adam@389: (* Copyright (c) 2008-2012, Adam Chlipala
adamc@2:  * 
adamc@2:  * This work is licensed under a
adamc@2:  * Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
adamc@2:  * Unported License.
adamc@2:  * The license text is available at:
adamc@2:  *   http://creativecommons.org/licenses/by-nc-nd/3.0/
adamc@2:  *)
adamc@2: 
adamc@85: Require Import Eqdep List.
adamc@2: 
adamc@38: Require Omega.
adamc@38: 
adamc@59: Set Implicit Arguments.
adamc@59: 
adamc@2: 
adam@389: (** 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. *)
adamc@66: Ltac inject H := injection H; clear H; intros; try subst.
adamc@51: 
adam@389: (** Try calling tactic function [f] on all hypotheses, keeping the first application that doesn't fail. *)
adamc@59: Ltac appHyps f :=
adamc@50:   match goal with
adamc@59:     | [ H : _ |- _ ] => f H
adamc@59:   end.
adamc@59: 
adam@389: (** Succeed iff [x] is in the list [ls], represented with left-associated nested tuples. *)
adamc@59: Ltac inList x ls :=
adamc@59:   match ls with
adamc@59:     | x => idtac
adamc@84:     | (_, x) => idtac
adamc@59:     | (?LS, _) => inList x LS
adamc@59:   end.
adamc@59: 
adam@389: (** Try calling tactic function [f] on every element of tupled list [ls], keeping the first call not to fail. *)
adamc@59: Ltac app f ls :=
adamc@59:   match ls with
adamc@59:     | (?LS, ?X) => f X || app f LS || fail 1
adamc@59:     | _ => f ls
adamc@59:   end.
adamc@59: 
adam@389: (** Run [f] on every element of [ls], not just the first that doesn't fail. *)
adamc@59: Ltac all f ls :=
adamc@59:   match ls with
adamc@59:     | (?LS, ?X) => f X; all f LS
adamc@59:     | (_, _) => fail 1
adamc@59:     | _ => f ls
adamc@59:   end.
adamc@59: 
adam@389: (** Workhorse tactic to simplify hypotheses for a variety of proofs.
adam@389:    * Argument [invOne] is a tuple-list of predicates for which we always do inversion automatically. *)
adamc@59: Ltac simplHyp invOne :=
adam@389:   (** Helper function to do inversion on certain hypotheses, where [H] is the hypothesis and [F] its head symbol *)
adamc@99:   let invert H F :=
adam@389:     (** We only proceed for those predicates in [invOne]. *)
adam@389:     inList F invOne;
adam@389:     (** This case covers an inversion that succeeds immediately, meaning no constructors of [F] applied. *)
adam@389:       (inversion H; fail)
adam@389:     (** Otherwise, we only proceed if inversion eliminates all but one constructor case. *)
adam@389:       || (inversion H; [idtac]; clear H; try subst) in
adam@389: 
adamc@59:   match goal with
adam@389:     (** Eliminate all existential hypotheses. *)
adamc@59:     | [ H : ex _ |- _ ] => destruct H
adamc@59: 
adam@389:     (** Find opportunities to take advantage of injectivity of data constructors, for several different arities. *)
adamc@204:     | [ H : ?F ?X = ?F ?Y |- ?G ] =>
adam@389:       (** This first branch of the [||] fails the whole attempt iff the arguments of the constructor applications are already easy to prove equal. *)
adamc@204:       (assert (X = Y); [ assumption | fail 1 ])
adam@389:       (** If we pass that filter, then we use injection on [H] and do some simplification as in [inject].
