# ListsWorking with Structured Data

Require Export Induction.

Module NatList.

Module NatList.

# Pairs of Numbers

Inductive natprod : Type :=

| pair : nat → nat → natprod.

| pair : nat → nat → natprod.

This declaration can be read: "There is just one way to
construct a pair of numbers: by applying the constructor pair to
two arguments of type nat."

Check (pair 3 5).

Here are two simple functions for extracting the first and
second components of a pair. The definitions also illustrate how
to do pattern matching on two-argument constructors.

Definition fst (p : natprod) : nat :=

match p with

| pair x y ⇒ x

end.

Definition snd (p : natprod) : nat :=

match p with

| pair x y ⇒ y

end.

Compute (fst (pair 3 5)).

(* ===> 3 *)

match p with

| pair x y ⇒ x

end.

Definition snd (p : natprod) : nat :=

match p with

| pair x y ⇒ y

end.

Compute (fst (pair 3 5)).

(* ===> 3 *)

Since pairs are used quite a bit, it is nice to be able to
write them with the standard mathematical notation (x,y) instead
of pair x y. We can tell Coq to allow this with a Notation
declaration.

Notation "( x , y )" := (pair x y).

The new pair notation can be used both in expressions and in
pattern matches (indeed, we've actually seen this already in the
Basics chapter, in the definition of the minus function —
this works because the pair notation is also provided as part of
the standard library):

Compute (fst (3,5)).

Definition fst' (p : natprod) : nat :=

match p with

| (x,y) ⇒ x

end.

Definition snd' (p : natprod) : nat :=

match p with

| (x,y) ⇒ y

end.

Definition swap_pair (p : natprod) : natprod :=

match p with

| (x,y) ⇒ (y,x)

end.

Definition fst' (p : natprod) : nat :=

match p with

| (x,y) ⇒ x

end.

Definition snd' (p : natprod) : nat :=

match p with

| (x,y) ⇒ y

end.

Definition swap_pair (p : natprod) : natprod :=

match p with

| (x,y) ⇒ (y,x)

end.

Let's try to prove a few simple facts about pairs.
If we state things in a particular (and slightly peculiar) way, we
can complete proofs with just reflexivity (and its built-in
simplification):

Theorem surjective_pairing' : ∀ (n m : nat),

(n,m) = (fst (n,m), snd (n,m)).

Proof.

reflexivity. Qed.

(n,m) = (fst (n,m), snd (n,m)).

Proof.

reflexivity. Qed.

But reflexivity is not enough if we state the lemma in a more
natural way:

Theorem surjective_pairing_stuck : ∀ (p : natprod),

p = (fst p, snd p).

Proof.

simpl. (* Doesn't reduce anything! *)

Abort.

p = (fst p, snd p).

Proof.

simpl. (* Doesn't reduce anything! *)

Abort.

We have to expose the structure of p so that simpl can
perform the pattern match in fst and snd. We can do this with
destruct.

Theorem surjective_pairing : ∀ (p : natprod),

p = (fst p, snd p).

Proof.

intros p. destruct p as [n m]. simpl. reflexivity. Qed.

p = (fst p, snd p).

Proof.

intros p. destruct p as [n m]. simpl. reflexivity. Qed.

Notice that, unlike its behavior with nats, destruct
generates just one subgoal here. That's because natprods can
only be constructed in one way.

#### Exercise: 1 star (snd_fst_is_swap)

Theorem snd_fst_is_swap : ∀ (p : natprod),

(snd p, fst p) = swap_pair p.

Proof.

(* FILL IN HERE *) Admitted.

☐
(snd p, fst p) = swap_pair p.

Proof.

(* FILL IN HERE *) Admitted.

Theorem fst_swap_is_snd : ∀ (p : natprod),

fst (swap_pair p) = snd p.

Proof.

(* FILL IN HERE *) Admitted.

☐
fst (swap_pair p) = snd p.

Proof.

(* FILL IN HERE *) Admitted.

# Lists of Numbers

*lists*of numbers like this: "A list is either the empty list or else a pair of a number and another list."

Inductive natlist : Type :=

| nil : natlist

| cons : nat → natlist → natlist.

| nil : natlist

| cons : nat → natlist → natlist.

For example, here is a three-element list:

Definition mylist := cons 1 (cons 2 (cons 3 nil)).

As with pairs, it is more convenient to write lists in
familiar programming notation. The following declarations
allow us to use :: as an infix cons operator and square
brackets as an "outfix" notation for constructing lists.

Notation "x :: l" := (cons x l)

(at level 60, right associativity).

Notation "[ ]" := nil.

Notation "[ x ; .. ; y ]" := (cons x .. (cons y nil) ..).

(at level 60, right associativity).

Notation "[ ]" := nil.

Notation "[ x ; .. ; y ]" := (cons x .. (cons y nil) ..).

It is not necessary to understand the details of these
declarations, but in case you are interested, here is roughly
what's going on. The right associativity annotation tells Coq
how to parenthesize expressions involving several uses of :: so
that, for example, the next three declarations mean exactly the
same thing:

Definition mylist1 := 1 :: (2 :: (3 :: nil)).

