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(* Title: BDD
Author: Veronika Ortner and Norbert Schirmer, 2004
Maintainer: Norbert Schirmer, norbert.schirmer at web de
License: LGPL
*)
(*
LevellistProof.thy
Copyright (C) 2004-2008 Veronika Ortner and Norbert Schirmer
Some rights reserved, TU Muenchen
This library is free software; you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as
published by the Free Software Foundation; either version 2.1 of the
License, or (at your option) any later version.
This library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with this library; if not, write to the Free Software
Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307
USA
*)
section \<open>Proof of Procedure Levellist\<close>
theory LevellistProof imports ProcedureSpecs Simpl.HeapList begin
hide_const (open) DistinctTreeProver.set_of tree.Node tree.Tip
lemma (in Levellist_impl) Levellist_modifies:
shows "\<forall>\<sigma>. \<Gamma>\<turnstile>{\<sigma>} \<acute>levellist :== PROC Levellist (\<acute>p, \<acute>m, \<acute>levellist)
{t. t may_only_modify_globals \<sigma> in [mark,next]}"
apply (hoare_rule HoarePartial.ProcRec1)
apply (vcg spec=modifies)
done
(*a well formed levellist is a list that contains all nodes with variable
i on position i
because the elements of levellist can contain old elements before the call of Levellist,
subdag_eq t pt can not be postulated for all elements of the sublists. One has to make
shure that the initial call of Levellist is parameterized with a levellist with empty sublists.
Otherwise some problems could arise in the call of Reduce!
(\<exists> ptt. (Dag pt low high ptt \<and> subdag_eq (Node lt p rt) ptt \<and> pt\<rightarrow>var = i))
consts wf_levellist :: "dag \<Rightarrow> ref list list \<Rightarrow> ref list list \<Rightarrow>
(ref \<Rightarrow> nat) \<Rightarrow> (ref \<Rightarrow> ref) \<Rightarrow> (ref \<Rightarrow> ref) \<Rightarrow> bool"
defs wf_levellist_def: "wf_levellist t levellist_old levellist_new var low high \<equiv>
case t of Tip \<Rightarrow> levellist_old = levellist_new
| (Node lt p rt) \<Rightarrow>
(\<forall> q. q \<in> set_of t \<longrightarrow> q \<in> set (levellist_new ! (q\<rightarrow>var))) \<and>
(\<forall> i \<le> p\<rightarrow>var. (\<exists> prx. (levellist_new ! i) = prx@(levellist_old ! i)
\<and> (\<forall> pt \<in> set prx. pt \<in> set_of t \<and> pt\<rightarrow>var = i))) \<and>
(\<forall> i. (p\<rightarrow>var) < i \<longrightarrow> (levellist_new ! i) = (levellist_old ! i)) \<and>
(length levellist_new = length levellist_old)"
*)
lemma all_stop_cong: "(\<forall>x. P x) = (\<forall>x. P x)"
by simp
lemma Dag_RefD:
"\<lbrakk>Dag p l r t; p\<noteq>Null\<rbrakk> \<Longrightarrow>
\<exists>lt rt. t=Node lt p rt \<and> Dag (l p) l r lt \<and> Dag (r p) l r rt"
by simp
lemma Dag_unique_ex_conjI:
"\<lbrakk>Dag p l r t; P t\<rbrakk> \<Longrightarrow> (\<exists>t. Dag p l r t \<and> P t)"
by simp
(* FIXME: To BinDag *)
lemma dag_Null [simp]: "dag Null l r = Tip"
by (simp add: dag_def)
definition first:: "ref list \<Rightarrow> ref" where
"first ps = (case ps of [] \<Rightarrow> Null | (p#rs) \<Rightarrow> p)"
lemma first_simps [simp]:
"first [] = Null"
"first (r#rs) = r"
by (simp_all add: first_def)
definition Levellist:: "ref list \<Rightarrow> (ref \<Rightarrow> ref) \<Rightarrow> (ref list list) \<Rightarrow> bool" where
"Levellist hds next ll \<longleftrightarrow> (map first ll = hds) \<and>
(\<forall>i < length hds. List (hds ! i) next (ll!i))"
lemma Levellist_unique:
assumes ll: "Levellist hds next ll"
assumes ll': "Levellist hds next ll'"
shows "ll=ll'"
proof -
from ll have "length ll = length hds"
by (clarsimp simp add: Levellist_def)
moreover
from ll' have "length ll' = length hds"
by (clarsimp simp add: Levellist_def)
ultimately have leq: "length ll = length ll'" by simp
show ?thesis
proof (rule nth_equalityI [OF leq, rule_format])
fix i
assume "i < length ll"
with ll ll'
show "ll!i = ll'!i"
apply (clarsimp simp add: Levellist_def)
apply (erule_tac x=i in allE)
apply (erule_tac x=i in allE)
apply simp
by (erule List_unique)
qed
qed
lemma Levellist_unique_ex_conj_simp [simp]:
"Levellist hds next ll \<Longrightarrow> (\<exists>ll. Levellist hds next ll \<and> P ll) = P ll"
by (auto dest: Levellist_unique)
lemma in_set_concat_idx:
"x \<in> set (concat xss) \<Longrightarrow> \<exists>i < length xss. x \<in> set (xss!i)"
apply (induct xss)
apply simp
apply clarsimp
apply (erule disjE)
apply (rule_tac x=0 in exI)
apply simp
apply auto
done
definition wf_levellist :: "dag \<Rightarrow> ref list list \<Rightarrow> ref list list \<Rightarrow>
(ref \<Rightarrow> nat) \<Rightarrow> bool" where
"wf_levellist t levellist_old levellist_new var =
(case t of Tip \<Rightarrow> levellist_old = levellist_new
| (Node lt p rt) \<Rightarrow>
(\<forall> q. q \<in> set_of t \<longrightarrow> q \<in> set (levellist_new ! (var q))) \<and>
(\<forall> i \<le> var p. (\<exists> prx. (levellist_new ! i) = prx@(levellist_old ! i)
\<and> (\<forall> pt \<in> set prx. pt \<in> set_of t \<and> var pt = i))) \<and>
(\<forall> i. (var p) < i \<longrightarrow> (levellist_new ! i) = (levellist_old ! i)) \<and>
(length levellist_new = length levellist_old))"
lemma wf_levellist_subset:
assumes wf_ll: "wf_levellist t ll ll' var"
shows "set (concat ll') \<subseteq> set (concat ll) \<union> set_of t"
proof (cases t)
case Tip with wf_ll show ?thesis by (simp add: wf_levellist_def)
next
case (Node lt p rt)
show ?thesis
proof -
{
fix n
