add tactic aliases: trans -> transitivity; etrans -> etransitivity

algebra/agree.v

@@ -40,8 +40,8 @@ Proof.
     + 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.
+      * trans (agree_is_valid y n'). by apply Hxy. by apply Hyz.
+      * trans (y n'). by apply Hxy. by apply Hyz, Hxy.
   - intros n x y Hxy; split; intros; apply Hxy; auto.
   - intros n c; apply and_wlog_r; intros;
       symmetry; apply (chain_cauchy c); naive_solver.
@@ -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'];
+  - etrans; [apply Hxy|apply Hy]; eauto using agree_valid_le.
+  - etrans; [apply Hxy|symmetry; apply Hy, Hy'];
       eauto using agree_valid_le.
 Qed.
 Instance: Proper (dist n ==> dist n ==> dist n) (@op (agree A) _).

algebra/auth.v

@@ -44,7 +44,7 @@ Proof.
   - intros n; split.
     + by intros ?; split.
     + by intros ?? [??]; split; symmetry.
-    + intros ??? [??] [??]; split; etransitivity; eauto.
+    + intros ??? [??] [??]; split; etrans; eauto.
   - by intros ? [??] [??] [??]; split; apply dist_S.
   - intros n c; split. apply (conv_compl n (chain_map authoritative c)).
     apply (conv_compl n (chain_map own c)).

algebra/cmra.v

@@ -250,7 +250,7 @@ Qed.
 Global Instance cmra_included_preorder: PreOrder (@included A _ _).
 Proof.
   split; red; intros until 0; rewrite !cmra_included_includedN; first done.
-  intros; etransitivity; eauto.
+  intros; etrans; eauto.
 Qed.
 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.
@@ -391,7 +391,7 @@ Section identity_updates.
   Lemma cmra_update_empty x : x ~~> ∅.
   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.
+  Proof. split; [intros; trans ∅|]; auto using cmra_update_empty. Qed.
 End identity_updates.
 End cmra.
 

algebra/cmra_big_op.v

@@ -23,7 +23,7 @@ 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 trans (big_op xs2).
 Qed.
 Global Instance big_op_proper : Proper ((≡) ==> (≡)) big_op.
 Proof. by induction 1; simpl; repeat apply (_ : Proper (_ ==> _ ==> _) op). Qed.

algebra/cofe.v

@@ -99,13 +99,13 @@ Section cofe.
     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 y z; rewrite !equiv_dist; intros; trans 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 trans x1; [|trans y1].
+    - by trans x2; [|trans y2].
   Qed.
   Global Instance dist_proper n : Proper ((≡) ==> (≡) ==> iff) (@dist A _ n).
   Proof.
@@ -217,7 +217,7 @@ Section cofe_mor.
     - intros n; split.
       + by intros f x.
       + by intros f g ? x.
-      + by intros f g h ?? x; transitivity (g x).
+      + by intros f g h ?? x; trans (g x).
     - by intros n f g ? x; apply dist_S.
     - intros n c x; simpl.
       by rewrite (conv_compl n (fun_chain c x)) /=.
@@ -352,7 +352,7 @@ Section later.
     - intros [|n]; [by split|split]; unfold dist, later_dist.
       + by intros [x].
       + by intros [x] [y].
-      + by intros [x] [y] [z] ??; transitivity y.
+      + by intros [x] [y] [z] ??; trans y.
     - intros [|n] [x] [y] ?; [done|]; unfold dist, later_dist; by apply dist_S.
     - intros [|n] c; [done|by apply (conv_compl n (later_chain c))].
   Qed.

algebra/cofe_solver.v

@@ -71,7 +71,7 @@ Proof.
   - intros k; split.
     + by intros X n.
     + by intros X Y ? n.
-    + by intros X Y Z ?? n; transitivity (Y n).
+    + by intros X Y Z ?? n; trans (Y n).
   - intros k X Y HXY n; apply dist_S.
     by rewrite -(g_tower X) (HXY (S n)) g_tower.
   - intros n c k; rewrite /= (conv_compl n (tower_chain c k)).
@@ -209,7 +209,7 @@ Proof.
   - 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)))).
+    trans (map (ff n, gg n) (X (S (n + k)))).
     { 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.
@@ -234,6 +234,6 @@ Proof.
     apply (contractive_ne map); split => Y /=.
     + apply dist_le with n; last omega.
       rewrite f_tower. apply dist_S. by rewrite embed_tower.
-    + etransitivity; [apply embed_ne, equiv_dist, g_tower|apply embed_tower].
+    + etrans; [apply embed_ne, equiv_dist, g_tower|apply embed_tower].
 Qed.
 End solver. End solver.

