Use scheme - then + then * for bullets.

algebra/agree.v

@@ -33,17 +33,17 @@ Qed.
 Definition agree_cofe_mixin : CofeMixin (agree A).
 Proof.
   split.
-  * intros x y; split.
+  - intros x y; split.
     + by intros Hxy n; split; intros; apply Hxy.
     + by intros Hxy; split; intros; apply Hxy with n.
-  * split.
+  - split.
     + by split.
     + by intros x y Hxy; split; intros; symmetry; apply Hxy; auto; apply Hxy.
     + intros x y z Hxy Hyz; split; intros n'; intros.
-      - 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;
+      * 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;
       symmetry; apply (chain_cauchy c); naive_solver.
 Qed.
 Canonical Structure agreeC := CofeT agree_cofe_mixin.
@@ -74,8 +74,8 @@ Proof.
   intros n x y1 y2 [Hy' Hy]; split; [|done].
   split; intros (?&?&Hxy); repeat (intro || split);
     try apply Hy'; eauto using agree_valid_le.
-  * etransitivity; [apply Hxy|apply Hy]; eauto using agree_valid_le.
-  * etransitivity; [apply Hxy|symmetry; apply Hy, Hy'];
+  - etransitivity; [apply Hxy|apply Hy]; eauto using agree_valid_le.
+  - etransitivity; [apply Hxy|symmetry; apply Hy, Hy'];
       eauto using agree_valid_le.
 Qed.
 Instance: Proper (dist n ==> dist n ==> dist n) (@op (agree A) _).
@@ -95,13 +95,13 @@ Qed.
 Definition agree_cmra_mixin : CMRAMixin (agree A).
 Proof.
   split; try (apply _ || done).
-  * by intros n x1 x2 Hx y1 y2 Hy.
-  * intros n x [? Hx]; split; [by apply agree_valid_S|intros n' ?].
+  - by intros n x1 x2 Hx y1 y2 Hy.
+  - intros n x [? Hx]; split; [by apply agree_valid_S|intros n' ?].
     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.
+  - intros x; apply agree_idemp.
+  - by intros x y n [(?&?&?) ?].
+  - by intros x y n; rewrite agree_includedN.
 Qed.
 Lemma agree_op_inv (x1 x2 : agree A) n : ✓{n} (x1 ⋅ x2) → x1 ≡{n}≡ x2.
 Proof. intros Hxy; apply Hxy. Qed.
@@ -113,8 +113,8 @@ Qed.
 Definition agree_cmra_extend_mixin : CMRAExtendMixin (agree A).
 Proof.
   intros n x y1 y2 Hval Hx; exists (x,x); simpl; split.
-  * by rewrite agree_idemp.
-  * by move: Hval; rewrite Hx; move=> /agree_op_inv->; rewrite agree_idemp.
+  - by rewrite agree_idemp.
+  - by move: Hval; rewrite Hx; move=> /agree_op_inv->; rewrite agree_idemp.
 Qed.
 Canonical Structure agreeRA : cmraT :=
   CMRAT agree_cofe_mixin agree_cmra_mixin agree_cmra_extend_mixin.

