# TacticsMore Basic Tactics

- how to use auxiliary lemmas in both "forward-" and "backward-style" proofs;
- how to reason about data constructors -- in particular, how to use the fact that they are injective and disjoint;
- how to strengthen an induction hypothesis, and when such strengthening is required; and
- more details on how to reason by case analysis.

# The apply Tactic

*exactly*the same as some hypothesis in the context or some previously proved lemma.

Here, we could finish with "rewrite → eq. reflexivity." as we
have done several times before. Or we can finish in a single step
by using apply:

apply eq. Qed.

The apply tactic also works with

*conditional*hypotheses and lemmas: if the statement being applied is an implication, then the premises of this implication will be added to the list of subgoals needing to be proved. apply also works with*conditional*hypotheses:
Theorem silly2 : ∀ (n m o p : nat),

n = m →

(n = m → [n;o] = [m;p]) →

[n;o] = [m;p].

Proof.

intros n m o p eq

apply eq

n = m →

(n = m → [n;o] = [m;p]) →

[n;o] = [m;p].

Proof.

intros n m o p eq

_{1}eq_{2}.apply eq

_{2}. apply eq_{1}. Qed.
Typically, when we use apply H, the statement H will
begin with a ∀ that introduces some
When Coq matches the current goal against the conclusion of H,
it will try to find appropriate values for these variables. For
example, when we do apply eq

*universally quantified variables*._{2}in the following proof, the universal variable q in eq_{2}gets instantiated with n, and r gets instantiated with m.
Theorem silly2a : ∀ (n m : nat),

(n,n) = (m,m) →

(∀ (q r : nat), (q,q) = (r,r) → [q] = [r]) →

[n] = [m].

Proof.

intros n m eq

apply eq

(n,n) = (m,m) →

(∀ (q r : nat), (q,q) = (r,r) → [q] = [r]) →

[n] = [m].

Proof.

intros n m eq

_{1}eq_{2}.apply eq

_{2}. apply eq_{1}. Qed.#### Exercise: 2 stars, standard, optional (silly_ex)

Complete the following proof using only intros and apply.
Theorem silly_ex : ∀ p,

(∀ n, even n = true → even (S n) = false) →

(∀ n, even n = false → odd n = true) →

even p = true →

odd (S p) = true.

Proof.

(* FILL IN HERE *) Admitted.

☐

(∀ n, even n = true → even (S n) = false) →

(∀ n, even n = false → odd n = true) →

even p = true →

odd (S p) = true.

Proof.

(* FILL IN HERE *) Admitted.

☐

Here we cannot use apply directly...

Fail apply H.

but we can use the symmetry tactic, which switches the left
and right sides of an equality in the goal.

symmetry. apply H. Qed.

#### Exercise: 2 stars, standard (apply_exercise1)

You can use apply with previously defined theorems, not just hypotheses in the context. Use Search to find a previously-defined theorem about rev from Lists. Use that theorem as part of your (relatively short) solution to this exercise. You do not need induction.
Theorem rev_exercise1 : ∀ (l l' : list nat),

l = rev l' →

l' = rev l.

Proof.

(* FILL IN HERE *) Admitted.

☐

l = rev l' →

l' = rev l.

Proof.

(* FILL IN HERE *) Admitted.

☐

#### Exercise: 1 star, standard, optional (apply_rewrite)

Briefly explain the difference between the tactics apply and rewrite. What are the situations where both can usefully be applied?
(* FILL IN HERE *)

☐

☐

# The apply with Tactic

Example trans_eq_example : ∀ (a b c d e f : nat),

[a;b] = [c;d] →

[c;d] = [e;f] →

[a;b] = [e;f].

Proof.

intros a b c d e f eq

rewrite → eq

[a;b] = [c;d] →

[c;d] = [e;f] →

[a;b] = [e;f].

Proof.

intros a b c d e f eq

_{1}eq_{2}.rewrite → eq

_{1}. rewrite → eq_{2}. reflexivity. Qed.
Since this is a common pattern, we might like to pull it out as a
lemma that records, once and for all, the fact that equality is
transitive.

Theorem trans_eq : ∀ (X:Type) (n m o : X),

n = m → m = o → n = o.

Proof.

intros X n m o eq

reflexivity. Qed.

n = m → m = o → n = o.

Proof.

intros X n m o eq

_{1}eq_{2}. rewrite → eq_{1}. rewrite → eq_{2}.reflexivity. Qed.

Now, we should be able to use trans_eq to prove the above
example. However, to do this we need a slight refinement of the
apply tactic.

Example trans_eq_example' : ∀ (a b c d e f : nat),

[a;b] = [c;d] →

[c;d] = [e;f] →

[a;b] = [e;f].

Proof.

intros a b c d e f eq

[a;b] = [c;d] →

[c;d] = [e;f] →

[a;b] = [e;f].

