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

theories/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 ]}.

theories/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 ???.

theories/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) _).

theories/list.v

@@ -371,7 +371,7 @@ Section setoid.
     - intros l; induction l; constructor; auto.
     - induction 1; constructor; auto.
     - intros l1 l2 l3 Hl; revert l3.
-      induction Hl; inversion_clear 1; constructor; try etransitivity; eauto.
+      induction Hl; inversion_clear 1; constructor; try etrans; eauto.
   Qed.
   Global Instance cons_proper : Proper ((≡) ==> (≡) ==> (≡)) (@cons A).
   Proof. by constructor. Qed.
@@ -1719,7 +1719,7 @@ Proof. revert i. by induction l; intros [|?]; simpl; constructor. Qed.
 Lemma sublist_foldr_delete l is : foldr delete l is `sublist` l.
 Proof.
   induction is as [|i is IH]; simpl; [done |].
-  transitivity (foldr delete l is); auto using sublist_delete.
+  trans (foldr delete l is); auto using sublist_delete.
 Qed.
 Lemma sublist_alt l1 l2 : l1 `sublist` l2 ↔ ∃ is, l1 = foldr delete l2 is.
 Proof.
@@ -1749,7 +1749,7 @@ Proof.
     + by rewrite !Permutation_middle, Permutation_swap.
   - intros l3 ?. destruct (IH2 l3) as (l3'&?&?); trivial.
     destruct (IH1 l3') as (l3'' &?&?); trivial. exists l3''.
-    split. done. etransitivity; eauto.
+    split. done. etrans; eauto.
 Qed.
 Lemma sublist_Permutation l1 l2 l3 :
   l1 `sublist` l2 → l2 ≡ₚ l3 → ∃ l4, l1 ≡ₚ l4 ∧ l4 `sublist` l3.
@@ -1770,7 +1770,7 @@ Proof.
     + exists (x :: y :: l1''). by repeat constructor.
   - intros l1 ?. destruct (IH1 l1) as (l3'&?&?); trivial.
     destruct (IH2 l3') as (l3'' &?&?); trivial. exists l3''.
-    split; [|done]. etransitivity; eauto.
+    split; [|done]. etrans; eauto.
 Qed.
 
