# HG changeset patch
# User popescua
# Date 1350385786 -7200
# Node ID 41ee3bfccb4de4532771d98ebf0443a7104c938f
# Parent  4a15873c4ec915cc50b3e3602fdcc3a338f0fcb5
changed name of BNF/Example directory from Infinite_Derivation_Trees to Derivation_Trees

diff -r 4a15873c4ec9 -r 41ee3bfccb4d src/HOL/BNF/Examples/Derivation_Trees/Gram_Lang.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/BNF/Examples/Derivation_Trees/Gram_Lang.thy	Tue Oct 16 13:09:46 2012 +0200
@@ -0,0 +1,1374 @@
+(*  Title:      HOL/BNF/Examples/Infinite_Derivation_Trees/Gram_Lang.thy
+    Author:     Andrei Popescu, TU Muenchen
+    Copyright   2012
+
+Language of a grammar.
+*)
+
+header {* Language of a Grammar *}
+
+theory Gram_Lang
+imports Tree
+begin
+
+
+consts P :: "(N \<times> (T + N) set) set"
+axiomatization where
+    finite_N: "finite (UNIV::N set)"
+and finite_in_P: "\<And> n tns. (n,tns) \<in> P \<longrightarrow> finite tns"
+and used: "\<And> n. \<exists> tns. (n,tns) \<in> P"
+
+
+subsection{* Tree basics: frontier, interior, etc. *}
+
+lemma Tree_cong:
+assumes "root tr = root tr'" and "cont tr = cont tr'"
+shows "tr = tr'"
+by (metis Node_root_cont assms)
+
+inductive finiteT where
+Node: "\<lbrakk>finite as; (finiteT^#) as\<rbrakk> \<Longrightarrow> finiteT (Node a as)"
+monos lift_mono
+
+lemma finiteT_induct[consumes 1, case_names Node, induct pred: finiteT]:
+assumes 1: "finiteT tr"
+and IH: "\<And>as n. \<lbrakk>finite as; (\<phi>^#) as\<rbrakk> \<Longrightarrow> \<phi> (Node n as)"
+shows "\<phi> tr"
+using 1 apply(induct rule: finiteT.induct)
+apply(rule IH) apply assumption apply(elim mono_lift) by simp
+
+
+(* Frontier *)
+
+inductive inFr :: "N set \<Rightarrow> Tree \<Rightarrow> T \<Rightarrow> bool" where
+Base: "\<lbrakk>root tr \<in> ns; Inl t \<in> cont tr\<rbrakk> \<Longrightarrow> inFr ns tr t"
+|
+Ind: "\<lbrakk>root tr \<in> ns; Inr tr1 \<in> cont tr; inFr ns tr1 t\<rbrakk> \<Longrightarrow> inFr ns tr t"
+
+definition "Fr ns tr \<equiv> {t. inFr ns tr t}"
+
+lemma inFr_root_in: "inFr ns tr t \<Longrightarrow> root tr \<in> ns"
+by (metis inFr.simps)
+
+lemma inFr_mono:
+assumes "inFr ns tr t" and "ns \<subseteq> ns'"
+shows "inFr ns' tr t"
+using assms apply(induct arbitrary: ns' rule: inFr.induct)
+using Base Ind by (metis inFr.simps set_mp)+
+
+lemma inFr_Ind_minus:
+assumes "inFr ns1 tr1 t" and "Inr tr1 \<in> cont tr"
+shows "inFr (insert (root tr) ns1) tr t"
+using assms apply(induct rule: inFr.induct)
+  apply (metis inFr.simps insert_iff)
+  by (metis inFr.simps inFr_mono insertI1 subset_insertI)
+
+(* alternative definition *)
+inductive inFr2 :: "N set \<Rightarrow> Tree \<Rightarrow> T \<Rightarrow> bool" where
+Base: "\<lbrakk>root tr \<in> ns; Inl t \<in> cont tr\<rbrakk> \<Longrightarrow> inFr2 ns tr t"
+|
+Ind: "\<lbrakk>Inr tr1 \<in> cont tr; inFr2 ns1 tr1 t\<rbrakk>
+      \<Longrightarrow> inFr2 (insert (root tr) ns1) tr t"
+
+lemma inFr2_root_in: "inFr2 ns tr t \<Longrightarrow> root tr \<in> ns"
+apply(induct rule: inFr2.induct) by auto
+
+lemma inFr2_mono:
+assumes "inFr2 ns tr t" and "ns \<subseteq> ns'"
+shows "inFr2 ns' tr t"
+using assms apply(induct arbitrary: ns' rule: inFr2.induct)
+using Base Ind
+apply (metis subsetD) by (metis inFr2.simps insert_absorb insert_subset)
+
+lemma inFr2_Ind:
+assumes "inFr2 ns tr1 t" and "root tr \<in> ns" and "Inr tr1 \<in> cont tr"
+shows "inFr2 ns tr t"
+using assms apply(induct rule: inFr2.induct)
+  apply (metis inFr2.simps insert_absorb)
+  by (metis inFr2.simps insert_absorb)
+
+lemma inFr_inFr2:
+"inFr = inFr2"
+apply (rule ext)+  apply(safe)
+  apply(erule inFr.induct)
+    apply (metis (lifting) inFr2.Base)
+    apply (metis (lifting) inFr2_Ind)
+  apply(erule inFr2.induct)
+    apply (metis (lifting) inFr.Base)
+    apply (metis (lifting) inFr_Ind_minus)
+done
+
+lemma not_root_inFr:
+assumes "root tr \<notin> ns"
+shows "\<not> inFr ns tr t"
+by (metis assms inFr_root_in)
+
+theorem not_root_Fr:
+assumes "root tr \<notin> ns"
+shows "Fr ns tr = {}"
+using not_root_inFr[OF assms] unfolding Fr_def by auto
+
+
+(* Interior *)
+
+inductive inItr :: "N set \<Rightarrow> Tree \<Rightarrow> N \<Rightarrow> bool" where
+Base: "root tr \<in> ns \<Longrightarrow> inItr ns tr (root tr)"
+|
+Ind: "\<lbrakk>root tr \<in> ns; Inr tr1 \<in> cont tr; inItr ns tr1 n\<rbrakk> \<Longrightarrow> inItr ns tr n"
+
+definition "Itr ns tr \<equiv> {n. inItr ns tr n}"
+
+lemma inItr_root_in: "inItr ns tr n \<Longrightarrow> root tr \<in> ns"
+by (metis inItr.simps)
+
+lemma inItr_mono:
+assumes "inItr ns tr n" and "ns \<subseteq> ns'"
+shows "inItr ns' tr n"
+using assms apply(induct arbitrary: ns' rule: inItr.induct)
+using Base Ind by (metis inItr.simps set_mp)+
+
+
+(* The subtree relation *)
+
+inductive subtr where
+Refl: "root tr \<in> ns \<Longrightarrow> subtr ns tr tr"
+|
+Step: "\<lbrakk>root tr3 \<in> ns; subtr ns tr1 tr2; Inr tr2 \<in> cont tr3\<rbrakk> \<Longrightarrow> subtr ns tr1 tr3"
+
+lemma subtr_rootL_in:
+assumes "subtr ns tr1 tr2"
+shows "root tr1 \<in> ns"
+using assms apply(induct rule: subtr.induct) by auto
+
+lemma subtr_rootR_in:
+assumes "subtr ns tr1 tr2"
+shows "root tr2 \<in> ns"
+using assms apply(induct rule: subtr.induct) by auto
+
+lemmas subtr_roots_in = subtr_rootL_in subtr_rootR_in
+
+lemma subtr_mono:
+assumes "subtr ns tr1 tr2" and "ns \<subseteq> ns'"
+shows "subtr ns' tr1 tr2"
+using assms apply(induct arbitrary: ns' rule: subtr.induct)
+using Refl Step by (metis subtr.simps set_mp)+
+
+lemma subtr_trans_Un:
+assumes "subtr ns12 tr1 tr2" and "subtr ns23 tr2 tr3"
+shows "subtr (ns12 \<union> ns23) tr1 tr3"
+proof-
+  have "subtr ns23 tr2 tr3  \<Longrightarrow>
+        (\<forall> ns12 tr1. subtr ns12 tr1 tr2 \<longrightarrow> subtr (ns12 \<union> ns23) tr1 tr3)"
+  apply(induct  rule: subtr.induct, safe)
+    apply (metis subtr_mono sup_commute sup_ge2)
+    by (metis (lifting) Step UnI2)
+  thus ?thesis using assms by auto
+qed
+
+lemma subtr_trans:
+assumes "subtr ns tr1 tr2" and "subtr ns tr2 tr3"
+shows "subtr ns tr1 tr3"
+using subtr_trans_Un[OF assms] by simp
+
+lemma subtr_StepL:
+assumes r: "root tr1 \<in> ns" and tr12: "Inr tr1 \<in> cont tr2" and s: "subtr ns tr2 tr3"
+shows "subtr ns tr1 tr3"
+apply(rule subtr_trans[OF _ s])
+apply(rule Step[of tr2 ns tr1 tr1])
+apply(rule subtr_rootL_in[OF s])
+apply(rule Refl[OF r])
+apply(rule tr12)
+done
+
+(* alternative definition: *)
+inductive subtr2 where
+Refl: "root tr \<in> ns \<Longrightarrow> subtr2 ns tr tr"
+|
+Step: "\<lbrakk>root tr1 \<in> ns; Inr tr1 \<in> cont tr2; subtr2 ns tr2 tr3\<rbrakk> \<Longrightarrow> subtr2 ns tr1 tr3"
+
+lemma subtr2_rootL_in:
+assumes "subtr2 ns tr1 tr2"
+shows "root tr1 \<in> ns"
+using assms apply(induct rule: subtr2.induct) by auto
+
+lemma subtr2_rootR_in:
+assumes "subtr2 ns tr1 tr2"
+shows "root tr2 \<in> ns"
+using assms apply(induct rule: subtr2.induct) by auto
+
+lemmas subtr2_roots_in = subtr2_rootL_in subtr2_rootR_in
+
+lemma subtr2_mono:
+assumes "subtr2 ns tr1 tr2" and "ns \<subseteq> ns'"
+shows "subtr2 ns' tr1 tr2"
+using assms apply(induct arbitrary: ns' rule: subtr2.induct)
+using Refl Step by (metis subtr2.simps set_mp)+
+
+lemma subtr2_trans_Un:
+assumes "subtr2 ns12 tr1 tr2" and "subtr2 ns23 tr2 tr3"
+shows "subtr2 (ns12 \<union> ns23) tr1 tr3"
+proof-
+  have "subtr2 ns12 tr1 tr2  \<Longrightarrow>
+        (\<forall> ns23 tr3. subtr2 ns23 tr2 tr3 \<longrightarrow> subtr2 (ns12 \<union> ns23) tr1 tr3)"
+  apply(induct  rule: subtr2.induct, safe)
+    apply (metis subtr2_mono sup_commute sup_ge2)
+    by (metis Un_iff subtr2.simps)
+  thus ?thesis using assms by auto
+qed
+
+lemma subtr2_trans:
+assumes "subtr2 ns tr1 tr2" and "subtr2 ns tr2 tr3"
+shows "subtr2 ns tr1 tr3"
+using subtr2_trans_Un[OF assms] by simp
+
+lemma subtr2_StepR:
+assumes r: "root tr3 \<in> ns" and tr23: "Inr tr2 \<in> cont tr3" and s: "subtr2 ns tr1 tr2"
+shows "subtr2 ns tr1 tr3"
+apply(rule subtr2_trans[OF s])
+apply(rule Step[of _ _ tr3])
+apply(rule subtr2_rootR_in[OF s])
+apply(rule tr23)
+apply(rule Refl[OF r])
+done
+
+lemma subtr_subtr2:
+"subtr = subtr2"
+apply (rule ext)+  apply(safe)
+  apply(erule subtr.induct)
+    apply (metis (lifting) subtr2.Refl)
+    apply (metis (lifting) subtr2_StepR)
+  apply(erule subtr2.induct)
+    apply (metis (lifting) subtr.Refl)
+    apply (metis (lifting) subtr_StepL)
+done
+
+lemma subtr_inductL[consumes 1, case_names Refl Step]:
+assumes s: "subtr ns tr1 tr2" and Refl: "\<And>ns tr. \<phi> ns tr tr"
+and Step:
+"\<And>ns tr1 tr2 tr3.
+   \<lbrakk>root tr1 \<in> ns; Inr tr1 \<in> cont tr2; subtr ns tr2 tr3; \<phi> ns tr2 tr3\<rbrakk> \<Longrightarrow> \<phi> ns tr1 tr3"
+shows "\<phi> ns tr1 tr2"
+using s unfolding subtr_subtr2 apply(rule subtr2.induct)
+using Refl Step unfolding subtr_subtr2 by auto
+
+lemma subtr_UNIV_inductL[consumes 1, case_names Refl Step]:
+assumes s: "subtr UNIV tr1 tr2" and Refl: "\<And>tr. \<phi> tr tr"
+and Step:
+"\<And>tr1 tr2 tr3.
+   \<lbrakk>Inr tr1 \<in> cont tr2; subtr UNIV tr2 tr3; \<phi> tr2 tr3\<rbrakk> \<Longrightarrow> \<phi> tr1 tr3"
+shows "\<phi> tr1 tr2"
+using s apply(induct rule: subtr_inductL)
+apply(rule Refl) using Step subtr_mono by (metis subset_UNIV)
+
+(* Subtree versus frontier: *)
+lemma subtr_inFr:
+assumes "inFr ns tr t" and "subtr ns tr tr1"
+shows "inFr ns tr1 t"
+proof-
+  have "subtr ns tr tr1 \<Longrightarrow> (\<forall> t. inFr ns tr t \<longrightarrow> inFr ns tr1 t)"
+  apply(induct rule: subtr.induct, safe) by (metis inFr.Ind)
+  thus ?thesis using assms by auto
+qed
+
+corollary Fr_subtr:
+"Fr ns tr = \<Union> {Fr ns tr' | tr'. subtr ns tr' tr}"
+unfolding Fr_def proof safe
+  fix t assume t: "inFr ns tr t"  hence "root tr \<in> ns" by (rule inFr_root_in)
+  thus "t \<in> \<Union>{{t. inFr ns tr' t} |tr'. subtr ns tr' tr}"
+  apply(intro UnionI[of "{t. inFr ns tr t}" _ t]) using t subtr.Refl by auto
+qed(metis subtr_inFr)
+
+lemma inFr_subtr:
+assumes "inFr ns tr t"
+shows "\<exists> tr'. subtr ns tr' tr \<and> Inl t \<in> cont tr'"
+using assms apply(induct rule: inFr.induct) apply safe
+  apply (metis subtr.Refl)
+  by (metis (lifting) subtr.Step)
+
+corollary Fr_subtr_cont:
+"Fr ns tr = \<Union> {Inl -` cont tr' | tr'. subtr ns tr' tr}"
+unfolding Fr_def
+apply safe
+apply (frule inFr_subtr)
+apply auto
+by (metis inFr.Base subtr_inFr subtr_rootL_in)
+
+(* Subtree versus interior: *)
+lemma subtr_inItr:
+assumes "inItr ns tr n" and "subtr ns tr tr1"
+shows "inItr ns tr1 n"
+proof-
+  have "subtr ns tr tr1 \<Longrightarrow> (\<forall> t. inItr ns tr n \<longrightarrow> inItr ns tr1 n)"
+  apply(induct rule: subtr.induct, safe) by (metis inItr.Ind)
+  thus ?thesis using assms by auto
+qed
+
+corollary Itr_subtr:
+"Itr ns tr = \<Union> {Itr ns tr' | tr'. subtr ns tr' tr}"
+unfolding Itr_def apply safe
+apply (metis (lifting, mono_tags) UnionI inItr_root_in mem_Collect_eq subtr.Refl)
+by (metis subtr_inItr)
+
+lemma inItr_subtr:
+assumes "inItr ns tr n"
+shows "\<exists> tr'. subtr ns tr' tr \<and> root tr' = n"
+using assms apply(induct rule: inItr.induct) apply safe
+  apply (metis subtr.Refl)
+  by (metis (lifting) subtr.Step)
+
+corollary Itr_subtr_cont:
+"Itr ns tr = {root tr' | tr'. subtr ns tr' tr}"
+unfolding Itr_def apply safe
+  apply (metis (lifting, mono_tags) inItr_subtr)
+  by (metis inItr.Base subtr_inItr subtr_rootL_in)
+
+
+subsection{* The immediate subtree function *}
+
+(* production of: *)
+abbreviation "prodOf tr \<equiv> (id \<oplus> root) ` (cont tr)"
+(* subtree of: *)
+definition "subtrOf tr n \<equiv> SOME tr'. Inr tr' \<in> cont tr \<and> root tr' = n"
+
+lemma subtrOf:
+assumes n: "Inr n \<in> prodOf tr"
+shows "Inr (subtrOf tr n) \<in> cont tr \<and> root (subtrOf tr n) = n"
+proof-
+  obtain tr' where "Inr tr' \<in> cont tr \<and> root tr' = n"
+  using n unfolding image_def by (metis (lifting) Inr_oplus_elim assms)
+  thus ?thesis unfolding subtrOf_def by(rule someI)
+qed
+
+lemmas Inr_subtrOf = subtrOf[THEN conjunct1]
+lemmas root_subtrOf[simp] = subtrOf[THEN conjunct2]
+
+lemma Inl_prodOf: "Inl -` (prodOf tr) = Inl -` (cont tr)"
+proof safe
+  fix t ttr assume "Inl t = (id \<oplus> root) ttr" and "ttr \<in> cont tr"
+  thus "t \<in> Inl -` cont tr" by(cases ttr, auto)
+next
+  fix t assume "Inl t \<in> cont tr" thus "t \<in> Inl -` prodOf tr"
+  by (metis (lifting) id_def image_iff sum_map.simps(1) vimageI2)
+qed
+
+lemma root_prodOf:
+assumes "Inr tr' \<in> cont tr"
+shows "Inr (root tr') \<in> prodOf tr"
+by (metis (lifting) assms image_iff sum_map.simps(2))
+
+
+subsection{* Derivation trees *}
+
+coinductive dtree where
+Tree: "\<lbrakk>(root tr, (id \<oplus> root) ` (cont tr)) \<in> P; inj_on root (Inr -` cont tr);
+        lift dtree (cont tr)\<rbrakk> \<Longrightarrow> dtree tr"
+monos lift_mono
+
+(* destruction rules: *)
+lemma dtree_P:
+assumes "dtree tr"
+shows "(root tr, (id \<oplus> root) ` (cont tr)) \<in> P"
+using assms unfolding dtree.simps by auto
+
+lemma dtree_inj_on:
+assumes "dtree tr"
+shows "inj_on root (Inr -` cont tr)"
+using assms unfolding dtree.simps by auto
+
+lemma dtree_inj[simp]:
+assumes "dtree tr" and "Inr tr1 \<in> cont tr" and "Inr tr2 \<in> cont tr"
+shows "root tr1 = root tr2 \<longleftrightarrow> tr1 = tr2"
+using assms dtree_inj_on unfolding inj_on_def by auto
+
+lemma dtree_lift:
+assumes "dtree tr"
+shows "lift dtree (cont tr)"
+using assms unfolding dtree.simps by auto
+
+
+(* coinduction:*)
+lemma dtree_coind[elim, consumes 1, case_names Hyp]:
+assumes phi: "\<phi> tr"
+and Hyp:
+"\<And> tr. \<phi> tr \<Longrightarrow>
+       (root tr, image (id \<oplus> root) (cont tr)) \<in> P \<and>
+       inj_on root (Inr -` cont tr) \<and>
+       lift (\<lambda> tr. \<phi> tr \<or> dtree tr) (cont tr)"
+shows "dtree tr"
+apply(rule dtree.coinduct[of \<phi> tr, OF phi])
+using Hyp by blast
+
+lemma dtree_raw_coind[elim, consumes 1, case_names Hyp]:
+assumes phi: "\<phi> tr"
+and Hyp:
+"\<And> tr. \<phi> tr \<Longrightarrow>
+       (root tr, image (id \<oplus> root) (cont tr)) \<in> P \<and>
+       inj_on root (Inr -` cont tr) \<and>
+       lift \<phi> (cont tr)"
+shows "dtree tr"
+using phi apply(induct rule: dtree_coind)
+using Hyp mono_lift
+by (metis (mono_tags) mono_lift)
+
+lemma dtree_subtr_inj_on:
+assumes d: "dtree tr1" and s: "subtr ns tr tr1"
+shows "inj_on root (Inr -` cont tr)"
+using s d apply(induct rule: subtr.induct)
+apply (metis (lifting) dtree_inj_on) by (metis dtree_lift lift_def)
+
+lemma dtree_subtr_P:
+assumes d: "dtree tr1" and s: "subtr ns tr tr1"
+shows "(root tr, (id \<oplus> root) ` cont tr) \<in> P"
+using s d apply(induct rule: subtr.induct)
+apply (metis (lifting) dtree_P) by (metis dtree_lift lift_def)
+
+lemma subtrOf_root[simp]:
+assumes tr: "dtree tr" and cont: "Inr tr' \<in> cont tr"
+shows "subtrOf tr (root tr') = tr'"
+proof-
+  have 0: "Inr (subtrOf tr (root tr')) \<in> cont tr" using Inr_subtrOf
+  by (metis (lifting) cont root_prodOf)
+  have "root (subtrOf tr (root tr')) = root tr'"
+  using root_subtrOf by (metis (lifting) cont root_prodOf)
+  thus ?thesis unfolding dtree_inj[OF tr 0 cont] .
+qed
+
+lemma surj_subtrOf:
+assumes "dtree tr" and 0: "Inr tr' \<in> cont tr"
+shows "\<exists> n. Inr n \<in> prodOf tr \<and> subtrOf tr n = tr'"
+apply(rule exI[of _ "root tr'"])
+using root_prodOf[OF 0] subtrOf_root[OF assms] by simp
+
+lemma dtree_subtr:
+assumes "dtree tr1" and "subtr ns tr tr1"
+shows "dtree tr"
+proof-
+  have "(\<exists> ns tr1. dtree tr1 \<and> subtr ns tr tr1) \<Longrightarrow> dtree tr"
+  proof (induct rule: dtree_raw_coind)
+    case (Hyp tr)
+    then obtain ns tr1 where tr1: "dtree tr1" and tr_tr1: "subtr ns tr tr1" by auto
+    show ?case unfolding lift_def proof safe
+      show "(root tr, (id \<oplus> root) ` cont tr) \<in> P" using dtree_subtr_P[OF tr1 tr_tr1] .
+    next
+      show "inj_on root (Inr -` cont tr)" using dtree_subtr_inj_on[OF tr1 tr_tr1] .
