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| (* Title: BDD | |
| Author: Veronika Ortner and Norbert Schirmer, 2004 | |
| Maintainer: Norbert Schirmer, norbert.schirmer at web de | |
| License: LGPL | |
| *) | |
| (* | |
| RepointProof.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 Repoint\<close> | |
| theory RepointProof imports ProcedureSpecs begin | |
| hide_const (open) DistinctTreeProver.set_of tree.Node tree.Tip | |
| lemma (in Repoint_impl) Repoint_modifies: | |
| shows "\<forall>\<sigma>. \<Gamma>\<turnstile>{\<sigma>} \<acute>p :== PROC Repoint (\<acute>p) | |
| {t. t may_only_modify_globals \<sigma> in [low,high]}" | |
| apply (hoare_rule HoarePartial.ProcRec1) | |
| apply (vcg spec=modifies) | |
| done | |
| lemma low_high_exchange_dag: | |
| assumes pt_same: "\<forall>pt. pt \<notin> set_of lt \<longrightarrow> low pt = lowa pt \<and> high pt = higha pt" | |
| assumes pt_changed: "\<forall>pt \<in> set_of lt. lowa pt = (rep \<propto> low) pt \<and> | |
| higha pt = (rep \<propto> high) pt" | |
| assumes rep_pt: "\<forall>pt \<in> set_of rt. rep pt = pt" | |
| shows "\<And>q. Dag q (rep \<propto> low) (rep \<propto> high) rt \<Longrightarrow> | |
| Dag q (rep \<propto> lowa) (rep \<propto> higha) rt" | |
| using rep_pt | |
| proof (induct rt) | |
| case Tip thus ?case by simp | |
| next | |
| case (Node lrt q' rrt) | |
| have "Dag q (rep \<propto> low) (rep \<propto> high) (Node lrt q' rrt)" by fact | |
| then obtain | |
| q': "q = q'" and | |
| q_notNull: "q \<noteq> Null" and | |
| lrt: "Dag ((rep \<propto> low) q) (rep \<propto> low) (rep \<propto> high) lrt" and | |
| rrt: "Dag ((rep \<propto> high) q) (rep \<propto> low) (rep \<propto> high) rrt" | |
| by auto | |
| have rlowa_rlow: "((rep \<propto> lowa) q) = ((rep \<propto> low) q)" | |
| proof (cases "q \<in> set_of lt") | |
| case True | |
| note q_in_lt=this | |
| with pt_changed have lowa_q: "lowa q = (rep \<propto> low) q" | |
| by simp | |
| thus "(rep \<propto> lowa) q = (rep \<propto> low) q" | |
| proof (cases "low q = Null") | |
| case True | |
| with lowa_q have "lowa q = Null" | |
| by (simp add: null_comp_def) | |
| with True show ?thesis | |
| by (simp add: null_comp_def) | |
| next | |
| assume lq_nNull: "low q \<noteq> Null" | |
| show ?thesis | |
| proof (cases "(rep \<propto> low) q = Null") | |
| case True | |
| with lowa_q have "lowa q = Null" | |
| by simp | |
| with True show ?thesis | |
| by (simp add: null_comp_def) | |
| next | |
| assume rlq_nNull: "(rep \<propto> low) q \<noteq> Null" | |
| with lrt lowa_q have "lowa q \<in> set_of lrt" | |
| by auto | |
| with Node.prems Node have "lowa q \<in> set_of (Node lrt q' rrt)" | |
| by simp | |
| with Node.prems have "rep (lowa q) = lowa q" | |
| by auto | |
| with lowa_q rlq_nNull show ?thesis | |
| by (simp add: null_comp_def) | |
| qed | |
| qed | |
| next | |
| assume q_notin_lt: " q \<notin> set_of lt" | |
| with pt_same have "low q = lowa q" | |
| by auto | |
| thus ?thesis | |
| by (simp add: null_comp_def) | |
