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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" />
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async></script>
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<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.
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<div><textarea id='coq-ta-1'>
From mathcomp Require Import mini_ssreflect.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
</textarea></div>
<div><p>
<hr/>
<div class="slide">
<p>
<h2>
Exercise 1:
</h2>
<p>
<ul class="doclist">
<li> Improve the last symplification function of lesson 5 by adding
the simplification rule $$x - 0 = x$$
</li>
</ul>
<div>
</div>
<div><textarea id='coq-ta-2'>
Module Ex1.
Inductive expr :=
| Zero
| Minus (x : expr) (y : expr)
| Var (n : nat).
Fixpoint simplify e :=
match e with
| Minus x y =>
match simplify x, simplify y with
| Var n, Var m =>
match n == m with
| true => Zero
| false => Minus (Var n) (Var m)
end
| (* Fill me *)
a, Zero => a
| a, b => Minus a b
end
| y => y
end.
Fixpoint interp (e : expr) (c : list nat) :=
match e with
| Zero => 0
| Minus x y => (interp x c) - (interp y c)
| Var x => nth 0 c x
end.
Lemma simplify_correct (e : expr) (c : list nat) : interp e c = interp (simplify e) c.
Proof.
elim: e => //= x Hx y Hy.
case: (simplify x) Hx => [|x1 x2|n] -> ; case: (simplify y) Hy => [|y1 y2|m] -> //.
(* fill in *)
1,2: by rewrite subn0.
by case: eqP => [->|//]; rewrite subnn.
Qed.
Lemma test (x : nat) : x - x - 0 = 0.
Proof.
pose AST : expr := Minus (Minus (Var 0) (Var 0)) Zero.
pose CTX : list nat := [:: x].
rewrite -[LHS]/(interp AST CTX).
rewrite simplify_correct.
rewrite /=.
by [].
Qed.
End Ex1.
</textarea></div>
<div><p>
</div>
<hr/>
<div class="slide">
<p>
<h2>
Exercise 2:
</h2>
<p>
<ul class="doclist">
<li> Improve on exercise 1 by adding
to the <tt>expr</tt> inductive type a constructor for multiplication and
the simplification rules $$x * 0 = 0$$ and $$0 * x = 0$$
</li>
</ul>
<div>
</div>
<div><textarea id='coq-ta-3'>
Module Ex2.
Inductive expr :=
| Zero
| (* fill in *)
Mult (x : expr) (y : expr)
| Minus (x : expr) (y : expr)
| Var (n : nat).
Fixpoint simplify e :=
match e with
| Minus x y =>
match simplify x, simplify y with
| Var n, Var m =>
match n == m with
| true => Zero
| false => Minus (Var n) (Var m)
end
| (* Fill me, from ex 1*)
a, Zero => a
| a, b => Minus a b
end
| (* fill me *)
Mult x y =>
match simplify x, simplify y with
| Zero, _ => Zero
| _, Zero => Zero
| a, b => Mult a b
end
| y => y
end.
Fixpoint interp (e : expr) (c : list nat) :=
match e with
| Zero => 0
| Minus x y => (interp x c) - (interp y c)
| (* fill in *)
Mult x y => (interp x c) * (interp y c)
| Var x => nth 0 c x
end.
Lemma simplify_correct (e : expr) (c : list nat) : interp e c = interp (simplify e) c.
Proof.
elim: e => //= x Hx y Hy.
- case: (simplify x) Hx => [|x1 x2|x1 x2|n] -> ; case: (simplify y) Hy => [|y1 y2|y1 y2|m] -> //.
(* fill in *)
1,2,3: by rewrite muln0.
- case: (simplify x) Hx => [|x1 x2|x1 x2|n] -> ; case: (simplify y) Hy => [|y1 y2|y1 y2|m] -> //.
(* fill in, like in exercise 1 *)
1,2,3: by rewrite subn0.
by case: eqP => [->|//]; rewrite subnn.
Qed.
Lemma test (x : nat) (y : nat) : x - x - (y * 0) = 0.
Proof.
pose AST : expr := (* fill me *)
Minus (Minus (Var 0) (Var 0)) (Mult (Var 1) Zero).
pose CTX : list nat := (* fill me *)
[:: x; y].
rewrite -[LHS]/(interp AST CTX).
rewrite simplify_correct.
rewrite /=.
by [].
Qed.
End Ex2.
</textarea></div>
<div><p>
</div>
<hr/>
<div class="slide">
<p>
<h2>
Exercise 3:
</h2>
<p>
<ul class="doclist">
<li> Improve on exercise 2 by adding
to the <tt>expr</tt> data type a constructor for <tt>One</tt> and
the simplification rules $$x * 1 = x$$ and $$1 * x = x$$
</li>
</ul>
<div>
</div>
<div><textarea id='coq-ta-4'>
Module Ex3.
Inductive expr :=
| Zero
| (* fill in *)
One
| (* fill in, as in exercise 2 *)
Mult (x : expr) (y : expr)
| Minus (x : expr) (y : expr)
| Var (n : nat).
Fixpoint simplify e :=
match e with
| Minus x y =>
match simplify x, simplify y with
| Var n, Var m =>
match n == m with
| true => Zero
| false => Minus (Var n) (Var m)
end
| (* Fill me, from ex 1*)
a, Zero => a
| a, b => Minus a b
end
| (* fill me *)
Mult x y =>
match simplify x, simplify y with
| Zero, _ => Zero
| _, Zero => Zero
| One, b => b
| a, One => a
| a, b => Mult a b
end
| y => y
end.
Fixpoint interp (e : expr) (c : list nat) :=
match e with
| Zero => 0
| (* fill in *)
One => 1
| Minus x y => (interp x c) - (interp y c)
| (* fill in *)
Mult x y => (interp x c) * (interp y c)
| Var x => nth 0 c x
end.
Lemma simplify_correct (e : expr) (c : list nat) : interp e c = interp (simplify e) c.
Proof.
elim: e => //= x Hx y Hy.
- case: (simplify x) Hx => [||x1 x2|x1 x2|n] -> ; case: (simplify y) Hy => [||y1 y2|y1 y2|m] -> //.
(* fill in *)
1,2,3: by rewrite mul1n.
1,3,5: by rewrite muln0.
1,2,3: by rewrite muln1.
- case: (simplify x) Hx => [||x1 x2|x1 x2|n] -> ; case: (simplify y) Hy => [||y1 y2|y1 y2|m] -> //.
(* fill in *)
1,2,3: by rewrite subn0.
by case: eqP => [->|//]; rewrite subnn.
Qed.
Lemma test (x : nat) (y : nat) : x - x - (y * 0) = 0.
Proof.
pose AST : expr := (* fill me *)
Minus (Minus (Var 0) (Var 0)) (Mult (Var 1) Zero).
pose CTX : list nat := (* fill me *)
[:: x; y].
rewrite -[LHS]/(interp AST CTX).
rewrite simplify_correct.
rewrite /=.
by [].
Qed.
End Ex3.
(*
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----------------------------------------------------------
*)
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