Shorter names for common math notions.

Also do some minor clean up.

Shorter names for common math notions.
Also do some minor clean up.
44b18f4d · Robbert Krebbers · 7ebc1859 · 44b18f4d · 44b18f4d · 44b18f4d
Commit 44b18f4d authored Feb 11, 2016 by Robbert Krebbers
17 changed files
--- a/theories/base.v
+++ b/theories/base.v
--- a/theories/countable.v
+++ b/theories/countable.v
@@ -15,13 +15,13 @@ Definition encode_nat `{Countable A} (x : A) : nat :=
  pred (Pos.to_nat (encode x)).
 Definition decode_nat `{Countable A} (i : nat) : option A :=
  decode (Pos.of_nat (S i)).
-Instance encode_injective `{Countable A} : Injective (=) (=) encode.
+Instance encode_inj `{Countable A} : Inj (=) (=) encode.
 Proof.
-  intros x y Hxy; apply (injective Some).
+  intros x y Hxy; apply (inj Some).
  by rewrite <-(decode_encode x), Hxy, decode_encode.
 Qed.
-Instance encode_nat_injective `{Countable A} : Injective (=) (=) encode_nat.
-Proof. unfold encode_nat; intros x y Hxy; apply (injective encode); lia. Qed.
+Instance encode_nat_inj `{Countable A} : Inj (=) (=) encode_nat.
+Proof. unfold encode_nat; intros x y Hxy; apply (inj encode); lia. Qed.
 Lemma decode_encode_nat `{Countable A} x : decode_nat (encode_nat x) = Some x.
 Proof.
  pose proof (Pos2Nat.is_pos (encode x)).
@@ -70,11 +70,11 @@ Section choice.
  Definition choice (HA : ∃ x, P x) : { x | P x } := _↾choose_correct HA.
 End choice.

-Lemma surjective_cancel `{Countable A} `{∀ x y : B, Decision (x = y)}
-  (f : A → B) `{!Surjective (=) f} : { g : B → A & Cancel (=) f g }.
+Lemma surj_cancel `{Countable A} `{∀ x y : B, Decision (x = y)}
+  (f : A → B) `{!Surj (=) f} : { g : B → A & Cancel (=) f g }.
 Proof.
-  exists (λ y, choose (λ x, f x = y) (surjective f y)).
-  intros y. by rewrite (choose_correct (λ x, f x = y) (surjective f y)).
+  exists (λ y, choose (λ x, f x = y) (surj f y)).
+  intros y. by rewrite (choose_correct (λ x, f x = y) (surj f y)).
 Qed.

 (** * Instances *)
@@ -197,7 +197,7 @@ Lemma list_encode_app' `{Countable A} (l1 l2 : list A) acc :
 Proof.
  revert acc; induction l1; simpl; auto.
  induction l2 as [|x l IH]; intros acc; simpl; [by rewrite ?(left_id_L _ _)|].
-  by rewrite !(IH (Nat.iter _ _ _)), (associative_L _), x0_iter_x1.
+  by rewrite !(IH (Nat.iter _ _ _)), (assoc_L _), x0_iter_x1.
 Qed.
 Program Instance list_countable `{Countable A} : Countable (list A) :=
  {| encode := list_encode 1; decode := list_decode [] 0 |}.
@@ -211,7 +211,7 @@ Next Obligation.
  { by intros help l; rewrite help, (right_id_L _ _). }
  induction l as [|x l IH] using @rev_ind; intros acc; [done|].
  rewrite list_encode_app'; simpl; rewrite <-x0_iter_x1, decode_iter; simpl.
-  by rewrite decode_encode_nat; simpl; rewrite IH, <-(associative_L _).
+  by rewrite decode_encode_nat; simpl; rewrite IH, <-(assoc_L _).
 Qed.
 Lemma list_encode_app `{Countable A} (l1 l2 : list A) :
  encode (l1 ++ l2)%list = encode l1 ++ encode l2.

--- a/theories/decidable.v
+++ b/theories/decidable.v
@@ -12,7 +12,7 @@ Proof. firstorder. Qed.

 Lemma Is_true_reflect (b : bool) : reflect b b.
 Proof. destruct b. by left. right. intros []. Qed.
-Instance: Injective (=) (↔) Is_true.
+Instance: Inj (=) (↔) Is_true.
 Proof. intros [] []; simpl; intuition. Qed.

