# PolyPolymorphism and Higher-Order Functions

(* Suppress some annoying warnings from Coq: *)

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

Require Export Lists.

(*** Polymorphism *)

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

Require Export Lists.

(*** Polymorphism *)

In this chapter we continue our development of basic
concepts of functional programming. The critical new ideas are

*polymorphism*(abstracting functions over the types of the data they manipulate) and*higher-order functions*(treating functions as data). We begin with polymorphism.## Polymorphic Lists

*could*just define a new inductive datatype for each of these, for example...

Inductive boollist : Type :=

| bool_nil : boollist

| bool_cons : bool → boollist → boollist.

| bool_nil : boollist

| bool_cons : bool → boollist → boollist.

... but this would quickly become tedious, partly because we
have to make up different constructor names for each datatype, but
mostly because we would also need to define new versions of all
our list manipulating functions (length, rev, etc.) for each
new datatype definition.
To avoid all this repetition, Coq supports

*polymorphic*inductive type definitions. For example, here is a*polymorphic list*datatype.
Inductive list (X:Type) : Type :=

| nil : list X

| cons : X → list X → list X.

| nil : list X

| cons : X → list X → list X.

This is exactly like the definition of natlist from the
previous chapter, except that the nat argument to the cons
constructor has been replaced by an arbitrary type X, a binding
for X has been added to the header, and the occurrences of
natlist in the types of the constructors have been replaced by
list X. (We can re-use the constructor names nil and cons
because the earlier definition of natlist was inside of a
Module definition that is now out of scope.)
What sort of thing is list itself? One good way to think
about it is that list is a

*function*from Types to Inductive definitions; or, to put it another way, list is a function from Types to Types. For any particular type X, the type list X is an Inductively defined set of lists whose elements are of type X.
Check list.

(* ===> list : Type -> Type *)

(* ===> list : Type -> Type *)

The parameter X in the definition of list becomes a parameter
to the constructors nil and cons — that is, nil and cons
are now polymorphic constructors, that need to be supplied with
the type of the list they are building. As an example, nil nat
constructs the empty list of type nat.

Check (nil nat).

(* ===> nil nat : list nat *)

(* ===> nil nat : list nat *)

Similarly, cons nat adds an element of type nat to a list of
type list nat. Here is an example of forming a list containing
just the natural number 3.

Check (cons nat 3 (nil nat)).

(* ===> cons nat 3 (nil nat) : list nat *)

(* ===> cons nat 3 (nil nat) : list nat *)

What might the type of nil be? We can read off the type list X
from the definition, but this omits the binding for X which is
the parameter to list. Type → list X does not explain the
meaning of X. (X : Type) → list X comes closer. Coq's
notation for this situation is ∀X : Type, list X.

Check nil.

(* ===> nil : forall X : Type, list X *)

(* ===> nil : forall X : Type, list X *)

Similarly, the type of cons from the definition looks like
X → list X → list X, but using this convention to explain the
meaning of X results in the type ∀X, X → list X → list
X.

Check cons.

(* ===> cons : forall X : Type, X -> list X -> list X *)

(* ===> cons : forall X : Type, X -> list X -> list X *)

(Side note on notation: In .v files, the "forall" quantifier
is spelled out in letters. In the generated HTML files and in the
way various IDEs show .v files (with certain settings of their
display controls), ∀is usually typeset as the usual
mathematical "upside down A," but you'll still see the spelled-out
"forall" in a few places. This is just a quirk of typesetting:
there is no difference in meaning.)
Having to supply a type argument for each use of a list
constructor may seem an awkward burden, but we will soon see
ways of reducing that burden.

Check (cons nat 2 (cons nat 1 (nil nat))).

(We've written nil and cons explicitly here because we haven't
yet defined the [] and :: notations for the new version of
lists. We'll do that in a bit.)
We can now go back and make polymorphic versions of all the
list-processing functions that we wrote before. Here is repeat,
for example:

Fixpoint repeat (X : Type) (x : X) (count : nat) : list X :=

match count with

| 0 ⇒ nil X

| S count' ⇒ cons X x (repeat X x count')

end.

match count with

| 0 ⇒ nil X

| S count' ⇒ cons X x (repeat X x count')

end.

As with nil and cons, we can use repeat by applying it
first to a type and then to an element of this type (and a number):

Example test_repeat1 :

repeat nat 4 2 = cons nat 4 (cons nat 4 (nil nat)).

Proof. reflexivity. Qed.

repeat nat 4 2 = cons nat 4 (cons nat 4 (nil nat)).

Proof. reflexivity. Qed.

To use repeat to build other kinds of lists, we simply
instantiate it with an appropriate type parameter:

Example test_repeat2 :

repeat bool false 1 = cons bool false (nil bool).

Proof. reflexivity. Qed.

Module MumbleGrumble.

repeat bool false 1 = cons bool false (nil bool).

Proof. reflexivity. Qed.

Module MumbleGrumble.

Inductive mumble : Type :=

| a : mumble

| b : mumble → nat → mumble

| c : mumble.