| qed | |
| have rhigha_rhigh: "((rep \<propto> higha) q) = ((rep \<propto> high) q)" | |
| proof (cases "q \<in> set_of lt") | |
| case True | |
| note q_in_lt=this | |
| with pt_changed have higha_q: "higha q = (rep \<propto> high) q" | |
| by simp | |
| thus ?thesis | |
| proof (cases "high q = Null") | |
| case True | |
| with higha_q have "higha q = Null" | |
| by (simp add: null_comp_def) | |
| with True show ?thesis | |
| by (simp add: null_comp_def) | |
| next | |
| assume hq_nNull: "high q \<noteq> Null" | |
| show ?thesis | |
| proof (cases "(rep \<propto> high) q = Null") | |
| case True | |
| with higha_q have "higha q = Null" | |
| by simp | |
| with True show ?thesis | |
| by (simp add: null_comp_def) | |
| next | |
| assume rhq_nNull: "(rep \<propto> high) q \<noteq> Null" | |
| with rrt higha_q have "higha q \<in> set_of rrt" | |
| by auto | |
| with Node.prems Node have "higha q \<in> set_of (Node lrt q' rrt)" | |
| by simp | |
| with Node.prems have "rep (higha q) = higha q" | |
| by auto | |
| with higha_q rhq_nNull show ?thesis | |
| by (simp add: null_comp_def) | |
| qed | |
| qed | |
| next | |
| assume q_notin_lt: " q \<notin> set_of lt" | |
| with pt_same have "high q = higha q" | |
| by auto | |
| thus ?thesis | |
| by (simp add: null_comp_def) | |
| qed | |
| with rrt have rhigha_mixed_dag: | |
| "Dag ((rep \<propto> higha) q) (rep \<propto> low) (rep \<propto> high) rrt" | |
| by simp | |
| from lrt rlowa_rlow have rlowa_mixed_dag: | |
| " Dag ((rep \<propto> lowa) q) (rep \<propto> low) (rep \<propto> high) lrt" | |
| by simp | |
| from Node.prems have lrt_rep_eq: " \<forall>pt\<in>set_of lrt. rep pt = pt" | |
| by simp | |
| from Node.prems have rrt_rep_eq: "\<forall>pt\<in>set_of rrt. rep pt = pt" | |
| by simp | |
| from rlowa_mixed_dag lrt_rep_eq have lowa_lrt: | |
| " Dag ((rep \<propto> lowa) q) (rep \<propto> lowa) (rep \<propto> higha) lrt" | |
| apply - | |
| apply (rule Node.hyps) | |
| apply auto | |
| done | |
| from rhigha_mixed_dag rrt_rep_eq have higha_rrt: | |
| " Dag ((rep \<propto> higha) q) (rep \<propto> lowa) (rep \<propto> higha) rrt" | |
| apply - | |
| apply (rule Node.hyps) | |
| apply auto | |
| done | |
| with lowa_lrt q' q_notNull | |
| show " Dag q (rep \<propto> lowa) (rep \<propto> higha) (Node lrt q' rrt)" | |
| by simp | |
| qed | |
| (*lemma Repoint_spec : | |
| includes Repoint_impl | |
| shows | |
| "\<forall>\<sigma> rept. \<Gamma>\<turnstile> \<lbrace>\<sigma>. (Dag ((\<^bsup>\<sigma>\<^esup>rep \<propto> id) \<^bsup>\<sigma>\<^esup>p) (\<^bsup>\<sigma>\<^esup>rep \<propto> \<^bsup>\<sigma>\<^esup>low) (\<^bsup>\<sigma>\<^esup>rep \<propto> \<^bsup>\<sigma>\<^esup>high) rept) | |
| \<and> (\<forall> no \<in> set_of rept. \<^bsup>\<sigma>\<^esup>rep no = no) \<rbrace> | |
| \<acute>p :== CALL Repoint (\<acute>p) | |
| \<lbrace>Dag \<acute>p \<acute>low \<acute>high rept \<and> | |
| (\<forall>pt. pt \<notin> set_of rept \<longrightarrow> \<^bsup>\<sigma>\<^esup>low pt = \<acute>low pt \<and> \<^bsup>\<sigma>\<^esup>high pt = \<acute>high pt)\<rbrace>" | |
