(** * Lists: Working with Structured Data *)
Require Export Induction.
Module NatList.
(* ###################################################### *)
(** * Pairs of Numbers *)
(** In an [Inductive] type definition, each constructor can take
any number of arguments -- none (as with [true] and [O]), one (as
with [S]), or more than one, as in this definition: *)
Inductive natprod : Type :=
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]." *)
(** We can construct an element of [natprod] like this: *)
Check (pair 3 5).
(** *** *)
(** Here are two simple function definitions 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.
Eval compute in (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 notation can be used both in expressions and in
pattern matches (indeed, we've seen it already in the previous
chapter -- this notation is provided as part of the standard
library): *)
Eval compute in (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.
(** *** *)
(** Let's try and prove a few simple facts about pairs. If we
state the lemmas in a particular (and slightly peculiar) way, we
can prove them with just reflexivity (and its built-in
simplification): *)
Theorem surjective_pairing' : forall (n m : nat),
(n,m) = (fst (n,m), snd (n,m)).
Proof.
reflexivity. Qed.
(** Note that [reflexivity] is not enough if we state the lemma in a
more natural way: *)
Theorem surjective_pairing_stuck : forall (p : natprod),
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].
Notice that, unlike for [nat]s, [destruct] doesn't generate an
extra subgoal here. That's because [natprod]s can only be
constructed in one way. *)
Theorem surjective_pairing : forall (p : natprod),
p = (fst p, snd p).
Proof.
intros p. destruct p as [n m]. simpl. reflexivity. Qed.
(** **** Exercise: 1 star (snd_fst_is_swap) *)
Theorem snd_fst_is_swap : forall (p : natprod),
(snd p, fst p) = swap_pair p.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 1 star, optional (fst_swap_is_snd) *)
Theorem fst_swap_is_snd : forall (p : natprod),
fst (swap_pair p) = snd p.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ###################################################### *)
(** * Lists of Numbers *)
(** Generalizing the definition of pairs a little, we can
describe the type of _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.
(** 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 two 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) ..).
(** It is not necessary to fully understand 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].
(** 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,
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])].
(By the way, it's worth noting in passing that 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. *)
(** *** Repeat *)
(** 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]. *)
Fixpoint repeat (n count : nat) : natlist :=
match count with
| O => nil
| S count' => n :: (repeat n count')
end.
(** *** Length *)
(** The [length] function calculates the length of a list. *)
Fixpoint length (l:natlist) : nat :=
match l with
| nil => O
| h :: t => S (length t)
end.
(** *** Append *)
(** The [app] ("append") function concatenates two lists. *)
Fixpoint app (l1 l2 : natlist) : natlist :=
match l1 with
| nil => l2
| h :: t => h :: (app t l2)
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.
(** Here are two smaller examples of programming with lists.
The [hd] function returns the first element (the "head") of the
list, while [tl] returns everything but the first
element (the "tail").
Of course, the empty list has no first element, so we
must pass a default value to be returned in that case. *)
(** *** 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_hd1: hd 0 [1;2;3] = 1.
Proof. reflexivity. Qed.
Example test_hd2: hd 0 [] = 0.
Proof. reflexivity. Qed.
Example test_tl: tl [1;2;3] = [2;3].
Proof. reflexivity. Qed.
(** **** Exercise: 2 stars (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 :=
(* FILL IN HERE *) admit.
Example test_nonzeros: nonzeros [0;1;0;2;3;0;0] = [1;2;3].
(* FILL IN HERE *) Admitted.
Fixpoint oddmembers (l:natlist) : natlist :=
(* FILL IN HERE *) admit.
Example test_oddmembers: oddmembers [0;1;0;2;3;0;0] = [1;3].
(* FILL IN HERE *) Admitted.
Fixpoint countoddmembers (l:natlist) : nat :=
(* FILL IN HERE *) admit.
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.
Note: one natural and elegant way of writing [alternate] will fail
to satisfy Coq's requirement that all [Fixpoint] definitions be
"obviously terminating." If you find yourself in this rut, look
for a slightly more verbose solution that considers elements of
both lists at the same time. (One possible solution requires
defining a new kind of pairs, but this is not the only way.) *)
Fixpoint alternate (l1 l2 : natlist) : natlist :=
(* FILL IN HERE *) admit.
