(** * Auto: More Automation *)
Require Import Coq.omega.Omega.
Require Import Maps.
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.
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. *)
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;
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; assumption. }
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_WhileEnd *)
- (* b evaluates to false *)
reflexivity.
- (* b evaluates to true (contradiction) *)
rewrite H in H2. inversion H2.
(* E_WhileLoop *)
- (* 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.
intros P Q R H1 H2 H3.
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. (* does nothing: auto doesn't destruct hypotheses! *)
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. *)
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.
(** Now let's take a first pass over [ceval_deterministic] to simplify
the proof script.
*)
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;
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_WhileEnd *)
+ (* b evaluates to true (contradiction) *)
rewrite H in H2. inversion H2.
(* E_WhileLoop *)
- (* 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,
c / st \\ st1 ->
c / st \\ 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_WhileEnd *)
+ (* b evaluates to true (contradiction) *)
rewrite H in H2. inversion H2.
(* E_WhileLoop *)
- (* 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 Coq's tactic programming
language, 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;
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_WhileEnd *)
+ (* b evaluates to true (contradiction) *)
rwinv H H2.
(* E_WhileLoop *)
- (* 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 is a bit better, but not much. 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.
(** The [match goal] tactic 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,
c / st \\ st1 ->
c / st \\ 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_WhileLoop *)
+ (* 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,
c / st \\ st1 ->
c / st \\ 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_WhileLoop *)
+ (* 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,
c / st \\ st1 ->
c / st \\ 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 robust in the face of modest changes to our language.
For example, we can add a [REPEAT] command to the language. *)
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. *)
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) (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 (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''.
Notation "c1 '/' st '\\' st'" := (ceval st c1 st')
(at level 40, st at level 39).
(** Our first attempt at the proof is disappointing: the [E_RepeatEnd]
and [E_RepeatLoop] cases are not handled by our previous
automation. *)
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;
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.
(** To fix this, we just have to swap the invocations of [find_eqn]
and [find_rwinv]. *)
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;
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. *)
(* ----------------------------------------------------------------- *)
(** *** [eapply] and [eauto] *)
(** 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:
(X ::= ANum 2;;
IFB BLe (AId X) (ANum 1)
THEN Y ::= ANum 3
ELSE Z ::= ANum 4
FI)
/ empty_state
\\ (t_update (t_update empty_state X 2) Z 4).
Proof.
(* We supply the intermediate state [st']... *)
apply E_Seq with (t_update empty_state 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'',
c1 / st \\ st' ->
c2 / st' \\ st'' ->
(c1 ;; c2) / st \\ 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:
(X ::= ANum 2;;
IFB BLe (AId X) (ANum 1)
THEN Y ::= ANum 3
ELSE Z ::= ANum 4
FI)
/ empty_state
\\ (t_update (t_update empty_state X 2) Z 4).
Proof.
eapply E_Seq. (* 1 *)
- apply E_Ass. (* 2 *)
reflexivity. (* 3 *)
- (* 4 *) apply E_IfFalse. reflexivity. apply E_Ass. reflexivity.
Qed.
(** The tactic [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 [e...] variants. For example,
here's a proof using [eauto]: *)
Hint Constructors ceval.
Hint Transparent state.
Hint Transparent total_map.
Definition st12 := t_update (t_update empty_state X 1) Y 2.
Definition st21 := t_update (t_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. eauto. Qed.
(** The [eauto] tactic works just like [auto], except that it uses
[eapply] instead of [apply]. *)
(** $Date: 2016-10-18 15:42:43 -0400 (Tue, 18 Oct 2016) $ *)