Definition mylist2 := 1 :: 2 :: 3 :: nil.

Definition mylist3 := [1;2;3].

Definition mylist2 := 1 :: 2 :: 3 :: nil.

Definition mylist3 := [1;2;3].

The at level 60 part tells Coq how to parenthesize
expressions that involve both :: and some other infix operator.
For example, since we defined + as infix notation for the plus
function at level 50,
(Expressions like "1 + 2 :: [3]" can be a little confusing when
you read them in a .v file. The inner brackets, around 3, indicate
a list, but the outer brackets, which are invisible in the HTML
rendering, are there to instruct the "coqdoc" tool that the bracketed
part should be displayed as Coq code rather than running text.)
The second and third Notation declarations above introduce the
standard square-bracket notation for lists; the right-hand side of
the third one illustrates Coq's syntax for declaring n-ary
notations and translating them to nested sequences of binary
constructors.
A number of functions are useful for manipulating lists.
For example, the repeat function takes a number n and a
count and returns a list of length count where every element
is n.

Notation "x + y" := (plus x y)

(at level 50, left associativity).

the + operator will bind tighter than ::, so 1 + 2 :: [3]
will be parsed, as we'd expect, as (1 + 2) :: [3] rather than 1
+ (2 :: [3]).
(at level 50, left associativity).

### Repeat

Fixpoint repeat (n count : nat) : natlist :=

match count with

| O ⇒ nil

| S count' ⇒ n :: (repeat n count')

end.

match count with

| O ⇒ nil

| S count' ⇒ n :: (repeat n count')

end.

Fixpoint length (l:natlist) : nat :=

match l with

| nil ⇒ O

| h :: t ⇒ S (length t)

end.

match l with

| nil ⇒ O

| h :: t ⇒ S (length t)

end.

Fixpoint app (l

match l

| nil ⇒ l

| h :: t ⇒ h :: (app t l

end.

_{1}l_{2}: natlist) : natlist :=match l

_{1}with| nil ⇒ l

_{2}| h :: t ⇒ h :: (app t l

_{2})end.

Actually, app will be used a lot in some parts of what
follows, so it is convenient to have an infix operator for it.

Notation "x ++ y" := (app x y)

(right associativity, at level 60).

Example test_app1: [1;2;3] ++ [4;5] = [1;2;3;4;5].

Proof. reflexivity. Qed.

Example test_app2: nil ++ [4;5] = [4;5].

Proof. reflexivity. Qed.

Example test_app3: [1;2;3] ++ nil = [1;2;3].

Proof. reflexivity. Qed.

(right associativity, at level 60).

Example test_app1: [1;2;3] ++ [4;5] = [1;2;3;4;5].

Proof. reflexivity. Qed.

Example test_app2: nil ++ [4;5] = [4;5].

Proof. reflexivity. Qed.

Example test_app3: [1;2;3] ++ nil = [1;2;3].

Proof. reflexivity. Qed.

### Head (with default) and Tail

Definition hd (default:nat) (l:natlist) : nat :=

match l with

| nil ⇒ default

| h :: t ⇒ h

end.

Definition tl (l:natlist) : natlist :=

match l with

| nil ⇒ nil

| h :: t ⇒ t

end.

Example test_hd

Proof. reflexivity. Qed.

Example test_hd

Proof. reflexivity. Qed.

Example test_tl: tl [1;2;3] = [2;3].

Proof. reflexivity. Qed.

match l with

| nil ⇒ default

| h :: t ⇒ h

end.

Definition tl (l:natlist) : natlist :=

match l with

| nil ⇒ nil

| h :: t ⇒ t

end.

Example test_hd

_{1}: hd 0 [1;2;3] = 1.Proof. reflexivity. Qed.

Example test_hd

_{2}: hd 0 [] = 0.Proof. reflexivity. Qed.

Example test_tl: tl [1;2;3] = [2;3].

Proof. reflexivity. Qed.

### Exercises

#### Exercise: 2 stars, recommended (list_funs)

Complete the definitions of nonzeros, oddmembers and countoddmembers below. Have a look at the tests to understand what these functions should do.
Fixpoint nonzeros (l:natlist) : natlist

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

Example test_nonzeros:

nonzeros [0;1;0;2;3;0;0] = [1;2;3].

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.test_nonzeros *)

Fixpoint oddmembers (l:natlist) : natlist

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

Example test_oddmembers:

oddmembers [0;1;0;2;3;0;0] = [1;3].

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.test_oddmembers *)

Definition countoddmembers (l:natlist) : nat

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

Example test_countoddmembers1:

countoddmembers [1;0;3;1;4;5] = 4.

(* FILL IN HERE *) Admitted.

Example test_countoddmembers2:

countoddmembers [0;2;4] = 0.

(* FILL IN HERE *) Admitted.

Example test_countoddmembers3:

countoddmembers nil = 0.

(* FILL IN HERE *) Admitted.