Proof.

intros a b c d e f eq

_{1}eq_{2}.
If we simply tell Coq apply trans_eq at this point, it can
tell (by matching the goal against the conclusion of the lemma)
that it should instantiate X with [nat], n with [a,b], and
o with [e,f]. However, the matching process doesn't determine
an instantiation for m: we have to supply one explicitly by
adding "with (m:=[c,d])" to the invocation of apply.

Actually, the name m in the with clause is not required,
since Coq is often smart enough to figure out which variable we
are instantiating. We could instead simply write apply trans_eq
with [c;d].
Coq also has a built-in tactic transitivity that
accomplishes the same purpose as applying trans_eq. The tactic
requires us to state the instantiation we want, just like apply
with does.

Example trans_eq_example'' : ∀ (a b c d e f : nat),

[a;b] = [c;d] →

[c;d] = [e;f] →

[a;b] = [e;f].

Proof.

intros a b c d e f eq

transitivity [c;d].

apply eq

[a;b] = [c;d] →

[c;d] = [e;f] →

[a;b] = [e;f].

Proof.

intros a b c d e f eq

_{1}eq_{2}.transitivity [c;d].

apply eq

_{1}. apply eq_{2}. Qed.
Example trans_eq_exercise : ∀ (n m o p : nat),

m = (minustwo o) →

(n + p) = m →

(n + p) = (minustwo o).

Proof.

(* FILL IN HERE *) Admitted.

☐

m = (minustwo o) →

(n + p) = m →

(n + p) = (minustwo o).

Proof.

(* FILL IN HERE *) Admitted.

☐

# The injection and discriminate Tactics

Inductive nat : Type :=

| O

| S (n : nat). It is obvious from this definition that every number has one of two forms: either it is the constructor O or it is built by applying the constructor S to another number. But there is more here than meets the eye: implicit in the definition are two additional facts:

- The constructor S is
*injective*(or*one-to-one*). That is, if S n = S m, it must also be that n = m. - The constructors O and S are
*disjoint*. That is, O is not equal to S n for any n.

*prove*the injectivity of S by using the pred function defined in Basics.v.

Theorem S_injective : ∀ (n m : nat),

S n = S m →

n = m.

Proof.

intros n m H

assert (H

rewrite H

Qed.

S n = S m →

n = m.

Proof.

intros n m H

_{1}.assert (H

_{2}: n = pred (S n)). { reflexivity. }rewrite H

_{2}. rewrite H_{1}. simpl. reflexivity.Qed.

This technique can be generalized to any constructor by
writing the equivalent of pred -- i.e., writing a function that
"undoes" one application of the constructor.
As a more convenient alternative, Coq provides a tactic called
injection that allows us to exploit the injectivity of any
constructor. Here is an alternate proof of the above theorem
using injection:

By writing injection H as Hmn at this point, we are asking Coq
to generate all equations that it can infer from H using the
injectivity of constructors (in the present example, the equation
n = m). Each such equation is added as a hypothesis (called
Hmn in this case) into the context.

injection H as Hnm. apply Hnm.

Qed.

Qed.

Here's a more interesting example that shows how injection can
derive multiple equations at once.

Theorem injection_ex

[n;m] = [o;o] →

n = m.

Proof.

intros n m o H.

(* WORKED IN CLASS *)

injection H as H

rewrite H

Qed.

_{1}: ∀ (n m o : nat),[n;m] = [o;o] →

n = m.

Proof.

intros n m o H.

(* WORKED IN CLASS *)

injection H as H

_{1}H_{2}.rewrite H

_{1}. rewrite H_{2}. reflexivity.Qed.

Example injection_ex

x :: y :: l = z :: j →

j = z :: l →

x = y.

Proof.

(* FILL IN HERE *) Admitted.

☐

_{3}: ∀ (X : Type) (x y z : X) (l j : list X),x :: y :: l = z :: j →

j = z :: l →

x = y.

Proof.

(* FILL IN HERE *) Admitted.

☐

*assumed*that two such terms are equal, we are justified in concluding anything we want, since the assumption is nonsensical.

Theorem discriminate_ex

false = true →

n = m.

Proof.

intros n m contra. discriminate contra. Qed.

Theorem discriminate_ex

S n = O →

2 + 2 = 5.

Proof.

intros n contra. discriminate contra. Qed.

_{1}: ∀ (n m : nat),false = true →

n = m.

Proof.

intros n m contra. discriminate contra. Qed.

Theorem discriminate_ex

_{2}: ∀ (n : nat),S n = O →

2 + 2 = 5.

Proof.

intros n contra. discriminate contra. Qed.

These examples are instances of a logical principle known as the
If you find the principle of explosion confusing, remember
that these proofs are
We'll explore the principle of explosion in more detail in the
next chapter.
#### Exercise: 1 star, standard (discriminate_ex

*principle of explosion*, which asserts that a contradictory hypothesis entails anything (even manifestly false things!).*not*showing that the conclusion of the statement holds. Rather, they are showing that,*if*the nonsensical situation described by the premise did somehow hold,*then*the nonsensical conclusion would also follow, because we'd be living in an inconsistent universe where every statement is true.#### Exercise: 1 star, standard (discriminate_ex_{3})

Example discriminate_ex

∀ (X : Type) (x y z : X) (l j : list X),

x :: y :: l = [] →

x = z.

