# AutoMore Automation

Set Warnings "-notation-overridden,-parsing".

From Coq Require Import Lia.

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
There's another major category of automation we haven't discussed
much yet, namely built-in decision procedures for specific kinds
of problems: lia 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.

*Programming Language Foundations*.Theorem ceval_deterministic: ∀ c st st

_{1}st

_{2},

st =[ c ]=> st

_{1}→

st =[ c ]=> st

_{2}→

st

_{1}= st

_{2}.

Proof.

intros c st st

_{1}st

_{2}E

_{1}E

_{2};

generalize dependent st

_{2};

induction E

_{1}; intros st

_{2}E

_{2}; inversion E

_{2}; subst.

- (* E_Skip *) reflexivity.

- (* E_Ass *) reflexivity.

- (* E_Seq *)

rewrite (IHE1_1 st'0 H

_{1}) in ×.

apply IHE1_2. assumption.

(* E_IfTrue *)

- (* b evaluates to true *)

apply IHE1. assumption.

- (* b evaluates to false (contradiction) *)

rewrite H in H

_{5}. discriminate.

(* E_IfFalse *)

- (* b evaluates to true (contradiction) *)

rewrite H in H

_{5}. discriminate.

- (* b evaluates to false *)

apply IHE1. assumption.

(* E_WhileFalse *)

- (* b evaluates to false *)

reflexivity.

- (* b evaluates to true (contradiction) *)

rewrite H in H

_{2}. discriminate.

(* E_WhileTrue *)

- (* b evaluates to false (contradiction) *)

rewrite H in H

_{4}. discriminate.

- (* b evaluates to true *)

rewrite (IHE1_1 st'0 H

_{3}) in ×.

apply IHE1_2. assumption. Qed.

# The auto Tactic

Example auto_example_1 : ∀ (P Q R: Prop),

(P → Q) → (Q → R) → P → R.

Proof.

intros P Q R H

_{1}H

_{2}H

_{3}.

apply H

_{2}. apply H

_{1}. 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 larger example showing auto's power:

- intros and
- apply (of hypotheses from the local context, 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.

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 : ∀ (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.
If we want to see which facts auto is using, we can use
info_auto instead.

We can extend the hint database just for the purposes of one
application of auto by writing "auto using ...".

Lemma le_antisym : ∀ n m: nat, (n ≤ m ∧ m ≤ n) → n = m.

Proof. intros. lia. Qed.

Example auto_example_6 : ∀ n m p : nat,

(n ≤ p → (n ≤ m ∧ m ≤ n)) →

n ≤ p →

n = m.

Proof.

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 : core.
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 : core.
to tell Coq to do a Hint Resolve for
It is also sometimes necessary to add

Hint Unfold d : core.
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
details.

Hint Resolve T : core.

Hint Constructors c : core.

*all*of the constructors from the inductive definition of c.Hint Unfold d : core.

Hint Resolve le_antisym : core.

Example auto_example_6' : ∀ n m p : nat,

(n≤ p → (n ≤ m ∧ m ≤ n)) →

n ≤ p →

n = m.

Proof.

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 : core.

Example auto_example_7' : ∀ x,

(x ≤ 42 ∧ 42 ≤ x) → is_fortytwo x.

Proof.

auto. (* try also: info_auto. *)

Qed.

Let's take a first pass over ceval_deterministic to simplify the
proof script.

Theorem ceval_deterministic': ∀ c st st

_{1}st

_{2},

st =[ c ]=> st

_{1}→

st =[ c ]=> st

_{2}→

st

_{1}= st

_{2}.

Proof.

intros c st st

_{1}st

_{2}E

_{1}E

_{2}.

generalize dependent st

_{2};

induction E

_{1}; intros st

_{2}E

_{2}; inversion E

_{2}; subst; auto.

- (* E_Seq *)

rewrite (IHE1_1 st'0 H

_{1}) in ×.

auto.

- (* E_IfTrue *)

+ (* b evaluates to false (contradiction) *)

rewrite H in H

_{5}. discriminate.

- (* E_IfFalse *)

+ (* b evaluates to true (contradiction) *)

rewrite H in H

_{5}. discriminate.

- (* E_WhileFalse *)

+ (* b evaluates to true (contradiction) *)

rewrite H in H

_{2}. discriminate.

(* E_WhileTrue *)

- (* b evaluates to false (contradiction) *)

rewrite H in H

_{4}. discriminate.

- (* b evaluates to true *)

rewrite (IHE1_1 st'0 H

_{3}) in ×.