algebra/auth.v

@@ -39,14 +39,14 @@ Instance auth_compl : Compl (auth A) := λ c,
 Definition auth_cofe_mixin : CofeMixin (auth A).
 Proof.
   split.
-  * intros x y; unfold dist, auth_dist, equiv, auth_equiv.
+  - intros x y; unfold dist, auth_dist, equiv, auth_equiv.
     rewrite !equiv_dist; naive_solver.
-  * intros n; split.
+  - intros n; split.
     + by intros ?; split.
     + 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).
+  - 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).
 Qed.
 Canonical Structure authC := CofeT auth_cofe_mixin.
@@ -99,24 +99,24 @@ Proof. destruct x as [[]]; naive_solver eauto using cmra_validN_includedN. Qed.
 Definition auth_cmra_mixin : CMRAMixin (auth A).
 Proof.
   split.
-  * by intros n x y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'.
-  * by intros n y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'.
-  * intros n [x a] [y b] [Hx Ha]; simpl in *;
+  - by intros n x y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'.
+  - by intros n y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'.
+  - intros n [x a] [y b] [Hx Ha]; simpl in *;
       destruct Hx; intros ?; cofe_subst; auto.
-  * by intros n x1 x2 [Hx Hx'] y1 y2 [Hy Hy'];
+  - by intros n x1 x2 [Hx Hx'] y1 y2 [Hy Hy'];
       split; simpl; rewrite ?Hy ?Hy' ?Hx ?Hx'.
-  * intros n [[] ?] ?; naive_solver eauto using cmra_includedN_S, cmra_validN_S.
-  * by split; simpl; rewrite assoc.
-  * by split; simpl; rewrite comm.
-  * by split; simpl; rewrite ?cmra_unit_l.
-  * by split; simpl; rewrite ?cmra_unit_idemp.
-  * intros n ??; rewrite! auth_includedN; intros [??].
+  - intros n [[] ?] ?; naive_solver eauto using cmra_includedN_S, cmra_validN_S.
+  - by split; simpl; rewrite assoc.
+  - by split; simpl; rewrite comm.
+  - by split; simpl; rewrite ?cmra_unit_l.
+  - by split; simpl; rewrite ?cmra_unit_idemp.
+  - intros n ??; rewrite! auth_includedN; intros [??].
     by split; simpl; apply cmra_unit_preservingN.
-  * assert (∀ n (a b1 b2 : A), b1 ⋅ b2 ≼{n} a → b1 ≼{n} a).
+  - assert (∀ n (a b1 b2 : A), b1 ⋅ b2 ≼{n} a → b1 ≼{n} a).
     { intros n a b1 b2 <-; apply cmra_includedN_l. }
    intros n [[a1| |] b1] [[a2| |] b2];
      naive_solver eauto using cmra_validN_op_l, cmra_validN_includedN.
-  * by intros n ??; rewrite auth_includedN;
+  - by intros n ??; rewrite auth_includedN;
       intros [??]; split; simpl; apply cmra_op_minus.
 Qed.
 Definition auth_cmra_extend_mixin : CMRAExtendMixin (auth A).
@@ -150,9 +150,9 @@ Context `{Empty A, !CMRAIdentity A}.
 Global Instance auth_cmra_identity : CMRAIdentity authRA.
 Proof.
   split; simpl.
-  * by apply (@cmra_empty_valid A _).
-  * by intros x; constructor; rewrite /= left_id.
-  * apply _.
+  - by apply (@cmra_empty_valid A _).
+  - by intros x; constructor; rewrite /= left_id.
+  - apply _.
 Qed.
 Lemma auth_frag_op a b : ◯ (a ⋅ b) ≡ ◯ a ⋅ ◯ b.
 Proof. done. Qed.
@@ -221,9 +221,9 @@ Instance auth_map_cmra_monotone {A B : cmraT} (f : A → B) :
   CMRAMonotone f → CMRAMonotone (auth_map f).
 Proof.
   split.
-  * by intros n [x a] [y b]; rewrite !auth_includedN /=;
+  - by intros n [x a] [y b]; rewrite !auth_includedN /=;
       intros [??]; split; simpl; apply: includedN_preserving.
-  * intros n [[a| |] b]; rewrite /= /cmra_validN;
+  - intros n [[a| |] b]; rewrite /= /cmra_validN;
       naive_solver eauto using @includedN_preserving, @validN_preserving.
 Qed.
 Definition authC_map {A B} (f : A -n> B) : authC A -n> authC B :=

algebra/cmra.v

@@ -243,8 +243,8 @@ Qed.
 Global Instance cmra_includedN_preorder n : PreOrder (@includedN A _ _ n).
 Proof.
   split.
-  * by intros x; exists (unit x); rewrite cmra_unit_r.
-  * intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2).
+  - by intros x; exists (unit x); rewrite cmra_unit_r.
+  - intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2).
     by rewrite assoc -Hy -Hz.
 Qed.
 Global Instance cmra_included_preorder: PreOrder (@included A _ _).
@@ -349,8 +349,8 @@ Proof. split. by intros x y. intros x y y' ?? z ?; naive_solver. Qed.
 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 z ?; exists y; split; [done|apply (Hx z)].
+  - by intros Hx z n ?; destruct (Hx z n) as (?&<-&?).
 Qed.
 Lemma cmra_updateP_id (P : A → Prop) x : P x → x ~~>: P.
 Proof. by intros ? z n ?; exists x. Qed.
@@ -402,8 +402,8 @@ 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.
-  * move=> n x Hx /=. by apply validN_preserving, validN_preserving.
+  - move=> n x y Hxy /=. by apply includedN_preserving, includedN_preserving.
+  - move=> n x Hx /=. by apply validN_preserving, validN_preserving.
 Qed.
 