+    next
+      fix tr' assume tr': "Inr tr' \<in> cont tr"
+      have tr_tr1: "subtr (ns \<union> {root tr'}) tr tr1" using subtr_mono[OF tr_tr1] by auto
+      have "subtr (ns \<union> {root tr'}) tr' tr1" using subtr_StepL[OF _ tr' tr_tr1] by auto
+      thus "\<exists>ns' tr1. dtree tr1 \<and> subtr ns' tr' tr1" using tr1 by blast
+    qed
+  qed
+  thus ?thesis using assms by auto
+qed
+
+
+subsection{* Default trees *}
+
+(* Pick a left-hand side of a production for each nonterminal *)
+definition S where "S n \<equiv> SOME tns. (n,tns) \<in> P"
+
+lemma S_P: "(n, S n) \<in> P"
+using used unfolding S_def by(rule someI_ex)
+
+lemma finite_S: "finite (S n)"
+using S_P finite_in_P by auto
+
+
+(* The default tree of a nonterminal *)
+definition deftr :: "N \<Rightarrow> Tree" where
+"deftr \<equiv> unfold id S"
+
+lemma deftr_simps[simp]:
+"root (deftr n) = n"
+"cont (deftr n) = image (id \<oplus> deftr) (S n)"
+using unfold(1)[of id S n] unfold(2)[of S n id, OF finite_S]
+unfolding deftr_def by simp_all
+
+lemmas root_deftr = deftr_simps(1)
+lemmas cont_deftr = deftr_simps(2)
+
+lemma root_o_deftr[simp]: "root o deftr = id"
+by (rule ext, auto)
+
+lemma dtree_deftr: "dtree (deftr n)"
+proof-
+  {fix tr assume "\<exists> n. tr = deftr n" hence "dtree tr"
+   apply(induct rule: dtree_raw_coind) apply safe
+   unfolding deftr_simps image_compose[symmetric] sum_map.comp id_o
+   root_o_deftr sum_map.id image_id id_apply apply(rule S_P)
+   unfolding inj_on_def lift_def by auto
+  }
+  thus ?thesis by auto
+qed
+
+
+subsection{* Hereditary substitution *}
+
+(* Auxiliary concept: The root-ommiting frontier: *)
+definition "inFrr ns tr t \<equiv> \<exists> tr'. Inr tr' \<in> cont tr \<and> inFr ns tr' t"
+definition "Frr ns tr \<equiv> {t. \<exists> tr'. Inr tr' \<in> cont tr \<and> t \<in> Fr ns tr'}"
+
+context
+fixes tr0 :: Tree
+begin
+
+definition "hsubst_r tr \<equiv> root tr"
+definition "hsubst_c tr \<equiv> if root tr = root tr0 then cont tr0 else cont tr"
+
+(* Hereditary substitution: *)
+definition hsubst :: "Tree \<Rightarrow> Tree" where
+"hsubst \<equiv> unfold hsubst_r hsubst_c"
+
+lemma finite_hsubst_c: "finite (hsubst_c n)"
+unfolding hsubst_c_def by (metis (full_types) finite_cont)
+
+lemma root_hsubst[simp]: "root (hsubst tr) = root tr"
+using unfold(1)[of hsubst_r hsubst_c tr] unfolding hsubst_def hsubst_r_def by simp
+
+lemma root_o_subst[simp]: "root o hsubst = root"
+unfolding comp_def root_hsubst ..
+
+lemma cont_hsubst_eq[simp]:
+assumes "root tr = root tr0"
+shows "cont (hsubst tr) = (id \<oplus> hsubst) ` (cont tr0)"
+apply(subst id_o[symmetric, of id]) unfolding id_o
+using unfold(2)[of hsubst_c tr hsubst_r, OF finite_hsubst_c]
+unfolding hsubst_def hsubst_c_def using assms by simp
+
+lemma hsubst_eq:
+assumes "root tr = root tr0"
+shows "hsubst tr = hsubst tr0"
+apply(rule Tree_cong) using assms cont_hsubst_eq by auto
+
+lemma cont_hsubst_neq[simp]:
+assumes "root tr \<noteq> root tr0"
+shows "cont (hsubst tr) = (id \<oplus> hsubst) ` (cont tr)"
+apply(subst id_o[symmetric, of id]) unfolding id_o
+using unfold(2)[of hsubst_c tr hsubst_r, OF finite_hsubst_c]
+unfolding hsubst_def hsubst_c_def using assms by simp
+
+lemma Inl_cont_hsubst_eq[simp]:
+assumes "root tr = root tr0"
+shows "Inl -` cont (hsubst tr) = Inl -` (cont tr0)"
+unfolding cont_hsubst_eq[OF assms] by simp
+
+lemma Inr_cont_hsubst_eq[simp]:
+assumes "root tr = root tr0"
+shows "Inr -` cont (hsubst tr) = hsubst ` Inr -` cont tr0"
+unfolding cont_hsubst_eq[OF assms] by simp
+
+lemma Inl_cont_hsubst_neq[simp]:
+assumes "root tr \<noteq> root tr0"
+shows "Inl -` cont (hsubst tr) = Inl -` (cont tr)"
+unfolding cont_hsubst_neq[OF assms] by simp
+
+lemma Inr_cont_hsubst_neq[simp]:
+assumes "root tr \<noteq> root tr0"
+shows "Inr -` cont (hsubst tr) = hsubst ` Inr -` cont tr"
+unfolding cont_hsubst_neq[OF assms] by simp
+
+lemma dtree_hsubst:
+assumes tr0: "dtree tr0" and tr: "dtree tr"
+shows "dtree (hsubst tr)"
+proof-
+  {fix tr1 have "(\<exists> tr. dtree tr \<and> tr1 = hsubst tr) \<Longrightarrow> dtree tr1"
+   proof (induct rule: dtree_raw_coind)
+     case (Hyp tr1) then obtain tr
+     where dtr: "dtree tr" and tr1: "tr1 = hsubst tr" by auto
+     show ?case unfolding lift_def tr1 proof safe
+       show "(root (hsubst tr), prodOf (hsubst tr)) \<in> P"
+       unfolding tr1 apply(cases "root tr = root tr0")
+       using  dtree_P[OF dtr] dtree_P[OF tr0]
+       by (auto simp add: image_compose[symmetric] sum_map.comp)
+       show "inj_on root (Inr -` cont (hsubst tr))"
+       apply(cases "root tr = root tr0") using dtree_inj_on[OF dtr] dtree_inj_on[OF tr0]
+       unfolding inj_on_def by (auto, blast)
+       fix tr' assume "Inr tr' \<in> cont (hsubst tr)"
+       thus "\<exists>tra. dtree tra \<and> tr' = hsubst tra"
+       apply(cases "root tr = root tr0", simp_all)
+         apply (metis dtree_lift lift_def tr0)
+         by (metis dtr dtree_lift lift_def)
+     qed
+   qed
+  }
+  thus ?thesis using assms by blast
+qed
+
+lemma Frr: "Frr ns tr = {t. inFrr ns tr t}"
+unfolding inFrr_def Frr_def Fr_def by auto
+
+lemma inFr_hsubst_imp:
+assumes "inFr ns (hsubst tr) t"
+shows "t \<in> Inl -` (cont tr0) \<or> inFrr (ns - {root tr0}) tr0 t \<or>
+       inFr (ns - {root tr0}) tr t"
+proof-
+  {fix tr1
+   have "inFr ns tr1 t \<Longrightarrow>
+   (\<And> tr. tr1 = hsubst tr \<Longrightarrow> (t \<in> Inl -` (cont tr0) \<or> inFrr (ns - {root tr0}) tr0 t \<or>
+                              inFr (ns - {root tr0}) tr t))"
+   proof(induct rule: inFr.induct)
+     case (Base tr1 ns t tr)
+     hence rtr: "root tr1 \<in> ns" and t_tr1: "Inl t \<in> cont tr1" and tr1: "tr1 = hsubst tr"
+     by auto
+     show ?case
+     proof(cases "root tr1 = root tr0")
+       case True
+       hence "t \<in> Inl -` (cont tr0)" using t_tr1 unfolding tr1 by auto
+       thus ?thesis by simp
+     next
+       case False
+       hence "inFr (ns - {root tr0}) tr t" using t_tr1 unfolding tr1 apply simp
+       by (metis Base.prems Diff_iff root_hsubst inFr.Base rtr singletonE)
+       thus ?thesis by simp
+     qed
+   next
+     case (Ind tr1 ns tr1' t) note IH = Ind(4)
+     have rtr1: "root tr1 \<in> ns" and tr1'_tr1: "Inr tr1' \<in> cont tr1"
+     and t_tr1': "inFr ns tr1' t" and tr1: "tr1 = hsubst tr" using Ind by auto
+     have rtr1: "root tr1 = root tr" unfolding tr1 by simp
+     show ?case
+     proof(cases "root tr1 = root tr0")
+       case True
+       then obtain tr' where tr'_tr0: "Inr tr' \<in> cont tr0" and tr1': "tr1' = hsubst tr'"
+       using tr1'_tr1 unfolding tr1 by auto
+       show ?thesis using IH[OF tr1'] proof (elim disjE)
+         assume "inFr (ns - {root tr0}) tr' t"
+         thus ?thesis using tr'_tr0 unfolding inFrr_def by auto
+       qed auto
+     next
+       case False
+       then obtain tr' where tr'_tr: "Inr tr' \<in> cont tr" and tr1': "tr1' = hsubst tr'"
+       using tr1'_tr1 unfolding tr1 by auto
+       show ?thesis using IH[OF tr1'] proof (elim disjE)
+         assume "inFr (ns - {root tr0}) tr' t"
+         thus ?thesis using tr'_tr unfolding inFrr_def
+         by (metis Diff_iff False Ind(1) empty_iff inFr2_Ind inFr_inFr2 insert_iff rtr1)
+       qed auto
+     qed
+   qed
+  }
+  thus ?thesis using assms by auto
+qed
+
+lemma inFr_hsubst_notin:
+assumes "inFr ns tr t" and "root tr0 \<notin> ns"
+shows "inFr ns (hsubst tr) t"
+using assms apply(induct rule: inFr.induct)
+apply (metis Inl_cont_hsubst_neq inFr2.Base inFr_inFr2 root_hsubst vimageD vimageI2)
+by (metis (lifting) Inr_cont_hsubst_neq inFr.Ind rev_image_eqI root_hsubst vimageD vimageI2)
+
+lemma inFr_hsubst_minus:
+assumes "inFr (ns - {root tr0}) tr t"
+shows "inFr ns (hsubst tr) t"
+proof-
+  have 1: "inFr (ns - {root tr0}) (hsubst tr) t"
+  using inFr_hsubst_notin[OF assms] by simp
+  show ?thesis using inFr_mono[OF 1] by auto
+qed
+
+lemma inFr_self_hsubst:
+assumes "root tr0 \<in> ns"
+shows
+"inFr ns (hsubst tr0) t \<longleftrightarrow>
+ t \<in> Inl -` (cont tr0) \<or> inFrr (ns - {root tr0}) tr0 t"
+(is "?A \<longleftrightarrow> ?B \<or> ?C")
+apply(intro iffI)
+apply (metis inFr_hsubst_imp Diff_iff inFr_root_in insertI1) proof(elim disjE)
+  assume ?B thus ?A apply(intro inFr.Base) using assms by auto
+next
+  assume ?C then obtain tr where
+  tr_tr0: "Inr tr \<in> cont tr0" and t_tr: "inFr (ns - {root tr0}) tr t"
+  unfolding inFrr_def by auto
+  def tr1 \<equiv> "hsubst tr"
+  have 1: "inFr ns tr1 t" using t_tr unfolding tr1_def using inFr_hsubst_minus by auto
+  have "Inr tr1 \<in> cont (hsubst tr0)" unfolding tr1_def using tr_tr0 by auto
+  thus ?A using 1 inFr.Ind assms by (metis root_hsubst)
+qed
+
+theorem Fr_self_hsubst:
+assumes "root tr0 \<in> ns"
+shows "Fr ns (hsubst tr0) = Inl -` (cont tr0) \<union> Frr (ns - {root tr0}) tr0"
+using inFr_self_hsubst[OF assms] unfolding Frr Fr_def by auto
+
+end (* context *)
+
+
+subsection{* Regular trees *}
+
+hide_const regular
+
+definition "reg f tr \<equiv> \<forall> tr'. subtr UNIV tr' tr \<longrightarrow> tr' = f (root tr')"
+definition "regular tr \<equiv> \<exists> f. reg f tr"
+
+lemma reg_def2: "reg f tr \<longleftrightarrow> (\<forall> ns tr'. subtr ns tr' tr \<longrightarrow> tr' = f (root tr'))"
+unfolding reg_def using subtr_mono by (metis subset_UNIV)
+
+lemma regular_def2: "regular tr \<longleftrightarrow> (\<exists> f. reg f tr \<and> (\<forall> n. root (f n) = n))"
+unfolding regular_def proof safe
+  fix f assume f: "reg f tr"
+  def g \<equiv> "\<lambda> n. if inItr UNIV tr n then f n else deftr n"
+  show "\<exists>g. reg g tr \<and> (\<forall>n. root (g n) = n)"
+  apply(rule exI[of _ g])
+  using f deftr_simps(1) unfolding g_def reg_def apply safe
+    apply (metis (lifting) inItr.Base subtr_inItr subtr_rootL_in)
+    by (metis (full_types) inItr_subtr)
+qed auto
+
+lemma reg_root:
+assumes "reg f tr"
+shows "f (root tr) = tr"
+using assms unfolding reg_def
+by (metis (lifting) iso_tuple_UNIV_I subtr.Refl)
+
+
+lemma reg_Inr_cont:
+assumes "reg f tr" and "Inr tr' \<in> cont tr"
+shows "reg f tr'"
+by (metis (lifting) assms iso_tuple_UNIV_I reg_def subtr.Step)
+
+lemma reg_subtr:
+assumes "reg f tr" and "subtr ns tr' tr"
+shows "reg f tr'"
+using assms unfolding reg_def using subtr_trans[of UNIV tr] UNIV_I
+by (metis UNIV_eq_I UnCI Un_upper1 iso_tuple_UNIV_I subtr_mono subtr_trans)
+
+lemma regular_subtr:
+assumes r: "regular tr" and s: "subtr ns tr' tr"
+shows "regular tr'"
+using r reg_subtr[OF _ s] unfolding regular_def by auto
+
+lemma subtr_deftr:
+assumes "subtr ns tr' (deftr n)"
+shows "tr' = deftr (root tr')"
+proof-
+  {fix tr have "subtr ns tr' tr \<Longrightarrow> (\<forall> n. tr = deftr n \<longrightarrow> tr' = deftr (root tr'))"
+   apply (induct rule: subtr.induct)
+   proof(metis (lifting) deftr_simps(1), safe)
+     fix tr3 ns tr1 tr2 n
+     assume 1: "root (deftr n) \<in> ns" and 2: "subtr ns tr1 tr2"
+     and IH: "\<forall>n. tr2 = deftr n \<longrightarrow> tr1 = deftr (root tr1)"
+     and 3: "Inr tr2 \<in> cont (deftr n)"
+     have "tr2 \<in> deftr ` UNIV"
+     using 3 unfolding deftr_simps image_def
+     by (metis (lifting, full_types) 3 CollectI Inr_oplus_iff cont_deftr
+         iso_tuple_UNIV_I)
+     then obtain n where "tr2 = deftr n" by auto
+     thus "tr1 = deftr (root tr1)" using IH by auto
+   qed
+  }
+  thus ?thesis using assms by auto
+qed
+
+lemma reg_deftr: "reg deftr (deftr n)"
+unfolding reg_def using subtr_deftr by auto
+
+lemma dtree_subtrOf_Union:
+assumes "dtree tr"
+shows "\<Union>{K tr' |tr'. Inr tr' \<in> cont tr} =
+       \<Union>{K (subtrOf tr n) |n. Inr n \<in> prodOf tr}"
+unfolding Union_eq Bex_def mem_Collect_eq proof safe
+  fix x xa tr'
+  assume x: "x \<in> K tr'" and tr'_tr: "Inr tr' \<in> cont tr"
+  show "\<exists>X. (\<exists>n. X = K (subtrOf tr n) \<and> Inr n \<in> prodOf tr) \<and> x \<in> X"
+  apply(rule exI[of _ "K (subtrOf tr (root tr'))"]) apply(intro conjI)
+    apply(rule exI[of _ "root tr'"]) apply (metis (lifting) root_prodOf tr'_tr)
+    by (metis (lifting) assms subtrOf_root tr'_tr x)
+next
+  fix x X n ttr
+  assume x: "x \<in> K (subtrOf tr n)" and n: "Inr n = (id \<oplus> root) ttr" and ttr: "ttr \<in> cont tr"
+  show "\<exists>X. (\<exists>tr'. X = K tr' \<and> Inr tr' \<in> cont tr) \<and> x \<in> X"
+  apply(rule exI[of _ "K (subtrOf tr n)"]) apply(intro conjI)
+    apply(rule exI[of _ "subtrOf tr n"]) apply (metis imageI n subtrOf ttr)
+    using x .
+qed
+
+
+
+
+subsection {* Paths in a regular tree *}
+
+inductive path :: "(N \<Rightarrow> Tree) \<Rightarrow> N list \<Rightarrow> bool" for f where
+Base: "path f [n]"
+|
+Ind: "\<lbrakk>path f (n1 # nl); Inr (f n1) \<in> cont (f n)\<rbrakk>
+      \<Longrightarrow> path f (n # n1 # nl)"
+
+lemma path_NE:
+assumes "path f nl"
+shows "nl \<noteq> Nil"
+using assms apply(induct rule: path.induct) by auto
+
+lemma path_post:
+assumes f: "path f (n # nl)" and nl: "nl \<noteq> []"
+shows "path f nl"
+proof-
+  obtain n1 nl1 where nl: "nl = n1 # nl1" using nl by (cases nl, auto)
+  show ?thesis using assms unfolding nl using path.simps by (metis (lifting) list.inject)
+qed
+
+lemma path_post_concat:
+assumes "path f (nl1 @ nl2)" and "nl2 \<noteq> Nil"
+shows "path f nl2"
+using assms apply (induct nl1)
+apply (metis append_Nil) by (metis Nil_is_append_conv append_Cons path_post)
+
+lemma path_concat:
+assumes "path f nl1" and "path f ((last nl1) # nl2)"
+shows "path f (nl1 @ nl2)"
+using assms apply(induct rule: path.induct) apply simp
+by (metis append_Cons last.simps list.simps(3) path.Ind)
+
+lemma path_distinct:
+assumes "path f nl"
+shows "\<exists> nl'. path f nl' \<and> hd nl' = hd nl \<and> last nl' = last nl \<and>
+              set nl' \<subseteq> set nl \<and> distinct nl'"
+using assms proof(induct rule: length_induct)
+  case (1 nl)  hence p_nl: "path f nl" by simp
+  then obtain n nl1 where nl: "nl = n # nl1" by (metis list.exhaust path_NE)
+  show ?case
+  proof(cases nl1)
+    case Nil
+    show ?thesis apply(rule exI[of _ nl]) using path.Base unfolding nl Nil by simp
+  next
+    case (Cons n1 nl2)
+    hence p1: "path f nl1" by (metis list.simps(3) nl p_nl path_post)
+    show ?thesis
+    proof(cases "n \<in> set nl1")
+      case False
+      obtain nl1' where p1': "path f nl1'" and hd_nl1': "hd nl1' = hd nl1" and
+      l_nl1': "last nl1' = last nl1" and d_nl1': "distinct nl1'"
+      and s_nl1': "set nl1' \<subseteq> set nl1"
+      using 1(1)[THEN allE[of _ nl1]] p1 unfolding nl by auto
+      obtain nl2' where nl1': "nl1' = n1 # nl2'" using path_NE[OF p1'] hd_nl1'
+      unfolding Cons by(cases nl1', auto)
+      show ?thesis apply(intro exI[of _ "n # nl1'"]) unfolding nl proof safe
+        show "path f (n # nl1')" unfolding nl1'
+        apply(rule path.Ind, metis nl1' p1')
+        by (metis (lifting) Cons list.inject nl p1 p_nl path.simps path_NE)
+      qed(insert l_nl1' Cons nl1' s_nl1' d_nl1' False, auto)
+    next
+      case True
+      then obtain nl11 nl12 where nl1: "nl1 = nl11 @ n # nl12"
+      by (metis split_list)
+      have p12: "path f (n # nl12)"
+      apply(rule path_post_concat[of _ "n # nl11"]) using p_nl[unfolded nl nl1] by auto
+      obtain nl12' where p1': "path f nl12'" and hd_nl12': "hd nl12' = n" and
+      l_nl12': "last nl12' = last (n # nl12)" and d_nl12': "distinct nl12'"
+      and s_nl12': "set nl12' \<subseteq> {n} \<union> set nl12"
+      using 1(1)[THEN allE[of _ "n # nl12"]] p12 unfolding nl nl1 by auto
+      thus ?thesis apply(intro exI[of _ nl12']) unfolding nl nl1 by auto
+    qed
+  qed
+qed
+
+lemma path_subtr:
+assumes f: "\<And> n. root (f n) = n"
+and p: "path f nl"
+shows "subtr (set nl) (f (last nl)) (f (hd nl))"
+using p proof (induct rule: path.induct)
+  case (Ind n1 nl n)  let ?ns1 = "insert n1 (set nl)"
+  have "path f (n1 # nl)"
+  and "subtr ?ns1 (f (last (n1 # nl))) (f n1)"
+  and fn1: "Inr (f n1) \<in> cont (f n)" using Ind by simp_all
+  hence fn1_flast:  "subtr (insert n ?ns1) (f (last (n1 # nl))) (f n1)"
+  by (metis subset_insertI subtr_mono)
+  have 1: "last (n # n1 # nl) = last (n1 # nl)" by auto
+  have "subtr (insert n ?ns1) (f (last (n1 # nl))) (f n)"
+  using f subtr.Step[OF _ fn1_flast fn1] by auto
+  thus ?case unfolding 1 by simp
+qed (metis f hd.simps last_ConsL last_in_set not_Cons_self2 subtr.Refl)
+
+lemma reg_subtr_path_aux:
+assumes f: "reg f tr" and n: "subtr ns tr1 tr"
+shows "\<exists> nl. path f nl \<and> f (hd nl) = tr \<and> f (last nl) = tr1 \<and> set nl \<subseteq> ns"
+using n f proof(induct rule: subtr.induct)
+  case (Refl tr ns)
+  thus ?case
+  apply(intro exI[of _ "[root tr]"]) apply simp by (metis (lifting) path.Base reg_root)
+next
+  case (Step tr ns tr2 tr1)
+  hence rtr: "root tr \<in> ns" and tr1_tr: "Inr tr1 \<in> cont tr"
+  and tr2_tr1: "subtr ns tr2 tr1" and tr: "reg f tr" by auto
+  have tr1: "reg f tr1" using reg_subtr[OF tr] rtr tr1_tr
+  by (metis (lifting) Step.prems iso_tuple_UNIV_I reg_def subtr.Step)
+  obtain nl where nl: "path f nl" and f_nl: "f (hd nl) = tr1"
+  and last_nl: "f (last nl) = tr2" and set: "set nl \<subseteq> ns" using Step(3)[OF tr1] by auto
+  have 0: "path f (root tr # nl)" apply (subst path.simps)
+  using f_nl nl reg_root tr tr1_tr by (metis hd.simps neq_Nil_conv)
+  show ?case apply(rule exI[of _ "(root tr) # nl"])
+  using 0 reg_root tr last_nl nl path_NE rtr set by auto
+qed
+
+lemma reg_subtr_path:
+assumes f: "reg f tr" and n: "subtr ns tr1 tr"
+shows "\<exists> nl. distinct nl \<and> path f nl \<and> f (hd nl) = tr \<and> f (last nl) = tr1 \<and> set nl \<subseteq> ns"
+using reg_subtr_path_aux[OF assms] path_distinct[of f]
+by (metis (lifting) order_trans)
+
+lemma subtr_iff_path:
+assumes r: "reg f tr" and f: "\<And> n. root (f n) = n"
+shows "subtr ns tr1 tr \<longleftrightarrow>
+       (\<exists> nl. distinct nl \<and> path f nl \<and> f (hd nl) = tr \<and> f (last nl) = tr1 \<and> set nl \<subseteq> ns)"
+proof safe
+  fix nl assume p: "path f nl" and nl: "set nl \<subseteq> ns"
+  have "subtr (set nl) (f (last nl)) (f (hd nl))"
+  apply(rule path_subtr) using p f by simp_all
+  thus "subtr ns (f (last nl)) (f (hd nl))"
+  using subtr_mono nl by auto
+qed(insert reg_subtr_path[OF r], auto)
+
+lemma inFr_iff_path:
+assumes r: "reg f tr" and f: "\<And> n. root (f n) = n"
+shows
+"inFr ns tr t \<longleftrightarrow>
+ (\<exists> nl tr1. distinct nl \<and> path f nl \<and> f (hd nl) = tr \<and> f (last nl) = tr1 \<and>
+            set nl \<subseteq> ns \<and> Inl t \<in> cont tr1)"
+apply safe
+apply (metis (no_types) inFr_subtr r reg_subtr_path)
+by (metis f inFr.Base path_subtr subtr_inFr subtr_mono subtr_rootL_in)
+
+
+
+subsection{* The regular cut of a tree *}
+
+context fixes tr0 :: Tree
+begin
+
+(* Picking a subtree of a certain root: *)
+definition "pick n \<equiv> SOME tr. subtr UNIV tr tr0 \<and> root tr = n"
+
+lemma pick:
+assumes "inItr UNIV tr0 n"
+shows "subtr UNIV (pick n) tr0 \<and> root (pick n) = n"
+proof-
+  have "\<exists> tr. subtr UNIV tr tr0 \<and> root tr = n"
+  using assms by (metis (lifting) inItr_subtr)
+  thus ?thesis unfolding pick_def by(rule someI_ex)
+qed
+
+lemmas subtr_pick = pick[THEN conjunct1]
+lemmas root_pick = pick[THEN conjunct2]
+
+lemma dtree_pick:
+assumes tr0: "dtree tr0" and n: "inItr UNIV tr0 n"
+shows "dtree (pick n)"
+using dtree_subtr[OF tr0 subtr_pick[OF n]] .