assume "n \<in> set (concat ll')"
from in_set_concat_idx [OF this]
obtain i where i_bound: "i < length ll'" and n_in: "n \<in> set (ll' ! i)"
by blast
have "n \<in> set (concat ll) \<union> set_of t"
proof (cases "i \<le> var p")
case True
with wf_ll obtain prx where
ll'_ll: "ll' ! i = prx @ ll ! i" and
prx: "\<forall>pt \<in> set prx. pt \<in> set_of t" and
leq: "length ll' = length ll"
apply (clarsimp simp add: wf_levellist_def Node)
apply (erule_tac x="i" in allE)
apply clarsimp
done
show ?thesis
proof (cases "n \<in> set prx")
case True
with prx have "n \<in> set_of t"
by simp
thus ?thesis by simp
next
case False
with n_in ll'_ll
have "n \<in> set (ll ! i)"
by simp
with i_bound leq
have "n \<in> set (concat ll)"
by auto
thus ?thesis by simp
qed
next
case False
with wf_ll obtain "ll'!i = ll!i" "length ll' = length ll"
by (auto simp add: wf_levellist_def Node)
with n_in i_bound
have "n \<in> set (concat ll)"
by auto
thus ?thesis by simp
qed
}
thus ?thesis by auto
qed
qed
(*
next
show "set (concat ll) \<union> set_of t \<subseteq> set (concat ll')"
proof -
{
fix n
assume "n \<in> set (concat ll)"
from in_set_concat_idx [OF this]
obtain i where i_bound: "i < length ll" and n_in: "n \<in> set (ll ! i)"
by blast
with wf_ll
obtain "n \<in> set (ll' ! i)" "length ll = length ll'"
apply (clarsimp simp add: wf_levellist_def Node)
apply (case_tac "i \<le> var p")
apply fastforce
apply fastforce
done
with i_bound have "n \<in> set (concat ll')"
by auto
}
moreover
{
fix n
assume "n \<in> set_of t"
with wf_ll obtain "n \<in> set (ll' ! var n)" "length ll' = length ll"
by (auto simp add: wf_levellist_def Node)
with root
next
proof (cases prx)
case Nil
with ll'_ll i_bound leq n_in
have "n \<in> set (concat ll)"
by auto
thus ?thesis by simp
next
case (Cons p prx')
show ?thesis
apply auto
*)
(*
consts wf_levellist :: "dag \<Rightarrow> ref list \<Rightarrow> ref list \<Rightarrow>
(ref \<Rightarrow> ref) \<Rightarrow> (ref \<Rightarrow> ref) \<Rightarrow>
(ref \<Rightarrow> nat) \<Rightarrow> bool"
defs wf_levellist_def:
"wf_levellist t levellist_old levellist_new next_old next_new var \<equiv>
case t of Tip \<Rightarrow> levellist_old = levellist_new
| (Node lt p rt) \<Rightarrow>
(\<forall> q. q \<in> set_of t \<longrightarrow> (\<exists>ns. List (levellist_new ! (var q)) next_new ns \<and>
q \<in> set ns)) \<and>
(\<forall> i \<le> var p. (\<exists>ns_new ns_old.
List (levellist_new ! i) next_new ns_new \<and>
List (levellist_old ! i) next_old ns_old \<and>
(\<exists> prx. ns_new = prx@ns_old
\<and> (\<forall> pt \<in> set prx. pt \<in> set_of t \<and> var pt = i)))) \<and>
(\<forall> i. (var p) < i \<longrightarrow> (\<exists>ns_new ns_old.
List (levellist_new ! i) next_new ns_new \<and>
List (levellist_old ! i) next_old ns_old \<and>
ns_new = ns_old)) \<and>
(length levellist_new = length levellist_old)"
*)
lemma Levellist_ext_to_all: "((\<exists>ll. Levellist hds next ll \<and> P ll) \<longrightarrow> Q)
=
(\<forall>ll. Levellist hds next ll \<and> P ll \<longrightarrow> Q)"
apply blast
done
lemma Levellist_length: "Levellist hds p ll \<Longrightarrow> length ll = length hds"
by (auto simp add: Levellist_def)
lemma map_update:
"\<And>i. i < length xss \<Longrightarrow> map f (xss[i := xs]) = (map f xss) [i := f xs]"
apply (induct xss)
apply simp
apply (case_tac i)
apply simp
apply simp
done
lemma (in Levellist_impl) Levellist_spec_total':
shows "\<forall>ll \<sigma> t. \<Gamma>,\<Theta>\<turnstile>\<^sub>t
\<lbrace>\<sigma>. Dag \<acute>p \<acute>low \<acute>high t \<and> (\<acute>p \<noteq> Null \<longrightarrow> (\<acute>p\<rightarrow>\<acute>var) < length \<acute>levellist) \<and>
ordered t \<acute>var \<and> Levellist \<acute>levellist \<acute>next ll \<and>
(\<forall>n \<in> set_of t.
(if \<acute>mark n = \<acute>m
then n \<in> set (ll ! \<acute>var n) \<and>
(\<forall>nt p. Dag n \<acute>low \<acute>high nt \<and> p \<in> set_of nt
\<longrightarrow> \<acute>mark p = \<acute>m)
else n \<notin> set (concat ll)))\<rbrace>
\<acute>levellist :== PROC Levellist (\<acute>p, \<acute>m, \<acute>levellist)
\<lbrace>\<exists>ll'. Levellist \<acute>levellist \<acute>next ll' \<and> wf_levellist t ll ll' \<^bsup>\<sigma>\<^esup>var \<and>
wf_marking t \<^bsup>\<sigma>\<^esup>mark \<acute>mark \<^bsup>\<sigma>\<^esup>m \<and>
(\<forall>p. p \<notin> set_of t \<longrightarrow> \<^bsup>\<sigma>\<^esup>next p = \<acute>next p)
\<rbrace>"
apply (hoare_rule HoareTotal.ProcRec1
[where r="measure (\<lambda>(s,p). size (dag \<^bsup>s\<^esup>p \<^bsup>s\<^esup>low \<^bsup>s\<^esup>high))"])
apply vcg
apply (rule conjI)
apply clarify
apply (rule conjI)
apply clarify
apply (clarsimp simp del: BinDag.set_of.simps split del: if_split)
defer
apply (rule impI)
apply (clarsimp simp del: BinDag.set_of.simps split del: if_split)
defer
apply (clarsimp simp add: wf_levellist_def wf_marking_def) (* p=Null*)
apply (simp only: Levellist_ext_to_all )
proof -
fix ll var low high mark "next" nexta p levellist m lt rt
assume pnN: "p \<noteq> Null"
assume mark_p: "mark p = (\<not> m)"
assume lt: "Dag (low p) low high lt"
assume rt: "Dag (high p) low high rt"
from pnN lt rt have Dag_p: "Dag p low high (Node lt p rt)" by simp
from Dag_p rt
have size_rt_dec: "size (dag (high p) low high) < size (dag p low high)"
by (simp only: Dag_dag) simp
from Dag_p lt
have size_lt_dec: "size (dag (low p) low high) < size (dag p low high)"
by (simp only: Dag_dag) simp
assume ll: "Levellist levellist next ll"
assume marked_child_ll:
"\<forall>n \<in> set_of (Node lt p rt).
