Turn equality and metric on res into an inductive.

This way, they are non-delta unfoldable constants, which showed a positive
impact on the performance of setoid rewriting. We may want to do this for
other cmra/cofe structures too.

iris/model.v

@@ -25,8 +25,8 @@ Qed.
 
 (* Resources *)
 Record res (Σ : iParam) (A : cofeT) := Res {
-  wld : gmap positive (agree (later A));
-  pst : excl (leibnizC (istate Σ));
+  wld : mapRA positive (agreeRA (laterC A));
+  pst : exclRA (leibnizC (istate Σ));
   gst : icmra Σ (laterC A);
 }.
 Add Printing Constructor res.
@@ -43,46 +43,50 @@ Section res.
 Context (Σ : iParam) (A : cofeT).
 Implicit Types r : res Σ A.
 
-Instance res_equiv : Equiv (res Σ A) := λ r1 r2,
-  wld r1 ≡ wld r2 ∧ pst r1 ≡ pst r2 ∧ gst r1 ≡ gst r2.
-Instance res_dist : Dist (res Σ A) := λ n r1 r2,
-  wld r1 ={n}= wld r2 ∧ pst r1 ={n}= pst r2 ∧ gst r1 ={n}= gst r2.
+Inductive res_equiv' (r1 r2 : res Σ A) := Res_equiv :
+  wld r1 ≡ wld r2 → pst r1 ≡ pst r2 → gst r1 ≡ gst r2 → res_equiv' r1 r2.
+Instance res_equiv : Equiv (res Σ A) := res_equiv'.
+Inductive res_dist' (n : nat) (r1 r2 : res Σ A) := Res_dist :
+  wld r1 ={n}= wld r2 → pst r1 ={n}= pst r2 → gst r1 ={n}= gst r2 →
+  res_dist' n r1 r2.
+Instance res_dist : Dist (res Σ A) := res_dist'.
 Global Instance Res_ne n :
   Proper (dist n ==> dist n ==> dist n ==> dist n) (@Res Σ A).
 Proof. done. Qed.
 Global Instance Res_proper : Proper ((≡) ==> (≡) ==> (≡) ==> (≡)) (@Res Σ A).
 Proof. done. Qed.
 Global Instance wld_ne n : Proper (dist n ==> dist n) (@wld Σ A).
-Proof. by intros r1 r2 (?&?&?). Qed.
+Proof. by destruct 1. Qed.
 Global Instance wld_proper : Proper ((≡) ==> (≡)) (@wld Σ A).
-Proof. by intros r1 r2 (?&?&?). Qed.
+Proof. by destruct 1. Qed.
 Global Instance pst_ne n : Proper (dist n ==> dist n) (@pst Σ A).
-Proof. by intros r1 r2 (?&?&?). Qed.
+Proof. by destruct 1. Qed.
 Global Instance pst_ne' n : Proper (dist (S n) ==> (≡)) (@pst Σ A).
 Proof.
-  intros σ σ' (_&?&_); apply (timeless _), dist_le with (S n); auto with lia.
+  intros σ σ' [???]; apply (timeless _), dist_le with (S n); auto with lia.
 Qed.
 Global Instance pst_proper : Proper ((≡) ==> (≡)) (@pst Σ A).
-Proof. by intros r1 r2 (?&?&?). Qed.
+Proof. by destruct 1. Qed.
 Global Instance gst_ne n : Proper (dist n ==> dist n) (@gst Σ A).
-Proof. by intros r1 r2 (?&?&?). Qed.
+Proof. by destruct 1. Qed.
 Global Instance gst_proper : Proper ((≡) ==> (≡)) (@gst Σ A).
-Proof. by intros r1 r2 (?&?&?). Qed.
+Proof. by destruct 1. Qed.
 Instance res_compl : Compl (res Σ A) := λ c,
   Res (compl (chain_map wld c))
       (compl (chain_map pst c)) (compl (chain_map gst c)).
 Definition res_cofe_mixin : CofeMixin (res Σ A).
 Proof.
   split.
-  * intros w1 w2; unfold equiv, res_equiv, dist, res_dist.
-    rewrite !equiv_dist; naive_solver.
+  * intros w1 w2; split.
+    + by destruct 1; constructor; apply equiv_dist.
+    + by intros Hw; constructor; apply equiv_dist=>n; destruct (Hw n).
   * intros n; split.
     + done.
-    + by intros ?? (?&?&?); split_ands'.
-    + intros ??? (?&?&?) (?&?&?); split_ands'; etransitivity; eauto.
-  * by intros n ?? (?&?&?); split_ands'; apply dist_S.
+    + by destruct 1; constructor.
+    + do 2 destruct 1; constructor; etransitivity; eauto.
+  * by destruct 1; constructor; apply dist_S.
   * done.
-  * intros c n; split_ands'.
+  * intros c n; constructor.
     + apply (conv_compl (chain_map wld c) n).
     + apply (conv_compl (chain_map pst c) n).
     + apply (conv_compl (chain_map gst c) n).
@@ -90,15 +94,14 @@ Qed.
 Canonical Structure resC : cofeT := CofeT res_cofe_mixin.
 Global Instance res_timeless r :
   Timeless (wld r) → Timeless (gst r) → Timeless r.
-Proof. by intros ??? (?&?&?); split_ands'; try apply (timeless _). Qed.
+Proof. by destruct 3; constructor; try apply (timeless _). Qed.
 
