logical-foundations/Auto.v

(** * Auto: More Automation *)

Set Warnings "-notation-overridden,-parsing".
From Coq Require Import omega.Omega.
From LF Require Import Maps.
From LF Require Import Imp.

(** Up to now, we've used the more manual part of Coq's tactic
    facilities.  In this chapter, we'll learn more about some of Coq's
    powerful automation features: proof search via the [auto] tactic,
    automated forward reasoning via the [Ltac] hypothesis matching
    machinery, and deferred instantiation of existential variables
    using [eapply] and [eauto].  Using these features together with
    Ltac's scripting facilities will enable us to make our proofs
    startlingly short!  Used properly, they can also make proofs more
    maintainable and robust to changes in underlying definitions.  A
    deeper treatment of [auto] and [eauto] can be found in the
    [UseAuto] chapter in _Programming Language Foundations_.

    There's another major category of automation we haven't discussed
    much yet, namely built-in decision procedures for specific kinds
    of problems: [omega] is one example, but there are others.  This
    topic will be deferred for a while longer.

    Our motivating example will be this proof, repeated with just a
    few small changes from the [Imp] chapter.  We will simplify
    this proof in several stages. *)

(** First, define a little Ltac macro to compress a common
    pattern into a single command. *)
Ltac inv H := inversion H; subst; clear H.

Theorem ceval_deterministic: forall c st st1 st2,
    st =[ c ]=> st1  ->
    st =[ c ]=> st2 ->
    st1 = st2.
Proof.
  intros c st st1 st2 E1 E2;
  generalize dependent st2;
  induction E1; intros st2 E2; inv E2.
  - (* E_Skip *) reflexivity.
  - (* E_Ass *) reflexivity.
  - (* E_Seq *)
    assert (st' = st'0) as EQ1.
    { (* Proof of assertion *) apply IHE1_1; apply H1. }
    subst st'0.
    apply IHE1_2. assumption.
  (* E_IfTrue *)
  - (* b evaluates to true *)
    apply IHE1. assumption.
  - (* b evaluates to false (contradiction) *)
    rewrite H in H5. inversion H5.
  (* E_IfFalse *)
  - (* b evaluates to true (contradiction) *)
    rewrite H in H5. inversion H5.
  - (* b evaluates to false *)
    apply IHE1. assumption.
  (* E_WhileFalse *)
  - (* b evaluates to false *)
    reflexivity.
  - (* b evaluates to true (contradiction) *)
    rewrite H in H2. inversion H2.
  (* E_WhileTrue *)
  - (* b evaluates to false (contradiction) *)
    rewrite H in H4. inversion H4.
  - (* b evaluates to true *)
    assert (st' = st'0) as EQ1.
    { (* Proof of assertion *) apply IHE1_1; assumption. }
    subst st'0.
    apply IHE1_2. assumption.  Qed.

(* ################################################################# *)
(** * 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 : forall (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' : forall (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). *)

(** Using [auto] is always "safe" in the sense that it will never fail
    and will never change the proof state: either it completely solves
    the current goal, or it does nothing. *)

(** Here is a more interesting example showing [auto]'s power: *)

Example auto_example_2 : forall 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 : forall (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.

(** When searching for potential proofs of the current goal,
    [auto] considers the hypotheses in the current context together
    with a _hint database_ of other lemmas and constructors.  Some
    common lemmas about equality and logical operators are installed
    in this hint database by default. *)

Example auto_example_4 : forall P Q R : Prop,
  Q ->
  (Q -> R) ->
  P \/ (Q /\ R).
Proof. auto. Qed.

(** We can extend the hint database just for the purposes of one
    application of [auto] by writing "[auto using ...]". *)

Lemma le_antisym : forall n m: nat, (n <= m /\ m <= n) -> n = m.
Proof. intros. omega. Qed.

Example auto_example_6 : forall n m p : nat,
  (n <= p -> (n <= m /\ m <= n)) ->
  n <= p ->
  n = m.
Proof.
  intros.
  auto using le_antisym.
Qed.

(** Of course, in any given development there will probably be
    some specific constructors and lemmas that are used very often in
    proofs.  We can add these to the global hint database by writing

      Hint Resolve T.

    at the top level, where [T] is a top-level theorem or a
    constructor of an inductively defined proposition (i.e., anything
    whose type is an implication).  As a shorthand, we can write

      Hint Constructors c.

    to tell Coq to do a [Hint Resolve] for _all_ of the constructors
    from the inductive definition of [c].

