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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>
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<script src="FileSaver.js" type="text/javascript"></script>
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<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.
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<hl />
</p>
<div><textarea id='coq-ta-1'>
(* ignore these directives *)
From mathcomp Require Import mini_ssreflect mini_ssrfun mini_ssrbool.
From mathcomp Require Import mini_eqtype mini_ssrnat mini_div mini_seq.
From mathcomp Require Import mini_prime mini_sum.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Notation "n < m" := (S n <= m) (only printing).
</textarea></div>
<div><p>
<h3>
Exercise 1:
</h3>
<p>
<ul class="doclist">
<li> We define binary trees in the following way:
</li>
</ul>
<div>
</div>
<div><textarea id='coq-ta-2'>
Inductive bintree :=
| Leaf
| Node (l r : bintree).
</textarea></div>
<div><p>
</div>
<p>
<ul class="doclist">
<li> Define a function <tt>eq_bintree</tt> such that the lemma <tt>eq_bintreeP</tt> holds,
and prove it.
</li>
</ul>
<div>
</div>
<div><textarea id='coq-ta-3'>
Fixpoint eq_bintree (t1 t2 : bintree) :=
match t1, t2 with
| Leaf, Leaf => true
| Node l1 r1, Node l2 r2 => eq_bintree l1 l2 && eq_bintree r1 r2
| _, _ => false
end.
Lemma eq_bintreeP (t1 t2 : bintree) : reflect (t1 = t2) (eq_bintree t1 t2).
Proof.
prove_reflect=> [|<-//]; last first.
by elim: t1 => [|l1 IHl1 r1 IHr1]//=; rewrite IHl1 IHr1.
elim: t1 t2 => [|l1 IHl1 r1 IHr1] [|l2 r2]//=.
by move=> /andP[eql1l2 eqr1r2]; rewrite (IHl1 l2)// (IHr1 r2).
Qed.
</textarea></div>
<div><p>
</div>
<p>
<ul class="doclist">
<li> We define the depth of a tree as a function of type <tt>bintree -> nat</tt> as follows:
</li>
</ul>
<div>
</div>
<div><textarea id='coq-ta-4'>
Fixpoint depth (t : bintree) : nat :=
match t with
| Leaf => 0
| Node l r => S (maxn (depth l) (depth r))
end.
</textarea></div>
<div><p>
</div>
<p>
<ul class="doclist">
<li> We define balanced trees as the following predicate:
</li>
</ul>
<div>
</div>
<div><textarea id='coq-ta-5'>
Fixpoint balanced (t : bintree) : bool :=
match t with
| Leaf => true
| Node l r => balanced l && balanced r && (depth l == depth r)
end.
</textarea></div>
<div><p>
</div>
<p>
<ul class="doclist">
<li> prove that balanced trees of equal depth are in fact equal:
</li>
</ul>
<div>
</div>
<div><textarea id='coq-ta-6'>
Lemma balanced_eq_depth_eq (t1 t2 : bintree) :
balanced t1 -> balanced t2 -> depth t1 = depth t2 -> t1 = t2.
Proof.
suff equ12 d u1 u2 : balanced u1 -> balanced u2 ->
depth u1 = d -> depth u2 = d -> u1 = u2.
by move=> bt1 bt2 dt12; apply: (equ12 (depth t2)).
elim: d u1 u2 => [|d IHd] [|l1 r1] [|l2 r2]//=.
move=> /andP[/andP[bl1 br1 /eqP eqdlr1]].
move=> /andP[/andP[bl2 br2 /eqP eqdlr2]].
rewrite eqdlr1 eqdlr2 !maxnn => - [eq_dr1_d] [eq_dr2_d].
by rewrite (IHd r1 r2)// (IHd l1 l2) ?eqdlr1 ?eqdlr2.
Qed.
</textarea></div>
<div><p>
</div>
<p>
<h3>
Exercise 2:
</h3>
<p>
Let \(a \ge 0\) and \(n \ge 1\) be natural numbers.
<p>
<ul class="doclist">
<li> Show that \[a ^ n − 1 = (a - 1) \sum_{i = 0}^{n-1} a^i.\]
</li>
<li> Hint <tt>Search _ sum in MC</tt>.