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

Example test_nonzeros:

nonzeros [0;1;0;2;3;0;0] = [1;2;3].

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.test_nonzeros *)

Fixpoint oddmembers (l:natlist) : natlist

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

Example test_oddmembers:

oddmembers [0;1;0;2;3;0;0] = [1;3].

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.test_oddmembers *)

Definition countoddmembers (l:natlist) : nat

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

Example test_countoddmembers1:

countoddmembers [1;0;3;1;4;5] = 4.

(* FILL IN HERE *) Admitted.

Example test_countoddmembers2:

countoddmembers [0;2;4] = 0.

(* FILL IN HERE *) Admitted.

Example test_countoddmembers3:

countoddmembers nil = 0.

(* FILL IN HERE *) Admitted.

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

Complete the definition of alternate, which "zips up" two lists into one, alternating between elements taken from the first list and elements from the second. See the tests below for more specific examples.
Fixpoint alternate (l

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

Example test_alternate1:

alternate [1;2;3] [4;5;6] = [1;4;2;5;3;6].

(* FILL IN HERE *) Admitted.

Example test_alternate2:

alternate [1] [4;5;6] = [1;4;5;6].

(* FILL IN HERE *) Admitted.

Example test_alternate3:

alternate [1;2;3] [4] = [1;4;2;3].

(* FILL IN HERE *) Admitted.

Example test_alternate4:

alternate [] [20;30] = [20;30].

(* FILL IN HERE *) Admitted.

☐
_{1}l_{2}: natlist) : natlist(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Example test_alternate1:

alternate [1;2;3] [4;5;6] = [1;4;2;5;3;6].

(* FILL IN HERE *) Admitted.

Example test_alternate2:

alternate [1] [4;5;6] = [1;4;5;6].

(* FILL IN HERE *) Admitted.

Example test_alternate3:

alternate [1;2;3] [4] = [1;4;2;3].

(* FILL IN HERE *) Admitted.

Example test_alternate4:

alternate [] [20;30] = [20;30].

(* FILL IN HERE *) Admitted.

### Bags via Lists

Definition bag := natlist.

#### Exercise: 3 stars, recommended (bag_functions)

Complete the following definitions for the functions count, sum, add, and member for bags.
Fixpoint count (v:nat) (s:bag) : nat

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

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

All these proofs can be done just by reflexivity.

Example test_count1: count 1 [1;2;3;1;4;1] = 3.

(* FILL IN HERE *) Admitted.

Example test_count2: count 6 [1;2;3;1;4;1] = 0.

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.test_count2 *)

(* FILL IN HERE *) Admitted.

Example test_count2: count 6 [1;2;3;1;4;1] = 0.

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.test_count2 *)

Multiset sum is similar to set union: sum a b contains all
the elements of a and of b. (Mathematicians usually define
union on multisets a little bit differently — using max instead
of sum — which is why we don't use that name for this operation.)
For sum we're giving you a header that does not give explicit
names to the arguments. Moreover, it uses the keyword
Definition instead of Fixpoint, so even if you had names for
the arguments, you wouldn't be able to process them recursively.
The point of stating the question this way is to encourage you to
think about whether sum can be implemented in another way —
perhaps by using functions that have already been defined.

Definition sum : bag → bag → bag

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

Example test_sum1: count 1 (sum [1;2;3] [1;4;1]) = 3.

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.test_sum1 *)

Definition add (v:nat) (s:bag) : bag

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

Example test_add1: count 1 (add 1 [1;4;1]) = 3.

(* FILL IN HERE *) Admitted.

Example test_add2: count 5 (add 1 [1;4;1]) = 0.

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.test_add1 *)

(* GRADE_THEOREM 0.5: NatList.test_add2 *)

Definition member (v:nat) (s:bag) : bool

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

Example test_member1: member 1 [1;4;1] = true.

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.test_member1 *)

(* GRADE_THEOREM 0.5: NatList.test_member2 *)

Example test_member2: member 2 [1;4;1] = false.

(* FILL IN HERE *) Admitted.

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

Example test_sum1: count 1 (sum [1;2;3] [1;4;1]) = 3.

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.test_sum1 *)

Definition add (v:nat) (s:bag) : bag

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

Example test_add1: count 1 (add 1 [1;4;1]) = 3.

(* FILL IN HERE *) Admitted.

Example test_add2: count 5 (add 1 [1;4;1]) = 0.

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.test_add1 *)

(* GRADE_THEOREM 0.5: NatList.test_add2 *)

Definition member (v:nat) (s:bag) : bool

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

Example test_member1: member 1 [1;4;1] = true.

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.test_member1 *)

(* GRADE_THEOREM 0.5: NatList.test_member2 *)

Example test_member2: member 2 [1;4;1] = false.

(* FILL IN HERE *) Admitted.

#### Exercise: 3 stars, optional (bag_more_functions)

Here are some more bag functions for you to practice with.
Fixpoint remove_one (v:nat) (s:bag) : bag

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

Example test_remove_one1:

count 5 (remove_one 5 [2;1;5;4;1]) = 0.