algebra/dra.v

@@ -60,7 +60,7 @@ Proof.
   - 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.
+    split; [|intros; trans y]; tauto.
 Qed.
 Instance dra_valid_proper' : Proper ((≡) ==> iff) (valid : A → Prop).
 Proof. by split; apply dra_valid_proper. Qed.

algebra/excl.v

@@ -56,7 +56,7 @@ Proof.
   - intros n; split.
     + by intros [x| |]; constructor.
     + by destruct 1; constructor.
-    + destruct 1; inversion_clear 1; constructor; etransitivity; eauto.
+    + destruct 1; inversion_clear 1; constructor; etrans; eauto.
   - by inversion_clear 1; constructor; apply dist_S.
   - intros n c; unfold compl, excl_compl.
     destruct (Some_dec (maybe Excl (c 1))) as [[x Hx]|].

algebra/fin_maps.v

@@ -22,7 +22,7 @@ Proof.
   - intros n; split.
     + by intros m k.
     + by intros m1 m2 ? k.
-    + by intros m1 m2 m3 ?? k; transitivity (m2 !! k).
+    + by intros m1 m2 m3 ?? k; trans (m2 !! k).
   - by intros n m1 m2 ? k; apply dist_S.
   - 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.

algebra/iprod.v

@@ -35,7 +35,7 @@ Section iprod_cofe.
     - intros n; split.
       + by intros f x.
       + by intros f g ? x.
-      + by intros f g h ?? x; transitivity (g x).
+      + by intros f g h ?? x; trans (g x).
     - intros n f g Hfg x; apply dist_S, Hfg.
     - intros n c x.
       rewrite /compl /iprod_compl (conv_compl n (iprod_chain c x)).

algebra/option.v

@@ -29,7 +29,7 @@ Proof.
   - intros n; split.
     + by intros [x|]; constructor.
     + by destruct 1; constructor.
-    + destruct 1; inversion_clear 1; constructor; etransitivity; eauto.
+    + destruct 1; inversion_clear 1; constructor; etrans; eauto.
   - by inversion_clear 1; constructor; apply dist_S.
   - intros n c; unfold compl, option_compl.
     destruct (Some_dec (c 1)) as [[x Hx]|].

algebra/sts.v

@@ -224,7 +224,7 @@ Proof.
   split.
   - by intros []; constructor.
   - by destruct 1; constructor.
-  - destruct 1; inversion_clear 1; constructor; etransitivity; eauto.
+  - destruct 1; inversion_clear 1; constructor; etrans; eauto.
 Qed.
 Global Instance sts_dra : DRA (car sts).
 Proof.
@@ -366,7 +366,7 @@ Lemma sts_op_frag S1 S2 T1 T2 :
 Proof.
   intros HT HS1 HS2. rewrite /sts_frag.
   (* FIXME why does rewrite not work?? *)
-  etransitivity; last eapply to_validity_op; try done; [].
+  etrans; last eapply to_validity_op; try done; [].
   intros Hval. constructor; last set_solver. eapply closed_ne, Hval.
 Qed.
 

algebra/upred.v

 From algebra Require Export cmra.
-Local Hint Extern 1 (_ ≼ _) => etransitivity; [eassumption|].
-Local Hint Extern 1 (_ ≼ _) => etransitivity; [|eassumption].
+Local Hint Extern 1 (_ ≼ _) => etrans; [eassumption|].
+Local Hint Extern 1 (_ ≼ _) => etrans; [|eassumption].
 Local Hint Extern 10 (_ ≤ _) => omega.
 