Proof.

(* FILL IN HERE *) Admitted.

☐

_{3}:∀ (X : Type) (x y z : X) (l j : list X),

x :: y :: l = [] →

x = z.

Proof.

(* FILL IN HERE *) Admitted.

☐

We can proceed by case analysis on n. The first case is
trivial.

destruct n as [| n'] eqn:E.

- (* n = 0 *)

intros H. reflexivity.

- (* n = 0 *)

intros H. reflexivity.

However, the second one doesn't look so simple: assuming 0
=? (S n') = true, we must show S n' = 0! The way forward is to
observe that the assumption itself is nonsensical:

- (* n = S n' *)

simpl.

simpl.

If we use discriminate on this hypothesis, Coq confirms
that the subgoal we are working on is impossible and removes it
from further consideration.

intros H. discriminate H.

Qed.

Qed.

The injectivity of constructors allows us to reason that
∀ (n m : nat), S n = S m → n = m. The converse of this
implication is an instance of a more general fact about both
constructors and functions, which we will find convenient
below:

Theorem f_equal : ∀ (A B : Type) (f: A → B) (x y: A),

x = y → f x = f y.

Proof. intros A B f x y eq. rewrite eq. reflexivity. Qed.

Theorem eq_implies_succ_equal : ∀ (n m : nat),

n = m → S n = S m.

Proof. intros n m H. apply f_equal. apply H. Qed.

x = y → f x = f y.

Proof. intros A B f x y eq. rewrite eq. reflexivity. Qed.

Theorem eq_implies_succ_equal : ∀ (n m : nat),

n = m → S n = S m.

Proof. intros n m H. apply f_equal. apply H. Qed.

Indeed, there is also a tactic named `f_equal` that can
prove such theorems directly. Given a goal of the form f a

_{1}... an = g b_{1}... bn, the tactic f_equal will produce subgoals of the form f = g, a_{1}= b_{1}, ..., an = bn. At the same time, any of these subgoals that are simple enough (e.g., immediately provable by reflexivity) will be automatically discharged by f_equal.
Theorem eq_implies_succ_equal' : ∀ (n m : nat),

n = m → S n = S m.

Proof. intros n m H. f_equal. apply H. Qed.

n = m → S n = S m.

Proof. intros n m H. f_equal. apply H. Qed.

# Using Tactics on Hypotheses

Theorem S_inj : ∀ (n m : nat) (b : bool),

((S n) =? (S m)) = b →

(n =? m) = b.

Proof.

intros n m b H. simpl in H. apply H. Qed.

((S n) =? (S m)) = b →

(n =? m) = b.

Proof.

intros n m b H. simpl in H. apply H. Qed.

Similarly, apply L in H matches some conditional statement
L (of the form X → Y, say) against a hypothesis H in the
context. However, unlike ordinary apply (which rewrites a goal
matching Y into a subgoal X), apply L in H matches H
against X and, if successful, replaces it with Y.
In other words, apply L in H gives us a form of "forward
reasoning": given X → Y and a hypothesis matching X, it
produces a hypothesis matching Y.
By contrast, apply L is "backward reasoning": it says that if we
know X → Y and we are trying to prove Y, it suffices to prove
X.
Here is a variant of a proof from above, using forward reasoning
throughout instead of backward reasoning.

Theorem silly4 : ∀ (n m p q : nat),

(n = m → p = q) →

m = n →

q = p.

Proof.

intros n m p q EQ H.

symmetry in H. apply EQ in H. symmetry in H.

apply H. Qed.

(n = m → p = q) →

m = n →

q = p.

Proof.

intros n m p q EQ H.

symmetry in H. apply EQ in H. symmetry in H.

apply H. Qed.

Forward reasoning starts from what is
The informal proofs seen in math or computer science classes tend
to use forward reasoning. By contrast, idiomatic use of Coq
generally favors backward reasoning, though in some situations the
forward style can be easier to think about.

*given*(premises, previously proven theorems) and iteratively draws conclusions from them until the goal is reached. Backward reasoning starts from the*goal*and iteratively reasons about what would imply the goal, until premises or previously proven theorems are reached.# Specializing Hypotheses

Theorem specialize_example: ∀ n,

(∀ m, m×n = 0)

→ n = 0.

Proof.

intros n H.

specialize H with (m := 1).

simpl in H.

rewrite add_comm in H.

simpl in H.

apply H. Qed.

(∀ m, m×n = 0)

→ n = 0.

Proof.

intros n H.

specialize H with (m := 1).

simpl in H.

rewrite add_comm in H.

simpl in H.

apply H. Qed.

Using specialize before apply gives us yet another way to
control where apply does its work.

Example trans_eq_example''' : ∀ (a b c d e f : nat),

[a;b] = [c;d] →

[c;d] = [e;f] →

[a;b] = [e;f].