(* FILL IN HERE *) Admitted.

Example test_remove_one2:

count 5 (remove_one 5 [2;1;4;1]) = 0.

(* FILL IN HERE *) Admitted.

Example test_remove_one3:

count 4 (remove_one 5 [2;1;4;5;1;4]) = 2.

(* FILL IN HERE *) Admitted.

Example test_remove_one4:

count 5 (remove_one 5 [2;1;5;4;5;1;4]) = 1.

(* FILL IN HERE *) Admitted.

Fixpoint remove_all (v:nat) (s:bag) : bag

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

Example test_remove_all1: count 5 (remove_all 5 [2;1;5;4;1]) = 0.

(* FILL IN HERE *) Admitted.

Example test_remove_all2: count 5 (remove_all 5 [2;1;4;1]) = 0.

(* FILL IN HERE *) Admitted.

Example test_remove_all3: count 4 (remove_all 5 [2;1;4;5;1;4]) = 2.

(* FILL IN HERE *) Admitted.

Example test_remove_all4: count 5 (remove_all 5 [2;1;5;4;5;1;4;5;1;4]) = 0.

(* FILL IN HERE *) Admitted.

Fixpoint subset (s

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

Example test_subset1: subset [1;2] [2;1;4;1] = true.

(* FILL IN HERE *) Admitted.

Example test_subset2: subset [1;2;2] [2;1;4;1] = false.

(* FILL IN HERE *) Admitted.

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

Example test_remove_one1:

count 5 (remove_one 5 [2;1;5;4;1]) = 0.

(* FILL IN HERE *) Admitted.

Example test_remove_one2:

count 5 (remove_one 5 [2;1;4;1]) = 0.

(* FILL IN HERE *) Admitted.

Example test_remove_one3:

count 4 (remove_one 5 [2;1;4;5;1;4]) = 2.

(* FILL IN HERE *) Admitted.

Example test_remove_one4:

count 5 (remove_one 5 [2;1;5;4;5;1;4]) = 1.

(* FILL IN HERE *) Admitted.

Fixpoint remove_all (v:nat) (s:bag) : bag

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

Example test_remove_all1: count 5 (remove_all 5 [2;1;5;4;1]) = 0.

(* FILL IN HERE *) Admitted.

Example test_remove_all2: count 5 (remove_all 5 [2;1;4;1]) = 0.

(* FILL IN HERE *) Admitted.

Example test_remove_all3: count 4 (remove_all 5 [2;1;4;5;1;4]) = 2.

(* FILL IN HERE *) Admitted.

Example test_remove_all4: count 5 (remove_all 5 [2;1;5;4;5;1;4;5;1;4]) = 0.

(* FILL IN HERE *) Admitted.

Fixpoint subset (s

_{1}:bag) (s_{2}:bag) : bool(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Example test_subset1: subset [1;2] [2;1;4;1] = true.

(* FILL IN HERE *) Admitted.

Example test_subset2: subset [1;2;2] [2;1;4;1] = false.

(* FILL IN HERE *) Admitted.

#### Exercise: 3 stars, recommended (bag_theorem)

Write down an interesting theorem bag_theorem about bags involving the functions count and add, and prove it. Note that, since this problem is somewhat open-ended, it's possible that you may come up with a theorem which is true, but whose proof requires techniques you haven't learned yet. Feel free to ask for help if you get stuck!
(*

Theorem bag_theorem : ...

Proof.

...

Qed.

*)

☐
Theorem bag_theorem : ...

Proof.

...

Qed.

*)

# Reasoning About Lists

Theorem nil_app : ∀ l:natlist,

[] ++ l = l.

Proof. reflexivity. Qed.

[] ++ l = l.

Proof. reflexivity. Qed.

... because the [] is substituted into the
"scrutinee" (the expression whose value is being "scrutinized" by
the match) in the definition of app, allowing the match itself
to be simplified.
Also, as with numbers, it is sometimes helpful to perform case
analysis on the possible shapes (empty or non-empty) of an unknown
list.

Theorem tl_length_pred : ∀ l:natlist,

pred (length l) = length (tl l).

Proof.

intros l. destruct l as [| n l'].

- (* l = nil *)

reflexivity.

- (* l = cons n l' *)

reflexivity. Qed.

pred (length l) = length (tl l).

Proof.

intros l. destruct l as [| n l'].

- (* l = nil *)

reflexivity.

- (* l = cons n l' *)

reflexivity. Qed.

Here, the nil case works because we've chosen to define
tl nil = nil. Notice that the as annotation on the destruct
tactic here introduces two names, n and l', corresponding to
the fact that the cons constructor for lists takes two
arguments (the head and tail of the list it is constructing).
Usually, though, interesting theorems about lists require
induction for their proofs.

### Micro-Sermon

## Induction on Lists

*only*possible shapes that elements of an inductively defined set can have, and this fact directly gives rise to a way of reasoning about inductively defined sets: a number is either O or else it is S applied to some

*smaller*number; a list is either nil or else it is cons applied to some number and some

*smaller*list; etc. So, if we have in mind some proposition P that mentions a list l and we want to argue that P holds for

*all*lists, we can reason as follows:

- First, show that P is true of l when l is nil.
- Then show that P is true of l when l is cons n l' for some number n and some smaller list l', assuming that P is true for l'.