| apply (hoare_rule CallRec1_SamePost) | |
| apply (vcg) | |
| apply (rule conjI) | |
| apply clarify | |
| prefer 2 | |
| apply (intro impI allI ) | |
| apply (simp add: null_comp_def) | |
| apply (rule conjI) | |
| prefer 2 | |
| apply (clarsimp) | |
| apply clarify | |
| *) | |
| lemma (in Repoint_impl) Repoint_spec': | |
| shows | |
| "\<forall>\<sigma>. \<Gamma>\<turnstile> {\<sigma>} | |
| \<acute>p :== PROC Repoint (\<acute>p) | |
| \<lbrace>\<forall> rept. ((Dag ((\<^bsup>\<sigma>\<^esup>rep \<propto> id) \<^bsup>\<sigma>\<^esup>p) (\<^bsup>\<sigma>\<^esup>rep \<propto> \<^bsup>\<sigma>\<^esup>low) (\<^bsup>\<sigma>\<^esup>rep \<propto> \<^bsup>\<sigma>\<^esup>high) rept) | |
| \<and> (\<forall> no \<in> set_of rept. \<^bsup>\<sigma>\<^esup>rep no = no)) | |
| \<longrightarrow> Dag \<acute>p \<acute>low \<acute>high rept \<and> | |
| (\<forall>pt. pt \<notin> set_of rept \<longrightarrow> \<^bsup>\<sigma>\<^esup>low pt = \<acute>low pt \<and> \<^bsup>\<sigma>\<^esup>high pt = \<acute>high pt)\<rbrace>" | |
| apply (hoare_rule HoarePartial.ProcRec1) | |
| apply vcg | |
| apply (rule conjI) | |
| apply clarify | |
| prefer 2 | |
| apply (intro impI allI ) | |
| apply (simp add: null_comp_def) | |
| apply (rule conjI) | |
| prefer 2 | |
| apply (clarsimp) | |
| apply clarify | |
| proof - | |
| fix low high p rep lowa higha pa lowb highb pb rept | |
| assume p_nNull: "p \<noteq> Null" | |
| assume rp_nNull: " rep p \<noteq> Null" | |
| assume rec_spec_lrept: | |
| "\<forall>rept. Dag ((rep \<propto> id) (low (rep p))) (rep \<propto> low) (rep \<propto> high) rept | |
| \<and> (\<forall>no\<in>set_of rept. rep no = no) | |
| \<longrightarrow> Dag pa lowa higha rept \<and> | |
| (\<forall>pt. pt \<notin> set_of rept \<longrightarrow> low pt = lowa pt \<and> high pt = higha pt)" | |
| assume rec_spec_rrept: | |
| "\<forall>rept. Dag ((rep \<propto> id) (higha (rep p))) (rep \<propto> lowa(rep p := pa)) (rep \<propto> higha) rept | |
| \<and> (\<forall>no\<in>set_of rept. rep no = no) | |
| \<longrightarrow> Dag pb lowb highb rept \<and> | |
| (\<forall>pt. pt \<notin> set_of rept \<longrightarrow> (lowa(rep p := pa)) pt = lowb pt \<and> higha pt = highb pt)" | |
| assume rept_dag: "Dag ((rep \<propto> id) p) (rep \<propto> low) (rep \<propto> high) rept" | |
| assume rno_rept: "\<forall>no\<in>set_of rept. rep no = no" | |
| show " Dag (rep p) lowb (highb(rep p := pb)) rept \<and> | |
| (\<forall>pt. pt \<notin> set_of rept \<longrightarrow> low pt = lowb pt \<and> high pt = (highb(rep p := pb)) pt)" | |
| proof - | |
| from rp_nNull rept_dag p_nNull obtain lrept rrept where | |
| rept_def: "rept = Node lrept (rep p) rrept" | |
| by auto | |
| with rept_dag p_nNull have lrept_dag: | |
| "Dag ((rep \<propto> low) (rep p)) (rep \<propto> low) (rep \<propto> high) lrept" | |
| by simp | |
| from rept_def rept_dag p_nNull have rrept_dag: | |
| "Dag ((rep \<propto> high) (rep p)) (rep \<propto> low) (rep \<propto> high) rrept" | |