if mark n = m
then n \<in> set (ll ! var n) \<and>
(\<forall>nt p. Dag n low high nt \<and> p \<in> set_of nt \<longrightarrow> mark p = m)
else n \<notin> set (concat ll)"
with mark_p have p_notin_ll: "p \<notin> set (concat ll)"
by auto
assume varsll': "var p < length levellist"
with ll have varsll: "var p < length ll"
by (simp add: Levellist_length)
assume orderedt: "ordered (Node lt p rt) var"
show "(low p \<noteq> Null \<longrightarrow> var (low p) < length levellist) \<and>
ordered lt var \<and>
(\<forall>n \<in> set_of lt.
if mark n = m
then n \<in> set (ll ! var n) \<and>
(\<forall>nt p. Dag n low high nt \<and> p \<in> set_of nt \<longrightarrow> mark p = m)
else n \<notin> set (concat ll)) \<and>
size (dag (low p) low high) < size (dag p low high) \<and>
(\<forall>marka nexta levellist lla.
Levellist levellist nexta lla \<and>
wf_levellist lt ll lla var \<and> wf_marking lt mark marka m \<and>
(\<forall>p. p \<notin> set_of lt \<longrightarrow> next p = nexta p)\<longrightarrow>
(high p \<noteq> Null \<longrightarrow> var (high p) < length levellist) \<and>
ordered rt var \<and>
(\<exists>lla. Levellist levellist nexta lla \<and>
(\<forall>n \<in> set_of rt.
if marka n = m
then n \<in> set (lla ! var n) \<and>
(\<forall>nt p. Dag n low high nt \<and> p \<in> set_of nt \<longrightarrow>
marka p = m)
else n \<notin> set (concat lla)) \<and>
size (dag (high p) low high) < size (dag p low high) \<and>
(\<forall>markb nextb levellist llb.
Levellist levellist nextb llb \<and>
wf_levellist rt lla llb var \<and>
wf_marking rt marka markb m \<and>
(\<forall>p. p \<notin> set_of rt \<longrightarrow> nexta p = nextb p) \<longrightarrow>
(\<exists>ll'. Levellist (levellist[var p := p])
(nextb(p := levellist ! var p)) ll' \<and>
wf_levellist (Node lt p rt) ll ll' var \<and>
wf_marking (Node lt p rt) mark (markb(p := m)) m \<and>
(\<forall>pa. pa \<notin> set_of (Node lt p rt) \<longrightarrow>
next pa =
(if pa = p then levellist ! var p
else nextb pa))))))"
proof (cases "lt")
case Tip
note lt_Tip = this
show ?thesis
proof (cases "rt")
case Tip
show ?thesis
using size_rt_dec Tip lt_Tip Tip lt rt
apply clarsimp
subgoal premises prems for marka nexta levellista lla markb nextb levellistb llb
proof -
have lla: "Levellist levellista nexta lla" by fact
have llb: "Levellist levellistb nextb llb" by fact
have wfll_lt: "wf_levellist Tip ll lla var"
"wf_marking Tip mark marka m" by fact+
then have ll_lla: "ll = lla"
by (simp add: wf_levellist_def)
moreover
with wfll_lt lt_Tip lt have "marka = mark"
by (simp add: wf_marking_def)
moreover
have wfll_rt:"wf_levellist Tip lla llb var"
"wf_marking Tip marka markb m" by fact+
then have lla_llb: "lla = llb"
by (simp add: wf_levellist_def)
moreover
with wfll_rt Tip rt have "markb = marka"
by (simp add: wf_marking_def)
moreover
from varsll llb ll_lla lla_llb
obtain "var p < length levellistb" "var p < length llb"
by (simp add: Levellist_length)
with llb pnN
have llc: "Levellist (levellistb[var p := p]) (nextb(p := levellistb ! var p))
(llb[var p := p # llb ! var p])"
apply (clarsimp simp add: Levellist_def map_update)
apply (erule_tac x=i in allE)
apply clarsimp
apply (subgoal_tac "p \<notin> set (llb ! i) ")
prefer 2
using p_notin_ll ll_lla lla_llb
apply simp
apply (case_tac "i=var p")
apply simp
apply simp
done
ultimately
show ?thesis
using lt_Tip Tip varsll
apply (clarsimp simp add: wf_levellist_def wf_marking_def)
proof -
fix i
assume varsllb: "var p < length llb"
assume "i \<le> var p"
show "\<exists>prx. llb[var p := p#llb!var p]!i = prx @ llb!i \<and>
(\<forall>pt\<in>set prx. pt = p \<and> var pt = i)"
proof (cases "i = var p")
case True
with pnN lt rt varsllb lt_Tip Tip show ?thesis
apply -
apply (rule_tac x="[p]" in exI)
apply (simp add: subdag_eq_def)
done
next
assume "i \<noteq> var p"
with varsllb show ?thesis
apply -
apply (rule_tac x="[]" in exI)
apply (simp add: subdag_eq_def)
done
qed
qed
qed
done
next
case (Node dag1 a dag2)
have rt_node: "rt = Node dag1 a dag2" by fact
with rt have high_p: "high p = a"
by simp
have s: "\<And>nexta. (\<forall>p. next p = nexta p) = (next = nexta)"
by auto
show ?thesis
using size_rt_dec size_lt_dec rt_node lt_Tip Tip lt rt
apply (clarsimp simp del: set_of_Node split del: if_split simp add: s)
subgoal premises prems for marka levellista lla
proof -
have lla: "Levellist levellista next lla" by fact
have wfll_lt:"wf_levellist Tip ll lla var"
"wf_marking Tip mark marka m" by fact+
from this have ll_lla: "ll = lla"
by (simp add: wf_levellist_def)
moreover
from wfll_lt lt_Tip lt have marklrec: "marka = mark"
by (simp add: wf_marking_def)
from orderedt varsll lla ll_lla rt_node lt_Tip high_p
have var_highp_bound: "var (high p) < length levellista"
by (auto simp add: Levellist_length)
from orderedt high_p rt_node lt_Tip
have ordered_rt: "ordered (Node dag1 (high p) dag2) var"
by simp
from high_p marklrec marked_child_ll lt rt lt_Tip rt_node ll_lla
have mark_rt: "(\<forall>n\<in>set_of (Node dag1 (high p) dag2).
