Simplify CMRAMonotone.

It now also contains a non-expansiveness proof.

algebra/agree.v

@@ -97,11 +97,6 @@ Proof.
   split; [|by intros ?; exists y].
   by intros [z Hz]; rewrite Hz assoc agree_idemp.
 Qed.
-Lemma agree_includedN n (x y : agree A) : x ≼{n} y ↔ y ≡{n}≡ x ⋅ y.
-Proof.
-  split; [|by intros ?; exists y].
-  by intros [z Hz]; rewrite Hz assoc agree_idemp.
-Qed.
 Lemma agree_op_inv n (x1 x2 : agree A) : ✓{n} (x1 ⋅ x2) → x1 ≡{n}≡ x2.
 Proof. intros Hxy; apply Hxy. Qed.
 Lemma agree_valid_includedN n (x y : agree A) : ✓{n} y → x ≼{n} y → x ≡{n}≡ y.
@@ -160,20 +155,20 @@ Proof. done. Qed.
 
 Section agree_map.
   Context {A B : cofeT} (f : A → B) `{Hf: ∀ n, Proper (dist n ==> dist n) f}.
-  Global Instance agree_map_ne n : Proper (dist n ==> dist n) (agree_map f).
+  Instance agree_map_ne n : Proper (dist n ==> dist n) (agree_map f).
   Proof. by intros x1 x2 Hx; split; simpl; intros; [apply Hx|apply Hf, Hx]. Qed.
-  Global Instance agree_map_proper :
-    Proper ((≡) ==> (≡)) (agree_map f) := ne_proper _.
+  Instance agree_map_proper : Proper ((≡) ==> (≡)) (agree_map f) := ne_proper _.
   Lemma agree_map_ext (g : A → B) x :
     (∀ x, f x ≡ g x) → agree_map f x ≡ agree_map g x.
   Proof. by intros Hfg; split; simpl; intros; rewrite ?Hfg. Qed.
   Global Instance agree_map_monotone : CMRAMonotone (agree_map f).
   Proof.
-    split; [|by intros n x [? Hx]; split; simpl; [|by intros n' ?; rewrite Hx]].
-    intros n x y; rewrite !agree_includedN; intros Hy; rewrite Hy.
-    split; last done; split; simpl; last tauto.
-    by intros (?&?&Hxy); repeat split; intros;
-       try apply Hxy; try apply Hf; eauto using @agree_valid_le.
+    split; first apply _.
+    - by intros n x [? Hx]; split; simpl; [|by intros n' ?; rewrite Hx].
+    - intros x y; rewrite !agree_included=> ->.
+      split; last done; split; simpl; last tauto.
+      by intros (?&?&Hxy); repeat split; intros;
+        try apply Hxy; try apply Hf; eauto using @agree_valid_le.
   Qed.
 End agree_map.
 

algebra/auth.v

@@ -98,12 +98,6 @@ Proof.
   split; [intros [[z1 z2] Hz]; split; [exists z1|exists z2]; apply Hz|].
   intros [[z1 Hz1] [z2 Hz2]]; exists (Auth z1 z2); split; auto.
 Qed.
-Lemma auth_includedN n (x y : auth A) :
-  x ≼{n} y ↔ authoritative x ≼{n} authoritative y ∧ own x ≼{n} own y.
-Proof.
-  split; [intros [[z1 z2] Hz]; split; [exists z1|exists z2]; apply Hz|].
-  intros [[z1 Hz1] [z2 Hz2]]; exists (Auth z1 z2); split; auto.
-Qed.
 Lemma authoritative_validN n (x : auth A) : ✓{n} x → ✓{n} authoritative x.
 Proof. by destruct x as [[]]. Qed.
 Lemma own_validN n (x : auth A) : ✓{n} x → ✓{n} own x.
@@ -212,7 +206,6 @@ Proof.
   intros. apply auth_update=>n af ? EQ; split; last by apply cmra_valid_validN.
   by rewrite -(local_updateN L) // EQ -(local_updateN L) // -EQ.
 Qed.
-
 End cmra.
 