 (** We introduce [decide_rel] to avoid inefficienct computation due to eager

--- a/theories/error.v
+++ b/theories/error.v
@@ -47,7 +47,7 @@ Lemma error_fmap_bind {S E A B C} (f : A → B) (g : B → error S E C) x s :
  ((f <$> x) ≫= g) s = (x ≫= g ∘ f) s.
 Proof. by compute; destruct (x s) as [|[??]]. Qed.

-Lemma error_associative {S E A B C} (f : A → error S E B) (g : B → error S E C) x s :
+Lemma error_assoc {S E A B C} (f : A → error S E B) (g : B → error S E C) x s :
  ((x ≫= f) ≫= g) s = (a ← x; f a ≫= g) s.
 Proof. by compute; destruct (x s) as [|[??]]. Qed.
 Lemma error_of_option_bind {S E A B} (f : A → option B) o e :
@@ -114,7 +114,7 @@ Tactic Notation "error_proceed" :=
  | H : (gets _ ≫= _) _ = _ |- _ => rewrite error_left_gets in H
  | H : (modify _ ≫= _) _ = _ |- _ => rewrite error_left_modify in H
  | H : ((_ <$> _) ≫= _) _ = _ |- _ => rewrite error_fmap_bind in H
-  | H : ((_ ≫= _) ≫= _) _ = _ |- _ => rewrite error_associative in H
+  | H : ((_ ≫= _) ≫= _) _ = _ |- _ => rewrite error_assoc in H
  | H : (error_guard _ _ _) _ = _ |- _ =>
     let H' := fresh in apply error_guard_ret in H; destruct H as [H' H]
  | _ => progress simplify_equality'

--- a/theories/fin_collections.v
+++ b/theories/fin_collections.v
@@ -108,7 +108,7 @@ Lemma size_union_alt X Y : size (X ∪ Y) = size X + size (Y ∖ X).
 Proof.
  rewrite <-size_union by solve_elem_of.
  setoid_replace (Y ∖ X) with ((Y ∪ X) ∖ X) by solve_elem_of.
-  rewrite <-union_difference, (commutative (∪)); solve_elem_of.
+  rewrite <-union_difference, (comm (∪)); solve_elem_of.
 Qed.
 Lemma subseteq_size X Y : X ⊆ Y → size X ≤ size Y.
 Proof. intros. rewrite (union_difference X Y), size_union_alt by done. lia. Qed.

--- a/theories/fin_maps.v
+++ b/theories/fin_maps.v
@@ -820,28 +820,28 @@ Proof.
  intros ??. apply map_eq. intros.
  by rewrite !(lookup_merge f), lookup_empty, (right_id_L None f).
 Qed.
-Lemma merge_commutative m1 m2 :
+Lemma merge_comm m1 m2 :
  (∀ i, f (m1 !! i) (m2 !! i) = f (m2 !! i) (m1 !! i)) →
  merge f m1 m2 = merge f m2 m1.
 Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
-Global Instance: Commutative (=) f → Commutative (=) (merge f).
+Global Instance: Comm (=) f → Comm (=) (merge f).
 Proof.
-  intros ???. apply merge_commutative. intros. by apply (commutative f).
+  intros ???. apply merge_comm. intros. by apply (comm f).
 Qed.
-Lemma merge_associative m1 m2 m3 :
+Lemma merge_assoc m1 m2 m3 :
  (∀ i, f (m1 !! i) (f (m2 !! i) (m3 !! i)) =
        f (f (m1 !! i) (m2 !! i)) (m3 !! i)) →
  merge f m1 (merge f m2 m3) = merge f (merge f m1 m2) m3.
 Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
-Global Instance: Associative (=) f → Associative (=) (merge f).
+Global Instance: Assoc (=) f → Assoc (=) (merge f).
 Proof.
-  intros ????. apply merge_associative. intros. by apply (associative_L f).
+  intros ????. apply merge_assoc. intros. by apply (assoc_L f).
 Qed.
-Lemma merge_idempotent m1 :
+Lemma merge_idemp m1 :
  (∀ i, f (m1 !! i) (m1 !! i) = m1 !! i) → merge f m1 m1 = m1.
 Proof. intros. apply map_eq. intros. by rewrite !(lookup_merge f). Qed.
-Global Instance: Idempotent (=) f → Idempotent (=) (merge f).
-Proof. intros ??. apply merge_idempotent. intros. by apply (idempotent f). Qed.
+Global Instance: IdemP (=) f → IdemP (=) (merge f).
+Proof. intros ??. apply merge_idemp. intros. by apply (idemp f). Qed.
 End merge.