Inductive grumble (X:Type) : Type :=

| d : mumble → grumble X

| e : X → grumble X.

| a : mumble

| b : mumble → nat → mumble

| c : mumble.

Inductive grumble (X:Type) : Type :=

| d : mumble → grumble X

| e : X → grumble X.

Which of the following are well-typed elements of grumble X for
some type X?

☐

- d (b a 5)
- d mumble (b a 5)
- d bool (b a 5)
- e bool true
- e mumble (b c 0)
- e bool (b c 0)
- c

☐

End MumbleGrumble.

### Type Annotation Inference

Fixpoint repeat' X x count : list X :=

match count with

| 0 ⇒ nil X

| S count' ⇒ cons X x (repeat' X x count')

end.

match count with

| 0 ⇒ nil X

| S count' ⇒ cons X x (repeat' X x count')

end.

Indeed it will. Let's see what type Coq has assigned to repeat':

Check repeat'.

(* ===> forall X : Type, X -> nat -> list X *)

Check repeat.

(* ===> forall X : Type, X -> nat -> list X *)

(* ===> forall X : Type, X -> nat -> list X *)

Check repeat.

(* ===> forall X : Type, X -> nat -> list X *)

It has exactly the same type type as repeat. Coq was able
to use
This powerful facility means we don't always have to write
explicit type annotations everywhere, although explicit type
annotations are still quite useful as documentation and sanity
checks, so we will continue to use them most of the time. You
should try to find a balance in your own code between too many
type annotations (which can clutter and distract) and too
few (which forces readers to perform type inference in their heads
in order to understand your code).

*type inference*to deduce what the types of X, x, and count must be, based on how they are used. For example, since X is used as an argument to cons, it must be a Type, since cons expects a Type as its first argument; matching count with 0 and S means it must be a nat; and so on.### Type Argument Synthesis

*unify*all locally available information — the type of the function being applied, the types of the other arguments, and the type expected by the context in which the application appears — to determine what concrete type should replace the _.

repeat' X x count : list X :=

we can also replace the types with _
repeat' (X : _) (x : _) (count : _) : list X :=

to tell Coq to attempt to infer the missing information.
Fixpoint repeat'' X x count : list X :=

match count with

| 0 ⇒ nil _

| S count' ⇒ cons _ x (repeat'' _ x count')

end.

match count with

| 0 ⇒ nil _

| S count' ⇒ cons _ x (repeat'' _ x count')

end.

In this instance, we don't save much by writing _ instead of
X. But in many cases the difference in both keystrokes and
readability is nontrivial. For example, suppose we want to write
down a list containing the numbers 1, 2, and 3. Instead of
writing this...

Definition list123 :=

cons nat 1 (cons nat 2 (cons nat 3 (nil nat))).

cons nat 1 (cons nat 2 (cons nat 3 (nil nat))).

...we can use argument synthesis to write this:

Definition list123' :=

cons _ 1 (cons _ 2 (cons _ 3 (nil _))).

cons _ 1 (cons _ 2 (cons _ 3 (nil _))).

### Implicit Arguments

*always*to infer the type argument(s) of a given function.

Arguments nil {X}.

Arguments cons {X} _ _.

Arguments repeat {X} x count.

Arguments cons {X} _ _.

Arguments repeat {X} x count.

Now, we don't have to supply type arguments at all:

Definition list123'' := cons 1 (cons 2 (cons 3 nil)).

Alternatively, we can declare an argument to be implicit
when defining the function itself, by surrounding it in curly
braces instead of parens. For example:

Fixpoint repeat''' {X : Type} (x : X) (count : nat) : list X :=

match count with

| 0 ⇒ nil

| S count' ⇒ cons x (repeat''' x count')

end.

match count with

| 0 ⇒ nil

| S count' ⇒ cons x (repeat''' x count')

end.

(Note that we didn't even have to provide a type argument to the
recursive call to repeat'''; indeed, it would be invalid to
provide one!)
We will use the latter style whenever possible, but we will
continue to use explicit Argument declarations for Inductive
constructors. The reason for this is that marking the parameter
of an inductive type as implicit causes it to become implicit for
the type itself, not just for its constructors. For instance,
consider the following alternative definition of the list
type:

Inductive list' {X:Type} : Type :=

| nil' : list'

| cons' : X → list' → list'.

| nil' : list'

| cons' : X → list' → list'.

Because X is declared as implicit for the
Let's finish by re-implementing a few other standard list
functions on our new polymorphic lists...

*entire*inductive definition including list' itself, we now have to write just list' whether we are talking about lists of numbers or booleans or anything else, rather than list' nat or list' bool or whatever; this is a step too far.
Fixpoint app {X : Type} (l

: (list X) :=

match l

| nil ⇒ l

| cons h t ⇒ cons h (app t l

end.

Fixpoint rev {X:Type} (l:list X) : list X :=

match l with

| nil ⇒ nil

| cons h t ⇒ app (rev t) (cons h nil)

end.

Fixpoint length {X : Type} (l : list X) : nat :=

match l with

| nil ⇒ 0

| cons _ l' ⇒ S (length l')

end.

