--- a/src/HOL/MetisExamples/BT.thy Wed Oct 21 16:54:04 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,243 +0,0 @@
-(* Title: HOL/MetisTest/BT.thy
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory
-
-Testing the metis method
-*)
-
-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"
- appnd :: "'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
- "appnd Lf t = t"
- "appnd (Br a t1 t2) t = Br a (appnd t1 t) (appnd t2 t)"
-
-
-text {* \medskip BT simplification *}
-
-declare [[ atp_problem_prefix = "BT__n_leaves_reflect" ]]
-lemma n_leaves_reflect: "n_leaves (reflect t) = n_leaves t"
- apply (induct t)
- apply (metis add_right_cancel n_leaves.simps(1) reflect.simps(1))
- apply (metis add_commute n_leaves.simps(2) reflect.simps(2))
- done
-
-declare [[ atp_problem_prefix = "BT__n_nodes_reflect" ]]
-lemma n_nodes_reflect: "n_nodes (reflect t) = n_nodes t"
- apply (induct t)
- apply (metis reflect.simps(1))
- apply (metis n_nodes.simps(2) nat_add_commute reflect.simps(2))
- done
-
-declare [[ atp_problem_prefix = "BT__depth_reflect" ]]
-lemma depth_reflect: "depth (reflect t) = depth t"
- apply (induct t)
- apply (metis depth.simps(1) reflect.simps(1))
- apply (metis depth.simps(2) min_max.sup_commute reflect.simps(2))
- done
-
-text {*
- The famous relationship between the numbers of leaves and nodes.
-*}
-
-declare [[ atp_problem_prefix = "BT__n_leaves_nodes" ]]
-lemma n_leaves_nodes: "n_leaves t = Suc (n_nodes t)"
- apply (induct t)
- apply (metis n_leaves.simps(1) n_nodes.simps(1))
- apply auto
- done
-
-declare [[ atp_problem_prefix = "BT__reflect_reflect_ident" ]]
-lemma reflect_reflect_ident: "reflect (reflect t) = t"
- apply (induct t)
- apply (metis add_right_cancel reflect.simps(1));
- apply (metis reflect.simps(2))
- done
-
-declare [[ atp_problem_prefix = "BT__bt_map_ident" ]]
-lemma bt_map_ident: "bt_map (%x. x) = (%y. y)"
-apply (rule ext)
-apply (induct_tac y)
- apply (metis bt_map.simps(1))
-txt{*BUG involving flex-flex pairs*}
-(* apply (metis bt_map.simps(2)) *)
-apply auto
-done
-
-
-declare [[ atp_problem_prefix = "BT__bt_map_appnd" ]]
-lemma bt_map_appnd: "bt_map f (appnd t u) = appnd (bt_map f t) (bt_map f u)"
-apply (induct t)
- apply (metis appnd.simps(1) bt_map.simps(1))
- apply (metis appnd.simps(2) bt_map.simps(2)) (*slow!!*)
-done
-
-
-declare [[ atp_problem_prefix = "BT__bt_map_compose" ]]
-lemma bt_map_compose: "bt_map (f o g) t = bt_map f (bt_map g t)"
-apply (induct t)
- apply (metis bt_map.simps(1))
-txt{*Metis runs forever*}
-(* apply (metis bt_map.simps(2) o_apply)*)
-apply auto
-done
-
-
-declare [[ atp_problem_prefix = "BT__bt_map_reflect" ]]
-lemma bt_map_reflect: "bt_map f (reflect t) = reflect (bt_map f t)"
- apply (induct t)
- apply (metis add_right_cancel bt_map.simps(1) reflect.simps(1))
- apply (metis add_right_cancel bt_map.simps(2) reflect.simps(2))
- done
-
-declare [[ atp_problem_prefix = "BT__preorder_bt_map" ]]
-lemma preorder_bt_map: "preorder (bt_map f t) = map f (preorder t)"
- apply (induct t)
- apply (metis bt_map.simps(1) map.simps(1) preorder.simps(1))
- apply simp
- done
-
-declare [[ atp_problem_prefix = "BT__inorder_bt_map" ]]