Proof.

intros a b c d e f eq

specialize trans_eq with (m:=[c;d]) as H.

apply H.

apply eq

apply eq

[a;b] = [c;d] →

[c;d] = [e;f] →

[a;b] = [e;f].

Proof.

intros a b c d e f eq

_{1}eq_{2}.specialize trans_eq with (m:=[c;d]) as H.

apply H.

apply eq

_{1}.apply eq

_{2}. Qed.
Note:

- We can specialize facts in the global context, not just local hypotheses.
- The as... clause at the end tells specialize how to name the new hypothesis in this case.

# Varying the Induction Hypothesis

Theorem double_injective: ∀ n m,

double n = double m →

n = m. The way we start this proof is a bit delicate: if we begin it with

intros n. induction n. then all is well. But if we begin it with introducing both variables

intros n m. induction n. we get stuck in the middle of the inductive case...

Theorem double_injective_FAILED : ∀ n m,

double n = double m →

n = m.

Proof.

intros n m. induction n as [| n' IHn'].

- (* n = O *) simpl. intros eq. destruct m as [| m'] eqn:E.

+ (* m = O *) reflexivity.

+ (* m = S m' *) discriminate eq.

- (* n = S n' *) intros eq. destruct m as [| m'] eqn:E.

+ (* m = O *) discriminate eq.

+ (* m = S m' *) f_equal.

double n = double m →

n = m.

Proof.

intros n m. induction n as [| n' IHn'].

- (* n = O *) simpl. intros eq. destruct m as [| m'] eqn:E.

+ (* m = O *) reflexivity.

+ (* m = S m' *) discriminate eq.

- (* n = S n' *) intros eq. destruct m as [| m'] eqn:E.

+ (* m = O *) discriminate eq.

+ (* m = S m' *) f_equal.

At this point, the induction hypothesis (IHn') does

*not*give us n' = m' -- there is an extra S in the way -- so the goal is not provable.
Abort.

What went wrong?
The problem is that, at the point where we invoke the
induction hypothesis, we have already introduced m into the
context -- intuitively, we have told Coq, "Let's consider some
particular n and m..." and we now have to prove that, if
double n = double m for
The next tactic, induction n says to Coq: We are going to show
the goal by induction on n. That is, we are going to prove, for
holds, by showing
If we look closely at the second statement, it is saying something
rather strange: that, for a
then we can prove
To see why this is strange, let's think of a particular m --
say, 5. The statement is then saying that, if we know
then we can prove
But knowing Q doesn't give us any help at all with proving R!
If we tried to prove R from Q, we would start with something
like "Suppose double (S n) = 10..." but then we'd be stuck:
knowing that double (S n) is 10 tells us nothing helpful about
whether double n is 10 (indeed, it strongly suggests that
double n is
Trying to carry out this proof by induction on n when m is
already in the context doesn't work because we are then trying to
prove a statement involving
A successful proof of double_injective leaves m universally
quantified in the goal statement at the point where the
induction tactic is invoked on n:

*these particular*n and m, then n = m.*all*n, that the proposition- P n = "if double n = double m, then n = m"

- P O
- P n → P (S n)

*particular*m, if we know- "if double n = double m then n = m"

- "if double (S n) = double m then S n = m".

- Q = "if double n = 10 then n = 5"

- R = "if double (S n) = 10 then S n = 5".

*not*10!!), so Q is useless.*every*n but just a*particular*m.
Theorem double_injective : ∀ n m,

double n = double m →

n = m.

Proof.

intros n. induction n as [| n' IHn'].

- (* n = O *) simpl. intros m eq. destruct m as [| m'] eqn:E.

+ (* m = O *) reflexivity.

+ (* m = S m' *) discriminate eq.

- (* n = S n' *)

double n = double m →

n = m.

Proof.

intros n. induction n as [| n' IHn'].

- (* n = O *) simpl. intros m eq. destruct m as [| m'] eqn:E.

+ (* m = O *) reflexivity.

+ (* m = S m' *) discriminate eq.

- (* n = S n' *)

Notice that both the goal and the induction hypothesis are
different this time: the goal asks us to prove something more
general (i.e., we must prove the statement for

*every*m), but the IH is correspondingly more flexible, allowing us to choose any m we like when we apply the IH.intros m eq.

Now we've chosen a particular m and introduced the assumption
that double n = double m. Since we are doing a case analysis on
n, we also need a case analysis on m to keep the two "in sync."

destruct m as [| m'] eqn:E.

+ (* m = O *)

+ (* m = O *)

The 0 case is trivial:

discriminate eq.

+ (* m = S m' *)

f_equal.

Since we are now in the second branch of the destruct m, the
m' mentioned in the context is the predecessor of the m we
started out talking about. Since we are also in the S branch of
the induction, this is perfect: if we instantiate the generic m
in the IH with the current m' (this instantiation is performed
automatically by the apply in the next step), then IHn' gives
us exactly what we need to finish the proof.

apply IHn'. simpl in eq. injection eq as goal. apply goal. Qed.