    It is also sometimes necessary to add

      Hint Unfold d.

    where [d] is a defined symbol, so that [auto] knows to expand uses
    of [d], thus enabling further possibilities for applying lemmas that
    it knows about. *)

(** 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' : forall 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: forall x,
  (x <= 42 /\ 42 <= x) -> is_fortytwo x.
Proof.
  auto.  (* does nothing *)
Abort.

Hint Unfold is_fortytwo.

Example auto_example_7' : forall x,
  (x <= 42 /\ 42 <= x) -> is_fortytwo x.
Proof. auto. Qed.

(** Let's take a first pass over [ceval_deterministic] to simplify the
    proof script. *)

Theorem ceval_deterministic': forall c st st1 st2,
    st =[ c ]=> st1  ->
    st =[ c ]=> st2 ->
    st1 = st2.
Proof.
  intros c st st1 st2 E1 E2.
  generalize dependent st2;
       induction E1; intros st2 E2; inv E2; auto.
  - (* E_Seq *)
    assert (st' = st'0) as EQ1 by auto.
    subst st'0.
    auto.
  - (* E_IfTrue *)
    + (* b evaluates to false (contradiction) *)
      rewrite H in H5. inversion H5.
  - (* E_IfFalse *)
    + (* b evaluates to true (contradiction) *)
      rewrite H in H5. inversion H5.
  - (* E_WhileFalse *)
    + (* b evaluates to true (contradiction) *)
      rewrite H in H2. inversion H2.
  (* E_WhileTrue *)
  - (* b evaluates to false (contradiction) *)
    rewrite H in H4. inversion H4.
  - (* b evaluates to true *)
    assert (st' = st'0) as EQ1 by auto.
    subst st'0.
    auto.
Qed.

(** When we are using a particular tactic many times in a proof, we
    can use a variant of the [Proof] command to make that tactic into
    a default within the proof.  Saying [Proof with t] (where [t] is
    an arbitrary tactic) allows us to use [t1...] as a shorthand for
    [t1;t] within the proof.  As an illustration, here is an alternate
    version of the previous proof, using [Proof with auto]. *)

Theorem ceval_deterministic'_alt: forall c st st1 st2,
    st =[ c ]=> st1 ->
    st =[ c ]=> st2 ->
    st1 = st2.
Proof with auto.
  intros c st st1 st2 E1 E2;
  generalize dependent st2;
  induction E1;
           intros st2 E2; inv E2...
  - (* E_Seq *)
    assert (st' = st'0) as EQ1...
    subst st'0...
  - (* E_IfTrue *)
    + (* b evaluates to false (contradiction) *)
      rewrite H in H5. inversion H5.
  - (* E_IfFalse *)
    + (* b evaluates to true (contradiction) *)
      rewrite H in H5. inversion H5.
  - (* E_WhileFalse *)
    + (* b evaluates to true (contradiction) *)
      rewrite H in H2. inversion H2.
  (* E_WhileTrue *)
  - (* b evaluates to false (contradiction) *)
    rewrite H in H4. inversion H4.
  - (* b evaluates to true *)
    assert (st' = st'0) as EQ1...
    subst st'0...
Qed.

(* ################################################################# *)
(** * Searching For Hypotheses *)

(** The proof has become simpler, but there is still an annoying
    amount of repetition. Let's start by tackling the contradiction
    cases. Each of them occurs in a situation where we have both

      H1: beval st b = false

    and

      H2: beval st b = true

    as hypotheses.  The contradiction is evident, but demonstrating it
    is a little complicated: we have to locate the two hypotheses [H1]
    and [H2] and do a [rewrite] following by an [inversion].  We'd
    like to automate this process.

    (In fact, Coq has a built-in tactic [congruence] that will do the
    job in this case.  But we'll ignore the existence of this tactic
    for now, in order to demonstrate how to build forward search
    tactics by hand.)

    As a first step, we can abstract out the piece of script in
    question by writing a little function in Ltac. *)

Ltac rwinv H1 H2 := rewrite H1 in H2; inv H2.

