Commit 5de5b522 authored by Björn Brandenburg's avatar Björn Brandenburg

add a function for computing a sequence's supremum

spell checker: add 'supremum' to dictionary
parent 2dc14ac2
......@@ -39,4 +39,5 @@ Layland
Liu
sequentiality
equalities
extremum
\ No newline at end of file
extremum
supremum
......@@ -13,5 +13,6 @@ Require Export rt.util.epsilon.
Require Export rt.util.search_arg.
Require Export rt.util.rel.
Require Export rt.util.minmax.
Require Export rt.util.supremum.
Require Export rt.util.nondecreasing.
Require Export rt.util.rewrite_facilities.
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:
forall 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: forall 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); rewrite /choose_superior //.
by destruct (R a s).
Qed.
(** Conversely, if [supremum] finds nothing, then the list is empty. *)
Lemma supremum_none: forall 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); 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:
forall 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); rewrite /choose_superior.
{ elim: (R a s) => EQ.
- left; apply /eqP.
by injection EQ.
- right; by apply IN_TAIL. }
{ 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:
forall x s,
supremum s = Some x ->
forall 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 with (y := b) => //.
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.
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