Put step-indexes first.

035f0b29 · Robbert Krebbers · 88679d3e · 035f0b29 · 035f0b29 · 035f0b29
Commit 035f0b29 authored Feb 18, 2016 by Robbert Krebbers
17 changed files
--- a/algebra/agree.v
+++ b/algebra/agree.v
@@ -16,7 +16,7 @@ Context {A : cofeT}.

 Instance agree_validN : ValidN (agree A) := λ n x,
  agree_is_valid x n ∧ ∀ n', n' ≤ n → x n' ≡{n'}≡ x n.
-Lemma agree_valid_le (x : agree A) n n' :
+Lemma agree_valid_le n n' (x : agree A) :
  agree_is_valid x n → n' ≤ n → agree_is_valid x n'.
 Proof. induction 2; eauto using agree_valid_S. Qed.
 Instance agree_equiv : Equiv (agree A) := λ x y,
@@ -43,14 +43,14 @@ Proof.
      * transitivity (agree_is_valid y n'). by apply Hxy. by apply Hyz.
      * transitivity (y n'). by apply Hxy. by apply Hyz, Hxy.
  - intros n x y Hxy; split; intros; apply Hxy; auto.
-  - intros c n; apply and_wlog_r; intros;
+  - intros n c; apply and_wlog_r; intros;
      symmetry; apply (chain_cauchy c); naive_solver.
 Qed.
 Canonical Structure agreeC := CofeT agree_cofe_mixin.

-Lemma agree_car_ne (x y : agree A) n : ✓{n} x → x ≡{n}≡ y → x n ≡{n}≡ y n.
+Lemma agree_car_ne n (x y : agree A) : ✓{n} x → x ≡{n}≡ y → x n ≡{n}≡ y n.
 Proof. by intros [??] Hxy; apply Hxy. Qed.
-Lemma agree_cauchy (x : agree A) n i : ✓{n} x → i ≤ n → x i ≡{i}≡ x n.
+Lemma agree_cauchy n (x : agree A) i : ✓{n} x → i ≤ n → x i ≡{i}≡ x n.
 Proof. by intros [? Hx]; apply Hx. Qed.

 Program Instance agree_op : Op (agree A) := λ x y,
@@ -87,7 +87,7 @@ Proof.
    repeat match goal with H : agree_is_valid _ _ |- _ => clear H end;
    by cofe_subst; rewrite !agree_idemp.
 Qed.
-Lemma agree_includedN (x y : agree A) n : x ≼{n} y ↔ y ≡{n}≡ x ⋅ y.
+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.
@@ -100,12 +100,12 @@ Proof.
    rewrite (Hx n'); last auto.
    symmetry; apply dist_le with n; try apply Hx; auto.
  - intros x; apply agree_idemp.
-  - by intros x y n [(?&?&?) ?].
-  - by intros x y n; rewrite agree_includedN.
+  - by intros n x y [(?&?&?) ?].
+  - by intros n x y; rewrite agree_includedN.
 Qed.
-Lemma agree_op_inv (x1 x2 : agree A) n : ✓{n} (x1 ⋅ x2) → x1 ≡{n}≡ x2.
+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 (x y : agree A) n : ✓{n} y → x ≼{n} y → x ≡{n}≡ y.
+Lemma agree_valid_includedN n (x y : agree A) : ✓{n} y → x ≼{n} y → x ≡{n}≡ y.
 Proof.
  move=> Hval [z Hy]; move: Hval; rewrite Hy.
  by move=> /agree_op_inv->; rewrite agree_idemp.
@@ -161,7 +161,7 @@ Section agree_map.
  Global Instance agree_map_monotone : CMRAMonotone (agree_map f).
  Proof.
    split; [|by intros n x [? Hx]; split; simpl; [|by intros n' ?; rewrite Hx]].
-    intros x y n; rewrite !agree_includedN; intros Hy; rewrite Hy.
+    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.

--- a/algebra/auth.v
+++ b/algebra/auth.v
@@ -46,8 +46,8 @@ Proof.
    + by intros ?? [??]; split; symmetry.
    + intros ??? [??] [??]; split; etransitivity; eauto.
  - by intros ? [??] [??] [??]; split; apply dist_S.
-  - intros c n; split. apply (conv_compl (chain_map authoritative c) n).
-    apply (conv_compl (chain_map own c) n).
+  - intros n c; split. apply (conv_compl n (chain_map authoritative c)).
+    apply (conv_compl n (chain_map own c)).
 Qed.
 Canonical Structure authC := CofeT auth_cofe_mixin.
 Global Instance auth_timeless (x : auth A) :
@@ -163,7 +163,7 @@ Lemma auth_update a a' b b' :
  (∀ n af, ✓{n} a → a ≡{n}≡ a' ⋅ af → b ≡{n}≡ b' ⋅ af ∧ ✓{n} b) →
  ● a ⋅ ◯ a' ~~> ● b ⋅ ◯ b'.
 Proof.
-  move=> Hab [[?| |] bf1] n // =>-[[bf2 Ha] ?]; do 2 red; simpl in *.
+  move=> Hab n [[?| |] bf1] // =>-[[bf2 Ha] ?]; do 2 red; simpl in *.
  destruct (Hab n (bf1 ⋅ bf2)) as [Ha' ?]; auto.
  { by rewrite Ha left_id assoc. }
  split; [by rewrite Ha' left_id assoc; apply cmra_includedN_l|done].

