Inductive color : Set := Red | Black | Green.
Print color.

Inductive rb : color -> Prop :=
  | rbR : rb Red
  | rbB : rb Black.

Definition rbf c : bool :=
  match c with
  | Red => true
  | Black => true
  | _ => false
  end.

Lemma rb_rbf_by_destruct_inv : 
   forall c, rb c -> rbf c = true.
Proof.
 intros c r. destruct c; simpl; try reflexivity. 
inversion r.
Qed.

Lemma rb_rbf_by_case : 
   forall c, rb c -> rbf c = true.
Proof.
 intros c r. revert r. case c; simpl; try reflexivity. 
intro r. inversion r.
Qed.

(* The right way! *)
Lemma rb_rbf_by_clever_case : 
   forall c, rb c -> rbf c = true.
Proof.
 intros c r. case r; reflexivity.
Qed.

Print rb_rbf_by_clever_case.
Print rb_rbf_by_case.

Inductive isred : color -> Prop :=
  | rr : isred Red.

Lemma isred_rbf : 
   forall c, isred c -> rbf c = true.
Proof.
intros c i. case i. reflexivity.
Qed.

Section to_be_some_color.

Variable x: color.

Inductive isx : color -> Prop :=
  | ix : isx x.

Variable P : color -> Prop.

Hypothesis px : P x.

Lemma isx_P : 
   forall c, isx c -> P c.
Proof.
intros c i. case i. assumption.
Qed.

End to_be_some_color.

Print isx_P.

Print eq_ind.

Print eq.