AltAutoMore Automation

Set Warnings "-notation-overridden,-parsing".
From Coq Require Import omega.Omega.
From LF Require Import IndProp.
Here's a function that rewrites regular expressions into potentially simpler forms without changing their matching behavior.
Fixpoint re_opt_e {T:Type} (re: reg_exp T) : reg_exp T :=
  match re with
  | App EmptyStr re2 ⇒ re_opt_e re2
  | App re1 re2 ⇒ App (re_opt_e re1) (re_opt_e re2)
  | Union re1 re2 ⇒ Union (re_opt_e re1) (re_opt_e re2)
  | Star re ⇒ Star (re_opt_e re)
  | _ ⇒ re
  end.
We would like to show the equivalence of re's with their "optimized" form. One direction of this equivalence looks like this (the other is similar).
Lemma re_opt_e_match : ∀T (re: reg_exp T) s,
    s =~ re → s =~ re_opt_e re.
Proof.
  intros T re s M.
  induction M
    as [| x'
        | s1 re1 s2 re2 Hmatch1 IH1 Hmatch2 IH2
        | s1 re1 re2 Hmatch IH | re1 s2 re2 Hmatch IH
        | re | s1 s2 re Hmatch1 IH1 Hmatch2 IH2].
  - (* MEmpty *) simpl. apply MEmpty.
  - (* MChar *) simpl. apply MChar.
  - (* MApp *) simpl.
    destruct re1.
    + apply MApp. apply IH1. apply IH2.
    + inversion Hmatch1. simpl. apply IH2.
    + apply MApp. apply IH1. apply IH2.
    + apply MApp. apply IH1. apply IH2.
    + apply MApp. apply IH1. apply IH2.
    + apply MApp. apply IH1. apply IH2.
  - (* MUnionL *) simpl. apply MUnionL. apply IH.
  - (* MUnionR *) simpl. apply MUnionR. apply IH.
  - (* MStar0 *) simpl. apply MStar0.
  - (* MStarApp *) simpl. apply MStarApp. apply IH1. apply IH2.
Qed.

Coq Automation

That last proof was getting a little repetitive. Time to learn a few more Coq tricks...

Tacticals

Tacticals is Coq's term for tactics that take other tactics as arguments — "higher-order tactics," if you will.

The try Tactical

If T is a tactic, then try T is a tactic that is just like T except that, if T fails, try T successfully does nothing at all (instead of failing).
Theorem silly2 : ∀(P : Prop), P → P.
Proof.
  intros P HP.
  try reflexivity. (* just reflexivity would have failed *)
  apply HP. (* we can still finish the proof in some other way *)
Qed.

The ; Tactical (Simple Form)

In its most common form, the ; tactical takes two tactics as arguments. The compound tactic T;T' first performs T and then performs T' on each subgoal generated by T.
For example:
Lemma foo : ∀n, 0 <=? n = true.
Proof.
  intros.
  destruct n eqn:E.
    (* Leaves two subgoals, which are discharged identically...  *)
    - (* n=0 *) simpl. reflexivity.
    - (* n=Sn' *) simpl. reflexivity.
Qed.

We can simplify this proof using the ; tactical:
Lemma foo' : ∀n, 0 <=? n = true.
Proof.
  intros.
  (* destruct the current goal *)
  destruct n;
  (* then simpl each resulting subgoal *)
  simpl;
  (* and do reflexivity on each resulting subgoal *)
  reflexivity.
Qed.