--- a/algebra/cmra.v
+++ b/algebra/cmra.v
@@ -147,11 +147,11 @@ Class LocalUpdate {A : cmraT} (Lv : A → Prop) (L : A → A) := {
 Arguments local_updateN {_ _} _ {_} _ _ _ _ _.

 (** * Frame preserving updates *)
-Definition cmra_updateP {A : cmraT} (x : A) (P : A → Prop) := ∀ z n,
+Definition cmra_updateP {A : cmraT} (x : A) (P : A → Prop) := ∀ n z,
  ✓{n} (x ⋅ z) → ∃ y, P y ∧ ✓{n} (y ⋅ z).
 Instance: Params (@cmra_updateP) 1.
 Infix "~~>:" := cmra_updateP (at level 70).
-Definition cmra_update {A : cmraT} (x y : A) := ∀ z n,
+Definition cmra_update {A : cmraT} (x y : A) := ∀ n z,
  ✓{n} (x ⋅ z) → ✓{n} (y ⋅ z).
 Infix "~~>" := cmra_update (at level 70).
 Instance: Params (@cmra_update) 1.
@@ -202,23 +202,23 @@ Qed.
 Global Instance cmra_update_proper :
  Proper ((≡) ==> (≡) ==> iff) (@cmra_update A).
 Proof.
-  intros x1 x2 Hx y1 y2 Hy; split=>? z n; [rewrite -Hx -Hy|rewrite Hx Hy]; auto.
+  intros x1 x2 Hx y1 y2 Hy; split=>? n z; [rewrite -Hx -Hy|rewrite Hx Hy]; auto.
 Qed.
 Global Instance cmra_updateP_proper :
  Proper ((≡) ==> pointwise_relation _ iff ==> iff) (@cmra_updateP A).
 Proof.
-  intros x1 x2 Hx P1 P2 HP; split=>Hup z n;
+  intros x1 x2 Hx P1 P2 HP; split=>Hup n z;
    [rewrite -Hx; setoid_rewrite <-HP|rewrite Hx; setoid_rewrite HP]; auto.
 Qed.

 (** ** Validity *)
 Lemma cmra_valid_validN x : ✓ x ↔ ∀ n, ✓{n} x.
 Proof. done. Qed.
-Lemma cmra_validN_le x n n' : ✓{n} x → n' ≤ n → ✓{n'} x.
+Lemma cmra_validN_le n n' x : ✓{n} x → n' ≤ n → ✓{n'} x.
 Proof. induction 2; eauto using cmra_validN_S. Qed.
 Lemma cmra_valid_op_l x y : ✓ (x ⋅ y) → ✓ x.
 Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_l. Qed.
-Lemma cmra_validN_op_r x y n : ✓{n} (x ⋅ y) → ✓{n} y.
+Lemma cmra_validN_op_r n x y : ✓{n} (x ⋅ y) → ✓{n} y.
 Proof. rewrite (comm _ x); apply cmra_validN_op_l. Qed.
 Lemma cmra_valid_op_r x y : ✓ (x ⋅ y) → ✓ y.
 Proof. rewrite !cmra_valid_validN; eauto using cmra_validN_op_r. Qed.
@@ -228,7 +228,7 @@ Lemma cmra_unit_r x : x ⋅ unit x ≡ x.
 Proof. by rewrite (comm _ x) cmra_unit_l. Qed.
 Lemma cmra_unit_unit x : unit x ⋅ unit x ≡ unit x.
 Proof. by rewrite -{2}(cmra_unit_idemp x) cmra_unit_r. Qed.
-Lemma cmra_unit_validN x n : ✓{n} x → ✓{n} unit x.
+Lemma cmra_unit_validN n x : ✓{n} x → ✓{n} unit x.
 Proof. rewrite -{1}(cmra_unit_l x); apply cmra_validN_op_l. Qed.
 Lemma cmra_unit_valid x : ✓ x → ✓ unit x.
 Proof. rewrite -{1}(cmra_unit_l x); apply cmra_valid_op_l. Qed.
@@ -237,7 +237,7 @@ Proof. rewrite -{1}(cmra_unit_l x); apply cmra_valid_op_l. Qed.
 Lemma cmra_included_includedN x y : x ≼ y ↔ ∀ n, x ≼{n} y.
 Proof.
  split; [by intros [z Hz] n; exists z; rewrite Hz|].
-  intros Hxy; exists (y ⩪ x); apply equiv_dist; intros n.
+  intros Hxy; exists (y ⩪ x); apply equiv_dist=> n.
  symmetry; apply cmra_op_minus, Hxy.
 Qed.
 Global Instance cmra_includedN_preorder n : PreOrder (@includedN A _ _ n).
@@ -252,14 +252,14 @@ Proof.
  split; red; intros until 0; rewrite !cmra_included_includedN; first done.
  intros; etransitivity; eauto.
 Qed.
-Lemma cmra_validN_includedN x y n : ✓{n} y → x ≼{n} y → ✓{n} x.
+Lemma cmra_validN_includedN n x y : ✓{n} y → x ≼{n} y → ✓{n} x.
 Proof. intros Hyv [z ?]; cofe_subst y; eauto using cmra_validN_op_l. Qed.
-Lemma cmra_validN_included x y n : ✓{n} y → x ≼ y → ✓{n} x.
+Lemma cmra_validN_included n x y : ✓{n} y → x ≼ y → ✓{n} x.
 Proof. rewrite cmra_included_includedN; eauto using cmra_validN_includedN. Qed.