Example test_rev1 :

rev (cons 1 (cons 2 nil)) = (cons 2 (cons 1 nil)).

Proof. reflexivity. Qed.

Example test_rev2:

rev (cons true nil) = cons true nil.

Proof. reflexivity. Qed.

Example test_length1: length (cons 1 (cons 2 (cons 3 nil))) = 3.

Proof. reflexivity. Qed.

_{1}l_{2}: list X): (list X) :=

match l

_{1}with| nil ⇒ l

_{2}| cons h t ⇒ cons h (app t l

_{2})end.

Fixpoint rev {X:Type} (l:list X) : list X :=

match l with

| nil ⇒ nil

| cons h t ⇒ app (rev t) (cons h nil)

end.

Fixpoint length {X : Type} (l : list X) : nat :=

match l with

| nil ⇒ 0

| cons _ l' ⇒ S (length l')

end.

Example test_rev1 :

rev (cons 1 (cons 2 nil)) = (cons 2 (cons 1 nil)).

Proof. reflexivity. Qed.

Example test_rev2:

rev (cons true nil) = cons true nil.

Proof. reflexivity. Qed.

Example test_length1: length (cons 1 (cons 2 (cons 3 nil))) = 3.

Proof. reflexivity. Qed.

### Supplying Type Arguments Explicitly

Fail Definition mynil := nil.

(The Fail qualifier that appears before Definition can be
used with
Here, Coq gives us an error because it doesn't know what type
argument to supply to nil. We can help it by providing an
explicit type declaration (so that Coq has more information
available when it gets to the "application" of nil):

*any*command, and is used to ensure that that command indeed fails when executed. If the command does fail, Coq prints the corresponding error message, but continues processing the rest of the file.)
Definition mynil : list nat := nil.

Alternatively, we can force the implicit arguments to be explicit by
prefixing the function name with @.

Check @nil.

Definition mynil' := @nil nat.

Definition mynil' := @nil nat.

Using argument synthesis and implicit arguments, we can
define convenient notation for lists, as before. Since we have
made the constructor type arguments implicit, Coq will know to
automatically infer these when we use the notations.

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

(at level 60, right associativity).

Notation "[ ]" := nil.

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

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

(at level 60, right associativity).

(at level 60, right associativity).

Notation "[ ]" := nil.

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

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

(at level 60, right associativity).

Now lists can be written just the way we'd hope:

Definition list123''' := [1; 2; 3].

### Exercises

#### Exercise: 2 stars, optional (poly_exercises)

Here are a few simple exercises, just like ones in the Lists chapter, for practice with polymorphism. Complete the proofs below.
Theorem app_nil_r : ∀ (X:Type), ∀ l:list X,

l ++ [] = l.

Proof.

(* FILL IN HERE *) Admitted.

Theorem app_assoc : ∀ A (l m n:list A),

l ++ m ++ n = (l ++ m) ++ n.

Proof.

(* FILL IN HERE *) Admitted.

Lemma app_length : ∀ (X:Type) (l

length (l

Proof.

(* FILL IN HERE *) Admitted.

☐
l ++ [] = l.

Proof.

(* FILL IN HERE *) Admitted.

Theorem app_assoc : ∀ A (l m n:list A),

l ++ m ++ n = (l ++ m) ++ n.

Proof.

(* FILL IN HERE *) Admitted.

Lemma app_length : ∀ (X:Type) (l

_{1}l_{2}: list X),length (l

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

(* FILL IN HERE *) Admitted.

Theorem rev_app_distr: ∀ X (l

rev (l

Proof.

(* FILL IN HERE *) Admitted.

Theorem rev_involutive : ∀ X : Type, ∀ l : list X,

rev (rev l) = l.

Proof.

(* FILL IN HERE *) Admitted.

☐
_{1}l_{2}: list X),rev (l

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

(* FILL IN HERE *) Admitted.

Theorem rev_involutive : ∀ X : Type, ∀ l : list X,

rev (rev l) = l.

Proof.

(* FILL IN HERE *) Admitted.

## Polymorphic Pairs

*polymorphic pairs*, often called

*products*:

Inductive prod (X Y : Type) : Type :=

| pair : X → Y → prod X Y.

Arguments pair {X} {Y} _ _.

| pair : X → Y → prod X Y.

Arguments pair {X} {Y} _ _.

As with lists, we make the type arguments implicit and define the
familiar concrete notation.

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

We can also use the Notation mechanism to define the standard
notation for product

*types*:
Notation "X * Y" := (prod X Y) : type_scope.

(The annotation : type_scope tells Coq that this abbreviation
should only be used when parsing types. This avoids a clash with
the multiplication symbol.)
It is easy at first to get (x,y) and X*Y confused.
Remember that (x,y) is a
The first and second projection functions now look pretty
much as they would in any functional programming language.

*value*built from two other values, while X*Y is a*type*built from two other types. If x has type X and y has type Y, then (x,y) has type X*Y.
Definition fst {X Y : Type} (p : X * Y) : X :=

match p with

| (x, y) ⇒ x

end.