The thing to take away from all this is that you need to be
careful, when using induction, that you are not trying to prove
something too specific: When proving a property quantified over
variables n and m by induction on n, it is sometimes crucial
to leave m generic.
The following exercise, which further strengthens the link between
=? and =, follows the same pattern.

#### Exercise: 2 stars, standard (eqb_true)

#### Exercise: 2 stars, advanced (eqb_true_informal)

Give a careful informal proof of eqb_true, stating the induction hypothesis explicitly and being as explicit as possible about quantifiers, everywhere.
(* FILL IN HERE *)

(* Do not modify the following line: *)

Definition manual_grade_for_informal_proof : option (nat×string) := None.

☐

(* Do not modify the following line: *)

Definition manual_grade_for_informal_proof : option (nat×string) := None.

☐

#### Exercise: 3 stars, standard, especially useful (plus_n_n_injective)

In addition to being careful about how you use intros, practice using "in" variants in this proof. (Hint: use plus_n_Sm.)*rearrangement*of quantified variables is needed. Suppose, for example, that we wanted to prove double_injective by induction on m instead of n.

Theorem double_injective_take2_FAILED : ∀ n m,

double n = double m →

n = m.

Proof.

intros n m. induction m as [| m' IHm'].

- (* m = O *) simpl. intros eq. destruct n as [| n'] eqn:E.

+ (* n = O *) reflexivity.

+ (* n = S n' *) discriminate eq.

- (* m = S m' *) intros eq. destruct n as [| n'] eqn:E.

+ (* n = O *) discriminate eq.

+ (* n = S n' *) f_equal.

(* We are stuck here, just like before. *)

Abort.

double n = double m →

n = m.

Proof.

intros n m. induction m as [| m' IHm'].

- (* m = O *) simpl. intros eq. destruct n as [| n'] eqn:E.

+ (* n = O *) reflexivity.

+ (* n = S n' *) discriminate eq.

- (* m = S m' *) intros eq. destruct n as [| n'] eqn:E.

+ (* n = O *) discriminate eq.

+ (* n = S n' *) f_equal.

(* We are stuck here, just like before. *)

Abort.

The problem is that, to do induction on m, we must first
introduce n. (If we simply say induction m without
introducing anything first, Coq will automatically introduce n
for us!)
What can we do about this? One possibility is to rewrite the
statement of the lemma so that m is quantified before n. This
works, but it's not nice: We don't want to have to twist the
statements of lemmas to fit the needs of a particular strategy for
proving them! Rather we want to state them in the clearest and
most natural way.
What we can do instead is to first introduce all the quantified
variables and then

*re-generalize*one or more of them, selectively taking variables out of the context and putting them back at the beginning of the goal. The generalize dependent tactic does this.
Theorem double_injective_take2 : ∀ n m,

double n = double m →

n = m.

Proof.

intros n m.

(* n and m are both in the context *)

generalize dependent n.

(* Now n is back in the goal and we can do induction on

m and get a sufficiently general IH. *)

induction m as [| m' IHm'].

- (* m = O *) simpl. intros n eq. destruct n as [| n'] eqn:E.

+ (* n = O *) reflexivity.

+ (* n = S n' *) discriminate eq.

- (* m = S m' *) intros n eq. destruct n as [| n'] eqn:E.

+ (* n = O *) discriminate eq.

+ (* n = S n' *) f_equal.

apply IHm'. injection eq as goal. apply goal. Qed.

double n = double m →

n = m.

Proof.

intros n m.

(* n and m are both in the context *)

generalize dependent n.

(* Now n is back in the goal and we can do induction on

m and get a sufficiently general IH. *)

induction m as [| m' IHm'].

- (* m = O *) simpl. intros n eq. destruct n as [| n'] eqn:E.

+ (* n = O *) reflexivity.

+ (* n = S n' *) discriminate eq.

- (* m = S m' *) intros n eq. destruct n as [| n'] eqn:E.

+ (* n = O *) discriminate eq.

+ (* n = S n' *) f_equal.

apply IHm'. injection eq as goal. apply goal. Qed.

Let's look at an informal proof of this theorem. Note that
the proposition we prove by induction leaves n quantified,
corresponding to the use of generalize dependent in our formal
proof.

*Theorem*: For any nats n and m, if double n = double m, then n = m.*Proof*: Let m be a nat. We prove by induction on m that, for any n, if double n = double m then n = m.- First, suppose m = 0, and suppose n is a number such
that double n = double m. We must show that n = 0.
- Second, suppose m = S m' and that n is again a number such
that double n = double m. We must show that n = S m', with
the induction hypothesis that for every number s, if double s =
double m' then s = m'.

#### Exercise: 3 stars, standard, especially useful (gen_dep_practice)

Prove this by induction on l.
Theorem nth_error_after_last: ∀ (n : nat) (X : Type) (l : list X),

length l = n →

nth_error l n = None.

Proof.

(* FILL IN HERE *) Admitted.

☐

length l = n →

nth_error l n = None.

Proof.

(* FILL IN HERE *) Admitted.