 Instance res_op : Op (res Σ A) := λ r1 r2,
   Res (wld r1 ⋅ wld r2) (pst r1 ⋅ pst r2) (gst r1 ⋅ gst r2).
 Global Instance res_empty : Empty (res Σ A) := Res ∅ ∅ ∅.
 Instance res_unit : Unit (res Σ A) := λ r,
   Res (unit (wld r)) (unit (pst r)) (unit (gst r)).
-Instance res_valid : Valid (res Σ A) := λ r,
-  ✓ (wld r) ∧ ✓ (pst r) ∧ ✓ (gst r).
+Instance res_valid : Valid (res Σ A) := λ r, ✓ (wld r) ∧ ✓ (pst r) ∧ ✓ (gst r).
 Instance res_validN : ValidN (res Σ A) := λ n r,
   ✓{n} (wld r) ∧ ✓{n} (pst r) ∧ ✓{n} (gst r).
 Instance res_minus : Minus (res Σ A) := λ r1 r2,
@@ -120,25 +123,25 @@ Qed.
 Definition res_cmra_mixin : CMRAMixin (res Σ A).
 Proof.
   split.
-  * by intros n x [???] ? (?&?&?); split_ands'; simpl in *; cofe_subst.
-  * by intros n [???] ? (?&?&?); split_ands'; simpl in *; cofe_subst.
-  * by intros n [???] ? (?&?&?) (?&?&?); split_ands'; simpl in *; cofe_subst.
-  * by intros n [???] ? (?&?&?) [???] ? (?&?&?);
-      split_ands'; simpl in *; cofe_subst.
+  * by intros n x [???] ? [???]; constructor; simpl in *; cofe_subst.
+  * by intros n [???] ? [???]; constructor; simpl in *; cofe_subst.
+  * by intros n [???] ? [???] (?&?&?); split_ands'; simpl in *; cofe_subst.
+  * by intros n [???] ? [???] [???] ? [???];
+      constructor; simpl in *; cofe_subst.
   * done.
   * by intros n ? (?&?&?); split_ands'; apply cmra_valid_S.
   * intros r; unfold valid, res_valid, validN, res_validN.
     rewrite !cmra_valid_validN; naive_solver.
-  * intros ???; split_ands'; simpl; apply (associative _).
-  * intros ??; split_ands'; simpl; apply (commutative _).
-  * intros ?; split_ands'; simpl; apply ra_unit_l.
-  * intros ?; split_ands'; simpl; apply ra_unit_idempotent.
+  * intros ???; constructor; simpl; apply (associative _).
+  * intros ??; constructor; simpl; apply (commutative _).
+  * intros ?; constructor; simpl; apply ra_unit_l.
+  * intros ?; constructor; simpl; apply ra_unit_idempotent.
   * intros n r1 r2; rewrite !res_includedN.
     by intros (?&?&?); split_ands'; apply cmra_unit_preserving.
   * intros n r1 r2 (?&?&?);
       split_ands'; simpl in *; eapply cmra_valid_op_l; eauto.
   * intros n r1 r2; rewrite res_includedN; intros (?&?&?).
-    by split_ands'; apply cmra_op_minus.
+    by constructor; apply cmra_op_minus.
 Qed.
 Global Instance res_ra_empty : RAIdentity (res Σ A).
 Proof.
@@ -147,7 +150,7 @@ Qed.
 