 Section more_merge.
@@ -1033,19 +1033,19 @@ Global Instance: LeftId (@eq (M A)) ∅ (union_with f).
 Proof. unfold union_with, map_union_with. apply _. Qed.
 Global Instance: RightId (@eq (M A)) ∅ (union_with f).
 Proof. unfold union_with, map_union_with. apply _. Qed.
-Lemma union_with_commutative m1 m2 :
+Lemma union_with_comm m1 m2 :
  (∀ i x y, m1 !! i = Some x → m2 !! i = Some y → f x y = f y x) →
  union_with f m1 m2 = union_with f m2 m1.
 Proof.
-  intros. apply (merge_commutative _). intros i.
+  intros. apply (merge_comm _). intros i.
  destruct (m1 !! i) eqn:?, (m2 !! i) eqn:?; simpl; eauto.
 Qed.
-Global Instance: Commutative (=) f → Commutative (@eq (M A)) (union_with f).
-Proof. intros ???. apply union_with_commutative. eauto. Qed.
-Lemma union_with_idempotent m :
+Global Instance: Comm (=) f → Comm (@eq (M A)) (union_with f).
+Proof. intros ???. apply union_with_comm. eauto. Qed.
+Lemma union_with_idemp m :
  (∀ i x, m !! i = Some x → f x x = Some x) → union_with f m m = m.
 Proof.
-  intros. apply (merge_idempotent _). intros i.
+  intros. apply (merge_idemp _). intros i.
  destruct (m !! i) eqn:?; simpl; eauto.
 Qed.
 Lemma alter_union_with (g : A → A) m1 m2 i :
@@ -1100,14 +1100,14 @@ End union_with.
 (** ** Properties of the [union] operation *)
 Global Instance: LeftId (@eq (M A)) ∅ (∪) := _.
 Global Instance: RightId (@eq (M A)) ∅ (∪) := _.
-Global Instance: Associative (@eq (M A)) (∪).
+Global Instance: Assoc (@eq (M A)) (∪).
 Proof.
  intros A m1 m2 m3. unfold union, map_union, union_with, map_union_with.
-  apply (merge_associative _). intros i.
+  apply (merge_assoc _). intros i.
  by destruct (m1 !! i), (m2 !! i), (m3 !! i).
 Qed.
-Global Instance: Idempotent (@eq (M A)) (∪).
-Proof. intros A ?. by apply union_with_idempotent. Qed.
+Global Instance: IdemP (@eq (M A)) (∪).
+Proof. intros A ?. by apply union_with_idemp. Qed.
 Lemma lookup_union_Some_raw {A} (m1 m2 : M A) i x :
  (m1 ∪ m2) !! i = Some x ↔
    m1 !! i = Some x ∨ (m1 !! i = None ∧ m2 !! i = Some x).
@@ -1140,9 +1140,9 @@ Proof. intro. rewrite lookup_union_Some_raw; intuition. Qed.
 Lemma lookup_union_Some_r {A} (m1 m2 : M A) i x :
  m1 ⊥ₘ m2 → m2 !! i = Some x → (m1 ∪ m2) !! i = Some x.
 Proof. intro. rewrite lookup_union_Some; intuition. Qed.
-Lemma map_union_commutative {A} (m1 m2 : M A) : m1 ⊥ₘ m2 → m1 ∪ m2 = m2 ∪ m1.
+Lemma map_union_comm {A} (m1 m2 : M A) : m1 ⊥ₘ m2 → m1 ∪ m2 = m2 ∪ m1.
 Proof.
-  intros Hdisjoint. apply (merge_commutative (union_with (λ x _, Some x))).
+  intros Hdisjoint. apply (merge_comm (union_with (λ x _, Some x))).
  intros i. specialize (Hdisjoint i).
  destruct (m1 !! i), (m2 !! i); compute; naive_solver.
 Qed.
@@ -1160,7 +1160,7 @@ Proof.
 Qed.
 Lemma map_union_subseteq_r {A} (m1 m2 : M A) : m1 ⊥ₘ m2 → m2 ⊆ m1 ∪ m2.
 Proof.
-  intros. rewrite map_union_commutative by done. by apply map_union_subseteq_l.
+  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.
@@ -1175,7 +1175,7 @@ Qed.
 Lemma map_union_preserving_r {A} (m1 m2 m3 : M A) :
  m2 ⊥ₘ m3 → m1 ⊆ m2 → m1 ∪ m3 ⊆ m2 ∪ m3.
 Proof.
-  intros. rewrite !(map_union_commutative _ m3)
+  intros. rewrite !(map_union_comm _ m3)
    by eauto using map_disjoint_weaken_l.
  by apply map_union_preserving_l.
 Qed.
@@ -1189,19 +1189,19 @@ Qed.
 Lemma map_union_reflecting_r {A} (m1 m2 m3 : M A) :
  m1 ⊥ₘ m3 → m2 ⊥ₘ m3 → m1 ∪ m3 ⊆ m2 ∪ m3 → m1 ⊆ m2.
 Proof.
-  intros ??. rewrite !(map_union_commutative _ m3) by done.
+  intros ??. rewrite !(map_union_comm _ m3) by done.
  by apply map_union_reflecting_l.
 Qed.
 Lemma map_union_cancel_l {A} (m1 m2 m3 : M A) :
  m1 ⊥ₘ m3 → m2 ⊥ₘ m3 → m3 ∪ m1 = m3 ∪ m2 → m1 = m2.
 Proof.
-  intros. apply (anti_symmetric (⊆));
+  intros. apply (anti_symm (⊆));
    apply map_union_reflecting_l with m3; auto using (reflexive_eq (R:=(⊆))).
 Qed.
 Lemma map_union_cancel_r {A} (m1 m2 m3 : M A) :
  m1 ⊥ₘ m3 → m2 ⊥ₘ m3 → m1 ∪ m3 = m2 ∪ m3 → m1 = m2.
 Proof.
-  intros. apply (anti_symmetric (⊆));
+  intros. apply (anti_symm (⊆));
    apply map_union_reflecting_r with m3; auto using (reflexive_eq (R:=(⊆))).
 Qed.
 Lemma map_disjoint_union_l {A} (m1 m2 m3 : M A) :
@@ -1231,7 +1231,7 @@ Qed.
 Lemma insert_union_singleton_r {A} (m : M A) i x :
  m !! i = None → <[i:=x]>m = m ∪ {[i ↦ x]}.
 Proof.
-  intro. rewrite insert_union_singleton_l, map_union_commutative; [done |].
+  intro. rewrite insert_union_singleton_l, map_union_comm; [done |].
  by apply map_disjoint_singleton_l.
 Qed.
 Lemma map_disjoint_insert_l {A} (m1 m2 : M A) i x :
@@ -1254,12 +1254,12 @@ Lemma map_disjoint_insert_r_2 {A} (m1 m2 : M A) i x :
 Proof. by rewrite map_disjoint_insert_r. Qed.
 Lemma insert_union_l {A} (m1 m2 : M A) i x :
  <[i:=x]>(m1 ∪ m2) = <[i:=x]>m1 ∪ m2.
-Proof. by rewrite !insert_union_singleton_l, (associative_L (∪)). Qed.
+Proof. by rewrite !insert_union_singleton_l, (assoc_L (∪)). Qed.
 Lemma insert_union_r {A} (m1 m2 : M A) i x :
  m1 !! i = None → <[i:=x]>(m1 ∪ m2) = m1 ∪ <[i:=x]>m2.
 Proof.
-  intro. rewrite !insert_union_singleton_l, !(associative_L (∪)).
-  rewrite (map_union_commutative m1); [done |].
+  intro. rewrite !insert_union_singleton_l, !(assoc_L (∪)).
+  rewrite (map_union_comm m1); [done |].
  by apply map_disjoint_singleton_r.
 Qed.
 Lemma foldr_insert_union {A} (m : M A) l :

