear H; intro H
+clear x; specialize (H x')
 end
 end.
+(** Version of [guess] that leaves the original [H] intact *)
 Ltac guessKeep v H :=
 let H' := fresh "H'" in
 generalize H; intro H'; guess v H'.

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comparison src/CpdtTactics.v @ 389:76e1dfcb86eb