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 *)
(** A [bag] (or [multiset]) is like a set, but each element can appear
multiple times instead of just once. One reasonable
implementation of bags is to represent a bag of numbers as a
list. *)
Definition bag := natlist.
(** **** Exercise: 3 stars (bag_functions) *)
(** Complete the following definitions for the functions
[count], [sum], [add], and [member] for bags. *)
Fixpoint count (v:nat) (s:bag) : nat :=
(* FILL IN HERE *) admit.
(** 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.
(** 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, 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 :=
(* FILL IN HERE *) admit.
Example test_sum1: count 1 (sum [1;2;3] [1;4;1]) = 3.
(* FILL IN HERE *) Admitted.
Definition add (v:nat) (s:bag) : bag :=
(* FILL IN HERE *) admit.
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.
Definition member (v:nat) (s:bag) : bool :=
(* FILL IN HERE *) admit.
Example test_member1: member 1 [1;4;1] = true.
(* FILL IN HERE *) Admitted.
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 :=
(* When remove_one is applied to a bag without the number to remove,
it should return the same bag unchanged. *)
(* FILL IN HERE *) admit.
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 :=
(* FILL IN HERE *) admit.
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 (s1:bag) (s2:bag) : bool :=
(* FILL IN HERE *) admit.
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 (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! *)
(* FILL IN HERE *)
(** [] *)
(* ###################################################### *)
(** * Reasoning About Lists *)
(** Just as with numbers, simple facts about list-processing
functions can sometimes be proved entirely by simplification. For
example, the simplification performed by [reflexivity] is enough
for this theorem... *)
Theorem nil_app : forall l:natlist,
[] ++ l = l.
Proof. reflexivity. Qed.
(** ... because the [[]] is substituted into the match position
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 : forall l:natlist,
pred (length l) = length (tl l).
Proof.
intros l. destruct l as [| n l'].
Case "l = nil".
reflexivity.
Case "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 *)
(** Simply reading example proof scripts will not get you very far!
It is very important to work through the details of each one,
using Coq and thinking about what each step achieves. Otherwise
it is more or less guaranteed that the exercises will make no
sense... *)
(* ###################################################### *)
(** ** Induction on Lists *)
(** Proofs by induction over datatypes like [natlist] are
perhaps a little less familiar than standard natural number
induction, but the basic idea is equally simple. Each [Inductive]
declaration defines a set of data values that can be built up from
the declared constructors: a boolean can be either [true] or
[false]; a number can be either [O] or [S] applied to a number; a
list can be either [nil] or [cons] applied to a number and a list.
Moreover, applications of the declared constructors to one another
are the _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'].
Since larger lists can only be built up from smaller ones,
eventually reaching [nil], these two things together establish the
truth of [P] for all lists [l]. Here's a concrete example: *)
Theorem app_assoc : forall l1 l2 l3 : natlist,
(l1 ++ l2) ++ l3 = l1 ++ (l2 ++ l3).
Proof.
intros l1 l2 l3. induction l1 as [| n l1'].
Case "l1 = nil".
reflexivity.
Case "l1 = cons n l1'".
simpl. rewrite -> IHl1'. reflexivity. Qed.
(** 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. *)
(** *** Informal version *)
(** _Theorem_: For all lists [l1], [l2], and [l3],
[(l1 ++ l2) ++ l3 = l1 ++ (l2 ++ l3)].
_Proof_: By induction on [l1].
- First, suppose [l1 = []]. We must show
([] ++ l2) ++ l3 = [] ++ (l2 ++ l3),
which follows directly from the definition of [++].
- Next, suppose [l1 = n::l1'], with
(l1' ++ l2) ++ l3 = l1' ++ (l2 ++ l3)
(the induction hypothesis). We must show
((n :: l1') ++ l2) ++ l3 = (n :: l1') ++ (l2 ++ l3).