if marka n = m
then n \<in> set (lla ! var n) \<and>
(\<forall>nt p. Dag n low high nt \<and> p \<in> set_of nt \<longrightarrow> marka p = m)
else n \<notin> set (concat lla))"
apply (simp only: BinDag.set_of.simps)
apply clarify
apply (drule_tac x=n in bspec)
apply blast
apply assumption
done
show ?thesis
apply (rule conjI)
apply (rule var_highp_bound)
apply (rule conjI)
apply (rule ordered_rt)
apply (rule conjI)
apply (rule mark_rt)
apply clarify
apply clarsimp
subgoal premises prems for markb nextb levellistb llb
proof -
have llb: "Levellist levellistb nextb llb" by fact
have wfll_rt: "wf_levellist (Node dag1 (high p) dag2) lla llb var" by fact
have wfmarking_rt: "wf_marking (Node dag1 (high p) dag2) marka markb m" by fact
from wfll_rt varsll llb ll_lla
obtain var_p_bounds: "var p < length levellistb" "var p < length llb"
by (simp add: Levellist_length wf_levellist_def)
with p_notin_ll ll_lla wfll_rt
have p_notin_llb: "\<forall>i < length llb. p \<notin> set (llb ! i)"
apply -
apply (intro allI impI)
apply (clarsimp simp add: wf_levellist_def)
apply (case_tac "i \<le> var (high p)")
apply (drule_tac x=i in spec)
using orderedt rt_node lt_Tip high_p
apply clarsimp
apply (drule_tac x=i in spec)
apply (drule_tac x=i in spec)
apply clarsimp
done
with llb pnN var_p_bounds
have llc: "Levellist (levellistb[var p := p])
(nextb(p := levellistb ! var p))
(llb[var p := p # llb ! var p])"
apply (clarsimp simp add: Levellist_def map_update)
apply (erule_tac x=i in allE)
apply (erule_tac x=i in allE)
apply clarsimp
apply (case_tac "i=var p")
apply simp
apply simp
done
then show ?thesis
apply simp
using wfll_rt wfmarking_rt
lt_Tip rt_node varsll orderedt lt rt pnN ll_lla marklrec
apply (clarsimp simp add: wf_levellist_def wf_marking_def)
apply (intro conjI)
apply (rule allI)
apply (rule conjI)
apply (erule_tac x="q" in allE)
apply (case_tac "var p = var q")
apply fastforce
apply fastforce
apply (case_tac "var p = var q")
apply hypsubst_thin
apply fastforce
apply fastforce
apply (rule allI)
apply (rotate_tac 4)
apply (erule_tac x="i" in allE)
apply (case_tac "i=var p")
apply simp
apply (case_tac "var (high p) < i")
apply simp
apply simp
apply (erule exE)
apply (rule_tac x="prx" in exI)
apply (intro conjI)
apply simp
apply clarify
apply (rotate_tac 15)
apply (erule_tac x="pt" in ballE)
apply fastforce
apply fastforce
done
qed
done
qed
done
qed
next
case (Node llt l rlt)
have lt_Node: "lt = Node llt l rlt" by fact
from orderedt lt varsll' lt_Node
obtain ordered_lt:
"ordered lt var" "(low p \<noteq> Null \<longrightarrow> var (low p) < length levellist)"
by (cases rt) auto
from lt lt_Node marked_child_ll
have mark_lt: "\<forall>n\<in>set_of lt.
if mark n = m
then n \<in> set (ll ! var n) \<and>
(\<forall>nt p. Dag n low high nt \<and> p \<in> set_of nt \<longrightarrow> mark p = m)
else n \<notin> set (concat ll)"
apply (simp only: BinDag.set_of.simps)
apply clarify
apply (drule_tac x=n in bspec)
apply blast
apply assumption
done
show ?thesis
apply (intro conjI ordered_lt mark_lt size_lt_dec)
apply (clarify)
apply (simp add: size_rt_dec split del: if_split)
apply (simp only: Levellist_ext_to_all)
subgoal premises prems for marka nexta levellista lla
proof -
have lla: "Levellist levellista nexta lla" by fact
have wfll_lt: "wf_levellist lt ll lla var" by fact
have wfmarking_lt:"wf_marking lt mark marka m" by fact
from wfll_lt lt_Node
have lla_eq_ll: "length lla = length ll"
by (simp add: wf_levellist_def)
with ll lla have lla_eq_ll': "length levellista = length levellist"
by (simp add: Levellist_length)
with orderedt rt lt_Node lt varsll'
obtain ordered_rt:
"ordered rt var" "(high p \<noteq> Null \<longrightarrow> var (high p) < length levellista)"
by (cases rt) auto
from wfll_lt lt_Node
have nodes_in_lla: "\<forall> q. q \<in> set_of lt \<longrightarrow> q \<in> set (lla ! (q\<rightarrow>var))"
by (simp add: wf_levellist_def)
from wfll_lt lt_Node lt
have lla_st: "(\<forall>i \<le> (low p)\<rightarrow>var.
(\<exists>prx. (lla ! i) = prx@(ll ! i) \<and>
(\<forall>pt \<in> set prx. pt \<in> set_of lt \<and> pt\<rightarrow>var = i)))"
by (simp add: wf_levellist_def)
from wfll_lt lt_Node lt
have lla_nc: "\<forall>i. ((low p)\<rightarrow>var) < i \<longrightarrow> (lla ! i) = (ll ! i)"
by (simp add: wf_levellist_def)
from wfmarking_lt lt_Node lt
have mot_nc: "\<forall> n. n \<notin> set_of lt \<longrightarrow> mark n = marka n"
by (simp add: wf_marking_def)
from wfmarking_lt lt_Node lt
have mit_marked: "\<forall>n. n \<in> set_of lt \<longrightarrow> marka n = m"
by (simp add: wf_marking_def)
from marked_child_ll nodes_in_lla mot_nc mit_marked lla_st
have mark_rt: "\<forall>n\<in>set_of rt.
if marka n = m
then n \<in> set (lla ! var n) \<and>
(\<forall>nt p. Dag n low high nt \<and> p \<in> set_of nt \<longrightarrow> marka p = m)
else n \<notin> set (concat lla)"
apply -
apply (rule ballI)
apply (drule_tac x="n" in bspec)
apply (simp)
proof -
fix n
assume nodes_in_lla: "\<forall>q. q \<in> set_of lt \<longrightarrow> q \<in> set (lla ! var q)"
assume mot_nc: "\<forall>n. n \<notin> set_of lt \<longrightarrow> mark n = marka n"
assume mit_marked: "\<forall>n. n \<in> set_of lt \<longrightarrow> marka n = m"
assume marked_child_ll: "if mark n = m
then n \<in> set (ll ! var n) \<and>
(\<forall>nt p. Dag n low high nt \<and> p \<in> set_of nt \<longrightarrow> mark p = m)
else n \<notin> set (concat ll)"
assume lla_st: "\<forall>i\<le>var (low p).