-Lemma cmra_includedN_S x y n : x ≼{S n} y → x ≼{n} y.
+Lemma cmra_includedN_S n x y : x ≼{S n} y → x ≼{n} y.
 Proof. by intros [z Hz]; exists z; apply dist_S. Qed.
-Lemma cmra_includedN_le x y n n' : x ≼{n} y → n' ≤ n → x ≼{n'} y.
+Lemma cmra_includedN_le n n' x y : x ≼{n} y → n' ≤ n → x ≼{n'} y.
 Proof. induction 2; auto using cmra_includedN_S. Qed.

 Lemma cmra_includedN_l n x y : x ≼{n} x ⋅ y.
@@ -284,7 +284,7 @@ Proof. by intros; rewrite -!(comm _ z); apply cmra_preservingN_l. Qed.
 Lemma cmra_preserving_r x y z : x ≼ y → x ⋅ z ≼ y ⋅ z.
 Proof. by intros; rewrite -!(comm _ z); apply cmra_preserving_l. Qed.

-Lemma cmra_included_dist_l x1 x2 x1' n :
+Lemma cmra_included_dist_l n x1 x2 x1' :
  x1 ≼ x2 → x1' ≡{n}≡ x1 → ∃ x2', x1' ≼ x2' ∧ x2' ≡{n}≡ x2.
 Proof.
  intros [z Hx2] Hx1; exists (x1' ⋅ z); split; auto using cmra_included_l.
@@ -318,7 +318,7 @@ Qed.
 (** ** RAs with an empty element *)
 Section identity.
  Context `{Empty A, !CMRAIdentity A}.
-  Lemma cmra_empty_leastN  n x : ∅ ≼{n} x.
+  Lemma cmra_empty_leastN n x : ∅ ≼{n} x.
  Proof. by exists x; rewrite left_id. Qed.
  Lemma cmra_empty_least x : ∅ ≼ x.
  Proof. by exists x; rewrite left_id. Qed.
@@ -350,14 +350,14 @@ Lemma cmra_update_updateP x y : x ~~> y ↔ x ~~>: (y =).
 Proof.
  split.
  - by intros Hx z ?; exists y; split; [done|apply (Hx z)].
-  - by intros Hx z n ?; destruct (Hx z n) as (?&<-&?).
+  - by intros Hx n z ?; destruct (Hx n z) as (?&<-&?).
 Qed.
 Lemma cmra_updateP_id (P : A → Prop) x : P x → x ~~>: P.
-Proof. by intros ? z n ?; exists x. Qed.
+Proof. by intros ? n z ?; exists x. Qed.
 Lemma cmra_updateP_compose (P Q : A → Prop) x :
  x ~~>: P → (∀ y, P y → y ~~>: Q) → x ~~>: Q.
 Proof.
-  intros Hx Hy z n ?. destruct (Hx z n) as (y&?&?); auto. by apply (Hy y).
+  intros Hx Hy n z ?. destruct (Hx n z) as (y&?&?); auto. by apply (Hy y).
 Qed.
 Lemma cmra_updateP_compose_l (Q : A → Prop) x y : x ~~> y → y ~~>: Q → x ~~>: Q.
 Proof.
@@ -370,9 +370,9 @@ Proof. eauto using cmra_updateP_compose, cmra_updateP_id. Qed.
 Lemma cmra_updateP_op (P1 P2 Q : A → Prop) x1 x2 :
  x1 ~~>: P1 → x2 ~~>: P2 → (∀ y1 y2, P1 y1 → P2 y2 → Q (y1 ⋅ y2)) → x1 ⋅ x2 ~~>: Q.
 Proof.
-  intros Hx1 Hx2 Hy z n ?.
-  destruct (Hx1 (x2 ⋅ z) n) as (y1&?&?); first by rewrite assoc.
-  destruct (Hx2 (y1 ⋅ z) n) as (y2&?&?);
+  intros Hx1 Hx2 Hy n z ?.
+  destruct (Hx1 n (x2 ⋅ z)) as (y1&?&?); first by rewrite assoc.
+  destruct (Hx2 n (y1 ⋅ z)) as (y2&?&?);
    first by rewrite assoc (comm _ x2) -assoc.
  exists (y1 ⋅ y2); split; last rewrite (comm _ y1) -assoc; auto.