 Section cmra_monotone.
@@ -527,20 +527,20 @@ Section prod.
   Definition prod_cmra_mixin : CMRAMixin (A * B).
   Proof.
     split; try apply _.
-    * by intros n x y1 y2 [Hy1 Hy2]; split; rewrite /= ?Hy1 ?Hy2.
-    * by intros n y1 y2 [Hy1 Hy2]; split; rewrite /= ?Hy1 ?Hy2.
-    * by intros n y1 y2 [Hy1 Hy2] [??]; split; rewrite /= -?Hy1 -?Hy2.
-    * by intros n x1 x2 [Hx1 Hx2] y1 y2 [Hy1 Hy2];
+    - by intros n x y1 y2 [Hy1 Hy2]; split; rewrite /= ?Hy1 ?Hy2.
+    - by intros n y1 y2 [Hy1 Hy2]; split; rewrite /= ?Hy1 ?Hy2.
+    - by intros n y1 y2 [Hy1 Hy2] [??]; split; rewrite /= -?Hy1 -?Hy2.
+    - by intros n x1 x2 [Hx1 Hx2] y1 y2 [Hy1 Hy2];
         split; rewrite /= ?Hx1 ?Hx2 ?Hy1 ?Hy2.
-    * by intros n x [??]; split; apply cmra_validN_S.
-    * by split; rewrite /= assoc.
-    * by split; rewrite /= comm.
-    * by split; rewrite /= cmra_unit_l.
-    * by split; rewrite /= cmra_unit_idemp.
-    * intros n x y; rewrite !prod_includedN.
+    - by intros n x [??]; split; apply cmra_validN_S.
+    - by split; rewrite /= assoc.
+    - by split; rewrite /= comm.
+    - by split; rewrite /= cmra_unit_l.
+    - by split; rewrite /= cmra_unit_idemp.
+    - 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 [??]; split; simpl in *; eauto using cmra_validN_op_l.
+    - intros x y n; rewrite prod_includedN; intros [??].
       by split; apply cmra_op_minus.
   Qed.
   Definition prod_cmra_extend_mixin : CMRAExtendMixin (A * B).
@@ -556,9 +556,9 @@ Section prod.
     CMRAIdentity A → CMRAIdentity B → CMRAIdentity prodRA.
   Proof.
     split.
-    * split; apply cmra_empty_valid.
-    * by split; rewrite /=left_id.
-    * by intros ? [??]; split; apply (timeless _).
+    - split; apply cmra_empty_valid.
+    - by split; rewrite /=left_id.
+    - 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.
@@ -579,7 +579,7 @@ 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.
+  - intros n x y; rewrite !prod_includedN; intros [??]; simpl.
     by split; apply includedN_preserving.
-  * by intros n x [??]; split; simpl; apply validN_preserving.
+  - by intros n x [??]; split; simpl; apply validN_preserving.
 Qed.

algebra/cmra_big_op.v

@@ -21,9 +21,9 @@ Proof. done. Qed.
 Global Instance big_op_permutation : Proper ((≡ₚ) ==> (≡)) big_op.
 Proof.
   induction 1 as [|x xs1 xs2 ? IH|x y xs|xs1 xs2 xs3]; simpl; auto.
-  * by rewrite IH.
-  * by rewrite !assoc (comm _ x).
-  * by transitivity (big_op xs2).
+  - by rewrite IH.
+  - by rewrite !assoc (comm _ x).
+  - by transitivity (big_op xs2).
 Qed.
 Global Instance big_op_proper : Proper ((≡) ==> (≡)) big_op.
 Proof. by induction 1; simpl; repeat apply (_ : Proper (_ ==> _ ==> _) op). Qed.
@@ -35,10 +35,10 @@ Qed.
 Lemma big_op_contains xs ys : xs `contains` ys → big_op xs ≼ big_op ys.
 Proof.
   induction 1 as [|x xs ys|x y xs|x xs ys|xs ys zs]; rewrite //=.
-  * by apply cmra_preserving_l.
-  * by rewrite !assoc (comm _ y).
-  * by transitivity (big_op ys); last apply cmra_included_r.
-  * by transitivity (big_op ys).
+  - by apply cmra_preserving_l.
+  - by rewrite !assoc (comm _ y).
+  - by transitivity (big_op ys); last apply cmra_included_r.
+  - by transitivity (big_op ys).
 Qed.
 Lemma big_op_delete xs i x :
   xs !! i = Some x → x ⋅ big_op (delete i xs) ≡ big_op xs.