+
+definition "regOf_r n \<equiv> root (pick n)"
+definition "regOf_c n \<equiv> (id \<oplus> root) ` cont (pick n)"
+
+(* The regular tree of a function: *)
+definition regOf :: "N \<Rightarrow> Tree" where
+"regOf \<equiv> unfold regOf_r regOf_c"
+
+lemma finite_regOf_c: "finite (regOf_c n)"
+unfolding regOf_c_def by (metis finite_cont finite_imageI)
+
+lemma root_regOf_pick: "root (regOf n) = root (pick n)"
+using unfold(1)[of regOf_r regOf_c n] unfolding regOf_def regOf_r_def by simp
+
+lemma root_regOf[simp]:
+assumes "inItr UNIV tr0 n"
+shows "root (regOf n) = n"
+unfolding root_regOf_pick root_pick[OF assms] ..
+
+lemma cont_regOf[simp]:
+"cont (regOf n) = (id \<oplus> (regOf o root)) ` cont (pick n)"
+apply(subst id_o[symmetric, of id]) unfolding sum_map.comp[symmetric]
+unfolding image_compose unfolding regOf_c_def[symmetric]
+using unfold(2)[of regOf_c n regOf_r, OF finite_regOf_c]
+unfolding regOf_def ..
+
+lemma Inl_cont_regOf[simp]:
+"Inl -` (cont (regOf n)) = Inl -` (cont (pick n))"
+unfolding cont_regOf by simp
+
+lemma Inr_cont_regOf:
+"Inr -` (cont (regOf n)) = (regOf \<circ> root) ` (Inr -` cont (pick n))"
+unfolding cont_regOf by simp
+
+lemma subtr_regOf:
+assumes n: "inItr UNIV tr0 n" and "subtr UNIV tr1 (regOf n)"
+shows "\<exists> n1. inItr UNIV tr0 n1 \<and> tr1 = regOf n1"
+proof-
+  {fix tr ns assume "subtr UNIV tr1 tr"
+   hence "tr = regOf n \<longrightarrow> (\<exists> n1. inItr UNIV tr0 n1 \<and> tr1 = regOf n1)"
+   proof (induct rule: subtr_UNIV_inductL)
+     case (Step tr2 tr1 tr)
+     show ?case proof
+       assume "tr = regOf n"
+       then obtain n1 where tr2: "Inr tr2 \<in> cont tr1"
+       and tr1_tr: "subtr UNIV tr1 tr" and n1: "inItr UNIV tr0 n1" and tr1: "tr1 = regOf n1"
+       using Step by auto
+       obtain tr2' where tr2: "tr2 = regOf (root tr2')"
+       and tr2': "Inr tr2' \<in> cont (pick n1)"
+       using tr2 Inr_cont_regOf[of n1]
+       unfolding tr1 image_def o_def using vimage_eq by auto
+       have "inItr UNIV tr0 (root tr2')"
+       using inItr.Base inItr.Ind n1 pick subtr_inItr tr2' by (metis iso_tuple_UNIV_I)
+       thus "\<exists>n2. inItr UNIV tr0 n2 \<and> tr2 = regOf n2" using tr2 by blast
+     qed
+   qed(insert n, auto)
+  }
+  thus ?thesis using assms by auto
+qed
+
+lemma root_regOf_root:
+assumes n: "inItr UNIV tr0 n" and t_tr: "t_tr \<in> cont (pick n)"
+shows "(id \<oplus> (root \<circ> regOf \<circ> root)) t_tr = (id \<oplus> root) t_tr"
+using assms apply(cases t_tr)
+  apply (metis (lifting) sum_map.simps(1))
+  using pick regOf_def regOf_r_def unfold(1)
+      inItr.Base o_apply subtr_StepL subtr_inItr sum_map.simps(2)
+  by (metis UNIV_I)
+
+lemma regOf_P:
+assumes tr0: "dtree tr0" and n: "inItr UNIV tr0 n"
+shows "(n, (id \<oplus> root) ` cont (regOf n)) \<in> P" (is "?L \<in> P")
+proof-
+  have "?L = (n, (id \<oplus> root) ` cont (pick n))"
+  unfolding cont_regOf image_compose[symmetric] sum_map.comp id_o o_assoc
+  unfolding Pair_eq apply(rule conjI[OF refl]) apply(rule image_cong[OF refl])
+  by(rule root_regOf_root[OF n])
+  moreover have "... \<in> P" by (metis (lifting) dtree_pick root_pick dtree_P n tr0)
+  ultimately show ?thesis by simp
+qed
+
+lemma dtree_regOf:
+assumes tr0: "dtree tr0" and "inItr UNIV tr0 n"
+shows "dtree (regOf n)"
+proof-
+  {fix tr have "\<exists> n. inItr UNIV tr0 n \<and> tr = regOf n \<Longrightarrow> dtree tr"
+   proof (induct rule: dtree_raw_coind)
+     case (Hyp tr)
+     then obtain n where n: "inItr UNIV tr0 n" and tr: "tr = regOf n" by auto
+     show ?case unfolding lift_def apply safe
+     apply (metis (lifting) regOf_P root_regOf n tr tr0)
+     unfolding tr Inr_cont_regOf unfolding inj_on_def apply clarsimp using root_regOf
+     apply (metis UNIV_I inItr.Base n pick subtr2.simps subtr_inItr subtr_subtr2)
+     by (metis n subtr.Refl subtr_StepL subtr_regOf tr UNIV_I)
+   qed
+  }
+  thus ?thesis using assms by blast
+qed
+
+(* The regular cut of a tree: *)
+definition "rcut \<equiv> regOf (root tr0)"
+
+theorem reg_rcut: "reg regOf rcut"
+unfolding reg_def rcut_def
+by (metis inItr.Base root_regOf subtr_regOf UNIV_I)
+
+lemma rcut_reg:
+assumes "reg regOf tr0"
+shows "rcut = tr0"
+using assms unfolding rcut_def reg_def by (metis subtr.Refl UNIV_I)
+
+theorem rcut_eq: "rcut = tr0 \<longleftrightarrow> reg regOf tr0"
+using reg_rcut rcut_reg by metis
+
+theorem regular_rcut: "regular rcut"
+using reg_rcut unfolding regular_def by blast
+
+theorem Fr_rcut: "Fr UNIV rcut \<subseteq> Fr UNIV tr0"
+proof safe
+  fix t assume "t \<in> Fr UNIV rcut"
+  then obtain tr where t: "Inl t \<in> cont tr" and tr: "subtr UNIV tr (regOf (root tr0))"
+  using Fr_subtr[of UNIV "regOf (root tr0)"] unfolding rcut_def
+  by (metis (full_types) Fr_def inFr_subtr mem_Collect_eq)
+  obtain n where n: "inItr UNIV tr0 n" and tr: "tr = regOf n" using tr
+  by (metis (lifting) inItr.Base subtr_regOf UNIV_I)
+  have "Inl t \<in> cont (pick n)" using t using Inl_cont_regOf[of n] unfolding tr
+  by (metis (lifting) vimageD vimageI2)
+  moreover have "subtr UNIV (pick n) tr0" using subtr_pick[OF n] ..
+  ultimately show "t \<in> Fr UNIV tr0" unfolding Fr_subtr_cont by auto
+qed
+
+theorem dtree_rcut:
+assumes "dtree tr0"
+shows "dtree rcut"
+unfolding rcut_def using dtree_regOf[OF assms inItr.Base] by simp
+
+theorem root_rcut[simp]: "root rcut = root tr0"
+unfolding rcut_def
+by (metis (lifting) root_regOf inItr.Base reg_def reg_root subtr_rootR_in)
+
+end (* context *)
+
+
+subsection{* Recursive description of the regular tree frontiers *}
+
+lemma regular_inFr:
+assumes r: "regular tr" and In: "root tr \<in> ns"
+and t: "inFr ns tr t"
+shows "t \<in> Inl -` (cont tr) \<or>
+       (\<exists> tr'. Inr tr' \<in> cont tr \<and> inFr (ns - {root tr}) tr' t)"
+(is "?L \<or> ?R")
+proof-
+  obtain f where r: "reg f tr" and f: "\<And>n. root (f n) = n"
+  using r unfolding regular_def2 by auto
+  obtain nl tr1 where d_nl: "distinct nl" and p: "path f nl" and hd_nl: "f (hd nl) = tr"
+  and l_nl: "f (last nl) = tr1" and s_nl: "set nl \<subseteq> ns" and t_tr1: "Inl t \<in> cont tr1"
+  using t unfolding inFr_iff_path[OF r f] by auto
+  obtain n nl1 where nl: "nl = n # nl1" by (metis (lifting) p path.simps)
+  hence f_n: "f n = tr" using hd_nl by simp
+  have n_nl1: "n \<notin> set nl1" using d_nl unfolding nl by auto
+  show ?thesis
+  proof(cases nl1)
+    case Nil hence "tr = tr1" using f_n l_nl unfolding nl by simp
+    hence ?L using t_tr1 by simp thus ?thesis by simp
+  next
+    case (Cons n1 nl2) note nl1 = Cons
+    have 1: "last nl1 = last nl" "hd nl1 = n1" unfolding nl nl1 by simp_all
+    have p1: "path f nl1" and n1_tr: "Inr (f n1) \<in> cont tr"
+    using path.simps[of f nl] p f_n unfolding nl nl1 by auto
+    have r1: "reg f (f n1)" using reg_Inr_cont[OF r n1_tr] .
+    have 0: "inFr (set nl1) (f n1) t" unfolding inFr_iff_path[OF r1 f]
+    apply(intro exI[of _ nl1], intro exI[of _ tr1])
+    using d_nl unfolding 1 l_nl unfolding nl using p1 t_tr1 by auto
+    have root_tr: "root tr = n" by (metis f f_n)
+    have "inFr (ns - {root tr}) (f n1) t" apply(rule inFr_mono[OF 0])
+    using s_nl unfolding root_tr unfolding nl using n_nl1 by auto
+    thus ?thesis using n1_tr by auto
+  qed
+qed
+
+theorem regular_Fr:
+assumes r: "regular tr" and In: "root tr \<in> ns"
+shows "Fr ns tr =
+       Inl -` (cont tr) \<union>
+       \<Union> {Fr (ns - {root tr}) tr' | tr'. Inr tr' \<in> cont tr}"
+unfolding Fr_def
+using In inFr.Base regular_inFr[OF assms] apply safe
+apply (simp, metis (full_types) mem_Collect_eq)
+apply simp
+by (simp, metis (lifting) inFr_Ind_minus insert_Diff)
+
+
+subsection{* The generated languages *}
+
+(* The (possibly inifinite tree) generated language *)
+definition "L ns n \<equiv> {Fr ns tr | tr. dtree tr \<and> root tr = n}"
+
+(* The regular-tree generated language *)
+definition "Lr ns n \<equiv> {Fr ns tr | tr. dtree tr \<and> root tr = n \<and> regular tr}"
+
+theorem L_rec_notin:
+assumes "n \<notin> ns"
+shows "L ns n = {{}}"
+using assms unfolding L_def apply safe
+  using not_root_Fr apply force
+  apply(rule exI[of _ "deftr n"])
+  by (metis (no_types) dtree_deftr not_root_Fr root_deftr)
+
+theorem Lr_rec_notin:
+assumes "n \<notin> ns"
+shows "Lr ns n = {{}}"
+using assms unfolding Lr_def apply safe
+  using not_root_Fr apply force
+  apply(rule exI[of _ "deftr n"])
+  by (metis (no_types) regular_def dtree_deftr not_root_Fr reg_deftr root_deftr)
+
+lemma dtree_subtrOf:
+assumes "dtree tr" and "Inr n \<in> prodOf tr"
+shows "dtree (subtrOf tr n)"
+by (metis assms dtree_lift lift_def subtrOf)
+
+theorem Lr_rec_in:
+assumes n: "n \<in> ns"
+shows "Lr ns n \<subseteq>
+{Inl -` tns \<union> (\<Union> {K n' | n'. Inr n' \<in> tns}) | tns K.
+    (n,tns) \<in> P \<and>
+    (\<forall> n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> Lr (ns - {n}) n')}"
+(is "Lr ns n \<subseteq> {?F tns K | tns K. (n,tns) \<in> P \<and> ?\<phi> tns K}")
+proof safe
+  fix ts assume "ts \<in> Lr ns n"
+  then obtain tr where dtr: "dtree tr" and r: "root tr = n" and tr: "regular tr"
+  and ts: "ts = Fr ns tr" unfolding Lr_def by auto
+  def tns \<equiv> "(id \<oplus> root) ` (cont tr)"
+  def K \<equiv> "\<lambda> n'. Fr (ns - {n}) (subtrOf tr n')"
+  show "\<exists>tns K. ts = ?F tns K \<and> (n, tns) \<in> P \<and> ?\<phi> tns K"
+  apply(rule exI[of _ tns], rule exI[of _ K]) proof(intro conjI allI impI)
+    show "ts = Inl -` tns \<union> \<Union>{K n' |n'. Inr n' \<in> tns}"
+    unfolding ts regular_Fr[OF tr n[unfolded r[symmetric]]]
+    unfolding tns_def K_def r[symmetric]
+    unfolding Inl_prodOf dtree_subtrOf_Union[OF dtr] ..
+    show "(n, tns) \<in> P" unfolding tns_def r[symmetric] using dtree_P[OF dtr] .
+    fix n' assume "Inr n' \<in> tns" thus "K n' \<in> Lr (ns - {n}) n'"
+    unfolding K_def Lr_def mem_Collect_eq apply(intro exI[of _ "subtrOf tr n'"])
+    using dtr tr apply(intro conjI refl)  unfolding tns_def
+      apply(erule dtree_subtrOf[OF dtr])
+      apply (metis subtrOf)
+      by (metis Inr_subtrOf UNIV_I regular_subtr subtr.simps)
+  qed
+qed
+
+lemma hsubst_aux:
+fixes n ftr tns
+assumes n: "n \<in> ns" and tns: "finite tns" and
+1: "\<And> n'. Inr n' \<in> tns \<Longrightarrow> dtree (ftr n')"
+defines "tr \<equiv> Node n ((id \<oplus> ftr) ` tns)"  defines "tr' \<equiv> hsubst tr tr"
+shows "Fr ns tr' = Inl -` tns \<union> \<Union>{Fr (ns - {n}) (ftr n') |n'. Inr n' \<in> tns}"
+(is "_ = ?B") proof-
+  have rtr: "root tr = n" and ctr: "cont tr = (id \<oplus> ftr) ` tns"
+  unfolding tr_def using tns by auto
+  have Frr: "Frr (ns - {n}) tr = \<Union>{Fr (ns - {n}) (ftr n') |n'. Inr n' \<in> tns}"
+  unfolding Frr_def ctr by auto
+  have "Fr ns tr' = Inl -` (cont tr) \<union> Frr (ns - {n}) tr"
+  using Fr_self_hsubst[OF n[unfolded rtr[symmetric]]] unfolding tr'_def rtr ..
+  also have "... = ?B" unfolding ctr Frr by simp
+  finally show ?thesis .
+qed
+
+theorem L_rec_in:
+assumes n: "n \<in> ns"
+shows "
+{Inl -` tns \<union> (\<Union> {K n' | n'. Inr n' \<in> tns}) | tns K.
+    (n,tns) \<in> P \<and>
+    (\<forall> n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> L (ns - {n}) n')}
+ \<subseteq> L ns n"
+proof safe
+  fix tns K
+  assume P: "(n, tns) \<in> P" and 0: "\<forall>n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> L (ns - {n}) n'"
+  {fix n' assume "Inr n' \<in> tns"
+   hence "K n' \<in> L (ns - {n}) n'" using 0 by auto
+   hence "\<exists> tr'. K n' = Fr (ns - {n}) tr' \<and> dtree tr' \<and> root tr' = n'"
+   unfolding L_def mem_Collect_eq by auto
+  }
+  then obtain ftr where 0: "\<And> n'. Inr n' \<in> tns \<Longrightarrow>
+  K n' = Fr (ns - {n}) (ftr n') \<and> dtree (ftr n') \<and> root (ftr n') = n'"
+  by metis
+  def tr \<equiv> "Node n ((id \<oplus> ftr) ` tns)"  def tr' \<equiv> "hsubst tr tr"
+  have rtr: "root tr = n" and ctr: "cont tr = (id \<oplus> ftr) ` tns"
+  unfolding tr_def by (simp, metis P cont_Node finite_imageI finite_in_P)
+  have prtr: "prodOf tr = tns" apply(rule Inl_Inr_image_cong)
+  unfolding ctr apply simp apply simp apply safe
+  using 0 unfolding image_def apply force apply simp by (metis 0 vimageI2)
+  have 1: "{K n' |n'. Inr n' \<in> tns} = {Fr (ns - {n}) (ftr n') |n'. Inr n' \<in> tns}"
+  using 0 by auto
+  have dtr: "dtree tr" apply(rule dtree.Tree)
+    apply (metis (lifting) P prtr rtr)
+    unfolding inj_on_def ctr lift_def using 0 by auto
+  hence dtr': "dtree tr'" unfolding tr'_def by (metis dtree_hsubst)
+  have tns: "finite tns" using finite_in_P P by simp
+  have "Inl -` tns \<union> \<Union>{Fr (ns - {n}) (ftr n') |n'. Inr n' \<in> tns} \<in> L ns n"
+  unfolding L_def mem_Collect_eq apply(intro exI[of _ tr'] conjI)
+  using dtr' 0 hsubst_aux[OF assms tns, of ftr] unfolding tr_def tr'_def by auto
+  thus "Inl -` tns \<union> \<Union>{K n' |n'. Inr n' \<in> tns} \<in> L ns n" unfolding 1 .
+qed
+
+lemma card_N: "(n::N) \<in> ns \<Longrightarrow> card (ns - {n}) < card ns"
+by (metis finite_N Diff_UNIV Diff_infinite_finite card_Diff1_less finite.emptyI)
+
+function LL where
+"LL ns n =
+ (if n \<notin> ns then {{}} else
+ {Inl -` tns \<union> (\<Union> {K n' | n'. Inr n' \<in> tns}) | tns K.
+    (n,tns) \<in> P \<and>
+    (\<forall> n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> LL (ns - {n}) n')})"
+by(pat_completeness, auto)
+termination apply(relation "inv_image (measure card) fst")
+using card_N by auto
+
+declare LL.simps[code]  (* TODO: Does code generation for LL work? *)
+declare LL.simps[simp del]
+
+theorem Lr_LL: "Lr ns n \<subseteq> LL ns n"
+proof (induct ns arbitrary: n rule: measure_induct[of card])
+  case (1 ns n) show ?case proof(cases "n \<in> ns")
+    case False thus ?thesis unfolding Lr_rec_notin[OF False] by (simp add: LL.simps)
+  next
+    case True show ?thesis apply(rule subset_trans)
+    using Lr_rec_in[OF True] apply assumption
+    unfolding LL.simps[of ns n] using True 1 card_N proof clarsimp
+      fix tns K
+      assume "n \<in> ns" hence c: "card (ns - {n}) < card ns" using card_N by blast
+      assume "(n, tns) \<in> P"
+      and "\<forall>n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> Lr (ns - {n}) n'"
+      thus "\<exists>tnsa Ka.
+             Inl -` tns \<union> \<Union>{K n' |n'. Inr n' \<in> tns} =
+             Inl -` tnsa \<union> \<Union>{Ka n' |n'. Inr n' \<in> tnsa} \<and>
+             (n, tnsa) \<in> P \<and> (\<forall>n'. Inr n' \<in> tnsa \<longrightarrow> Ka n' \<in> LL (ns - {n}) n')"
+      apply(intro exI[of _ tns] exI[of _ K]) using c 1 by auto
+    qed
+  qed
+qed
+
+theorem LL_L: "LL ns n \<subseteq> L ns n"
+proof (induct ns arbitrary: n rule: measure_induct[of card])
+  case (1 ns n) show ?case proof(cases "n \<in> ns")
+    case False thus ?thesis unfolding L_rec_notin[OF False] by (simp add: LL.simps)
+  next
+    case True show ?thesis apply(rule subset_trans)
+    prefer 2 using L_rec_in[OF True] apply assumption
+    unfolding LL.simps[of ns n] using True 1 card_N proof clarsimp
+      fix tns K
+      assume "n \<in> ns" hence c: "card (ns - {n}) < card ns" using card_N by blast
+      assume "(n, tns) \<in> P"
+      and "\<forall>n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> LL (ns - {n}) n'"
+      thus "\<exists>tnsa Ka.
+             Inl -` tns \<union> \<Union>{K n' |n'. Inr n' \<in> tns} =
+             Inl -` tnsa \<union> \<Union>{Ka n' |n'. Inr n' \<in> tnsa} \<and>
+             (n, tnsa) \<in> P \<and> (\<forall>n'. Inr n' \<in> tnsa \<longrightarrow> Ka n' \<in> L (ns - {n}) n')"
+      apply(intro exI[of _ tns] exI[of _ K]) using c 1 by auto
+    qed
+  qed
+qed
+
+(* The subsumpsion relation between languages *)
+definition "subs L1 L2 \<equiv> \<forall> ts2 \<in> L2. \<exists> ts1 \<in> L1. ts1 \<subseteq> ts2"
+
+lemma incl_subs[simp]: "L2 \<subseteq> L1 \<Longrightarrow> subs L1 L2"
+unfolding subs_def by auto
+
+lemma subs_refl[simp]: "subs L1 L1" unfolding subs_def by auto
+
+lemma subs_trans: "\<lbrakk>subs L1 L2; subs L2 L3\<rbrakk> \<Longrightarrow> subs L1 L3"
+unfolding subs_def by (metis subset_trans)
+
+(* Language equivalence *)
+definition "leqv L1 L2 \<equiv> subs L1 L2 \<and> subs L2 L1"
+
+lemma subs_leqv[simp]: "leqv L1 L2 \<Longrightarrow> subs L1 L2"
+unfolding leqv_def by auto
+
+lemma subs_leqv_sym[simp]: "leqv L1 L2 \<Longrightarrow> subs L2 L1"
+unfolding leqv_def by auto
+
+lemma leqv_refl[simp]: "leqv L1 L1" unfolding leqv_def by auto
+
+lemma leqv_trans:
+assumes 12: "leqv L1 L2" and 23: "leqv L2 L3"
+shows "leqv L1 L3"
+using assms unfolding leqv_def by (metis (lifting) subs_trans)
+
+lemma leqv_sym: "leqv L1 L2 \<Longrightarrow> leqv L2 L1"
+unfolding leqv_def by auto
+
+lemma leqv_Sym: "leqv L1 L2 \<longleftrightarrow> leqv L2 L1"
+unfolding leqv_def by auto
+
+lemma Lr_incl_L: "Lr ns ts \<subseteq> L ns ts"
+unfolding Lr_def L_def by auto
+
+lemma Lr_subs_L: "subs (Lr UNIV ts) (L UNIV ts)"
+unfolding subs_def proof safe
+  fix ts2 assume "ts2 \<in> L UNIV ts"
+  then obtain tr where ts2: "ts2 = Fr UNIV tr" and dtr: "dtree tr" and rtr: "root tr = ts"
+  unfolding L_def by auto
+  thus "\<exists>ts1\<in>Lr UNIV ts. ts1 \<subseteq> ts2"
+  apply(intro bexI[of _ "Fr UNIV (rcut tr)"])
+  unfolding Lr_def L_def using Fr_rcut dtree_rcut root_rcut regular_rcut by auto
+qed
+
+theorem Lr_leqv_L: "leqv (Lr UNIV ts) (L UNIV ts)"
+using Lr_subs_L unfolding leqv_def by (metis (lifting) Lr_incl_L incl_subs)
+
+theorem LL_leqv_L: "leqv (LL UNIV ts) (L UNIV ts)"
+by (metis (lifting) LL_L Lr_LL Lr_subs_L incl_subs leqv_def subs_trans)
+
+theorem LL_leqv_Lr: "leqv (LL UNIV ts) (Lr UNIV ts)"
+using Lr_leqv_L LL_leqv_L by (metis leqv_Sym leqv_trans)
+
+
+end
diff -r 4a15873c4ec9 -r 41ee3bfccb4d src/HOL/BNF/Examples/Derivation_Trees/Parallel.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/BNF/Examples/Derivation_Trees/Parallel.thy	Tue Oct 16 13:09:46 2012 +0200
@@ -0,0 +1,152 @@
+(*  Title:      HOL/BNF/Examples/Infinite_Derivation_Trees/Parallel.thy
+    Author:     Andrei Popescu, TU Muenchen
+    Copyright   2012
+
+Parallel composition.