 Qed.
@@ -389,7 +389,7 @@ Proof. intro. auto. Qed.
 Section identity_updates.
  Context `{Empty A, !CMRAIdentity A}.
  Lemma cmra_update_empty x : x ~~> ∅.
-  Proof. intros z n; rewrite left_id; apply cmra_validN_op_r. Qed.
+  Proof. intros n z; rewrite left_id; apply cmra_validN_op_r. Qed.
  Lemma cmra_update_empty_alt y : ∅ ~~> y ↔ ∀ x, x ~~> y.
  Proof. split; [intros; transitivity ∅|]; auto using cmra_update_empty. Qed.
 End identity_updates.
@@ -472,7 +472,7 @@ Section discrete.
  Definition discrete_cmra_mixin : CMRAMixin A.
  Proof.
    by destruct ra; split; unfold Proper, respectful, includedN;
-      try setoid_rewrite <-(timeless_iff _ _ _ _).
+      try setoid_rewrite <-(timeless_iff _ _).
  Qed.
  Definition discrete_extend_mixin : CMRAExtendMixin A.
  Proof.
@@ -483,10 +483,10 @@ Section discrete.
    CMRAT (cofe_mixin A) discrete_cmra_mixin discrete_extend_mixin.
  Lemma discrete_updateP (x : discreteRA) (P : A → Prop) :
    (∀ z, ✓ (x ⋅ z) → ∃ y, P y ∧ ✓ (y ⋅ z)) → x ~~>: P.
-  Proof. intros Hvalid z n; apply Hvalid. Qed.
+  Proof. intros Hvalid n z; apply Hvalid. Qed.
  Lemma discrete_update (x y : discreteRA) :
    (∀ z, ✓ (x ⋅ z) → ✓ (y ⋅ z)) → x ~~> y.
-  Proof. intros Hvalid z n; apply Hvalid. Qed.
+  Proof. intros Hvalid n z; apply Hvalid. Qed.
  Lemma discrete_valid (x : discreteRA) : v x → validN_valid x.
  Proof. move=>Hx n. exact Hx. Qed.
 End discrete.
@@ -540,7 +540,7 @@ Section prod.
    - intros n x y; rewrite !prod_includedN.
      by intros [??]; split; apply cmra_unit_preservingN.
    - intros n x y [??]; split; simpl in *; eauto using cmra_validN_op_l.
-    - intros x y n; rewrite prod_includedN; intros [??].
+    - intros n x y; rewrite prod_includedN; intros [??].
      by split; apply cmra_op_minus.
  Qed.
  Definition prod_cmra_extend_mixin : CMRAExtendMixin (A * B).
@@ -561,12 +561,12 @@ Section prod.
    - by intros ? [??]; split; apply (timeless _).
  Qed.
  Lemma prod_update x y : x.1 ~~> y.1 → x.2 ~~> y.2 → x ~~> y.
-  Proof. intros ?? z n [??]; split; simpl in *; auto. Qed.
+  Proof. intros ?? n z [??]; split; simpl in *; auto. Qed.
  Lemma prod_updateP P1 P2 (Q : A * B → Prop)  x :
    x.1 ~~>: P1 → x.2 ~~>: P2 → (∀ a b, P1 a → P2 b → Q (a,b)) → x ~~>: Q.
  Proof.
-    intros Hx1 Hx2 HP z n [??]; simpl in *.
-    destruct (Hx1 (z.1) n) as (a&?&?), (Hx2 (z.2) n) as (b&?&?); auto.
+    intros Hx1 Hx2 HP n z [??]; simpl in *.
+    destruct (Hx1 n (z.1)) as (a&?&?), (Hx2 n (z.2)) as (b&?&?); auto.
    exists (a,b); repeat split; auto.
  Qed.
  Lemma prod_updateP' P1 P2 x :