| by simp | |
| from rno_rept rept_def have rno_lrept: "\<forall> no \<in> set_of lrept. rep no = no" | |
| by auto | |
| from rno_rept rept_def have rno_rrept: "\<forall> no \<in> set_of rrept. rep no = no" | |
| by auto | |
| have repoint_post_low: | |
| " Dag pa lowa higha lrept \<and> | |
| (\<forall>pt. pt \<notin> set_of lrept \<longrightarrow> low pt = lowa pt \<and> high pt = higha pt)" | |
| proof - | |
| from lrept_dag have " Dag ((rep \<propto> id) (low (rep p))) (rep \<propto> low) (rep \<propto> high) lrept" | |
| by (simp add: id_trans) | |
| with rec_spec_lrept rno_lrept show ?thesis | |
| apply - | |
| apply (erule_tac x=lrept in allE) | |
| apply (erule impE) | |
| apply simp | |
| apply assumption | |
| done | |
| qed | |
| hence low_lowa_nc: "(\<forall>pt. pt \<notin> set_of lrept \<longrightarrow> low pt = lowa pt \<and> high pt = higha pt)" | |
| by simp | |
| from lrept_dag repoint_post_low obtain | |
| pa_def: "pa = (rep \<propto> low) (rep p)" and | |
| lowa_higha_def: "(\<forall> no \<in> set_of lrept. lowa no = (rep \<propto> low) no \<and> higha no = (rep \<propto> high) no)" | |
| apply - | |
| apply (drule Dags_eq_hp_eq) | |
| apply auto | |
| done | |
| from rept_dag have rept_DAG: "DAG rept" | |
| by (rule Dag_is_DAG) | |
| with rept_def have rp_notin_lrept: "rep p \<notin> set_of lrept" | |
| by simp | |
| from rept_DAG rept_def have rp_notin_rrept: "rep p \<notin> set_of rrept" | |
| by simp | |
| have "Dag ((rep \<propto> id) (higha (rep p))) (rep \<propto> lowa(rep p := pa)) (rep \<propto> higha) rrept" | |
| proof - | |
| from low_lowa_nc rp_notin_lrept have "(rep \<propto> high) (rep p) = (rep \<propto> higha) (rep p)" | |
| by (auto simp add: null_comp_def) | |
| with rrept_dag have higha_mixed_rrept: | |
| "Dag ((rep \<propto> id) (higha (rep p))) (rep \<propto> low) (rep \<propto> high) rrept" | |
| by (simp add: id_trans) | |
| thm low_high_exchange_dag | |
| with low_lowa_nc lowa_higha_def rno_rrept have lowa_higha_rrept: | |
| "Dag ((rep \<propto> id) (higha (rep p))) (rep \<propto> lowa) (rep \<propto> higha) rrept" | |
| apply - | |
| apply (rule low_high_exchange_dag) | |
| apply auto | |
| done | |
| have "Dag ((rep \<propto> id) (higha (rep p))) (rep \<propto> lowa) (rep \<propto> higha) rrept = | |
| Dag ((rep \<propto> id) (higha (rep p))) (rep \<propto> lowa(rep p := pa)) (rep \<propto> higha) rrept" | |
| proof - | |
| have "\<forall> no \<in> set_of rrept. (rep \<propto> lowa) no = (rep \<propto> lowa(rep p := pa)) no \<and> | |
| (rep \<propto> higha) no = (rep \<propto> higha) no" | |
| proof | |
| fix no | |
| assume no_in_rrept: "no \<in> set_of rrept" | |
| with rp_notin_rrept have "no \<noteq> rep p" | |
| by blast | |
| thus "(rep \<propto> lowa) no = (rep \<propto> lowa(rep p := pa)) no \<and> | |
| (rep \<propto> higha) no = (rep \<propto> higha) no" | |
| by (simp add: null_comp_def) | |
| qed | |
| thus ?thesis | |
| by (rule heaps_eq_Dag_eq) | |
| qed | |
| with lowa_higha_rrept show ?thesis | |