+*)
+
+header {* Parallel Composition *}
+
+theory Parallel
+imports Tree
+begin
+
+no_notation plus_class.plus (infixl "+" 65)
+no_notation Sublist.parallel (infixl "\<parallel>" 50)
+    
+consts Nplus :: "N \<Rightarrow> N \<Rightarrow> N" (infixl "+" 60)
+
+axiomatization where
+    Nplus_comm: "(a::N) + b = b + (a::N)"
+and Nplus_assoc: "((a::N) + b) + c = a + (b + c)"
+
+section{* Parallel composition *}
+
+fun par_r where "par_r (tr1,tr2) = root tr1 + root tr2"
+fun par_c where
+"par_c (tr1,tr2) =
+ Inl ` (Inl -` (cont tr1 \<union> cont tr2)) \<union>
+ Inr ` (Inr -` cont tr1 \<times> Inr -` cont tr2)"
+
+declare par_r.simps[simp del]  declare par_c.simps[simp del]
+
+definition par :: "Tree \<times> Tree \<Rightarrow> Tree" where
+"par \<equiv> unfold par_r par_c"
+
+abbreviation par_abbr (infixr "\<parallel>" 80) where "tr1 \<parallel> tr2 \<equiv> par (tr1, tr2)"
+
+lemma finite_par_c: "finite (par_c (tr1, tr2))"
+unfolding par_c.simps apply(rule finite_UnI)
+  apply (metis finite_Un finite_cont finite_imageI finite_vimageI inj_Inl)
+  apply(intro finite_imageI finite_cartesian_product finite_vimageI)
+  using finite_cont by auto
+
+lemma root_par: "root (tr1 \<parallel> tr2) = root tr1 + root tr2"
+using unfold(1)[of par_r par_c "(tr1,tr2)"] unfolding par_def par_r.simps by simp
+
+lemma cont_par:
+"cont (tr1 \<parallel> tr2) = (id \<oplus> par) ` par_c (tr1,tr2)"
+using unfold(2)[of par_c "(tr1,tr2)" par_r, OF finite_par_c]
+unfolding par_def ..
+
+lemma Inl_cont_par[simp]:
+"Inl -` (cont (tr1 \<parallel> tr2)) = Inl -` (cont tr1 \<union> cont tr2)"
+unfolding cont_par par_c.simps by auto
+
+lemma Inr_cont_par[simp]:
+"Inr -` (cont (tr1 \<parallel> tr2)) = par ` (Inr -` cont tr1 \<times> Inr -` cont tr2)"
+unfolding cont_par par_c.simps by auto
+
+lemma Inl_in_cont_par:
+"Inl t \<in> cont (tr1 \<parallel> tr2) \<longleftrightarrow> (Inl t \<in> cont tr1 \<or> Inl t \<in> cont tr2)"
+using Inl_cont_par[of tr1 tr2] unfolding vimage_def by auto
+
+lemma Inr_in_cont_par:
+"Inr t \<in> cont (tr1 \<parallel> tr2) \<longleftrightarrow> (t \<in> par ` (Inr -` cont tr1 \<times> Inr -` cont tr2))"
+using Inr_cont_par[of tr1 tr2] unfolding vimage_def by auto
+
+
+section{* =-coinductive proofs *}
+
+(* Detailed proofs of commutativity and associativity: *)
+theorem par_com: "tr1 \<parallel> tr2 = tr2 \<parallel> tr1"
+proof-
+  let ?\<phi> = "\<lambda> trA trB. \<exists> tr1 tr2. trA = tr1 \<parallel> tr2 \<and> trB = tr2 \<parallel> tr1"
+  {fix trA trB
+   assume "?\<phi> trA trB" hence "trA = trB"
+   proof (induct rule: Tree_coind, safe)
+     fix tr1 tr2
+     show "root (tr1 \<parallel> tr2) = root (tr2 \<parallel> tr1)"
+     unfolding root_par by (rule Nplus_comm)
+   next
+     fix tr1 tr2 :: Tree
+     let ?trA = "tr1 \<parallel> tr2"  let ?trB = "tr2 \<parallel> tr1"
+     show "(?\<phi> ^#2) (cont ?trA) (cont ?trB)"
+     unfolding lift2_def proof(intro conjI allI impI)
+       fix n show "Inl n \<in> cont (tr1 \<parallel> tr2) \<longleftrightarrow> Inl n \<in> cont (tr2 \<parallel> tr1)"
+       unfolding Inl_in_cont_par by auto
+     next
+       fix trA' assume "Inr trA' \<in> cont ?trA"
+       then obtain tr1' tr2' where "trA' = tr1' \<parallel> tr2'"
+       and "Inr tr1' \<in> cont tr1" and "Inr tr2' \<in> cont tr2"
+       unfolding Inr_in_cont_par by auto
+       thus "\<exists> trB'. Inr trB' \<in> cont ?trB \<and> ?\<phi> trA' trB'"
+       apply(intro exI[of _ "tr2' \<parallel> tr1'"]) unfolding Inr_in_cont_par by auto
+     next
+       fix trB' assume "Inr trB' \<in> cont ?trB"
+       then obtain tr1' tr2' where "trB' = tr2' \<parallel> tr1'"
+       and "Inr tr1' \<in> cont tr1" and "Inr tr2' \<in> cont tr2"
+       unfolding Inr_in_cont_par by auto
+       thus "\<exists> trA'. Inr trA' \<in> cont ?trA \<and> ?\<phi> trA' trB'"
+       apply(intro exI[of _ "tr1' \<parallel> tr2'"]) unfolding Inr_in_cont_par by auto
+     qed
+   qed
+  }
+  thus ?thesis by blast
+qed
+
+theorem par_assoc: "(tr1 \<parallel> tr2) \<parallel> tr3 = tr1 \<parallel> (tr2 \<parallel> tr3)"
+proof-
+  let ?\<phi> =
+  "\<lambda> trA trB. \<exists> tr1 tr2 tr3. trA = (tr1 \<parallel> tr2) \<parallel> tr3 \<and>
+                             trB = tr1 \<parallel> (tr2 \<parallel> tr3)"
+  {fix trA trB
+   assume "?\<phi> trA trB" hence "trA = trB"
+   proof (induct rule: Tree_coind, safe)
+     fix tr1 tr2 tr3
+     show "root ((tr1 \<parallel> tr2) \<parallel> tr3) = root (tr1 \<parallel> (tr2 \<parallel> tr3))"
+     unfolding root_par by (rule Nplus_assoc)
+   next
+     fix tr1 tr2 tr3
+     let ?trA = "(tr1 \<parallel> tr2) \<parallel> tr3"  let ?trB = "tr1 \<parallel> (tr2 \<parallel> tr3)"
+     show "(?\<phi> ^#2) (cont ?trA) (cont ?trB)"
+     unfolding lift2_def proof(intro conjI allI impI)
+       fix n show "Inl n \<in> (cont ?trA) \<longleftrightarrow> Inl n \<in> (cont ?trB)"
+       unfolding Inl_in_cont_par by simp
+     next
+       fix trA' assume "Inr trA' \<in> cont ?trA"
+       then obtain tr1' tr2' tr3' where "trA' = (tr1' \<parallel> tr2') \<parallel> tr3'"
+       and "Inr tr1' \<in> cont tr1" and "Inr tr2' \<in> cont tr2"
+       and "Inr tr3' \<in> cont tr3" unfolding Inr_in_cont_par by auto
+       thus "\<exists> trB'. Inr trB' \<in> cont ?trB \<and> ?\<phi> trA' trB'"
+       apply(intro exI[of _ "tr1' \<parallel> (tr2' \<parallel> tr3')"])
+       unfolding Inr_in_cont_par by auto
+     next
+       fix trB' assume "Inr trB' \<in> cont ?trB"
+       then obtain tr1' tr2' tr3' where "trB' = tr1' \<parallel> (tr2' \<parallel> tr3')"
+       and "Inr tr1' \<in> cont tr1" and "Inr tr2' \<in> cont tr2"
+       and "Inr tr3' \<in> cont tr3" unfolding Inr_in_cont_par by auto
+       thus "\<exists> trA'. Inr trA' \<in> cont ?trA \<and> ?\<phi> trA' trB'"
+       apply(intro exI[of _ "(tr1' \<parallel> tr2') \<parallel> tr3'"])
+       unfolding Inr_in_cont_par by auto
+     qed
+   qed
+  }
+  thus ?thesis by blast
+qed
+
+
+
+
+
+end
\ No newline at end of file
diff -r 4a15873c4ec9 -r 41ee3bfccb4d src/HOL/BNF/Examples/Derivation_Trees/Prelim.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/BNF/Examples/Derivation_Trees/Prelim.thy	Tue Oct 16 13:09:46 2012 +0200
@@ -0,0 +1,67 @@
+(*  Title:      HOL/BNF/Examples/Infinite_Derivation_Trees/Prelim.thy
+    Author:     Andrei Popescu, TU Muenchen
+    Copyright   2012
+
+Preliminaries.
+*)
+
+header {* Preliminaries *}
+
+theory Prelim
+imports "../../BNF"
+begin
+
+declare fset_to_fset[simp]
+
+lemma fst_snd_convol_o[simp]: "<fst o s, snd o s> = s"
+apply(rule ext) by (simp add: convol_def)
+
+abbreviation sm_abbrev (infix "\<oplus>" 60)
+where "f \<oplus> g \<equiv> Sum_Type.sum_map f g"
+
+lemma sum_map_InlD: "(f \<oplus> g) z = Inl x \<Longrightarrow> \<exists>y. z = Inl y \<and> f y = x"
+by (cases z) auto
+
+lemma sum_map_InrD: "(f \<oplus> g) z = Inr x \<Longrightarrow> \<exists>y. z = Inr y \<and> g y = x"
+by (cases z) auto
+
+abbreviation sum_case_abbrev ("[[_,_]]" 800)
+where "[[f,g]] \<equiv> Sum_Type.sum_case f g"
+
+lemma inj_Inl[simp]: "inj Inl" unfolding inj_on_def by auto
+lemma inj_Inr[simp]: "inj Inr" unfolding inj_on_def by auto
+
+lemma Inl_oplus_elim:
+assumes "Inl tr \<in> (id \<oplus> f) ` tns"
+shows "Inl tr \<in> tns"
+using assms apply clarify by (case_tac x, auto)
+
+lemma Inl_oplus_iff[simp]: "Inl tr \<in> (id \<oplus> f) ` tns \<longleftrightarrow> Inl tr \<in> tns"
+using Inl_oplus_elim
+by (metis id_def image_iff sum_map.simps(1))
+
+lemma Inl_m_oplus[simp]: "Inl -` (id \<oplus> f) ` tns = Inl -` tns"
+using Inl_oplus_iff unfolding vimage_def by auto
+
+lemma Inr_oplus_elim:
+assumes "Inr tr \<in> (id \<oplus> f) ` tns"
+shows "\<exists> n. Inr n \<in> tns \<and> f n = tr"
+using assms apply clarify by (case_tac x, auto)
+
+lemma Inr_oplus_iff[simp]:
+"Inr tr \<in> (id \<oplus> f) ` tns \<longleftrightarrow> (\<exists> n. Inr n \<in> tns \<and> f n = tr)"
+apply (rule iffI)
+ apply (metis Inr_oplus_elim)
+by (metis image_iff sum_map.simps(2))
+
+lemma Inr_m_oplus[simp]: "Inr -` (id \<oplus> f) ` tns = f ` (Inr -` tns)"
+using Inr_oplus_iff unfolding vimage_def by auto
+
+lemma Inl_Inr_image_cong:
+assumes "Inl -` A = Inl -` B" and "Inr -` A = Inr -` B"
+shows "A = B"
+apply safe using assms apply(case_tac x, auto) by(case_tac x, auto)
+
+
+
+end
\ No newline at end of file
diff -r 4a15873c4ec9 -r 41ee3bfccb4d src/HOL/BNF/Examples/Derivation_Trees/Tree.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/BNF/Examples/Derivation_Trees/Tree.thy	Tue Oct 16 13:09:46 2012 +0200
@@ -0,0 +1,192 @@
+(*  Title:      HOL/BNF/Examples/Infinite_Derivation_Trees/Tree.thy
+    Author:     Andrei Popescu, TU Muenchen
+    Copyright   2012
+
+Trees with nonterminal internal nodes and terminal leaves.
+*)
+
+header {* Trees with Nonterminal Internal Nodes and Terminal Leaves *}
+
+theory Tree
+imports Prelim
+begin
+
+hide_fact (open) Quotient_Product.prod_rel_def
+
+typedecl N
+typedecl T
+
+codata Tree = NNode (root: N) (ccont: "(T + Tree) fset")
+
+
+section {* Sugar notations for Tree *}
+
+definition
+"llift2 \<phi> as1 as2 \<longleftrightarrow>
+ (\<forall> n. Inl n \<in> fset as1 \<longleftrightarrow> Inl n \<in> fset as2) \<and>
+ (\<forall> tr1. Inr tr1 \<in> fset as1 \<longrightarrow> (\<exists> tr2. Inr tr2 \<in> fset as2 \<and> \<phi> tr1 tr2)) \<and>
+ (\<forall> tr2. Inr tr2 \<in> fset as2 \<longrightarrow> (\<exists> tr1. Inr tr1 \<in> fset as1 \<and> \<phi> tr1 tr2))"
+
+lemma pre_Tree_rel: "pre_Tree_rel \<phi> (n1,as1) (n2,as2) \<longleftrightarrow> n1 = n2 \<and> llift2 \<phi> as1 as2"
+unfolding llift2_def pre_Tree_rel_def sum_rel_def[abs_def] prod_rel_def fset_rel_def split_conv
+apply (auto split: sum.splits)
+apply (metis sumE)
+apply (metis sumE)
+apply (metis sumE)
+apply (metis sumE)
+apply (metis sumE sum.simps(1,2,4))
+apply (metis sumE sum.simps(1,2,4))
+done
+
+
+subsection{* Coinduction *}
+
+theorem Tree_coind_NNode[elim, consumes 1, case_names NNode, induct pred: "HOL.eq"]:
+assumes phi: "\<phi> tr1 tr2" and
+NNode: "\<And> n1 n2 as1 as2.
+          \<lbrakk>\<phi> (NNode n1 as1) (NNode n2 as2)\<rbrakk> \<Longrightarrow>
+          n1 = n2 \<and> llift2 \<phi> as1 as2"
+shows "tr1 = tr2"
+apply(rule mp[OF Tree.dtor_coinduct[of \<phi> tr1 tr2] phi]) proof clarify
+  fix tr1 tr2  assume \<phi>: "\<phi> tr1 tr2"
+  show "pre_Tree_rel \<phi> (Tree_dtor tr1) (Tree_dtor tr2)"
+  apply(cases rule: Tree.ctor_exhaust[of tr1], cases rule: Tree.ctor_exhaust[of tr2])
+  apply (simp add: Tree.dtor_ctor)
+  apply(case_tac x, case_tac xa, simp)
+  unfolding pre_Tree_rel apply(rule NNode) using \<phi> unfolding NNode_def by simp
+qed
+
+theorem TTree_coind[elim, consumes 1, case_names LLift]:
+assumes phi: "\<phi> tr1 tr2" and
+LLift: "\<And> tr1 tr2. \<phi> tr1 tr2 \<Longrightarrow> root tr1 = root tr2 \<and> llift2 \<phi> (ccont tr1) (ccont tr2)"
+shows "tr1 = tr2"
+using phi apply(induct rule: Tree_coind_NNode)
+using LLift by (metis Tree.sels)
+
+
+subsection{* The characteristic theorems transported from fset to set *}
+
+definition "Node n as \<equiv> NNode n (the_inv fset as)"
+definition "cont \<equiv> fset o ccont"
+definition "unfold rt ct \<equiv> Tree_unfold rt (the_inv fset o ct)"
+definition "corec rt qt ct dt \<equiv> Tree_corec rt qt (the_inv fset o ct) (the_inv fset o dt)"
+
+definition lift ("_ ^#" 200) where
+"lift \<phi> as \<longleftrightarrow> (\<forall> tr. Inr tr \<in> as \<longrightarrow> \<phi> tr)"
+
+definition lift2 ("_ ^#2" 200) where
+"lift2 \<phi> as1 as2 \<longleftrightarrow>
+ (\<forall> n. Inl n \<in> as1 \<longleftrightarrow> Inl n \<in> as2) \<and>
+ (\<forall> tr1. Inr tr1 \<in> as1 \<longrightarrow> (\<exists> tr2. Inr tr2 \<in> as2 \<and> \<phi> tr1 tr2)) \<and>
+ (\<forall> tr2. Inr tr2 \<in> as2 \<longrightarrow> (\<exists> tr1. Inr tr1 \<in> as1 \<and> \<phi> tr1 tr2))"
+
+definition liftS ("_ ^#s" 200) where
+"liftS trs = {as. Inr -` as \<subseteq> trs}"
+
+lemma lift2_llift2:
+"\<lbrakk>finite as1; finite as2\<rbrakk> \<Longrightarrow>
+ lift2 \<phi> as1 as2 \<longleftrightarrow> llift2 \<phi> (the_inv fset as1) (the_inv fset as2)"
+unfolding lift2_def llift2_def by auto
+
+lemma llift2_lift2:
+"llift2 \<phi> as1 as2 \<longleftrightarrow> lift2 \<phi> (fset as1) (fset as2)"
+using lift2_llift2 by (metis finite_fset fset_cong fset_to_fset)
+
+lemma mono_lift:
+assumes "(\<phi>^#) as"
+and "\<And> tr. \<phi> tr \<Longrightarrow> \<phi>' tr"
+shows "(\<phi>'^#) as"
+using assms unfolding lift_def[abs_def] by blast
+
+lemma mono_liftS:
+assumes "trs1 \<subseteq> trs2 "
+shows "(trs1 ^#s) \<subseteq> (trs2 ^#s)"
+using assms unfolding liftS_def[abs_def] by blast
+
+lemma lift_mono:
+assumes "\<phi> \<le> \<phi>'"
+shows "(\<phi>^#) \<le> (\<phi>'^#)"
+using assms unfolding lift_def[abs_def] by blast
+
+lemma mono_lift2:
+assumes "(\<phi>^#2) as1 as2"
+and "\<And> tr1 tr2. \<phi> tr1 tr2 \<Longrightarrow> \<phi>' tr1 tr2"
+shows "(\<phi>'^#2) as1 as2"
+using assms unfolding lift2_def[abs_def] by blast
+
+lemma lift2_mono:
+assumes "\<phi> \<le> \<phi>'"
+shows "(\<phi>^#2) \<le> (\<phi>'^#2)"
+using assms unfolding lift2_def[abs_def] by blast
+
+lemma finite_cont[simp]: "finite (cont tr)"
+unfolding cont_def by auto
+
+theorem Node_root_cont[simp]:
+"Node (root tr) (cont tr) = tr"
+using Tree.collapse unfolding Node_def cont_def
+by (metis cont_def finite_cont fset_cong fset_to_fset o_def)
+
+theorem Tree_simps[simp]:
+assumes "finite as" and "finite as'"
+shows "Node n as = Node n' as' \<longleftrightarrow> n = n' \<and> as = as'"
+using assms Tree.inject unfolding Node_def
+by (metis fset_to_fset)
+
+theorem Tree_cases[elim, case_names Node Choice]:
+assumes Node: "\<And> n as. \<lbrakk>finite as; tr = Node n as\<rbrakk> \<Longrightarrow> phi"
+shows phi
+apply(cases rule: Tree.exhaust[of tr])
+using Node unfolding Node_def
+by (metis Node Node_root_cont finite_cont)
+
+theorem Tree_sel_ctor[simp]:
+"root (Node n as) = n"
+"finite as \<Longrightarrow> cont (Node n as) = as"
+unfolding Node_def cont_def by auto
+
+theorems root_Node = Tree_sel_ctor(1)
+theorems cont_Node = Tree_sel_ctor(2)
+
+theorem Tree_coind_Node[elim, consumes 1, case_names Node]:
+assumes phi: "\<phi> tr1 tr2" and
+Node:
+"\<And> n1 n2 as1 as2.
+   \<lbrakk>finite as1; finite as2; \<phi> (Node n1 as1) (Node n2 as2)\<rbrakk>
+   \<Longrightarrow> n1 = n2 \<and> (\<phi>^#2) as1 as2"
+shows "tr1 = tr2"
+using phi apply(induct rule: Tree_coind_NNode)
+unfolding llift2_lift2 apply(rule Node)
+unfolding Node_def
+apply (metis finite_fset)
+apply (metis finite_fset)
+by (metis finite_fset fset_cong fset_to_fset)
+
+theorem Tree_coind[elim, consumes 1, case_names Lift, induct pred: "HOL.eq"]:
+assumes phi: "\<phi> tr1 tr2" and
+Lift: "\<And> tr1 tr2. \<phi> tr1 tr2 \<Longrightarrow>
+                  root tr1 = root tr2 \<and> (\<phi>^#2) (cont tr1) (cont tr2)"
+shows "tr1 = tr2"
+using phi apply(induct rule: TTree_coind)
+unfolding llift2_lift2 apply(rule Lift[unfolded cont_def comp_def]) .
+
+theorem unfold:
+"root (unfold rt ct b) = rt b"
+"finite (ct b) \<Longrightarrow> cont (unfold rt ct b) = image (id \<oplus> unfold rt ct) (ct b)"
+using Tree.sel_unfold[of rt "the_inv fset \<circ> ct" b] unfolding unfold_def
+apply - apply metis
+unfolding cont_def comp_def
+by (metis (no_types) fset_to_fset map_fset_image)
+
+theorem corec:
+"root (corec rt qt ct dt b) = rt b"
+"\<lbrakk>finite (ct b); finite (dt b)\<rbrakk> \<Longrightarrow>
+ cont (corec rt qt ct dt b) =
+ (if qt b then ct b else image (id \<oplus> corec rt qt ct dt) (dt b))"
+using Tree.sel_corec[of rt qt "the_inv fset \<circ> ct" "the_inv fset \<circ> dt" b] unfolding corec_def
+apply -
+apply simp
+unfolding cont_def comp_def id_def
+by (metis (no_types) fset_to_fset map_fset_image)
+
+end
diff -r 4a15873c4ec9 -r 41ee3bfccb4d src/HOL/BNF/Examples/Infinite_Derivation_Trees/Gram_Lang.thy
--- a/src/HOL/BNF/Examples/Infinite_Derivation_Trees/Gram_Lang.thy	Mon Oct 15 19:03:02 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1374 +0,0 @@
-(*  Title:      HOL/BNF/Examples/Infinite_Derivation_Trees/Gram_Lang.thy
-    Author:     Andrei Popescu, TU Muenchen
-    Copyright   2012
-
-Language of a grammar.