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 t

_{1}... as a shorthand for t_{1};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 st

_{1}st

_{2},

st =[ c ]=> st

_{1}→

st =[ c ]=> st

_{2}→

st

_{1}= st

_{2}.

Proof with auto.

intros c st st

generalize dependent st

induction E

intros st

- (* E_Seq *)

rewrite (IHE1_1 st'0 H

- (* E_IfTrue *)

+ (* b evaluates to false (contradiction) *)

rewrite H in H

- (* E_IfFalse *)

+ (* b evaluates to true (contradiction) *)

rewrite H in H

- (* E_WhileFalse *)

+ (* b evaluates to true (contradiction) *)

rewrite H in H

(* E_WhileTrue *)

- (* b evaluates to false (contradiction) *)

rewrite H in H

- (* b evaluates to true *)

rewrite (IHE1_1 st'0 H

Qed.

intros c st st

_{1}st_{2}E_{1}E_{2};generalize dependent st

_{2};induction E

_{1};intros st

_{2}E_{2}; inversion E_{2}; subst...- (* E_Seq *)

rewrite (IHE1_1 st'0 H

_{1}) in ×...- (* E_IfTrue *)

+ (* b evaluates to false (contradiction) *)

rewrite H in H

_{5}. discriminate.- (* E_IfFalse *)

+ (* b evaluates to true (contradiction) *)

rewrite H in H

_{5}. discriminate.- (* E_WhileFalse *)

+ (* b evaluates to true (contradiction) *)

rewrite H in H

_{2}. discriminate.(* E_WhileTrue *)

- (* b evaluates to false (contradiction) *)

rewrite H in H

_{4}. discriminate.- (* b evaluates to true *)

rewrite (IHE1_1 st'0 H

_{3}) in ×...Qed.

# Searching For Hypotheses

H

_{1}: beval st b = false

H

_{2}: beval st b = true

_{1}and H

_{2}and do a rewrite following by a discriminate. We'd like to automate this process.

Ltac rwd H

_{1}H

_{2}:= rewrite H

_{1}in H

_{2}; discriminate.

Theorem ceval_deterministic'': ∀ c st st

_{1}st

_{2},

st =[ c ]=> st

_{1}→

st =[ c ]=> st

_{2}→

st

_{1}= st

_{2}.

Proof.

intros c st st

_{1}st

_{2}E

_{1}E

_{2}.

generalize dependent st

_{2};

induction E

_{1}; intros st

_{2}E

_{2}; inversion E

_{2}; subst; auto.

- (* E_Seq *)

rewrite (IHE1_1 st'0 H

_{1}) in ×.

auto.

- (* E_IfTrue *)

+ (* b evaluates to false (contradiction) *)

rwd H H

_{5}.

- (* E_IfFalse *)

+ (* b evaluates to true (contradiction) *)

rwd H H

_{5}.

- (* E_WhileFalse *)

+ (* b evaluates to true (contradiction) *)

rwd H H

_{2}.

(* E_WhileTrue *)

- (* b evaluates to false (contradiction) *)

rwd H H

_{4}.

- (* b evaluates to true *)

rewrite (IHE1_1 st'0 H

_{3}) in ×.

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

_{1}and H_{2}to their names and applies the rwd tactic to H_{1}and H_{2}.Theorem ceval_deterministic''': ∀ c st st

_{1}st

_{2},

st =[ c ]=> st

_{1}→

st =[ c ]=> st

_{2}→

st

_{1}= st

_{2}.

Proof.

intros c st st

_{1}st

_{2}E

_{1}E

_{2}.

generalize dependent st

_{2};

induction E

_{1}; intros st

_{2}E

_{2}; inversion E

_{2}; subst; try find_rwd; auto.

- (* E_Seq *)

rewrite (IHE1_1 st'0 H

_{1}) in ×.

auto.

- (* E_WhileTrue *)

+ (* b evaluates to true *)

rewrite (IHE1_1 st'0 H

_{3}) in ×.

auto. Qed.

Let's see about the remaining cases. Each of them involves
applying rewrting an hypothesis after feeding it with the required
condition. We can automate the task of finding the relevant
hypotheses to rewrite with.

Ltac find_eqn :=

match goal with

H

_{1}: ∀ x, ?P x → ?L = ?R,

H

_{2}: ?P ?X

⊢ _ ⇒ rewrite (H

_{1}X H

_{2}) in ×

end.