Definition snd {X Y : Type} (p : X * Y) : Y :=

match p with

| (x, y) ⇒ y

end.

match p with

| (x, y) ⇒ x

end.

Definition snd {X Y : Type} (p : X * Y) : Y :=

match p with

| (x, y) ⇒ y

end.

The following function takes two lists and combines them
into a list of pairs. In other functional languages, it is often
called zip; we call it combine for consistency with Coq's
standard library.

Fixpoint combine {X Y : Type} (lx : list X) (ly : list Y)

: list (X*Y) :=

match lx, ly with

| [], _ ⇒ []

| _, [] ⇒ []

| x :: tx, y :: ty ⇒ (x, y) :: (combine tx ty)

end.

: list (X*Y) :=

match lx, ly with

| [], _ ⇒ []

| _, [] ⇒ []

| x :: tx, y :: ty ⇒ (x, y) :: (combine tx ty)

end.

#### Exercise: 1 star, optional (combine_checks)

Try answering the following questions on paper and checking your answers in coq:- What is the type of combine (i.e., what does Check @combine print?)
- What does
Compute (combine [1;2] [false;false;true;true]).print?

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

The function split is the right inverse of combine: it takes a list of pairs and returns a pair of lists. In many functional languages, it is called unzip.
Fixpoint split {X Y : Type} (l : list (X*Y))

: (list X) * (list Y)

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

Example test_split:

split [(1,false);(2,false)] = ([1;2],[false;false]).

Proof.

(* FILL IN HERE *) Admitted.

☐
: (list X) * (list Y)

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

Example test_split:

split [(1,false);(2,false)] = ([1;2],[false;false]).

Proof.

(* FILL IN HERE *) Admitted.

## Polymorphic Options

*polymorphic options*, which generalize natoption from the previous chapter:

Inductive option (X:Type) : Type :=

| Some : X → option X

| None : option X.

Arguments Some {X} _.

Arguments None {X}.

| Some : X → option X

| None : option X.

Arguments Some {X} _.

Arguments None {X}.

We can now rewrite the nth_error function so that it works
with any type of lists.

Fixpoint nth_error {X : Type} (l : list X) (n : nat)

: option X :=

match l with

| [] ⇒ None

| a :: l' ⇒ if beq_nat n O then Some a else nth_error l' (pred n)

end.

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

: option X :=

match l with

| [] ⇒ None

| a :: l' ⇒ if beq_nat n O then Some a else nth_error l' (pred n)

end.

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

Proof. reflexivity. Qed.

Example test_nth_error2 : nth_error [[1];[2]] 1 = Some [2].
Proof. reflexivity. Qed.

Example test_nth_error3 : nth_error [true] 2 = None.
Proof. reflexivity. Qed.

#### Exercise: 1 star, optional (hd_error_poly)

Complete the definition of a polymorphic version of the hd_error function from the last chapter. Be sure that it passes the unit tests below.
Definition hd_error {X : Type} (l : list X) : option X

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

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

Once again, to force the implicit arguments to be explicit,
we can use @ before the name of the function.

Check @hd_error.

Example test_hd_error1 : hd_error [1;2] = Some 1.

(* FILL IN HERE *) Admitted.

Example test_hd_error2 : hd_error [[1];[2]] = Some [1].

(* FILL IN HERE *) Admitted.

☐
Example test_hd_error1 : hd_error [1;2] = Some 1.

(* FILL IN HERE *) Admitted.

Example test_hd_error2 : hd_error [[1];[2]] = Some [1].

(* FILL IN HERE *) Admitted.

# Functions as Data

## Higher-Order Functions

*higher-order*functions. Here's a simple one:

Definition doit3times {X:Type} (f:X→X) (n:X) : X :=

f (f (f n)).

f (f (f n)).

The argument f here is itself a function (from X to
X); the body of doit3times applies f three times to some
value n.

Check @doit3times.

(* ===> doit3times : forall X : Type, (X -> X) -> X -> X *)

Example test_doit3times: doit3times minustwo 9 = 3.

Proof. reflexivity. Qed.

Example test_doit3times': doit3times negb true = false.

Proof. reflexivity. Qed.

(* ===> doit3times : forall X : Type, (X -> X) -> X -> X *)

Example test_doit3times: doit3times minustwo 9 = 3.

Proof. reflexivity. Qed.

Example test_doit3times': doit3times negb true = false.

Proof. reflexivity. Qed.

## Filter

*predicate*on X (a function from X to bool) and "filtering" the list, returning a new list containing just those elements for which the predicate returns true.

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

: (list X) :=

match l with

| [] ⇒ []

| h :: t ⇒ if test h then h :: (filter test t)

else filter test t

end.

: (list X) :=

match l with

| [] ⇒ []

| h :: t ⇒ if test h then h :: (filter test t)

else filter test t

end.

For example, if we apply filter to the predicate evenb
and a list of numbers l, it returns a list containing just the
even members of l.

Example test_filter1: filter evenb [1;2;3;4] = [2;4].