-lemma inorder_bt_map: "inorder (bt_map f t) = map f (inorder t)"
- apply (induct t)
- apply (metis bt_map.simps(1) inorder.simps(1) map.simps(1))
- apply simp
- done
-
-declare [[ atp_problem_prefix = "BT__postorder_bt_map" ]]
-lemma postorder_bt_map: "postorder (bt_map f t) = map f (postorder t)"
- apply (induct t)
- apply (metis bt_map.simps(1) map.simps(1) postorder.simps(1))
- apply simp
- done
-
-declare [[ atp_problem_prefix = "BT__depth_bt_map" ]]
-lemma depth_bt_map [simp]: "depth (bt_map f t) = depth t"
- apply (induct t)
- apply (metis bt_map.simps(1) depth.simps(1))
- apply simp
- done
-
-declare [[ atp_problem_prefix = "BT__n_leaves_bt_map" ]]
-lemma n_leaves_bt_map [simp]: "n_leaves (bt_map f t) = n_leaves t"
- apply (induct t)
- apply (metis One_nat_def Suc_eq_plus1 bt_map.simps(1) less_add_one less_antisym linorder_neq_iff n_leaves.simps(1))
- apply (metis bt_map.simps(2) n_leaves.simps(2))
- done
-
-
-declare [[ atp_problem_prefix = "BT__preorder_reflect" ]]
-lemma preorder_reflect: "preorder (reflect t) = rev (postorder t)"
- apply (induct t)
- apply (metis postorder.simps(1) preorder.simps(1) reflect.simps(1) rev_is_Nil_conv)
- apply (metis append_Nil Cons_eq_append_conv postorder.simps(2) preorder.simps(2) reflect.simps(2) rev.simps(2) rev_append rev_rev_ident)
- done
-
-declare [[ atp_problem_prefix = "BT__inorder_reflect" ]]
-lemma inorder_reflect: "inorder (reflect t) = rev (inorder t)"
- apply (induct t)
- apply (metis inorder.simps(1) reflect.simps(1) rev.simps(1))
- apply simp
- done
-
-declare [[ atp_problem_prefix = "BT__postorder_reflect" ]]
-lemma postorder_reflect: "postorder (reflect t) = rev (preorder t)"
- apply (induct t)
- apply (metis postorder.simps(1) preorder.simps(1) reflect.simps(1) rev.simps(1))
- apply (metis Cons_eq_appendI postorder.simps(2) preorder.simps(2) reflect.simps(2) rev.simps(2) rev_append self_append_conv2)
- done
-
-text {*
- Analogues of the standard properties of the append function for lists.
-*}
-
-declare [[ atp_problem_prefix = "BT__appnd_assoc" ]]
-lemma appnd_assoc [simp]:
- "appnd (appnd t1 t2) t3 = appnd t1 (appnd t2 t3)"
- apply (induct t1)
- apply (metis appnd.simps(1))
- apply (metis appnd.simps(2))
- done
-
-declare [[ atp_problem_prefix = "BT__appnd_Lf2" ]]
-lemma appnd_Lf2 [simp]: "appnd t Lf = t"
- apply (induct t)
- apply (metis appnd.simps(1))
- apply (metis appnd.simps(2))
- done
-
-declare [[ atp_problem_prefix = "BT__depth_appnd" ]]
- declare max_add_distrib_left [simp]
-lemma depth_appnd [simp]: "depth (appnd t1 t2) = depth t1 + depth t2"
- apply (induct t1)
- apply (metis add_0 appnd.simps(1) depth.simps(1))
-apply (simp add: );
- done
-
-declare [[ atp_problem_prefix = "BT__n_leaves_appnd" ]]
-lemma n_leaves_appnd [simp]:
- "n_leaves (appnd t1 t2) = n_leaves t1 * n_leaves t2"
- apply (induct t1)
- apply (metis One_nat_def appnd.simps(1) less_irrefl less_linear n_leaves.simps(1) nat_mult_1)
- apply (simp add: left_distrib)
- done
-
-declare [[ atp_problem_prefix = "BT__bt_map_appnd" ]]
-lemma (*bt_map_appnd:*)
- "bt_map f (appnd t1 t2) = appnd (bt_map f t1) (bt_map f t2)"
- apply (induct t1)
- apply (metis appnd.simps(1) bt_map_appnd)
- apply (metis bt_map_appnd)
- done
-
-end