 Record uPred (M : cmraT) : Type := IProp {
@@ -40,7 +40,7 @@ Section cofe.
     - intros n; split.
       + by intros P x i.
       + by intros P Q HPQ x i ??; symmetry; apply HPQ.
-      + by intros P Q Q' HP HQ i x ??; transitivity (Q i x);[apply HP|apply HQ].
+      + by intros P Q Q' HP HQ i x ??; trans (Q i x);[apply HP|apply HQ].
     - intros n P Q HPQ i x ??; apply HPQ; auto.
     - intros n c i x ??; symmetry; apply (chain_cauchy c i (S n)); auto.
   Qed.
@@ -243,8 +243,8 @@ Global Instance entails_proper :
   Proper ((≡) ==> (≡) ==> iff) ((⊑) : relation (uPred M)).
 Proof.
   move => P1 P2 /equiv_spec [HP1 HP2] Q1 Q2 /equiv_spec [HQ1 HQ2]; split; intros.
-  - by transitivity P1; [|transitivity Q1].
-  - by transitivity P2; [|transitivity Q2].
+  - by trans P1; [|trans Q1].
+  - by trans P2; [|trans Q2].
 Qed.
 
 (** Non-expansiveness and setoid morphisms *)
@@ -734,7 +734,7 @@ Proof. by rewrite /uPred_iff later_and !later_impl. Qed.
 Lemma löb_strong P Q : (P ∧ ▷Q) ⊑ Q → P ⊑ Q.
 Proof.
   intros Hlöb. apply impl_entails.
-  etransitivity; last by eapply löb.
+  etrans; last by eapply löb.
   apply impl_intro_l, impl_intro_l. rewrite right_id -{2}Hlöb.
   apply and_intro; first by eauto.
   by rewrite {1}(later_intro P) later_impl impl_elim_r.

algebra/upred_big_op.v

@@ -61,14 +61,14 @@ Proof.
   induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto.
   - by rewrite IH.
   - by rewrite !assoc (comm _ P).
-  - etransitivity; eauto.
+  - etrans; eauto.
 Qed.
 Global Instance big_sep_perm : Proper ((≡ₚ) ==> (≡)) (@uPred_big_sep M).
 Proof.
   induction 1 as [|P Ps Qs ? IH|P Q Ps|]; simpl; auto.
   - by rewrite IH.
   - by rewrite !assoc (comm _ P).
-  - etransitivity; eauto.
+  - etrans; eauto.
 Qed.
 
 Lemma big_and_app Ps Qs : (Π∧ (Ps ++ Qs))%I ≡ (Π∧ Ps ∧ Π∧ Qs)%I.
@@ -103,7 +103,7 @@ Section gmap.
     m2 ⊆ m1 → (∀ x k, m2 !! k = Some x → Φ k x ⊑ Ψ k x) →
     (Π★{map m1} Φ) ⊑ (Π★{map m2} Ψ).
   Proof.
-    intros HX HΦ. transitivity (Π★{map m2} Φ)%I.
+    intros HX HΦ. trans (Π★{map m2} Φ)%I.
     - by apply big_sep_contains, fmap_contains, map_to_list_contains.
     - apply big_sep_mono', Forall2_fmap, Forall2_Forall.
       apply Forall_forall=> -[i x] ? /=. by apply HΦ, elem_of_map_to_list.
@@ -163,7 +163,7 @@ Section gset.
   Lemma big_sepS_mono Φ Ψ X Y :
     Y ⊆ X → (∀ x, x ∈ Y → Φ x ⊑ Ψ x) → (Π★{set X} Φ) ⊑ (Π★{set Y} Ψ).
   Proof.
-    intros HX HΦ. transitivity (Π★{set Y} Φ)%I.
+    intros HX HΦ. trans (Π★{set Y} Φ)%I.
     - by apply big_sep_contains, fmap_contains, elements_contains.
     - apply big_sep_mono', Forall2_fmap, Forall2_Forall.
       apply Forall_forall=> x ? /=. by apply HΦ, elem_of_elements.