Proof. reflexivity. Qed.

Definition length_is_1 {X : Type} (l : list X) : bool :=

beq_nat (length l) 1.

Example test_filter2:

filter length_is_1

[ [1; 2]; [3]; [4]; [5;6;7]; []; [8] ]

= [ [3]; [4]; [8] ].

Proof. reflexivity. Qed.

Proof. reflexivity. Qed.

Definition length_is_1 {X : Type} (l : list X) : bool :=

beq_nat (length l) 1.

Example test_filter2:

filter length_is_1

[ [1; 2]; [3]; [4]; [5;6;7]; []; [8] ]

= [ [3]; [4]; [8] ].

Proof. reflexivity. Qed.

Definition countoddmembers' (l:list nat) : nat :=

length (filter oddb l).

Example test_countoddmembers'1: countoddmembers' [1;0;3;1;4;5] = 4.

Proof. reflexivity. Qed.

Example test_countoddmembers'2: countoddmembers' [0;2;4] = 0.

Proof. reflexivity. Qed.

Example test_countoddmembers'3: countoddmembers' nil = 0.

Proof. reflexivity. Qed.

length (filter oddb l).

Example test_countoddmembers'1: countoddmembers' [1;0;3;1;4;5] = 4.

Proof. reflexivity. Qed.

Example test_countoddmembers'2: countoddmembers' [0;2;4] = 0.

Proof. reflexivity. Qed.

Example test_countoddmembers'3: countoddmembers' nil = 0.

Proof. reflexivity. Qed.

## Anonymous Functions

Example test_anon_fun':

doit3times (fun n ⇒ n * n) 2 = 256.

Proof. reflexivity. Qed.

doit3times (fun n ⇒ n * n) 2 = 256.

Proof. reflexivity. Qed.

The expression (fun n ⇒ n * n) can be read as "the function
that, given a number n, yields n * n."
Here is the filter example, rewritten to use an anonymous
function.

Example test_filter2':

filter (fun l ⇒ beq_nat (length l) 1)

[ [1; 2]; [3]; [4]; [5;6;7]; []; [8] ]

= [ [3]; [4]; [8] ].

Proof. reflexivity. Qed.

filter (fun l ⇒ beq_nat (length l) 1)

[ [1; 2]; [3]; [4]; [5;6;7]; []; [8] ]

= [ [3]; [4]; [8] ].

Proof. reflexivity. Qed.

#### Exercise: 2 stars (filter_even_gt_{7})

Use filter (instead of Fixpoint) to write a Coq function
filter_even_gt_{7}that takes a list of natural numbers as input and returns a list of just those that are even and greater than 7.

Definition filter_even_gt

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

Example test_filter_even_gt7_1 :

filter_even_gt

(* FILL IN HERE *) Admitted.

Example test_filter_even_gt7_2 :

filter_even_gt

(* FILL IN HERE *) Admitted.

☐
_{7}(l : list nat) : list nat(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.

Example test_filter_even_gt7_1 :

filter_even_gt

_{7}[1;2;6;9;10;3;12;8] = [10;12;8].(* FILL IN HERE *) Admitted.

Example test_filter_even_gt7_2 :

filter_even_gt

_{7}[5;2;6;19;129] = [].(* FILL IN HERE *) Admitted.

#### Exercise: 3 stars (partition)

Use filter to write a Coq function partition:
partition : ∀ X : Type,

(X → bool) → list X → list X * list X

Given a set X, a test function of type X → bool and a list
X, partition should return a pair of lists. The first member of
the pair is the sublist of the original list containing the
elements that satisfy the test, and the second is the sublist
containing those that fail the test. The order of elements in the
two sublists should be the same as their order in the original
list.
(X → bool) → list X → list X * list X

Definition partition {X : Type}

(test : X → bool)

(l : list X)

: list X * list X

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

Example test_partition1: partition oddb [1;2;3;4;5] = ([1;3;5], [2;4]).

(* FILL IN HERE *) Admitted.

Example test_partition2: partition (fun x ⇒ false) [5;9;0] = ([], [5;9;0]).

(* FILL IN HERE *) Admitted.

☐
(test : X → bool)

(l : list X)

: list X * list X

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

Example test_partition1: partition oddb [1;2;3;4;5] = ([1;3;5], [2;4]).

(* FILL IN HERE *) Admitted.

Example test_partition2: partition (fun x ⇒ false) [5;9;0] = ([], [5;9;0]).

(* FILL IN HERE *) Admitted.

Fixpoint map {X Y:Type} (f:X→Y) (l:list X) : (list Y) :=

match l with

| [] ⇒ []

| h :: t ⇒ (f h) :: (map f t)

end.

match l with

| [] ⇒ []

| h :: t ⇒ (f h) :: (map f t)

end.

It takes a function f and a list l = [n

_{1}, n_{2}, n_{3}, ...] and returns the list [f n_{1}, f n_{2}, f n_{3},...] , where f has been applied to each element of l in turn. For example:
Example test_map1: map (fun x ⇒ plus 3 x) [2;0;2] = [5;3;5].

