(* 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 *}
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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.
*}
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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_add_numeral_1 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
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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.
*}
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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
ML {*AtpWrapper.problem_name := "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