Theorem app_assoc : ∀ l

(l

Proof.

intros l

- (* l

reflexivity.

- (* l

simpl. rewrite → IHl1'. reflexivity. Qed.

_{1}l_{2}l_{3}: natlist,(l

_{1}++ l_{2}) ++ l_{3}= l_{1}++ (l_{2}++ l_{3}).Proof.

intros l

_{1}l_{2}l_{3}. induction l_{1}as [| n l_{1}' IHl1'].- (* l

_{1}= nil *)reflexivity.

- (* l

_{1}= cons n l_{1}' *)simpl. rewrite → IHl1'. reflexivity. Qed.

Notice that, as when doing induction on natural numbers, the
as... clause provided to the induction tactic gives a name to
the induction hypothesis corresponding to the smaller list l
For comparison, here is an informal proof of the same theorem.
For a slightly more involved example of inductive proof over
lists, suppose we use app to define a list-reversing function
rev:

_{1}' in the cons case. Once again, this Coq proof is not especially illuminating as a static written document — it is easy to see what's going on if you are reading the proof in an interactive Coq session and you can see the current goal and context at each point, but this state is not visible in the written-down parts of the Coq proof. So a natural-language proof — one written for human readers — will need to include more explicit signposts; in particular, it will help the reader stay oriented if we remind them exactly what the induction hypothesis is in the second case.*Theorem*: For all lists l_{1}, l_{2}, and l_{3}, (l_{1}++ l_{2}) ++ l_{3}= l_{1}++ (l_{2}++ l_{3}).*Proof*: By induction on l_{1}.- First, suppose l
_{1}= []. We must show([] ++ lwhich follows directly from the definition of ++._{2}) ++ l_{3}= [] ++ (l_{2}++ l_{3}), - Next, suppose l
_{1}= n::l_{1}', with(l(the induction hypothesis). We must show_{1}' ++ l_{2}) ++ l_{3}= l_{1}' ++ (l_{2}++ l_{3})((n :: lBy the definition of ++, this follows from_{1}') ++ l_{2}) ++ l_{3}= (n :: l_{1}') ++ (l_{2}++ l_{3}).n :: ((lwhich is immediate from the induction hypothesis. ☐_{1}' ++ l_{2}) ++ l_{3}) = n :: (l_{1}' ++ (l_{2}++ l_{3})),

### Reversing a List

Fixpoint rev (l:natlist) : natlist :=

match l with

| nil ⇒ nil

| h :: t ⇒ rev t ++ [h]

end.

Example test_rev1: rev [1;2;3] = [3;2;1].

Proof. reflexivity. Qed.

Example test_rev2: rev nil = nil.

Proof. reflexivity. Qed.

match l with

| nil ⇒ nil

| h :: t ⇒ rev t ++ [h]

end.

Example test_rev1: rev [1;2;3] = [3;2;1].

Proof. reflexivity. Qed.

Example test_rev2: rev nil = nil.

Proof. reflexivity. Qed.

### Properties of rev

Theorem rev_length_firsttry : ∀ l : natlist,

length (rev l) = length l.

Proof.

intros l. induction l as [| n l' IHl'].

- (* l = *)

reflexivity.

- (* l = n :: l' *)

(* This is the tricky case. Let's begin as usual

by simplifying. *)

simpl.

(* Now we seem to be stuck: the goal is an equality

involving ++, but we don't have any useful equations

in either the immediate context or in the global

environment! We can make a little progress by using

the IH to rewrite the goal... *)

rewrite <- IHl'.

(* ... but now we can't go any further. *)

Abort.

length (rev l) = length l.

Proof.

intros l. induction l as [| n l' IHl'].

- (* l = *)

reflexivity.

- (* l = n :: l' *)

(* This is the tricky case. Let's begin as usual

by simplifying. *)

simpl.

(* Now we seem to be stuck: the goal is an equality

involving ++, but we don't have any useful equations

in either the immediate context or in the global

environment! We can make a little progress by using

the IH to rewrite the goal... *)

rewrite <- IHl'.

(* ... but now we can't go any further. *)

Abort.

So let's take the equation relating ++ and length that
would have enabled us to make progress and prove it as a separate
lemma.

Theorem app_length : ∀ l

length (l

Proof.

(* WORKED IN CLASS *)

intros l

- (* l

reflexivity.

- (* l

simpl. rewrite → IHl1'. reflexivity. Qed.

_{1}l_{2}: natlist,length (l

_{1}++ l_{2}) = (length l_{1}) + (length l_{2}).Proof.

(* WORKED IN CLASS *)

intros l

_{1}l_{2}. induction l_{1}as [| n l_{1}' IHl1'].- (* l

_{1}= nil *)reflexivity.

- (* l

_{1}= cons *)simpl. rewrite → IHl1'. reflexivity. Qed.