Using try and ; together, we can get rid of the repetition in the proof that was bothering us a little while ago.
Lemma re_opt_e_match' : ∀T (re: reg_exp T) s,
    s =~ re → s =~ re_opt_e re.
Proof.
  intros T re s M.
  induction M
    as [| x'
        | s1 re1 s2 re2 Hmatch1 IH1 Hmatch2 IH2
        | s1 re1 re2 Hmatch IH | re1 s2 re2 Hmatch IH
        | re | s1 s2 re Hmatch1 IH1 Hmatch2 IH2];
    (* Do the simpl for every case here: *)
    simpl.
  - (* MEmpty *) apply MEmpty.
  - (* MChar *) apply MChar.
  - (* MApp *)
    destruct re1;
    (* Most cases follow by the same formula.
       Notice that apply MApp gives two subgoals:
       try apply H1 is run on both of them and
       succeeds on the first but not the second;
       apply H2 is then run on this remaining goal. *)
    try (apply MApp; try apply IH1; apply IH2).
    (* The interesting case, on which try... does nothing,
       is when re1 = EmptyStr. In this case, we have
       to appeal to the fact that re1 matches only the
       empty string: *)
    inversion Hmatch1. simpl. apply IH2.
  - (* MUnionL *) apply MUnionL. apply IH.
  - (* MUnionR *) apply MUnionR. apply IH.
  - (* MStar0 *) apply MStar0.
  - (* MStarApp *) apply MStarApp. apply IH1. apply IH2.
Qed.

The repeat Tactical

The repeat tactical takes another tactic and keeps applying this tactic until it fails. Here is an example showing that 10 is in a long list using repeat.
Theorem In10 : In 10 [1;2;3;4;5;6;7;8;9;10].
Proof.
  repeat (try (left; reflexivity); right).
Qed.

A Few More Handy Tactics

By the way, here are some miscellaneous tactics that you may find convenient as we continue.
  • clear H: Delete hypothesis H from the context.
  • rename... into...: Change the name of a hypothesis in the proof context. For example, if the context includes a variable named x, then rename x into y will change all occurrences of x to y.
  • subst x: Find an assumption x = e or e = x in the context, replace x with e throughout the context and current goal, and clear the assumption.
  • subst: Substitute away all assumptions of the form x = e or e = x.
We'll see examples as we go along.

Defining New Tactics

Coq also provides several ways of "programming" tactic scripts:
  • Ltac: scripting language for tactics (good for more sophisticated proof engineering)
  • OCaml tactic scripting API (only for wizards).
Here is a simple Ltac example:
Ltac impl_and_try c := simpl; try c.

Decision Procedures

The Omega Tactic


Example silly_presburger_example : ∀m n o p,
  m + n ≤ n + o ∧ o + 3 = p + 3 →
  m ≤ p.
Proof.
  intros. omega.
Qed.

Search Tactics

The constructor tactic.

A simple first example of a search tactic is constructor, which tries to find a constructor c (from some Inductive definition in the current environment) that can be applied to solve the current goal. If one is found, behave like apply c.
Example constructor_example: ∀(n:nat),
    even (n+n).
Proof.
  induction n; simpl.
  - constructor. (* applies ev_0 *)
  - rewrite plus_comm. simpl. constructor. (* applies ev_SS *) auto.
Qed.
This saves us from needing to remember the names of our constructors. Warning: if more than one constructor can apply, constructor picks the first one (in the order in which they were defined in the Inductive) which is not necessarily the one we want!

The auto Tactic

Thus far, our proof scripts mostly apply relevant hypotheses or lemmas by name, and one at a time.
Example auto_example_1 : ∀(P Q R: Prop),
  (P → Q) → (Q → R) → P → R.
Proof.
  intros P Q R H1 H2 H3.
  apply H2. apply H1. assumption.
Qed.
The auto tactic frees us from this drudgery by searching for a sequence of applications that will prove the goal:
Example auto_example_1' : ∀(P Q R: Prop),
  (P → Q) → (Q → R) → P → R.
Proof.
  auto.
Qed.

The auto tactic solves goals that are solvable by any combination of
  • intros and
  • apply (of hypotheses from the local context, by default).
Here is a more interesting example showing auto's power:
Example auto_example_2 : ∀P Q R S T U : Prop,
  (P → Q) →
  (P → R) →
  (T → R) →
  (S → T → U) →
  ((P→Q) → (P→S)) →
  T →
  P →
  U.
Proof. auto. Qed.