Theorem ceval_deterministic'': forall c st st1 st2,
    st =[ c ]=> st1  ->
    st =[ c ]=> st2 ->
    st1 = st2.
Proof.
  intros c st st1 st2 E1 E2.
  generalize dependent st2;
  induction E1; intros st2 E2; inv E2; auto.
  - (* E_Seq *)
    assert (st' = st'0) as EQ1 by auto.
    subst st'0.
    auto.
  - (* E_IfTrue *)
    + (* b evaluates to false (contradiction) *)
      rwinv H H5.
  - (* E_IfFalse *)
    + (* b evaluates to true (contradiction) *)
      rwinv H H5.
  - (* E_WhileFalse *)
    + (* b evaluates to true (contradiction) *)
      rwinv H H2.
  (* E_WhileTrue *)
  - (* b evaluates to false (contradiction) *)
    rwinv H H4.
  - (* b evaluates to true *)
    assert (st' = st'0) as EQ1 by auto.
    subst st'0.
    auto. Qed.

(** That was a bit better, but we really want Coq to discover the
    relevant hypotheses for us.  We can do this by using the [match
    goal] facility of Ltac. *)

Ltac find_rwinv :=
  match goal with
    H1: ?E = true,
    H2: ?E = false
    |- _ => rwinv H1 H2
  end.

(** This [match goal] looks for two distinct hypotheses that
    have the form of equalities, with the same arbitrary expression
    [E] on the left and with conflicting boolean values on the right.
    If such hypotheses are found, it binds [H1] and [H2] to their
    names and applies the [rwinv] tactic to [H1] and [H2].

    Adding this tactic to the ones that we invoke in each case of the
    induction handles all of the contradictory cases. *)

Theorem ceval_deterministic''': forall c st st1 st2,
    st =[ c ]=> st1  ->
    st =[ c ]=> st2 ->
    st1 = st2.
Proof.
  intros c st st1 st2 E1 E2.
  generalize dependent st2;
  induction E1; intros st2 E2; inv E2; try find_rwinv; auto.
  - (* E_Seq *)
    assert (st' = st'0) as EQ1 by auto.
    subst st'0.
    auto.
  - (* E_WhileTrue *)
    + (* b evaluates to true *)
      assert (st' = st'0) as EQ1 by auto.
      subst st'0.
      auto. Qed.

(** Let's see about the remaining cases. Each of them involves
    applying a conditional hypothesis to extract an equality.
    Currently we have phrased these as assertions, so that we have to
    predict what the resulting equality will be (although we can then
    use [auto] to prove it).  An alternative is to pick the relevant
    hypotheses to use and then [rewrite] with them, as follows: *)

Theorem ceval_deterministic'''': forall c st st1 st2,
    st =[ c ]=> st1  ->
    st =[ c ]=> st2 ->
    st1 = st2.
Proof.
  intros c st st1 st2 E1 E2.
  generalize dependent st2;
  induction E1; intros st2 E2; inv E2; try find_rwinv; auto.
  - (* E_Seq *)
    rewrite (IHE1_1 st'0 H1) in *. auto.
  - (* E_WhileTrue *)
    + (* b evaluates to true *)
      rewrite (IHE1_1 st'0 H3) in *. auto. Qed.

(** Now we can automate the task of finding the relevant hypotheses to
    rewrite with. *)

Ltac find_eqn :=
  match goal with
    H1: forall x, ?P x -> ?L = ?R,
    H2: ?P ?X
    |- _ => rewrite (H1 X H2) in *
  end.

(** The pattern [forall x, ?P x -> ?L = ?R] matches any hypothesis of
    the form "for all [x], _some property of [x]_ implies _some
    equality_."  The property of [x] is bound to the pattern variable
    [P], and the left- and right-hand sides of the equality are bound
    to [L] and [R].  The name of this hypothesis is bound to [H1].
    Then the pattern [?P ?X] matches any hypothesis that provides
    evidence that [P] holds for some concrete [X].  If both patterns
    succeed, we apply the [rewrite] tactic (instantiating the
    quantified [x] with [X] and providing [H2] as the required
    evidence for [P X]) in all hypotheses and the goal.

    One problem remains: in general, there may be several pairs of
    hypotheses that have the right general form, and it seems tricky
    to pick out the ones we actually need.  A key trick is to realize
    that we can _try them all_!  Here's how this works:

    - each execution of [match goal] will keep trying to find a valid
      pair of hypotheses until the tactic on the RHS of the match
      succeeds; if there are no such pairs, it fails;
    - [rewrite] will fail given a trivial equation of the form [X = X];
    - we can wrap the whole thing in a [repeat], which will keep doing
      useful rewrites until only trivial ones are left. *)

Theorem ceval_deterministic''''': forall c st st1 st2,
    st =[ c ]=> st1  ->
    st =[ c ]=> st2 ->
    st1 = st2.
Proof.
  intros c st st1 st2 E1 E2.
  generalize dependent st2;
  induction E1; intros st2 E2; inv E2; try find_rwinv;
    repeat find_eqn; auto.
Qed.