--- a/algebra/cofe.v
+++ b/algebra/cofe.v
@@ -54,7 +54,7 @@ Record CofeMixin A `{Equiv A, Compl A} := {
  mixin_equiv_dist x y : x ≡ y ↔ ∀ n, x ≡{n}≡ y;
  mixin_dist_equivalence n : Equivalence (dist n);
  mixin_dist_S n x y : x ≡{S n}≡ y → x ≡{n}≡ y;
-  mixin_conv_compl (c : chain A) n : compl c ≡{n}≡ c (S n)
+  mixin_conv_compl n c : compl c ≡{n}≡ c (S n)
 }.
 Class Contractive `{Dist A, Dist B} (f : A -> B) :=
  contractive n x y : (∀ i, i < n → x ≡{i}≡ y) → f x ≡{n}≡ f y.
@@ -86,7 +86,7 @@ Section cofe_mixin.
  Proof. apply (mixin_dist_equivalence _ (cofe_mixin A)). Qed.
  Lemma dist_S n x y : x ≡{S n}≡ y → x ≡{n}≡ y.
  Proof. apply (mixin_dist_S _ (cofe_mixin A)). Qed.
-  Lemma conv_compl (c : chain A) n : compl c ≡{n}≡ c (S n).
+  Lemma conv_compl n (c : chain A) : compl c ≡{n}≡ c (S n).
  Proof. apply (mixin_conv_compl _ (cofe_mixin A)). Qed.
 End cofe_mixin.

@@ -113,7 +113,7 @@ Section cofe.
  Qed.
  Global Instance dist_proper_2 n x : Proper ((≡) ==> iff) (dist n x).
  Proof. by apply dist_proper. Qed.
-  Lemma dist_le (x y : A) n n' : x ≡{n}≡ y → n' ≤ n → x ≡{n'}≡ y.
+  Lemma dist_le n n' x y : x ≡{n}≡ y → n' ≤ n → x ≡{n'}≡ y.
  Proof. induction 2; eauto using dist_S. Qed.
  Instance ne_proper {B : cofeT} (f : A → B)
    `{!∀ n, Proper (dist n ==> dist n) f} : Proper ((≡) ==> (≡)) f | 100.
@@ -147,7 +147,7 @@ Next Obligation. by intros ? A ? B f Hf c n i ?; apply Hf, chain_cauchy. Qed.
 (** Timeless elements *)
 Class Timeless {A : cofeT} (x : A) := timeless y : x ≡{0}≡ y → x ≡ y.
 Arguments timeless {_} _ {_} _ _.
-Lemma timeless_iff {A : cofeT} (x y : A) n : Timeless x → x ≡ y ↔ x ≡{n}≡ y.
+Lemma timeless_iff {A : cofeT} n (x : A) `{!Timeless x} y : x ≡ y ↔ x ≡{n}≡ y.
 Proof.
  split; intros; [by apply equiv_dist|].
  apply (timeless _), dist_le with n; auto with lia.
@@ -168,14 +168,14 @@ Section fixpoint.
  Context {A : cofeT} `{Inhabited A} (f : A → A) `{!Contractive f}.
  Lemma fixpoint_unfold : fixpoint f ≡ f (fixpoint f).
  Proof.
-    apply equiv_dist=>n; rewrite /fixpoint (conv_compl (fixpoint_chain f) n) //.
+    apply equiv_dist=>n; rewrite /fixpoint (conv_compl n (fixpoint_chain f)) //.
    induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S.
  Qed.
  Lemma fixpoint_ne (g : A → A) `{!Contractive g} n :
    (∀ z, f z ≡{n}≡ g z) → fixpoint f ≡{n}≡ fixpoint g.
  Proof.
    intros Hfg. rewrite /fixpoint
-      (conv_compl (fixpoint_chain f) n) (conv_compl (fixpoint_chain g) n) /=.
+      (conv_compl n (fixpoint_chain f)) (conv_compl n (fixpoint_chain g)) /=.
    induction n as [|n IH]; simpl in *; [by rewrite !Hfg|].
    rewrite Hfg; apply contractive_S, IH; auto using dist_S.
  Qed.
@@ -206,21 +206,21 @@ Section cofe_mor.
  Program Instance cofe_mor_compl : Compl (cofeMor A B) := λ c,
    {| cofe_mor_car x := compl (fun_chain c x) |}.
  Next Obligation.
-    intros c n x y Hx. by rewrite (conv_compl (fun_chain c x) n)
-      (conv_compl (fun_chain c y) n) /= Hx.
+    intros c n x y Hx. by rewrite (conv_compl n (fun_chain c x))
+      (conv_compl n (fun_chain c y)) /= Hx.
  Qed.
  Definition cofe_mor_cofe_mixin : CofeMixin (cofeMor A B).
  Proof.
    split.
    - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
-      intros Hfg k; apply equiv_dist; intros n; apply Hfg.
+      intros Hfg k; apply equiv_dist=> n; apply Hfg.
    - intros n; split.
      + by intros f x.
      + by intros f g ? x.
      + by intros f g h ?? x; transitivity (g x).
    - by intros n f g ? x; apply dist_S.
-    - intros c n x; simpl.
-      by rewrite (conv_compl (fun_chain c x) n) /=.
+    - intros n c x; simpl.
+      by rewrite (conv_compl n (fun_chain c x)) /=.
  Qed.
  Canonical Structure cofe_mor : cofeT := CofeT cofe_mor_cofe_mixin.