☐

# Unfolding Definitions

...and try to prove a simple fact about square...

...we appear to be stuck: simpl doesn't simplify anything, and
since we haven't proved any other facts about square, there is
nothing we can apply or rewrite with.
To make progress, we can manually unfold the definition of
square:

Now we have plenty to work with: both sides of the equality are
expressions involving multiplication, and we have lots of facts
about multiplication at our disposal. In particular, we know that
it is commutative and associative, and from these it is not hard
to finish the proof.

rewrite mult_assoc.

assert (H : n × m × n = n × n × m).

{ rewrite mul_comm. apply mult_assoc. }

rewrite H. rewrite mult_assoc. reflexivity.

Qed.

assert (H : n × m × n = n × n × m).

{ rewrite mul_comm. apply mult_assoc. }

rewrite H. rewrite mult_assoc. reflexivity.

Qed.

At this point, a bit deeper discussion of unfolding and
simplification is in order.
We already have observed that tactics like simpl, reflexivity,
and apply will often unfold the definitions of functions
automatically when this allows them to make progress. For
example, if we define foo m to be the constant 5...

.... then the simpl in the following proof (or the
reflexivity, if we omit the simpl) will unfold foo m to
(fun x ⇒ 5) m and further simplify this expression to just
5.

But this automatic unfolding is somewhat conservative. For
example, if we define a slightly more complicated function
involving a pattern match...

...then the analogous proof will get stuck:

Fact silly_fact_2_FAILED : ∀ m, bar m + 1 = bar (m + 1) + 1.

Proof.

intros m.

simpl. (* Does nothing! *)

Abort.

Proof.

intros m.

simpl. (* Does nothing! *)

Abort.

The reason that simpl doesn't make progress here is that it
notices that, after tentatively unfolding bar m, it is left with
a match whose scrutinee, m, is a variable, so the match cannot
be simplified further. It is not smart enough to notice that the
two branches of the match are identical, so it gives up on
unfolding bar m and leaves it alone.
Similarly, tentatively unfolding bar (m+1) leaves a match
whose scrutinee is a function application (that cannot itself be
simplified, even after unfolding the definition of +), so
simpl leaves it alone.
At this point, there are two ways to make progress. One is to use
destruct m to break the proof into two cases, each focusing on a
more concrete choice of m (O vs S _). In each case, the
match inside of bar can now make progress, and the proof is
easy to complete.

Fact silly_fact_2 : ∀ m, bar m + 1 = bar (m + 1) + 1.

Proof.

intros m.

destruct m eqn:E.

- simpl. reflexivity.

- simpl. reflexivity.

Qed.

Proof.

intros m.

destruct m eqn:E.

- simpl. reflexivity.

- simpl. reflexivity.

Qed.

This approach works, but it depends on our recognizing that the
match hidden inside bar is what was preventing us from making
progress.
A more straightforward way forward is to explicitly tell Coq to
unfold bar.

Now it is apparent that we are stuck on the match expressions on
both sides of the =, and we can use destruct to finish the
proof without thinking so hard.

destruct m eqn:E.

- reflexivity.

- reflexivity.

Qed.

- reflexivity.

- reflexivity.

Qed.

# Using destruct on Compound Expressions

*expression*. We can also do this with destruct.

Definition sillyfun (n : nat) : bool :=

if n =? 3 then false

else if n =? 5 then false

else false.

Theorem sillyfun_false : ∀ (n : nat),

sillyfun n = false.

Proof.

intros n. unfold sillyfun.

destruct (n =? 3) eqn:E

- (* n =? 3 = true *) reflexivity.

- (* n =? 3 = false *) destruct (n =? 5) eqn:E

+ (* n =? 5 = true *) reflexivity.

+ (* n =? 5 = false *) reflexivity. Qed.

if n =? 3 then false

else if n =? 5 then false

else false.

Theorem sillyfun_false : ∀ (n : nat),

sillyfun n = false.

Proof.

intros n. unfold sillyfun.

destruct (n =? 3) eqn:E

_{1}.- (* n =? 3 = true *) reflexivity.

- (* n =? 3 = false *) destruct (n =? 5) eqn:E

_{2}.+ (* n =? 5 = true *) reflexivity.

+ (* n =? 5 = false *) reflexivity. Qed.

After unfolding sillyfun in the above proof, we find that
we are stuck on if (n =? 3) then ... else .... But either
n is equal to 3 or it isn't, so we can use destruct (eqb
n 3) to let us reason about the two cases.
In general, the destruct tactic can be used to perform case
analysis of the results of arbitrary computations. If e is an
expression whose type is some inductively defined type T, then,
for each constructor c of T, destruct e generates a subgoal
in which all occurrences of e (in the goal and in the context)
are replaced by c.

#### Exercise: 3 stars, standard (combine_split)

Here is an implementation of the split function mentioned in chapter Poly:
Fixpoint split {X Y : Type} (l : list (X×Y))

: (list X) × (list Y) :=

match l with

| [] ⇒ ([], [])

| (x, y) :: t ⇒

match split t with

| (lx, ly) ⇒ (x :: lx, y :: ly)

end

end.