Note that, to make the lemma as general as possible, we
quantify over
Now we can complete the original proof.

*all*natlists, not just those that result from an application of rev. This should seem natural, because the truth of the goal clearly doesn't depend on the list having been reversed. Moreover, it is easier to prove the more general property.
Theorem rev_length : ∀ l : natlist,

length (rev l) = length l.

Proof.

intros l. induction l as [| n l' IHl'].

- (* l = nil *)

reflexivity.

- (* l = cons *)

simpl. rewrite → app_length, plus_comm.

simpl. rewrite → IHl'. reflexivity. Qed.

length (rev l) = length l.

Proof.

intros l. induction l as [| n l' IHl'].

- (* l = nil *)

reflexivity.

- (* l = cons *)

simpl. rewrite → app_length, plus_comm.

simpl. rewrite → IHl'. reflexivity. Qed.

For comparison, here are informal proofs of these two theorems:
The style of these proofs is rather longwinded and pedantic.
After the first few, we might find it easier to follow proofs that
give fewer details (which can easily work out in our own minds or
on scratch paper if necessary) and just highlight the non-obvious
steps. In this more compressed style, the above proof might look
like this:
Which style is preferable in a given situation depends on
the sophistication of the expected audience and how similar the
proof at hand is to ones that the audience will already be
familiar with. The more pedantic style is a good default for our
present purposes.

*Theorem*: For all lists l_{1}and l_{2}, length (l_{1}++ l_{2}) = length l_{1}+ length l_{2}.*Proof*: By induction on l_{1}.- First, suppose l
_{1}= []. We must showlength ([] ++ lwhich follows directly from the definitions of length and ++._{2}) = length [] + length l_{2}, - Next, suppose l
_{1}= n::l_{1}', withlength (lWe must show_{1}' ++ l_{2}) = length l_{1}' + length l_{2}.length ((n::lThis follows directly from the definitions of length and ++ together with the induction hypothesis. ☐_{1}') ++ l_{2}) = length (n::l_{1}') + length l_{2}).

*Theorem*: For all lists l, length (rev l) = length l.*Proof*: By induction on l.- First, suppose l = []. We must show
length (rev []) = length [],which follows directly from the definitions of length and rev.
- Next, suppose l = n::l', with
length (rev l') = length l'.We must showlength (rev (n :: l')) = length (n :: l').By the definition of rev, this follows fromlength ((rev l') ++ [n]) = S (length l')which, by the previous lemma, is the same aslength (rev l') + length [n] = S (length l').This follows directly from the induction hypothesis and the definition of length. ☐

*Theorem*: For all lists l, length (rev l) = length l.*Proof*: First, observe that length (l ++ [n]) = S (length l) for any l (this follows by a straightforward induction on l). The main property again follows by induction on l, using the observation together with the induction hypothesis in the case where l = n'::l'. ☐## Search

(* Search rev. *)

Keep Search in mind as you do the following exercises and
throughout the rest of the book; it can save you a lot of time!
If you are using ProofGeneral, you can run Search with C-c
C-a C-a. Pasting its response into your buffer can be
accomplished with C-c C-;.

Theorem app_nil_r : ∀ l : natlist,

l ++ [] = l.

Proof.

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.app_nil_r *)

Theorem rev_app_distr: ∀ l

rev (l

Proof.

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.rev_app_distr *)

Theorem rev_involutive : ∀ l : natlist,

rev (rev l) = l.

Proof.

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.rev_involutive *)

l ++ [] = l.

Proof.

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.app_nil_r *)

Theorem rev_app_distr: ∀ l

_{1}l_{2}: natlist,rev (l

_{1}++ l_{2}) = rev l_{2}++ rev l_{1}.Proof.

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.rev_app_distr *)

Theorem rev_involutive : ∀ l : natlist,

rev (rev l) = l.

Proof.

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.rev_involutive *)

There is a short solution to the next one. If you find yourself
getting tangled up, step back and try to look for a simpler
way.

Theorem app_assoc4 : ∀ l

l

Proof.

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.app_assoc4 *)

_{1}l_{2}l_{3}l_{4}: natlist,l

_{1}++ (l_{2}++ (l_{3}++ l_{4})) = ((l_{1}++ l_{2}) ++ l_{3}) ++ l_{4}.Proof.

(* FILL IN HERE *) Admitted.

(* GRADE_THEOREM 0.5: NatList.app_assoc4 *)

An exercise about your implementation of nonzeros:

Lemma nonzeros_app : ∀ l

nonzeros (l

Proof.

(* FILL IN HERE *) Admitted.

☐
_{1}l_{2}: natlist,nonzeros (l

_{1}++ l_{2}) = (nonzeros l_{1}) ++ (nonzeros l_{2}).Proof.

(* FILL IN HERE *) Admitted.

#### Exercise: 2 stars (beq_natlist)

Fill in the definition of beq_natlist, which compares lists of numbers for equality. Prove that beq_natlist l l yields true for every list l.
Fixpoint beq_natlist (l

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

Example test_beq_natlist1 :

(beq_natlist nil nil = true).