@@ -278,8 +278,8 @@ Section product.
      rewrite !equiv_dist; naive_solver.
    - apply _.
    - by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S.
-    - intros c n; split. apply (conv_compl (chain_map fst c) n).
-      apply (conv_compl (chain_map snd c) n).
+    - intros n c; split. apply (conv_compl n (chain_map fst c)).
+      apply (conv_compl n (chain_map snd c)).
  Qed.
  Canonical Structure prodC : cofeT := CofeT prod_cofe_mixin.
  Global Instance pair_timeless (x : A) (y : B) :
@@ -311,7 +311,7 @@ Section discrete_cofe.
    - intros x y; split; [done|intros Hn; apply (Hn 0)].
    - done.
    - done.
-    - intros c n. rewrite /compl /discrete_compl /=.
+    - intros n c. rewrite /compl /discrete_compl /=.
      symmetry; apply (chain_cauchy c 0 (S n)); omega.
  Qed.
  Definition discreteC : cofeT := CofeT discrete_cofe_mixin.
@@ -354,7 +354,7 @@ Section later.
      + by intros [x] [y].
      + by intros [x] [y] [z] ??; transitivity y.
    - intros [|n] [x] [y] ?; [done|]; unfold dist, later_dist; by apply dist_S.
-    - intros c [|n]; [done|by apply (conv_compl (later_chain c) n)].
+    - intros [|n] c; [done|by apply (conv_compl n (later_chain c))].
  Qed.
  Canonical Structure laterC : cofeT := CofeT later_cofe_mixin.
  Global Instance Next_contractive : Contractive (@Next A).

--- a/algebra/cofe_solver.v
+++ b/algebra/cofe_solver.v
@@ -60,8 +60,8 @@ Program Instance tower_compl : Compl tower := λ c,
  {| tower_car n := compl (tower_chain c n) |}.
 Next Obligation.
  intros c k; apply equiv_dist=> n.
-  by rewrite (conv_compl (tower_chain c k) n)
-    (conv_compl (tower_chain c (S k)) n) /= (g_tower (c (S n)) k).
+  by rewrite (conv_compl n (tower_chain c k))
+    (conv_compl n (tower_chain c (S k))) /= (g_tower (c (S n)) k).
 Qed.
 Definition tower_cofe_mixin : CofeMixin tower.
 Proof.
@@ -74,7 +74,7 @@ Proof.
    + by intros X Y Z ?? n; transitivity (Y n).
  - intros k X Y HXY n; apply dist_S.
    by rewrite -(g_tower X) (HXY (S n)) g_tower.
-  - intros c n k; rewrite /= (conv_compl (tower_chain c k) n).
+  - intros n c k; rewrite /= (conv_compl n (tower_chain c k)).
    apply (chain_cauchy c); lia.
 Qed.
 Definition T : cofeT := CofeT tower_cofe_mixin.
@@ -189,8 +189,8 @@ Qed.
 Definition unfold (X : T) : F T T := compl (unfold_chain X).
 Instance unfold_ne : Proper (dist n ==> dist n) unfold.
 Proof.
-  intros n X Y HXY. by rewrite /unfold (conv_compl (unfold_chain X) n)
-    (conv_compl (unfold_chain Y) n) /= (HXY (S (S n))).
+  intros n X Y HXY. by rewrite /unfold (conv_compl n (unfold_chain X))
+    (conv_compl n (unfold_chain Y)) /= (HXY (S (S n))).
 Qed.

 Program Definition fold (X : F T T) : T :=
@@ -210,7 +210,7 @@ Proof.
    rewrite equiv_dist; intros n k; unfold unfold, fold; simpl.
    rewrite -g_tower -(gg_tower _ n); apply (_ : Proper (_ ==> _) (g _)).
    transitivity (map (ff n, gg n) (X (S (n + k)))).
-    { rewrite /unfold (conv_compl (unfold_chain X) n).
+    { rewrite /unfold (conv_compl n (unfold_chain X)).
      rewrite -(chain_cauchy (unfold_chain X) n (S (n + k))) /=; last lia.
      rewrite -(dist_le _ _ _ _ (f_tower (n + k) _)); last lia.
      rewrite f_S -!map_comp; apply (contractive_ne map); split=> Y.
@@ -229,7 +229,7 @@ Proof.
    rewrite (map_ff_gg _ _ _ H).
    apply (_ : Proper (_ ==> _) (gg _)); by destruct H.
  - intros X; rewrite equiv_dist=> n /=.
-    rewrite /unfold /= (conv_compl (unfold_chain (fold X)) n) /=.
+    rewrite /unfold /= (conv_compl n (unfold_chain (fold X))) /=.
    rewrite g_S -!map_comp -{2}(map_id _ _ X).
    apply (contractive_ne map); split => Y /=.
    + apply dist_le with n; last omega.

