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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>
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Finally, you can
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your working copy of the file, e.g., for sending it to teachers.
<hl />
</p>
<div><textarea id='coq-ta-1'>
From mathcomp Require Import mini_ssreflect mini_ssrfun mini_ssrbool.
From mathcomp Require Import mini_eqtype mini_ssrnat mini_seq mini_div.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Notation "n < m" := (S n <= m) (only printing).
</textarea></div>
<div><p>
<p>
what follows is a slide, it creates an index item next to the scroll bar,
just move the mouse there.
<hr/>
<div class="slide">
<p>
<h1>
Induction and structured proofs
</h1>
<p>
<h3>
Exercise 1
</h3>
<p>
Let us prove directly formula
\[ \sum_{i=0}^{n-1} (2 i + 1) = n ^ 2 \]
from lesson 1, slightly modified.
<p>
Let us first recall custom sum operator:
<div>
</div>
<div><textarea id='coq-ta-2'>
Definition sum m n F := (foldr (fun i a => F i + a) 0 (iota m (n - m))).
Notation "\sum_ ( m <= i < n ) F" := (sum m n (fun i => F))
(at level 41, F at level 41, i, m, n at level 50,
format "'[' \sum_ ( m <= i < n ) '/ ' F ']'").
</textarea></div>
<div><p>
</div>
<p>
<ul class="doclist">
<li> First prove a very useful lemma about <tt>iota</tt> (you can use <tt>iota_add</tt>) (Easy)
</li>
</ul>
<div>
</div>
<div><textarea id='coq-ta-3'>
Lemma iotaSr m n : iota m (S n) = iota m n ++ [:: m + n].
Proof. by rewrite -addn1 iota_add /=. Qed.
</textarea></div>
<div><p>
</div>
<p>
<ul class="doclist">
<li> Then prove a very useful lemma about summation. (Medium)
</li>
</ul>
<div>
</div>
<div><textarea id='coq-ta-4'>
Lemma sum_recr m n F : m <= n ->
\sum_(m <= i < n.+1) F i = \sum_(m <= i < n) F i + F n.
Proof.
pose addFi i a := F i + a; move=> le_mn.
have foldr0 s x : foldr addFi x s = foldr addFi 0 s + x.
Admitted.
</textarea></div>
<div><p>
</div>
<p>
<ul class="doclist">
<li> Now use the previous result to get the main result (Easy - Medium)
</li>
</ul>
<div>
</div>
<div><textarea id='coq-ta-5'>
Lemma sum_odds n : \sum_(0 <= i < n) (2 * i + 1) = n ^ 2.
Proof.
Admitted.
</textarea></div>
<div><p>
</div>
<p><br/><p>
</div>
<hr/>
<p>
<h3>
Exercise 2
</h3>
<p>
Prove that the equation \[ 8y = 6x + 1 \] has no solution. (Easy)
<p>
<ul class="doclist">
<li> Hint 1: take the modulo 2 of the equation (using <tt>suff</tt>).
</li>
<li> Hint 2: <tt>Search _ "contra" in MC</tt> and use the view <tt>eqP</tt>.
</li>
<li> Hint 3: <tt>Search _ modn addn in MC</tt> and <tt>Search _ modn muln in MC</tt>
</li>
</ul>
<br/><div> </div>
<div><textarea id='coq-ta-6'>
Lemma ex2 x y : 8 * y != 6 * x + 1.
Proof.
Admitted.
</textarea></div>
<div><p>
</div>
<hr/>
<p>
<h3>
Exercise 3:
</h3>
<p>
The ultimate Goal of this exercise is to find the solutions of the equation
\[ 2^n = a^2 + b^2,\] where n is fixed and a and b unkwown.
We hence study the following predicate:
<div> </div>
<div><textarea id='coq-ta-7'>
Definition sol n a b := [&& a > 0, b > 0 & 2 ^ n == a ^ 2 + b ^ 2].
</textarea></div>
<div><p>
</div>
<p>
<ul class="doclist">
<li> We give the set of solutions for \(n = 0\) or \(1\)
</li>
</ul>
<div> </div>
<div><textarea id='coq-ta-8'>
(* no solution for `n = 0` *)
Lemma sol0 a b : ~~ sol 0 a b.
Proof. by move: a b => [|[|a]] []. Qed.
(* The only solution for `n = 1` is `(a = 1, b = 1)` *)
Lemma sol1 a b : sol 1 a b = (a == 1) && (b == 1).
Proof. by move: a b => [|[|[|a]]] [|[]]. Qed.
</textarea></div>
<div><p>
</div>
<p>
<ul class="doclist">
<li> Now prove a little lemma that will guarantee that a and b are even. (Medium - Hard)
<ul class="doclist">
<li> Hint: <tt>Search _ (_ ^ 2) (_ + _) in MC</tt>.
</li>
<li> Hint: <tt>Search ((_ * _) ^ _) in MC</tt>.
</li>
<li> ...
</li>
<li> Hint: <tt>About divn_eq</tt> to substitute a and b (rewrite {1}(div_eq ...))
</li>
<li> Hint: <tt>Search _ modn odd in MC</tt>.
<p>
</li>
</ul>
</li>
<li> Do it on paper first!
</li>
</ul>
<div> </div>
<div><textarea id='coq-ta-9'>
Lemma mod4Dsqr_even a b : (a ^ 2 + b ^ 2) %% 4 = 0 -> (~~ odd a) && (~~ odd b).
Proof.
have sqr_x2Dy_mod4 x y : (x * 2 + y) ^ 2 = y ^ 2 %[mod 4].
Admitted.
</textarea></div>
<div><p>
</div>
<p>
<ul class="doclist">
<li> Deduce that if n is greater than 2 and a and b are solutions, then they are even. (HARD)
</li>
</ul>
<div> </div>
<div><textarea id='coq-ta-10'>
Lemma sol_are_even n a b : n > 1 -> sol n a b -> (~~ odd a) && (~~ odd b).
Proof.
Admitted.
</textarea></div>
<div><p>
</div>
<p>
<ul class="doclist">
<li> Prove that the solutions for n are the halves of the solutions for n + 2.
<ul class="doclist">
<li> Hint: <tt>Search _ odd double in MC</tt> and <tt>Search _ "eq" "mul" in MC</tt>.
</li>
</ul>
</li>
</ul>
<div> </div>
<div><textarea id='coq-ta-11'>
Lemma sol_add2 n a b : sol (2 + n) a b -> sol n (half a) (half b).
Proof.
Admitted.
</textarea></div>
<div><p>
</div>
<p>
<ul class="doclist">
<li> Prove there are no solutions for n even (Easy)
<ul class="doclist">
<li> Hint: Use <tt>sol0</tt> and <tt>solSS</tt>.
</li>
</ul>
</li>
</ul>
<div> </div>
<div><textarea id='coq-ta-12'>
Lemma sol_even a b n : ~~ sol (2 * n) a b.
Proof.
Admitted.
</textarea></div>
<div><p>
</div>
<p>
<ul class="doclist">
<li> Certify the only solution when n is odd. (SUPER HARD)
<ul class="doclist">
<li> Hint 1: Use <tt>sol1</tt>, <tt>solSS</tt> and <tt>sol_are_even</tt>.
</li>
<li> Hint 2: Really sketch it on paper first!
</li>
</ul>
</li>
</ul>
<div> </div>
<div><textarea id='coq-ta-13'>
Lemma sol_odd a b n : sol (2 * n + 1) a b = (a == 2 ^ n) && (b == 2 ^ n).
Proof.
Admitted.
</textarea></div>
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