The pattern ∀ 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 H_{1}. 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 H_{2}as the required evidence for P X) in all hypotheses and the goal.Theorem ceval_deterministic''''': ∀ c st st

_{1}st

_{2},

st =[ c ]=> st

_{1}→

st =[ c ]=> st

_{2}→

st

_{1}= st

_{2}.

Proof.

intros c st st

_{1}st

_{2}E

_{1}E

_{2}.

generalize dependent st

_{2};

induction E

_{1}; intros st

_{2}E

_{2}; inversion E

_{2}; subst; try find_rwd;

try 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

| CAss (x : string) (a : aexp)

| CSeq (c

_{1}c

_{2}: com)

| CIf (b : bexp) (c

_{1}c

_{2}: 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 "'repeat' x 'until' y 'end'" :=

(CRepeat x y)

(in custom com at level 0,

x at level 99, y at level 99).

Notation "'skip'" :=

CSkip (in custom com at level 0).

Notation "x := y" :=

(CAss x y)

(in custom com at level 0, x constr at level 0,

y at level 85, no associativity).

Notation "x ; y" :=

(CSeq x y)

(in custom com at level 90, right associativity).

Notation "'if' x 'then' y 'else' z 'end'" :=

(CIf x y z)

(in custom com at level 89, x at level 99,

y at level 99, z at level 99).

Notation "'while' x 'do' y 'end'" :=

(CWhile x y)

(in custom com at level 89, x at level 99, y at level 99).

Reserved Notation "st '=[' c ']=>' st'"

(at level 40, c custom com at level 99, st' constr at next level).

Inductive ceval : com → state → state → Prop :=

| E_Skip : ∀ st,

st =[ skip ]=> st

| E_Ass : ∀ st a

_{1}n x,

aeval st a

_{1}= n →

st =[ x := a

_{1}]=> (x !-> n ; st)

| E_Seq : ∀ c

_{1}c

_{2}st st' st'',

st =[ c

_{1}]=> st' →

st' =[ c

_{2}]=> st'' →

st =[ c

_{1}; c

_{2}]=> st''

| E_IfTrue : ∀ st st' b c

_{1}c

_{2},

beval st b = true →

st =[ c

_{1}]=> st' →

st =[ if b then c

_{1}else c

_{2}end ]=> st'

| E_IfFalse : ∀ st st' b c

_{1}c

_{2},

beval st b = false →

st =[ c

_{2}]=> st' →

st =[ if b then c

_{1}else c

_{2}end ]=> st'

| E_WhileFalse : ∀ b st c,

beval st b = false →

st =[ while b do c end ]=> st

| E_WhileTrue : ∀ st st' st'' b c,

beval st b = true →

st =[ c ]=> st' →

st' =[ while b do c end ]=> st'' →

st =[ while b do c end ]=> st''

| E_RepeatEnd : ∀ st st' b c,

st =[ c ]=> st' →

beval st' b = true →

st =[ repeat c until b end ]=> st'

| E_RepeatLoop : ∀ st st' st'' b c,

st =[ c ]=> st' →

beval st' b = false →

st' =[ repeat c until b end ]=> st'' →

st =[ repeat c until b end ]=> st''

where "st =[ c ]=> st'" := (ceval c st st').

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: ∀ c st st

_{1}st

_{2},

st =[ c ]=> st

_{1}→

st =[ c ]=> st

_{2}→

st

_{1}= st

_{2}.

Proof.

intros c st st

_{1}st

_{2}E

_{1}E

_{2}.

generalize dependent st

_{2};

induction E

_{1};

intros st

_{2}E

_{2}; inversion E

_{2}; subst; try find_rwd; try find_eqn; auto.

- (* E_RepeatEnd *)

+ (* b evaluates to false (contradiction) *)

find_rwd.

(* oops: why didn't find_rwd solve this for us already?

answer: we did things in the wrong order. *)

- (* E_RepeatLoop *)

+ (* b evaluates to true (contradiction) *)

find_rwd.

Qed.

Fortunately, to fix this, we just have to swap the invocations of
find_eqn and find_rwd.

Theorem ceval_deterministic': ∀ c st st

_{1}st

_{2},

st =[ c ]=> st

_{1}→

st =[ c ]=> st

_{2}→

st

_{1}= st

_{2}.

