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From prelude Require Export base tactics.

Section definitions.
  Context {A T : Type} `{ a b : A, Decision (a = b)}.
  Global Instance fn_insert : Insert A T (A  T) :=
    λ a t f b, if decide (a = b) then t else f b.
  Global Instance fn_alter : Alter A T (A  T) :=
    λ (g : T  T) a f b, if decide (a = b) then g (f a) else f b.
End definitions.

(* For now, we only have the properties here that do not need a notion
   of equality of functions. *)

Section functions.
  Context {A T : Type} `{ a b : A, Decision (a = b)}.

  Lemma fn_lookup_insert (f : A  T) a t : <[a:=t]>f a = t.
  Proof. unfold insert, fn_insert. by destruct (decide (a = a)). Qed.
  Lemma fn_lookup_insert_rev  (f : A  T) a t1 t2 :
    <[a:=t1]>f a = t2  t1 = t2.
  Proof. rewrite fn_lookup_insert. congruence. Qed.
  Lemma fn_lookup_insert_ne (f : A  T) a b t : a  b  <[a:=t]>f b = f b.
  Proof. unfold insert, fn_insert. by destruct (decide (a = b)). Qed.

  Lemma fn_lookup_alter (g : T  T) (f : A  T) a : alter g a f a = g (f a).
  Proof. unfold alter, fn_alter. by destruct (decide (a = a)). Qed.
  Lemma fn_lookup_alter_ne (g : T  T) (f : A  T) a b :
    a  b  alter g a f b = f b.
  Proof. unfold alter, fn_alter. by destruct (decide (a = b)). Qed.
End functions.