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(** Constructions of semi-lattices. *)
Require Import Coqlib.
Require Import Maps.
(** * Signatures of semi-lattices *)
(** A semi-lattice is a type [t] equipped with a decidable equality [eq],
a partial order [ge], a smallest element [bot], and an upper
bound operation [lub]. Note that we do not demand that [lub] computes
the least upper bound. *)
Module Type SEMILATTICE.
Variable t: Set.
Variable eq: forall (x y: t), {x=y} + {x<>y}.
Variable ge: t -> t -> Prop.
Hypothesis ge_refl: forall x, ge x x.
Hypothesis ge_trans: forall x y z, ge x y -> ge y z -> ge x z.
Variable bot: t.
Hypothesis ge_bot: forall x, ge x bot.
Variable lub: t -> t -> t.
Hypothesis lub_commut: forall x y, lub x y = lub y x.
Hypothesis ge_lub_left: forall x y, ge (lub x y) x.
End SEMILATTICE.
(** A semi-lattice ``with top'' is similar, but also has a greatest
element [top]. *)
Module Type SEMILATTICE_WITH_TOP.
Variable t: Set.
Variable eq: forall (x y: t), {x=y} + {x<>y}.
Variable ge: t -> t -> Prop.
Hypothesis ge_refl: forall x, ge x x.
Hypothesis ge_trans: forall x y z, ge x y -> ge y z -> ge x z.
Variable bot: t.
Hypothesis ge_bot: forall x, ge x bot.
Variable top: t.
Hypothesis ge_top: forall x, ge top x.
Variable lub: t -> t -> t.
Hypothesis lub_commut: forall x y, lub x y = lub y x.
Hypothesis ge_lub_left: forall x y, ge (lub x y) x.
End SEMILATTICE_WITH_TOP.
(** * Semi-lattice over maps *)
(** Given a semi-lattice with top [L], the following functor implements
a semi-lattice structure over finite maps from positive numbers to [L.t].
The default value for these maps is either [L.top] or [L.bot]. *)
Module LPMap(L: SEMILATTICE_WITH_TOP) <: SEMILATTICE_WITH_TOP.
Inductive t_ : Set :=
| Bot_except: PTree.t L.t -> t_
| Top_except: PTree.t L.t -> t_.
Definition t: Set := t_.
Lemma eq: forall (x y: t), {x=y} + {x<>y}.
Proof.
assert (forall m1 m2: PTree.t L.t, {m1=m2} + {m1<>m2}).
apply PTree.eq. exact L.eq.
decide equality.
Qed.
Definition get (p: positive) (x: t) : L.t :=
match x with
| Bot_except m =>
match m!p with None => L.bot | Some x => x end
| Top_except m =>
match m!p with None => L.top | Some x => x end
end.
Definition set (p: positive) (v: L.t) (x: t) : t :=
match x with
| Bot_except m =>
Bot_except (PTree.set p v m)
| Top_except m =>
Top_except (PTree.set p v m)
end.
Lemma gss:
forall p v x,
get p (set p v x) = v.
Proof.
intros. destruct x; simpl; rewrite PTree.gss; auto.
Qed.
Lemma gso:
forall p q v x,
p <> q -> get p (set q v x) = get p x.
Proof.
intros. destruct x; simpl; rewrite PTree.gso; auto.
Qed.
Definition ge (x y: t) : Prop :=
forall p, L.ge (get p x) (get p y).
Lemma ge_refl: forall x, ge x x.
Proof.
unfold ge; intros. apply L.ge_refl.
Qed.
Lemma ge_trans: forall x y z, ge x y -> ge y z -> ge x z.
Proof.
unfold ge; intros. apply L.ge_trans with (get p y); auto.
Qed.
Definition bot := Bot_except (PTree.empty L.t).
Lemma get_bot: forall p, get p bot = L.bot.
Proof.
unfold bot; intros; simpl. rewrite PTree.gempty. auto.
Qed.
Lemma ge_bot: forall x, ge x bot.
Proof.
unfold ge; intros. rewrite get_bot. apply L.ge_bot.
Qed.
Definition top := Top_except (PTree.empty L.t).
Lemma get_top: forall p, get p top = L.top.
Proof.
unfold top; intros; simpl. rewrite PTree.gempty. auto.
Qed.
Lemma ge_top: forall x, ge top x.
Proof.
unfold ge; intros. rewrite get_top. apply L.ge_top.
Qed.
Definition lub (x y: t) : t :=
match x, y with
| Bot_except m, Bot_except n =>
Bot_except
(PTree.combine
(fun a b =>
match a, b with
| Some u, Some v => Some (L.lub u v)
| None, _ => b
| _, None => a
end)
m n)
| Bot_except m, Top_except n =>
Top_except
(PTree.combine
(fun a b =>
match a, b with
| Some u, Some v => Some (L.lub u v)
| None, _ => b
| _, None => None
end)
m n)
| Top_except m, Bot_except n =>
Top_except
(PTree.combine
(fun a b =>
match a, b with
| Some u, Some v => Some (L.lub u v)
| None, _ => None
| _, None => a
end)
m n)
| Top_except m, Top_except n =>
Top_except
(PTree.combine
(fun a b =>
match a, b with
| Some u, Some v => Some (L.lub u v)
| _, _ => None
end)
m n)
end.