 Arguments authRA : clear implicits.
@@ -234,14 +227,13 @@ Proof.
   intros f g Hf [??] [??] [??]; split; [by apply excl_map_cmra_ne|by apply Hf].
 Qed.
 Instance auth_map_cmra_monotone {A B : cmraT} (f : A → B) :
-  (∀ n, Proper (dist n ==> dist n) f) →
   CMRAMonotone f → CMRAMonotone (auth_map f).
 Proof.
-  split.
-  - by intros n [x a] [y b]; rewrite !auth_includedN /=;
-      intros [??]; split; simpl; apply: includedN_preserving.
-  - intros n [[a| |] b]; rewrite /= /cmra_validN;
-      naive_solver eauto using @includedN_preserving, @validN_preserving.
+  split; try apply _.
+  - intros n [[a| |] b]; rewrite /= /cmra_validN /=; try
+      naive_solver eauto using includedN_preserving, validN_preserving.
+  - by intros [x a] [y b]; rewrite !auth_included /=;
+      intros [??]; split; simpl; apply: included_preserving.
 Qed.
 Definition authC_map {A B} (f : A -n> B) : authC A -n> authC B :=
   CofeMor (auth_map f).

algebra/cmra.v

@@ -142,8 +142,9 @@ Class CMRADiscrete (A : cmraT) : Prop := {
 
 (** * Morphisms *)
 Class CMRAMonotone {A B : cmraT} (f : A → B) := {
-  includedN_preserving n x y : x ≼{n} y → f x ≼{n} f y;
-  validN_preserving n x : ✓{n} x → ✓{n} f x
+  cmra_monotone_ne n :> Proper (dist n ==> dist n) f;
+  validN_preserving n x : ✓{n} x → ✓{n} f x;
+  included_preserving x y : x ≼ y → f x ≼ f y
 }.
 
 (** * Local updates *)
@@ -430,26 +431,28 @@ End cmra.
 
 (** * Properties about monotone functions *)
 Instance cmra_monotone_id {A : cmraT} : CMRAMonotone (@id A).
-Proof. by split. Qed.
+Proof. repeat split; by try apply _. Qed.
 Instance cmra_monotone_compose {A B C : cmraT} (f : A → B) (g : B → C) :
   CMRAMonotone f → CMRAMonotone g → CMRAMonotone (g ∘ f).
 Proof.
   split.
-  - move=> n x y Hxy /=. by apply includedN_preserving, includedN_preserving.
+  - apply _. 
   - move=> n x Hx /=. by apply validN_preserving, validN_preserving.
+  - move=> x y Hxy /=. by apply included_preserving, included_preserving.
 Qed.
 
 Section cmra_monotone.
   Context {A B : cmraT} (f : A → B) `{!CMRAMonotone f}.
-  Lemma included_preserving x y : x ≼ y → f x ≼ f y.
+  Global Instance cmra_monotone_proper : Proper ((≡) ==> (≡)) f := ne_proper _.
+  Lemma includedN_preserving n x y : x ≼{n} y → f x ≼{n} f y.
   Proof.
-    rewrite !cmra_included_includedN; eauto using includedN_preserving.
+    intros [z ->].
+    apply cmra_included_includedN, included_preserving, cmra_included_l.
   Qed.
   Lemma valid_preserving x : ✓ x → ✓ f x.
   Proof. rewrite !cmra_valid_validN; eauto using validN_preserving. Qed.
 End cmra_monotone.
 
-
 (** * Transporting a CMRA equality *)
 Definition cmra_transport {A B : cmraT} (H : A = B) (x : A) : B :=
   eq_rect A id x _ H.
@@ -607,8 +610,8 @@ Arguments prodRA : clear implicits.
 Instance prod_map_cmra_monotone {A A' B B' : cmraT} (f : A → A') (g : B → B') :
   CMRAMonotone f → CMRAMonotone g → CMRAMonotone (prod_map f g).
 Proof.
-  split.
-  - intros n x y; rewrite !prod_includedN; intros [??]; simpl.
-    by split; apply includedN_preserving.
+  split; first apply _.
   - by intros n x [??]; split; simpl; apply validN_preserving.
+  - intros x y; rewrite !prod_included=> -[??] /=.
+    by split; apply included_preserving.
 Qed.