--- a/algebra/excl.v
+++ b/algebra/excl.v
@@ -27,7 +27,7 @@ Inductive excl_dist `{Dist A} : Dist (excl A) :=
  | ExclUnit_dist n : ExclUnit ≡{n}≡ ExclUnit
  | ExclBot_dist n : ExclBot ≡{n}≡ ExclBot.
 Existing Instance excl_dist.
-Global Instance Excl_ne : Proper (dist n ==> dist n) (@Excl A).
+Global Instance Excl_ne n : Proper (dist n ==> dist n) (@Excl A).
 Proof. by constructor. Qed.
 Global Instance Excl_proper : Proper ((≡) ==> (≡)) (@Excl A).
 Proof. by constructor. Qed.
@@ -58,13 +58,13 @@ Proof.
    + by destruct 1; constructor.
    + destruct 1; inversion_clear 1; constructor; etransitivity; eauto.
  - by inversion_clear 1; constructor; apply dist_S.
-  - intros c n; unfold compl, excl_compl.
+  - intros n c; unfold compl, excl_compl.
    destruct (Some_dec (maybe Excl (c 1))) as [[x Hx]|].
    { assert (c 1 = Excl x) by (by destruct (c 1); simplify_eq/=).
      assert (∃ y, c (S n) = Excl y) as [y Hy].
      { feed inversion (chain_cauchy c 0 (S n)); eauto with lia congruence. }
      rewrite Hy; constructor.
-      by rewrite (conv_compl (excl_chain c x Hx) n) /= Hy. }
+      by rewrite (conv_compl n (excl_chain c x Hx)) /= Hy. }
    feed inversion (chain_cauchy c 0 (S n)); first lia;
       constructor; destruct (c 1); simplify_eq/=.
 Qed.
@@ -161,9 +161,9 @@ Proof. split. by intros n y1 y2 Hy. by intros n [a| |] [b'| |]. Qed.

 (** Updates *)
 Lemma excl_update (x : A) y : ✓ y → Excl x ~~> y.
-Proof. by destruct y; intros ? [?| |]. Qed.
+Proof. destruct y; by intros ?? [?| |]. Qed.
 Lemma excl_updateP (P : excl A → Prop) x y : ✓ y → P y → Excl x ~~>: P.
-Proof. intros ?? z n ?; exists y. by destruct y, z as [?| |]. Qed.
+Proof. intros ?? n z ?; exists y. by destruct y, z as [?| |]. Qed.
 End excl.

 Arguments exclC : clear implicits.

--- a/algebra/fin_maps.v
+++ b/algebra/fin_maps.v
@@ -24,7 +24,7 @@ Proof.
    + by intros m1 m2 ? k.
    + by intros m1 m2 m3 ?? k; transitivity (m2 !! k).
  - by intros n m1 m2 ? k; apply dist_S.
-  - intros c n k; rewrite /compl /map_compl lookup_imap.
+  - intros n c k; rewrite /compl /map_compl lookup_imap.
    feed inversion (λ H, chain_cauchy c 0 (S n) H k); simpl; auto with lia.
    by rewrite conv_compl /=; apply reflexive_eq.
 Qed.
@@ -61,7 +61,7 @@ Proof.
    [by constructor|by apply lookup_ne].
 Qed.

-Global Instance map_timeless `{∀ a : A, Timeless a} (m : gmap K A) : Timeless m.
+Global Instance map_timeless `{∀ a : A, Timeless a} m : Timeless m.
 Proof. by intros m' ? i; apply (timeless _). Qed.

