complete one_shot

algebra/one_shot.v

@@ -2,6 +2,8 @@ From iris.algebra Require Export cmra.
 From iris.algebra Require Import upred.
 Local Arguments validN _ _ _ !_ /.
 Local Arguments valid _ _  !_ /.
+Local Arguments cmra_validN _ _ !_ /.
+Local Arguments cmra_valid _  !_ /.
 
 (* TODO: Really, we should have sums, and then this should be just "excl unit + A". *)
 Inductive one_shot {A : Type} :=
@@ -92,3 +94,198 @@ End cofe.
 
 Arguments one_shotC : clear implicits.
 
+(* Functor on COFEs *)
+Definition one_shot_map {A B} (f : A → B) (x : one_shot A) : one_shot B :=
+  match x with
+  | OneShotPending => OneShotPending
+  | Shot a => Shot (f a)
+  | OneShotUnit => OneShotUnit
+  | OneShotBot => OneShotBot
+  end.
+Instance: Params (@one_shot_map) 2.
+
+Lemma one_shot_map_id {A} (x : one_shot A) : one_shot_map id x = x.
+Proof. by destruct x. Qed.
+Lemma one_shot_map_compose {A B C} (f : A → B) (g : B → C) (x : one_shot A) :
+  one_shot_map (g ∘ f) x = one_shot_map g (one_shot_map f x).
+Proof. by destruct x. Qed.
+Lemma one_shot_map_ext {A B : cofeT} (f g : A → B) x :
+  (∀ x, f x ≡ g x) → one_shot_map f x ≡ one_shot_map g x.
+Proof. by destruct x; constructor. Qed.
+Instance one_shot_map_cmra_ne {A B : cofeT} n :
+  Proper ((dist n ==> dist n) ==> dist n ==> dist n) (@one_shot_map A B).
+Proof. intros f f' Hf; destruct 1; constructor; by try apply Hf. Qed.
+Definition one_shotC_map {A B} (f : A -n> B) : one_shotC A -n> one_shotC B :=
+  CofeMor (one_shot_map f).
+Instance one_shotC_map_ne A B n : Proper (dist n ==> dist n) (@one_shotC_map A B).
+Proof. intros f f' Hf []; constructor; by try apply Hf. Qed.
+
+Section cmra.
+Context {A : cmraT}.
+Implicit Types a b : A.
+Implicit Types x y : one_shot A.
+
+(* CMRA *)
+Instance one_shot_valid : Valid (one_shot A) := λ x,
+  match x with
+  | OneShotPending => True
+  | Shot a => ✓ a
+  | OneShotUnit => True
+  | OneShotBot => False
+  end.
+Instance one_shot_validN : ValidN (one_shot A) := λ n x,
+  match x with
+  | OneShotPending => True
+  | Shot a => ✓{n} a
+  | OneShotUnit => True
+  | OneShotBot => False
+  end.
+Global Instance one_shot_empty : Empty (one_shot A) := OneShotUnit.
+Instance one_shot_core : Core (one_shot A) := λ x,
+  match x with
+  | Shot a => Shot (core a)
+  | OneShotBot => OneShotBot
+  | _ => ∅
+  end.
+Instance one_shot_op : Op (one_shot A) := λ x y,
+  match x, y with
+  | Shot a, Shot b => Shot (a ⋅ b)
+  | Shot a, OneShotUnit | OneShotUnit, Shot a => Shot a
+  | OneShotUnit, OneShotPending | OneShotPending, OneShotUnit => OneShotPending
+  | OneShotUnit, OneShotUnit => OneShotUnit
+  | _, _ => OneShotBot
+  end.
+
+Lemma Shot_op a b : Shot a ⋅ Shot b = Shot (a ⋅ b).
+Proof. done. Qed.
+Lemma Shot_incl a b : Shot a ≼ Shot b ↔ a ≼ b.
+Proof.
+  split; intros [c H].
+  - destruct c; inversion_clear H; first by eexists.
+    by rewrite (_ : b ≡ a).
+  - exists (Shot c). constructor. done.
+Qed. 
+
+Definition one_shot_cmra_mixin : CMRAMixin (one_shot A).
+Proof.
+  split.
+  - intros n []; destruct 1; constructor; by cofe_subst. 
+  - intros ? [|a| |] [|b| |] H; inversion_clear H; constructor; by f_equiv.
+  - intros ? [|a| |] [|b| |] H; inversion_clear H; cofe_subst; done.
+  - intros [|a| |]; rewrite /= ?cmra_valid_validN; naive_solver eauto using O.
+  - intros n [|a| |]; simpl; auto using cmra_validN_S.
+  - intros [|a1| |] [|a2| |] [|a3| |]; constructor; by rewrite ?assoc.
+  - intros [|a1| |] [|a2| |]; constructor; by rewrite 1?comm.
+  - intros [|a| |]; constructor; []. exact: cmra_core_l.
+  - intros [|a| |]; constructor; []. exact: cmra_core_idemp.
+  - intros [|a1| |] [|a2| |]; simpl;
+      try solve [ by exists OneShotUnit; constructor
+                | by exists OneShotBot; constructor
+                | by intros [[|a3| |] H]; inversion_clear H ].
+    + intros H%Shot_incl. apply Shot_incl, cmra_core_preserving. done.
+    + intros _. exists (Shot (core a2)). by constructor.
+  - intros n [|a1| |] [|a2| |]; simpl; eauto using cmra_validN_op_l; done.
