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(* Title: ZF/Induct/Binary_Trees.thy
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1992 University of Cambridge
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*)
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header {* Binary trees *}
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haftmann@16417
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theory Binary_Trees imports Main begin
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subsection {* Datatype definition *}
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consts
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bt :: "i => i"
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datatype "bt(A)" =
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Lf | Br ("a \<in> A", "t1 \<in> bt(A)", "t2 \<in> bt(A)")
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declare bt.intros [simp]
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lemma Br_neq_left: "l \<in> bt(A) ==> Br(x, l, r) \<noteq> l"
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by (induct arbitrary: x r set: bt) auto
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lemma Br_iff: "Br(a, l, r) = Br(a', l', r') \<longleftrightarrow> a = a' & l = l' & r = r'"
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-- "Proving a freeness theorem."
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by (fast elim!: bt.free_elims)
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inductive_cases BrE: "Br(a, l, r) \<in> bt(A)"
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-- "An elimination rule, for type-checking."
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text {*
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\medskip Lemmas to justify using @{term bt} in other recursive type
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definitions.
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*}
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lemma bt_mono: "A \<subseteq> B ==> bt(A) \<subseteq> bt(B)"
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apply (unfold bt.defs)
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apply (rule lfp_mono)
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apply (rule bt.bnd_mono)+
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apply (rule univ_mono basic_monos | assumption)+
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done
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lemma bt_univ: "bt(univ(A)) \<subseteq> univ(A)"
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apply (unfold bt.defs bt.con_defs)
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apply (rule lfp_lowerbound)
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apply (rule_tac [2] A_subset_univ [THEN univ_mono])
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apply (fast intro!: zero_in_univ Inl_in_univ Inr_in_univ Pair_in_univ)
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done
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lemma bt_subset_univ: "A \<subseteq> univ(B) ==> bt(A) \<subseteq> univ(B)"
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apply (rule subset_trans)
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apply (erule bt_mono)
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apply (rule bt_univ)
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done
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lemma bt_rec_type:
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"[| t \<in> bt(A);
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c \<in> C(Lf);
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!!x y z r s. [| x \<in> A; y \<in> bt(A); z \<in> bt(A); r \<in> C(y); s \<in> C(z) |] ==>
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h(x, y, z, r, s) \<in> C(Br(x, y, z))
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|] ==> bt_rec(c, h, t) \<in> C(t)"
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-- {* Type checking for recursor -- example only; not really needed. *}
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apply (induct_tac t)
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apply simp_all
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done
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subsection {* Number of nodes, with an example of tail-recursion *}
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consts n_nodes :: "i => i"
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primrec
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"n_nodes(Lf) = 0"
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"n_nodes(Br(a, l, r)) = succ(n_nodes(l) #+ n_nodes(r))"
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lemma n_nodes_type [simp]: "t \<in> bt(A) ==> n_nodes(t) \<in> nat"
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by (induct set: bt) auto
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consts n_nodes_aux :: "i => i"
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primrec
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"n_nodes_aux(Lf) = (\<lambda>k \<in> nat. k)"
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"n_nodes_aux(Br(a, l, r)) =
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(\<lambda>k \<in> nat. n_nodes_aux(r) ` (n_nodes_aux(l) ` succ(k)))"
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lemma n_nodes_aux_eq:
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"t \<in> bt(A) ==> k \<in> nat ==> n_nodes_aux(t)`k = n_nodes(t) #+ k"
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apply (induct arbitrary: k set: bt)
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apply simp
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apply (atomize, simp)
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done
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definition
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n_nodes_tail :: "i => i" where
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"n_nodes_tail(t) == n_nodes_aux(t) ` 0"
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lemma "t \<in> bt(A) ==> n_nodes_tail(t) = n_nodes(t)"
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by (simp add: n_nodes_tail_def n_nodes_aux_eq)
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subsection {* Number of leaves *}
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consts
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n_leaves :: "i => i"
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primrec
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"n_leaves(Lf) = 1"
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"n_leaves(Br(a, l, r)) = n_leaves(l) #+ n_leaves(r)"
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lemma n_leaves_type [simp]: "t \<in> bt(A) ==> n_leaves(t) \<in> nat"
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by (induct set: bt) auto
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subsection {* Reflecting trees *}
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consts
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bt_reflect :: "i => i"
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primrec
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"bt_reflect(Lf) = Lf"
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"bt_reflect(Br(a, l, r)) = Br(a, bt_reflect(r), bt_reflect(l))"
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lemma bt_reflect_type [simp]: "t \<in> bt(A) ==> bt_reflect(t) \<in> bt(A)"
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by (induct set: bt) auto
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text {*
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\medskip Theorems about @{term n_leaves}.
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*}
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lemma n_leaves_reflect: "t \<in> bt(A) ==> n_leaves(bt_reflect(t)) = n_leaves(t)"
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by (induct set: bt) (simp_all add: add_commute)
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lemma n_leaves_nodes: "t \<in> bt(A) ==> n_leaves(t) = succ(n_nodes(t))"
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by (induct set: bt) simp_all
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text {*
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Theorems about @{term bt_reflect}.
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*}
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lemma bt_reflect_bt_reflect_ident: "t \<in> bt(A) ==> bt_reflect(bt_reflect(t)) = t"
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by (induct set: bt) simp_all
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end
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