barrier/barrier.v

@@ -157,7 +157,7 @@ Section proof.
     { by eapply (saved_prop_alloc _ P). }
     rewrite pvs_frame_l. apply pvs_strip_pvs. rewrite sep_exist_l.
     apply exist_elim=>i.
-    transitivity (pvs ⊤ ⊤ (heap_ctx heapN ★ ▷ (barrier_inv l P (State Low {[ i ]}))  ★ saved_prop_own i P)).
+    trans (pvs ⊤ ⊤ (heap_ctx heapN ★ ▷ (barrier_inv l P (State Low {[ i ]}))  ★ saved_prop_own i P)).
     - rewrite -pvs_intro. rewrite [(_ ★ heap_ctx _)%I]comm -!assoc. apply sep_mono_r.
       rewrite {1}[saved_prop_own _ _]always_sep_dup !assoc. apply sep_mono_l.
       rewrite /barrier_inv /waiting -later_intro. apply sep_mono_r.
@@ -215,7 +215,7 @@ Section proof.
     apply const_elim_sep_l=>Hs. destruct p; last done.
     rewrite {1}/barrier_inv =>/={Hs}. rewrite later_sep.
     eapply wp_store; eauto with I ndisj.
-    rewrite -!assoc. apply sep_mono_r. etransitivity; last eapply later_mono.
+    rewrite -!assoc. apply sep_mono_r. etrans; last eapply later_mono.
     { (* Is this really the best way to strip the later? *)
       erewrite later_sep. apply sep_mono_r. apply later_intro. }
     apply wand_intro_l. rewrite -(exist_intro (State High I)).
@@ -256,7 +256,7 @@ Section proof.
     apply const_elim_sep_l=>Hs.
     rewrite {1}/barrier_inv =>/=. rewrite later_sep.
     eapply wp_load; eauto with I ndisj.
-    rewrite -!assoc. apply sep_mono_r. etransitivity; last eapply later_mono.
+    rewrite -!assoc. apply sep_mono_r. etrans; last eapply later_mono.
     { (* Is this really the best way to strip the later? *)
       erewrite later_sep. apply sep_mono_r. rewrite !assoc. erewrite later_sep.
       apply sep_mono_l, later_intro. }
@@ -294,7 +294,7 @@ Section proof.
     rewrite [(sts_own _ _ _ ★ _)%I]sep_elim_r [(sts_ctx _ _ _ ★ _)%I]sep_elim_r.
     rewrite !assoc [(_ ★ saved_prop_own i Q)%I]comm !assoc saved_prop_agree.
     wp_op>; last done. intros _.
-    etransitivity; last eapply later_mono.
+    etrans; last eapply later_mono.
     { (* Is this really the best way to strip the later? *)
       erewrite later_sep. apply sep_mono; last apply later_intro.
       rewrite ->later_sep. apply sep_mono_l. rewrite ->later_sep. done. }

heap_lang/heap.v

@@ -67,7 +67,7 @@ Section heap.
     authG heap_lang Σ heapRA → nclose N ⊆ E →
     ownP σ ⊑ (|={E}=> ∃ _ : heapG Σ, heap_ctx N ∧ Π★{map σ} heap_mapsto).
   Proof.
-    intros. rewrite -{1}(from_to_heap σ). etransitivity.
+    intros. rewrite -{1}(from_to_heap σ). etrans.
     { rewrite [ownP _]later_intro.
       apply (auth_alloc (ownP ∘ of_heap) E N (to_heap σ)); last done.
       apply to_heap_valid. }
@@ -103,7 +103,7 @@ Section heap.
     P ⊑ || Alloc e @ E {{ Φ }}.
   Proof.
     rewrite /heap_ctx /heap_inv /heap_mapsto=> ?? Hctx HP.
-    transitivity (|={E}=> auth_own heap_name ∅ ★ P)%I.
+    trans (|={E}=> auth_own heap_name ∅ ★ P)%I.
     { by rewrite -pvs_frame_r -(auth_empty _ E) left_id. }
     apply wp_strip_pvs, (auth_fsa heap_inv (wp_fsa (Alloc e)))
       with N heap_name ∅; simpl; eauto with I.

heap_lang/wp_tactics.v

@@ -13,19 +13,19 @@ Ltac wp_strip_later :=
     | |- _ ⊑ ▷ _ => apply later_intro
     | |- _ ⊑ _ => reflexivity
     end
-  in revert_intros ltac:(etransitivity; [|go]).
+  in revert_intros ltac:(etrans; [|go]).
 