+  - intros n [|a| |] y1 y2 Hx Hx'; last 2 first.
+    + by exists (∅, ∅); destruct y1, y2; inversion_clear Hx'.
+    + by exists (OneShotBot, OneShotBot); destruct y1, y2; inversion_clear Hx'.
+    + destruct y1, y2; try (exfalso; by inversion_clear Hx').
+      * by exists (OneShotPending, OneShotUnit).
+      * by exists (OneShotUnit, OneShotPending).
+    +  destruct y1 as [|b1 | |], y2 as [|b2| |]; try (exfalso; by inversion_clear Hx').
+       * apply (inj Shot) in Hx'.
+         destruct (cmra_extend n a b1 b2) as ([z1 z2]&?&?&?); auto.
+         exists (Shot z1, Shot z2). by repeat constructor.
+       * exists (Shot a, ∅). inversion_clear Hx'. by repeat constructor.
+       * exists (∅, Shot a). inversion_clear Hx'. by repeat constructor.
+Qed.
+Canonical Structure one_shotR : cmraT := CMRAT one_shot_cofe_mixin one_shot_cmra_mixin.
+Global Instance one_shot_cmra_unit : CMRAUnit one_shotR.
+Proof. split. done. by intros []. apply _. Qed.
+Global Instance one_shot_cmra_discrete : CMRADiscrete A → CMRADiscrete one_shotR.
+Proof.
+  split; first apply _.
+  intros [|a| |]; simpl; auto using cmra_discrete_valid.
+Qed.
+
+Lemma one_shot_validN_inv_l n y : ✓{n} (OneShotPending ⋅ y) → y = ∅.
+Proof.
+  destruct y as [|b| |]; [done| |done|done]. destruct 1.
+Qed.
+Lemma one_shot_valid_inv_l y : ✓ (OneShotPending ⋅ y) → y = ∅.
+Proof. intros. by apply one_shot_validN_inv_l with 0, cmra_valid_validN. Qed.
+Lemma one_shot_bot_largest y : y ≼ OneShotBot.
+Proof.
+  destruct y; exists OneShotBot; constructor.
+Qed.
+
+(** Internalized properties *)
+Lemma one_shot_equivI {M} (x y : one_shot A) :
+  (x ≡ y) ⊣⊢ (match x, y with
+               | OneShotPending, OneShotPending => True
+               | Shot a, Shot b => a ≡ b
+               | OneShotUnit, OneShotUnit => True
+               | OneShotBot, OneShotBot => True
+               | _, _ => False
+               end : uPred M).
+Proof.
+  uPred.unseal; do 2 split; first by destruct 1.
+  by destruct x, y; try destruct 1; try constructor.
+Qed.
+Lemma one_shot_validI {M} (x : one_shot A) :
+  (✓ x) ⊣⊢ (match x with
+            | Shot a => ✓ a 
+            | OneShotBot => False
+            | _ => True
+            end : uPred M).
+Proof. uPred.unseal. by destruct x. Qed.
+
+(** Updates *)
+Lemma one_shot_update_shoot (a : A) : ✓ a → OneShotPending ~~> Shot a.
+Proof.
+  move=> ? n y /one_shot_validN_inv_l ->. by apply cmra_valid_validN.
+Qed.
+Lemma one_shot_update (a1 a2 : A) : a1 ~~> a2 → Shot a1 ~~> Shot a2.
+Proof.
+  intros Ha n [|b| |] ?; simpl; auto.
+  apply cmra_validN_op_l with (core a1), Ha. by rewrite cmra_core_r.
+Qed.
+
+End cmra.
+
+Arguments one_shotR : clear implicits.
+
+(* Functor *)
+Instance one_shot_map_cmra_monotone {A B : cmraT} (f : A → B) :
+  CMRAMonotone f → CMRAMonotone (one_shot_map f).
+Proof.
+  split; try apply _.
+  - intros n [|a| |]; simpl; auto using validN_preserving.
+  - intros [|a1| |] [|a2| |] [[|a3| |] Hx];
+      inversion Hx; setoid_subst; try apply cmra_unit_least;
+        try apply one_shot_bot_largest; auto; [].
+    destruct (included_preserving f a1 (a1 ⋅ a3)) as [b ?].
+    { by apply cmra_included_l. }
+    by exists (Shot b); constructor.
+Qed.
+
+Program Definition one_shotRF (F : rFunctor) : rFunctor := {|
+  rFunctor_car A B := one_shotR (rFunctor_car F A B);
+  rFunctor_map A1 A2 B1 B2 fg := one_shotC_map (rFunctor_map F fg)
+|}.
+Next Obligation.
+  by intros F A1 A2 B1 B2 n f g Hfg; apply one_shotC_map_ne, rFunctor_ne.
+Qed.
+Next Obligation.
+  intros F A B x. rewrite /= -{2}(one_shot_map_id x).
+  apply one_shot_map_ext=>y; apply rFunctor_id.
+Qed.
+Next Obligation.
+  intros F A1 A2 A3 B1 B2 B3 f g f' g' x. rewrite /= -one_shot_map_compose.
+  apply one_shot_map_ext=>y; apply rFunctor_compose.
+Qed.
+
+Instance one_shotRF_contractive F :
+  rFunctorContractive F → rFunctorContractive (one_shotRF F).
+Proof.
+  by intros ? A1 A2 B1 B2 n f g Hfg; apply one_shotC_map_ne, rFunctor_contractive.
+Qed.