Proof. reflexivity. Qed.

Proof. reflexivity. Qed.

The element types of the input and output lists need not be
the same, since map takes

*two*type arguments, X and Y; it can thus be applied to a list of numbers and a function from numbers to booleans to yield a list of booleans:
Example test_map2:

map oddb [2;1;2;5] = [false;true;false;true].

Proof. reflexivity. Qed.

map oddb [2;1;2;5] = [false;true;false;true].

Proof. reflexivity. Qed.

It can even be applied to a list of numbers and
a function from numbers to

*lists*of booleans to yield a*list of lists*of booleans:
Example test_map3:

map (fun n ⇒ [evenb n;oddb n]) [2;1;2;5]

= [[true;false];[false;true];[true;false];[false;true]].

Proof. reflexivity. Qed.

map (fun n ⇒ [evenb n;oddb n]) [2;1;2;5]

= [[true;false];[false;true];[true;false];[false;true]].

Proof. reflexivity. Qed.

### Exercises

#### Exercise: 3 stars (map_rev)

Show that map and rev commute. You may need to define an auxiliary lemma.
Theorem map_rev : ∀ (X Y : Type) (f : X → Y) (l : list X),

map f (rev l) = rev (map f l).

Proof.

(* FILL IN HERE *) Admitted.

☐
map f (rev l) = rev (map f l).

Proof.

(* FILL IN HERE *) Admitted.

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

The function map maps a list X to a list Y using a function of type X → Y. We can define a similar function, flat_map, which maps a list X to a list Y using a function f of type X → list Y. Your definition should work by 'flattening' the results of f, like so:
flat_map (fun n ⇒ [n;n+1;n+2]) [1;5;10]

= [1; 2; 3; 5; 6; 7; 10; 11; 12].

= [1; 2; 3; 5; 6; 7; 10; 11; 12].

Fixpoint flat_map {X Y:Type} (f:X → list Y) (l:list X)

: (list Y)

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

Example test_flat_map1:

flat_map (fun n ⇒ [n;n;n]) [1;5;4]

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

(* FILL IN HERE *) Admitted.

☐
: (list Y)

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

Example test_flat_map1:

flat_map (fun n ⇒ [n;n;n]) [1;5;4]

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

(* FILL IN HERE *) Admitted.

Definition option_map {X Y : Type} (f : X → Y) (xo : option X)

: option Y :=

match xo with

| None ⇒ None

| Some x ⇒ Some (f x)

end.

: option Y :=

match xo with

| None ⇒ None

| Some x ⇒ Some (f x)

end.

#### Exercise: 2 stars, optional (implicit_args)

The definitions and uses of filter and map use implicit arguments in many places. Replace the curly braces around the implicit arguments with parentheses, and then fill in explicit type parameters where necessary and use Coq to check that you've done so correctly. (This exercise is not to be turned in; it is probably easiest to do it on a*copy*of this file that you can throw away afterwards.) ☐

## Fold

Fixpoint fold {X Y:Type} (f: X→Y→Y) (l:list X) (b:Y)

: Y :=

match l with

| nil ⇒ b

| h :: t ⇒ f h (fold f t b)

end.

: Y :=

match l with

| nil ⇒ b

| h :: t ⇒ f h (fold f t b)

end.

Intuitively, the behavior of the fold operation is to
insert a given binary operator f between every pair of elements
in a given list. For example, fold plus [1;2;3;4] intuitively
means 1+2+3+4. To make this precise, we also need a "starting
element" that serves as the initial second input to f. So, for
example,

fold plus [1;2;3;4] 0

yields
1 + (2 + (3 + (4 + 0))).

Some more examples:
Check (fold andb).

(* ===> fold andb : list bool -> bool -> bool *)

Example fold_example1 :

fold mult [1;2;3;4] 1 = 24.

Example fold_example2 :

fold andb [true;true;false;true] true = false.

Example fold_example3 :

fold app [[1];[];[2;3];[4]] [] = [1;2;3;4].

(* ===> fold andb : list bool -> bool -> bool *)

Example fold_example1 :

fold mult [1;2;3;4] 1 = 24.

Proof. reflexivity. Qed.

Example fold_example2 :

fold andb [true;true;false;true] true = false.

Proof. reflexivity. Qed.

Example fold_example3 :

fold app [[1];[];[2;3];[4]] [] = [1;2;3;4].

Proof. reflexivity. Qed.

#### Exercise: 1 star, advanced (fold_types_different)

Observe that the type of fold is parameterized by*two*type variables, X and Y, and the parameter f is a binary operator that takes an X and a Y and returns a Y. Can you think of a situation where it would be useful for X and Y to be different?

(* FILL IN HERE *)

☐
## Functions That Construct Functions

*returning*functions as the results of other functions. To begin, here is a function that takes a value x (drawn from some type X) and returns a function from nat to X that yields x whenever it is called, ignoring its nat argument.

Definition constfun {X: Type} (x: X) : nat→X :=

fun (k:nat) ⇒ x.