--- a/theories/finite.v
+++ b/theories/finite.v
@@ -71,27 +71,27 @@ Proof.
  unfold card; intros. destruct finA as [[|x ?] ??]; simpl in *; [exfalso;lia|].
  constructor; exact x.
 Qed.
-Lemma finite_injective_contains `{finA: Finite A} `{finB: Finite B} (f: A → B)
-  `{!Injective (=) (=) f} : f <$> enum A `contains` enum B.
+Lemma finite_inj_contains `{finA: Finite A} `{finB: Finite B} (f: A → B)
+  `{!Inj (=) (=) f} : f <$> enum A `contains` enum B.
 Proof.
  intros. destruct finA, finB. apply NoDup_contains; auto using NoDup_fmap_2.
 Qed.
-Lemma finite_injective_Permutation `{Finite A} `{Finite B} (f : A → B)
-  `{!Injective (=) (=) f} : card A = card B → f <$> enum A ≡ₚ enum B.
+Lemma finite_inj_Permutation `{Finite A} `{Finite B} (f : A → B)
+  `{!Inj (=) (=) f} : card A = card B → f <$> enum A ≡ₚ enum B.
 Proof.
  intros. apply contains_Permutation_length_eq.
  * by rewrite fmap_length.
-  * by apply finite_injective_contains.
+  * by apply finite_inj_contains.
 Qed.
-Lemma finite_injective_surjective `{Finite A} `{Finite B} (f : A → B)
-  `{!Injective (=) (=) f} : card A = card B → Surjective (=) f.
+Lemma finite_inj_surj `{Finite A} `{Finite B} (f : A → B)
+  `{!Inj (=) (=) f} : card A = card B → Surj (=) f.
 Proof.
  intros HAB y. destruct (elem_of_list_fmap_2 f (enum A) y) as (x&?&?); eauto.
-  rewrite finite_injective_Permutation; auto using elem_of_enum.
+  rewrite finite_inj_Permutation; auto using elem_of_enum.
 Qed.