Proof search could, in principle, take an arbitrarily long time, so there are limits to how far auto will search by default.
Example auto_example_3 : ∀(P Q R S T U: Prop),
  (P → Q) →
  (Q → R) →
  (R → S) →
  (S → T) →
  (T → U) →
  P →
  U.
Proof.
  (* When it cannot solve the goal, auto does nothing *)
  auto.
  (* Optional argument says how deep to search (default is 5) *)
  auto 6.
Qed.

auto considers the hypotheses in the current context together with a hint database of other lemmas and constructors. Some common facts about equality and logical operators are installed in the hint database by default.
Example auto_example_4 : ∀P Q R : Prop,
  Q →
  (Q → R) →
  P ∨ (Q ∧ R).
Proof. auto. Qed.
If we want to see which facts auto is using, we can use info_auto instead.
Example auto_example_5: 2 = 2.
Proof.
  (* auto subsumes reflexivity because eq_refl is in hint database *)
  info_auto.
Qed.

We can extend the hint database just for the purposes of one application of auto by writing "auto using ...".
Lemma le_antisym : ∀n m: nat, (n ≤ m ∧ m ≤ n) → n = m.
Proof. intros. omega. Qed.
Example auto_example_6 : ∀n m p : nat,
  (n ≤ p → (n ≤ m ∧ m ≤ n)) →
  n ≤ p →
  n = m.
Proof.
  intros.
  auto using le_antisym.
Qed.

We can also permanently extend the hint database:
  • Hint Resolve T.
    Add theorem or constructor T to the global DB
  • Hint Constructors c.
    Add all constructors of c to the global DB
  • Hint Unfold d.
    Automatically expand defined symbol d during auto
It is also possible to define specialized hint databases that can be activated only when needed. See the Coq reference manual for more.

Hint Resolve le_antisym.
Example auto_example_6' : ∀n m p : nat,
  (n≤ p → (n ≤ m ∧ m ≤ n)) →
  n ≤ p →
  n = m.
Proof.
  intros.
  auto. (* picks up hint from database *)
Qed.

Definition is_fortytwo x := (x = 42).
Example auto_example_7: ∀x,
  (x ≤ 42 ∧ 42 ≤ x) → is_fortytwo x.
Proof.
  auto. (* does nothing *)
Abort.
Hint Unfold is_fortytwo.
Example auto_example_7' : ∀x,
  (x ≤ 42 ∧ 42 ≤ x) → is_fortytwo x.
Proof. info_auto. Qed.

The eapply and eauto variants

Consider this example:
Example trans_example1: ∀a b c d,
    a ≤ b + b*c →
    (1+c)*b ≤ d →
    a ≤ d.
Proof.
  intros a b c d H1 H2.
  apply le_trans with (b+ b*c). (* <-- We must supply the intermediate value *)
  + apply H1.
  + simpl in H2. rewrite mult_comm. apply H2.
Qed.
If we leave out the with, this step fails, because Coq cannot find an instance for the variable n. But this is silly! The appropriate value for n will become obvious in the very next step.

With eapply, we can eliminate this silliness:
Example trans_example1': ∀a b c d,
    a ≤ b + b*c →
    (1+c)*b ≤ d →
    a ≤ d.
Proof.
  intros a b c d H1 H2.
  eapply le_trans. (* 1 *)
  + apply H1. (* 2 *)
  + simpl in H2. rewrite mult_comm. apply H2.
Qed.

Several of the tactics that we've seen so far, including ∃, constructor, and auto, have e... variants. For example, here's a proof using eauto:
Example trans_example2: ∀a b c d,
    a ≤ b + b*c →
    b + b*c ≤ d →
    a ≤ d.
Proof.
  intros a b c d H1 H2.
  info_eauto using le_trans.
Qed.
The eauto tactic works just like auto, except that it uses eapply instead of apply.
Pro tip: One might think that, since eapply and eauto are more powerful than apply and auto, it would be a good idea to use them all the time. Unfortunately, they are also significantly slower — especially eauto. Coq experts tend to use apply and auto most of the time, only switching to the e variants when the ordinary variants don't do the job.