Definition ftrue := constfun true.

Example constfun_example1 : ftrue 0 = true.

Example constfun_example2 : (constfun 5) 99 = 5.

fun (k:nat) ⇒ x.

Definition ftrue := constfun true.

Example constfun_example1 : ftrue 0 = true.

Proof. reflexivity. Qed.

Example constfun_example2 : (constfun 5) 99 = 5.

Proof. reflexivity. Qed.

In fact, the multiple-argument functions we have already
seen are also examples of passing functions as data. To see why,
recall the type of plus.

Check plus.

(* ==> nat -> nat -> nat *)

(* ==> nat -> nat -> nat *)

Each → in this expression is actually a

*binary*operator on types. This operator is*right-associative*, so the type of plus is really a shorthand for nat → (nat → nat) — i.e., it can be read as saying that "plus is a one-argument function that takes a nat and returns a one-argument function that takes another nat and returns a nat." In the examples above, we have always applied plus to both of its arguments at once, but if we like we can supply just the first. This is called*partial application*.
Definition plus3 := plus 3.

Check plus3.

Example test_plus3 : plus3 4 = 7.

Proof. reflexivity. Qed.

Example test_plus3' : doit3times plus3 0 = 9.

Proof. reflexivity. Qed.

Example test_plus3'' : doit3times (plus 3) 0 = 9.

Proof. reflexivity. Qed.

Check plus3.

Example test_plus3 : plus3 4 = 7.

Proof. reflexivity. Qed.

Example test_plus3' : doit3times plus3 0 = 9.

Proof. reflexivity. Qed.

Example test_plus3'' : doit3times (plus 3) 0 = 9.

Proof. reflexivity. Qed.

Module Exercises.

#### Exercise: 2 stars (fold_length)

Many common functions on lists can be implemented in terms of fold. For example, here is an alternative definition of length:
Definition fold_length {X : Type} (l : list X) : nat :=

fold (fun _ n ⇒ S n) l 0.

Example test_fold_length1 : fold_length [4;7;0] = 3.

fold (fun _ n ⇒ S n) l 0.

Example test_fold_length1 : fold_length [4;7;0] = 3.

Proof. reflexivity. Qed.

Prove the correctness of fold_length.

Theorem fold_length_correct : ∀ X (l : list X),

fold_length l = length l.

Proof.

(* FILL IN HERE *) Admitted.

☐
fold_length l = length l.

Proof.

(* FILL IN HERE *) Admitted.

Definition fold_map {X Y:Type} (f : X → Y) (l : list X) : list Y

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

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

Write down a theorem fold_map_correct in Coq stating that
fold_map is correct, and prove it.

(* FILL IN HERE *)

☐
#### Exercise: 2 stars, advanced (currying)

In Coq, a function f : A → B → C really has the type A → (B → C). That is, if you give f a value of type A, it will give you function f' : B → C. If you then give f' a value of type B, it will return a value of type C. This allows for partial application, as in plus3. Processing a list of arguments with functions that return functions is called*currying*, in honor of the logician Haskell Curry.

*uncurrying*. With an uncurried binary function, both arguments must be given at once as a pair; there is no partial application.

Definition prod_curry {X Y Z : Type}

(f : X * Y → Z) (x : X) (y : Y) : Z := f (x, y).

(f : X * Y → Z) (x : X) (y : Y) : Z := f (x, y).

As an exercise, define its inverse, prod_uncurry. Then prove
the theorems below to show that the two are inverses.

Definition prod_uncurry {X Y Z : Type}

(f : X → Y → Z) (p : X * Y) : Z

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

(f : X → Y → Z) (p : X * Y) : Z

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

As a (trivial) example of the usefulness of currying, we can use it
to shorten one of the examples that we saw above:

Example test_map1': map (plus 3) [2;0;2] = [5;3;5].

Proof. reflexivity. Qed.

Proof. reflexivity. Qed.

Thought exercise: before running the following commands, can you
calculate the types of prod_curry and prod_uncurry?

Check @prod_curry.

Check @prod_uncurry.

Theorem uncurry_curry : ∀ (X Y Z : Type)

(f : X → Y → Z)

x y,

prod_curry (prod_uncurry f) x y = f x y.

Proof.

(* FILL IN HERE *) Admitted.

Theorem curry_uncurry : ∀ (X Y Z : Type)

(f : (X * Y) → Z) (p : X * Y),

prod_uncurry (prod_curry f) p = f p.

Proof.

(* FILL IN HERE *) Admitted.

☐
Check @prod_uncurry.

Theorem uncurry_curry : ∀ (X Y Z : Type)

(f : X → Y → Z)

x y,

prod_curry (prod_uncurry f) x y = f x y.

Proof.

(* FILL IN HERE *) Admitted.

Theorem curry_uncurry : ∀ (X Y Z : Type)

(f : (X * Y) → Z) (p : X * Y),

prod_uncurry (prod_curry f) p = f p.

Proof.

(* FILL IN HERE *) Admitted.