| by simp | |
| qed | |
| with rec_spec_rrept rno_rrept have repoint_rrept: "Dag pb lowb highb rrept \<and> | |
| (\<forall>pt. pt \<notin> set_of rrept \<longrightarrow> | |
| (lowa(rep p := pa)) pt = lowb pt \<and> higha pt = highb pt)" | |
| apply - | |
| apply (erule_tac x=rrept in allE) | |
| apply (erule impE) | |
| apply simp | |
| apply assumption | |
| done | |
| then have ab_nc: "(\<forall>pt. pt \<notin> set_of rrept \<longrightarrow> | |
| (lowa(rep p := pa)) pt = lowb pt \<and> higha pt = highb pt)" | |
| by simp | |
| from repoint_rrept rrept_dag obtain | |
| pb_def: "pb = ((rep \<propto> high) (rep p))" and | |
| lowb_highb_def: "(\<forall> no \<in> set_of rrept. lowb no = (rep \<propto> low) no \<and> highb no = (rep \<propto> high) no)" | |
| apply - | |
| apply (drule Dags_eq_hp_eq) | |
| apply auto | |
| done | |
| have rept_end_dag: " Dag (rep p) lowb (highb(rep p := pb)) rept" | |
| proof - | |
| have "\<forall> no \<in> set_of rept. lowb no = (rep \<propto> low) no \<and> (highb(rep p := pb)) no = (rep \<propto> high) no" | |
| proof | |
| fix no | |
| assume no_in_rept: " no \<in> set_of rept" | |
| show "lowb no = (rep \<propto> low) no \<and> (highb(rep p := pb)) no = (rep \<propto> high) no" | |
| proof (cases "no \<in> set_of rrept") | |
| case True | |
| with lowb_highb_def pb_def show ?thesis | |
| by simp | |
| next | |
| assume no_notin_rrept: " no \<notin> set_of rrept" | |
| show ?thesis | |
| proof (cases "no \<in> set_of lrept") | |
| case True | |
| with no_notin_rrept rp_notin_lrept ab_nc | |
| have ab_nc_no: "lowa no = lowb no \<and> higha no = highb no" | |
| apply - | |
| apply (erule_tac x=no in allE) | |
| apply (erule impE) | |
| apply simp | |
| apply (subgoal_tac "no \<noteq> rep p") | |
| apply simp | |
| apply blast | |
| done | |
| from lowa_higha_def True have | |
| "lowa no = (rep \<propto> low) no \<and> higha no = (rep \<propto> high) no" | |
| by auto | |
| with ab_nc_no have "lowb no = (rep \<propto> low) no \<and> highb no =(rep \<propto> high) no" | |
| by simp | |
| with rp_notin_lrept True show ?thesis | |
| apply (subgoal_tac "no \<noteq> rep p") | |
| apply simp | |
| apply blast | |
| done | |
| next | |
| assume no_notin_lrept: " no \<notin> set_of lrept" | |
| with no_in_rept rept_def no_notin_rrept have no_rp: "no = rep p" | |
| by simp | |
| with rp_notin_lrept low_lowa_nc have a_nc: | |
| "low no = lowa no \<and> high no = higha no" | |
| by auto | |
| from rp_notin_rrept no_rp ab_nc have | |
| "(lowa(rep p := pa)) no = lowb no \<and> higha no = highb no" | |
| by auto | |
| with a_nc pa_def no_rp have "(rep \<propto> low) no = lowb no \<and> high no = highb no" | |
| by auto | |
| with pb_def no_rp show ?thesis | |
| by simp | |
| qed | |
| qed | |
| qed | |
| with rept_dag have " Dag (rep p) lowb (highb(rep p := pb)) rept = | |
| Dag (rep p) (rep \<propto> low) (rep \<propto> high) rept" | |
| apply - | |
| thm heaps_eq_Dag_eq | |
| apply (rule heaps_eq_Dag_eq) | |
| apply auto | |