 (** Properties of the [contains] predicate *)
@@ -1816,10 +1816,10 @@ Proof. intro. apply contains_Permutation_length_le. lia. Qed.
 Global Instance: Proper ((≡ₚ) ==> (≡ₚ) ==> iff) (@contains A).
 Proof.
   intros l1 l2 ? k1 k2 ?. split; intros.
-  - transitivity l1. by apply Permutation_contains.
-    transitivity k1. done. by apply Permutation_contains.
-  - transitivity l2. by apply Permutation_contains.
-    transitivity k2. done. by apply Permutation_contains.
+  - trans l1. by apply Permutation_contains.
+    trans k1. done. by apply Permutation_contains.
+  - trans l2. by apply Permutation_contains.
+    trans k2. done. by apply Permutation_contains.
 Qed.
 Global Instance: AntiSymm (≡ₚ) (@contains A).
 Proof. red. auto using contains_Permutation_length_le, contains_length. Qed.
@@ -1842,9 +1842,9 @@ Proof.
     - intros x l1 l3 ? (l2&?&?). exists (x :: l2). by repeat constructor.
     - intros l1 l3 l5 ? (l2&?&?) ? (l4&?&?).
       destruct (Permutation_sublist l2 l3 l4) as (l3'&?&?); trivial.
-      exists l3'. split; etransitivity; eauto. }
+      exists l3'. split; etrans; eauto. }
   intros (l2&?&?).
-  transitivity l2; auto using sublist_contains, Permutation_contains.
+  trans l2; auto using sublist_contains, Permutation_contains.
 Qed.
 Lemma contains_sublist_r l1 l3 :
   l1 `contains` l3 ↔ ∃ l2, l1 ≡ₚ l2 ∧ l2 `sublist` l3.
@@ -1863,7 +1863,7 @@ Proof. rewrite !(comm (++) _ k). apply contains_skips_l. Qed.
 Lemma contains_app l1 l2 k1 k2 :
   l1 `contains` l2 → k1 `contains` k2 → l1 ++ k1 `contains` l2 ++ k2.
 Proof.
-  transitivity (l1 ++ k2); auto using contains_skips_l, contains_skips_r.
+  trans (l1 ++ k2); auto using contains_skips_l, contains_skips_r.
 Qed.
 Lemma contains_cons_r x l k :
   l `contains` x :: k ↔ l `contains` k ∨ ∃ l', l ≡ₚ x :: l' ∧ l' `contains` k.
@@ -1975,7 +1975,7 @@ Section contains_dec.
     - simplify_option_eq; eauto using Permutation_swap.
     - destruct (IH1 k1) as (k2&?&?); trivial.
       destruct (IH2 k2) as (k3&?&?); trivial.
-      exists k3. split; eauto. by transitivity k2.
+      exists k3. split; eauto. by trans k2.
   Qed.
   Lemma list_remove_Some l k x : list_remove x l = Some k → l ≡ₚ x :: k.
   Proof.
@@ -2493,7 +2493,7 @@ Section Forall2_order.
   Global Instance: Symmetric R → Symmetric (Forall2 R).
   Proof. intros. induction 1; constructor; auto. Qed.
   Global Instance: Transitive R → Transitive (Forall2 R).
-  Proof. intros ????. apply Forall2_transitive. by apply @transitivity. Qed.
+  Proof. intros ????. apply Forall2_transitive. by apply @trans. Qed.
   Global Instance: Equivalence R → Equivalence (Forall2 R).
   Proof. split; apply _. Qed.
   Global Instance: PreOrder R → PreOrder (Forall2 R).
@@ -2768,14 +2768,14 @@ Section bind.
     - by apply contains_app.
     - by rewrite !(assoc_L (++)), (comm (++) (f _)).
     - by apply contains_inserts_l.
-    - etransitivity; eauto.
+    - etrans; eauto.
   Qed.
   Global Instance bind_Permutation: Proper ((≡ₚ) ==> (≡ₚ)) (mbind f).
   Proof.
     induction 1; csimpl; auto.
     - by f_equiv.
     - by rewrite !(assoc_L (++)), (comm (++) (f _)).
-    - etransitivity; eauto.
+    - etrans; eauto.
   Qed.
   Lemma bind_cons x l : (x :: l) ≫= f = f x ++ l ≫= f.
   Proof. done. Qed.
@@ -2998,7 +2998,7 @@ Lemma foldr_permutation {A B} (R : relation B) `{!Equivalence R}
     (f : A → B → B) (b : B) `{!Proper ((=) ==> R ==> R) f}
     (Hf : ∀ a1 a2 b, R (f a1 (f a2 b)) (f a2 (f a1 b))) :
   Proper ((≡ₚ) ==> R) (foldr f b).
-Proof. induction 1; simpl; [done|by f_equiv|apply Hf|etransitivity; eauto]. Qed.
+Proof. induction 1; simpl; [done|by f_equiv|apply Hf|etrans; eauto]. Qed.
 
 (** ** Properties of the [zip_with] and [zip] functions *)
 Section zip_with.

theories/numbers.v

@@ -243,7 +243,7 @@ Proof.
   intros [??] ?.
   destruct (decide (y = 1)); subst; [rewrite Z.quot_1_r; auto |].
   destruct (decide (x = 0)); subst; [rewrite Z.quot_0_l; auto with lia |].
-  split. apply Z.quot_pos; lia. transitivity x; auto. apply Z.quot_lt; lia.
+  split. apply Z.quot_pos; lia. trans x; auto. apply Z.quot_lt; lia.
 Qed.
 
 (* Note that we cannot disable simpl for [Z.of_nat] as that would break
@@ -396,7 +396,7 @@ Lemma Qcplus_pos_pos (x y : Qc) : 0 < x → 0 < y → 0 < x + y.
 Proof. auto using Qcplus_pos_nonneg, Qclt_le_weak. Qed.
 Lemma Qcplus_nonneg_nonneg (x y : Qc) : 0 ≤ x → 0 ≤ y → 0 ≤ x + y.
 Proof.
-  intros. transitivity (x + 0); [by rewrite Qcplus_0_r|].
+  intros. trans (x + 0); [by rewrite Qcplus_0_r|].
   by apply Qcplus_le_mono_l.
 Qed.
 Lemma Qcplus_neg_nonpos (x y : Qc) : x < 0 → y ≤ 0 → x + y < 0.
@@ -410,7 +410,7 @@ Lemma Qcplus_neg_neg (x y : Qc) : x < 0 → y < 0 → x + y < 0.
 Proof. auto using Qcplus_nonpos_neg, Qclt_le_weak. Qed.
 Lemma Qcplus_nonpos_nonpos (x y : Qc) : x ≤ 0 → y ≤ 0 → x + y ≤ 0.
 Proof.
-  intros. transitivity (x + 0); [|by rewrite Qcplus_0_r].
+  intros. trans (x + 0); [|by rewrite Qcplus_0_r].
   by apply Qcplus_le_mono_l.
 Qed.
 Lemma Qcmult_le_mono_nonneg_l x y z : 0 ≤ z → x ≤ y → z * x ≤ z * y.
@@ -436,7 +436,7 @@ Proof.
 Qed.
 Lemma Qcmult_nonneg_nonneg x y : 0 ≤ x → 0 ≤ y → 0 ≤ x * y.
 Proof.
-  intros. transitivity (0 * y); [by rewrite Qcmult_0_l|].
+  intros. trans (0 * y); [by rewrite Qcmult_0_l|].
   by apply Qcmult_le_mono_nonneg_r.
 Qed.
 

