(*  Title:      ZF/Induct/Binary_Trees.thy

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1992  University of Cambridge

 *)

 header {* Binary trees *}

 theory Binary_Trees imports Main begin

 subsection {* Datatype definition *}

 consts

   bt :: "i => i"

 datatype "bt(A)" =

   Lf | Br ("a \<in> A", "t1 \<in> bt(A)", "t2 \<in> bt(A)")

 declare bt.intros [simp]

 lemma Br_neq_left: "l \<in> bt(A) ==> Br(x, l, r) \<noteq> l"

   by (induct arbitrary: x r set: bt) auto

 lemma Br_iff: "Br(a, l, r) = Br(a', l', r') \<longleftrightarrow> a = a' & l = l' & r = r'"

   -- "Proving a freeness theorem."

   by (fast elim!: bt.free_elims)

 inductive_cases BrE: "Br(a, l, r) \<in> bt(A)"

   -- "An elimination rule, for type-checking."

 text {*

   \medskip Lemmas to justify using @{term bt} in other recursive type

   definitions.

 *}

 lemma bt_mono: "A \<subseteq> B ==> bt(A) \<subseteq> bt(B)"

   apply (unfold bt.defs)

   apply (rule lfp_mono)

     apply (rule bt.bnd_mono)+

   apply (rule univ_mono basic_monos | assumption)+

   done

 lemma bt_univ: "bt(univ(A)) \<subseteq> univ(A)"

   apply (unfold bt.defs bt.con_defs)

   apply (rule lfp_lowerbound)

    apply (rule_tac [2] A_subset_univ [THEN univ_mono])

   apply (fast intro!: zero_in_univ Inl_in_univ Inr_in_univ Pair_in_univ)

   done

 lemma bt_subset_univ: "A \<subseteq> univ(B) ==> bt(A) \<subseteq> univ(B)"

   apply (rule subset_trans)

    apply (erule bt_mono)

   apply (rule bt_univ)

   done

 lemma bt_rec_type:

   "[| t \<in> bt(A);

     c \<in> C(Lf);

     !!x y z r s. [| x \<in> A;  y \<in> bt(A);  z \<in> bt(A);  r \<in> C(y);  s \<in> C(z) |] ==>

     h(x, y, z, r, s) \<in> C(Br(x, y, z))

   |] ==> bt_rec(c, h, t) \<in> C(t)"

   -- {* Type checking for recursor -- example only; not really needed. *}

   apply (induct_tac t)

    apply simp_all

   done

 subsection {* Number of nodes, with an example of tail-recursion *}

 consts  n_nodes :: "i => i"

 primrec

   "n_nodes(Lf) = 0"

   "n_nodes(Br(a, l, r)) = succ(n_nodes(l) #+ n_nodes(r))"

 lemma n_nodes_type [simp]: "t \<in> bt(A) ==> n_nodes(t) \<in> nat"

   by (induct set: bt) auto

 consts  n_nodes_aux :: "i => i"

 primrec

   "n_nodes_aux(Lf) = (\<lambda>k \<in> nat. k)"

   "n_nodes_aux(Br(a, l, r)) =

       (\<lambda>k \<in> nat. n_nodes_aux(r) `  (n_nodes_aux(l) ` succ(k)))"

 lemma n_nodes_aux_eq:

     "t \<in> bt(A) ==> k \<in> nat ==> n_nodes_aux(t)`k = n_nodes(t) #+ k"

   apply (induct arbitrary: k set: bt)

    apply simp

   apply (atomize, simp)

   done

 definition

   n_nodes_tail :: "i => i"  where

   "n_nodes_tail(t) == n_nodes_aux(t) ` 0"

 lemma "t \<in> bt(A) ==> n_nodes_tail(t) = n_nodes(t)"

   by (simp add: n_nodes_tail_def n_nodes_aux_eq)

 subsection {* Number of leaves *}

 consts

   n_leaves :: "i => i"

 primrec

   "n_leaves(Lf) = 1"

   "n_leaves(Br(a, l, r)) = n_leaves(l) #+ n_leaves(r)"

 lemma n_leaves_type [simp]: "t \<in> bt(A) ==> n_leaves(t) \<in> nat"

   by (induct set: bt) auto

 subsection {* Reflecting trees *}

 consts

   bt_reflect :: "i => i"

 primrec

   "bt_reflect(Lf) = Lf"

   "bt_reflect(Br(a, l, r)) = Br(a, bt_reflect(r), bt_reflect(l))"

 lemma bt_reflect_type [simp]: "t \<in> bt(A) ==> bt_reflect(t) \<in> bt(A)"

   by (induct set: bt) auto

 text {*

   \medskip Theorems about @{term n_leaves}.

 *}

 lemma n_leaves_reflect: "t \<in> bt(A) ==> n_leaves(bt_reflect(t)) = n_leaves(t)"

   by (induct set: bt) (simp_all add: add_commute)

 lemma n_leaves_nodes: "t \<in> bt(A) ==> n_leaves(t) = succ(n_nodes(t))"

   by (induct set: bt) simp_all

 text {*

   Theorems about @{term bt_reflect}.

 *}

 lemma bt_reflect_bt_reflect_ident: "t \<in> bt(A) ==> bt_reflect(bt_reflect(t)) = t"

   by (induct set: bt) simp_all

 end