(* Title: HOL/BNF/Examples/TreeFI.thy
Author: Dmitriy Traytel, TU Muenchen
Author: Andrei Popescu, TU Muenchen
Copyright 2012
Finitely branching possibly infinite trees.
*)
header {* Finitely Branching Possibly Infinite Trees *}
theory TreeFI
imports ListF
begin
codatatype 'a treeFI = Tree (lab: 'a) (sub: "'a treeFI listF")
lemma pre_treeFI_listF_set[simp]: "pre_treeFI_set2 (i, xs) = listF_set xs"
unfolding pre_treeFI_set2_def collect_def[abs_def] prod_set_defs
by (auto simp add: listF.set_map')
lemma dtor[simp]: "treeFI_dtor tr = (lab tr, sub tr)"
unfolding lab_def sub_def treeFI_case_def
by (metis fst_def pair_collapse snd_def)
definition pair_fun (infixr "\<odot>" 50) where
"f \<odot> g \<equiv> \<lambda>x. (f x, g x)"
(* Tree reverse:*)
definition "trev \<equiv> treeFI_unfold lab (lrev o sub)"
lemma trev_simps1[simp]: "lab (trev t) = lab t"
unfolding trev_def by simp
lemma trev_simps2[simp]: "sub (trev t) = listF_map trev (lrev (sub t))"
unfolding trev_def by simp
lemma treeFI_coinduct:
assumes *: "phi x y"
and step: "\<And>a b. phi a b \<Longrightarrow>
lab a = lab b \<and>
lengthh (sub a) = lengthh (sub b) \<and>
(\<forall>i < lengthh (sub a). phi (nthh (sub a) i) (nthh (sub b) i))"
shows "x = y"
proof (rule mp[OF treeFI.dtor_map_coinduct, of phi, OF _ *])
fix a b :: "'a treeFI"
let ?zs = "zipp (sub a) (sub b)"
let ?z = "(lab a, ?zs)"
assume "phi a b"
with step have step': "lab a = lab b" "lengthh (sub a) = lengthh (sub b)"
"\<forall>i < lengthh (sub a). phi (nthh (sub a) i) (nthh (sub b) i)" by auto
hence "pre_treeFI_map id fst ?z = treeFI_dtor a" "pre_treeFI_map id snd ?z = treeFI_dtor b"
unfolding pre_treeFI_map_def by auto
moreover have "\<forall>(x, y) \<in> pre_treeFI_set2 ?z. phi x y"
proof safe
fix z1 z2
assume "(z1, z2) \<in> pre_treeFI_set2 ?z"
hence "(z1, z2) \<in> listF_set ?zs" by auto
hence "\<exists>i < lengthh ?zs. nthh ?zs i = (z1, z2)" by auto
with step'(2) obtain i where "i < lengthh (sub a)"
"nthh (sub a) i = z1" "nthh (sub b) i = z2" by auto
with step'(3) show "phi z1 z2" by auto
qed
ultimately show "\<exists>z.
(pre_treeFI_map id fst z = treeFI_dtor a \<and>
pre_treeFI_map id snd z = treeFI_dtor b) \<and>
(\<forall>x y. (x, y) \<in> pre_treeFI_set2 z \<longrightarrow> phi x y)" by blast
qed
lemma trev_trev: "trev (trev tr) = tr"
by (rule treeFI_coinduct[of "%a b. trev (trev b) = a"]) auto
end