</li>
<li> Hint do as many <tt>have</tt> as needed.
</li>
</ul>
<div>
</div>
<div><textarea id='coq-ta-7'>
Lemma subX1_factor a n : 1 <= n ->
a ^ n - 1 = (a - 1) * \sum_(0 <= i < n) a ^ i.
Proof.
case: n => [//|n] _; rewrite mulnBl mul1n muln_sumr.
rewrite sum_recr//= sum_recl//= addnC -expnS expn0.
by rewrite (eqn_sum (fun i => a ^ (S i))) ?subnDr// => i; rewrite expnS.
Qed.
</textarea></div>
<div><p>
</div>
<p>
Let \(a , n \ge 2\) be natural numbers.
<p>
<ul class="doclist">
<li> Show that if \(a ^ n − 1\) is prime, then \(a = 2\) and \(n\) is prime.
Complete the following proof script
</li>
</ul>
<p>
<div>
</div>
<div><textarea id='coq-ta-8'>
Lemma subX1_prime (a n : nat) : 2 <= a -> 2 <= n ->
prime (a ^ n - 1) -> (a == 2) && prime n.
Proof.
move=> a_ge2 n_ge2 /primeP [_ mP].
have n_gt0 : S 0 <= n by rewrite (leq_trans _ n_ge2).
have a_gt0 : S 0 <= a by rewrite (leq_trans _ a_ge2).
have dvdam : a - 1 %| a ^ n - 1 by rewrite subX1_factor// dvdn_mulr.
have neq_m : a - 1 != a ^ n - 1.
rewrite -(eqn_add2r 1) !subnK// ?expn_gt0 ?a_gt0//.
by rewrite -{1}[a]expn1 ltn_eqF// ltn_exp2l//.
have a_eq2: a = 2.
have := mP _ dvdam.
by rewrite -(eqn_add2r 1) subnK// (negPf neq_m) orbF => /eqP.
move: mP; rewrite a_eq2 /= => mP.
apply/primeP; split => // d /dvdnP[k eq_n].
case d_neq1 : (d == 1) => //=.
have d_gt0 : d > 0.
by move: n_gt0; rewrite eq_n !lt0n; apply: contra_neq => ->; rewrite muln0.
have k_gt0 : k > 0.
by move: n_gt0; rewrite eq_n !lt0n; apply: contra_neq => ->; rewrite mul0n.
have d_gt1 : d > 1 by rewrite ltn_neqAle eq_sym d_neq1/=.
have dvddm : 2 ^ d - 1 %| 2 ^ n - 1.
by rewrite eq_n mulnC expnM (subX1_factor (_ ^ _))// dvdn_mulr.
have := mP _ dvddm; rewrite gtn_eqF/=; last first.
by rewrite ltn_subRL; have: 2 ^ 1 < 2 ^ d by rewrite ltn_exp2l.
by rewrite -(eqn_add2r 1) !subnK// ?expn_gt0// eqn_exp2l.
Qed.
</textarea></div>
<div><p>
</div>
<p>
<ul class="doclist">
<li> We write \(M_n = 2 ^ n − 1\) the \(n^{\textrm{th}}\) Mersenne number.
Show that \(M_{100}\) is not prime.
<p>
WARNING: do not substitute <tt>n = 100</tt> in an expression where you have <tt>2 ^ n</tt>,
otherwise the computation will take <quote>forever</quote> and possibly crash your jscoq
(in which case you will need to <quote>Reset worker</quote> or reload the page), hence
you must use the previous lemma.
</li>
</ul>
<div>
</div>
<div><textarea id='coq-ta-9'>
Lemma M12_not_prime n : n = 100 -> ~~ prime (2 ^ n - 1).
Proof.
by move=> eq_n; apply: (contraNN (subX1_prime _ _)); rewrite ?eq_n.
Qed.
</textarea></div>
<div><p>
</div>
<p>
<div>
When you are done, click the Download link at the top of the page
and send us your homework by email: Assia.Mahboubi@inria.fr
</div>
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