--- a/src/HOL/Metis_Examples/BT.thy Wed Sep 08 13:25:22 2010 +0200
+++ b/src/HOL/Metis_Examples/BT.thy Wed Sep 08 19:21:46 2010 +0200
@@ -15,53 +15,41 @@
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"
- appnd :: "'a bt => 'a bt => 'a bt"
+primrec n_nodes :: "'a bt => nat" where
+ "n_nodes Lf = 0"
+| "n_nodes (Br a t1 t2) = Suc (n_nodes t1 + n_nodes t2)"
+
+primrec n_leaves :: "'a bt => nat" where
+ "n_leaves Lf = Suc 0"
+| "n_leaves (Br a t1 t2) = n_leaves t1 + n_leaves t2"
-primrec
- "n_nodes Lf = 0"
- "n_nodes (Br a t1 t2) = Suc (n_nodes t1 + n_nodes t2)"
+primrec depth :: "'a bt => nat" where
+ "depth Lf = 0"
+| "depth (Br a t1 t2) = Suc (max (depth t1) (depth 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 :: "'a bt => 'a bt" where
+ "reflect Lf = Lf"
+| "reflect (Br a t1 t2) = Br a (reflect t2) (reflect t1)"
-primrec
- "reflect Lf = Lf"
- "reflect (Br a t1 t2) = Br a (reflect t2) (reflect t1)"
-
-primrec
+primrec bt_map :: "('a => 'b) => ('a bt => 'b bt)" where
"bt_map f Lf = Lf"
- "bt_map f (Br a t1 t2) = Br (f a) (bt_map f t1) (bt_map f t2)"
+| "bt_map f (Br a t1 t2) = Br (f a) (bt_map f t1) (bt_map f t2)"
-primrec
+primrec preorder :: "'a bt => 'a list" where
"preorder Lf = []"
- "preorder (Br a t1 t2) = [a] @ (preorder t1) @ (preorder t2)"
+| "preorder (Br a t1 t2) = [a] @ (preorder t1) @ (preorder t2)"
-primrec
+primrec inorder :: "'a bt => 'a list" where
"inorder Lf = []"
- "inorder (Br a t1 t2) = (inorder t1) @ [a] @ (inorder t2)"
+| "inorder (Br a t1 t2) = (inorder t1) @ [a] @ (inorder t2)"
-primrec
+primrec postorder :: "'a bt => 'a list" where
"postorder Lf = []"
- "postorder (Br a t1 t2) = (postorder t1) @ (postorder t2) @ [a]"
+| "postorder (Br a t1 t2) = (postorder t1) @ (postorder t2) @ [a]"
-primrec
- "appnd Lf t = t"
- "appnd (Br a t1 t2) t = Br a (appnd t1 t) (appnd t2 t)"
-
+primrec append :: "'a bt => 'a bt => 'a bt" where
+ "append Lf t = t"
+| "append (Br a t1 t2) t = Br a (append t1 t) (append t2 t)"
text {* \medskip BT simplification *}
@@ -135,12 +123,12 @@
apply (metis bt_map.simps(1))
by (metis bt_map.simps(2))
-declare [[ sledgehammer_problem_prefix = "BT__bt_map_appnd" ]]
+declare [[ sledgehammer_problem_prefix = "BT__bt_map_append" ]]
-lemma bt_map_appnd: "bt_map f (appnd t u) = appnd (bt_map f t) (bt_map f u)"
+lemma bt_map_append: "bt_map f (append t u) = append (bt_map f t) (bt_map f u)"
apply (induct t)
- apply (metis appnd.simps(1) bt_map.simps(1))
-by (metis appnd.simps(2) bt_map.simps(2))
+ apply (metis append.simps(1) bt_map.simps(1))
+by (metis append.simps(2) bt_map.simps(2))
declare [[ sledgehammer_problem_prefix = "BT__bt_map_compose" ]]
@@ -219,8 +207,8 @@
apply (induct t)
apply (metis Nil_is_rev_conv postorder.simps(1) preorder.simps(1)
reflect.simps(1))
-by (metis append.simps(1) append.simps(2) postorder.simps(2) preorder.simps(2)
- reflect.simps(2) rev.simps(2) rev_append rev_swap)
+apply simp
+done
declare [[ sledgehammer_problem_prefix = "BT__inorder_reflect" ]]
@@ -245,44 +233,44 @@
Analogues of the standard properties of the append function for lists.
*}
-declare [[ sledgehammer_problem_prefix = "BT__appnd_assoc" ]]
+declare [[ sledgehammer_problem_prefix = "BT__append_assoc" ]]
-lemma appnd_assoc [simp]: "appnd (appnd t1 t2) t3 = appnd t1 (appnd t2 t3)"
+lemma append_assoc [simp]: "append (append t1 t2) t3 = append t1 (append t2 t3)"
apply (induct t1)
- apply (metis appnd.simps(1))
-by (metis appnd.simps(2))
+ apply (metis append.simps(1))
+by (metis append.simps(2))
-declare [[ sledgehammer_problem_prefix = "BT__appnd_Lf2" ]]
+declare [[ sledgehammer_problem_prefix = "BT__append_Lf2" ]]
-lemma appnd_Lf2 [simp]: "appnd t Lf = t"
+lemma append_Lf2 [simp]: "append t Lf = t"
apply (induct t)
- apply (metis appnd.simps(1))
-by (metis appnd.simps(2))
+ apply (metis append.simps(1))
+by (metis append.simps(2))
declare max_add_distrib_left [simp]
-declare [[ sledgehammer_problem_prefix = "BT__depth_appnd" ]]
+declare [[ sledgehammer_problem_prefix = "BT__depth_append" ]]
-lemma depth_appnd [simp]: "depth (appnd t1 t2) = depth t1 + depth t2"
+lemma depth_append [simp]: "depth (append t1 t2) = depth t1 + depth t2"
apply (induct t1)
- apply (metis appnd.simps(1) depth.simps(1) plus_nat.simps(1))
+ apply (metis append.simps(1) depth.simps(1) plus_nat.simps(1))
by simp
-declare [[ sledgehammer_problem_prefix = "BT__n_leaves_appnd" ]]
+declare [[ sledgehammer_problem_prefix = "BT__n_leaves_append" ]]
-lemma n_leaves_appnd [simp]:
- "n_leaves (appnd t1 t2) = n_leaves t1 * n_leaves t2"
+lemma n_leaves_append [simp]:
+ "n_leaves (append t1 t2) = n_leaves t1 * n_leaves t2"
apply (induct t1)
- apply (metis appnd.simps(1) n_leaves.simps(1) nat_mult_1 plus_nat.simps(1)
+ apply (metis append.simps(1) n_leaves.simps(1) nat_mult_1 plus_nat.simps(1)
semiring_norm(111))
by (simp add: left_distrib)
-declare [[ sledgehammer_problem_prefix = "BT__bt_map_appnd" ]]
+declare [[ sledgehammer_problem_prefix = "BT__bt_map_append" ]]
-lemma (*bt_map_appnd:*)
- "bt_map f (appnd t1 t2) = appnd (bt_map f t1) (bt_map f t2)"
+lemma (*bt_map_append:*)
+ "bt_map f (append t1 t2) = append (bt_map f t1) (bt_map f t2)"
apply (induct t1)
- apply (metis appnd.simps(1) bt_map.simps(1))
-by (metis bt_map_appnd)
+ apply (metis append.simps(1) bt_map.simps(1))
+by (metis bt_map_append)
end