AutoMore Automation
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: ∀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 and eauto tactics
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.
intros P Q R H1 H2 H3.
auto.
Qed.
The auto tactic solves goals that are solvable by any combination of
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:
- intros,
- apply (with a local hypothesis, by default).
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.
Search can 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.
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.
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 : ∀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. (* 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
It is also sometimes necessary to add
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.
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' : ∀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.
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 : ∃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.
Example auto_example_8' : ∃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. info_eauto. 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': ∀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: ∀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.
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 Hypotheses
Ltac rwinv H1 H2 := rewrite H1 in H2; inv H2.
Theorem ceval_deterministic'': ∀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.
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.
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''': ∀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.
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'''': ∀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: ∀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''''': ∀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 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.
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 : ∀st,
ceval st SKIP st
| E_Ass : ∀st a1 n X,
aeval st a1 = n →
ceval st (X ::= a1) (t_update st X n)
| E_Seq : ∀c1 c2 st st' st'',
ceval st c1 st' →
ceval st' c2 st'' →
ceval st (c1 ; c2) st''
| E_IfTrue : ∀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 : ∀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 : ∀b1 st c1,
beval st b1 = false →
ceval st (WHILE b1 DO c1 END) st
| E_WhileLoop : ∀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 : ∀st st' b1 c1,
ceval st c1 st' →
beval st' b1 = true →
ceval st (CRepeat c1 b1) st'
| E_RepeatLoop : ∀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).
Theorem ceval_deterministic: ∀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.
Theorem ceval_deterministic': ∀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 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.