algebra/cofe.v

@@ -97,15 +97,15 @@ Section cofe.
   Global Instance cofe_equivalence : Equivalence ((≡) : relation A).
   Proof.
     split.
-    * by intros x; rewrite equiv_dist.
-    * by intros x y; rewrite !equiv_dist.
-    * by intros x y z; rewrite !equiv_dist; intros; transitivity y.
+    - by intros x; rewrite equiv_dist.
+    - by intros x y; rewrite !equiv_dist.
+    - by intros x y z; rewrite !equiv_dist; intros; transitivity y.
   Qed.
   Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (@dist A _ n).
   Proof.
     intros x1 x2 ? y1 y2 ?; split; intros.
-    * by transitivity x1; [|transitivity y1].
-    * by transitivity x2; [|transitivity y2].
+    - by transitivity x1; [|transitivity y1].
+    - by transitivity x2; [|transitivity y2].
   Qed.
   Global Instance dist_proper n : Proper ((≡) ==> (≡) ==> iff) (@dist A _ n).
   Proof.
@@ -158,8 +158,8 @@ Program Definition fixpoint_chain {A : cofeT} `{Inhabited A} (f : A → A)
   `{!Contractive f} : chain A := {| chain_car i := Nat.iter (S i) f inhabitant |}.
 Next Obligation.
   intros A ? f ? n. induction n as [|n IH]; intros [|i] ?; simpl; try omega.
-  * apply (contractive_0 f).
-  * apply (contractive_S f), IH; auto with omega.
+  - apply (contractive_0 f).
+  - apply (contractive_S f), IH; auto with omega.
 Qed.
 Program Definition fixpoint {A : cofeT} `{Inhabited A} (f : A → A)
   `{!Contractive f} : A := compl (fixpoint_chain f).
@@ -212,14 +212,14 @@ Section cofe_mor.
   Definition cofe_mor_cofe_mixin : CofeMixin (cofeMor A B).
   Proof.
     split.
-    * intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
+    - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
       intros Hfg k; apply equiv_dist; intros n; apply Hfg.
-    * intros n; split.
+    - 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 intros n f g ? x; apply dist_S.
+    - intros c n x; simpl.
       by rewrite (conv_compl (fun_chain c x) n) /=.
   Qed.
   Canonical Structure cofe_mor : cofeT := CofeT cofe_mor_cofe_mixin.
@@ -274,11 +274,11 @@ Section product.
   Definition prod_cofe_mixin : CofeMixin (A * B).
   Proof.
     split.
-    * intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation.
+    - intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation.
       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 _.
+    - 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).
   Qed.
   Canonical Structure prodC : cofeT := CofeT prod_cofe_mixin.
@@ -308,10 +308,10 @@ Section discrete_cofe.
   Definition discrete_cofe_mixin : CofeMixin A.
   Proof.
     split.
-    * intros x y; split; [done|intros Hn; apply (Hn 0)].
-    * done.
-    * done.
-    * intros c n. rewrite /compl /discrete_compl /=.
+    - intros x y; split; [done|intros Hn; apply (Hn 0)].
+    - done.
+    - done.
+    - intros c n. rewrite /compl /discrete_compl /=.
       symmetry; apply (chain_cauchy c 0 (S n)); omega.
   Qed.
   Definition discreteC : cofeT := CofeT discrete_cofe_mixin.
@@ -347,14 +347,14 @@ Section later.
   Definition later_cofe_mixin : CofeMixin (later A).
   Proof.
     split.
-    * intros x y; unfold equiv, later_equiv; rewrite !equiv_dist.
+    - intros x y; unfold equiv, later_equiv; rewrite !equiv_dist.
       split. intros Hxy [|n]; [done|apply Hxy]. intros Hxy n; apply (Hxy (S n)).
-    * intros [|n]; [by split|split]; unfold dist, later_dist.
+    - intros [|n]; [by split|split]; unfold dist, later_dist.
       + by intros [x].
       + 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] [x] [y] ?; [done|]; unfold dist, later_dist; by apply dist_S.
+    - intros c [|n]; [done|by apply (conv_compl (later_chain c) n)].
   Qed.
   Canonical Structure laterC : cofeT := CofeT later_cofe_mixin.
   Global Instance Next_contractive : Contractive (@Next A).