(* FILL IN HERE *) Admitted.

Example test_beq_natlist2 :

beq_natlist [1;2;3] [1;2;3] = true.

(* FILL IN HERE *) Admitted.

Example test_beq_natlist3 :

beq_natlist [1;2;3] [1;2;4] = false.

(* FILL IN HERE *) Admitted.

Theorem beq_natlist_refl : ∀ l:natlist,

true = beq_natlist l l.

Proof.

(* FILL IN HERE *) Admitted.

☐
_{1}l_{2}: natlist) : bool(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Example test_beq_natlist1 :

(beq_natlist nil nil = true).

(* FILL IN HERE *) Admitted.

Example test_beq_natlist2 :

beq_natlist [1;2;3] [1;2;3] = true.

(* FILL IN HERE *) Admitted.

Example test_beq_natlist3 :

beq_natlist [1;2;3] [1;2;4] = false.

(* FILL IN HERE *) Admitted.

Theorem beq_natlist_refl : ∀ l:natlist,

true = beq_natlist l l.

Proof.

(* FILL IN HERE *) Admitted.

## List Exercises, Part 2

#### Exercise: 1 star (count_member_nonzero)

Theorem count_member_nonzero : ∀ (s : bag),

leb 1 (count 1 (1 :: s)) = true.

Proof.

(* FILL IN HERE *) Admitted.

☐
leb 1 (count 1 (1 :: s)) = true.

Proof.

(* FILL IN HERE *) Admitted.

Theorem ble_n_Sn : ∀ n,

leb n (S n) = true.

Proof.

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

- (* 0 *)

simpl. reflexivity.

- (* S n' *)

simpl. rewrite IHn'. reflexivity. Qed.

leb n (S n) = true.

Proof.

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

- (* 0 *)

simpl. reflexivity.

- (* S n' *)

simpl. rewrite IHn'. reflexivity. Qed.

Theorem remove_decreases_count: ∀ (s : bag),

leb (count 0 (remove_one 0 s)) (count 0 s) = true.

Proof.

(* FILL IN HERE *) Admitted.

☐
leb (count 0 (remove_one 0 s)) (count 0 s) = true.

Proof.

(* FILL IN HERE *) Admitted.

#### Exercise: 3 stars, optional (bag_count_sum)

Write down an interesting theorem bag_count_sum about bags involving the functions count and sum, and prove it using Coq. (You may find that the difficulty of the proof depends on how you defined count!)
(* FILL IN HERE *)

☐
#### Exercise: 4 stars, advanced (rev_injective)

Prove that the rev function is injective — that is,
∀ (l

(There is a hard way and an easy way to do this.)
_{1}l_{2}: natlist), rev l_{1}= rev l_{2}→ l_{1}= l_{2}.
(* FILL IN HERE *)

☐
# Options

Fixpoint nth_bad (l:natlist) (n:nat) : nat :=

match l with

| nil ⇒ 42 (* arbitrary! *)

| a :: l' ⇒ match beq_nat n O with

| true ⇒ a

| false ⇒ nth_bad l' (pred n)

end

end.

match l with

| nil ⇒ 42 (* arbitrary! *)

| a :: l' ⇒ match beq_nat n O with

| true ⇒ a

| false ⇒ nth_bad l' (pred n)

end

end.

This solution is not so good: If nth_bad returns 42, we
can't tell whether that value actually appears on the input
without further processing. A better alternative is to change the
return type of nth_bad to include an error value as a possible
outcome. We call this type natoption.

Inductive natoption : Type :=

| Some : nat → natoption

| None : natoption.

| Some : nat → natoption

| None : natoption.

We can then change the above definition of nth_bad to
return None when the list is too short and Some a when the
list has enough members and a appears at position n. We call
this new function nth_error to indicate that it may result in an
error.

Fixpoint nth_error (l:natlist) (n:nat) : natoption :=

match l with

| nil ⇒ None

| a :: l' ⇒ match beq_nat n O with

| true ⇒ Some a

| false ⇒ nth_error l' (pred n)

end

end.

Example test_nth_error1 : nth_error [4;5;6;7] 0 = Some 4.

match l with

| nil ⇒ None

| a :: l' ⇒ match beq_nat n O with

| true ⇒ Some a

| false ⇒ nth_error l' (pred n)

end

end.

Example test_nth_error1 : nth_error [4;5;6;7] 0 = Some 4.

Proof. reflexivity. Qed.

Example test_nth_error2 : nth_error [4;5;6;7] 3 = Some 7.
Proof. reflexivity. Qed.

Example test_nth_error3 : nth_error [4;5;6;7] 9 = None.
Proof. reflexivity. Qed.

(In the HTML version, the boilerplate proofs of these
examples are elided. Click on a box if you want to see one.)
This example is also an opportunity to introduce one more small
feature of Coq's programming language: conditional
expressions...