-*)
-
-header {* Language of a Grammar *}
-
-theory Gram_Lang
-imports Tree
-begin
-
-
-consts P :: "(N \<times> (T + N) set) set"
-axiomatization where
-    finite_N: "finite (UNIV::N set)"
-and finite_in_P: "\<And> n tns. (n,tns) \<in> P \<longrightarrow> finite tns"
-and used: "\<And> n. \<exists> tns. (n,tns) \<in> P"
-
-
-subsection{* Tree basics: frontier, interior, etc. *}
-
-lemma Tree_cong:
-assumes "root tr = root tr'" and "cont tr = cont tr'"
-shows "tr = tr'"
-by (metis Node_root_cont assms)
-
-inductive finiteT where
-Node: "\<lbrakk>finite as; (finiteT^#) as\<rbrakk> \<Longrightarrow> finiteT (Node a as)"
-monos lift_mono
-
-lemma finiteT_induct[consumes 1, case_names Node, induct pred: finiteT]:
-assumes 1: "finiteT tr"
-and IH: "\<And>as n. \<lbrakk>finite as; (\<phi>^#) as\<rbrakk> \<Longrightarrow> \<phi> (Node n as)"
-shows "\<phi> tr"
-using 1 apply(induct rule: finiteT.induct)
-apply(rule IH) apply assumption apply(elim mono_lift) by simp
-
-
-(* Frontier *)
-
-inductive inFr :: "N set \<Rightarrow> Tree \<Rightarrow> T \<Rightarrow> bool" where
-Base: "\<lbrakk>root tr \<in> ns; Inl t \<in> cont tr\<rbrakk> \<Longrightarrow> inFr ns tr t"
-|
-Ind: "\<lbrakk>root tr \<in> ns; Inr tr1 \<in> cont tr; inFr ns tr1 t\<rbrakk> \<Longrightarrow> inFr ns tr t"
-
-definition "Fr ns tr \<equiv> {t. inFr ns tr t}"
-
-lemma inFr_root_in: "inFr ns tr t \<Longrightarrow> root tr \<in> ns"
-by (metis inFr.simps)
-
-lemma inFr_mono:
-assumes "inFr ns tr t" and "ns \<subseteq> ns'"
-shows "inFr ns' tr t"
-using assms apply(induct arbitrary: ns' rule: inFr.induct)
-using Base Ind by (metis inFr.simps set_mp)+
-
-lemma inFr_Ind_minus:
-assumes "inFr ns1 tr1 t" and "Inr tr1 \<in> cont tr"
-shows "inFr (insert (root tr) ns1) tr t"
-using assms apply(induct rule: inFr.induct)
-  apply (metis inFr.simps insert_iff)
-  by (metis inFr.simps inFr_mono insertI1 subset_insertI)
-
-(* alternative definition *)
-inductive inFr2 :: "N set \<Rightarrow> Tree \<Rightarrow> T \<Rightarrow> bool" where
-Base: "\<lbrakk>root tr \<in> ns; Inl t \<in> cont tr\<rbrakk> \<Longrightarrow> inFr2 ns tr t"
-|
-Ind: "\<lbrakk>Inr tr1 \<in> cont tr; inFr2 ns1 tr1 t\<rbrakk>
-      \<Longrightarrow> inFr2 (insert (root tr) ns1) tr t"
-
-lemma inFr2_root_in: "inFr2 ns tr t \<Longrightarrow> root tr \<in> ns"
-apply(induct rule: inFr2.induct) by auto
-
-lemma inFr2_mono:
-assumes "inFr2 ns tr t" and "ns \<subseteq> ns'"
-shows "inFr2 ns' tr t"
-using assms apply(induct arbitrary: ns' rule: inFr2.induct)
-using Base Ind
-apply (metis subsetD) by (metis inFr2.simps insert_absorb insert_subset)
-
-lemma inFr2_Ind:
-assumes "inFr2 ns tr1 t" and "root tr \<in> ns" and "Inr tr1 \<in> cont tr"
-shows "inFr2 ns tr t"
-using assms apply(induct rule: inFr2.induct)
-  apply (metis inFr2.simps insert_absorb)
-  by (metis inFr2.simps insert_absorb)
-
-lemma inFr_inFr2:
-"inFr = inFr2"
-apply (rule ext)+  apply(safe)
-  apply(erule inFr.induct)
-    apply (metis (lifting) inFr2.Base)
-    apply (metis (lifting) inFr2_Ind)
-  apply(erule inFr2.induct)
-    apply (metis (lifting) inFr.Base)
-    apply (metis (lifting) inFr_Ind_minus)
-done
-
-lemma not_root_inFr:
-assumes "root tr \<notin> ns"
-shows "\<not> inFr ns tr t"
-by (metis assms inFr_root_in)
-
-theorem not_root_Fr:
-assumes "root tr \<notin> ns"
-shows "Fr ns tr = {}"
-using not_root_inFr[OF assms] unfolding Fr_def by auto
-
-
-(* Interior *)
-
-inductive inItr :: "N set \<Rightarrow> Tree \<Rightarrow> N \<Rightarrow> bool" where
-Base: "root tr \<in> ns \<Longrightarrow> inItr ns tr (root tr)"
-|
-Ind: "\<lbrakk>root tr \<in> ns; Inr tr1 \<in> cont tr; inItr ns tr1 n\<rbrakk> \<Longrightarrow> inItr ns tr n"
-
-definition "Itr ns tr \<equiv> {n. inItr ns tr n}"
-
-lemma inItr_root_in: "inItr ns tr n \<Longrightarrow> root tr \<in> ns"
-by (metis inItr.simps)
-
-lemma inItr_mono:
-assumes "inItr ns tr n" and "ns \<subseteq> ns'"
-shows "inItr ns' tr n"
-using assms apply(induct arbitrary: ns' rule: inItr.induct)
-using Base Ind by (metis inItr.simps set_mp)+
-
-
-(* The subtree relation *)
-
-inductive subtr where
-Refl: "root tr \<in> ns \<Longrightarrow> subtr ns tr tr"
-|
-Step: "\<lbrakk>root tr3 \<in> ns; subtr ns tr1 tr2; Inr tr2 \<in> cont tr3\<rbrakk> \<Longrightarrow> subtr ns tr1 tr3"
-
-lemma subtr_rootL_in:
-assumes "subtr ns tr1 tr2"
-shows "root tr1 \<in> ns"
-using assms apply(induct rule: subtr.induct) by auto
-
-lemma subtr_rootR_in:
-assumes "subtr ns tr1 tr2"
-shows "root tr2 \<in> ns"
-using assms apply(induct rule: subtr.induct) by auto
-
-lemmas subtr_roots_in = subtr_rootL_in subtr_rootR_in
-
-lemma subtr_mono:
-assumes "subtr ns tr1 tr2" and "ns \<subseteq> ns'"
-shows "subtr ns' tr1 tr2"
-using assms apply(induct arbitrary: ns' rule: subtr.induct)
-using Refl Step by (metis subtr.simps set_mp)+
-
-lemma subtr_trans_Un:
-assumes "subtr ns12 tr1 tr2" and "subtr ns23 tr2 tr3"
-shows "subtr (ns12 \<union> ns23) tr1 tr3"
-proof-
-  have "subtr ns23 tr2 tr3  \<Longrightarrow>
-        (\<forall> ns12 tr1. subtr ns12 tr1 tr2 \<longrightarrow> subtr (ns12 \<union> ns23) tr1 tr3)"
-  apply(induct  rule: subtr.induct, safe)
-    apply (metis subtr_mono sup_commute sup_ge2)
-    by (metis (lifting) Step UnI2)
-  thus ?thesis using assms by auto
-qed
-
-lemma subtr_trans:
-assumes "subtr ns tr1 tr2" and "subtr ns tr2 tr3"
-shows "subtr ns tr1 tr3"
-using subtr_trans_Un[OF assms] by simp
-
-lemma subtr_StepL:
-assumes r: "root tr1 \<in> ns" and tr12: "Inr tr1 \<in> cont tr2" and s: "subtr ns tr2 tr3"
-shows "subtr ns tr1 tr3"
-apply(rule subtr_trans[OF _ s])
-apply(rule Step[of tr2 ns tr1 tr1])
-apply(rule subtr_rootL_in[OF s])
-apply(rule Refl[OF r])
-apply(rule tr12)
-done
-
-(* alternative definition: *)
-inductive subtr2 where
-Refl: "root tr \<in> ns \<Longrightarrow> subtr2 ns tr tr"
-|
-Step: "\<lbrakk>root tr1 \<in> ns; Inr tr1 \<in> cont tr2; subtr2 ns tr2 tr3\<rbrakk> \<Longrightarrow> subtr2 ns tr1 tr3"
-
-lemma subtr2_rootL_in:
-assumes "subtr2 ns tr1 tr2"
-shows "root tr1 \<in> ns"
-using assms apply(induct rule: subtr2.induct) by auto
-
-lemma subtr2_rootR_in:
-assumes "subtr2 ns tr1 tr2"
-shows "root tr2 \<in> ns"
-using assms apply(induct rule: subtr2.induct) by auto
-
-lemmas subtr2_roots_in = subtr2_rootL_in subtr2_rootR_in
-
-lemma subtr2_mono:
-assumes "subtr2 ns tr1 tr2" and "ns \<subseteq> ns'"
-shows "subtr2 ns' tr1 tr2"
-using assms apply(induct arbitrary: ns' rule: subtr2.induct)
-using Refl Step by (metis subtr2.simps set_mp)+
-
-lemma subtr2_trans_Un:
-assumes "subtr2 ns12 tr1 tr2" and "subtr2 ns23 tr2 tr3"
-shows "subtr2 (ns12 \<union> ns23) tr1 tr3"
-proof-
-  have "subtr2 ns12 tr1 tr2  \<Longrightarrow>
-        (\<forall> ns23 tr3. subtr2 ns23 tr2 tr3 \<longrightarrow> subtr2 (ns12 \<union> ns23) tr1 tr3)"
-  apply(induct  rule: subtr2.induct, safe)
-    apply (metis subtr2_mono sup_commute sup_ge2)
-    by (metis Un_iff subtr2.simps)
-  thus ?thesis using assms by auto
-qed
-
-lemma subtr2_trans:
-assumes "subtr2 ns tr1 tr2" and "subtr2 ns tr2 tr3"
-shows "subtr2 ns tr1 tr3"
-using subtr2_trans_Un[OF assms] by simp
-
-lemma subtr2_StepR:
-assumes r: "root tr3 \<in> ns" and tr23: "Inr tr2 \<in> cont tr3" and s: "subtr2 ns tr1 tr2"
-shows "subtr2 ns tr1 tr3"
-apply(rule subtr2_trans[OF s])
-apply(rule Step[of _ _ tr3])
-apply(rule subtr2_rootR_in[OF s])
-apply(rule tr23)
-apply(rule Refl[OF r])
-done
-
-lemma subtr_subtr2:
-"subtr = subtr2"
-apply (rule ext)+  apply(safe)
-  apply(erule subtr.induct)
-    apply (metis (lifting) subtr2.Refl)
-    apply (metis (lifting) subtr2_StepR)
-  apply(erule subtr2.induct)
-    apply (metis (lifting) subtr.Refl)
-    apply (metis (lifting) subtr_StepL)
-done
-
-lemma subtr_inductL[consumes 1, case_names Refl Step]:
-assumes s: "subtr ns tr1 tr2" and Refl: "\<And>ns tr. \<phi> ns tr tr"
-and Step:
-"\<And>ns tr1 tr2 tr3.
-   \<lbrakk>root tr1 \<in> ns; Inr tr1 \<in> cont tr2; subtr ns tr2 tr3; \<phi> ns tr2 tr3\<rbrakk> \<Longrightarrow> \<phi> ns tr1 tr3"
-shows "\<phi> ns tr1 tr2"
-using s unfolding subtr_subtr2 apply(rule subtr2.induct)
-using Refl Step unfolding subtr_subtr2 by auto
-
-lemma subtr_UNIV_inductL[consumes 1, case_names Refl Step]:
-assumes s: "subtr UNIV tr1 tr2" and Refl: "\<And>tr. \<phi> tr tr"
-and Step:
-"\<And>tr1 tr2 tr3.
-   \<lbrakk>Inr tr1 \<in> cont tr2; subtr UNIV tr2 tr3; \<phi> tr2 tr3\<rbrakk> \<Longrightarrow> \<phi> tr1 tr3"
-shows "\<phi> tr1 tr2"
-using s apply(induct rule: subtr_inductL)
-apply(rule Refl) using Step subtr_mono by (metis subset_UNIV)
-
-(* Subtree versus frontier: *)
-lemma subtr_inFr:
-assumes "inFr ns tr t" and "subtr ns tr tr1"
-shows "inFr ns tr1 t"
-proof-
-  have "subtr ns tr tr1 \<Longrightarrow> (\<forall> t. inFr ns tr t \<longrightarrow> inFr ns tr1 t)"
-  apply(induct rule: subtr.induct, safe) by (metis inFr.Ind)
-  thus ?thesis using assms by auto
-qed
-
-corollary Fr_subtr:
-"Fr ns tr = \<Union> {Fr ns tr' | tr'. subtr ns tr' tr}"
-unfolding Fr_def proof safe
-  fix t assume t: "inFr ns tr t"  hence "root tr \<in> ns" by (rule inFr_root_in)
-  thus "t \<in> \<Union>{{t. inFr ns tr' t} |tr'. subtr ns tr' tr}"
-  apply(intro UnionI[of "{t. inFr ns tr t}" _ t]) using t subtr.Refl by auto
-qed(metis subtr_inFr)
-
-lemma inFr_subtr:
-assumes "inFr ns tr t"
-shows "\<exists> tr'. subtr ns tr' tr \<and> Inl t \<in> cont tr'"
-using assms apply(induct rule: inFr.induct) apply safe
-  apply (metis subtr.Refl)
-  by (metis (lifting) subtr.Step)
-
-corollary Fr_subtr_cont:
-"Fr ns tr = \<Union> {Inl -` cont tr' | tr'. subtr ns tr' tr}"
-unfolding Fr_def
-apply safe
-apply (frule inFr_subtr)
-apply auto
-by (metis inFr.Base subtr_inFr subtr_rootL_in)
-
-(* Subtree versus interior: *)
-lemma subtr_inItr:
-assumes "inItr ns tr n" and "subtr ns tr tr1"
-shows "inItr ns tr1 n"
-proof-
-  have "subtr ns tr tr1 \<Longrightarrow> (\<forall> t. inItr ns tr n \<longrightarrow> inItr ns tr1 n)"
-  apply(induct rule: subtr.induct, safe) by (metis inItr.Ind)
-  thus ?thesis using assms by auto
-qed
-
-corollary Itr_subtr:
-"Itr ns tr = \<Union> {Itr ns tr' | tr'. subtr ns tr' tr}"
-unfolding Itr_def apply safe
-apply (metis (lifting, mono_tags) UnionI inItr_root_in mem_Collect_eq subtr.Refl)
-by (metis subtr_inItr)
-
-lemma inItr_subtr:
-assumes "inItr ns tr n"
-shows "\<exists> tr'. subtr ns tr' tr \<and> root tr' = n"
-using assms apply(induct rule: inItr.induct) apply safe
-  apply (metis subtr.Refl)
-  by (metis (lifting) subtr.Step)
-
-corollary Itr_subtr_cont:
-"Itr ns tr = {root tr' | tr'. subtr ns tr' tr}"
-unfolding Itr_def apply safe
-  apply (metis (lifting, mono_tags) inItr_subtr)
-  by (metis inItr.Base subtr_inItr subtr_rootL_in)
-
-
-subsection{* The immediate subtree function *}
-
-(* production of: *)
-abbreviation "prodOf tr \<equiv> (id \<oplus> root) ` (cont tr)"
-(* subtree of: *)
-definition "subtrOf tr n \<equiv> SOME tr'. Inr tr' \<in> cont tr \<and> root tr' = n"
-
-lemma subtrOf:
-assumes n: "Inr n \<in> prodOf tr"
-shows "Inr (subtrOf tr n) \<in> cont tr \<and> root (subtrOf tr n) = n"
-proof-
-  obtain tr' where "Inr tr' \<in> cont tr \<and> root tr' = n"
-  using n unfolding image_def by (metis (lifting) Inr_oplus_elim assms)
-  thus ?thesis unfolding subtrOf_def by(rule someI)
-qed
-
-lemmas Inr_subtrOf = subtrOf[THEN conjunct1]
-lemmas root_subtrOf[simp] = subtrOf[THEN conjunct2]
-
-lemma Inl_prodOf: "Inl -` (prodOf tr) = Inl -` (cont tr)"
-proof safe
-  fix t ttr assume "Inl t = (id \<oplus> root) ttr" and "ttr \<in> cont tr"
-  thus "t \<in> Inl -` cont tr" by(cases ttr, auto)
-next
-  fix t assume "Inl t \<in> cont tr" thus "t \<in> Inl -` prodOf tr"
-  by (metis (lifting) id_def image_iff sum_map.simps(1) vimageI2)
-qed
-
-lemma root_prodOf:
-assumes "Inr tr' \<in> cont tr"
-shows "Inr (root tr') \<in> prodOf tr"
-by (metis (lifting) assms image_iff sum_map.simps(2))
-
-
-subsection{* Derivation trees *}
-
-coinductive dtree where
-Tree: "\<lbrakk>(root tr, (id \<oplus> root) ` (cont tr)) \<in> P; inj_on root (Inr -` cont tr);
-        lift dtree (cont tr)\<rbrakk> \<Longrightarrow> dtree tr"
-monos lift_mono
-
-(* destruction rules: *)
-lemma dtree_P:
-assumes "dtree tr"
-shows "(root tr, (id \<oplus> root) ` (cont tr)) \<in> P"
-using assms unfolding dtree.simps by auto
-
-lemma dtree_inj_on:
-assumes "dtree tr"
-shows "inj_on root (Inr -` cont tr)"
-using assms unfolding dtree.simps by auto
-
-lemma dtree_inj[simp]:
-assumes "dtree tr" and "Inr tr1 \<in> cont tr" and "Inr tr2 \<in> cont tr"
-shows "root tr1 = root tr2 \<longleftrightarrow> tr1 = tr2"
-using assms dtree_inj_on unfolding inj_on_def by auto
-
-lemma dtree_lift:
-assumes "dtree tr"
-shows "lift dtree (cont tr)"
-using assms unfolding dtree.simps by auto
-
-
-(* coinduction:*)
-lemma dtree_coind[elim, consumes 1, case_names Hyp]:
-assumes phi: "\<phi> tr"
-and Hyp:
-"\<And> tr. \<phi> tr \<Longrightarrow>
-       (root tr, image (id \<oplus> root) (cont tr)) \<in> P \<and>
-       inj_on root (Inr -` cont tr) \<and>
-       lift (\<lambda> tr. \<phi> tr \<or> dtree tr) (cont tr)"
-shows "dtree tr"
-apply(rule dtree.coinduct[of \<phi> tr, OF phi])
-using Hyp by blast
-
-lemma dtree_raw_coind[elim, consumes 1, case_names Hyp]:
-assumes phi: "\<phi> tr"
-and Hyp:
-"\<And> tr. \<phi> tr \<Longrightarrow>
-       (root tr, image (id \<oplus> root) (cont tr)) \<in> P \<and>
-       inj_on root (Inr -` cont tr) \<and>
-       lift \<phi> (cont tr)"
-shows "dtree tr"
-using phi apply(induct rule: dtree_coind)
-using Hyp mono_lift
-by (metis (mono_tags) mono_lift)
-
-lemma dtree_subtr_inj_on:
-assumes d: "dtree tr1" and s: "subtr ns tr tr1"
-shows "inj_on root (Inr -` cont tr)"
-using s d apply(induct rule: subtr.induct)
-apply (metis (lifting) dtree_inj_on) by (metis dtree_lift lift_def)
-
-lemma dtree_subtr_P:
-assumes d: "dtree tr1" and s: "subtr ns tr tr1"
-shows "(root tr, (id \<oplus> root) ` cont tr) \<in> P"
-using s d apply(induct rule: subtr.induct)
-apply (metis (lifting) dtree_P) by (metis dtree_lift lift_def)
-
-lemma subtrOf_root[simp]:
-assumes tr: "dtree tr" and cont: "Inr tr' \<in> cont tr"
-shows "subtrOf tr (root tr') = tr'"
-proof-
-  have 0: "Inr (subtrOf tr (root tr')) \<in> cont tr" using Inr_subtrOf
-  by (metis (lifting) cont root_prodOf)
-  have "root (subtrOf tr (root tr')) = root tr'"
-  using root_subtrOf by (metis (lifting) cont root_prodOf)
-  thus ?thesis unfolding dtree_inj[OF tr 0 cont] .
-qed
-
-lemma surj_subtrOf:
-assumes "dtree tr" and 0: "Inr tr' \<in> cont tr"
-shows "\<exists> n. Inr n \<in> prodOf tr \<and> subtrOf tr n = tr'"
-apply(rule exI[of _ "root tr'"])
-using root_prodOf[OF 0] subtrOf_root[OF assms] by simp
-
-lemma dtree_subtr:
-assumes "dtree tr1" and "subtr ns tr tr1"
-shows "dtree tr"
-proof-
-  have "(\<exists> ns tr1. dtree tr1 \<and> subtr ns tr tr1) \<Longrightarrow> dtree tr"
-  proof (induct rule: dtree_raw_coind)
-    case (Hyp tr)
-    then obtain ns tr1 where tr1: "dtree tr1" and tr_tr1: "subtr ns tr tr1" by auto
-    show ?case unfolding lift_def proof safe
-      show "(root tr, (id \<oplus> root) ` cont tr) \<in> P" using dtree_subtr_P[OF tr1 tr_tr1] .
-    next
-      show "inj_on root (Inr -` cont tr)" using dtree_subtr_inj_on[OF tr1 tr_tr1] .