algebra/cofe_solver.v

@@ -43,8 +43,8 @@ Qed.
 Lemma fg {k} (x : A (S (S k))) : f (S k) (g (S k) x) ≡{k}≡ x.
 Proof.
   induction k as [|k IH]; simpl.
-  * rewrite f_S g_S -{2}(map_id _ _ x) -map_comp. apply (contractive_0 map).
-  * rewrite f_S g_S -{2}(map_id _ _ x) -map_comp. by apply (contractive_S map).
+  - rewrite f_S g_S -{2}(map_id _ _ x) -map_comp. apply (contractive_0 map).
+  - rewrite f_S g_S -{2}(map_id _ _ x) -map_comp. by apply (contractive_S map).
 Qed.
 
 Record tower := {
@@ -66,15 +66,15 @@ Qed.
 Definition tower_cofe_mixin : CofeMixin tower.
 Proof.
   split.
-  * intros X Y; split; [by intros HXY n k; apply equiv_dist|].
+  - intros X Y; split; [by intros HXY n k; apply equiv_dist|].
     intros HXY k; apply equiv_dist; intros n; apply HXY.
-  * intros k; split.
+  - intros k; split.
     + by intros X n.
     + by intros X Y ? n.
     + by intros X Y Z ?? n; transitivity (Y n).
-  * intros k X Y HXY n; apply dist_S.
+  - 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 c n k; rewrite /= (conv_compl (tower_chain c k) n).
     apply (chain_cauchy c); lia.
 Qed.
 Definition T : cofeT := CofeT tower_cofe_mixin.
@@ -136,12 +136,12 @@ Lemma g_embed_coerce {k i} (x : A k) :
   g i (embed_coerce (S i) x) ≡ embed_coerce i x.
 Proof.
   unfold embed_coerce; destruct (le_lt_dec (S i) k), (le_lt_dec i k); simpl.
-  * symmetry; by erewrite (@gg_gg _ _ 1 (k - S i)); simpl.
-  * exfalso; lia.
-  * assert (i = k) by lia; subst.
+  - symmetry; by erewrite (@gg_gg _ _ 1 (k - S i)); simpl.
+  - exfalso; lia.
+  - assert (i = k) by lia; subst.
     rewrite (ff_ff _ (eq_refl (1 + (0 + k)))) /= gf.
     by rewrite (gg_gg _ (eq_refl (0 + (0 + k)))).
-  * assert (H : 1 + ((i - k) + k) = S i) by lia.
+  - assert (H : 1 + ((i - k) + k) = S i) by lia.
     rewrite (ff_ff _ H) /= -{2}(gf (ff (i - k) x)) g_coerce.
     by erewrite coerce_proper by done.
 Qed.
@@ -156,22 +156,22 @@ Lemma embed_f k (x : A k) : embed (S k) (f k x) ≡ embed k x.
 Proof.
   rewrite equiv_dist=> n i; rewrite /embed /= /embed_coerce.
   destruct (le_lt_dec i (S k)), (le_lt_dec i k); simpl.
-  * assert (H : S k = S (k - i) + (0 + i)) by lia; rewrite (gg_gg _ H) /=.
+  - assert (H : S k = S (k - i) + (0 + i)) by lia; rewrite (gg_gg _ H) /=.
     by erewrite g_coerce, gf, coerce_proper by done.
-  * assert (S k = 0 + (0 + i)) as H by lia.
+  - assert (S k = 0 + (0 + i)) as H by lia.
     rewrite (gg_gg _ H); simplify_equality'.
     by rewrite (ff_ff _ (eq_refl (1 + (0 + k)))).
-  * exfalso; lia.
-  * assert (H : (i - S k) + (1 + k) = i) by lia; rewrite (ff_ff _ H) /=.
+  - exfalso; lia.
+  - assert (H : (i - S k) + (1 + k) = i) by lia; rewrite (ff_ff _ H) /=.
     by erewrite coerce_proper by done.
 Qed.
 Lemma embed_tower k (X : T) : embed (S k) (X (S k)) ≡{k}≡ X.
 Proof.
   intros i; rewrite /= /embed_coerce.
   destruct (le_lt_dec i (S k)) as [H|H]; simpl.
-  * rewrite -(gg_tower i (S k - i) X).
+  - rewrite -(gg_tower i (S k - i) X).
     apply (_ : Proper (_ ==> _) (gg _)); by destruct (eq_sym _).
-  * rewrite (ff_tower k (i - S k) X). by destruct (Nat.sub_add _ _ _).
+  - rewrite (ff_tower k (i - S k) X). by destruct (Nat.sub_add _ _ _).
 Qed.
 