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

Recall the definition of the nth_error function:
Fixpoint nth_error {X : Type} (l : list X) (n : nat) : option X :=

match l with

| [] ⇒ None

| a :: l' ⇒ if beq_nat n O then Some a else nth_error l' (pred n)

end.

Write an informal proof of the following theorem:
match l with

| [] ⇒ None

| a :: l' ⇒ if beq_nat n O then Some a else nth_error l' (pred n)

end.

∀ X n l, length l = n → @nth_error X l n = None

(* FILL IN HERE *)☐

#### Exercise: 4 stars, advanced (church_numerals)

This exercise explores an alternative way of defining natural numbers, using the so-called*Church numerals*, named after mathematician Alonzo Church. We can represent a natural number n as a function that takes a function f as a parameter and returns f iterated n times.

Module Church.

Definition nat := ∀ X : Type, (X → X) → X → X.

Definition nat := ∀ X : Type, (X → X) → X → X.

Let's see how to write some numbers with this notation. Iterating
a function once should be the same as just applying it. Thus:

Definition one : nat :=

fun (X : Type) (f : X → X) (x : X) ⇒ f x.

fun (X : Type) (f : X → X) (x : X) ⇒ f x.

Similarly, two should apply f twice to its argument:

Definition two : nat :=

fun (X : Type) (f : X → X) (x : X) ⇒ f (f x).

fun (X : Type) (f : X → X) (x : X) ⇒ f (f x).

Defining zero is somewhat trickier: how can we "apply a function
zero times"? The answer is actually simple: just return the
argument untouched.

Definition zero : nat :=

fun (X : Type) (f : X → X) (x : X) ⇒ x.

fun (X : Type) (f : X → X) (x : X) ⇒ x.

More generally, a number n can be written as fun X f x ⇒ f (f
... (f x) ...), with n occurrences of f. Notice in
particular how the doit3times function we've defined previously
is actually just the Church representation of 3.

Definition three : nat := @doit3times.

Complete the definitions of the following functions. Make sure
that the corresponding unit tests pass by proving them with
reflexivity.
Successor of a natural number:

Definition succ (n : nat) : nat

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

Example succ_1 : succ zero = one.

Proof. (* FILL IN HERE *) Admitted.

Example succ_2 : succ one = two.

Proof. (* FILL IN HERE *) Admitted.

Example succ_3 : succ two = three.

Proof. (* FILL IN HERE *) Admitted.

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

Example succ_1 : succ zero = one.

Proof. (* FILL IN HERE *) Admitted.

Example succ_2 : succ one = two.

Proof. (* FILL IN HERE *) Admitted.

Example succ_3 : succ two = three.

Proof. (* FILL IN HERE *) Admitted.

Addition of two natural numbers:

Definition plus (n m : nat) : nat

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

Example plus_1 : plus zero one = one.

Proof. (* FILL IN HERE *) Admitted.

Example plus_2 : plus two three = plus three two.

Proof. (* FILL IN HERE *) Admitted.

Example plus_3 :

plus (plus two two) three = plus one (plus three three).

Proof. (* FILL IN HERE *) Admitted.

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

Example plus_1 : plus zero one = one.

Proof. (* FILL IN HERE *) Admitted.

Example plus_2 : plus two three = plus three two.

Proof. (* FILL IN HERE *) Admitted.

Example plus_3 :

plus (plus two two) three = plus one (plus three three).

Proof. (* FILL IN HERE *) Admitted.

Multiplication:

Definition mult (n m : nat) : nat

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

Example mult_1 : mult one one = one.

Proof. (* FILL IN HERE *) Admitted.

Example mult_2 : mult zero (plus three three) = zero.

Proof. (* FILL IN HERE *) Admitted.

Example mult_3 : mult two three = plus three three.

Proof. (* FILL IN HERE *) Admitted.

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

Example mult_1 : mult one one = one.

Proof. (* FILL IN HERE *) Admitted.

Example mult_2 : mult zero (plus three three) = zero.

Proof. (* FILL IN HERE *) Admitted.

Example mult_3 : mult two three = plus three three.

Proof. (* FILL IN HERE *) Admitted.

Exponentiation:
(

*Hint*: Polymorphism plays a crucial role here. However, choosing the right type to iterate over can be tricky. If you hit a "Universe inconsistency" error, try iterating over a different type: nat itself is usually problematic.)
Definition exp (n m : nat) : nat

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

Example exp_1 : exp two two = plus two two.

Proof. (* FILL IN HERE *) Admitted.

Example exp_2 : exp three two = plus (mult two (mult two two)) one.

Proof. (* FILL IN HERE *) Admitted.

Example exp_3 : exp three zero = one.

Proof. (* FILL IN HERE *) Admitted.

End Church.

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

Example exp_1 : exp two two = plus two two.

Proof. (* FILL IN HERE *) Admitted.

Example exp_2 : exp three two = plus (mult two (mult two two)) one.

Proof. (* FILL IN HERE *) Admitted.

Example exp_3 : exp three zero = one.

Proof. (* FILL IN HERE *) Admitted.

End Church.

End Exercises.