(** * Auto: More Automation *)
Require Export Imp.
(** Up to now, we've continued to use a quite restricted set of
Coq's tactic facilities. In this chapter, we'll learn more about
two very powerful features of Coq's tactic language:
proof search via the [auto] and [eauto] tactics, and
automated forward reasoning via the [Ltac] hypothesis matching
machinery. 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 in the face
of incremental changes to underlying definitions.
There's a third major source of automation we haven't
fully studied yet, namely built-in decision procedures for specific
kinds of problems: [omega] is one example, but there are others.
This topic will be defered for a while longer.
*)
(** Our motivating example will be this proof, repeated with
just a few small changes from [Imp]. We will try to simplify
this proof in several stages. *)
Ltac inv H := inversion H; subst; clear H.
Theorem ceval_deterministic: forall c st st1 st2,
c / st || st1 ->
c / st || st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2;
generalize dependent st2;
ceval_cases (induction E1) Case;
intros st2 E2; inv E2.
Case "E_Skip". reflexivity.
Case "E_Ass". reflexivity.
Case "E_Seq".
assert (st' = st'0) as EQ1.
SCase "Proof of assertion". apply IHE1_1; assumption.
subst st'0.
apply IHE1_2. assumption.
Case "E_IfTrue".
SCase "b evaluates to true".
apply IHE1. assumption.
SCase "b evaluates to false (contradiction)".
rewrite H in H5. inversion H5.
Case "E_IfFalse".
SCase "b evaluates to true (contradiction)".
rewrite H in H5. inversion H5.
SCase "b evaluates to false".
apply IHE1. assumption.
Case "E_WhileEnd".
SCase "b evaluates to false".
reflexivity.
SCase "b evaluates to true (contradiction)".
rewrite H in H2. inversion H2.
Case "E_WhileLoop".
SCase "b evaluates to false (contradiction)".
rewrite H in H4. inversion H4.
SCase "b evaluates to true".
assert (st' = st'0) as EQ1.
SSCase "Proof of assertion". apply IHE1_1; assumption.
subst st'0.
apply IHE1_2. assumption. Qed.
(** * The [auto] and [eauto] tactics *)
(** Thus far, we have (nearly) always written proof scripts that
apply relevant hypothoses or lemmas by name. In particular, when
a chain of hypothesis applications is needed, we have specified
them explicitly. (The only exceptions introduced so far are using
[assumption] to find a matching unqualified hypothesis
or [(e)constructor] to find a matching constructor.) *)
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.
intros P Q R H1 H2 H3.
auto.
Qed.
(** The [auto] tactic solves goals that are solvable by any combination of
- [intros],
- [apply] (with a local hypothesis, by default).
The [eauto] tactic works just like [auto], except that it uses
[eapply] instead of [apply]. *)
(** 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.
*)
(** A more complicated example: *)
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.
(** Search can 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.
auto. (* When it cannot solve the goal, does nothing! *)
auto 6. (* Optional argument says how deep to search (default depth is 5) *)
Qed.
(** When searching for potential proofs of the current goal, [auto]
and [eauto] consider the hypotheses in the current context
together with a _hint database_ of other lemmas and constructors.
Some of the lemmas and constructors we've already seen -- e.g.,
[eq_refl], [conj], [or_introl], and [or_intror] -- 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.
(** If we want to see which facts [auto] is using, we can use [info_auto] instead. *)
Example auto_example_5: 2 = 2.
Proof.
info_auto. (* subsumes reflexivity because eq_refl is in hint database *)
Qed.
(** We can extend the hint database just for the purposes of one
application of [auto] or [eauto] 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. (* does nothing: auto doesn't destruct hypotheses! *)
auto using le_antisym.
Qed.
(** Of course, in any given development there will also be some of our
own 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] and enable further possibilities for applying
lemmas that it knows about. *)
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.
info_auto.
Qed.
Hint Constructors ceval.
Definition st12 := update (update empty_state X 1) Y 2.
Definition st21 := update (update empty_state X 2) Y 1.
Example auto_example_8 : exists s',
(IFB (BLe (AId X) (AId Y))
THEN (Z ::= AMinus (AId Y) (AId X))
ELSE (Y ::= APlus (AId X) (AId Z))
FI) / st21 || s'.
Proof.
eexists. info_auto.
Qed.
Example auto_example_8' : exists s',
(IFB (BLe (AId X) (AId Y))
THEN (Z ::= AMinus (AId Y) (AId X))
ELSE (Y ::= APlus (AId X) (AId Z))
FI) / st12 || s'.
Proof.
eexists. info_auto.
Qed.
(** Now let's take a pass over [ceval_deterministic] using [auto]
to simplify the proof script. We see that all simple sequences of hypothesis
applications and all uses of [reflexivity] can be replaced by [auto],
which we add to the default tactic to be applied to each case.
*)
Theorem ceval_deterministic': forall c st st1 st2,
c / st || st1 ->
c / st || st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2;
generalize dependent st2;
ceval_cases (induction E1) Case;
intros st2 E2; inv E2; auto.
Case "E_Seq".
assert (st' = st'0) as EQ1.
SCase "Proof of assertion". auto.
subst st'0.
auto.
Case "E_IfTrue".
SCase "b evaluates to false (contradiction)".
rewrite H in H5. inversion H5.
Case "E_IfFalse".
SCase "b evaluates to true (contradiction)".
rewrite H in H5. inversion H5.
Case "E_WhileEnd".
SCase "b evaluates to true (contradiction)".
rewrite H in H2. inversion H2.
Case "E_WhileLoop".
SCase "b evaluates to false (contradiction)".
rewrite H in H4. inversion H4.
