introduce [default] as abbreviation for [from_option id], and use it

theories/coPset.v

@@ -251,7 +251,7 @@ Fixpoint coPpick_raw (t : coPset_raw) : option positive :=
      | Some i => Some (i~0) | None => (~1) <$> coPpick_raw r
      end
   end.
-Definition coPpick (X : coPset) : positive := from_option id 1 (coPpick_raw (`X)).
+Definition coPpick (X : coPset) : positive := default 1 (coPpick_raw (`X)).
 
 Lemma coPpick_raw_elem_of t i : coPpick_raw t = Some i → e_of i t.
 Proof.

theories/finite.v

@@ -17,7 +17,7 @@ Definition card A `{Finite A} := length (enum A).
 
 Program Definition finite_countable `{Finite A} : Countable A := {|
   encode := λ x,
-    Pos.of_nat $ S $ from_option id 0 $ fst <$> list_find (x =) (enum A);
+    Pos.of_nat $ S $ default 0 $ fst <$> list_find (x =) (enum A);
   decode := λ p, enum A !! pred (Pos.to_nat p)
 |}.
 Arguments Pos.of_nat : simpl never.
@@ -134,7 +134,7 @@ 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 id x' (decode_nat (encode_nat y))).
+  exists (λ y : B, default x' (decode_nat (encode_nat y))).
   intros x. destruct (encode_decode B (encode_nat x)) as (y&Hy1&Hy2).
   { pose proof (encode_lt_card x); lia. }
   exists y. by rewrite Hy2, decode_encode_nat.

theories/gmap.v

@@ -131,7 +131,7 @@ Definition gmap_curry `{Countable K1, Countable K2} {A} :
 Definition gmap_uncurry `{Countable K1, Countable K2} {A} :
     gmap (K1 * K2) A → gmap K1 (gmap K2 A) :=
   map_fold (λ ii x, let '(i1,i2) := ii in
-    partial_alter (Some ∘ <[i2:=x]> ∘ from_option id ∅) i1) ∅.
+    partial_alter (Some ∘ <[i2:=x]> ∘ default ∅) i1) ∅.
 
 Section curry_uncurry.
   Context `{Countable K1, Countable K2} {A : Type}.

theories/list.v

@@ -484,7 +484,7 @@ Lemma list_lookup_middle l1 l2 x n :
   n = length l1 → (l1 ++ x :: l2) !! n = Some x.
 Proof. intros ->. by induction l1. Qed.
 
-Lemma nth_lookup l i d : nth i l d = from_option id d (l !! i).
+Lemma nth_lookup l i d : nth i l d = default d (l !! i).
 Proof. revert i. induction l as [|x l IH]; intros [|i]; simpl; auto. Qed.
 Lemma nth_lookup_Some l i d x : l !! i = Some x → nth i l d = x.
 Proof. rewrite nth_lookup. by intros ->. Qed.
@@ -3561,7 +3561,7 @@ Definition eval {A} (E : env A) : rlist nat → list A :=
   fix go t :=
   match t with
   | rnil => []
-  | rnode i => from_option id [] (E !! i)
+  | rnode i => default [] (E !! i)
   | rapp t1 t2 => go t1 ++ go t2
   end.
 
@@ -3595,7 +3595,7 @@ End quote.
 Section eval.
   Context {A} (E : env A).
 
-  Lemma eval_alt t : eval E t = to_list t ≫= from_option id [] ∘ (E !!).
+  Lemma eval_alt t : eval E t = to_list t ≫= default [] ∘ (E !!).
   Proof.
     induction t; csimpl.
     - done.

theories/option.v

@@ -26,6 +26,9 @@ Definition from_option {A B} (f : A → B) (y : B) (mx : option A) : B :=
 Instance: Params (@from_option) 3.
 Arguments from_option {_ _} _ _ !_ / : assert.
 
+(* The eliminator with the identity function. *)
+Notation default := (from_option id).
+
 (** An alternative, but equivalent, definition of equality on the option
 data type. This theorem is useful to prove that two options are the same. *)
 Lemma option_eq {A} (mx my: option A): mx = my ↔ ∀ x, mx = Some x ↔ my = Some x.