Module BAF := BA_Facts BA.
Include BAF.
(* Now, we prove the axioms about relativization. Again, this mostly involves packing and unpacking the canonTree type and pushing around mkCanon. *)
Definition rel (a x:t) : t.
case_eq (proj1_sig x); intros.
destruct b.
exact a.
exact (exist (fun t => canonicalTree t) (Leaf false) I).
refine (exist _ (relativ_tree (proj1_sig a) (proj1_sig x)) _).
apply relativ_almost_canon.
apply (proj2_sig a).
apply (proj2_sig x).
destruct x; simpl in *; subst; simpl in *; intuition.
Defined.
(* Axioms about splittability. These follow easily from relativization. *)
Definition leftTree : canonTree.
exists (Node (Leaf true) (Leaf false)).
simpl; intuition.
Defined.
Definition rightTree : canonTree.
exists (Node (Leaf false) (Leaf true)).
simpl; intuition.
Defined.
Definition split (x:canonTree) := (rel x leftTree, rel x rightTree).
(* 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).
Definition isTokenFactory (x:t) (n:nat) := isTokenFactory' (proj1_sig x) n.
Definition isToken (x:t) (n:nat) := isToken' (proj1_sig x) n.

Lemma isToken_canon : forall n tok,
isToken' tok n -> canonicalTree tok.