 Instance map_empty_timeless : Timeless (∅ : gmap K A).
@@ -140,7 +140,7 @@ Proof.
    by rewrite !lookup_unit; apply cmra_unit_preservingN.
  - intros n m1 m2 Hm i; apply cmra_validN_op_l with (m2 !! i).
    by rewrite -lookup_op.
-  - intros x y n; rewrite map_includedN_spec=> ? i.
+  - intros n x y; rewrite map_includedN_spec=> ? i.
    by rewrite lookup_op lookup_minus cmra_op_minus.
 Qed.
 Definition map_cmra_extend_mixin : CMRAExtendMixin (gmap K A).
@@ -248,8 +248,8 @@ Qed.
 Lemma map_insert_updateP (P : A → Prop) (Q : gmap K A → Prop) m i x :
  x ~~>: P → (∀ y, P y → Q (<[i:=y]>m)) → <[i:=x]>m ~~>: Q.
 Proof.
-  intros Hx%option_updateP' HP mf n Hm.
-  destruct (Hx (mf !! i) n) as ([y|]&?&?); try done.
+  intros Hx%option_updateP' HP n mf Hm.
+  destruct (Hx n (mf !! i)) as ([y|]&?&?); try done.
  { by generalize (Hm i); rewrite lookup_op; simplify_map_eq. }
  exists (<[i:=y]> m); split; first by auto.
  intros j; move: (Hm j)=>{Hm}; rewrite !lookup_op=>Hm.
@@ -276,8 +276,8 @@ Lemma map_singleton_updateP_empty `{Empty A, !CMRAIdentity A}
    (P : A → Prop) (Q : gmap K A → Prop) i :
  ∅ ~~>: P → (∀ y, P y → Q {[ i := y ]}) → ∅ ~~>: Q.
 Proof.
-  intros Hx HQ gf n Hg.
-  destruct (Hx (from_option ∅ (gf !! i)) n) as (y&?&Hy).
+  intros Hx HQ n gf Hg.
+  destruct (Hx n (from_option ∅ (gf !! i))) as (y&?&Hy).
  { move:(Hg i). rewrite !left_id.
    case _: (gf !! i); simpl; auto using cmra_empty_valid. }
  exists {[ i := y ]}; split; first by auto.
@@ -296,7 +296,7 @@ Context `{Fresh K (gset K), !FreshSpec K (gset K)}.
 Lemma map_updateP_alloc_strong (Q : gmap K A → Prop) (I : gset K) m x :
  ✓ x → (∀ i, m !! i = None → i ∉ I → Q (<[i:=x]>m)) → m ~~>: Q.
 Proof.
-  intros ? HQ mf n Hm. set (i := fresh (I ∪ dom (gset K) (m ⋅ mf))).
+  intros ? HQ n mf Hm. set (i := fresh (I ∪ dom (gset K) (m ⋅ mf))).
  assert (i ∉ I ∧ i ∉ dom (gset K) m ∧ i ∉ dom (gset K) mf) as [?[??]].
  { rewrite -not_elem_of_union -map_dom_op -not_elem_of_union; apply is_fresh. }
  exists (<[i:=x]>m); split.

--- a/algebra/iprod.v
+++ b/algebra/iprod.v
@@ -37,8 +37,8 @@ Section iprod_cofe.
      + by intros f g ? x.
      + by intros f g h ?? x; transitivity (g x).
    - intros n f g Hfg x; apply dist_S, Hfg.
-    - intros c n x.
-      rewrite /compl /iprod_compl (conv_compl (iprod_chain c x) n).
+    - intros n c x.
+      rewrite /compl /iprod_compl (conv_compl n (iprod_chain c x)).
      apply (chain_cauchy c); lia.
  Qed.
  Canonical Structure iprodC : cofeT := CofeT iprod_cofe_mixin.
@@ -55,7 +55,7 @@ Section iprod_cofe.
  (** Properties of iprod_insert. *)
  Context `{∀ x x' : A, Decision (x = x')}.

-  Global Instance iprod_insert_ne x n :
+  Global Instance iprod_insert_ne n x :
    Proper (dist n ==> dist n ==> dist n) (iprod_insert x).
  Proof.
    intros y1 y2 ? f1 f2 ? x'; rewrite /iprod_insert.
@@ -94,7 +94,7 @@ Section iprod_cofe.
  (** Properties of iprod_singletom. *)
  Context `{∀ x : A, Empty (B x)}.

-  Global Instance iprod_singleton_ne x n :
+  Global Instance iprod_singleton_ne n x :
    Proper (dist n ==> dist n) (iprod_singleton x).
  Proof. by intros y1 y2 Hy; rewrite /iprod_singleton Hy. Qed.
  Global Instance iprod_singleton_proper x :
@@ -182,7 +182,7 @@ Section iprod_cmra.
    y1 ~~>: P → (∀ y2, P y2 → Q (iprod_insert x y2 g)) →
    iprod_insert x y1 g ~~>: Q.
  Proof.
-    intros Hy1 HP gf n Hg. destruct (Hy1 (gf x) n) as (y2&?&?).
+    intros Hy1 HP n gf Hg. destruct (Hy1 n (gf x)) as (y2&?&?).
    { move: (Hg x). by rewrite iprod_lookup_op iprod_lookup_insert. }
    exists (iprod_insert x y2 g); split; [auto|].
    intros x'; destruct (decide (x' = x)) as [->|];
@@ -242,7 +242,7 @@ Section iprod_cmra.
  Lemma iprod_singleton_updateP_empty x (P : B x → Prop) (Q : iprod B → Prop) :
    ∅ ~~>: P → (∀ y2, P y2 → Q (iprod_singleton x y2)) → ∅ ~~>: Q.
  Proof.
-    intros Hx HQ gf n Hg. destruct (Hx (gf x) n) as (y2&?&?); first apply Hg.
+    intros Hx HQ n gf Hg. destruct (Hx n (gf x)) as (y2&?&?); first apply Hg.
    exists (iprod_singleton x y2); split; [by apply HQ|].
    intros x'; destruct (decide (x' = x)) as [->|].
    - by rewrite iprod_lookup_op iprod_lookup_singleton.

--- a/algebra/option.v
+++ b/algebra/option.v
@@ -31,12 +31,12 @@ Proof.
    + by destruct 1; constructor.
    + destruct 1; inversion_clear 1; constructor; etransitivity; eauto.
  - by inversion_clear 1; constructor; apply dist_S.