(* An association list has unique keys if the keys appear only once *)
Inductive unique_key : list (A * B) -> Prop :=
  | unique_key_nil : unique_key nil
  | unique_key_cons :
      forall (a : A) (b : B) l,
      (forall b : B, ~ In (a, b) l) ->
      unique_key l -> unique_key ((a, b) :: l).(* An association list has unique keys if the keys appear only once *)
Inductive unique_key : list (A * B) -> Prop :=
  | unique_key_nil : unique_key nil
  | unique_key_cons :
      forall (a : A) (b : B) l,
      (forall b : B, ~ In (a, b) l) ->
      unique_key l -> unique_key ((a, b) :: l).(* An association list has unique keys if the keys appear only once *)
Inductive unique_key : list (A * B) -> Prop :=
  | unique_key_nil : unique_key nil
  | unique_key_cons :
      forall (a : A) (b : B) l,
      (forall b : B, ~ In (a, b) l) ->
      unique_key l -> unique_key ((a, b) :: l).

Theorem unique_key_in_inv : forall a l1 l2 l, unique_key l -> In (a, l1) l -> In (a, l2) l -> l1 = l2.