-Lemma finite_surjective A `{Finite A} B `{Finite B} :
-  0 < card A ≤ card B → ∃ g : B → A, Surjective (=) g.
+Lemma finite_surj A `{Finite A} B `{Finite B} :
+  0 < card A ≤ card B → ∃ g : B → A, Surj (=) g.
 Proof.
  intros [??]. destruct (finite_inhabited A) as [x']; auto with lia.
  exists (λ y : B, from_option x' (decode_nat (encode_nat y))).
@@ -99,38 +99,38 @@ Proof.
  { pose proof (encode_lt_card x); lia. }
  exists y. by rewrite Hy2, decode_encode_nat.
 Qed.
-Lemma finite_injective A `{Finite A} B `{Finite B} :
-  card A ≤ card B ↔ ∃ f : A → B, Injective (=) (=) f.
+Lemma finite_inj A `{Finite A} B `{Finite B} :
+  card A ≤ card B ↔ ∃ f : A → B, Inj (=) (=) f.
 Proof.
  split.
  * intros. destruct (decide (card A = 0)) as [HA|?].
    { exists (card_0_inv B HA). intros y. apply (card_0_inv _ HA y). }
-    destruct (finite_surjective A B) as (g&?); auto with lia.
-    destruct (surjective_cancel g) as (f&?). exists f. apply cancel_injective.
+    destruct (finite_surj A B) as (g&?); auto with lia.
+    destruct (surj_cancel g) as (f&?). exists f. apply cancel_inj.
  * intros [f ?]. unfold card. rewrite <-(fmap_length f).
-    by apply contains_length, (finite_injective_contains f).
+    by apply contains_length, (finite_inj_contains f).
 Qed.
 Lemma finite_bijective A `{Finite A} B `{Finite B} :
-  card A = card B ↔ ∃ f : A → B, Injective (=) (=) f ∧ Surjective (=) f.
+  card A = card B ↔ ∃ f : A → B, Inj (=) (=) f ∧ Surj (=) f.
 Proof.
  split.
-  * intros; destruct (proj1 (finite_injective A B)) as [f ?]; auto with lia.
-    exists f; auto using (finite_injective_surjective f).
-  * intros (f&?&?). apply (anti_symmetric (≤)); apply finite_injective.
+  * intros; destruct (proj1 (finite_inj A B)) as [f ?]; auto with lia.
+    exists f; auto using (finite_inj_surj f).
+  * intros (f&?&?). apply (anti_symm (≤)); apply finite_inj.
    + by exists f.
-    + destruct (surjective_cancel f) as (g&?); eauto using cancel_injective.
+    + destruct (surj_cancel f) as (g&?); eauto using cancel_inj.
 Qed.
-Lemma injective_card `{Finite A} `{Finite B} (f : A → B)
-  `{!Injective (=) (=) f} : card A ≤ card B.
-Proof. apply finite_injective. eauto. Qed.
-Lemma surjective_card `{Finite A} `{Finite B} (f : A → B)
-  `{!Surjective (=) f} : card B ≤ card A.
+Lemma inj_card `{Finite A} `{Finite B} (f : A → B)
+  `{!Inj (=) (=) f} : card A ≤ card B.
+Proof. apply finite_inj. eauto. Qed.
+Lemma surj_card `{Finite A} `{Finite B} (f : A → B)
+  `{!Surj (=) f} : card B ≤ card A.
 Proof.
-  destruct (surjective_cancel f) as (g&?).
-  apply injective_card with g, cancel_injective.
+  destruct (surj_cancel f) as (g&?).
+  apply inj_card with g, cancel_inj.
 Qed.
 Lemma bijective_card `{Finite A} `{Finite B} (f : A → B)
-  `{!Injective (=) (=) f} `{!Surjective (=) f} : card A = card B.
+  `{!Inj (=) (=) f} `{!Surj (=) f} : card A = card B.
 Proof. apply finite_bijective. eauto. Qed.