theories/option.v

@@ -96,7 +96,7 @@ Section setoids.
     split.
     - by intros []; constructor.
     - by destruct 1; constructor.
-    - destruct 1; inversion 1; constructor; etransitivity; eauto.
+    - destruct 1; inversion 1; constructor; etrans; eauto.
   Qed.
   Global Instance Some_proper : Proper ((≡) ==> (≡)) (@Some A).
   Proof. by constructor. Qed.

theories/orders.v

@@ -29,13 +29,13 @@ Section orders.
   Proof. by intros [??] <-. Qed.
   Lemma strict_transitive_l `{!Transitive R} X Y Z : X ⊂ Y → Y ⊆ Z → X ⊂ Z.
   Proof.
-    intros [? HXY] ?. split; [by transitivity Y|].
-    contradict HXY. by transitivity Z.
+    intros [? HXY] ?. split; [by trans Y|].
+    contradict HXY. by trans Z.
   Qed.
   Lemma strict_transitive_r `{!Transitive R} X Y Z : X ⊆ Y → Y ⊂ Z → X ⊂ Z.
   Proof.
-    intros ? [? HYZ]. split; [by transitivity Y|].
-    contradict HYZ. by transitivity X.
+    intros ? [? HYZ]. split; [by trans Y|].
+    contradict HYZ. by trans X.
   Qed.
   Global Instance: Irreflexive (strict R).
   Proof. firstorder. Qed.
@@ -79,7 +79,7 @@ Section strict_orders.
   Proof. intros ->. apply (irreflexivity R). Qed.
   Lemma strict_anti_symm `{!StrictOrder R} X Y :
     X ⊂ Y → Y ⊂ X → False.
-  Proof. intros. apply (irreflexivity R X). by transitivity Y. Qed.
+  Proof. intros. apply (irreflexivity R X). by trans Y. Qed.
   Global Instance trichotomyT_dec `{!TrichotomyT R, !StrictOrder R} X Y :
       Decision (X ⊂ Y) :=
     match trichotomyT R X Y with
@@ -101,7 +101,7 @@ Ltac simplify_order := repeat
       assert (x = y) by (by apply (anti_symm R)); clear H1 H2
     | H2 : R y ?z |- _ =>
       unless (R x z) by done;
-      assert (R x z) by (by transitivity y)
+      assert (R x z) by (by trans y)
     end
   end.
 
@@ -319,13 +319,13 @@ Section preorder.
     split.
     - done.
     - by intros ?? [??].
-    - by intros X Y Z [??] [??]; split; transitivity Y.
+    - by intros X Y Z [??] [??]; split; trans Y.
   Qed.
   Global Instance: Proper ((≡) ==> (≡) ==> iff) ((⊆) : relation A).
   Proof.
     unfold equiv, preorder_equiv. intros X1 Y1 ? X2 Y2 ?. split; intro.
-    - transitivity X1. tauto. transitivity X2; tauto.
-    - transitivity Y1. tauto. transitivity Y2; tauto.
+    - trans X1. tauto. trans X2; tauto.
+    - trans Y1. tauto. trans Y2; tauto.
   Qed.
   Lemma subset_spec (X Y : A) : X ⊂ Y ↔ X ⊆ Y ∧ X ≢ Y.
   Proof.
@@ -376,9 +376,9 @@ Section join_semi_lattice.
 