\<exists>prx. lla ! i = prx @ ll ! i \<and>
(\<forall>pt\<in>set prx. pt \<in> set_of lt \<and> var pt = i)"
assume n_in_rt: " n \<in> set_of rt"
show n_in_lla_marked: "if marka n = m
then n \<in> set (lla ! var n) \<and>
(\<forall>nt p. Dag n low high nt \<and> p \<in> set_of nt \<longrightarrow> marka p = m)
else n \<notin> set (concat lla)"
proof (cases "n \<in> set_of lt")
case True
from True nodes_in_lla have n_in_ll: "n \<in> set (lla ! var n)"
by simp
moreover
from True wfmarking_lt
have "marka n = m"
apply (cases lt)
apply (auto simp add: wf_marking_def)
done
moreover
{
fix nt p
assume "Dag n low high nt"
with lt True have subset_nt_lt: "set_of nt \<subseteq> set_of lt"
by (rule dag_setof_subsetD)
moreover assume " p \<in> set_of nt"
ultimately have "p \<in> set_of lt"
by blast
with mit_marked have " marka p = m"
by simp
}
ultimately show ?thesis
using n_in_rt
apply clarsimp
done
next
assume n_notin_lt: "n \<notin> set_of lt"
show ?thesis
proof (cases "marka n = m")
case True
from n_notin_lt mot_nc have marka_eq_mark: "mark n = marka n"
by simp
from marka_eq_mark True have n_marked: "mark n = m"
by simp
from rt n_in_rt have nnN: "n \<noteq> Null"
apply -
apply (rule set_of_nn [rule_format])
apply fastforce
apply assumption
done
from marked_child_ll n_in_rt marka_eq_mark nnN n_marked
have n_in_ll: "n \<in> set (ll ! var n)"
by fastforce
from marked_child_ll n_in_rt marka_eq_mark nnN n_marked lt rt
have nt_mark: "\<forall>nt p. Dag n low high nt \<and> p \<in> set_of nt \<longrightarrow> mark p = m"
by simp
from nodes_in_lla n_in_ll lla_st
have n_in_lla: "n \<in> set (lla ! var n)"
proof (cases "var (low p) < (var n)")
case True
with lla_nc have "(lla ! var n) = (ll ! var n)"
by fastforce
with n_in_ll show ?thesis
by fastforce
next
assume varnslp: " \<not> var (low p) < var n"
with lla_st
have ll_in_lla: "\<exists>prx. lla ! (var n) = prx @ ll ! (var n)"
apply -
apply (erule_tac x="var n" in allE)
apply fastforce
done
with n_in_ll show ?thesis
by fastforce
qed
{
fix nt pt
assume nt_Dag: "Dag n low high nt"
assume pt_in_nt: "pt \<in> set_of nt"
have " marka pt = m"
proof (cases "pt \<in> set_of lt")
case True
with mit_marked show ?thesis
by fastforce
next
assume pt_notin_lt: " pt \<notin> set_of lt"
with mot_nc have "mark pt = marka pt"
by fastforce
with nt_mark nt_Dag pt_in_nt show ?thesis
by fastforce
qed
}
then have nt_marka:
"\<forall>nt pt. Dag n low high nt \<and> pt \<in> set_of nt \<longrightarrow> marka pt = m"
by fastforce
with n_in_lla nt_marka True show ?thesis
by fastforce
next
case False
note n_not_marka = this
with wfmarking_lt n_notin_lt
have "mark n \<noteq> m"
by (simp add: wf_marking_def lt_Node)
with marked_child_ll
have n_notin_ll: "n \<notin> set (concat ll)"
by simp
show ?thesis
proof (cases "n \<in> set (concat lla)")
case False with n_not_marka show ?thesis by simp
next
case True
with wf_levellist_subset [OF wfll_lt] n_notin_ll
have "n \<in> set_of lt"
by blast
with n_notin_lt have False by simp
thus ?thesis ..
qed
qed
qed
qed
show ?thesis
apply (intro conjI ordered_rt mark_rt)
apply clarify
subgoal premises prems for markb nextb levellistb llb
proof -
have llb: "Levellist levellistb nextb llb" by fact
have wfll_rt: "wf_levellist rt lla llb var" by fact
have wfmarking_rt: "wf_marking rt marka markb m" by fact
show ?thesis
proof (cases rt)
case Tip
from wfll_rt Tip have lla_llb: "lla = llb"
by (simp add: wf_levellist_def)
moreover
from wfmarking_rt Tip rt have "markb = marka"
by (simp add: wf_marking_def)
moreover
from wfll_lt varsll llb lla_llb
obtain var_p_bounds: "var p < length levellistb" "var p < length llb"
by (simp add: Levellist_length wf_levellist_def lt_Node Tip)
with p_notin_ll lla_llb wfll_lt
have p_notin_llb: "\<forall>i < length llb. p \<notin> set (llb ! i)"
apply -
apply (intro allI impI)
apply (clarsimp simp add: wf_levellist_def lt_Node)
apply (case_tac "i \<le> var l")
apply (drule_tac x=i in spec)
using orderedt Tip lt_Node
apply clarsimp
apply (drule_tac x=i in spec)
apply (drule_tac x=i in spec)
apply clarsimp
done
with llb pnN var_p_bounds
have llc: "Levellist (levellistb[var p := p])
(nextb(p := levellistb ! var p))
(llb[var p := p # llb ! var p])"
apply (clarsimp simp add: Levellist_def map_update)
apply (erule_tac x=i in allE)
apply (erule_tac x=i in allE)
apply clarsimp
apply (case_tac "i=var p")
apply simp
apply simp
done
ultimately show ?thesis
using Tip lt_Node varsll orderedt lt rt pnN wfll_lt wfmarking_lt
apply (clarsimp simp add: wf_levellist_def wf_marking_def)
apply (intro conjI)
apply (rule allI)
apply (rule conjI)
apply (erule_tac x="q" in allE)
apply (case_tac "var p = var q")
apply fastforce
apply fastforce
apply (case_tac "var p = var q")
apply hypsubst_thin
apply fastforce
apply fastforce
apply (rule allI)
apply (rotate_tac 4)
apply (erule_tac x="i" in allE)
apply (case_tac "i=var p")
apply simp
apply (case_tac "var (low p) < i")
apply simp
apply simp
apply (erule exE)
apply (rule_tac x="prx" in exI)
apply (intro conjI)
apply simp
apply clarify
apply (rotate_tac 15)
apply (erule_tac x="pt" in ballE)
apply fastforce
apply fastforce
done
next
case (Node lrt r rrt)
have rt_Node: "rt = Node lrt r rrt" by fact
from wfll_rt rt_Node
have llb_eq_lla: "length llb = length lla"
by (simp add: wf_levellist_def)
with llb lla
have llb_eq_lla': "length levellistb = length levellista"
by (simp add: Levellist_length)
from wfll_rt rt_Node
have nodes_in_llb: "\<forall>q. q \<in> set_of rt \<longrightarrow> q \<in> set (llb ! (q\<rightarrow>var))"
by (simp add: wf_levellist_def)
from wfll_rt rt_Node rt
have llb_st: "(\<forall> i \<le> (high p)\<rightarrow>var.