: (list X) × (list Y) :=

match l with

| [] ⇒ ([], [])

| (x, y) :: t ⇒

match split t with

| (lx, ly) ⇒ (x :: lx, y :: ly)

end

end.

Prove that split and combine are inverses in the following
sense:

Theorem combine_split : ∀ X Y (l : list (X × Y)) l

split l = (l

combine l

Proof.

(* FILL IN HERE *) Admitted.

☐

_{1}l_{2},split l = (l

_{1}, l_{2}) →combine l

_{1}l_{2}= l.Proof.

(* FILL IN HERE *) Admitted.

☐

Now suppose that we want to convince Coq that sillyfun1 n
yields true only when n is odd. If we start the proof like
this (with no eqn: on the destruct)...

Theorem sillyfun1_odd_FAILED : ∀ (n : nat),

sillyfun1 n = true →

odd n = true.

Proof.

intros n eq. unfold sillyfun1 in eq.

destruct (n =? 3).

(* stuck... *)

Abort.

sillyfun1 n = true →

odd n = true.

Proof.

intros n eq. unfold sillyfun1 in eq.

destruct (n =? 3).

(* stuck... *)

Abort.

... then we are stuck at this point because the context does
not contain enough information to prove the goal! The problem is
that the substitution performed by destruct is quite brutal --
in this case, it throws away every occurrence of n =? 3, but we
need to keep some memory of this expression and how it was
destructed, because we need to be able to reason that, since we
are assuming n =? 3 = true in this branch of the case analysis,
it must be that n = 3, from which it follows that n is odd.
What we want here is to substitute away all existing occurrences
of n =? 3, but at the same time add an equation to the context
that records which case we are in. This is precisely what the
eqn: qualifier does.

Theorem sillyfun1_odd : ∀ (n : nat),

sillyfun1 n = true →

odd n = true.

Proof.

intros n eq. unfold sillyfun1 in eq.

destruct (n =? 3) eqn:Heqe3.

sillyfun1 n = true →

odd n = true.

Proof.

intros n eq. unfold sillyfun1 in eq.

destruct (n =? 3) eqn:Heqe3.

Now we have the same state as at the point where we got
stuck above, except that the context contains an extra
equality assumption, which is exactly what we need to
make progress.

When we come to the second equality test in the body
of the function we are reasoning about, we can use
eqn: again in the same way, allowing us to finish the
proof.

destruct (n =? 5) eqn:Heqe5.

+ (* e

apply eqb_true in Heqe5.

rewrite → Heqe5. reflexivity.

+ (* e

+ (* e

_{5}= true *)apply eqb_true in Heqe5.

rewrite → Heqe5. reflexivity.

+ (* e

_{5}= false *) discriminate eq. Qed.
Theorem bool_fn_applied_thrice :

∀ (f : bool → bool) (b : bool),

f (f (f b)) = f b.

Proof.

(* FILL IN HERE *) Admitted.

☐

∀ (f : bool → bool) (b : bool),

f (f (f b)) = f b.

Proof.

(* FILL IN HERE *) Admitted.

☐

# Review

*automation*tactics that make Coq help us with low-level details. But basically we've got what we need to get work done.

- intros: move hypotheses/variables from goal to context
- reflexivity: finish the proof (when the goal looks like e =
e)
- apply: prove goal using a hypothesis, lemma, or constructor
- apply... in H: apply a hypothesis, lemma, or constructor to
a hypothesis in the context (forward reasoning)
- apply... with...: explicitly specify values for variables
that cannot be determined by pattern matching
- simpl: simplify computations in the goal
- simpl in H: ... or a hypothesis
- rewrite: use an equality hypothesis (or lemma) to rewrite
the goal
- rewrite ... in H: ... or a hypothesis
- symmetry: changes a goal of the form t=u into u=t
- symmetry in H: changes a hypothesis of the form t=u into
u=t
- transitivity y: prove a goal x=z by proving two new subgoals,
x=y and y=z
- unfold: replace a defined constant by its right-hand side in
the goal
- unfold... in H: ... or a hypothesis
- destruct... as...: case analysis on values of inductively
defined types
- destruct... eqn:...: specify the name of an equation to be
added to the context, recording the result of the case
analysis
- induction... as...: induction on values of inductively
defined types
- injection... as...: reason by injectivity on equalities
between values of inductively defined types
- discriminate: reason by disjointness of constructors on
equalities between values of inductively defined types
- assert (H: e) (or assert (e) as H): introduce a "local
lemma" e and call it H
- generalize dependent x: move the variable x (and anything
else that depends on it) from the context back to an explicit
hypothesis in the goal formula
- f_equal: change a goal of the form f x = f y into x = y

#### Exercise: 3 stars, advanced, optional (eqb_sym_informal)

Give an informal proof of this lemma that corresponds to your formal proof above:
(* FILL IN HERE *)

☐

☐

Theorem eqb_trans : ∀ n m p,

n =? m = true →

m =? p = true →

n =? p = true.