]]
By the definition of [++], this follows from
n :: ((l1' ++ l2) ++ l3) = n :: (l1' ++ (l2 ++ l3)),
which is immediate from the induction hypothesis. []
*)
(** *** Another example *)
(**
Here is a similar example to be worked together in class: *)
Theorem app_length : forall l1 l2 : natlist,
length (l1 ++ l2) = (length l1) + (length l2).
Proof.
(* WORKED IN CLASS *)
intros l1 l2. induction l1 as [| n l1'].
Case "l1 = nil".
reflexivity.
Case "l1 = cons".
simpl. rewrite -> IHl1'. reflexivity. Qed.
(** *** Reversing a list *)
(** For a slightly more involved example of an inductive proof
over lists, suppose we define a "cons on the right" function
[snoc] like this... *)
Fixpoint snoc (l:natlist) (v:nat) : natlist :=
match l with
| nil => [v]
| h :: t => h :: (snoc t v)
end.
(** ... and use it to define a list-reversing function [rev]
like this: *)
Fixpoint rev (l:natlist) : natlist :=
match l with
| nil => nil
| h :: t => snoc (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.
(** *** Proofs about reverse *)
(** Now let's prove some more list theorems using our newly
defined [snoc] and [rev]. For something a little more challenging
than the inductive proofs we've seen so far, let's prove that
reversing a list does not change its length. Our first attempt at
this proof gets stuck in the successor case... *)
Theorem rev_length_firsttry : forall l : natlist,
length (rev l) = length l.
Proof.
intros l. induction l as [| n l'].
Case "l = []".
reflexivity.
Case "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 [snoc], but we don't have any equations
in either the immediate context or the global
environment that have anything to do with [snoc]!
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 about [snoc] that would have
enabled us to make progress and prove it as a separate lemma.
*)
Theorem length_snoc : forall n : nat, forall l : natlist,
length (snoc l n) = S (length l).
Proof.
intros n l. induction l as [| n' l'].
Case "l = nil".
reflexivity.
Case "l = cons n' l'".
simpl. rewrite -> IHl'. reflexivity. Qed.
(**
Note that we make the lemma as _general_ as possible: in particular,
we quantify over _all_ [natlist]s, 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 much easier
to prove the more general property.
*)
(** Now we can complete the original proof. *)
Theorem rev_length : forall l : natlist,
length (rev l) = length l.
Proof.
intros l. induction l as [| n l'].
Case "l = nil".
reflexivity.
Case "l = cons".
simpl. rewrite -> length_snoc.
rewrite -> IHl'. reflexivity. Qed.
(** For comparison, here are informal proofs of these two theorems:
_Theorem_: For all numbers [n] and lists [l],
[length (snoc l n) = S (length l)].
_Proof_: By induction on [l].
- First, suppose [l = []]. We must show
length (snoc [] n) = S (length []),
which follows directly from the definitions of
[length] and [snoc].
- Next, suppose [l = n'::l'], with
length (snoc l' n) = S (length l').
We must show
length (snoc (n' :: l') n) = S (length (n' :: l')).
By the definitions of [length] and [snoc], this
follows from
S (length (snoc l' n)) = S (S (length l')),
]]
which is immediate from the induction hypothesis. [] *)
(** _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 show
length (rev (n :: l')) = length (n :: l').
By the definition of [rev], this follows from
length (snoc (rev l') n) = S (length l')
which, by the previous lemma, is the same as
S (length (rev l')) = S (length l').
This is immediate from the induction hypothesis. [] *)
(** Obviously, 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 (since we can easily work
them 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 more like this: *)
(** _Theorem_:
For all lists [l], [length (rev l) = length l].
_Proof_: First, observe that
length (snoc l n) = S (length l)
for any [l]. This follows by a straightforward induction on [l].
The main property now follows by another straightforward
induction on [l], using the observation together with the
induction hypothesis in the case where [l = n'::l']. [] *)
(** Which style is preferable in a given situation depends on
the sophistication of the expected audience and on 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
present purposes. *)
(* ###################################################### *)
(** ** [SearchAbout] *)
(** We've seen that proofs can make use of other theorems we've
already proved, using [rewrite], and later we will see other ways
of reusing previous theorems. But in order to refer to a theorem,
we need to know its name, and remembering the names of all the
theorems we might ever want to use can become quite difficult! It
is often hard even to remember what theorems have been proven,
much less what they are named.