-    next
-      fix tr' assume tr': "Inr tr' \<in> cont tr"
-      have tr_tr1: "subtr (ns \<union> {root tr'}) tr tr1" using subtr_mono[OF tr_tr1] by auto
-      have "subtr (ns \<union> {root tr'}) tr' tr1" using subtr_StepL[OF _ tr' tr_tr1] by auto
-      thus "\<exists>ns' tr1. dtree tr1 \<and> subtr ns' tr' tr1" using tr1 by blast
-    qed
-  qed
-  thus ?thesis using assms by auto
-qed
-
-
-subsection{* Default trees *}
-
-(* Pick a left-hand side of a production for each nonterminal *)
-definition S where "S n \<equiv> SOME tns. (n,tns) \<in> P"
-
-lemma S_P: "(n, S n) \<in> P"
-using used unfolding S_def by(rule someI_ex)
-
-lemma finite_S: "finite (S n)"
-using S_P finite_in_P by auto
-
-
-(* The default tree of a nonterminal *)
-definition deftr :: "N \<Rightarrow> Tree" where
-"deftr \<equiv> unfold id S"
-
-lemma deftr_simps[simp]:
-"root (deftr n) = n"
-"cont (deftr n) = image (id \<oplus> deftr) (S n)"
-using unfold(1)[of id S n] unfold(2)[of S n id, OF finite_S]
-unfolding deftr_def by simp_all
-
-lemmas root_deftr = deftr_simps(1)
-lemmas cont_deftr = deftr_simps(2)
-
-lemma root_o_deftr[simp]: "root o deftr = id"
-by (rule ext, auto)
-
-lemma dtree_deftr: "dtree (deftr n)"
-proof-
-  {fix tr assume "\<exists> n. tr = deftr n" hence "dtree tr"
-   apply(induct rule: dtree_raw_coind) apply safe
-   unfolding deftr_simps image_compose[symmetric] sum_map.comp id_o
-   root_o_deftr sum_map.id image_id id_apply apply(rule S_P)
-   unfolding inj_on_def lift_def by auto
-  }
-  thus ?thesis by auto
-qed
-
-
-subsection{* Hereditary substitution *}
-
-(* Auxiliary concept: The root-ommiting frontier: *)
-definition "inFrr ns tr t \<equiv> \<exists> tr'. Inr tr' \<in> cont tr \<and> inFr ns tr' t"
-definition "Frr ns tr \<equiv> {t. \<exists> tr'. Inr tr' \<in> cont tr \<and> t \<in> Fr ns tr'}"
-
-context
-fixes tr0 :: Tree
-begin
-
-definition "hsubst_r tr \<equiv> root tr"
-definition "hsubst_c tr \<equiv> if root tr = root tr0 then cont tr0 else cont tr"
-
-(* Hereditary substitution: *)
-definition hsubst :: "Tree \<Rightarrow> Tree" where
-"hsubst \<equiv> unfold hsubst_r hsubst_c"
-
-lemma finite_hsubst_c: "finite (hsubst_c n)"
-unfolding hsubst_c_def by (metis (full_types) finite_cont)
-
-lemma root_hsubst[simp]: "root (hsubst tr) = root tr"
-using unfold(1)[of hsubst_r hsubst_c tr] unfolding hsubst_def hsubst_r_def by simp
-
-lemma root_o_subst[simp]: "root o hsubst = root"
-unfolding comp_def root_hsubst ..
-
-lemma cont_hsubst_eq[simp]:
-assumes "root tr = root tr0"
-shows "cont (hsubst tr) = (id \<oplus> hsubst) ` (cont tr0)"
-apply(subst id_o[symmetric, of id]) unfolding id_o
-using unfold(2)[of hsubst_c tr hsubst_r, OF finite_hsubst_c]
-unfolding hsubst_def hsubst_c_def using assms by simp
-
-lemma hsubst_eq:
-assumes "root tr = root tr0"
-shows "hsubst tr = hsubst tr0"
-apply(rule Tree_cong) using assms cont_hsubst_eq by auto
-
-lemma cont_hsubst_neq[simp]:
-assumes "root tr \<noteq> root tr0"
-shows "cont (hsubst tr) = (id \<oplus> hsubst) ` (cont tr)"
-apply(subst id_o[symmetric, of id]) unfolding id_o
-using unfold(2)[of hsubst_c tr hsubst_r, OF finite_hsubst_c]
-unfolding hsubst_def hsubst_c_def using assms by simp
-
-lemma Inl_cont_hsubst_eq[simp]:
-assumes "root tr = root tr0"
-shows "Inl -` cont (hsubst tr) = Inl -` (cont tr0)"
-unfolding cont_hsubst_eq[OF assms] by simp
-
-lemma Inr_cont_hsubst_eq[simp]:
-assumes "root tr = root tr0"
-shows "Inr -` cont (hsubst tr) = hsubst ` Inr -` cont tr0"
-unfolding cont_hsubst_eq[OF assms] by simp
-
-lemma Inl_cont_hsubst_neq[simp]:
-assumes "root tr \<noteq> root tr0"
-shows "Inl -` cont (hsubst tr) = Inl -` (cont tr)"
-unfolding cont_hsubst_neq[OF assms] by simp
-
-lemma Inr_cont_hsubst_neq[simp]:
-assumes "root tr \<noteq> root tr0"
-shows "Inr -` cont (hsubst tr) = hsubst ` Inr -` cont tr"
-unfolding cont_hsubst_neq[OF assms] by simp
-
-lemma dtree_hsubst:
-assumes tr0: "dtree tr0" and tr: "dtree tr"
-shows "dtree (hsubst tr)"
-proof-
-  {fix tr1 have "(\<exists> tr. dtree tr \<and> tr1 = hsubst tr) \<Longrightarrow> dtree tr1"
-   proof (induct rule: dtree_raw_coind)
-     case (Hyp tr1) then obtain tr
-     where dtr: "dtree tr" and tr1: "tr1 = hsubst tr" by auto
-     show ?case unfolding lift_def tr1 proof safe
-       show "(root (hsubst tr), prodOf (hsubst tr)) \<in> P"
-       unfolding tr1 apply(cases "root tr = root tr0")
-       using  dtree_P[OF dtr] dtree_P[OF tr0]
-       by (auto simp add: image_compose[symmetric] sum_map.comp)
-       show "inj_on root (Inr -` cont (hsubst tr))"
-       apply(cases "root tr = root tr0") using dtree_inj_on[OF dtr] dtree_inj_on[OF tr0]
-       unfolding inj_on_def by (auto, blast)
-       fix tr' assume "Inr tr' \<in> cont (hsubst tr)"
-       thus "\<exists>tra. dtree tra \<and> tr' = hsubst tra"
-       apply(cases "root tr = root tr0", simp_all)
-         apply (metis dtree_lift lift_def tr0)
-         by (metis dtr dtree_lift lift_def)
-     qed
-   qed
-  }
-  thus ?thesis using assms by blast
-qed
-
-lemma Frr: "Frr ns tr = {t. inFrr ns tr t}"
-unfolding inFrr_def Frr_def Fr_def by auto
-
-lemma inFr_hsubst_imp:
-assumes "inFr ns (hsubst tr) t"
-shows "t \<in> Inl -` (cont tr0) \<or> inFrr (ns - {root tr0}) tr0 t \<or>
-       inFr (ns - {root tr0}) tr t"
-proof-
-  {fix tr1
-   have "inFr ns tr1 t \<Longrightarrow>
-   (\<And> tr. tr1 = hsubst tr \<Longrightarrow> (t \<in> Inl -` (cont tr0) \<or> inFrr (ns - {root tr0}) tr0 t \<or>
-                              inFr (ns - {root tr0}) tr t))"
-   proof(induct rule: inFr.induct)
-     case (Base tr1 ns t tr)
-     hence rtr: "root tr1 \<in> ns" and t_tr1: "Inl t \<in> cont tr1" and tr1: "tr1 = hsubst tr"
-     by auto
-     show ?case
-     proof(cases "root tr1 = root tr0")
-       case True
-       hence "t \<in> Inl -` (cont tr0)" using t_tr1 unfolding tr1 by auto
-       thus ?thesis by simp
-     next
-       case False
-       hence "inFr (ns - {root tr0}) tr t" using t_tr1 unfolding tr1 apply simp
-       by (metis Base.prems Diff_iff root_hsubst inFr.Base rtr singletonE)
-       thus ?thesis by simp
-     qed
-   next
-     case (Ind tr1 ns tr1' t) note IH = Ind(4)
-     have rtr1: "root tr1 \<in> ns" and tr1'_tr1: "Inr tr1' \<in> cont tr1"
-     and t_tr1': "inFr ns tr1' t" and tr1: "tr1 = hsubst tr" using Ind by auto
-     have rtr1: "root tr1 = root tr" unfolding tr1 by simp
-     show ?case
-     proof(cases "root tr1 = root tr0")
-       case True
-       then obtain tr' where tr'_tr0: "Inr tr' \<in> cont tr0" and tr1': "tr1' = hsubst tr'"
-       using tr1'_tr1 unfolding tr1 by auto
-       show ?thesis using IH[OF tr1'] proof (elim disjE)
-         assume "inFr (ns - {root tr0}) tr' t"
-         thus ?thesis using tr'_tr0 unfolding inFrr_def by auto
-       qed auto
-     next
-       case False
-       then obtain tr' where tr'_tr: "Inr tr' \<in> cont tr" and tr1': "tr1' = hsubst tr'"
-       using tr1'_tr1 unfolding tr1 by auto
-       show ?thesis using IH[OF tr1'] proof (elim disjE)
-         assume "inFr (ns - {root tr0}) tr' t"
-         thus ?thesis using tr'_tr unfolding inFrr_def
-         by (metis Diff_iff False Ind(1) empty_iff inFr2_Ind inFr_inFr2 insert_iff rtr1)
-       qed auto
-     qed
-   qed
-  }
-  thus ?thesis using assms by auto
-qed
-
-lemma inFr_hsubst_notin:
-assumes "inFr ns tr t" and "root tr0 \<notin> ns"
-shows "inFr ns (hsubst tr) t"
-using assms apply(induct rule: inFr.induct)
-apply (metis Inl_cont_hsubst_neq inFr2.Base inFr_inFr2 root_hsubst vimageD vimageI2)
-by (metis (lifting) Inr_cont_hsubst_neq inFr.Ind rev_image_eqI root_hsubst vimageD vimageI2)
-
-lemma inFr_hsubst_minus:
-assumes "inFr (ns - {root tr0}) tr t"
-shows "inFr ns (hsubst tr) t"
-proof-
-  have 1: "inFr (ns - {root tr0}) (hsubst tr) t"
-  using inFr_hsubst_notin[OF assms] by simp
-  show ?thesis using inFr_mono[OF 1] by auto
-qed
-
-lemma inFr_self_hsubst:
-assumes "root tr0 \<in> ns"
-shows
-"inFr ns (hsubst tr0) t \<longleftrightarrow>
- t \<in> Inl -` (cont tr0) \<or> inFrr (ns - {root tr0}) tr0 t"
-(is "?A \<longleftrightarrow> ?B \<or> ?C")
-apply(intro iffI)
-apply (metis inFr_hsubst_imp Diff_iff inFr_root_in insertI1) proof(elim disjE)
-  assume ?B thus ?A apply(intro inFr.Base) using assms by auto
-next
-  assume ?C then obtain tr where
-  tr_tr0: "Inr tr \<in> cont tr0" and t_tr: "inFr (ns - {root tr0}) tr t"
-  unfolding inFrr_def by auto
-  def tr1 \<equiv> "hsubst tr"
-  have 1: "inFr ns tr1 t" using t_tr unfolding tr1_def using inFr_hsubst_minus by auto
-  have "Inr tr1 \<in> cont (hsubst tr0)" unfolding tr1_def using tr_tr0 by auto
-  thus ?A using 1 inFr.Ind assms by (metis root_hsubst)
-qed
-
-theorem Fr_self_hsubst:
-assumes "root tr0 \<in> ns"
-shows "Fr ns (hsubst tr0) = Inl -` (cont tr0) \<union> Frr (ns - {root tr0}) tr0"
-using inFr_self_hsubst[OF assms] unfolding Frr Fr_def by auto
-
-end (* context *)
-
-
-subsection{* Regular trees *}
-
-hide_const regular
-
-definition "reg f tr \<equiv> \<forall> tr'. subtr UNIV tr' tr \<longrightarrow> tr' = f (root tr')"
-definition "regular tr \<equiv> \<exists> f. reg f tr"
-
-lemma reg_def2: "reg f tr \<longleftrightarrow> (\<forall> ns tr'. subtr ns tr' tr \<longrightarrow> tr' = f (root tr'))"
-unfolding reg_def using subtr_mono by (metis subset_UNIV)
-
-lemma regular_def2: "regular tr \<longleftrightarrow> (\<exists> f. reg f tr \<and> (\<forall> n. root (f n) = n))"
-unfolding regular_def proof safe
-  fix f assume f: "reg f tr"
-  def g \<equiv> "\<lambda> n. if inItr UNIV tr n then f n else deftr n"
-  show "\<exists>g. reg g tr \<and> (\<forall>n. root (g n) = n)"
-  apply(rule exI[of _ g])
-  using f deftr_simps(1) unfolding g_def reg_def apply safe
-    apply (metis (lifting) inItr.Base subtr_inItr subtr_rootL_in)
-    by (metis (full_types) inItr_subtr)
-qed auto
-
-lemma reg_root:
-assumes "reg f tr"
-shows "f (root tr) = tr"
-using assms unfolding reg_def
-by (metis (lifting) iso_tuple_UNIV_I subtr.Refl)
-
-
-lemma reg_Inr_cont:
-assumes "reg f tr" and "Inr tr' \<in> cont tr"
-shows "reg f tr'"
-by (metis (lifting) assms iso_tuple_UNIV_I reg_def subtr.Step)
-
-lemma reg_subtr:
-assumes "reg f tr" and "subtr ns tr' tr"
-shows "reg f tr'"
-using assms unfolding reg_def using subtr_trans[of UNIV tr] UNIV_I
-by (metis UNIV_eq_I UnCI Un_upper1 iso_tuple_UNIV_I subtr_mono subtr_trans)
-
-lemma regular_subtr:
-assumes r: "regular tr" and s: "subtr ns tr' tr"
-shows "regular tr'"
-using r reg_subtr[OF _ s] unfolding regular_def by auto
-
-lemma subtr_deftr:
-assumes "subtr ns tr' (deftr n)"
-shows "tr' = deftr (root tr')"
-proof-
-  {fix tr have "subtr ns tr' tr \<Longrightarrow> (\<forall> n. tr = deftr n \<longrightarrow> tr' = deftr (root tr'))"
-   apply (induct rule: subtr.induct)
-   proof(metis (lifting) deftr_simps(1), safe)
-     fix tr3 ns tr1 tr2 n
-     assume 1: "root (deftr n) \<in> ns" and 2: "subtr ns tr1 tr2"
-     and IH: "\<forall>n. tr2 = deftr n \<longrightarrow> tr1 = deftr (root tr1)"
-     and 3: "Inr tr2 \<in> cont (deftr n)"
-     have "tr2 \<in> deftr ` UNIV"
-     using 3 unfolding deftr_simps image_def
-     by (metis (lifting, full_types) 3 CollectI Inr_oplus_iff cont_deftr
-         iso_tuple_UNIV_I)
-     then obtain n where "tr2 = deftr n" by auto
-     thus "tr1 = deftr (root tr1)" using IH by auto
-   qed
-  }
-  thus ?thesis using assms by auto
-qed
-
-lemma reg_deftr: "reg deftr (deftr n)"
-unfolding reg_def using subtr_deftr by auto
-
-lemma dtree_subtrOf_Union:
-assumes "dtree tr"
-shows "\<Union>{K tr' |tr'. Inr tr' \<in> cont tr} =
-       \<Union>{K (subtrOf tr n) |n. Inr n \<in> prodOf tr}"
-unfolding Union_eq Bex_def mem_Collect_eq proof safe
-  fix x xa tr'
-  assume x: "x \<in> K tr'" and tr'_tr: "Inr tr' \<in> cont tr"
-  show "\<exists>X. (\<exists>n. X = K (subtrOf tr n) \<and> Inr n \<in> prodOf tr) \<and> x \<in> X"
-  apply(rule exI[of _ "K (subtrOf tr (root tr'))"]) apply(intro conjI)
-    apply(rule exI[of _ "root tr'"]) apply (metis (lifting) root_prodOf tr'_tr)
-    by (metis (lifting) assms subtrOf_root tr'_tr x)
-next
-  fix x X n ttr
-  assume x: "x \<in> K (subtrOf tr n)" and n: "Inr n = (id \<oplus> root) ttr" and ttr: "ttr \<in> cont tr"
-  show "\<exists>X. (\<exists>tr'. X = K tr' \<and> Inr tr' \<in> cont tr) \<and> x \<in> X"
-  apply(rule exI[of _ "K (subtrOf tr n)"]) apply(intro conjI)
-    apply(rule exI[of _ "subtrOf tr n"]) apply (metis imageI n subtrOf ttr)
-    using x .
-qed
-
-
-
-
-subsection {* Paths in a regular tree *}
-
-inductive path :: "(N \<Rightarrow> Tree) \<Rightarrow> N list \<Rightarrow> bool" for f where
-Base: "path f [n]"
-|
-Ind: "\<lbrakk>path f (n1 # nl); Inr (f n1) \<in> cont (f n)\<rbrakk>
-      \<Longrightarrow> path f (n # n1 # nl)"
-
-lemma path_NE:
-assumes "path f nl"
-shows "nl \<noteq> Nil"
-using assms apply(induct rule: path.induct) by auto
-
-lemma path_post:
-assumes f: "path f (n # nl)" and nl: "nl \<noteq> []"
-shows "path f nl"
-proof-
-  obtain n1 nl1 where nl: "nl = n1 # nl1" using nl by (cases nl, auto)
-  show ?thesis using assms unfolding nl using path.simps by (metis (lifting) list.inject)
-qed
-
-lemma path_post_concat:
-assumes "path f (nl1 @ nl2)" and "nl2 \<noteq> Nil"
-shows "path f nl2"
-using assms apply (induct nl1)
-apply (metis append_Nil) by (metis Nil_is_append_conv append_Cons path_post)
-
-lemma path_concat:
-assumes "path f nl1" and "path f ((last nl1) # nl2)"
-shows "path f (nl1 @ nl2)"
-using assms apply(induct rule: path.induct) apply simp
-by (metis append_Cons last.simps list.simps(3) path.Ind)
-
-lemma path_distinct:
-assumes "path f nl"
-shows "\<exists> nl'. path f nl' \<and> hd nl' = hd nl \<and> last nl' = last nl \<and>
-              set nl' \<subseteq> set nl \<and> distinct nl'"
-using assms proof(induct rule: length_induct)
-  case (1 nl)  hence p_nl: "path f nl" by simp
-  then obtain n nl1 where nl: "nl = n # nl1" by (metis list.exhaust path_NE)
-  show ?case
-  proof(cases nl1)
-    case Nil
-    show ?thesis apply(rule exI[of _ nl]) using path.Base unfolding nl Nil by simp
-  next
-    case (Cons n1 nl2)
-    hence p1: "path f nl1" by (metis list.simps(3) nl p_nl path_post)
-    show ?thesis
-    proof(cases "n \<in> set nl1")
-      case False
-      obtain nl1' where p1': "path f nl1'" and hd_nl1': "hd nl1' = hd nl1" and
-      l_nl1': "last nl1' = last nl1" and d_nl1': "distinct nl1'"
-      and s_nl1': "set nl1' \<subseteq> set nl1"
-      using 1(1)[THEN allE[of _ nl1]] p1 unfolding nl by auto
-      obtain nl2' where nl1': "nl1' = n1 # nl2'" using path_NE[OF p1'] hd_nl1'
-      unfolding Cons by(cases nl1', auto)
-      show ?thesis apply(intro exI[of _ "n # nl1'"]) unfolding nl proof safe
-        show "path f (n # nl1')" unfolding nl1'
-        apply(rule path.Ind, metis nl1' p1')
-        by (metis (lifting) Cons list.inject nl p1 p_nl path.simps path_NE)
-      qed(insert l_nl1' Cons nl1' s_nl1' d_nl1' False, auto)
-    next
-      case True
-      then obtain nl11 nl12 where nl1: "nl1 = nl11 @ n # nl12"
-      by (metis split_list)
-      have p12: "path f (n # nl12)"
-      apply(rule path_post_concat[of _ "n # nl11"]) using p_nl[unfolded nl nl1] by auto
-      obtain nl12' where p1': "path f nl12'" and hd_nl12': "hd nl12' = n" and
-      l_nl12': "last nl12' = last (n # nl12)" and d_nl12': "distinct nl12'"
-      and s_nl12': "set nl12' \<subseteq> {n} \<union> set nl12"
-      using 1(1)[THEN allE[of _ "n # nl12"]] p12 unfolding nl nl1 by auto
-      thus ?thesis apply(intro exI[of _ nl12']) unfolding nl nl1 by auto
-    qed
-  qed
-qed
-
-lemma path_subtr:
-assumes f: "\<And> n. root (f n) = n"
-and p: "path f nl"
-shows "subtr (set nl) (f (last nl)) (f (hd nl))"
-using p proof (induct rule: path.induct)
-  case (Ind n1 nl n)  let ?ns1 = "insert n1 (set nl)"
-  have "path f (n1 # nl)"
-  and "subtr ?ns1 (f (last (n1 # nl))) (f n1)"
-  and fn1: "Inr (f n1) \<in> cont (f n)" using Ind by simp_all
-  hence fn1_flast:  "subtr (insert n ?ns1) (f (last (n1 # nl))) (f n1)"
-  by (metis subset_insertI subtr_mono)
-  have 1: "last (n # n1 # nl) = last (n1 # nl)" by auto
-  have "subtr (insert n ?ns1) (f (last (n1 # nl))) (f n)"
-  using f subtr.Step[OF _ fn1_flast fn1] by auto
-  thus ?case unfolding 1 by simp
-qed (metis f hd.simps last_ConsL last_in_set not_Cons_self2 subtr.Refl)
-
-lemma reg_subtr_path_aux:
-assumes f: "reg f tr" and n: "subtr ns tr1 tr"
-shows "\<exists> nl. path f nl \<and> f (hd nl) = tr \<and> f (last nl) = tr1 \<and> set nl \<subseteq> ns"
-using n f proof(induct rule: subtr.induct)
-  case (Refl tr ns)
-  thus ?case
-  apply(intro exI[of _ "[root tr]"]) apply simp by (metis (lifting) path.Base reg_root)
-next
-  case (Step tr ns tr2 tr1)
-  hence rtr: "root tr \<in> ns" and tr1_tr: "Inr tr1 \<in> cont tr"
-  and tr2_tr1: "subtr ns tr2 tr1" and tr: "reg f tr" by auto
-  have tr1: "reg f tr1" using reg_subtr[OF tr] rtr tr1_tr
-  by (metis (lifting) Step.prems iso_tuple_UNIV_I reg_def subtr.Step)
-  obtain nl where nl: "path f nl" and f_nl: "f (hd nl) = tr1"
-  and last_nl: "f (last nl) = tr2" and set: "set nl \<subseteq> ns" using Step(3)[OF tr1] by auto
-  have 0: "path f (root tr # nl)" apply (subst path.simps)
-  using f_nl nl reg_root tr tr1_tr by (metis hd.simps neq_Nil_conv)
-  show ?case apply(rule exI[of _ "(root tr) # nl"])
-  using 0 reg_root tr last_nl nl path_NE rtr set by auto
-qed
-
-lemma reg_subtr_path:
-assumes f: "reg f tr" and n: "subtr ns tr1 tr"
-shows "\<exists> nl. distinct nl \<and> path f nl \<and> f (hd nl) = tr \<and> f (last nl) = tr1 \<and> set nl \<subseteq> ns"
-using reg_subtr_path_aux[OF assms] path_distinct[of f]
-by (metis (lifting) order_trans)
-
-lemma subtr_iff_path:
-assumes r: "reg f tr" and f: "\<And> n. root (f n) = n"
-shows "subtr ns tr1 tr \<longleftrightarrow>
-       (\<exists> nl. distinct nl \<and> path f nl \<and> f (hd nl) = tr \<and> f (last nl) = tr1 \<and> set nl \<subseteq> ns)"
-proof safe
-  fix nl assume p: "path f nl" and nl: "set nl \<subseteq> ns"
-  have "subtr (set nl) (f (last nl)) (f (hd nl))"
-  apply(rule path_subtr) using p f by simp_all
-  thus "subtr ns (f (last nl)) (f (hd nl))"
-  using subtr_mono nl by auto
-qed(insert reg_subtr_path[OF r], auto)
-
-lemma inFr_iff_path:
-assumes r: "reg f tr" and f: "\<And> n. root (f n) = n"
-shows
-"inFr ns tr t \<longleftrightarrow>
- (\<exists> nl tr1. distinct nl \<and> path f nl \<and> f (hd nl) = tr \<and> f (last nl) = tr1 \<and>
-            set nl \<subseteq> ns \<and> Inl t \<in> cont tr1)"
-apply safe
-apply (metis (no_types) inFr_subtr r reg_subtr_path)
-by (metis f inFr.Base path_subtr subtr_inFr subtr_mono subtr_rootL_in)
-
-
-
-subsection{* The regular cut of a tree *}
-
-context fixes tr0 :: Tree
-begin
-
-(* Picking a subtree of a certain root: *)
-definition "pick n \<equiv> SOME tr. subtr UNIV tr tr0 \<and> root tr = n"
-
-lemma pick:
-assumes "inItr UNIV tr0 n"
-shows "subtr UNIV (pick n) tr0 \<and> root (pick n) = n"
-proof-
-  have "\<exists> tr. subtr UNIV tr tr0 \<and> root tr = n"
-  using assms by (metis (lifting) inItr_subtr)
-  thus ?thesis unfolding pick_def by(rule someI_ex)
-qed
-
-lemmas subtr_pick = pick[THEN conjunct1]
-lemmas root_pick = pick[THEN conjunct2]
-
-lemma dtree_pick:
-assumes tr0: "dtree tr0" and n: "inItr UNIV tr0 n"
-shows "dtree (pick n)"
-using dtree_subtr[OF tr0 subtr_pick[OF n]] .