Proof.

(* FILL IN HERE *) Admitted.

☐

n =? m = true →

m =? p = true →

n =? p = true.

Proof.

(* FILL IN HERE *) Admitted.

☐

#### Exercise: 3 stars, advanced (split_combine)

We proved, in an exercise above, that combine is the inverse of split. Complete the definition of split_combine_statement below with a property that states that split is the inverse of combine. Then, prove that the property holds.
Definition split_combine_statement : Prop

(* (": Prop" means that we are giving a name to a

logical proposition here.) *)

(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Theorem split_combine : split_combine_statement.

Proof.

(* FILL IN HERE *) Admitted.

(* Do not modify the following line: *)

Definition manual_grade_for_split_combine : option (nat×string) := None.

☐

(* (": Prop" means that we are giving a name to a

logical proposition here.) *)

(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Theorem split_combine : split_combine_statement.

Proof.

(* FILL IN HERE *) Admitted.

(* Do not modify the following line: *)

Definition manual_grade_for_split_combine : option (nat×string) := None.

☐

Theorem filter_exercise : ∀ (X : Type) (test : X → bool)

(x : X) (l lf : list X),

filter test l = x :: lf →

test x = true.

Proof.

(* FILL IN HERE *) Admitted.

☐

(x : X) (l lf : list X),

filter test l = x :: lf →

test x = true.

Proof.

(* FILL IN HERE *) Admitted.

☐

#### Exercise: 4 stars, advanced, especially useful (forall_exists_challenge)

Define two recursive Fixpoints, forallb and existsb. The first checks whether every element in a list satisfies a given predicate:forallb odd [1;3;5;7;9] = true

forallb negb [false;false] = true

forallb even [0;2;4;5] = false

forallb (eqb 5) [] = true The second checks whether there exists an element in the list that satisfies a given predicate:

existsb (eqb 5) [0;2;3;6] = false

existsb (andb true) [true;true;false] = true

existsb odd [1;0;0;0;0;3] = true

existsb even [] = false Next, define a

*nonrecursive*version of existsb -- call it existsb' -- using forallb and negb.

Fixpoint forallb {X : Type} (test : X → bool) (l : list X) : bool

(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Example test_forallb_1 : forallb odd [1;3;5;7;9] = true.

Proof. (* FILL IN HERE *) Admitted.

Example test_forallb_2 : forallb negb [false;false] = true.

Proof. (* FILL IN HERE *) Admitted.

Example test_forallb_3 : forallb even [0;2;4;5] = false.

Proof. (* FILL IN HERE *) Admitted.

Example test_forallb_4 : forallb (eqb 5) [] = true.

Proof. (* FILL IN HERE *) Admitted.

Fixpoint existsb {X : Type} (test : X → bool) (l : list X) : bool

(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Example test_existsb_1 : existsb (eqb 5) [0;2;3;6] = false.

Proof. (* FILL IN HERE *) Admitted.

Example test_existsb_2 : existsb (andb true) [true;true;false] = true.

Proof. (* FILL IN HERE *) Admitted.

Example test_existsb_3 : existsb odd [1;0;0;0;0;3] = true.

Proof. (* FILL IN HERE *) Admitted.

Example test_existsb_4 : existsb even [] = false.

Proof. (* FILL IN HERE *) Admitted.

Definition existsb' {X : Type} (test : X → bool) (l : list X) : bool

(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Theorem existsb_existsb' : ∀ (X : Type) (test : X → bool) (l : list X),

existsb test l = existsb' test l.

Proof. (* FILL IN HERE *) Admitted.

☐

(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Example test_forallb_1 : forallb odd [1;3;5;7;9] = true.

Proof. (* FILL IN HERE *) Admitted.

Example test_forallb_2 : forallb negb [false;false] = true.

Proof. (* FILL IN HERE *) Admitted.

Example test_forallb_3 : forallb even [0;2;4;5] = false.

Proof. (* FILL IN HERE *) Admitted.

Example test_forallb_4 : forallb (eqb 5) [] = true.

Proof. (* FILL IN HERE *) Admitted.

Fixpoint existsb {X : Type} (test : X → bool) (l : list X) : bool

(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Example test_existsb_1 : existsb (eqb 5) [0;2;3;6] = false.

Proof. (* FILL IN HERE *) Admitted.

Example test_existsb_2 : existsb (andb true) [true;true;false] = true.

Proof. (* FILL IN HERE *) Admitted.

Example test_existsb_3 : existsb odd [1;0;0;0;0;3] = true.

Proof. (* FILL IN HERE *) Admitted.

Example test_existsb_4 : existsb even [] = false.

Proof. (* FILL IN HERE *) Admitted.

Definition existsb' {X : Type} (test : X → bool) (l : list X) : bool

(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Theorem existsb_existsb' : ∀ (X : Type) (test : X → bool) (l : list X),

existsb test l = existsb' test l.

Proof. (* FILL IN HERE *) Admitted.

☐

(* 2023-12-29 17:12 *)