   Hint Resolve subseteq_empty union_subseteq_l union_subseteq_r union_least.
   Lemma union_subseteq_l_transitive X1 X2 Y : X1 ⊆ X2 → X1 ⊆ X2 ∪ Y.
-  Proof. intros. transitivity X2; auto. Qed.
+  Proof. intros. trans X2; auto. Qed.
   Lemma union_subseteq_r_transitive X1 X2 Y : X1 ⊆ X2 → X1 ⊆ Y ∪ X2.
-  Proof. intros. transitivity X2; auto. Qed.
+  Proof. intros. trans X2; auto. Qed.
   Hint Resolve union_subseteq_l_transitive union_subseteq_r_transitive.
   Lemma union_preserving_l X Y1 Y2 : Y1 ⊆ Y2 → X ∪ Y1 ⊆ X ∪ Y2.
   Proof. auto. Qed.
@@ -436,7 +436,7 @@ Section join_semi_lattice.
   Proof.
     split.
     - intros HXY. split; apply equiv_empty;
-        by transitivity (X ∪ Y); [auto | rewrite HXY].
+        by trans (X ∪ Y); [auto | rewrite HXY].
     - intros [HX HY]. by rewrite HX, HY, (left_id _ _).
   Qed.
   Lemma empty_union_list Xs : ⋃ Xs ≡ ∅ ↔ Forall (≡ ∅) Xs.
@@ -502,9 +502,9 @@ Section meet_semi_lattice.
   Hint Resolve intersection_subseteq_l intersection_subseteq_r
     intersection_greatest.
   Lemma intersection_subseteq_l_transitive X1 X2 Y : X1 ⊆ X2 → X1 ∩ Y ⊆ X2.
-  Proof. intros. transitivity X1; auto. Qed.
+  Proof. intros. trans X1; auto. Qed.
   Lemma intersection_subseteq_r_transitive X1 X2 Y : X1 ⊆ X2 → Y ∩ X1 ⊆ X2.
-  Proof. intros. transitivity X1; auto. Qed.
+  Proof. intros. trans X1; auto. Qed.
   Hint Resolve intersection_subseteq_l_transitive
     intersection_subseteq_r_transitive.
   Lemma intersection_preserving_l X Y1 Y2 : Y1 ⊆ Y2 → X ∩ Y1 ⊆ X ∩ Y2.

theories/pretty.v

@@ -49,7 +49,7 @@ Proof.
     intros x; induction (N.lt_wf_0 x) as [x _ IH]; intros s.
     assert (x = 0 ∨ 0 < x)%N as [->|?] by lia; [by rewrite pretty_N_go_0|].
     rewrite pretty_N_go_step by done.
-    etransitivity; [|by eapply IH, N.div_lt]; simpl; lia. }
+    etrans; [|by eapply IH, N.div_lt]; simpl; lia. }
   intros x; induction (N.lt_wf_0 x) as [x _ IH]; intros y s s'.
   assert ((x = 0 ∨ 0 < x) ∧ (y = 0 ∨ 0 < y))%N as [[->|?] [->|?]] by lia;
     rewrite ?pretty_N_go_0, ?pretty_N_go_step, ?(pretty_N_go_step y) by done.

theories/relations.v

@@ -70,7 +70,7 @@ Section rtc.
   Lemma rtc_once x y : R x y → rtc R x y.
   Proof. eauto. Qed.
   Lemma rtc_r x y z : rtc R x y → R y z → rtc R x z.
-  Proof. intros. etransitivity; eauto. Qed.
+  Proof. intros. etrans; eauto. Qed.
   Lemma rtc_inv x z : rtc R x z → x = z ∨ ∃ y, R x y ∧ rtc R y z.
   Proof. inversion_clear 1; eauto. Qed.
   Lemma rtc_ind_l (P : A → Prop) (z : A)
@@ -152,7 +152,7 @@ Section rtc.
   Global Instance: Transitive (tc R).
   Proof. exact tc_transitive. Qed.
   Lemma tc_r x y z : tc R x y → R y z → tc R x z.
-  Proof. intros. etransitivity; eauto. Qed.
+  Proof. intros. etrans; eauto. Qed.
   Lemma tc_rtc_l x y z : rtc R x y → tc R y z → tc R x z.
   Proof. induction 1; eauto. Qed.
   Lemma tc_rtc_r x y z : tc R x y → rtc R y z → tc R x z.

theories/streams.v

@@ -42,7 +42,7 @@ Proof.
   split.
   - now cofix; intros [??]; constructor.
   - now cofix; intros ?? [??]; constructor.
-  - cofix; intros ??? [??] [??]; constructor; etransitivity; eauto.
+  - cofix; intros ??? [??] [??]; constructor; etrans; eauto.
 Qed.
 Global Instance scons_proper x : Proper ((≡) ==> (≡)) (scons x).
 Proof. by constructor. Qed.

theories/tactics.v

@@ -54,6 +54,10 @@ Ltac done :=
 Tactic Notation "by" tactic(tac) :=
   tac; done.
 
+(** Aliases for trans and etrans that are easier to type *)
+Tactic Notation "trans" constr(A) := transitivity A.
+Tactic Notation "etrans" := etransitivity.
+
 (** Tactics for splitting conjunctions:
 
 - [split_and] : split the goal if is syntactically of the shape [_ ∧ _]