-
-definition "regOf_r n \<equiv> root (pick n)"
-definition "regOf_c n \<equiv> (id \<oplus> root) ` cont (pick n)"
-
-(* The regular tree of a function: *)
-definition regOf :: "N \<Rightarrow> Tree" where
-"regOf \<equiv> unfold regOf_r regOf_c"
-
-lemma finite_regOf_c: "finite (regOf_c n)"
-unfolding regOf_c_def by (metis finite_cont finite_imageI)
-
-lemma root_regOf_pick: "root (regOf n) = root (pick n)"
-using unfold(1)[of regOf_r regOf_c n] unfolding regOf_def regOf_r_def by simp
-
-lemma root_regOf[simp]:
-assumes "inItr UNIV tr0 n"
-shows "root (regOf n) = n"
-unfolding root_regOf_pick root_pick[OF assms] ..
-
-lemma cont_regOf[simp]:
-"cont (regOf n) = (id \<oplus> (regOf o root)) ` cont (pick n)"
-apply(subst id_o[symmetric, of id]) unfolding sum_map.comp[symmetric]
-unfolding image_compose unfolding regOf_c_def[symmetric]
-using unfold(2)[of regOf_c n regOf_r, OF finite_regOf_c]
-unfolding regOf_def ..
-
-lemma Inl_cont_regOf[simp]:
-"Inl -` (cont (regOf n)) = Inl -` (cont (pick n))"
-unfolding cont_regOf by simp
-
-lemma Inr_cont_regOf:
-"Inr -` (cont (regOf n)) = (regOf \<circ> root) ` (Inr -` cont (pick n))"
-unfolding cont_regOf by simp
-
-lemma subtr_regOf:
-assumes n: "inItr UNIV tr0 n" and "subtr UNIV tr1 (regOf n)"
-shows "\<exists> n1. inItr UNIV tr0 n1 \<and> tr1 = regOf n1"
-proof-
-  {fix tr ns assume "subtr UNIV tr1 tr"
-   hence "tr = regOf n \<longrightarrow> (\<exists> n1. inItr UNIV tr0 n1 \<and> tr1 = regOf n1)"
-   proof (induct rule: subtr_UNIV_inductL)
-     case (Step tr2 tr1 tr)
-     show ?case proof
-       assume "tr = regOf n"
-       then obtain n1 where tr2: "Inr tr2 \<in> cont tr1"
-       and tr1_tr: "subtr UNIV tr1 tr" and n1: "inItr UNIV tr0 n1" and tr1: "tr1 = regOf n1"
-       using Step by auto
-       obtain tr2' where tr2: "tr2 = regOf (root tr2')"
-       and tr2': "Inr tr2' \<in> cont (pick n1)"
-       using tr2 Inr_cont_regOf[of n1]
-       unfolding tr1 image_def o_def using vimage_eq by auto
-       have "inItr UNIV tr0 (root tr2')"
-       using inItr.Base inItr.Ind n1 pick subtr_inItr tr2' by (metis iso_tuple_UNIV_I)
-       thus "\<exists>n2. inItr UNIV tr0 n2 \<and> tr2 = regOf n2" using tr2 by blast
-     qed
-   qed(insert n, auto)
-  }
-  thus ?thesis using assms by auto
-qed
-
-lemma root_regOf_root:
-assumes n: "inItr UNIV tr0 n" and t_tr: "t_tr \<in> cont (pick n)"
-shows "(id \<oplus> (root \<circ> regOf \<circ> root)) t_tr = (id \<oplus> root) t_tr"
-using assms apply(cases t_tr)
-  apply (metis (lifting) sum_map.simps(1))
-  using pick regOf_def regOf_r_def unfold(1)
-      inItr.Base o_apply subtr_StepL subtr_inItr sum_map.simps(2)
-  by (metis UNIV_I)
-
-lemma regOf_P:
-assumes tr0: "dtree tr0" and n: "inItr UNIV tr0 n"
-shows "(n, (id \<oplus> root) ` cont (regOf n)) \<in> P" (is "?L \<in> P")
-proof-
-  have "?L = (n, (id \<oplus> root) ` cont (pick n))"
-  unfolding cont_regOf image_compose[symmetric] sum_map.comp id_o o_assoc
-  unfolding Pair_eq apply(rule conjI[OF refl]) apply(rule image_cong[OF refl])
-  by(rule root_regOf_root[OF n])
-  moreover have "... \<in> P" by (metis (lifting) dtree_pick root_pick dtree_P n tr0)
-  ultimately show ?thesis by simp
-qed
-
-lemma dtree_regOf:
-assumes tr0: "dtree tr0" and "inItr UNIV tr0 n"
-shows "dtree (regOf n)"
-proof-
-  {fix tr have "\<exists> n. inItr UNIV tr0 n \<and> tr = regOf n \<Longrightarrow> dtree tr"
-   proof (induct rule: dtree_raw_coind)
-     case (Hyp tr)
-     then obtain n where n: "inItr UNIV tr0 n" and tr: "tr = regOf n" by auto
-     show ?case unfolding lift_def apply safe
-     apply (metis (lifting) regOf_P root_regOf n tr tr0)
-     unfolding tr Inr_cont_regOf unfolding inj_on_def apply clarsimp using root_regOf
-     apply (metis UNIV_I inItr.Base n pick subtr2.simps subtr_inItr subtr_subtr2)
-     by (metis n subtr.Refl subtr_StepL subtr_regOf tr UNIV_I)
-   qed
-  }
-  thus ?thesis using assms by blast
-qed
-
-(* The regular cut of a tree: *)
-definition "rcut \<equiv> regOf (root tr0)"
-
-theorem reg_rcut: "reg regOf rcut"
-unfolding reg_def rcut_def
-by (metis inItr.Base root_regOf subtr_regOf UNIV_I)
-
-lemma rcut_reg:
-assumes "reg regOf tr0"
-shows "rcut = tr0"
-using assms unfolding rcut_def reg_def by (metis subtr.Refl UNIV_I)
-
-theorem rcut_eq: "rcut = tr0 \<longleftrightarrow> reg regOf tr0"
-using reg_rcut rcut_reg by metis
-
-theorem regular_rcut: "regular rcut"
-using reg_rcut unfolding regular_def by blast
-
-theorem Fr_rcut: "Fr UNIV rcut \<subseteq> Fr UNIV tr0"
-proof safe
-  fix t assume "t \<in> Fr UNIV rcut"
-  then obtain tr where t: "Inl t \<in> cont tr" and tr: "subtr UNIV tr (regOf (root tr0))"
-  using Fr_subtr[of UNIV "regOf (root tr0)"] unfolding rcut_def
-  by (metis (full_types) Fr_def inFr_subtr mem_Collect_eq)
-  obtain n where n: "inItr UNIV tr0 n" and tr: "tr = regOf n" using tr
-  by (metis (lifting) inItr.Base subtr_regOf UNIV_I)
-  have "Inl t \<in> cont (pick n)" using t using Inl_cont_regOf[of n] unfolding tr
-  by (metis (lifting) vimageD vimageI2)
-  moreover have "subtr UNIV (pick n) tr0" using subtr_pick[OF n] ..
-  ultimately show "t \<in> Fr UNIV tr0" unfolding Fr_subtr_cont by auto
-qed
-
-theorem dtree_rcut:
-assumes "dtree tr0"
-shows "dtree rcut"
-unfolding rcut_def using dtree_regOf[OF assms inItr.Base] by simp
-
-theorem root_rcut[simp]: "root rcut = root tr0"
-unfolding rcut_def
-by (metis (lifting) root_regOf inItr.Base reg_def reg_root subtr_rootR_in)
-
-end (* context *)
-
-
-subsection{* Recursive description of the regular tree frontiers *}
-
-lemma regular_inFr:
-assumes r: "regular tr" and In: "root tr \<in> ns"
-and t: "inFr ns tr t"
-shows "t \<in> Inl -` (cont tr) \<or>
-       (\<exists> tr'. Inr tr' \<in> cont tr \<and> inFr (ns - {root tr}) tr' t)"
-(is "?L \<or> ?R")
-proof-
-  obtain f where r: "reg f tr" and f: "\<And>n. root (f n) = n"
-  using r unfolding regular_def2 by auto
-  obtain nl tr1 where d_nl: "distinct nl" and p: "path f nl" and hd_nl: "f (hd nl) = tr"
-  and l_nl: "f (last nl) = tr1" and s_nl: "set nl \<subseteq> ns" and t_tr1: "Inl t \<in> cont tr1"
-  using t unfolding inFr_iff_path[OF r f] by auto
-  obtain n nl1 where nl: "nl = n # nl1" by (metis (lifting) p path.simps)
-  hence f_n: "f n = tr" using hd_nl by simp
-  have n_nl1: "n \<notin> set nl1" using d_nl unfolding nl by auto
-  show ?thesis
-  proof(cases nl1)
-    case Nil hence "tr = tr1" using f_n l_nl unfolding nl by simp
-    hence ?L using t_tr1 by simp thus ?thesis by simp
-  next
-    case (Cons n1 nl2) note nl1 = Cons
-    have 1: "last nl1 = last nl" "hd nl1 = n1" unfolding nl nl1 by simp_all
-    have p1: "path f nl1" and n1_tr: "Inr (f n1) \<in> cont tr"
-    using path.simps[of f nl] p f_n unfolding nl nl1 by auto
-    have r1: "reg f (f n1)" using reg_Inr_cont[OF r n1_tr] .
-    have 0: "inFr (set nl1) (f n1) t" unfolding inFr_iff_path[OF r1 f]
-    apply(intro exI[of _ nl1], intro exI[of _ tr1])
-    using d_nl unfolding 1 l_nl unfolding nl using p1 t_tr1 by auto
-    have root_tr: "root tr = n" by (metis f f_n)
-    have "inFr (ns - {root tr}) (f n1) t" apply(rule inFr_mono[OF 0])
-    using s_nl unfolding root_tr unfolding nl using n_nl1 by auto
-    thus ?thesis using n1_tr by auto
-  qed
-qed
-
-theorem regular_Fr:
-assumes r: "regular tr" and In: "root tr \<in> ns"
-shows "Fr ns tr =
-       Inl -` (cont tr) \<union>
-       \<Union> {Fr (ns - {root tr}) tr' | tr'. Inr tr' \<in> cont tr}"
-unfolding Fr_def
-using In inFr.Base regular_inFr[OF assms] apply safe
-apply (simp, metis (full_types) mem_Collect_eq)
-apply simp
-by (simp, metis (lifting) inFr_Ind_minus insert_Diff)
-
-
-subsection{* The generated languages *}
-
-(* The (possibly inifinite tree) generated language *)
-definition "L ns n \<equiv> {Fr ns tr | tr. dtree tr \<and> root tr = n}"
-
-(* The regular-tree generated language *)
-definition "Lr ns n \<equiv> {Fr ns tr | tr. dtree tr \<and> root tr = n \<and> regular tr}"
-
-theorem L_rec_notin:
-assumes "n \<notin> ns"
-shows "L ns n = {{}}"
-using assms unfolding L_def apply safe
-  using not_root_Fr apply force
-  apply(rule exI[of _ "deftr n"])
-  by (metis (no_types) dtree_deftr not_root_Fr root_deftr)
-
-theorem Lr_rec_notin:
-assumes "n \<notin> ns"
-shows "Lr ns n = {{}}"
-using assms unfolding Lr_def apply safe
-  using not_root_Fr apply force
-  apply(rule exI[of _ "deftr n"])
-  by (metis (no_types) regular_def dtree_deftr not_root_Fr reg_deftr root_deftr)
-
-lemma dtree_subtrOf:
-assumes "dtree tr" and "Inr n \<in> prodOf tr"
-shows "dtree (subtrOf tr n)"
-by (metis assms dtree_lift lift_def subtrOf)
-
-theorem Lr_rec_in:
-assumes n: "n \<in> ns"
-shows "Lr ns n \<subseteq>
-{Inl -` tns \<union> (\<Union> {K n' | n'. Inr n' \<in> tns}) | tns K.
-    (n,tns) \<in> P \<and>
-    (\<forall> n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> Lr (ns - {n}) n')}"
-(is "Lr ns n \<subseteq> {?F tns K | tns K. (n,tns) \<in> P \<and> ?\<phi> tns K}")
-proof safe
-  fix ts assume "ts \<in> Lr ns n"
-  then obtain tr where dtr: "dtree tr" and r: "root tr = n" and tr: "regular tr"
-  and ts: "ts = Fr ns tr" unfolding Lr_def by auto
-  def tns \<equiv> "(id \<oplus> root) ` (cont tr)"
-  def K \<equiv> "\<lambda> n'. Fr (ns - {n}) (subtrOf tr n')"
-  show "\<exists>tns K. ts = ?F tns K \<and> (n, tns) \<in> P \<and> ?\<phi> tns K"
-  apply(rule exI[of _ tns], rule exI[of _ K]) proof(intro conjI allI impI)
-    show "ts = Inl -` tns \<union> \<Union>{K n' |n'. Inr n' \<in> tns}"
-    unfolding ts regular_Fr[OF tr n[unfolded r[symmetric]]]
-    unfolding tns_def K_def r[symmetric]
-    unfolding Inl_prodOf dtree_subtrOf_Union[OF dtr] ..
-    show "(n, tns) \<in> P" unfolding tns_def r[symmetric] using dtree_P[OF dtr] .
-    fix n' assume "Inr n' \<in> tns" thus "K n' \<in> Lr (ns - {n}) n'"
-    unfolding K_def Lr_def mem_Collect_eq apply(intro exI[of _ "subtrOf tr n'"])
-    using dtr tr apply(intro conjI refl)  unfolding tns_def
-      apply(erule dtree_subtrOf[OF dtr])
-      apply (metis subtrOf)
-      by (metis Inr_subtrOf UNIV_I regular_subtr subtr.simps)
-  qed
-qed
-
-lemma hsubst_aux:
-fixes n ftr tns
-assumes n: "n \<in> ns" and tns: "finite tns" and
-1: "\<And> n'. Inr n' \<in> tns \<Longrightarrow> dtree (ftr n')"
-defines "tr \<equiv> Node n ((id \<oplus> ftr) ` tns)"  defines "tr' \<equiv> hsubst tr tr"
-shows "Fr ns tr' = Inl -` tns \<union> \<Union>{Fr (ns - {n}) (ftr n') |n'. Inr n' \<in> tns}"
-(is "_ = ?B") proof-
-  have rtr: "root tr = n" and ctr: "cont tr = (id \<oplus> ftr) ` tns"
-  unfolding tr_def using tns by auto
-  have Frr: "Frr (ns - {n}) tr = \<Union>{Fr (ns - {n}) (ftr n') |n'. Inr n' \<in> tns}"
-  unfolding Frr_def ctr by auto
-  have "Fr ns tr' = Inl -` (cont tr) \<union> Frr (ns - {n}) tr"
-  using Fr_self_hsubst[OF n[unfolded rtr[symmetric]]] unfolding tr'_def rtr ..
-  also have "... = ?B" unfolding ctr Frr by simp
-  finally show ?thesis .
-qed
-
-theorem L_rec_in:
-assumes n: "n \<in> ns"
-shows "
-{Inl -` tns \<union> (\<Union> {K n' | n'. Inr n' \<in> tns}) | tns K.
-    (n,tns) \<in> P \<and>
-    (\<forall> n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> L (ns - {n}) n')}
- \<subseteq> L ns n"
-proof safe
-  fix tns K
-  assume P: "(n, tns) \<in> P" and 0: "\<forall>n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> L (ns - {n}) n'"
-  {fix n' assume "Inr n' \<in> tns"
-   hence "K n' \<in> L (ns - {n}) n'" using 0 by auto
-   hence "\<exists> tr'. K n' = Fr (ns - {n}) tr' \<and> dtree tr' \<and> root tr' = n'"
-   unfolding L_def mem_Collect_eq by auto
-  }
-  then obtain ftr where 0: "\<And> n'. Inr n' \<in> tns \<Longrightarrow>
-  K n' = Fr (ns - {n}) (ftr n') \<and> dtree (ftr n') \<and> root (ftr n') = n'"
-  by metis
-  def tr \<equiv> "Node n ((id \<oplus> ftr) ` tns)"  def tr' \<equiv> "hsubst tr tr"
-  have rtr: "root tr = n" and ctr: "cont tr = (id \<oplus> ftr) ` tns"
-  unfolding tr_def by (simp, metis P cont_Node finite_imageI finite_in_P)
-  have prtr: "prodOf tr = tns" apply(rule Inl_Inr_image_cong)
-  unfolding ctr apply simp apply simp apply safe
-  using 0 unfolding image_def apply force apply simp by (metis 0 vimageI2)
-  have 1: "{K n' |n'. Inr n' \<in> tns} = {Fr (ns - {n}) (ftr n') |n'. Inr n' \<in> tns}"
-  using 0 by auto
-  have dtr: "dtree tr" apply(rule dtree.Tree)
-    apply (metis (lifting) P prtr rtr)
-    unfolding inj_on_def ctr lift_def using 0 by auto
-  hence dtr': "dtree tr'" unfolding tr'_def by (metis dtree_hsubst)
-  have tns: "finite tns" using finite_in_P P by simp
-  have "Inl -` tns \<union> \<Union>{Fr (ns - {n}) (ftr n') |n'. Inr n' \<in> tns} \<in> L ns n"
-  unfolding L_def mem_Collect_eq apply(intro exI[of _ tr'] conjI)
-  using dtr' 0 hsubst_aux[OF assms tns, of ftr] unfolding tr_def tr'_def by auto
-  thus "Inl -` tns \<union> \<Union>{K n' |n'. Inr n' \<in> tns} \<in> L ns n" unfolding 1 .
-qed
-
-lemma card_N: "(n::N) \<in> ns \<Longrightarrow> card (ns - {n}) < card ns"
-by (metis finite_N Diff_UNIV Diff_infinite_finite card_Diff1_less finite.emptyI)
-
-function LL where
-"LL ns n =
- (if n \<notin> ns then {{}} else
- {Inl -` tns \<union> (\<Union> {K n' | n'. Inr n' \<in> tns}) | tns K.
-    (n,tns) \<in> P \<and>
-    (\<forall> n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> LL (ns - {n}) n')})"
-by(pat_completeness, auto)
-termination apply(relation "inv_image (measure card) fst")
-using card_N by auto
-
-declare LL.simps[code]  (* TODO: Does code generation for LL work? *)
-declare LL.simps[simp del]
-
-theorem Lr_LL: "Lr ns n \<subseteq> LL ns n"
-proof (induct ns arbitrary: n rule: measure_induct[of card])
-  case (1 ns n) show ?case proof(cases "n \<in> ns")
-    case False thus ?thesis unfolding Lr_rec_notin[OF False] by (simp add: LL.simps)
-  next
-    case True show ?thesis apply(rule subset_trans)
-    using Lr_rec_in[OF True] apply assumption
-    unfolding LL.simps[of ns n] using True 1 card_N proof clarsimp
-      fix tns K
-      assume "n \<in> ns" hence c: "card (ns - {n}) < card ns" using card_N by blast
-      assume "(n, tns) \<in> P"
-      and "\<forall>n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> Lr (ns - {n}) n'"
-      thus "\<exists>tnsa Ka.
-             Inl -` tns \<union> \<Union>{K n' |n'. Inr n' \<in> tns} =
-             Inl -` tnsa \<union> \<Union>{Ka n' |n'. Inr n' \<in> tnsa} \<and>
-             (n, tnsa) \<in> P \<and> (\<forall>n'. Inr n' \<in> tnsa \<longrightarrow> Ka n' \<in> LL (ns - {n}) n')"
-      apply(intro exI[of _ tns] exI[of _ K]) using c 1 by auto
-    qed
-  qed
-qed
-
-theorem LL_L: "LL ns n \<subseteq> L ns n"
-proof (induct ns arbitrary: n rule: measure_induct[of card])
-  case (1 ns n) show ?case proof(cases "n \<in> ns")
-    case False thus ?thesis unfolding L_rec_notin[OF False] by (simp add: LL.simps)
-  next
-    case True show ?thesis apply(rule subset_trans)
-    prefer 2 using L_rec_in[OF True] apply assumption
-    unfolding LL.simps[of ns n] using True 1 card_N proof clarsimp
-      fix tns K
-      assume "n \<in> ns" hence c: "card (ns - {n}) < card ns" using card_N by blast
-      assume "(n, tns) \<in> P"
-      and "\<forall>n'. Inr n' \<in> tns \<longrightarrow> K n' \<in> LL (ns - {n}) n'"
-      thus "\<exists>tnsa Ka.
-             Inl -` tns \<union> \<Union>{K n' |n'. Inr n' \<in> tns} =
-             Inl -` tnsa \<union> \<Union>{Ka n' |n'. Inr n' \<in> tnsa} \<and>
-             (n, tnsa) \<in> P \<and> (\<forall>n'. Inr n' \<in> tnsa \<longrightarrow> Ka n' \<in> L (ns - {n}) n')"
-      apply(intro exI[of _ tns] exI[of _ K]) using c 1 by auto
-    qed
-  qed
-qed
-
-(* The subsumpsion relation between languages *)
-definition "subs L1 L2 \<equiv> \<forall> ts2 \<in> L2. \<exists> ts1 \<in> L1. ts1 \<subseteq> ts2"
-
-lemma incl_subs[simp]: "L2 \<subseteq> L1 \<Longrightarrow> subs L1 L2"
-unfolding subs_def by auto
-
-lemma subs_refl[simp]: "subs L1 L1" unfolding subs_def by auto
-
-lemma subs_trans: "\<lbrakk>subs L1 L2; subs L2 L3\<rbrakk> \<Longrightarrow> subs L1 L3"
-unfolding subs_def by (metis subset_trans)
-
-(* Language equivalence *)
-definition "leqv L1 L2 \<equiv> subs L1 L2 \<and> subs L2 L1"
-
-lemma subs_leqv[simp]: "leqv L1 L2 \<Longrightarrow> subs L1 L2"
-unfolding leqv_def by auto
-
-lemma subs_leqv_sym[simp]: "leqv L1 L2 \<Longrightarrow> subs L2 L1"
-unfolding leqv_def by auto
-
-lemma leqv_refl[simp]: "leqv L1 L1" unfolding leqv_def by auto
-
-lemma leqv_trans:
-assumes 12: "leqv L1 L2" and 23: "leqv L2 L3"
-shows "leqv L1 L3"
-using assms unfolding leqv_def by (metis (lifting) subs_trans)
-
-lemma leqv_sym: "leqv L1 L2 \<Longrightarrow> leqv L2 L1"
-unfolding leqv_def by auto
-
-lemma leqv_Sym: "leqv L1 L2 \<longleftrightarrow> leqv L2 L1"
-unfolding leqv_def by auto
-
-lemma Lr_incl_L: "Lr ns ts \<subseteq> L ns ts"
-unfolding Lr_def L_def by auto
-
-lemma Lr_subs_L: "subs (Lr UNIV ts) (L UNIV ts)"
-unfolding subs_def proof safe
-  fix ts2 assume "ts2 \<in> L UNIV ts"
-  then obtain tr where ts2: "ts2 = Fr UNIV tr" and dtr: "dtree tr" and rtr: "root tr = ts"
-  unfolding L_def by auto
-  thus "\<exists>ts1\<in>Lr UNIV ts. ts1 \<subseteq> ts2"
-  apply(intro bexI[of _ "Fr UNIV (rcut tr)"])
-  unfolding Lr_def L_def using Fr_rcut dtree_rcut root_rcut regular_rcut by auto
-qed
-
-theorem Lr_leqv_L: "leqv (Lr UNIV ts) (L UNIV ts)"
-using Lr_subs_L unfolding leqv_def by (metis (lifting) Lr_incl_L incl_subs)
-
-theorem LL_leqv_L: "leqv (LL UNIV ts) (L UNIV ts)"
-by (metis (lifting) LL_L Lr_LL Lr_subs_L incl_subs leqv_def subs_trans)
-
-theorem LL_leqv_Lr: "leqv (LL UNIV ts) (Lr UNIV ts)"
-using Lr_leqv_L LL_leqv_L by (metis leqv_Sym leqv_trans)
-
-
-end
diff -r 4a15873c4ec9 -r 41ee3bfccb4d src/HOL/BNF/Examples/Infinite_Derivation_Trees/Parallel.thy
--- a/src/HOL/BNF/Examples/Infinite_Derivation_Trees/Parallel.thy	Mon Oct 15 19:03:02 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,152 +0,0 @@
-(*  Title:      HOL/BNF/Examples/Infinite_Derivation_Trees/Parallel.thy
-    Author:     Andrei Popescu, TU Muenchen
-    Copyright   2012
-
-Parallel composition.