(\<exists> prx. (llb ! i) = prx@(lla ! i) \<and>
(\<forall>pt \<in> set prx. pt \<in> set_of rt \<and> pt\<rightarrow>var = i)))"
by (simp add: wf_levellist_def)
from wfll_rt rt_Node rt
have llb_nc:
"\<forall>i. ((high p)\<rightarrow>var) < i \<longrightarrow> (llb ! i) = (lla ! i)"
by (simp add: wf_levellist_def)
from wfmarking_rt rt_Node rt
have mort_nc: "\<forall>n. n \<notin> set_of rt \<longrightarrow> marka n = markb n"
by (simp add: wf_marking_def)
from wfmarking_rt rt_Node rt
have mirt_marked: "\<forall>n. n \<in> set_of rt \<longrightarrow> markb n = m"
by (simp add: wf_marking_def)
with p_notin_ll wfll_rt wfll_lt
have p_notin_llb: "\<forall>i < length llb. p \<notin> set (llb ! i)"
apply -
apply (intro allI impI)
apply (clarsimp simp add: wf_levellist_def lt_Node rt_Node)
apply (case_tac "i \<le> var r")
apply (drule_tac x=i in spec)
using orderedt rt_Node lt_Node
apply clarsimp
apply (erule disjE)
apply clarsimp
apply (case_tac "i \<le> var l")
apply (drule_tac x=i in spec)
apply clarsimp
apply clarsimp
apply (subgoal_tac "llb ! i = lla ! i")
prefer 2
apply clarsimp
apply (case_tac "i \<le> var l")
apply (drule_tac x=i in spec, erule impE, assumption)
apply clarsimp
using orderedt rt_Node lt_Node
apply clarsimp
apply clarsimp
done
from wfll_lt wfll_rt varsll lla llb
obtain var_p_bounds: "var p < length levellistb" "var p < length llb"
by (simp add: Levellist_length wf_levellist_def lt_Node rt_Node)
with p_notin_llb llb pnN var_p_bounds
have llc: "Levellist (levellistb[var p := p])
(nextb(p := levellistb ! var p))
(llb[var p := p # llb ! var p])"
apply (clarsimp simp add: Levellist_def map_update)
apply (erule_tac x=i in allE)
apply (erule_tac x=i in allE)
apply clarsimp
apply (case_tac "i=var p")
apply simp
apply simp
done
then show ?thesis
proof (clarsimp)
show "wf_levellist (Node lt p rt) ll (llb[var p := p#llb ! var p]) var \<and>
wf_marking (Node lt p rt) mark (markb(p := m)) m"
proof -
have nodes_in_upllb: "\<forall> q. q \<in> set_of (Node lt p rt)
\<longrightarrow> q \<in> set (llb[var p :=p # llb ! var p] ! (var q))"
apply -
apply (rule allI)
apply (rule impI)
proof -
fix q
assume q_in_t: "q \<in> set_of (Node lt p rt)"
show q_in_upllb:
"q \<in> set (llb[var p :=p # llb ! var p] ! (var q))"
proof (cases "q \<in> set_of rt")
case True
with nodes_in_llb have q_in_llb: "q \<in> set (llb ! (var q))"
by fastforce
from orderedt rt_Node lt_Node lt rt
have ordered_rt: "ordered rt var"
by fastforce
from True rt ordered_rt rt_Node lt lt_Node have "var q \<le> var r"
apply -
apply (drule subnodes_ordered)
apply fastforce
apply fastforce
apply fastforce
done
with orderedt rt lt rt_Node lt_Node have "var q < var p"
by fastforce
then have
"llb[var p :=p#llb ! var p] ! var q =
llb ! var q"
by fastforce
with q_in_llb show ?thesis
by fastforce
next
assume q_notin_rt: "q \<notin> set_of rt"
show "q \<in> set (llb[var p :=p # llb ! var p] ! var q)"
proof (cases "q \<in> set_of lt")
case True
assume q_in_lt: "q \<in> set_of lt"
with nodes_in_lla have q_in_lla: "q \<in> set (lla ! (var q))"
by fastforce
from orderedt rt_Node lt_Node lt rt
have ordered_lt: "ordered lt var"
by fastforce
from q_in_lt lt ordered_lt rt_Node rt lt_Node
have "var q \<le> var l"
apply -
apply (drule subnodes_ordered)
apply fastforce
apply fastforce
apply fastforce
done
with orderedt rt lt rt_Node lt_Node have qsp: "var q < var p"
by fastforce
then show ?thesis
proof (cases "var q \<le> var (high p)")
case True
with llb_st
have "\<exists>prx. (llb ! (var q)) = prx@(lla ! (var q))"
by fastforce
with nodes_in_lla q_in_lla
have q_in_llb: "q \<in> set (llb ! (var q))"
by fastforce
from qsp
have "llb[var p :=p#llb ! var p]!var q =
llb ! (var q)"
by fastforce
with q_in_llb show ?thesis
by fastforce
next
assume "\<not> var q \<le> var (high p)"
with llb_nc have "llb ! (var q) = lla ! (var q)"
by fastforce
with q_in_lla have q_in_llb: "q \<in> set (llb ! (var q))"
by fastforce
from qsp have
"llb[var p :=p # llb ! var p] ! var q = llb ! (var q)"
by fastforce
with q_in_llb show ?thesis
by fastforce
qed
next
assume q_notin_lt: "q \<notin> set_of lt"
with q_notin_rt rt lt rt_Node lt_Node q_in_t have qp: "q = p"
by fastforce
with varsll lla_eq_ll llb_eq_lla have "var p < length llb"
by fastforce
with qp show ?thesis
by simp
qed
qed
qed
have prx_ll_st: "\<forall>i \<le> var p.
