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