Fixpoint nth_error' (l:natlist) (n:nat) : natoption :=

match l with

| nil ⇒ None

| a :: l' ⇒ if beq_nat n O then Some a

else nth_error' l' (pred n)

end.

match l with

| nil ⇒ None

| a :: l' ⇒ if beq_nat n O then Some a

else nth_error' l' (pred n)

end.

Coq's conditionals are exactly like those found in any other
language, with one small generalization. Since the boolean type
is not built in, Coq actually supports conditional expressions over
The function below pulls the nat out of a natoption, returning
a supplied default in the None case.

*any*inductively defined type with exactly two constructors. The guard is considered true if it evaluates to the first constructor in the Inductive definition and false if it evaluates to the second.
Definition option_elim (d : nat) (o : natoption) : nat :=

match o with

| Some n' ⇒ n'

| None ⇒ d

end.

match o with

| Some n' ⇒ n'

| None ⇒ d

end.

#### Exercise: 2 stars (hd_error)

Using the same idea, fix the hd function from earlier so we don't have to pass a default element for the nil case.
Definition hd_error (l : natlist) : natoption

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

Example test_hd_error1 : hd_error [] = None.

(* FILL IN HERE *) Admitted.

Example test_hd_error2 : hd_error [1] = Some 1.

(* FILL IN HERE *) Admitted.

Example test_hd_error3 : hd_error [5;6] = Some 5.

(* FILL IN HERE *) Admitted.

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

Example test_hd_error1 : hd_error [] = None.

(* FILL IN HERE *) Admitted.

Example test_hd_error2 : hd_error [1] = Some 1.

(* FILL IN HERE *) Admitted.

Example test_hd_error3 : hd_error [5;6] = Some 5.

(* FILL IN HERE *) Admitted.

Theorem option_elim_hd : ∀ (l:natlist) (default:nat),

hd default l = option_elim default (hd_error l).

Proof.

(* FILL IN HERE *) Admitted.

☐
hd default l = option_elim default (hd_error l).

Proof.

(* FILL IN HERE *) Admitted.

End NatList.

# Partial Maps

*partial map*data type, analogous to the map or dictionary data structures found in most programming languages.

Inductive id : Type :=

| Id : nat → id.

| Id : nat → id.

Internally, an id is just a number. Introducing a separate type
by wrapping each nat with the tag Id makes definitions more
readable and gives us the flexibility to change representations
later if we wish.
We'll also need an equality test for ids:

Definition beq_id (x

match x

| Id n

end.

_{1}x_{2}: id) :=match x

_{1}, x_{2}with| Id n

_{1}, Id n_{2}⇒ beq_nat n_{1}n_{2}end.

Theorem beq_id_refl : ∀ x, true = beq_id x x.

Proof.

(* FILL IN HERE *) Admitted.

☐
Proof.

(* FILL IN HERE *) Admitted.

Module PartialMap.

Export NatList.

Inductive partial_map : Type :=

| empty : partial_map

| record : id → nat → partial_map → partial_map.

Export NatList.

Inductive partial_map : Type :=

| empty : partial_map

| record : id → nat → partial_map → partial_map.

This declaration can be read: "There are two ways to construct a
partial_map: either using the constructor empty to represent an
empty partial map, or by applying the constructor record to
a key, a value, and an existing partial_map to construct a
partial_map with an additional key-to-value mapping."
The update function overrides the entry for a given key in a
partial map (or adds a new entry if the given key is not already
present).

Definition update (d : partial_map)

(x : id) (value : nat)

: partial_map :=

record x value d.

(x : id) (value : nat)

: partial_map :=

record x value d.

Last, the find function searches a partial_map for a given
key. It returns None if the key was not found and Some val if
the key was associated with val. If the same key is mapped to
multiple values, find will return the first one it
encounters.

Fixpoint find (x : id) (d : partial_map) : natoption :=

match d with

| empty ⇒ None

| record y v d' ⇒ if beq_id x y

then Some v

else find x d'

end.

match d with

| empty ⇒ None

| record y v d' ⇒ if beq_id x y

then Some v

else find x d'

end.

Theorem update_eq :

∀ (d : partial_map) (x : id) (v: nat),

find x (update d x v) = Some v.

Proof.

(* FILL IN HERE *) Admitted.

☐
∀ (d : partial_map) (x : id) (v: nat),

find x (update d x v) = Some v.

Proof.

(* FILL IN HERE *) Admitted.

Theorem update_neq :

∀ (d : partial_map) (x y : id) (o: nat),

beq_id x y = false → find x (update d y o) = find x d.

Proof.

(* FILL IN HERE *) Admitted.

☐
∀ (d : partial_map) (x y : id) (o: nat),

beq_id x y = false → find x (update d y o) = find x d.

Proof.

(* FILL IN HERE *) Admitted.

End PartialMap.

Inductive baz : Type :=

| Baz1 : baz → baz

| Baz2 : baz → bool → baz.

| Baz1 : baz → baz

| Baz2 : baz → bool → baz.

How
(* FILL IN HERE *)

☐

*many*elements does the type baz have? (Explain your answer in words, preferrably English.)☐