Lemma lub_commut:
forall x y, lub x y = lub y x.
Proof.
destruct x; destruct y; unfold lub; decEq;
apply PTree.combine_commut; intros;
(destruct i; destruct j; auto; decEq; apply L.lub_commut).
Qed.
Lemma ge_lub_left:
forall x y, ge (lub x y) x.
Proof.
unfold ge, get, lub; intros; destruct x; destruct y.
rewrite PTree.gcombine.
destruct t0!p. destruct t1!p. apply L.ge_lub_left.
apply L.ge_refl. destruct t1!p. apply L.ge_bot. apply L.ge_refl.
auto.
rewrite PTree.gcombine.
destruct t0!p. destruct t1!p. apply L.ge_lub_left.
apply L.ge_top. destruct t1!p. apply L.ge_bot. apply L.ge_bot.
auto.
rewrite PTree.gcombine.
destruct t0!p. destruct t1!p. apply L.ge_lub_left.
apply L.ge_refl. apply L.ge_refl. auto.
rewrite PTree.gcombine.
destruct t0!p. destruct t1!p. apply L.ge_lub_left.
apply L.ge_top. apply L.ge_refl.
auto.
Qed.
End LPMap.
(** * Flat semi-lattice *)
(** Given a type with decidable equality [X], the following functor
returns a semi-lattice structure over [X.t] complemented with
a top and a bottom element. The ordering is the flat ordering
[Bot < Inj x < Top]. *)
Module LFlat(X: EQUALITY_TYPE) <: SEMILATTICE_WITH_TOP.
Inductive t_ : Set :=
| Bot: t_
| Inj: X.t -> t_
| Top: t_.
Definition t : Set := t_.
Lemma eq: forall (x y: t), {x=y} + {x<>y}.
Proof.
decide equality. apply X.eq.
Qed.
Definition ge (x y: t) : Prop :=
match x, y with
| Top, _ => True
| _, Bot => True
| Inj a, Inj b => a = b
| _, _ => False
end.
Lemma ge_refl: forall x, ge x x.
Proof.
unfold ge; destruct x; auto.
Qed.
Lemma ge_trans: forall x y z, ge x y -> ge y z -> ge x z.
Proof.
unfold ge; destruct x; destruct y; try destruct z; intuition.
transitivity t1; auto.
Qed.
Definition bot: t := Bot.
Lemma ge_bot: forall x, ge x bot.
Proof.
destruct x; simpl; auto.
Qed.
Definition top: t := Top.
Lemma ge_top: forall x, ge top x.
Proof.
destruct x; simpl; auto.
Qed.
Definition lub (x y: t) : t :=
match x, y with
| Bot, _ => y
| _, Bot => x
| Top, _ => Top
| _, Top => Top
| Inj a, Inj b => if X.eq a b then Inj a else Top
end.
Lemma lub_commut: forall x y, lub x y = lub y x.
Proof.
destruct x; destruct y; simpl; auto.
case (X.eq t0 t1); case (X.eq t1 t0); intros; congruence.
Qed.
Lemma ge_lub_left: forall x y, ge (lub x y) x.
Proof.
destruct x; destruct y; simpl; auto.
case (X.eq t0 t1); simpl; auto.
Qed.
End LFlat.
(** * Boolean semi-lattice *)
(** This semi-lattice has only two elements, [bot] and [top], trivially
ordered. *)
Module LBoolean <: SEMILATTICE_WITH_TOP.
Definition t := bool.
Lemma eq: forall (x y: t), {x=y} + {x<>y}.
Proof. decide equality. Qed.
Definition ge (x y: t) : Prop := x = y \/ x = true.
Lemma ge_refl: forall x, ge x x.
Proof. unfold ge; tauto. Qed.
Lemma ge_trans: forall x y z, ge x y -> ge y z -> ge x z.
Proof. unfold ge; intuition congruence. Qed.
Definition bot := false.
Lemma ge_bot: forall x, ge x bot.
Proof. destruct x; compute; tauto. Qed.
Definition top := true.
Lemma ge_top: forall x, ge top x.
Proof. unfold ge, top; tauto. Qed.
Definition lub (x y: t) := x || y.
Lemma lub_commut: forall x y, lub x y = lub y x.
Proof. intros; unfold lub. apply orb_comm. Qed.
Lemma ge_lub_left: forall x y, ge (lub x y) x.
Proof. destruct x; destruct y; compute; tauto. Qed.
End LBoolean.
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