Proof.

intros c st st

_{1}st

_{2}E

_{1}E

_{2}.

generalize dependent st

_{2};

induction E

_{1};

intros st

_{2}E

_{2}; inversion E

_{2}; subst; try find_eqn; try find_rwd; 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.
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:

# Tactics eapply and eauto

Example ceval_example1:

empty_st =[

X := 2;

if (X ≤ 1)

then Y := 3

else Z := 4

end

]=> (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 : ∀ c

st =[ c

st' =[ c

st =[ c
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:

E_Seq : ∀ c

_{1}c_{2}st st' st'',st =[ c

_{1}]=> st' →st' =[ c

_{2}]=> st'' →st =[ c

_{1};; c_{2}]=> st''Example ceval'_example1:

empty_st =[

X := 2;

if (X ≤ 1)

then Y := 3

else Z := 4

end

]=> (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
Several of the tactics that we've seen so far, including ∃,
constructor, and auto, have similar variants. The eauto
tactic works like auto, except that it uses eapply instead of
apply. Tactic info_eauto shows us which tactics eauto uses
in its proof search.
Below is an example of eauto. Before using it, we need to give
some hints to auto about using the constructors of ceval
and the definitions of state and total_map as part of its
proof search.

*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.Hint Constructors ceval : core.

Hint Transparent state total_map : core.

Example eauto_example : ∃ s',

(Y !-> 1 ; X !-> 2) =[

if (X ≤ Y)

then Z := Y - X

else Y := X + Z

end

]=> s'.

Proof. info_eauto. Qed.

The eauto tactic works just like auto, except that it uses
eapply instead of apply; info_eauto shows us which facts
eauto uses.
Pro tip: One might think that, since eapply and eauto
are more powerful than apply and auto, we should just 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.
In order for Qed to succeed, all existential variables need to
be determined by the end of the proof. Otherwise Coq
will (rightly) refuse to accept the proof. Remember that the Coq
tactics build proof objects, and proof objects containing
existential variables are not complete.

# Constraints on Existential Variables

Lemma silly1 : ∀ (P : nat → nat → Prop) (Q : nat → Prop),

(∀ x y : nat, P x y) →

(∀ x y : nat, P x y → Q x) →

Q 42.

Proof.

intros P Q HP HQ. eapply HQ. apply HP.

Fail Qed.

Coq gives a warning after apply HP. ("All the remaining goals
are on the shelf," means that we've finished all our top-level
proof obligations but along the way we've put some aside to be
done later, and we have not finished those.) Trying to close the
proof with Qed gives an error.

Abort.

An additional constraint is that existential variables cannot be
instantiated with terms containing ordinary variables that did not
exist at the time the existential variable was created. (The
reason for this technical restriction is that allowing such
instantiation would lead to inconsistency of Coq's logic.)

Lemma silly2 :

∀ (P : nat → nat → Prop) (Q : nat → Prop),

(∃ y, P 42 y) →

(∀ x y : nat, P x y → Q x) →

Q 42.

Proof.

intros P Q HP HQ. eapply HQ. destruct HP as [y HP'].

Fail apply HP'.

The error we get, with some details elided, is:

cannot instantiate "?y" because "y" is not in its scope
In this case there is an easy fix: doing destruct HP

cannot instantiate "?y" because "y" is not in its scope

*before*doing eapply HQ.
Abort.

Lemma silly2_fixed :

∀ (P : nat → nat → Prop) (Q : nat → Prop),

(∃ y, P 42 y) →

(∀ x y : nat, P x y → Q x) →

Q 42.

Proof.

intros P Q HP HQ. destruct HP as [y HP'].

eapply HQ. apply HP'.

Qed.

Lemma silly2_fixed :

∀ (P : nat → nat → Prop) (Q : nat → Prop),

(∃ y, P 42 y) →

(∀ x y : nat, P x y → Q x) →

Q 42.

Proof.

intros P Q HP HQ. destruct HP as [y HP'].

eapply HQ. apply HP'.

Qed.

The apply HP' in the last step unifies the existential variable
in the goal with the variable y.
Note that the assumption tactic doesn't work in this case, since
it cannot handle existential variables. However, Coq also
provides an eassumption tactic that solves the goal if one of
the premises matches the goal up to instantiations of existential
variables. We can use it instead of apply HP' if we like.

Lemma silly2_eassumption : ∀ (P : nat → nat → Prop) (Q : nat → Prop),

(∃ y, P 42 y) →

(∀ x y : nat, P x y → Q x) →

Q 42.

Proof.

intros P Q HP HQ. destruct HP as [y HP']. eapply HQ. eassumption.

Qed.

The eauto tactic will use eapply and eassumption, streamlining
the proof even further.