 Definition res_cmra_extend_mixin : CMRAExtendMixin (res Σ A).
 Proof.
-  intros n r r1 r2 (?&?&?) (?&?&?); simpl in *.
+  intros n r r1 r2 (?&?&?) [???]; simpl in *.
   destruct (cmra_extend_op n (wld r) (wld r1) (wld r2)) as ([w w']&?&?&?),
     (cmra_extend_op n (pst r) (pst r1) (pst r2)) as ([σ σ']&?&?&?),
     (cmra_extend_op n (gst r) (gst r1) (gst r2)) as ([m m']&?&?&?); auto.
@@ -169,10 +172,10 @@ Definition res_map {Σ A B} (f : A -n> B) (r : res Σ A) : res Σ B :=
 Instance res_map_ne Σ (A B : cofeT) (f : A -n> B) :
   (∀ n, Proper (dist n ==> dist n) f) →
   ∀ n, Proper (dist n ==> dist n) (@res_map Σ _ _ f).
-Proof. by intros Hf n [] ? (?&?&?); split_ands'; simpl in *; cofe_subst. Qed.
+Proof. by intros Hf n [] ? [???]; constructor; simpl in *; cofe_subst. Qed.
 Lemma res_map_id {Σ A} (r : res Σ A) : res_map cid r ≡ r.
 Proof.
-  split_ands'; simpl; [|done|].
+  constructor; simpl; [|done|].
   * rewrite -{2}(map_fmap_id (wld r)); apply map_fmap_setoid_ext=> i y ? /=.
     rewrite -{2}(agree_map_id y); apply agree_map_ext=> y' /=.
     by rewrite later_map_id.
@@ -182,7 +185,7 @@ Qed.
 Lemma res_map_compose {Σ A B C} (f : A -n> B) (g : B -n> C) (r : res Σ A) :
   res_map (g ◎ f) r ≡ res_map g (res_map f r).
 Proof.
-  split_ands'; simpl; [|done|].
+  constructor; simpl; [|done|].
   * rewrite -map_fmap_compose; apply map_fmap_setoid_ext=> i y _ /=.
     rewrite -agree_map_compose; apply agree_map_ext=> y' /=.
     by rewrite later_map_compose.
@@ -201,7 +204,7 @@ Proof.
 Qed.
 Instance resRA_map_contractive {Σ A B} : Contractive (@resRA_map Σ A B).
 Proof.
-  intros n f g ? r; split_ands'; simpl; [|done|].
+  intros n f g ? r; constructor; simpl; [|done|].
   * by apply (mapRA_map_ne _ (agreeRA_map (laterC_map f))
       (agreeRA_map (laterC_map g))), agreeRA_map_ne, laterC_map_contractive.
   * by apply icmra_map_ne, laterC_map_contractive.
@@ -226,7 +229,7 @@ End iProp.
 
 (* Solution *)
 Definition iPreProp (Σ : iParam) : cofeT := iProp.result Σ.
-Notation res' Σ := (resRA Σ (iPreProp Σ)).
+Notation res' Σ := (res Σ (iPreProp Σ)).
 Notation icmra' Σ := (icmra Σ (laterC (iPreProp Σ))).
 Definition iProp (Σ : iParam) : cofeT := uPredC (resRA Σ (iPreProp Σ)).
 Definition iProp_unfold {Σ} : iProp Σ -n> iPreProp Σ := solution_fold _.