adam@389:          * 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. *)
adamc@204:       || (injection H;
adamc@204:         match goal with
adamc@204:           | [ |- X = Y -> G ] =>
adamc@204:             try clear H; intros; try subst
adamc@204:         end)
adamc@204:     | [ H : ?F ?X ?U = ?F ?Y ?V |- ?G ] =>
adamc@204:       (assert (X = Y); [ assumption
adamc@204:         | assert (U = V); [ assumption | fail 1 ] ])
adamc@204:       || (injection H;
adamc@204:         match goal with
adamc@204:           | [ |- U = V -> X = Y -> G ] =>
adamc@204:             try clear H; intros; try subst
adamc@204:         end)
adamc@59: 
adam@389:     (** Consider some different arities of a predicate [F] in a hypothesis that we might want to invert. *)
adamc@99:     | [ H : ?F _ |- _ ] => invert H F
adamc@99:     | [ H : ?F _ _ |- _ ] => invert H F
adamc@99:     | [ H : ?F _ _ _ |- _ ] => invert H F
adamc@99:     | [ H : ?F _ _ _ _ |- _ ] => invert H F
adamc@99:     | [ H : ?F _ _ _ _ _ |- _ ] => invert H F
adamc@176: 
adam@389:     (** Use an (axiom-dependent!) inversion principle for dependent pairs, from the standard library. *)
adamc@176:     | [ H : existT _ ?T _ = existT _ ?T _ |- _ ] => generalize (inj_pair2 _ _ _ _ _ H); clear H
adam@389: 
adam@389:     (** If we're not ready to use that principle yet, try the standard inversion, which often enables the previous rule. *)
adamc@176:     | [ H : existT _ _ _ = existT _ _ _ |- _ ] => inversion H; clear H
adamc@179: 
adam@389:     (** Similar logic to the cases for constructor injectivity above, but specialized to [Some], since the above cases won't deal with polymorphic constructors. *)
adamc@179:     | [ H : Some _ = Some _ |- _ ] => injection H; clear H
adamc@50:   end.
adamc@50: 
adam@389: (** Find some hypothesis to rewrite with, ensuring that [auto] proves all of the extra subgoals added by [rewrite]. *)
adamc@2: Ltac rewriteHyp :=
adamc@2:   match goal with
adam@389:     | [ H : _ |- _ ] => rewrite H by solve [ auto ]
adamc@2:   end.
adamc@2: 
adam@389: (** Combine [autorewrite] with automatic hypothesis rewrites. *)
adam@375: Ltac rewriterP := repeat (rewriteHyp; autorewrite with core in *).
adam@375: Ltac rewriter := autorewrite with core in *; rewriterP.
adamc@2: 
adam@389: (** This one is just so darned useful, let's add it as a hint here. *)
adam@375: Hint Rewrite app_ass.
adamc@2: 
adam@389: (** Devious marker predicate to use for encoding state within proof goals *)
adamc@59: Definition done (T : Type) (x : T) := True.
adamc@2: 
adam@389: (** Try a new instantiation of a universally quantified fact, proved by [e].
adam@389:    * [trace] is an accumulator recording which instantiations we choose. *)
adamc@59: Ltac inster e trace :=
adam@389:   (** Does [e] have any quantifiers left? *)
adamc@59:   match type of e with
adamc@59:     | forall x : _, _ =>
adam@389:       (** Yes, so let's pick the first context variable of the right type. *)
adamc@59:       match goal with
adamc@59:         | [ H : _ |- _ ] =>
adamc@59:           inster (e H) (trace, H)
adamc@59:         | _ => fail 2
adamc@59:       end
adamc@59:     | _ =>
adam@389:       (** No more quantifiers, so now we check if the trace we computed was already used. *)
adamc@59:       match trace with
adamc@59:         | (_, _) =>
adam@389:           (** We only reach this case if the trace is nonempty, ensuring that [inster] fails if no progress can be made. *)
adamc@59:           match goal with
adam@389:             | [ H : done (trace, _) |- _ ] =>
adam@389:               (** Uh oh, found a record of this trace in the context!  Abort to backtrack to try another trace. *)
adam@389:               fail 1
adamc@59:             | _ =>
adam@389:               (** What is the type of the proof [e] now? *)
adamc@59:               let T := type of e in
adamc@59:                 match type of T with
adamc@59:                   | Prop =>
adam@389:                     (** [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. *)
adamc@59:                     generalize e; intro;
adam@389:                       assert (done (trace, tt)) by constructor
adamc@59:                   | _ =>
adam@389:                     (** [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.) *)
adamc@59:                     all ltac:(fun X =>
adamc@59:                       match goal with
adamc@59:                         | [ H : done (_, X) |- _ ] => fail 1
adamc@59:                         | _ => idtac
adamc@59:                       end) trace;
adam@389:                     (** Pick a new name for our new instantiation. *)
adamc@59:                     let i := fresh "i" in (pose (i := e);
adam@389:                       assert (done (trace, i)) by constructor)
adamc@59:                 end
adamc@59:           end
adamc@59:       end
adamc@59:   end.
adamc@59: 
adam@389: (** After a round of application with the above, we will have a lot of junk [done] markers to clean up; hence this tactic. *)
adamc@65: Ltac un_done :=
adamc@65:   repeat match goal with
adamc@65:            | [ H : done _ |- _ ] => clear H
adamc@65:          end.