(** The big payoff in this approach is that our proof script should be
    more robust in the face of modest changes to our language.  To
    test this, let's try adding a [REPEAT] command to the language. *)

Module Repeat.

Inductive com : Type :=
  | CSkip
  | CAsgn (x : string) (a : aexp)
  | CSeq (c1 c2 : com)
  | CIf (b : bexp) (c1 c2 : com)
  | CWhile (b : bexp) (c : com)
  | CRepeat (c : com) (b : bexp).

(** [REPEAT] behaves like [WHILE], except that the loop guard is
    checked _after_ each execution of the body, with the loop
    repeating as long as the guard stays _false_.  Because of this,
    the body will always execute at least once. *)

Notation "'SKIP'" :=
  CSkip.
Notation "c1 ; c2" :=
  (CSeq c1 c2) (at level 80, right associativity).
Notation "X '::=' a" :=
  (CAsgn X a) (at level 60).
Notation "'WHILE' b 'DO' c 'END'" :=
  (CWhile b c) (at level 80, right associativity).
Notation "'TEST' e1 'THEN' e2 'ELSE' e3 'FI'" :=
  (CIf e1 e2 e3) (at level 80, right associativity).
Notation "'REPEAT' e1 'UNTIL' b2 'END'" :=
  (CRepeat e1 b2) (at level 80, right associativity).

Inductive ceval : state -> com -> state -> Prop :=
  | E_Skip : forall st,
      ceval st SKIP st
  | E_Ass  : forall st a1 n X,
      aeval st a1 = n ->
      ceval st (X ::= a1) (t_update st X n)
  | E_Seq : forall c1 c2 st st' st'',
      ceval st c1 st' ->
      ceval st' c2 st'' ->
      ceval st (c1 ; c2) st''
  | E_IfTrue : forall st st' b1 c1 c2,
      beval st b1 = true ->
      ceval st c1 st' ->
      ceval st (TEST b1 THEN c1 ELSE c2 FI) st'
  | E_IfFalse : forall st st' b1 c1 c2,
      beval st b1 = false ->
      ceval st c2 st' ->
      ceval st (TEST b1 THEN c1 ELSE c2 FI) st'
  | E_WhileFalse : forall b1 st c1,
      beval st b1 = false ->
      ceval st (WHILE b1 DO c1 END) st
  | E_WhileTrue : forall st st' st'' b1 c1,
      beval st b1 = true ->
      ceval st c1 st' ->
      ceval st' (WHILE b1 DO c1 END) st'' ->
      ceval st (WHILE b1 DO c1 END) st''
  | E_RepeatEnd : forall st st' b1 c1,
      ceval st c1 st' ->
      beval st' b1 = true ->
      ceval st (CRepeat c1 b1) st'
  | E_RepeatLoop : forall st st' st'' b1 c1,
      ceval st c1 st' ->
      beval st' b1 = false ->
      ceval st' (CRepeat c1 b1) st'' ->
      ceval st (CRepeat c1 b1) st''.

Notation "st '=[' c ']=>' st'" := (ceval st c st')
                                 (at level 40).

(** Our first attempt at the determinacy proof does not quite succeed:
    the [E_RepeatEnd] and [E_RepeatLoop] cases are not handled by our
    previous automation. *)

Theorem ceval_deterministic: forall c st st1 st2,
    st =[ c ]=> st1  ->
    st =[ c ]=> st2 ->
    st1 = st2.
Proof.
  intros c st st1 st2 E1 E2.
  generalize dependent st2;
  induction E1;
    intros st2 E2; inv E2; try find_rwinv; repeat find_eqn; auto.
  - (* E_RepeatEnd *)
    + (* b evaluates to false (contradiction) *)
       find_rwinv.
       (* oops: why didn't [find_rwinv] solve this for us already?
          answer: we did things in the wrong order. *)
  - (* E_RepeatLoop *)
     + (* b evaluates to true (contradiction) *)
        find_rwinv.
Qed.

