(* Token Factory definitions and proofs.
       We specify token factories and tokens
       inductively on trees, which makes
       the proofs fairly straightforward.
     *)

    Inductive isTokenFactory' : ShareTree -> nat -> Prop :=
      | isTokFac_0 : isTokenFactory' (Leaf true) O
      | isTokFac_S_true : forall t n,
          isTokenFactory' t (S n) ->
          isTokenFactory' (Node (Leaf true) t) (S n)
      | isTokFac_S_false : forall t n,
          isTokenFactory' t n ->
          isTokenFactory' (Node (Leaf false) t) (S n).(* Token Factory definitions and proofs.
       We specify token factories and tokens
       inductively on trees, which makes
       the proofs fairly straightforward.
     *)

    Inductive isTokenFactory' : ShareTree -> nat -> Prop :=
      | isTokFac_0 : isTokenFactory' (Leaf true) O
      | isTokFac_S_true : forall t n,
          isTokenFactory' t (S n) ->
          isTokenFactory' (Node (Leaf true) t) (S n)
      | isTokFac_S_false : forall t n,
          isTokenFactory' t n ->
          isTokenFactory' (Node (Leaf false) t) (S n).(* Token Factory definitions and proofs.
       We specify token factories and tokens
       inductively on trees, which makes
       the proofs fairly straightforward.
     *)

    Inductive isTokenFactory' : ShareTree -> nat -> Prop :=
      | isTokFac_0 : isTokenFactory' (Leaf true) O
      | isTokFac_S_true : forall t n,
          isTokenFactory' t (S n) ->
          isTokenFactory' (Node (Leaf true) t) (S n)
      | isTokFac_S_false : forall t n,
          isTokenFactory' t n ->
          isTokenFactory' (Node (Leaf false) t) (S n).(* Token Factory definitions and proofs.
       We specify token factories and tokens
       inductively on trees, which makes
       the proofs fairly straightforward.
     *)

    Inductive isTokenFactory' : ShareTree -> nat -> Prop :=
      | isTokFac_0 : isTokenFactory' (Leaf true) O
      | isTokFac_S_true : forall t n,
          isTokenFactory' t (S n) ->
          isTokenFactory' (Node (Leaf true) t) (S n)
      | isTokFac_S_false : forall t n,
          isTokenFactory' t n ->
          isTokenFactory' (Node (Leaf false) t) (S n).Inductive isToken' : ShareTree -> nat -> Prop :=
      | isTok_0 : isToken' (Leaf false) O
      | isTok_S_true : forall t n,
          isToken' t n ->
          isToken' (Node (Leaf true) t) (S n)
      | isTok_S_false : forall t n,
          isToken' t (S n) ->
          isToken' (Node (Leaf false) t) (S n).Inductive isToken' : ShareTree -> nat -> Prop :=
      | isTok_0 : isToken' (Leaf false) O
      | isTok_S_true : forall t n,
          isToken' t n ->
          isToken' (Node (Leaf true) t) (S n)
      | isTok_S_false : forall t n,
          isToken' t (S n) ->
          isToken' (Node (Leaf false) t) (S n).Inductive isToken' : ShareTree -> nat -> Prop :=
      | isTok_0 : isToken' (Leaf false) O
      | isTok_S_true : forall t n,
          isToken' t n ->
          isToken' (Node (Leaf true) t) (S n)
      | isTok_S_false : forall t n,
          isToken' t (S n) ->
          isToken' (Node (Leaf false) t) (S n).Inductive isToken' : ShareTree -> nat -> Prop :=
      | isTok_0 : isToken' (Leaf false) O
      | isTok_S_true : forall t n,
          isToken' t n ->
          isToken' (Node (Leaf true) t) (S n)
      | isTok_S_false : forall t n,
          isToken' t (S n) ->
          isToken' (Node (Leaf false) t) (S n).Definition isTokenFactory (x:t) (n:nat) := isTokenFactory' (proj1_sig x) n.Definition isToken (x:t) (n:nat) := isToken' (proj1_sig x) n.

(* Specifications: *)
Lemma isTokenFactory_canon : forall n fac, isTokenFactory' fac n -> canonicalTree fac.
Lemma isToken_canon : forall n tok, isToken' tok n -> canonicalTree tok.