adamc@65: 
adam@297: Require Import JMeq.
adam@297: 
adam@389: (** A more parameterized version of the famous [crush].  Extra arguments are:
adam@389:    * - A tuple-list of lemmas we try [inster]-ing 
adam@389:    * - A tuple-list of predicates we try inversion for *)
adamc@88: Ltac crush' lemmas invOne :=
adam@389:   (** A useful combination of standard automation *)
adam@389:   let sintuition := simpl in *; intuition; try subst;
adam@389:     repeat (simplHyp invOne; intuition; try subst); try congruence in
adamc@59: 
adam@389:   (** A fancier version of [rewriter] from above, which uses [crush'] to discharge side conditions *)
adam@389:   let rewriter := autorewrite with core in *;
adam@389:     repeat (match goal with
adam@389:               | [ H : ?P |- _ ] =>
adam@389:                 match P with
adam@389:                   | context[JMeq] => fail 1 (** JMeq is too fancy to deal with here. *)
adam@389:                   | _ => rewrite H by crush' lemmas invOne
adam@389:                 end
adam@389:             end; autorewrite with core in *) in
adam@389: 
adam@389:   (** Now the main sequence of heuristics: *)
adam@389:     (sintuition; rewriter;
adam@389:       match lemmas with
adam@389:         | false => idtac (** No lemmas?  Nothing to do here *)
adam@389:         | _ =>
adam@389:           (** Try a loop of instantiating lemmas... *)
adam@389:           repeat ((app ltac:(fun L => inster L L) lemmas
adam@389:           (** ...or instantiating hypotheses... *)
adam@389:             || appHyps ltac:(fun L => inster L L));
adam@389:           (** ...and then simplifying hypotheses. *)
adam@389:           repeat (simplHyp invOne; intuition)); un_done
adam@389:       end;
adam@389:       sintuition; rewriter; sintuition;
adam@389:       (** 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. *)
adam@389:       try omega; try (elimtype False; omega)).
adam@389: 
adam@389: (** [crush] instantiates [crush'] with the simplest possible parameters. *)
adamc@88: Ltac crush := crush' false fail.
adamc@85: 
adam@389: (** * Wrap Program's [dependent destruction] in a slightly more pleasant form *)
adam@389: 
adamc@213: Require Import Program.Equality.
adamc@85: 
adam@389: (** Run [dependent destruction] on [E] and look for opportunities to simplify the result.
adam@389:    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). *)
adamc@85: Ltac dep_destruct E :=
adamc@213:   let x := fresh "x" in
adamc@213:     remember E as x; simpl in x; dependent destruction x;
adamc@213:       try match goal with
adamc@213:             | [ H : _ = E |- _ ] => rewrite <- H in *; clear H
adamc@213:           end.
adamc@118: 
adam@389: (** Nuke all hypotheses that we can get away with, without invalidating the goal statement. *)
adamc@118: Ltac clear_all :=
adamc@118:   repeat match goal with
adamc@118:            | [ H : _ |- _ ] => clear H
adamc@118:          end.
adamc@256: 
adam@389: (** Instantiate a quantifier in a hypothesis [H] with value [v], or, if [v] doesn't have the right type, with a new unification variable.
adam@389:    * Also prove the lefthand sides of any implications that this exposes, simplifying [H] to leave out those implications. *)
adamc@262: Ltac guess v H :=
adamc@256:   repeat match type of H with
adamc@256:            | forall x : ?T, _ =>
adamc@256:              match type of T with
adamc@256:                | Prop =>
adamc@256:                  (let H' := fresh "H'" in
adamc@256:                    assert (H' : T); [
adamc@262:                      solve [ eauto 6 ]
adam@389:                      | specialize (H H'); clear H' ])
adamc@256:                  || fail 1
adamc@256:                | _ =>
adam@389:                  specialize (H v)
adamc@262:                  || let x := fresh "x" in
adamc@256:                    evar (x : T);
adam@389:                    let x' := eval unfold x in x in
adam@389:                      clear x; specialize (H x')
adamc@256:              end
adamc@256:          end.
adamc@256: 
adam@389: (** Version of [guess] that leaves the original [H] intact *)
adamc@262: Ltac guessKeep v H :=
adamc@256:   let H' := fresh "H'" in
adamc@262:     generalize H; intro H'; guess v H'.