(* Title: HOL/ex/BT.thy ID: $Id: BT.thy,v 1.13 2007/06/04 13:43:32 haftmann Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1995 University of Cambridge Binary trees *) header {* Binary trees *} theory BT imports Main begin datatype 'a bt = Lf | Br 'a "'a bt" "'a bt" consts n_nodes :: "'a bt => nat" n_leaves :: "'a bt => nat" depth :: "'a bt => nat" reflect :: "'a bt => 'a bt" bt_map :: "('a => 'b) => ('a bt => 'b bt)" preorder :: "'a bt => 'a list" inorder :: "'a bt => 'a list" postorder :: "'a bt => 'a list" append :: "'a bt => 'a bt => 'a bt" primrec "n_nodes Lf = 0" "n_nodes (Br a t1 t2) = Suc (n_nodes t1 + n_nodes t2)" primrec "n_leaves Lf = Suc 0" "n_leaves (Br a t1 t2) = n_leaves t1 + n_leaves t2" primrec "depth Lf = 0" "depth (Br a t1 t2) = Suc (max (depth t1) (depth t2))" primrec "reflect Lf = Lf" "reflect (Br a t1 t2) = Br a (reflect t2) (reflect t1)" primrec "bt_map f Lf = Lf" "bt_map f (Br a t1 t2) = Br (f a) (bt_map f t1) (bt_map f t2)" primrec "preorder Lf = []" "preorder (Br a t1 t2) = [a] @ (preorder t1) @ (preorder t2)" primrec "inorder Lf = []" "inorder (Br a t1 t2) = (inorder t1) @ [a] @ (inorder t2)" primrec "postorder Lf = []" "postorder (Br a t1 t2) = (postorder t1) @ (postorder t2) @ [a]" primrec "append Lf t = t" "append (Br a t1 t2) t = Br a (append t1 t) (append t2 t)" text {* \medskip BT simplification *} lemma n_leaves_reflect: "n_leaves (reflect t) = n_leaves t" apply (induct t) apply auto done lemma n_nodes_reflect: "n_nodes (reflect t) = n_nodes t" apply (induct t) apply auto done lemma depth_reflect: "depth (reflect t) = depth t" apply (induct t) apply auto done text {* The famous relationship between the numbers of leaves and nodes. *} lemma n_leaves_nodes: "n_leaves t = Suc (n_nodes t)" apply (induct t) apply auto done lemma reflect_reflect_ident: "reflect (reflect t) = t" apply (induct t) apply auto done lemma bt_map_reflect: "bt_map f (reflect t) = reflect (bt_map f t)" apply (induct t) apply simp_all done lemma preorder_bt_map: "preorder (bt_map f t) = map f (preorder t)" apply (induct t) apply simp_all done lemma inorder_bt_map: "inorder (bt_map f t) = map f (inorder t)" apply (induct t) apply simp_all done lemma postorder_bt_map: "postorder (bt_map f t) = map f (postorder t)" apply (induct t) apply simp_all done lemma depth_bt_map [simp]: "depth (bt_map f t) = depth t" apply (induct t) apply simp_all done lemma n_leaves_bt_map [simp]: "n_leaves (bt_map f t) = n_leaves t" apply (induct t) apply (simp_all add: left_distrib) done lemma preorder_reflect: "preorder (reflect t) = rev (postorder t)" apply (induct t) apply simp_all done lemma inorder_reflect: "inorder (reflect t) = rev (inorder t)" apply (induct t) apply simp_all done lemma postorder_reflect: "postorder (reflect t) = rev (preorder t)" apply (induct t) apply simp_all done text {* Analogues of the standard properties of the append function for lists. *} lemma append_assoc [simp]: "append (append t1 t2) t3 = append t1 (append t2 t3)" apply (induct t1) apply simp_all done lemma append_Lf2 [simp]: "append t Lf = t" apply (induct t) apply simp_all done lemma depth_append [simp]: "depth (append t1 t2) = depth t1 + depth t2" apply (induct t1) apply (simp_all add: max_add_distrib_left) done lemma n_leaves_append [simp]: "n_leaves (append t1 t2) = n_leaves t1 * n_leaves t2" apply (induct t1) apply (simp_all add: left_distrib) done lemma bt_map_append: "bt_map f (append t1 t2) = append (bt_map f t1) (bt_map f t2)" apply (induct t1) apply simp_all done end
lemma n_leaves_reflect:
n_leaves (reflect t) = n_leaves t
lemma n_nodes_reflect:
n_nodes (reflect t) = n_nodes t
lemma depth_reflect:
depth (reflect t) = depth t
lemma n_leaves_nodes:
n_leaves t = Suc (n_nodes t)
lemma reflect_reflect_ident:
reflect (reflect t) = t
lemma bt_map_reflect:
bt_map f (reflect t) = reflect (bt_map f t)
lemma preorder_bt_map:
preorder (bt_map f t) = map f (preorder t)
lemma inorder_bt_map:
inorder (bt_map f t) = map f (inorder t)
lemma postorder_bt_map:
postorder (bt_map f t) = map f (postorder t)
lemma depth_bt_map:
depth (bt_map f t) = depth t
lemma n_leaves_bt_map:
n_leaves (bt_map f t) = n_leaves t
lemma preorder_reflect:
preorder (reflect t) = rev (postorder t)
lemma inorder_reflect:
inorder (reflect t) = rev (inorder t)
lemma postorder_reflect:
postorder (reflect t) = rev (preorder t)
lemma append_assoc:
BT.append (BT.append t1.0 t2.0) t3.0 = BT.append t1.0 (BT.append t2.0 t3.0)
lemma append_Lf2:
BT.append t Lf = t
lemma depth_append:
depth (BT.append t1.0 t2.0) = depth t1.0 + depth t2.0
lemma n_leaves_append:
n_leaves (BT.append t1.0 t2.0) = n_leaves t1.0 * n_leaves t2.0
lemma bt_map_append:
bt_map f (BT.append t1.0 t2.0) = BT.append (bt_map f t1.0) (bt_map f t2.0)