From e7b134373bb3d79b195d7e403551247fbff0d7e2 Mon Sep 17 00:00:00 2001 From: Robbert Krebbers Date: Mon, 18 Sep 2017 21:54:12 +0200 Subject: [PATCH] Prove that generic trees are countable. These trees are useful to show that other types are countable. --- theories/countable.v | 55 ++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 55 insertions(+) diff --git a/theories/countable.v b/theories/countable.v index 11abff2..be1886a 100644 --- a/theories/countable.v +++ b/theories/countable.v @@ -287,3 +287,58 @@ Next Obligation. intros [p Hp]. unfold mguard, option_guard; simpl. case_match; [|done]. f_equal. by apply Qp_eq. Qed. + +(** ** Generic trees *) +Close Scope positive. + +Inductive gen_tree (T : Type) : Type := + | GenLeaf : T → gen_tree T + | GenNode : nat → list (gen_tree T) → gen_tree T. +Arguments GenLeaf {_} _ : assert. +Arguments GenNode {_} _ _ : assert. + +Instance gen_tree_dec `{EqDecision T} : EqDecision (gen_tree T). +Proof. + refine ( + fix go t1 t2 := + match t1, t2 with + | GenLeaf x1, GenLeaf x2 => cast_if (decide (x1 = x2)) + | GenNode n1 ts1, GenNode n2 ts2 => + cast_if_and (decide (n1 = n2)) (decide (ts1 = ts2)) + | _, _ => right _ + end); abstract congruence. +Defined. + +Fixpoint gen_tree_to_list {T} (t : gen_tree T) : list (nat * nat + T) := + match t with + | GenLeaf x => [inr x] + | GenNode n ts => (ts ≫= gen_tree_to_list) ++ [inl (length ts, n)] + end. + +Fixpoint gen_tree_of_list {T} + (k : list (gen_tree T)) (l : list (nat * nat + T)) : option (gen_tree T) := + match l with + | [] => head k + | inr x :: l => gen_tree_of_list (GenLeaf x :: k) l + | inl (len,n) :: l => + gen_tree_of_list (GenNode n (reverse (take len k)) :: drop len k) l + end. + +Lemma gen_tree_of_to_list {T} k l (t : gen_tree T) : + gen_tree_of_list k (gen_tree_to_list t ++ l) = gen_tree_of_list (t :: k) l. +Proof. + revert t k l; fix 1; intros [|n ts] k l; simpl; auto. + trans (gen_tree_of_list (reverse ts ++ k) ([inl (length ts, n)] ++ l)). + - rewrite <-(assoc_L _). revert k. generalize ([inl (length ts, n)] ++ l). + induction ts as [|t ts'' IH]; intros k ts'''; csimpl; auto. + rewrite reverse_cons, <-!(assoc_L _), gen_tree_of_to_list; simpl; auto. + - simpl. by rewrite take_app_alt, drop_app_alt, reverse_involutive + by (by rewrite reverse_length). +Qed. + +Program Instance gen_tree_countable `{Countable T} : Countable (gen_tree T) := + inj_countable gen_tree_to_list (gen_tree_of_list []) _. +Next Obligation. + intros T ?? t. + by rewrite <-(right_id_L [] _ (gen_tree_to_list _)), gen_tree_of_to_list. +Qed. -- GitLab