 Ltac wp_bind K :=
   lazymatch eval hnf in K with
   | [] => idtac
-  | _ => etransitivity; [|solve [ apply (wp_bind K) ]]; simpl
+  | _ => etrans; [|solve [ apply (wp_bind K) ]]; simpl
   end.
 Ltac wp_finish :=
   let rec go :=
   match goal with
-  | |- _ ⊑ ▷ _ => etransitivity; [|apply later_mono; go; reflexivity]
+  | |- _ ⊑ ▷ _ => etrans; [|apply later_mono; go; reflexivity]
   | |- _ ⊑ wp _ _ _ =>
-     etransitivity; [|eapply wp_value_pvs; reflexivity];
+     etrans; [|eapply wp_value_pvs; reflexivity];
      (* sometimes, we will have to do a final view shift, so only apply
      wp_value if we obtain a consecutive wp *)
      try (eapply pvs_intro;
@@ -38,7 +38,7 @@ Tactic Notation "wp_rec" ">" :=
   | |- _ ⊑ wp ?E ?e ?Q => reshape_expr e ltac:(fun K e' =>
     match eval cbv in e' with
     | App (Rec _ _ _) _ =>
-       wp_bind K; etransitivity; [|eapply wp_rec; reflexivity]; wp_finish
+       wp_bind K; etrans; [|eapply wp_rec; reflexivity]; wp_finish
     end)
   end.
 Tactic Notation "wp_rec" := wp_rec>; wp_strip_later.
@@ -48,7 +48,7 @@ Tactic Notation "wp_lam" ">" :=
   | |- _ ⊑ wp ?E ?e ?Q => reshape_expr e ltac:(fun K e' =>
     match eval cbv in e' with
     | App (Rec "" _ _) _ =>
-       wp_bind K; etransitivity; [|eapply wp_lam; reflexivity]; wp_finish
+       wp_bind K; etrans; [|eapply wp_lam; reflexivity]; wp_finish
     end)
   end.
 Tactic Notation "wp_lam" := wp_lam>; wp_strip_later.
@@ -66,9 +66,9 @@ Tactic Notation "wp_op" ">" :=
     | BinOp LeOp _ _ => wp_bind K; apply wp_le; wp_finish
     | BinOp EqOp _ _ => wp_bind K; apply wp_eq; wp_finish
     | BinOp _ _ _ =>
-       wp_bind K; etransitivity; [|eapply wp_bin_op; reflexivity]; wp_finish
+       wp_bind K; etrans; [|eapply wp_bin_op; reflexivity]; wp_finish
     | UnOp _ _ =>
-       wp_bind K; etransitivity; [|eapply wp_un_op; reflexivity]; wp_finish
+       wp_bind K; etrans; [|eapply wp_un_op; reflexivity]; wp_finish
     end)
   end.
 Tactic Notation "wp_op" := wp_op>; wp_strip_later.
@@ -79,7 +79,7 @@ Tactic Notation "wp_if" ">" :=
     match eval cbv in e' with
     | If _ _ _ =>
        wp_bind K;
-       etransitivity; [|apply wp_if_true || apply wp_if_false]; wp_finish
+       etrans; [|apply wp_if_true || apply wp_if_false]; wp_finish
     end)
   end.
 Tactic Notation "wp_if" := wp_if>; wp_strip_later.
@@ -97,5 +97,5 @@ Tactic Notation "wp" ">" tactic(tac) :=
 Tactic Notation "wp" tactic(tac) := (wp> tac); wp_strip_later.
 
 (* In case the precondition does not match *)
-Tactic Notation "ewp" tactic(tac) := wp (etransitivity; [|tac]).
-Tactic Notation "ewp" ">" tactic(tac) := wp> (etransitivity; [|tac]).
+Tactic Notation "ewp" tactic(tac) := wp (etrans; [|tac]).
+Tactic Notation "ewp" ">" tactic(tac) := wp> (etrans; [|tac]).

prelude/fin_map_dom.v

@@ -60,7 +60,7 @@ Lemma dom_insert_subseteq {A} (m : M A) i x : dom D m ⊆ dom D (<[i:=x]>m).
 Proof. rewrite (dom_insert _). set_solver. Qed.
 Lemma dom_insert_subseteq_compat_l {A} (m : M A) i x X :
   X ⊆ dom D m → X ⊆ dom D (<[i:=x]>m).
-Proof. intros. transitivity (dom D m); eauto using dom_insert_subseteq. Qed.
+Proof. intros. trans (dom D m); eauto using dom_insert_subseteq. Qed.
 Lemma dom_singleton {A} (i : K) (x : A) : dom D {[i := x]} ≡ {[ i ]}.
 Proof. rewrite <-insert_empty, dom_insert, dom_empty; set_solver. Qed.
 Lemma dom_delete {A} (m : M A) i : dom D (delete i m) ≡ dom D m ∖ {[ i ]}.