algebra/excl.v

@@ -27,6 +27,7 @@ Inductive excl_dist : Dist (excl A) :=
   | ExclUnit_dist n : ExclUnit ≡{n}≡ ExclUnit
   | ExclBot_dist n : ExclBot ≡{n}≡ ExclBot.
 Existing Instance excl_dist.
+
 Global Instance Excl_ne n : Proper (dist n ==> dist n) (@Excl A).
 Proof. by constructor. Qed.
 Global Instance Excl_proper : Proper ((≡) ==> (≡)) (@Excl A).
@@ -35,6 +36,7 @@ Global Instance Excl_inj : Inj (≡) (≡) (@Excl A).
 Proof. by inversion_clear 1. Qed.
 Global Instance Excl_dist_inj n : Inj (dist n) (dist n) (@Excl A).
 Proof. by inversion_clear 1. Qed.
+
 Program Definition excl_chain
     (c : chain (excl A)) (x : A) (H : maybe Excl (c 1) = Some x) : chain A :=
   {| chain_car n := match c n return _ with Excl y => y | _ => x end |}.
@@ -191,10 +193,10 @@ Proof. by intros f f' Hf; destruct 1; constructor; apply Hf. Qed.
 Instance excl_map_cmra_monotone {A B : cofeT} (f : A → B) :
   (∀ n, Proper (dist n ==> dist n) f) → CMRAMonotone (excl_map f).
 Proof.
-  split.
-  - intros n x y [z Hy]; exists (excl_map f z); rewrite Hy.
-    by destruct x, z; constructor.
+  split; try apply _.
   - by intros n [a| |].
+  - intros x y [z Hy]; exists (excl_map f z); apply equiv_dist=> n.
+    move: Hy=> /equiv_dist /(_ n) ->; by destruct x, z.
 Qed.
 Definition exclC_map {A B} (f : A -n> B) : exclC A -n> exclC B :=
   CofeMor (excl_map f).

algebra/fin_maps.v

@@ -231,8 +231,8 @@ Proof.
       [exists (x ⋅ y)|exists x]; eauto using cmra_included_l.
   - intros (y&Hi&?); rewrite map_includedN_spec=>j.
     destruct (decide (i = j)); simplify_map_eq.
-    + by rewrite Hi; apply Some_Some_includedN, cmra_included_includedN.
-    + apply None_includedN.
+    + rewrite Hi. by apply (includedN_preserving _), cmra_included_includedN.
+    + apply: cmra_empty_leastN.
 Qed.
 Lemma map_dom_op m1 m2 : dom (gset K) (m1 ⋅ m2) ≡ dom _ m1 ∪ dom _ m2.
 Proof.
@@ -338,10 +338,10 @@ Proof. by intros ? m m' Hm k; rewrite !lookup_fmap; apply option_fmap_ne. Qed.
 Instance map_fmap_cmra_monotone `{Countable K} {A B : cmraT} (f : A → B)
   `{!CMRAMonotone f} : CMRAMonotone (fmap f : gmap K A → gmap K B).
 Proof.
-  split.
-  - intros m1 m2 n; rewrite !map_includedN_spec; intros Hm i.
-    by rewrite !lookup_fmap; apply: includedN_preserving.
+  split; try apply _.
   - by intros n m ? i; rewrite lookup_fmap; apply validN_preserving.
+  - intros m1 m2; rewrite !map_included_spec=> Hm i.
+    by rewrite !lookup_fmap; apply: included_preserving.
 Qed.
 Definition mapC_map `{Countable K} {A B} (f: A -n> B) : mapC K A -n> mapC K B :=
   CofeMor (fmap f : mapC K A → mapC K B).

algebra/iprod.v

@@ -133,13 +133,6 @@ Section iprod_cmra.
     - intros Hh; exists (g ⩪ f)=> x; specialize (Hh x).
       by rewrite /op /iprod_op /minus /iprod_minus cmra_op_minus.
   Qed.
-  Lemma iprod_includedN_spec n (f g : iprod B) : f ≼{n} g ↔ ∀ x, f x ≼{n} g x.
-  Proof.
-    split.
-    - by intros [h Hh] x; exists (h x); rewrite /op /iprod_op (Hh x).
-    - intros Hh; exists (g ⩪ f)=> x; specialize (Hh x).
-      by rewrite /op /iprod_op /minus /iprod_minus cmra_op_minus'.
-  Qed.
 
   Definition iprod_cmra_mixin : CMRAMixin (iprod B).
   Proof.
@@ -283,10 +276,10 @@ Proof. by intros ? y1 y2 Hy x; rewrite /iprod_map (Hy x). Qed.
 Instance iprod_map_cmra_monotone {A} {B1 B2: A → cmraT} (f : ∀ x, B1 x → B2 x) :
   (∀ x, CMRAMonotone (f x)) → CMRAMonotone (iprod_map f).
 Proof.
-  split.
-  - intros n g1 g2; rewrite !iprod_includedN_spec=> Hf x.
-    rewrite /iprod_map; apply includedN_preserving, Hf.
+  split; first apply _.
   - intros n g Hg x; rewrite /iprod_map; apply validN_preserving, Hg.
+  - intros g1 g2; rewrite !iprod_included_spec=> Hf x.
+    rewrite /iprod_map; apply included_preserving, Hf.
 Qed.
 
