Inductive is_redblack : tree -> color -> nat -> Prop :=
 | IsRB_leaf: forall c, is_redblack EE c 0
 | IsRB_r: forall tl k tr n,
          is_redblack tl Red n ->
          is_redblack tr Red n ->
          is_redblack (T Red tl k tr) Black n
 | IsRB_b: forall c tl k tr n,
          is_redblack tl Black n ->
          is_redblack tr Black n ->
          is_redblack (T Black tl k tr) c (S n).Inductive is_redblack : tree -> color -> nat -> Prop :=
 | IsRB_leaf: forall c, is_redblack EE c 0
 | IsRB_r: forall tl k tr n,
          is_redblack tl Red n ->
          is_redblack tr Red n ->
          is_redblack (T Red tl k tr) Black n
 | IsRB_b: forall c tl k tr n,
          is_redblack tl Black n ->
          is_redblack tr Black n ->
          is_redblack (T Black tl k tr) c (S n).Inductive is_redblack : tree -> color -> nat -> Prop :=
 | IsRB_leaf: forall c, is_redblack EE c 0
 | IsRB_r: forall tl k tr n,
          is_redblack tl Red n ->
          is_redblack tr Red n ->
          is_redblack (T Red tl k tr) Black n
 | IsRB_b: forall c tl k tr n,
          is_redblack tl Black n ->
          is_redblack tr Black n ->
          is_redblack (T Black tl k tr) c (S n).Inductive is_redblack : tree -> color -> nat -> Prop :=
 | IsRB_leaf: forall c, is_redblack EE c 0
 | IsRB_r: forall tl k tr n,
          is_redblack tl Red n ->
          is_redblack tr Red n ->
          is_redblack (T Red tl k tr) Black n
 | IsRB_b: forall c tl k tr n,
          is_redblack tl Black n ->
          is_redblack tr Black n ->
          is_redblack (T Black tl k tr) c (S n).

Lemma is_redblack_Black_to_Red: forall s n, is_redblack s Black n -> exists n, is_redblack (makeBlack s) Red n.