Auto: More Automation
Set Warnings "notationoverridden,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 builtin 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.
 reflexivity.
 reflexivity.

assert (st' = st'0) as EQ1.
{ apply IHE1_1; apply H1. }
subst st'0.
apply IHE1_2. assumption.

apply IHE1. assumption.

rewrite H in H5. inversion H5.

rewrite H in H5. inversion H5.

apply IHE1. assumption.

reflexivity.

rewrite H in H2. inversion H2.

rewrite H in H4. inversion H4.

assert (st' = st'0) as EQ1.
{ apply IHE1_1; assumption. }
subst st'0.
apply IHE1_2. assumption. Qed.
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.
 reflexivity.
 reflexivity.

assert (st' = st'0) as EQ1.
{ apply IHE1_1; apply H1. }
subst st'0.
apply IHE1_2. assumption.

apply IHE1. assumption.

rewrite H in H5. inversion H5.

rewrite H in H5. inversion H5.

apply IHE1. assumption.

reflexivity.

rewrite H in H2. inversion H2.

rewrite H in H4. inversion H4.

assert (st' = st'0) as EQ1.
{ apply IHE1_1; assumption. }
subst st'0.
apply IHE1_2. assumption. Qed.
The auto Tactic
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:
The auto tactic solves goals that are solvable by any combination of
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:
 intros and
 apply (of hypotheses from the local context, by default).
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.
auto.
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.
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 toplevel 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. Qed.
Definition is_fortytwo x := (x = 42).
Example auto_example_7: forall x,
(x <= 42 /\ 42 <= x) > is_fortytwo x.
Proof.
auto. 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.

assert (st' = st'0) as EQ1 by auto.
subst st'0.
auto.

+
rewrite H in H5. inversion H5.

+
rewrite H in H5. inversion H5.

+
rewrite H in H2. inversion H2.

rewrite H in H4. inversion H4.

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...

assert (st' = st'0) as EQ1...
subst st'0...

+
rewrite H in H5. inversion H5.

+
rewrite H in H5. inversion H5.

+
rewrite H in H2. inversion H2.

rewrite H in H4. inversion H4.

assert (st' = st'0) as EQ1...
subst st'0...
Qed.
Searching For Hypotheses
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.

assert (st' = st'0) as EQ1 by auto.
subst st'0.
auto.

+
rwinv H H5.

+
rwinv H H5.

+
rwinv H H2.

rwinv H H4.

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.
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.

assert (st' = st'0) as EQ1 by auto.
subst st'0.
auto.

+
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.

rewrite (IHE1_1 st'0 H1) in *. auto.

+
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 righthand 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.

+
find_rwinv.

+
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 "hyperautomation"
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
Example ceval_example1:
empty_st =[
X ::= 2;;
TEST X <= 1
THEN Y ::= 3
ELSE Z ::= 4
FI
]=> (Z !> 4 ; X !> 2).
Proof.
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.  apply E_Ass. reflexivity.  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.