 (** Decidability of quantification over finite types *)
@@ -180,7 +180,7 @@ End enc_finite.

 Section bijective_finite.
  Context `{Finite A, ∀ x y : B, Decision (x = y)} (f : A → B) (g : B → A).
-  Context `{!Injective (=) (=) f, !Cancel (=) f g}.
+  Context `{!Inj (=) (=) f, !Cancel (=) f g}.

  Program Instance bijective_finite: Finite B := {| enum := f <$> enum A |}.
  Next Obligation. apply (NoDup_fmap_2 _), NoDup_enum. Qed.

--- a/theories/gmap.v
+++ b/theories/gmap.v
@@ -89,7 +89,7 @@ Proof.
  * done.
  * intros A f [m Hm] i; apply (lookup_partial_alter f m).
  * intros A f [m Hm] i j Hs; apply (lookup_partial_alter_ne f m).
-    by contradict Hs; apply (injective encode).
+    by contradict Hs; apply (inj encode).
  * intros A B f [m Hm] i; apply (lookup_fmap f m).
  * intros A [m Hm]; unfold map_to_list; simpl.
    apply bool_decide_unpack, map_Forall_to_list in Hm; revert Hm.

--- a/theories/list.v
+++ b/theories/list.v
--- a/theories/numbers.v
+++ b/theories/numbers.v
@@ -35,7 +35,7 @@ Instance nat_eq_dec: ∀ x y : nat, Decision (x = y) := eq_nat_dec.
 Instance nat_le_dec: ∀ x y : nat, Decision (x ≤ y) := le_dec.
 Instance nat_lt_dec: ∀ x y : nat, Decision (x < y) := lt_dec.
 Instance nat_inhabited: Inhabited nat := populate 0%nat.
-Instance: Injective (=) (=) S.
+Instance: Inj (=) (=) S.
 Proof. by injection 1. Qed.
 Instance: PartialOrder (≤).
 Proof. repeat split; repeat intro; auto with lia. Qed.
@@ -110,9 +110,9 @@ Instance positive_inhabited: Inhabited positive := populate 1.

 Instance maybe_xO : Maybe xO := λ p, match p with p~0 => Some p | _ => None end.
 Instance maybe_x1 : Maybe xI := λ p, match p with p~1 => Some p | _ => None end.
-Instance: Injective (=) (=) (~0).
+Instance: Inj (=) (=) (~0).
 Proof. by injection 1. Qed.
-Instance: Injective (=) (=) (~1).
+Instance: Inj (=) (=) (~1).
 Proof. by injection 1. Qed.

 (** Since [positive] represents lists of bits, we define list operations
@@ -141,9 +141,9 @@ Global Instance: LeftId (=) 1 (++).
 Proof. intros p. by induction p; intros; f_equal'. Qed.
 Global Instance: RightId (=) 1 (++).
 Proof. done. Qed.
-Global Instance: Associative (=) (++).
+Global Instance: Assoc (=) (++).
 Proof. intros ?? p. by induction p; intros; f_equal'. Qed.
-Global Instance: ∀ p : positive, Injective (=) (=) (++ p).
+Global Instance: ∀ p : positive, Inj (=) (=) (++ p).
 Proof. intros p ???. induction p; simplify_equality; auto. Qed.

 Lemma Preverse_go_app p1 p2 p3 :
@@ -184,7 +184,7 @@ Infix "`mod`" := N.modulo (at level 35) : N_scope.

 Arguments N.add _ _ : simpl never.

-Instance: Injective (=) (=) Npos.
+Instance: Inj (=) (=) Npos.
 Proof. by injection 1. Qed.