-*)
-
-header {* Parallel Composition *}
-
-theory Parallel
-imports Tree
-begin
-
-no_notation plus_class.plus (infixl "+" 65)
-no_notation Sublist.parallel (infixl "\<parallel>" 50)
-    
-consts Nplus :: "N \<Rightarrow> N \<Rightarrow> N" (infixl "+" 60)
-
-axiomatization where
-    Nplus_comm: "(a::N) + b = b + (a::N)"
-and Nplus_assoc: "((a::N) + b) + c = a + (b + c)"
-
-section{* Parallel composition *}
-
-fun par_r where "par_r (tr1,tr2) = root tr1 + root tr2"
-fun par_c where
-"par_c (tr1,tr2) =
- Inl ` (Inl -` (cont tr1 \<union> cont tr2)) \<union>
- Inr ` (Inr -` cont tr1 \<times> Inr -` cont tr2)"
-
-declare par_r.simps[simp del]  declare par_c.simps[simp del]
-
-definition par :: "Tree \<times> Tree \<Rightarrow> Tree" where
-"par \<equiv> unfold par_r par_c"
-
-abbreviation par_abbr (infixr "\<parallel>" 80) where "tr1 \<parallel> tr2 \<equiv> par (tr1, tr2)"
-
-lemma finite_par_c: "finite (par_c (tr1, tr2))"
-unfolding par_c.simps apply(rule finite_UnI)
-  apply (metis finite_Un finite_cont finite_imageI finite_vimageI inj_Inl)
-  apply(intro finite_imageI finite_cartesian_product finite_vimageI)
-  using finite_cont by auto
-
-lemma root_par: "root (tr1 \<parallel> tr2) = root tr1 + root tr2"
-using unfold(1)[of par_r par_c "(tr1,tr2)"] unfolding par_def par_r.simps by simp
-
-lemma cont_par:
-"cont (tr1 \<parallel> tr2) = (id \<oplus> par) ` par_c (tr1,tr2)"
-using unfold(2)[of par_c "(tr1,tr2)" par_r, OF finite_par_c]
-unfolding par_def ..
-
-lemma Inl_cont_par[simp]:
-"Inl -` (cont (tr1 \<parallel> tr2)) = Inl -` (cont tr1 \<union> cont tr2)"
-unfolding cont_par par_c.simps by auto
-
-lemma Inr_cont_par[simp]:
-"Inr -` (cont (tr1 \<parallel> tr2)) = par ` (Inr -` cont tr1 \<times> Inr -` cont tr2)"
-unfolding cont_par par_c.simps by auto
-
-lemma Inl_in_cont_par:
-"Inl t \<in> cont (tr1 \<parallel> tr2) \<longleftrightarrow> (Inl t \<in> cont tr1 \<or> Inl t \<in> cont tr2)"
-using Inl_cont_par[of tr1 tr2] unfolding vimage_def by auto
-
-lemma Inr_in_cont_par:
-"Inr t \<in> cont (tr1 \<parallel> tr2) \<longleftrightarrow> (t \<in> par ` (Inr -` cont tr1 \<times> Inr -` cont tr2))"
-using Inr_cont_par[of tr1 tr2] unfolding vimage_def by auto
-
-
-section{* =-coinductive proofs *}
-
-(* Detailed proofs of commutativity and associativity: *)
-theorem par_com: "tr1 \<parallel> tr2 = tr2 \<parallel> tr1"
-proof-
-  let ?\<phi> = "\<lambda> trA trB. \<exists> tr1 tr2. trA = tr1 \<parallel> tr2 \<and> trB = tr2 \<parallel> tr1"
-  {fix trA trB
-   assume "?\<phi> trA trB" hence "trA = trB"
-   proof (induct rule: Tree_coind, safe)
-     fix tr1 tr2
-     show "root (tr1 \<parallel> tr2) = root (tr2 \<parallel> tr1)"
-     unfolding root_par by (rule Nplus_comm)
-   next
-     fix tr1 tr2 :: Tree
-     let ?trA = "tr1 \<parallel> tr2"  let ?trB = "tr2 \<parallel> tr1"
-     show "(?\<phi> ^#2) (cont ?trA) (cont ?trB)"
-     unfolding lift2_def proof(intro conjI allI impI)
-       fix n show "Inl n \<in> cont (tr1 \<parallel> tr2) \<longleftrightarrow> Inl n \<in> cont (tr2 \<parallel> tr1)"
-       unfolding Inl_in_cont_par by auto
-     next
-       fix trA' assume "Inr trA' \<in> cont ?trA"
-       then obtain tr1' tr2' where "trA' = tr1' \<parallel> tr2'"
-       and "Inr tr1' \<in> cont tr1" and "Inr tr2' \<in> cont tr2"
-       unfolding Inr_in_cont_par by auto
-       thus "\<exists> trB'. Inr trB' \<in> cont ?trB \<and> ?\<phi> trA' trB'"
-       apply(intro exI[of _ "tr2' \<parallel> tr1'"]) unfolding Inr_in_cont_par by auto
-     next
-       fix trB' assume "Inr trB' \<in> cont ?trB"
-       then obtain tr1' tr2' where "trB' = tr2' \<parallel> tr1'"
-       and "Inr tr1' \<in> cont tr1" and "Inr tr2' \<in> cont tr2"
-       unfolding Inr_in_cont_par by auto
-       thus "\<exists> trA'. Inr trA' \<in> cont ?trA \<and> ?\<phi> trA' trB'"
-       apply(intro exI[of _ "tr1' \<parallel> tr2'"]) unfolding Inr_in_cont_par by auto
-     qed
-   qed
-  }
-  thus ?thesis by blast
-qed
-
-theorem par_assoc: "(tr1 \<parallel> tr2) \<parallel> tr3 = tr1 \<parallel> (tr2 \<parallel> tr3)"
-proof-
-  let ?\<phi> =
-  "\<lambda> trA trB. \<exists> tr1 tr2 tr3. trA = (tr1 \<parallel> tr2) \<parallel> tr3 \<and>
-                             trB = tr1 \<parallel> (tr2 \<parallel> tr3)"
-  {fix trA trB
-   assume "?\<phi> trA trB" hence "trA = trB"
-   proof (induct rule: Tree_coind, safe)
-     fix tr1 tr2 tr3
-     show "root ((tr1 \<parallel> tr2) \<parallel> tr3) = root (tr1 \<parallel> (tr2 \<parallel> tr3))"
-     unfolding root_par by (rule Nplus_assoc)
-   next
-     fix tr1 tr2 tr3
-     let ?trA = "(tr1 \<parallel> tr2) \<parallel> tr3"  let ?trB = "tr1 \<parallel> (tr2 \<parallel> tr3)"
-     show "(?\<phi> ^#2) (cont ?trA) (cont ?trB)"
-     unfolding lift2_def proof(intro conjI allI impI)
-       fix n show "Inl n \<in> (cont ?trA) \<longleftrightarrow> Inl n \<in> (cont ?trB)"
-       unfolding Inl_in_cont_par by simp
-     next
-       fix trA' assume "Inr trA' \<in> cont ?trA"
-       then obtain tr1' tr2' tr3' where "trA' = (tr1' \<parallel> tr2') \<parallel> tr3'"
-       and "Inr tr1' \<in> cont tr1" and "Inr tr2' \<in> cont tr2"
-       and "Inr tr3' \<in> cont tr3" unfolding Inr_in_cont_par by auto
-       thus "\<exists> trB'. Inr trB' \<in> cont ?trB \<and> ?\<phi> trA' trB'"
-       apply(intro exI[of _ "tr1' \<parallel> (tr2' \<parallel> tr3')"])
-       unfolding Inr_in_cont_par by auto
-     next
-       fix trB' assume "Inr trB' \<in> cont ?trB"
-       then obtain tr1' tr2' tr3' where "trB' = tr1' \<parallel> (tr2' \<parallel> tr3')"
-       and "Inr tr1' \<in> cont tr1" and "Inr tr2' \<in> cont tr2"
-       and "Inr tr3' \<in> cont tr3" unfolding Inr_in_cont_par by auto
-       thus "\<exists> trA'. Inr trA' \<in> cont ?trA \<and> ?\<phi> trA' trB'"
-       apply(intro exI[of _ "(tr1' \<parallel> tr2') \<parallel> tr3'"])
-       unfolding Inr_in_cont_par by auto
-     qed
-   qed
-  }
-  thus ?thesis by blast
-qed
-
-
-
-
-
-end
\ No newline at end of file
diff -r 4a15873c4ec9 -r 41ee3bfccb4d src/HOL/BNF/Examples/Infinite_Derivation_Trees/Prelim.thy
--- a/src/HOL/BNF/Examples/Infinite_Derivation_Trees/Prelim.thy	Mon Oct 15 19:03:02 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,67 +0,0 @@
-(*  Title:      HOL/BNF/Examples/Infinite_Derivation_Trees/Prelim.thy
-    Author:     Andrei Popescu, TU Muenchen
-    Copyright   2012
-
-Preliminaries.
-*)
-
-header {* Preliminaries *}
-
-theory Prelim
-imports "../../BNF"
-begin
-
-declare fset_to_fset[simp]
-
-lemma fst_snd_convol_o[simp]: "<fst o s, snd o s> = s"
-apply(rule ext) by (simp add: convol_def)
-
-abbreviation sm_abbrev (infix "\<oplus>" 60)
-where "f \<oplus> g \<equiv> Sum_Type.sum_map f g"
-
-lemma sum_map_InlD: "(f \<oplus> g) z = Inl x \<Longrightarrow> \<exists>y. z = Inl y \<and> f y = x"
-by (cases z) auto
-
-lemma sum_map_InrD: "(f \<oplus> g) z = Inr x \<Longrightarrow> \<exists>y. z = Inr y \<and> g y = x"
-by (cases z) auto
-
-abbreviation sum_case_abbrev ("[[_,_]]" 800)
-where "[[f,g]] \<equiv> Sum_Type.sum_case f g"
-
-lemma inj_Inl[simp]: "inj Inl" unfolding inj_on_def by auto
-lemma inj_Inr[simp]: "inj Inr" unfolding inj_on_def by auto
-
-lemma Inl_oplus_elim:
-assumes "Inl tr \<in> (id \<oplus> f) ` tns"
-shows "Inl tr \<in> tns"
-using assms apply clarify by (case_tac x, auto)
-
-lemma Inl_oplus_iff[simp]: "Inl tr \<in> (id \<oplus> f) ` tns \<longleftrightarrow> Inl tr \<in> tns"
-using Inl_oplus_elim
-by (metis id_def image_iff sum_map.simps(1))
-
-lemma Inl_m_oplus[simp]: "Inl -` (id \<oplus> f) ` tns = Inl -` tns"
-using Inl_oplus_iff unfolding vimage_def by auto
-
-lemma Inr_oplus_elim:
-assumes "Inr tr \<in> (id \<oplus> f) ` tns"
-shows "\<exists> n. Inr n \<in> tns \<and> f n = tr"
-using assms apply clarify by (case_tac x, auto)
-
-lemma Inr_oplus_iff[simp]:
-"Inr tr \<in> (id \<oplus> f) ` tns \<longleftrightarrow> (\<exists> n. Inr n \<in> tns \<and> f n = tr)"
-apply (rule iffI)
- apply (metis Inr_oplus_elim)
-by (metis image_iff sum_map.simps(2))
-
-lemma Inr_m_oplus[simp]: "Inr -` (id \<oplus> f) ` tns = f ` (Inr -` tns)"
-using Inr_oplus_iff unfolding vimage_def by auto
-
-lemma Inl_Inr_image_cong:
-assumes "Inl -` A = Inl -` B" and "Inr -` A = Inr -` B"
-shows "A = B"
-apply safe using assms apply(case_tac x, auto) by(case_tac x, auto)
-
-
-
-end
\ No newline at end of file
diff -r 4a15873c4ec9 -r 41ee3bfccb4d src/HOL/BNF/Examples/Infinite_Derivation_Trees/Tree.thy
--- a/src/HOL/BNF/Examples/Infinite_Derivation_Trees/Tree.thy	Mon Oct 15 19:03:02 2012 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,192 +0,0 @@
-(*  Title:      HOL/BNF/Examples/Infinite_Derivation_Trees/Tree.thy
-    Author:     Andrei Popescu, TU Muenchen
-    Copyright   2012
-
-Trees with nonterminal internal nodes and terminal leaves.
-*)
-
-header {* Trees with Nonterminal Internal Nodes and Terminal Leaves *}
-
-theory Tree
-imports Prelim
-begin
-
-hide_fact (open) Quotient_Product.prod_rel_def
-
-typedecl N
-typedecl T
-
-codata Tree = NNode (root: N) (ccont: "(T + Tree) fset")
-
-
-section {* Sugar notations for Tree *}
-
-definition
-"llift2 \<phi> as1 as2 \<longleftrightarrow>
- (\<forall> n. Inl n \<in> fset as1 \<longleftrightarrow> Inl n \<in> fset as2) \<and>
- (\<forall> tr1. Inr tr1 \<in> fset as1 \<longrightarrow> (\<exists> tr2. Inr tr2 \<in> fset as2 \<and> \<phi> tr1 tr2)) \<and>
- (\<forall> tr2. Inr tr2 \<in> fset as2 \<longrightarrow> (\<exists> tr1. Inr tr1 \<in> fset as1 \<and> \<phi> tr1 tr2))"
-
-lemma pre_Tree_rel: "pre_Tree_rel \<phi> (n1,as1) (n2,as2) \<longleftrightarrow> n1 = n2 \<and> llift2 \<phi> as1 as2"
-unfolding llift2_def pre_Tree_rel_def sum_rel_def[abs_def] prod_rel_def fset_rel_def split_conv
-apply (auto split: sum.splits)
-apply (metis sumE)
-apply (metis sumE)
-apply (metis sumE)
-apply (metis sumE)
-apply (metis sumE sum.simps(1,2,4))
-apply (metis sumE sum.simps(1,2,4))
-done
-
-
-subsection{* Coinduction *}
-
-theorem Tree_coind_NNode[elim, consumes 1, case_names NNode, induct pred: "HOL.eq"]:
-assumes phi: "\<phi> tr1 tr2" and
-NNode: "\<And> n1 n2 as1 as2.
-          \<lbrakk>\<phi> (NNode n1 as1) (NNode n2 as2)\<rbrakk> \<Longrightarrow>
-          n1 = n2 \<and> llift2 \<phi> as1 as2"
-shows "tr1 = tr2"
-apply(rule mp[OF Tree.dtor_coinduct[of \<phi> tr1 tr2] phi]) proof clarify
-  fix tr1 tr2  assume \<phi>: "\<phi> tr1 tr2"
-  show "pre_Tree_rel \<phi> (Tree_dtor tr1) (Tree_dtor tr2)"
-  apply(cases rule: Tree.ctor_exhaust[of tr1], cases rule: Tree.ctor_exhaust[of tr2])
-  apply (simp add: Tree.dtor_ctor)
-  apply(case_tac x, case_tac xa, simp)
-  unfolding pre_Tree_rel apply(rule NNode) using \<phi> unfolding NNode_def by simp
-qed
-
-theorem TTree_coind[elim, consumes 1, case_names LLift]:
-assumes phi: "\<phi> tr1 tr2" and
-LLift: "\<And> tr1 tr2. \<phi> tr1 tr2 \<Longrightarrow> root tr1 = root tr2 \<and> llift2 \<phi> (ccont tr1) (ccont tr2)"
-shows "tr1 = tr2"
-using phi apply(induct rule: Tree_coind_NNode)
-using LLift by (metis Tree.sels)
-
-
-subsection{* The characteristic theorems transported from fset to set *}
-
-definition "Node n as \<equiv> NNode n (the_inv fset as)"
-definition "cont \<equiv> fset o ccont"
-definition "unfold rt ct \<equiv> Tree_unfold rt (the_inv fset o ct)"
-definition "corec rt qt ct dt \<equiv> Tree_corec rt qt (the_inv fset o ct) (the_inv fset o dt)"
-
-definition lift ("_ ^#" 200) where
-"lift \<phi> as \<longleftrightarrow> (\<forall> tr. Inr tr \<in> as \<longrightarrow> \<phi> tr)"
-
-definition lift2 ("_ ^#2" 200) where
-"lift2 \<phi> as1 as2 \<longleftrightarrow>
- (\<forall> n. Inl n \<in> as1 \<longleftrightarrow> Inl n \<in> as2) \<and>
- (\<forall> tr1. Inr tr1 \<in> as1 \<longrightarrow> (\<exists> tr2. Inr tr2 \<in> as2 \<and> \<phi> tr1 tr2)) \<and>
- (\<forall> tr2. Inr tr2 \<in> as2 \<longrightarrow> (\<exists> tr1. Inr tr1 \<in> as1 \<and> \<phi> tr1 tr2))"
-
-definition liftS ("_ ^#s" 200) where
-"liftS trs = {as. Inr -` as \<subseteq> trs}"
-
-lemma lift2_llift2:
-"\<lbrakk>finite as1; finite as2\<rbrakk> \<Longrightarrow>
- lift2 \<phi> as1 as2 \<longleftrightarrow> llift2 \<phi> (the_inv fset as1) (the_inv fset as2)"
-unfolding lift2_def llift2_def by auto
-
-lemma llift2_lift2:
-"llift2 \<phi> as1 as2 \<longleftrightarrow> lift2 \<phi> (fset as1) (fset as2)"
-using lift2_llift2 by (metis finite_fset fset_cong fset_to_fset)
-
-lemma mono_lift:
-assumes "(\<phi>^#) as"
-and "\<And> tr. \<phi> tr \<Longrightarrow> \<phi>' tr"
-shows "(\<phi>'^#) as"
-using assms unfolding lift_def[abs_def] by blast
-
-lemma mono_liftS:
-assumes "trs1 \<subseteq> trs2 "
-shows "(trs1 ^#s) \<subseteq> (trs2 ^#s)"
-using assms unfolding liftS_def[abs_def] by blast
-
-lemma lift_mono:
-assumes "\<phi> \<le> \<phi>'"
-shows "(\<phi>^#) \<le> (\<phi>'^#)"
-using assms unfolding lift_def[abs_def] by blast
-
-lemma mono_lift2:
-assumes "(\<phi>^#2) as1 as2"
-and "\<And> tr1 tr2. \<phi> tr1 tr2 \<Longrightarrow> \<phi>' tr1 tr2"
-shows "(\<phi>'^#2) as1 as2"
-using assms unfolding lift2_def[abs_def] by blast
-
-lemma lift2_mono:
-assumes "\<phi> \<le> \<phi>'"
-shows "(\<phi>^#2) \<le> (\<phi>'^#2)"
-using assms unfolding lift2_def[abs_def] by blast
-
-lemma finite_cont[simp]: "finite (cont tr)"
-unfolding cont_def by auto
-
-theorem Node_root_cont[simp]:
-"Node (root tr) (cont tr) = tr"
-using Tree.collapse unfolding Node_def cont_def
-by (metis cont_def finite_cont fset_cong fset_to_fset o_def)
-
-theorem Tree_simps[simp]:
-assumes "finite as" and "finite as'"
-shows "Node n as = Node n' as' \<longleftrightarrow> n = n' \<and> as = as'"
-using assms Tree.inject unfolding Node_def
-by (metis fset_to_fset)
-
-theorem Tree_cases[elim, case_names Node Choice]:
-assumes Node: "\<And> n as. \<lbrakk>finite as; tr = Node n as\<rbrakk> \<Longrightarrow> phi"
-shows phi
-apply(cases rule: Tree.exhaust[of tr])
-using Node unfolding Node_def
-by (metis Node Node_root_cont finite_cont)
-
-theorem Tree_sel_ctor[simp]:
-"root (Node n as) = n"
-"finite as \<Longrightarrow> cont (Node n as) = as"
-unfolding Node_def cont_def by auto
-
-theorems root_Node = Tree_sel_ctor(1)
-theorems cont_Node = Tree_sel_ctor(2)
-
-theorem Tree_coind_Node[elim, consumes 1, case_names Node]:
-assumes phi: "\<phi> tr1 tr2" and
-Node:
-"\<And> n1 n2 as1 as2.
-   \<lbrakk>finite as1; finite as2; \<phi> (Node n1 as1) (Node n2 as2)\<rbrakk>
-   \<Longrightarrow> n1 = n2 \<and> (\<phi>^#2) as1 as2"
-shows "tr1 = tr2"
-using phi apply(induct rule: Tree_coind_NNode)
-unfolding llift2_lift2 apply(rule Node)
-unfolding Node_def
-apply (metis finite_fset)
-apply (metis finite_fset)
-by (metis finite_fset fset_cong fset_to_fset)
-
-theorem Tree_coind[elim, consumes 1, case_names Lift, induct pred: "HOL.eq"]:
-assumes phi: "\<phi> tr1 tr2" and
-Lift: "\<And> tr1 tr2. \<phi> tr1 tr2 \<Longrightarrow>
-                  root tr1 = root tr2 \<and> (\<phi>^#2) (cont tr1) (cont tr2)"
-shows "tr1 = tr2"
-using phi apply(induct rule: TTree_coind)
-unfolding llift2_lift2 apply(rule Lift[unfolded cont_def comp_def]) .
-
-theorem unfold:
-"root (unfold rt ct b) = rt b"
-"finite (ct b) \<Longrightarrow> cont (unfold rt ct b) = image (id \<oplus> unfold rt ct) (ct b)"
-using Tree.sel_unfold[of rt "the_inv fset \<circ> ct" b] unfolding unfold_def
-apply - apply metis
-unfolding cont_def comp_def
-by (metis (no_types) fset_to_fset map_fset_image)
-
-theorem corec:
-"root (corec rt qt ct dt b) = rt b"
-"\<lbrakk>finite (ct b); finite (dt b)\<rbrakk> \<Longrightarrow>
- cont (corec rt qt ct dt b) =
- (if qt b then ct b else image (id \<oplus> corec rt qt ct dt) (dt b))"
-using Tree.sel_corec[of rt qt "the_inv fset \<circ> ct" "the_inv fset \<circ> dt" b] unfolding corec_def
-apply -
-apply simp
-unfolding cont_def comp_def id_def
-by (metis (no_types) fset_to_fset map_fset_image)
-
-end