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exercise2-solution.html
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<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="content-type" content="text/html;charset=utf-8" />
<link rel="stylesheet" href="jscoq/node_modules/bootstrap/dist/css/bootstrap.min.css" />
<title>Machine-Checked Mathematics</title>
<link rel="stylesheet" href="local.css" />
<script src='https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML'
async></script>
<script src="Blob.js" type="text/javascript"></script>
<script src="FileSaver.js" type="text/javascript"></script>
</head>
<body>
<div id="ide-wrapper" class="toggled">
<div id="code-wrapper">
<div id="document">
<p>
Use ALT-(up-arrow) and ALT-(down-arrow) to process this document inside your browser, line-by-line.
Use ALT-(right-arrow) to go to the cursor.
You can
<span class="save-button" onClick="save_coq_snippets()">save your edits</span>
inside your browser and
<span class="save-button" onClick="load_coq_snippets()">load them back</span>.
<!-- (edits are also saved when you close the window) -->
Finally, you can
<span class="save-button" onClick="download_coq_snippets()">download</span>
your working copy of the file, e.g., for sending it to teachers.
<hl />
</p>
<div><textarea id='coq-ta-1'>
(* ignore these directives *)
From mathcomp Require Import mini_ssreflect.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Add Printing Coercion is_true.
Notation "x '= true'" := (is_true x) (x at level 100, at level 0, only printing).
Remove Printing If bool.
</textarea></div>
<div><p>
<h3>
Exercise 1:
</h3>
<p>
<ul class="doclist">
<li> Define a function double that takes a natural
number to its double.
</li>
<li> you can use the operations on nat that we defined in class
</li>
</ul>
</div>
<div><textarea id='coq-ta-2'>
Definition double (n : nat) : nat := (* fill this in *)
2 * n.
Eval lazy in double 3. (* = 6 *)
</textarea></div>
<div><p>
<h3>
Exercise 2:
</h3>
<p>
<ul class="doclist">
<li> Define a function that takes a natural number and tests
if it is zero. In that case the value of the function is 1,
otherwise it is the double of the given number.
</li>
<li> The skeleton of the function is given, fill in the blanks. </div>
<div><textarea id='coq-ta-3'>
Infix "==" := eqn. (* recall, infix == tests for equality *)
Definition double_z (n : nat) : nat :=
match
(* fill this in *)
n == 0
with
| true => (* fill this in*)
1
| false => (* fill this in *)
double n
end.
Eval lazy in double_z 0. (* = 1 *)
Eval lazy in double_z 4. (* = 8 *)
</textarea></div>
<div><p>
<h3>
Exercise 3:
</h3>
<p>
</li>
<li> Define boolean negation. We did peek at it during the lesson,
now try to write it yourself.
</li>
</ul>
</div>
<div><textarea id='coq-ta-4'>
Definition negb (b : bool) : bool := (* fill this in *)
match b with true => false | false => true end.
Eval lazy in negb true. (* = false *)
Eval lazy in negb false. (* = true *)
</textarea></div>
<div><p>
<h3>
Exercise 4:
</h3>
<p>
<ul class="doclist">
<li> write the Fibonacci function (growth of a rabbit population).
$$ \phi(n) = \begin{array}{|c}1~\mathrm{if}~n = 0 \\ 1~\mathrm{if}~n = 1\\ \phi(n-1) + \phi(n-2)~\mathrm{otherwise}\end{array} $$
</li>
</ul>
</div>
<div><textarea id='coq-ta-5'>
Fixpoint fibonacci (n : nat) : nat :=
match n with
| 0 => (* fill this in *)
1
| 1 => (* fill this in *)
1
| S p => (* fill this in, hint: think at the relation between n and p *)
fibonacci p + fibonacci (p - 1)
end.
Eval lazy in fibonacci 3. (* = 3*)
Eval lazy in fibonacci 7. (* = 21 *)
Eval lazy in fibonacci 10. (* = 89 *)
</textarea></div>
<div><p>
<h3>
Exercise 5:
</h3>
<p>
<ul class="doclist">
<li> Define the option container with constructors None and Some
</li>
<li> Define the <quote>projection</quote> default
</li>
</ul>
</div>
<div><textarea id='coq-ta-6'>
Inductive option A :=
| Some (* fill this in *)
(a : A)
| None.
(* Ignore these directives *)
Arguments Some {_}.
Arguments None {_}.
Definition default A (a : A) (x : option A) : A := (* fill this in *)
match x with Some v => v | None => a end.
Eval lazy in default 3 None. (* = 3 *)
Eval lazy in default 3 (Some 4). (* = 4 *)
</textarea></div>
<div><p>
<h3>
Exercise 6:
</h3>
Use the <tt>iter</tt> function below to define:
<p>
<ul class="doclist">
<li> addition over natural numbers.
</li>
<li> multiplication over natural unmbers.
</li>
</ul>
</div>
<div><textarea id='coq-ta-7'>
Fixpoint iter T (n : nat) (op : T -> T) (x : T) : T :=
match n with
| 0 => x
| S p => op (iter p op x)
end.
(* Ignore this directive *)
Arguments iter {T}.
Definition addn (n : nat) (m : nat) : nat := (* fill this in *)
iter n S m.
Eval lazy in addn 3 4.
Definition muln (n : nat) (m : nat) : nat := (* fill this in *)
iter n (addn m) 0.
Eval lazy in muln 3 4.
</textarea></div>
<div><p>
<h3>
Exercise 7:
</h3>
<p>
<ul class="doclist">
<li> prove the following statement by case split
</li>
</ul>
</div>
<div><textarea id='coq-ta-8'>
Lemma orbC (p : bool) (q : bool) : p || q = q || p.
Proof.
(* fill this in *)
by case: p; case: q.
Qed.
</textarea></div>
<div><p>
<h3>
Exercise 8:
</h3>
<p>
<ul class="doclist">
<li> look up what <tt>==></tt> is and prove the Peirce law
</li>
</ul>
</div>
<div><textarea id='coq-ta-9'>
Locate "fill this in".
Print (* the name behind ==> *)
implb.
Lemma Peirce (p : bool) (q : bool) : ((p ==> q) ==> p) ==> p.
Proof.
(* fill this in *)
by case: p; case: q.
Qed.
</textarea></div>
<div><p>
<h3>
Exercise 9:
</h3>
<p>
<ul class="doclist">
<li> what is <tt>(+)</tt> ?
</li>
<li> Hint: <tt>-></tt> in the goal stands for implication, use move to name the
assumption
</li>
<li> Hint: use Search to dind useful lemmas and use rewrite to use them
</li>
</ul>
</div>
<div><textarea id='coq-ta-10'>
Locate (* fill this in *)
"(+)".
Search _ (* fil this in*)
(_ (+) ~~ _).
Lemma find_me (p : bool) (q : bool) : ~~ p = q -> p (+) q.
Proof.
(* This is a new command: it names an assumption, so that you can
mention it later in your proof *)
move=> np_q.
(* fill this in *)
by rewrite -np_q addbN negb_add.
Qed.
</textarea></div>
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