 Definition iprodC_map {A} {B1 B2 : A → cofeT} (f : iprod (λ x, B1 x -n> B2 x)) :

algebra/option.v

@@ -72,6 +72,8 @@ Instance option_op : Op (option A) := union_with (λ x y, Some (x ⋅ y)).
 Instance option_minus : Minus (option A) :=
   difference_with (λ x y, Some (x ⩪ y)).
 
+Definition Some_op a b : Some (a ⋅ b) = Some a ⋅ Some b := eq_refl.
+
 Lemma option_included (mx my : option A) :
   mx ≼ my ↔ mx = None ∨ ∃ x y, mx = Some x ∧ my = Some y ∧ x ≼ y.
 Proof.
@@ -84,24 +86,6 @@ Proof.
   - intros [->|(x&y&->&->&z&Hz)]; try (by exists my; destruct my; constructor).
     by exists (Some z); constructor.
 Qed.
-Lemma option_includedN n (mx my : option A) :
-  mx ≼{n} my ↔ mx = None ∨ ∃ x y, mx = Some x ∧ my = Some y ∧ x ≼{n} y.
-Proof.
-  split.
-  - intros [mz Hmz].
-    destruct mx as [x|]; [right|by left].
-    destruct my as [y|]; [exists x, y|destruct mz; inversion_clear Hmz].
-    destruct mz as [z|]; inversion_clear Hmz; split_and?; auto;
-      cofe_subst; eauto using cmra_includedN_l.
-  - intros [->|(x&y&->&->&z&Hz)]; try (by exists my; destruct my; constructor).
-    by exists (Some z); constructor.
-Qed.
-
-Lemma None_includedN n (mx : option A) : None ≼{n} mx.
-Proof. rewrite option_includedN; auto. Qed.
-Lemma Some_Some_includedN n (x y : A) : x ≼{n} y → Some x ≼{n} Some y.
-Proof. rewrite option_includedN; eauto 10. Qed.
-Definition Some_op a b : Some (a ⋅ b) = Some a ⋅ Some b := eq_refl.
 
 Definition option_cmra_mixin  : CMRAMixin (option A).
 Proof.
@@ -140,6 +124,8 @@ Global Instance option_cmra_discrete : CMRADiscrete A → CMRADiscrete optionRA.
 Proof. split; [apply _|]. by intros [x|]; [apply (cmra_discrete_valid x)|]. Qed.
 
 (** Misc *)
+Global Instance Some_cmra_monotone : CMRAMonotone Some.
+Proof. split; [apply _|done|intros x y [z ->]; by exists (Some z)]. Qed.
 Lemma op_is_Some mx my : is_Some (mx ⋅ my) ↔ is_Some mx ∨ is_Some my.
 Proof.
   destruct mx, my; rewrite /op /option_op /= -!not_eq_None_Some; naive_solver.
@@ -192,10 +178,10 @@ Proof. by intros Hf; destruct 1; constructor; apply Hf. Qed.
 Instance option_fmap_cmra_monotone {A B : cmraT} (f: A → B) `{!CMRAMonotone f} :
   CMRAMonotone (fmap f : option A → option B).
 Proof.
-  split.
-  - intros n mx my; rewrite !option_includedN.
-    intros [->|(x&y&->&->&?)]; simpl; eauto 10 using @includedN_preserving.
-  - by intros n [x|] ?; rewrite /cmra_validN /=; try apply validN_preserving.
+  split; first apply _.
+  - intros n [x|] ?; rewrite /cmra_validN /=; by repeat apply validN_preserving.
+  - intros mx my; rewrite !option_included.
+    intros [->|(x&y&->&->&?)]; simpl; eauto 10 using @included_preserving.
 Qed.
 Definition optionC_map {A B} (f : A -n> B) : optionC A -n> optionC B :=
   CofeMor (fmap f : optionC A → optionC B).

program_logic/resources.v

@@ -205,10 +205,10 @@ Qed.
 Instance res_map_cmra_monotone {Λ Σ} {A B : cofeT} (f : A -n> B) :
   CMRAMonotone (@res_map Λ Σ _ _ f).
 Proof.
-  split.
-  - by intros n r1 r2; rewrite !res_includedN;
-      intros (?&?&?); split_and!; simpl; try apply includedN_preserving.
+  split; first apply _.
   - by intros n r (?&?&?); split_and!; simpl; try apply validN_preserving.
+  - by intros r1 r2; rewrite !res_included;
+      intros (?&?&?); split_and!; simpl; try apply included_preserving.
 Qed.
 Definition resC_map {Λ Σ A B} (f : A -n> B) : resC Λ Σ A -n> resC Λ Σ B :=
   CofeMor (res_map f : resRA Λ Σ A → resRA Λ Σ B).