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.