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exercise6-solution.html
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<title>Machine-Checked Mathematics</title>
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From mathcomp Require Import mini_ssreflect.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
</textarea></div>
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<hr/>
<div class="slide">
<p>
<h2>
Exercise 1:
</h2>
<p>
<ul class="doclist">
<li> Extend the definitions introduced in Lesson 6, so that
the proof of the <tt>test1</tt> lemma succeeds without changing
the proof script.
</li>
</ul>
<div>
</div>
<div><textarea id='coq-ta-2'>
Section PosNat.
Variable P : nat -> Prop.
Record pos_nat : Set := PosNat {val : nat; pos_val : 1 <= val}.
Coercion val : pos_nat >-> nat.
Hypothesis posP : forall x : pos_nat, P x.
Lemma pos_S (x : nat) : 1 <= S x.
Proof. by []. Qed.
Definition pos_nat_S (n : nat) : pos_nat := PosNat (pos_S n).
Canonical pos_nat_S.
Lemma pos_add (x y : pos_nat) : 1 <= x + y.
Proof. by rewrite addn_gt0; case: x => x ->. Qed.
Definition pos_nat_add (x y : pos_nat) : pos_nat := PosNat (pos_add x y).
Canonical pos_nat_add.
(* Something goes here *)
Lemma pos_mul (x y : pos_nat) : 1 <= x * y.
Proof. by rewrite muln_gt0; case: x => x ->; case: y => y ->. Qed.
Definition pos_nat_mul (x y : pos_nat) : pos_nat := PosNat (pos_mul x y).
Canonical pos_nat_mul.
Lemma test1 x y : P ((S x) * (3 + (S y))).
Proof.
exact: posP.
Qed.
End PosNat.
</textarea></div>
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<p>
Exercise 2.
We define a type <tt>tuple</tt> for polymorphic lists with a prescribed length, given
as a parameter. This type is a dependent pair.
This condition is expressed as a (boolean) constraint
on the size of a list, coerced to Prop via the usual <tt>is_true</tt> coercion.
The infix <tt>==</tt> notation refers to the notation available on instances of
the <tt>eqType</tt> structure, as defined by the library loaded in this section.
This structure is isomorphic to the one we used in Lesson 6. In particular, it is endowed with a reflection lemma <tt>eqP</tt>, relating the boolean test with
equality.
<p>
Fill the missing parts so that the <tt>test</tt> lemmas are proved without
changing their script.
<p>
<div>
</div>
<div><textarea id='coq-ta-3'>
About size.
About map.
Section Def.
Variables (n : nat) (T : Type).
Record tuple : Type := Tuple {tval : list T; valP : size tval == n}.
Coercion tval : tuple >-> list.
Lemma size_tuple (t : tuple) : size t = n.
(* Something goes here *)
Proof. apply/eqP. apply: valP.
Qed.
End Def.
Canonical nil_tuple T := @Tuple 0 T [::] erefl.
Canonical cons_tuple n T x (t : tuple n T) :=
Tuple (valP t : size (x :: t) == S n).
(* Something goes here, two things actually. *)
Section TupleTest.
Variable P : list nat -> Prop.
Variable hP : forall n, forall t : tuple (S n) nat, P t.
Lemma test2 : P (cons 2 nil).
Proof. apply: hP. Qed.
End TupleTest.
Section MapTuple.
Variables (T1 T2 : Type) (n : nat) (f : T1 -> T2).
Lemma map_tupleP (t : tuple n T1) : size (map f t) == n.
Proof.
(* Finish the proof, and may be more *)
by rewrite size_map size_tuple.
Qed.
Canonical map_tuple t := Tuple (map_tupleP t).
End MapTuple.
Section TupleTest.
Variable P : list bool -> Prop.
Variable hP : forall n, forall t : tuple (S n) bool, P t.
Lemma test3 (f : nat -> bool) : P (map f (cons 2 nil)).
Proof. apply: hP. Qed.
End TupleTest.
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