(\<exists>prx. llb[var p :=p#llb!var p]!i = prx@(ll!i) \<and>
(\<forall>pt \<in> set prx. pt \<in> set_of (Node lt p rt) \<and> var pt = i))"
apply -
apply (rule allI)
apply (rule impI)
proof -
fix i
assume isep: "i \<le> var p"
show "\<exists>prx. llb[var p :=p#llb!var p]!i = prx@ll!i \<and>
(\<forall>pt\<in>set prx. pt \<in> set_of (Node lt p rt) \<and> var pt = i)"
proof (cases "i = var p")
case True
with orderedt lt lt_Node rt rt_Node
have lpsp: "var (low p) < var p"
by fastforce
with orderedt lt lt_Node rt rt_Node
have hpsp: "var (high p) < var p"
by fastforce
with lpsp lla_nc
have llall: "lla ! var p = ll ! var p"
by fastforce
with hpsp llb_nc have "llb ! var p = ll ! var p"
by fastforce
with llb_eq_lla lla_eq_ll isep True varsll lt rt show ?thesis
apply -
apply (rule_tac x="[p]" in exI)
apply (rule conjI)
apply simp
apply (rule ballI)
apply fastforce
done
next
assume inp: " i \<noteq> var p"
show ?thesis
proof (cases "var (low p) < i")
case True
with lla_nc have llall: "lla ! i = ll ! i"
by fastforce
assume vpsi: "var (low p) < i"
show ?thesis
proof (cases "var (high p) < i")
case True
with llall llb_nc have "llb ! i = ll ! i"
by fastforce
with inp True vpsi varsll lt rt show ?thesis
apply -
apply (rule_tac x="[]" in exI)
apply (rule conjI)
apply simp
apply (rule ballI)
apply fastforce
done
next
assume isehp: " \<not> var (high p) < i"
with vpsi lla_nc have lla_ll: "lla ! i = ll ! i"
by fastforce
with isehp llb_st
have prx_lla: "\<exists>prx. llb ! i = prx @ lla ! i \<and>
(\<forall>pt\<in>set prx. pt \<in> set_of rt \<and> var pt = i)"
apply -
apply (erule_tac x="i" in allE)
apply simp
done
with lla_ll inp rt show ?thesis
apply -
apply (erule exE)
apply (rule_tac x="prx" in exI)
apply simp
done
qed
next
assume iselp: "\<not> var (low p) < i"
show ?thesis
proof (cases "var (high p) < i")
case True
with llb_nc have llb_ll: "llb ! i = lla ! i"
by fastforce
with iselp lla_st
have prx_ll: "\<exists>prx. lla ! i = prx @ ll ! i \<and>
(\<forall>pt\<in>set prx. pt \<in> set_of lt \<and> var pt = i)"
apply -
apply (erule_tac x="i" in allE)
apply simp
done
with llb_ll inp lt show ?thesis
apply -
apply (erule exE)
apply (rule_tac x="prx" in exI)
apply simp
done
next
assume isehp: " \<not> var (high p) < i"
from iselp lla_st
have prxl: "\<exists>prx. lla ! i = prx @ ll ! i \<and>
(\<forall>pt\<in>set prx. pt \<in> set_of lt \<and> var pt = i)"
by fastforce
from isehp llb_st
have prxh: "\<exists>prx. llb ! i = prx @ lla ! i \<and>
(\<forall>pt\<in>set prx. pt \<in> set_of rt \<and> var pt = i)"
by fastforce
with prxl inp lt pnN rt show ?thesis
apply -
apply (elim exE)
apply (rule_tac x="prxa @ prx" in exI)
apply simp
apply (elim conjE)
apply fastforce
done
qed
qed
qed
qed
have big_Nodes_nc: "\<forall>i. (p->var) < i
\<longrightarrow> (llb[var p :=p # llb ! var p]) ! i = ll ! i"
apply -
apply (rule allI)
apply (rule impI)
proof -
fix i
assume psi: "var p < i"
with orderedt lt rt lt_Node rt_Node have lpsi: "var (low p) < i"
by fastforce
with lla_nc have lla_ll: "lla ! i = ll ! i"
by fastforce
from psi orderedt lt rt lt_Node rt_Node have hpsi: "var (high p) < i"
by fastforce
with llb_nc have llb_lla: "llb ! i = lla ! i"
by fastforce
from psi
have upllb_llb: "llb[var p :=p#llb!var p]!i = llb!i"
by fastforce
from upllb_llb llb_lla lla_ll
show "llb[var p :=p # llb ! var p] ! i = ll ! i"
by fastforce
qed
from lla_eq_ll llb_eq_lla
have length_eq: "length (llb[var p :=p # llb ! var p]) = length ll"
by fastforce
from length_eq big_Nodes_nc prx_ll_st nodes_in_upllb
have wf_ll_upllb:
"wf_levellist (Node lt p rt) ll (llb[var p :=p # llb ! var p]) var"
by (simp add: wf_levellist_def)
have mark_nc:
"\<forall> n. n \<notin> set_of (Node lt p rt) \<longrightarrow> (markb(p:=m)) n = mark n"
apply -
apply (rule allI)
apply (rule impI)
proof -
fix n
assume nnit: "n \<notin> set_of (Node lt p rt)"
with lt rt have nnilt: " n \<notin> set_of lt"
by fastforce
from nnit lt rt have nnirt: " n \<notin> set_of rt"
by fastforce
with nnilt mot_nc mort_nc have mb_eq_m: "markb n = mark n"
by fastforce
from nnit have "n\<noteq>p"
by fastforce
then have upmarkb_markb: "(markb(p :=m)) n = markb n"
by fastforce
with mb_eq_m show "(markb(p :=m)) n = mark n"
by fastforce
qed
have mark_c: "\<forall> n. n \<in> set_of (Node lt p rt) \<longrightarrow> (markb(p :=m)) n = m"
apply -
apply (intro allI)
apply (rule impI)
proof -
fix n
assume nint: " n \<in> set_of (Node lt p rt)"
show "(markb(p :=m)) n = m"
proof (cases "n=p")
case True
then show ?thesis
by fastforce
next
assume nnp: " n \<noteq> p"
show ?thesis
proof (cases "n \<in> set_of rt")
case True
with mirt_marked have "markb n = m"
by fastforce
with nnp show ?thesis
by fastforce
next
assume nninrt: " n \<notin> set_of rt"
with nint nnp have ninlt: "n \<in> set_of lt"
by fastforce
with mit_marked have marka_m: "marka n = m"
by fastforce
from mort_nc nninrt have "marka n = markb n"
by fastforce
with marka_m have "markb n = m"
by fastforce
with nnp show ?thesis
by fastforce
qed
qed
qed
from mark_c mark_nc
have wf_mark: "wf_marking (Node lt p rt) mark (markb(p :=m)) m"
by (simp add: wf_marking_def)
with wf_ll_upllb show ?thesis
by fastforce
qed
qed
qed
qed
done
qed
done
qed
next
fix var low high p lt rt and levellist and
ll::"ref list list" and mark::"ref \<Rightarrow> bool" and "next"
assume pnN: "p \<noteq> Null"
assume ll: "Levellist levellist next ll"
assume vpsll: "var p < length levellist"
assume orderedt: "ordered (Node lt p rt) var"
assume marked_child_ll: "\<forall>n\<in>set_of (Node lt p rt).