 Program Definition unfold_chain (X : T) : chain (F T T) :=
@@ -206,7 +206,7 @@ Proof. by intros n X Y HXY k; rewrite /fold /= HXY. Qed.
 Theorem result : solution F.
 Proof.
   apply (Solution F T (CofeMor unfold) (CofeMor fold)).
-  * move=> X /=.
+  - move=> X /=.
     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)))).
@@ -228,7 +228,7 @@ Proof.
     assert (H: S n + k = n + S k) by lia.
     rewrite (map_ff_gg _ _ _ H).
     apply (_ : Proper (_ ==> _) (gg _)); by destruct H.
-  * intros X; rewrite equiv_dist=> n /=.
+  - intros X; rewrite equiv_dist=> n /=.
     rewrite /unfold /= (conv_compl (unfold_chain (fold X)) n) /=.
     rewrite g_S -!map_comp -{2}(map_id _ _ X).
     apply (contractive_ne map); split => Y /=.

algebra/dra.v

@@ -57,9 +57,9 @@ Instance validity_equiv : Equiv T := λ x y,
 Instance validity_equivalence : Equivalence ((≡) : relation T).
 Proof.
   split; unfold equiv, validity_equiv.
-  * by intros [x px ?]; simpl.
-  * intros [x px ?] [y py ?]; naive_solver.
-  * intros [x px ?] [y py ?] [z pz ?] [? Hxy] [? Hyz]; simpl in *.
+  - by intros [x px ?]; simpl.
+  - intros [x px ?] [y py ?]; naive_solver.
+  - intros [x px ?] [y py ?] [z pz ?] [? Hxy] [? Hyz]; simpl in *.
     split; [|intros; transitivity y]; tauto.
 Qed.
 Instance dra_valid_proper' : Proper ((≡) ==> iff) (valid : A → Prop).
@@ -69,8 +69,8 @@ Proof. by intros x1 x2 Hx; split; rewrite /= Hx. Qed.
 Instance: Proper ((≡) ==> (≡) ==> iff) (⊥).
 Proof.
   intros x1 x2 Hx y1 y2 Hy; split.
-  * by rewrite Hy (symmetry_iff (⊥) x1) (symmetry_iff (⊥) x2) Hx.
-  * by rewrite -Hy (symmetry_iff (⊥) x2) (symmetry_iff (⊥) x1) -Hx.
+  - by rewrite Hy (symmetry_iff (⊥) x1) (symmetry_iff (⊥) x2) Hx.
+  - by rewrite -Hy (symmetry_iff (⊥) x2) (symmetry_iff (⊥) x1) -Hx.
 Qed.
 Lemma dra_disjoint_rl x y z : ✓ x → ✓ y → ✓ z → y ⊥ z → x ⊥ y ⋅ z → x ⊥ y.
 Proof. intros ???. rewrite !(symmetry_iff _ x). by apply dra_disjoint_ll. Qed.
@@ -103,31 +103,31 @@ Solve Obligations with naive_solver auto using dra_minus_valid.
 Definition validity_ra : RA (discreteC T).
 Proof.
   split.
-  * intros ??? [? Heq]; split; simpl; [|by intros (?&?&?); rewrite Heq].
+  - intros ??? [? Heq]; split; simpl; [|by intros (?&?&?); rewrite Heq].
     split; intros (?&?&?); split_ands';
       first [rewrite ?Heq; tauto|rewrite -?Heq; tauto|tauto].
-  * by intros ?? [? Heq]; split; [done|]; simpl; intros ?; rewrite Heq.
-  * intros ?? [??]; naive_solver.
-  * intros x1 x2 [? Hx] y1 y2 [? Hy];
+  - by intros ?? [? Heq]; split; [done|]; simpl; intros ?; rewrite Heq.
+  - intros ?? [??]; naive_solver.
+  - intros x1 x2 [? Hx] y1 y2 [? Hy];
       split; simpl; [|by intros (?&?&?); rewrite Hx // Hy].
     split; intros (?&?&z&?&?); split_ands'; try tauto.
     + exists z. by rewrite -Hy // -Hx.
