# Library prosa.util.supremum

From mathcomp Require Import ssreflect ssrbool eqtype seq.

# Computation of a Sequence's Supremum

This module provides a simple function supremum for the computation of a maximal element of a sequence, according to any given relation R. If the relation R is reflexive, total, and transitive, the result of supremum is indeed the supremum of the set of items in the sequence.
Section SelectSupremum.
Consider any type of elements with decidable equality...
Context {T : eqType}.
...and any given relation R.
Variable R : rel T.
We first define a help function choose_superior that, given an element x and maybe a second element y, picks x if R x y holds, and y otherwise.
Definition choose_superior (x : T) (maybe_y : option T) : option T :=
match maybe_y with
| Some y ⇒ if R x y then Some x else Some y
| None ⇒ Some x
end.
The supremum of a given sequence s can then be computed by simply folding s with choose_superior.
Definition supremum (s : seq T) : option T := foldr choose_superior None s.
Next, we establish that supremum satisfies its specification. To this end, we first establish a few simple helper lemmas.
We start with observing how supremum can be unrolled one step.
Lemma supremum_unfold :
∀ head tail,
supremum (head :: tail) = choose_superior head (supremum tail).
Proof.
move⇒ head tail.
by rewrite [LHS]/supremum /foldr -/(foldr choose_superior None tail) -/(supremum tail).
Qed.
Next, we observe that supremum returns a result for any non-empty list.
Lemma supremum_exists : ∀ x s, x \in s → supremum s != None.
Proof.
move⇒ x s IN.
elim: s IN; first by done.
move⇒ a l _ _.
rewrite supremum_unfold.
destruct (supremum l) as [s|]; rewrite /choose_superior //.
by destruct (R a s).
Qed.
Conversely, if supremum finds nothing, then the list is empty.
Lemma supremum_none : ∀ s, supremum s = None → s = nil.
Proof.
move⇒ s.
elim: s; first by done.
move ⇒ a l IH.
rewrite supremum_unfold /choose_superior.
by destruct (supremum l) as [s|]; try destruct (R a s).
Qed.
Next, we observe that the value found by supremum comes indeed from the list that it was given.
Lemma supremum_in :
∀ x s,
supremum s = Some x →
x \in s.
Proof.
move⇒ x.
elim ⇒ // a l IN_TAIL IN.
rewrite in_cons; apply /orP.
move: IN; rewrite supremum_unfold.
destruct (supremum l) as [s|]; rewrite /choose_superior.
{ elim: (R a s) ⇒ EQ.
- left; apply /eqP.
by injection EQ.
- right; by apply IN_TAIL. }
{ move⇒ IN; left. apply /eqP.
by injection IN. }
Qed.
To prove that supremum indeed computes the given sequence's supremum, we need to make additional assumptions on R.
(1) R is reflexive.
Hypothesis H_R_reflexive : reflexive R.
(2) R is total.
Hypothesis H_R_total : total R.
(3) R is transitive.
Hypothesis H_R_transitive : transitive R.
Based on these assumptions, we show that the function supremum indeed computes an upper bound on all elements in the given sequence.
Lemma supremum_spec :
∀ x s,
supremum s = Some x →
∀ y,
y \in s → R x y.
Proof.
move⇒ x s SOME_x.
move: x SOME_x (supremum_in _ _ SOME_x).
elim: s; first by done.
move⇒ s1 sn IH z SOME_z IN_z_s y.
move: SOME_z. rewrite supremum_unfold /choose_superior ⇒ SOME_z.
destruct (supremum sn) as [b|] eqn:SUPR; last first.
{ apply supremum_none in SUPR; subst.
rewrite mem_seq1 ⇒ /eqP →.
by injection SOME_z ⇒ →. }
{ rewrite in_cons ⇒ /orP [/eqP EQy | INy]; last first.
- have R_by: R b y
by apply IH ⇒ //; apply supremum_in.
apply: H_R_transitive R_by.
destruct (R s1 b) eqn:R_s1b;
by injection SOME_z ⇒ <-.
- move: IN_z_s; rewrite in_cons ⇒ /orP [/eqP EQz | INz];
first by subst.
move: (H_R_total s1 b) ⇒ /orP [R_s1b|R_bs1].
+ move: SOME_z. rewrite R_s1b ⇒ SOME_z.
by injection SOME_z ⇒ EQ; subst.
+ by destruct (R s1 b); injection SOME_z ⇒ EQ; subst. }
Qed.
End SelectSupremum.