SCase "b evaluates to true".
assert (st' = st'0) as EQ1.
SSCase "Proof of assertion". auto.
subst st'0.
auto. Qed.
(** * Searching 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.
Note: In fact, Coq has a built-in tactic [congruence] that will do the
job. 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 small amount of paramerized Ltac. *)
Ltac rwinv H1 H2 := rewrite H1 in H2; inv H2.
Theorem ceval_deterministic'': forall c st st1 st2,
c / st || st1 ->
c / st || st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2;
generalize dependent st2;
ceval_cases (induction E1) Case;
intros st2 E2; inv E2; auto.
Case "E_Seq".
assert (st' = st'0) as EQ1.
SCase "Proof of assertion". auto.
subst st'0.
auto.
Case "E_IfTrue".
SCase "b evaluates to false (contradiction)".
rwinv H H5.
Case "E_IfFalse".
SCase "b evaluates to true (contradiction)".
rwinv H H5.
Case "E_WhileEnd".
SCase "b evaluates to true (contradiction)".
rwinv H H2.
Case "E_WhileLoop".
SCase "b evaluates to false (contradiction)".
rwinv H H4.
SCase "b evaluates to true".
assert (st' = st'0) as EQ1.
SSCase "Proof of assertion". auto.
subst st'0.
auto. Qed.
(** But this is not much better. We really want Coq to discover
the relevant hypotheses for us. We can do this by using the
[match goal with ... end] facility of Ltac. *)
Ltac find_rwinv :=
match goal with
H1: ?E = true, H2: ?E = false |- _ => rwinv H1 H2
end.
(** In words, this [match goal] looks for two (distinct) hypotheses that have
the form of equalities with the same arbitrary expression [E] on the
left and conflicting boolean values on the right; if such hypotheses are
found, it binds [H1] and [H2] to their names, and applies the tactic
after the [=>].
Adding this tactic to our default string handles all the contradiction cases. *)
Theorem ceval_deterministic''': forall c st st1 st2,
c / st || st1 ->
c / st || st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2;
generalize dependent st2;
ceval_cases (induction E1) Case;
intros st2 E2; inv E2; try find_rwinv; auto.
Case "E_Seq".
assert (st' = st'0) as EQ1.
SCase "Proof of assertion". auto.
subst st'0.
auto.
Case "E_WhileLoop".
SCase "b evaluates to true".
assert (st' = st'0) as EQ1.
SSCase "Proof of assertion". auto.
subst st'0.
auto. Qed.
(** Finally, 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,
c / st || st1 ->
c / st || st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2;
generalize dependent st2;
ceval_cases (induction E1) Case;
intros st2 E2; inv E2; try find_rwinv; auto.
Case "E_Seq".
rewrite (IHE1_1 st'0 H1) in *. auto.
Case "E_WhileLoop".
SCase "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.
(** But there are 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:
- [rewrite] will fail given a trivial equation of the form [X = X].
- 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.
- 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,
c / st || st1 ->
c / st || st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2;
generalize dependent st2;
ceval_cases (induction E1) Case;
intros st2 E2; inv E2; try find_rwinv; repeat find_eqn; auto.
Qed.
(** The big pay-off in this approach is that our proof script
should be robust in the face of modest changes to our language.
For example, we can add a [REPEAT] command to the language.
(This was an exercise in [Hoare.v].) *)
Module Repeat.
Inductive com : Type :=
| CSkip : com
| CAsgn : id -> aexp -> com
| CSeq : com -> com -> com
| CIf : bexp -> com -> com -> com
| CWhile : bexp -> com -> com
| CRepeat : com -> bexp -> com.
(** [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. *)
Tactic Notation "com_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "SKIP" | Case_aux c "::=" | Case_aux c ";"
| Case_aux c "IFB" | Case_aux c "WHILE"
| Case_aux c "CRepeat" ].
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 "'IFB' 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) (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 (IFB 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 (IFB b1 THEN c1 ELSE c2 FI) st'
| E_WhileEnd : forall b1 st c1,
beval st b1 = false ->
ceval st (WHILE b1 DO c1 END) st
| E_WhileLoop : 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''
.
Tactic Notation "ceval_cases" tactic(first) ident(c) :=
first;
[ Case_aux c "E_Skip" | Case_aux c "E_Ass"
| Case_aux c "E_Seq"
| Case_aux c "E_IfTrue" | Case_aux c "E_IfFalse"
| Case_aux c "E_WhileEnd" | Case_aux c "E_WhileLoop"
| Case_aux c "E_RepeatEnd" | Case_aux c "E_RepeatLoop"
].
Notation "c1 '/' st '||' st'" := (ceval st c1 st')
(at level 40, st at level 39).
Theorem ceval_deterministic: forall c st st1 st2,
c / st || st1 ->
c / st || st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2;
generalize dependent st2;
ceval_cases (induction E1) Case;
intros st2 E2; inv E2; try find_rwinv; repeat find_eqn; auto.
Case "E_RepeatEnd".
SCase "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. *)
case "E_RepeatLoop".
SCase "b evaluates to true (contradiction)".
find_rwinv.
Qed.
Theorem ceval_deterministic': forall c st st1 st2,
c / st || st1 ->
c / st || st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2;
generalize dependent st2;
ceval_cases (induction E1) Case;
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 do...
The details of using [match goal] are tricky, and debugging is
not pleasant at all. But it is well worth adding at least simple
uses to your proofs to avoid tedium and "future proof" your scripts.
*)
(* $Date: 2013-07-30 09:24:33 -0700 (Tue, 30 Jul 2013) $ *)