Coq's [SearchAbout] command is quite helpful with this. Typing
[SearchAbout foo] will cause Coq to display a list of all theorems
involving [foo]. For example, try uncommenting the following to
see a list of theorems that we have proved about [rev]: *)
(* SearchAbout rev. *)
(** Keep [SearchAbout] in mind as you do the following exercises and
throughout the rest of the course; it can save you a lot of time! *)
(** Also, if you are using ProofGeneral, you can run [SearchAbout]
with [C-c C-a C-a]. Pasting its response into your buffer can be
accomplished with [C-c C-;]. *)
(* ###################################################### *)
(** ** List Exercises, Part 1 *)
(** **** Exercise: 3 stars (list_exercises) *)
(** More practice with lists. *)
Theorem app_nil_end : forall l : natlist,
l ++ [] = l.
Proof.
(* FILL IN HERE *) Admitted.
Theorem rev_involutive : forall l : natlist,
rev (rev l) = l.
Proof.
(* FILL IN HERE *) Admitted.
(** There is a short solution to the next exercise. If you find
yourself getting tangled up, step back and try to look for a
simpler way. *)
Theorem app_assoc4 : forall l1 l2 l3 l4 : natlist,
l1 ++ (l2 ++ (l3 ++ l4)) = ((l1 ++ l2) ++ l3) ++ l4.
Proof.
(* FILL IN HERE *) Admitted.
Theorem snoc_append : forall (l:natlist) (n:nat),
snoc l n = l ++ [n].
Proof.
(* FILL IN HERE *) Admitted.
Theorem distr_rev : forall l1 l2 : natlist,
rev (l1 ++ l2) = (rev l2) ++ (rev l1).
Proof.
(* FILL IN HERE *) Admitted.
(** An exercise about your implementation of [nonzeros]: *)
Lemma nonzeros_app : forall l1 l2 : natlist,
nonzeros (l1 ++ l2) = (nonzeros l1) ++ (nonzeros l2).
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 (l1 l2 : natlist) : bool :=
(* FILL IN HERE *) admit.
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 : forall l:natlist,
true = beq_natlist l l.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ###################################################### *)
(** ** List Exercises, Part 2 *)
(** **** Exercise: 2 stars (list_design) *)
(** Design exercise:
- Write down a non-trivial theorem [cons_snoc_app]
involving [cons] ([::]), [snoc], and [app] ([++]).
- Prove it. *)
(* FILL IN HERE *)
(** [] *)
(** **** Exercise: 3 stars, advanced (bag_proofs) *)
(** Here are a couple of little theorems to prove about your
definitions about bags earlier in the file. *)
Theorem count_member_nonzero : forall (s : bag),
ble_nat 1 (count 1 (1 :: s)) = true.
Proof.
(* FILL IN HERE *) Admitted.
(** The following lemma about [ble_nat] might help you in the next proof. *)
Theorem ble_n_Sn : forall n,
ble_nat n (S n) = true.
Proof.
intros n. induction n as [| n'].
Case "0".
simpl. reflexivity.
Case "S n'".
simpl. rewrite IHn'. reflexivity. Qed.
Theorem remove_decreases_count: forall (s : bag),
ble_nat (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.*)
(* FILL IN HERE *)
(** [] *)
(** **** Exercise: 4 stars, advanced (rev_injective) *)
(** Prove that the [rev] function is injective, that is,
forall (l1 l2 : natlist), rev l1 = rev l2 -> l1 = l2.
There is a hard way and an easy way to solve this exercise.
*)
(* FILL IN HERE *)
(** [] *)
(* ###################################################### *)
(** * Options *)
(** One use of [natoption] is as a way of returning "error
codes" from functions. For example, suppose we want to write a
function that returns the [n]th element of some list. If we give
it type [nat -> natlist -> nat], then we'll have to return some
number when the list is too short! *)
Fixpoint index_bad (n:nat) (l:natlist) : nat :=
match l with
| nil => 42 (* arbitrary! *)
| a :: l' => match beq_nat n O with
| true => a
| false => index_bad (pred n) l'
end
end.