 Instance N_eq_dec: ∀ x y : N, Decision (x = y) := N.eq_dec.
@@ -220,9 +220,9 @@ Infix "`rem`" := Z.rem (at level 35) : Z_scope.
 Infix "≪" := Z.shiftl (at level 35) : Z_scope.
 Infix "≫" := Z.shiftr (at level 35) : Z_scope.

-Instance: Injective (=) (=) Zpos.
+Instance: Inj (=) (=) Zpos.
 Proof. by injection 1. Qed.
-Instance: Injective (=) (=) Zneg.
+Instance: Inj (=) (=) Zneg.
 Proof. by injection 1. Qed.

 Instance Z_eq_dec: ∀ x y : Z, Decision (x = y) := Z.eq_dec.
@@ -371,18 +371,18 @@ Lemma Qcplus_lt_mono_l (x y z : Qc) : x < y ↔ z + x < z + y.
 Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_l. Qed.
 Lemma Qcplus_lt_mono_r (x y z : Qc) : x < y ↔ x + z < y + z.
 Proof. by rewrite !Qclt_nge, <-Qcplus_le_mono_r. Qed.
-Instance: Injective (=) (=) Qcopp.
+Instance: Inj (=) (=) Qcopp.
 Proof.
  intros x y H. by rewrite <-(Qcopp_involutive x), H, Qcopp_involutive.
 Qed.
-Instance: ∀ z, Injective (=) (=) (Qcplus z).
+Instance: ∀ z, Inj (=) (=) (Qcplus z).
 Proof.
-  intros z x y H. by apply (anti_symmetric (≤));
+  intros z x y H. by apply (anti_symm (≤));
    rewrite (Qcplus_le_mono_l _ _ z), H.
 Qed.
-Instance: ∀ z, Injective (=) (=) (λ x, x + z).
+Instance: ∀ z, Inj (=) (=) (λ x, x + z).
 Proof.
-  intros z x y H. by apply (anti_symmetric (≤));
+  intros z x y H. by apply (anti_symm (≤));
    rewrite (Qcplus_le_mono_r _ _ z), H.
 Qed.
 Lemma Qcplus_pos_nonneg (x y : Qc) : 0 < x → 0 ≤ y → 0 < x + y.

--- a/theories/option.v
+++ b/theories/option.v
@@ -16,7 +16,7 @@ Lemma Some_ne_None {A} (a : A) : Some a ≠ None.
 Proof. congruence. Qed.
 Lemma eq_None_ne_Some {A} (x : option A) a : x = None → x ≠ Some a.
 Proof. congruence. Qed.
-Instance Some_inj {A} : Injective (=) (=) (@Some A).
+Instance Some_inj {A} : Inj (=) (=) (@Some A).
 Proof. congruence. Qed.

 (** The non dependent elimination principle on the option type. *)
@@ -232,14 +232,14 @@ Section option_union_intersection_difference.
  Proof. by intros [?|]. Qed.
  Global Instance: RightId (=) None (union_with f).
  Proof. by intros [?|]. Qed.
-  Global Instance: Commutative (=) f → Commutative (=) (union_with f).
-  Proof. by intros ? [?|] [?|]; compute; rewrite 1?(commutative f). Qed.
+  Global Instance: Comm (=) f → Comm (=) (union_with f).
+  Proof. by intros ? [?|] [?|]; compute; rewrite 1?(comm f). Qed.
  Global Instance: LeftAbsorb (=) None (intersection_with f).
  Proof. by intros [?|]. Qed.
  Global Instance: RightAbsorb (=) None (intersection_with f).
  Proof. by intros [?|]. Qed.
-  Global Instance: Commutative (=) f → Commutative (=) (intersection_with f).
-  Proof. by intros ? [?|] [?|]; compute; rewrite 1?(commutative f). Qed.
+  Global Instance: Comm (=) f → Comm (=) (intersection_with f).
+  Proof. by intros ? [?|] [?|]; compute; rewrite 1?(comm f). Qed.
  Global Instance: RightId (=) None (difference_with f).
  Proof. by intros [?|]. Qed.
 End option_union_intersection_difference.