(** Fortunately, to fix this, we just have to swap the invocations of
    [find_eqn] and [find_rwinv]. *)

Theorem ceval_deterministic': forall c st st1 st2,
    st =[ c ]=> st1  ->
    st =[ c ]=> st2 ->
    st1 = st2.
Proof.
  intros c st st1 st2 E1 E2.
  generalize dependent st2;
  induction E1;
    intros st2 E2; inv E2; repeat find_eqn; try find_rwinv; auto.
Qed.

End Repeat.

(** These examples just give a flavor of what "hyper-automation"
    can achieve in Coq.  The details of [match goal] are a bit
    tricky (and debugging scripts using it is, frankly, not very
    pleasant).  But it is well worth adding at least simple uses to
    your proofs, both to avoid tedium and to "future proof" them. *)

(* ================================================================= *)
(** ** The [eapply] and [eauto] variants *)

(** To close the chapter, we'll introduce one more convenient feature
    of Coq: its ability to delay instantiation of quantifiers.  To
    motivate this feature, recall this example from the [Imp]
    chapter: *)

Example ceval_example1:
  empty_st =[
    X ::= 2;;
    TEST X <= 1
      THEN Y ::= 3
      ELSE Z ::= 4
    FI
  ]=> (Z !-> 4 ; X !-> 2).
Proof.
  (* We supply the intermediate state [st']... *)
  apply E_Seq with (X !-> 2).
  - apply E_Ass. reflexivity.
  - apply E_IfFalse. reflexivity. apply E_Ass. reflexivity.
Qed.

(** In the first step of the proof, we had to explicitly provide a
    longish expression to help Coq instantiate a "hidden" argument to
    the [E_Seq] constructor.  This was needed because the definition
    of [E_Seq]...

          E_Seq : forall c1 c2 st st' st'',
            st  =[ c1 ]=> st'  ->
            st' =[ c2 ]=> st'' ->
            st  =[ c1 ;; c2 ]=> st''

   is quantified over a variable, [st'], that does not appear in its
   conclusion, so unifying its conclusion with the goal state doesn't
   help Coq find a suitable value for this variable.  If we leave
   out the [with], this step fails ("Error: Unable to find an
   instance for the variable [st']").

   What's silly about this error is that the appropriate value for [st']
   will actually become obvious in the very next step, where we apply
   [E_Ass].  If Coq could just wait until we get to this step, there
   would be no need to give the value explicitly.  This is exactly what
   the [eapply] tactic gives us: *)

Example ceval'_example1:
  empty_st =[
    X ::= 2;;
    TEST X <= 1
      THEN Y ::= 3
      ELSE Z ::= 4
    FI
  ]=> (Z !-> 4 ; X !-> 2).
Proof.
  eapply E_Seq. (* 1 *)
  - apply E_Ass. (* 2 *)
    reflexivity. (* 3 *)
  - (* 4 *) apply E_IfFalse. reflexivity. apply E_Ass. reflexivity.
Qed.

(** The [eapply H] tactic behaves just like [apply H] except
    that, after it finishes unifying the goal state with the
    conclusion of [H], it does not bother to check whether all the
    variables that were introduced in the process have been given
    concrete values during unification.

    If you step through the proof above, you'll see that the goal
    state at position [1] mentions the _existential variable_ [?st']
    in both of the generated subgoals.  The next step (which gets us
    to position [2]) replaces [?st'] with a concrete value.  This new
    value contains a new existential variable [?n], which is
    instantiated in its turn by the following [reflexivity] step,
    position [3].  When we start working on the second
    subgoal (position [4]), we observe that the occurrence of [?st']
    in this subgoal has been replaced by the value that it was given
    during the first subgoal. *)

(** Several of the tactics that we've seen so far, including [exists],
    [constructor], and [auto], have similar variants.  For example,
    here's a proof using [eauto]: *)

Hint Constructors ceval.
Hint Transparent state.
Hint Transparent total_map.

Definition st12 := (Y !-> 2 ; X !-> 1).
Definition st21 := (Y !-> 1 ; X !-> 2).

Example eauto_example : exists s',
  st21 =[
    TEST X <= Y
      THEN Z ::= Y - X
      ELSE Y ::= X + Z
    FI
  ]=> s'.
Proof. eauto. 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. *)


(* Wed Jan 9 12:02:47 EST 2019 *)