if mark n = mark p
then n \<in> set (ll ! var n) \<and>
(\<forall>nt pa. Dag n low high nt \<and> pa \<in> set_of nt \<longrightarrow> mark pa = mark p)
else n \<notin> set (concat ll)"
assume lt: "Dag (low p) low high lt"
assume rt: "Dag (high p) low high rt"
show "wf_levellist (Node lt p rt) ll ll var \<and>
wf_marking (Node lt p rt) mark mark (mark p)"
proof -
from marked_child_ll pnN lt rt have marked_st:
"(\<forall>pa. pa \<in> set_of (Node lt p rt) \<longrightarrow> mark pa = mark p)"
apply -
apply (drule_tac x="p" in bspec)
apply simp
apply (clarsimp)
apply (erule_tac x="(Node lt p rt)" in allE)
apply simp
done
have nodest_in_ll:
"\<forall>q. q \<in> set_of (Node lt p rt) \<longrightarrow> q \<in> set (ll ! var q)"
proof -
from marked_child_ll pnN have pinll: "p \<in> set (ll ! var p)"
apply -
apply (drule_tac x="p" in bspec)
apply simp
apply fastforce
done
from marked_st marked_child_ll lt rt show ?thesis
apply -
apply (rule allI)
apply (erule_tac x="q" in allE)
apply (rule impI)
apply (erule impE)
apply assumption
apply (drule_tac x="q" in bspec)
apply simp
apply fastforce
done
qed
have levellist_nc: "\<forall> i \<le> var p. (\<exists> prx. ll ! i = prx@(ll ! i) \<and>
(\<forall> pt \<in> set prx. pt \<in> set_of (Node lt p rt) \<and> var pt = i))"
apply -
apply (rule allI)
apply (rule impI)
apply (rule_tac x="[]" in exI)
apply fastforce
done
have ll_nc: "\<forall> i. (var p) < i \<longrightarrow> ll ! i = ll ! i"
by fastforce
have length_ll: "length ll = length ll"
by fastforce
with ll_nc levellist_nc nodest_in_ll
have wf: "wf_levellist (Node lt p rt) ll ll var"
by (simp add: wf_levellist_def)
have m_nc: "\<forall> n. n \<notin> set_of (Node lt p rt) \<longrightarrow> mark n = mark n"
by fastforce
from marked_st have "\<forall> n. n \<in> set_of (Node lt p rt) \<longrightarrow> mark n = mark p"
by fastforce
with m_nc have " wf_marking (Node lt p rt) mark mark (mark p)"
by (simp add: wf_marking_def)
with wf show ?thesis
by fastforce
qed
qed
lemma allD: "\<forall>ll. P ll \<Longrightarrow> P ll"
by blast
lemma replicate_spec: "\<lbrakk>\<forall>i < n. xs ! i = x; n=length xs\<rbrakk>
\<Longrightarrow> replicate (length xs) x = xs"
apply hypsubst_thin
apply (induct xs)
apply simp
apply force
done
lemma (in Levellist_impl) Levellist_spec_total:
shows "\<forall>\<sigma> t. \<Gamma>,\<Theta>\<turnstile>\<^sub>t
\<lbrace>\<sigma>. Dag \<acute>p \<acute>low \<acute>high t \<and> (\<forall>i < length \<acute>levellist. \<acute>levellist ! i = Null) \<and>
length \<acute>levellist = \<acute>p \<rightarrow> \<acute>var + 1 \<and>
ordered t \<acute>var \<and> (\<forall>n \<in> set_of t. \<acute>mark n = (\<not> \<acute>m) )\<rbrace>
\<acute>levellist :== PROC Levellist (\<acute>p, \<acute>m, \<acute>levellist)
\<lbrace>\<exists>ll. Levellist \<acute>levellist \<acute>next ll \<and> wf_ll t ll \<^bsup>\<sigma>\<^esup>var \<and>
length \<acute>levellist = \<^bsup>\<sigma>\<^esup>p \<rightarrow> \<^bsup>\<sigma>\<^esup>var + 1 \<and>
wf_marking t \<^bsup>\<sigma>\<^esup>mark \<acute>mark \<^bsup>\<sigma>\<^esup>m \<and>
(\<forall>p. p \<notin> set_of t \<longrightarrow> \<^bsup>\<sigma>\<^esup>next p = \<acute>next p)\<rbrace>"
apply (hoare_rule HoareTotal.conseq)
apply (rule_tac ll="replicate (\<^bsup>\<sigma>\<^esup>p\<rightarrow>\<^bsup>\<sigma>\<^esup>var + 1) []" in allD [OF Levellist_spec_total'])
apply (intro allI impI)
apply (rule_tac x=\<sigma> in exI)
apply (rule_tac x=t in exI)
apply (rule conjI)
apply (clarsimp split:if_split_asm simp del: concat_replicate_trivial)
apply (frule replicate_spec [symmetric])
apply (simp)
apply (clarsimp simp add: Levellist_def )
apply (case_tac i)
apply simp
apply simp
apply (simp add: Collect_conv_if split:if_split_asm)
apply vcg_step
apply (elim exE conjE)
apply (rule_tac x=ll' in exI)
apply simp
apply (thin_tac "\<forall>p. p \<notin> set_of t \<longrightarrow> next p = nexta p")
apply (simp add: wf_levellist_def wf_ll_def)
apply (case_tac "t = Tip")
apply simp
apply (rule conjI)
apply clarsimp
apply (case_tac k)
apply simp
apply simp
apply (subgoal_tac "length ll'=Suc (var Null)")
apply (simp add: Levellist_length)
apply fastforce
apply (split dag.splits)
apply simp
apply (elim conjE)
apply (intro conjI)
apply (rule allI)
apply (erule_tac x="pa" in allE)
apply clarify
prefer 2
apply (simp add: Levellist_length)
apply (rule allI)
apply (rule impI)
apply (rule ballI)
apply (rotate_tac 11)
apply (erule_tac x="k" in allE)
apply (rename_tac dag1 ref dag2 k pa)
apply (subgoal_tac "k <= var ref")
prefer 2
apply (subgoal_tac "ref = p")
apply simp
apply clarify
apply (erule_tac ?P = "Dag p low high (Node dag1 ref dag2)" in rev_mp)
apply (simp (no_asm))
apply (rotate_tac 14)
apply (erule_tac x=k in allE)
apply clarify
apply (erule_tac x=k in allE)
apply clarify
apply (case_tac k)
apply simp
apply simp
done
end
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