--- a/theories/orders.v
+++ b/theories/orders.v
--- a/theories/pmap.v
+++ b/theories/pmap.v
@@ -178,14 +178,14 @@ Proof.
    * rewrite elem_of_cons. intros [?|?]; simplify_equality.
      { left; exists 1. by rewrite (left_id_L 1 (++))%positive. }
      destruct (IHl (j~0) (Pto_list_raw j~1 r acc)) as [(i'&->&?)|?]; auto.
-      { left; exists (i' ~ 0). by rewrite Preverse_xO, (associative_L _). }
+      { left; exists (i' ~ 0). by rewrite Preverse_xO, (assoc_L _). }
      destruct (IHr (j~1) acc) as [(i'&->&?)|?]; auto.
-      left; exists (i' ~ 1). by rewrite Preverse_xI, (associative_L _).
+      left; exists (i' ~ 1). by rewrite Preverse_xI, (assoc_L _).
    * intros.
      destruct (IHl (j~0) (Pto_list_raw j~1 r acc)) as [(i'&->&?)|?]; auto.
-      { left; exists (i' ~ 0). by rewrite Preverse_xO, (associative_L _). }
+      { left; exists (i' ~ 0). by rewrite Preverse_xO, (assoc_L _). }
      destruct (IHr (j~1) acc) as [(i'&->&?)|?]; auto.
-      left; exists (i' ~ 1). by rewrite Preverse_xI, (associative_L _). }
+      left; exists (i' ~ 1). by rewrite Preverse_xI, (assoc_L _). }
  revert t j i acc. assert (∀ t j i acc,
    (i, x) ∈ acc → (i, x) ∈ Pto_list_raw j t acc) as help.
  { intros t; induction t as [|[y|] l IHl r IHr]; intros j i acc;
@@ -195,15 +195,15 @@ Proof.
  * done.
  * rewrite elem_of_cons. destruct i as [i|i|]; simplify_equality'.
    + right. apply help. specialize (IHr (j~1) i).
-      rewrite Preverse_xI, (associative_L _) in IHr. by apply IHr.
+      rewrite Preverse_xI, (assoc_L _) in IHr. by apply IHr.
    + right. specialize (IHl (j~0) i).
-      rewrite Preverse_xO, (associative_L _) in IHl. by apply IHl.
+      rewrite Preverse_xO, (assoc_L _) in IHl. by apply IHl.
    + left. by rewrite (left_id_L 1 (++))%positive.
  * destruct i as [i|i|]; simplify_equality'.
    + apply help. specialize (IHr (j~1) i).
-      rewrite Preverse_xI, (associative_L _) in IHr. by apply IHr.
+      rewrite Preverse_xI, (assoc_L _) in IHr. by apply IHr.
    + specialize (IHl (j~0) i).
-      rewrite Preverse_xO, (associative_L _) in IHl. by apply IHl.
+      rewrite Preverse_xO, (assoc_L _) in IHl. by apply IHl.
 Qed.
 Lemma Pto_list_nodup {A} j (t : Pmap_raw A) acc :
  (∀ i x, (i ++ Preverse j, x) ∈ acc → t !! i = None) →
@@ -222,18 +222,18 @@ Proof.
      discriminate (Hin Hj). }
   apply IHl.
   { intros i x. rewrite Pelem_of_to_list. intros [(?&Hi&?)|Hi].
-     + rewrite Preverse_xO, Preverse_xI, !(associative_L _) in Hi.
-       by apply (injective (++ _)) in Hi.
-     + apply (Hin (i~0) x). by rewrite Preverse_xO, (associative_L _) in Hi. }
+     + rewrite Preverse_xO, Preverse_xI, !(assoc_L _) in Hi.
+       by apply (inj (++ _)) in Hi.
+     + apply (Hin (i~0) x). by rewrite Preverse_xO, (assoc_L _) in Hi. }
   apply IHr; auto. intros i x Hi.
-   apply (Hin (i~1) x). by rewrite Preverse_xI, (associative_L _) in Hi.
+   apply (Hin (i~1) x). by rewrite Preverse_xI, (assoc_L _) in Hi.
 * apply IHl.
   { intros i x. rewrite Pelem_of_to_list. intros [(?&Hi&?)|Hi].
-     + rewrite Preverse_xO, Preverse_xI, !(associative_L _) in Hi.
-       by apply (injective (++ _)) in Hi.
-     + apply (Hin (i~0) x). by rewrite Preverse_xO, (associative_L _) in Hi. }
+     + rewrite Preverse_xO, Preverse_xI, !(assoc_L _) in Hi.
+       by apply (inj (++ _)) in Hi.
+     + apply (Hin (i~0) x). by rewrite Preverse_xO, (assoc_L _) in Hi. }
   apply IHr; auto. intros i x Hi.
-   apply (Hin (i~1