prelude/fin_maps.v

@@ -123,7 +123,7 @@ Section setoid.
     split.
     - by intros m i.
     - by intros m1 m2 ? i.
-    - by intros m1 m2 m3 ?? i; transitivity (m2 !! i).
+    - by intros m1 m2 m3 ?? i; trans (m2 !! i).
   Qed.
   Global Instance lookup_proper (i : K) :
     Proper ((≡) ==> (≡)) (lookup (M:=M A) i).
@@ -199,7 +199,7 @@ Proof.
   split; [intros m i; by destruct (m !! i); simpl|].
   intros m1 m2 m3 Hm12 Hm23 i; specialize (Hm12 i); specialize (Hm23 i).
   destruct (m1 !! i), (m2 !! i), (m3 !! i); simplify_eq/=;
-    done || etransitivity; eauto.
+    done || etrans; eauto.
 Qed.
 Global Instance: PartialOrder ((⊆) : relation (M A)).
 Proof.
@@ -1182,10 +1182,10 @@ Proof.
   intros. rewrite map_union_comm by done. by apply map_union_subseteq_l.
 Qed.
 Lemma map_union_subseteq_l_alt {A} (m1 m2 m3 : M A) : m1 ⊆ m2 → m1 ⊆ m2 ∪ m3.
-Proof. intros. transitivity m2; auto using map_union_subseteq_l. Qed.
+Proof. intros. trans m2; auto using map_union_subseteq_l. Qed.
 Lemma map_union_subseteq_r_alt {A} (m1 m2 m3 : M A) :
   m2 ⊥ₘ m3 → m1 ⊆ m3 → m1 ⊆ m2 ∪ m3.
-Proof. intros. transitivity m3; auto using map_union_subseteq_r. Qed.
+Proof. intros. trans m3; auto using map_union_subseteq_r. Qed.
 Lemma map_union_preserving_l {A} (m1 m2 m3 : M A) : m1 ⊆ m2 → m3 ∪ m1 ⊆ m3 ∪ m2.
 Proof.
   rewrite !map_subseteq_spec. intros ???.

prelude/lexico.v

@@ -42,7 +42,7 @@ Lemma prod_lexico_transitive `{Lexico A, Lexico B, !Transitive (@lexico A _)}
 Proof.
   intros Hx12 Hx23 ?; revert Hx12 Hx23. unfold lexico, prod_lexico.
   intros [|[??]] [?|[??]]; simplify_eq/=; auto.
-  by left; transitivity x2.
+  by left; trans x2.
 Qed.
 
 Instance prod_lexico_po `{Lexico A, Lexico B, !StrictOrder (@lexico A _)}
@@ -52,7 +52,7 @@ Proof.
   - intros [x y]. apply prod_lexico_irreflexive.
     by apply (irreflexivity lexico y).
   - intros [??] [??] [??] ??.
-    eapply prod_lexico_transitive; eauto. apply transitivity.
+    eapply prod_lexico_transitive; eauto. apply trans.
 Qed.
 Instance prod_lexico_trichotomyT `{Lexico A, tA : !TrichotomyT (@lexico A _)}
   `{Lexico B, tB : !TrichotomyT (@lexico B _)}: TrichotomyT (@lexico (A * B) _).
@@ -143,7 +143,7 @@ Instance sig_lexico_po `{Lexico A, !StrictOrder (@lexico A _)}
 Proof.
   unfold lexico, sig_lexico. split.
   - intros [x ?] ?. by apply (irreflexivity lexico x). 
-  - intros [x1 ?] [x2 ?] [x3 ?] ??. by transitivity x2.
+  - intros [x1 ?] [x2 ?] [x3 ?] ??. by trans x2.
 Qed.
 Instance sig_lexico_trichotomy `{Lexico A, tA : !TrichotomyT (@lexico A _)}
   (P : A → Prop) `{∀ x, ProofIrrel (P x)} : TrichotomyT (@lexico (sig P) _).