(** *** *)
(** On the other hand, if we give it type [nat -> natlist ->
natoption], then we can return [None] when the list is too short
and [Some a] when the list has enough members and [a] appears at
position [n]. *)
Inductive natoption : Type :=
| Some : nat -> natoption
| None : natoption.
Fixpoint index (n:nat) (l:natlist) : natoption :=
match l with
| nil => None
| a :: l' => match beq_nat n O with
| true => Some a
| false => index (pred n) l'
end
end.
Example test_index1 : index 0 [4;5;6;7] = Some 4.
Proof. reflexivity. Qed.
Example test_index2 : index 3 [4;5;6;7] = Some 7.
Proof. reflexivity. Qed.
Example test_index3 : index 10 [4;5;6;7] = None.
Proof. reflexivity. Qed.
(** This example is also an opportunity to introduce one more
small feature of Coq's programming language: conditional
expressions... *)
(** *** *)
Fixpoint index' (n:nat) (l:natlist) : natoption :=
match l with
| nil => None
| a :: l' => if beq_nat n O then Some a else index' (pred n) l'
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 allows conditional expressions over
_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. *)
(** The function below pulls the [nat] out of a [natoption], returning
a supplied default in the [None] case. *)
Definition option_elim (d : nat) (o : natoption) : nat :=
match o with
| Some n' => n'
| None => d
end.
(** **** Exercise: 2 stars (hd_opt) *)
(** 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_opt (l : natlist) : natoption :=
(* FILL IN HERE *) admit.
Example test_hd_opt1 : hd_opt [] = None.
(* FILL IN HERE *) Admitted.
Example test_hd_opt2 : hd_opt [1] = Some 1.
(* FILL IN HERE *) Admitted.
Example test_hd_opt3 : hd_opt [5;6] = Some 5.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 1 star, optional (option_elim_hd) *)
(** This exercise relates your new [hd_opt] to the old [hd]. *)
Theorem option_elim_hd : forall (l:natlist) (default:nat),
hd default l = option_elim default (hd_opt l).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ###################################################### *)
(** * Dictionaries *)
(** As a final illustration of how fundamental data structures
can be defined in Coq, here is the declaration of a simple
[dictionary] data type, using numbers for both the keys and the
values stored under these keys. (That is, a dictionary represents
a finite map from numbers to numbers.) *)
Module Dictionary.
Inductive dictionary : Type :=
| empty : dictionary
| record : nat -> nat -> dictionary -> dictionary.
(** This declaration can be read: "There are two ways to construct a
[dictionary]: either using the constructor [empty] to represent an
empty dictionary, or by applying the constructor [record] to
a key, a value, and an existing [dictionary] to construct a
[dictionary] with an additional key to value mapping." *)
Definition insert (key value : nat) (d : dictionary) : dictionary :=
(record key value d).
(** Here is a function [find] that searches a [dictionary] for a
given key. It evaluates evaluates to [None] if the key was not
found and [Some val] if the key was mapped to [val] in the
dictionary. If the same key is mapped to multiple values, [find]
will return the first one it finds. *)
Fixpoint find (key : nat) (d : dictionary) : natoption :=
match d with
| empty => None
| record k v d' => if (beq_nat key k)
then (Some v)
else (find key d')
end.
(** **** Exercise: 1 star (dictionary_invariant1) *)
(** Complete the following proof. *)
Theorem dictionary_invariant1' : forall (d : dictionary) (k v: nat),
(find k (insert k v d)) = Some v.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 1 star (dictionary_invariant2) *)
(** Complete the following proof. *)
Theorem dictionary_invariant2' : forall (d : dictionary) (m n o: nat),
beq_nat m n = false -> find m d = find m (insert n o d).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
End Dictionary.
End NatList.
(** $Date: 2014-12-31 11:17:56 -0500 (Wed, 31 Dec 2014) $ *)