From 4cda26dd8d42e6dee7ebeacbb71b3f3d37d54c98 Mon Sep 17 00:00:00 2001
From: Robbert Krebbers
Date: Wed, 10 Oct 2012 11:35:41 +0200
Subject: [PATCH] Various minor changes.
* Introduce a type class and notation for disjointness.
* Define unions of finite maps (a lot of theory has still to be
moved from memory to fin_maps).
* Prove the Hoare rule for function calls with arguments.
* Prove the Hoare rule to add sets of functions.
* Some additional theory on lifting of assertions.
---
theories/base.v | 7 +++++++
theories/fin_maps.v | 44 +++++++++++++++++++++++++++++++++-----------
2 files changed, 40 insertions(+), 11 deletions(-)
diff --git a/theories/base.v b/theories/base.v
index 7ce783d..38fdebf 100644
--- a/theories/base.v
+++ b/theories/base.v
@@ -160,6 +160,13 @@ Notation "(∉)" := (λ x X, x ∉ X) (only parsing) : C_scope.
Notation "( x ∉)" := (λ X, x ∉ X) (only parsing) : C_scope.
Notation "(∉ X )" := (λ x, x ∉ X) (only parsing) : C_scope.
+Class Disjoint A := disjoint : A → A → Prop.
+Instance: Params (@disjoint) 2.
+Infix "⊥" := disjoint (at level 70) : C_scope.
+Notation "(⊥)" := disjoint (only parsing) : C_scope.
+Notation "( X ⊥)" := (disjoint X) (only parsing) : C_scope.
+Notation "(⊥ X )" := (λ Y, disjoint Y X) (only parsing) : C_scope.
+
(** ** Operations on maps *)
(** In this section we define operational type classes for the operations
on maps. In the file [fin_maps] we will axiomatize finite maps.
diff --git a/theories/fin_maps.v b/theories/fin_maps.v
index d2d0113..073b46b 100644
--- a/theories/fin_maps.v
+++ b/theories/fin_maps.v
@@ -54,13 +54,23 @@ Instance finmap_singleton `{PartialAlter K M} {A}
Definition list_to_map `{Insert K M} {A} `{Empty (M A)}
(l : list (K * A)) : M A := insert_list l ∅.
-Instance finmap_union `{Merge M} : UnionWith M := λ A f,
+Instance finmap_union_with `{Merge M} : UnionWith M := λ A f,
merge (union_with f).
-Instance finmap_intersection `{Merge M} : IntersectionWith M := λ A f,
+Instance finmap_intersection_with `{Merge M} : IntersectionWith M := λ A f,
merge (intersection_with f).
-Instance finmap_difference `{Merge M} : DifferenceWith M := λ A f,
+Instance finmap_difference_with `{Merge M} : DifferenceWith M := λ A f,
merge (difference_with f).
+(** Two finite maps are disjoint if they do not have overlapping cells. *)
+Instance finmap_disjoint `{Lookup K M} {A} : Disjoint (M A) := λ m1 m2,
+ ∀ i, m1 !! i = None ∨ m2 !! i = None.
+
+(** The union of two finite maps only has a meaningful definition for maps
+that are disjoint. However, as working with partial functions is inconvenient
+in Coq, we define the union as a total function. In case both finite maps
+have a value at the same index, we take the value of the first map. *)
+Instance finmap_union `{Merge M} {A} : Union (M A) := union_with (λ x _ , x).
+
(** * General theorems *)
Section finmap.
Context `{FinMap K M} `{∀ i j : K, Decision (i = j)} {A : Type}.
@@ -385,30 +395,30 @@ End merge.
Section union_intersection.
Context (f : A → A → A).
- Lemma finmap_union_merge m1 m2 i x y :
+ Lemma finmap_union_with_merge m1 m2 i x y :
m1 !! i = Some x →
m2 !! i = Some y →
union_with f m1 m2 !! i = Some (f x y).
Proof.
- intros Hx Hy. unfold union_with, finmap_union.
+ intros Hx Hy. unfold union_with, finmap_union_with.
now rewrite (merge_spec _), Hx, Hy.
Qed.
- Lemma finmap_union_l m1 m2 i x :
+ Lemma finmap_union_with_l m1 m2 i x :
m1 !! i = Some x → m2 !! i = None → union_with f m1 m2 !! i = Some x.
Proof.
- intros Hx Hy. unfold union_with, finmap_union.
+ intros Hx Hy. unfold union_with, finmap_union_with.
now rewrite (merge_spec _), Hx, Hy.
Qed.
- Lemma finmap_union_r m1 m2 i y :
+ Lemma finmap_union_with_r m1 m2 i y :
m1 !! i = None → m2 !! i = Some y → union_with f m1 m2 !! i = Some y.
Proof.
- intros Hx Hy. unfold union_with, finmap_union.
+ intros Hx Hy. unfold union_with, finmap_union_with.
now rewrite (merge_spec _), Hx, Hy.
Qed.
- Lemma finmap_union_None m1 m2 i :
+ Lemma finmap_union_with_None m1 m2 i :
union_with f m1 m2 !! i = None ↔ m1 !! i = None ∧ m2 !! i = None.
Proof.
- unfold union_with, finmap_union. rewrite (merge_spec _).
+ unfold union_with, finmap_union_with. rewrite (merge_spec _).
destruct (m1 !! i), (m2 !! i); compute; intuition congruence.
Qed.
@@ -421,6 +431,18 @@ Section union_intersection.
Global Instance:
Idempotent (=) f → Idempotent (=) (union_with f : M A → M A → M A) := _.
End union_intersection.
+
+Lemma finmap_union_Some (m1 m2 : M A) i x :
+ (m1 ∪ m2) !! i = Some x ↔
+ m1 !! i = Some x ∨ (m1 !! i = None ∧ m2 !! i = Some x).
+Proof.
+ unfold union, finmap_union, union_with, finmap_union_with.
+ rewrite (merge_spec _).
+ destruct (m1 !! i), (m2 !! i); compute; try intuition congruence.
+Qed.
+Lemma finmap_union_None (m1 m2 : M A) b :
+ (m1 ∪ m2) !! b = None ↔ m1 !! b = None ∧ m2 !! b = None.
+Proof. apply finmap_union_with_None. Qed.
End finmap.
(** * The finite map tactic *)
--
2.26.2