     + exists z. by rewrite Hx ?Hy; tauto.
-  * intros [x px ?] [y py ?] [z pz ?]; split; simpl;
+  - intros [x px ?] [y py ?] [z pz ?]; split; simpl;
       [intuition eauto 2 using dra_disjoint_lr, dra_disjoint_rl
       |by intros; rewrite assoc].
-  * intros [x px ?] [y py ?]; split; naive_solver eauto using dra_comm.
-  * intros [x px ?]; split;
+  - intros [x px ?] [y py ?]; split; naive_solver eauto using dra_comm.
+  - intros [x px ?]; split;
       naive_solver eauto using dra_unit_l, dra_unit_disjoint_l.
-  * intros [x px ?]; split; naive_solver eauto using dra_unit_idemp.
-  * intros x y Hxy; exists (unit y ⩪ unit x).
+  - intros [x px ?]; split; naive_solver eauto using dra_unit_idemp.
+  - intros x y Hxy; exists (unit y ⩪ unit x).
     destruct x as [x px ?], y as [y py ?], Hxy as [[z pz ?] [??]]; simpl in *.
     assert (py → unit x ≼ unit y)
       by intuition eauto 10 using dra_unit_preserving.
     constructor; [|symmetry]; simpl in *;
       intuition eauto using dra_op_minus, dra_disjoint_minus, dra_unit_valid.
-  * by intros [x px ?] [y py ?] (?&?&?).
-  * intros [x px ?] [y py ?] [[z pz ?] [??]]; split; simpl in *;
+  - by intros [x px ?] [y py ?] (?&?&?).
+  - intros [x px ?] [y py ?] [[z pz ?] [??]]; split; simpl in *;
       intuition eauto 10 using dra_disjoint_minus, dra_op_minus.
 Qed.
 Definition validityRA : cmraT := discreteRA validity_ra.

algebra/excl.v

@@ -50,15 +50,15 @@ Instance excl_compl : Compl (excl A) := λ c,
 Definition excl_cofe_mixin : CofeMixin (excl A).
 Proof.
   split.
-  * intros mx my; split; [by destruct 1; constructor; apply equiv_dist|].
+  - intros mx my; split; [by destruct 1; constructor; apply equiv_dist|].
     intros Hxy; feed inversion (Hxy 1); subst; constructor; apply equiv_dist.
     by intros n; feed inversion (Hxy n).
-  * intros n; split.
+  - intros n; split.
     + by intros [x| |]; constructor.
     + 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.
+  - by inversion_clear 1; constructor; apply dist_S.
+  - intros c n; 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_equality').
       assert (∃ y, c (S n) = Excl y) as [y Hy].
@@ -102,18 +102,18 @@ Instance excl_minus : Minus (excl A) := λ x y,
 Definition excl_cmra_mixin : CMRAMixin (excl A).
 Proof.
   split.
-  * by intros n []; destruct 1; constructor.
-  * constructor.
-  * by destruct 1; intros ?.
-  * by destruct 1; inversion_clear 1; constructor.
-  * intros n [?| |]; simpl; auto with lia.
-  * by intros [?| |] [?| |] [?| |]; constructor.
-  * by intros [?| |] [?| |]; constructor.
-  * by intros [?| |]; constructor.
-  * constructor.
-  * by intros n [?| |] [?| |]; exists ∅.
-  * by intros n [?| |] [?| |].
-  * by intros n [?| |] [?| |] [[?| |] Hz]; inversion_clear Hz; constructor.
+  - by intros n []; destruct 1; constructor.
+  - constructor.