| done | |
| with rept_dag p_nNull show ?thesis | |
| by simp | |
| qed | |
| have "(\<forall>pt. pt \<notin> set_of rept \<longrightarrow> low pt = lowb pt \<and> high pt = (highb(rep p := pb)) pt)" | |
| proof (intro allI impI) | |
| fix pt | |
| assume pt_notin_rept: "pt \<notin> set_of rept" | |
| with rept_def obtain | |
| pt_notin_lrept: "pt \<notin> set_of lrept" and | |
| pt_notin_rrept: "pt \<notin> set_of rrept" and | |
| pt_neq_rp: "pt \<noteq> rep p" | |
| by simp | |
| with low_lowa_nc ab_nc show "low pt = lowb pt \<and> high pt = (highb(rep p := pb)) pt" | |
| by auto | |
| qed | |
| with rept_end_dag show ?thesis | |
| by simp | |
| qed | |
| qed | |
| lemma (in Repoint_impl) Repoint_spec: | |
| shows | |
| "\<forall>\<sigma> rept. \<Gamma>\<turnstile> \<lbrace>\<sigma>. Dag ((\<acute>rep \<propto> id) \<acute>p) (\<acute>rep \<propto> \<acute>low) (\<acute>rep \<propto> \<acute>high) rept | |
| \<and> (\<forall> no \<in> set_of rept. \<acute>rep no = no) \<rbrace> | |
| \<acute>p :== PROC Repoint (\<acute>p) | |
| \<lbrace>Dag \<acute>p \<acute>low \<acute>high rept \<and> | |
| (\<forall>pt. pt \<notin> set_of rept \<longrightarrow> \<^bsup>\<sigma>\<^esup>low pt = \<acute>low pt \<and> \<^bsup>\<sigma>\<^esup>high pt = \<acute>high pt)\<rbrace>" | |
| apply (hoare_rule HoarePartial.ProcRec1) | |
| apply vcg | |
| apply (rule conjI) | |
| prefer 2 | |
| apply (clarsimp simp add: null_comp_def) | |
| apply clarify | |
| apply (rule conjI) | |
| prefer 2 | |
| apply clarsimp | |
| apply clarify | |
| proof - | |
| fix rept low high rep p | |
| assume rept_dag: "Dag ((rep \<propto> id) p) (rep \<propto> low) (rep \<propto> high) rept" | |
| assume rno_rept: "\<forall>no\<in>set_of rept. rep no = no" | |
| assume p_nNull: "p \<noteq> Null" | |
| assume rp_nNull: " rep p \<noteq> Null" | |
| show "\<exists>lrept. | |
| Dag ((rep \<propto> id) (low (rep p))) (rep \<propto> low) (rep \<propto> high) lrept \<and> | |
| (\<forall>no\<in>set_of lrept. rep no = no) \<and> | |
| (\<forall>lowa higha pa. | |
| Dag pa lowa higha lrept \<and> | |
| (\<forall>pt. pt \<notin> set_of lrept \<longrightarrow> | |
| low pt = lowa pt \<and> high pt = higha pt) \<longrightarrow> | |
| (\<exists>rrept. | |
| Dag ((rep \<propto> id) (higha (rep p))) (rep \<propto> lowa(rep p := pa)) | |
| (rep \<propto> higha) rrept \<and> | |
| (\<forall>no\<in>set_of rrept. rep no = no) \<and> | |
| (\<forall>lowb highb pb. | |
| Dag pb lowb highb rrept \<and> | |
| (\<forall>pt. pt \<notin> set_of rrept \<longrightarrow> | |
| (lowa(rep p := pa)) pt = lowb pt \<and> | |
| higha pt = highb pt) \<longrightarrow> | |
| Dag (rep p) lowb (highb(rep p := pb)) rept \<and> | |
| (\<forall>pt. pt \<notin> set_of rept \<longrightarrow> | |
| low pt = lowb pt \<and> | |
| high pt = (highb(rep p := pb)) pt))))" | |
| proof - | |
| from rp_nNull rept_dag p_nNull obtain lrept rrept where | |
| rept_def: "rept = Node lrept (rep p) rrept" | |
| by auto | |
| with rept_dag p_nNull have lrept_dag: | |
| "Dag ((rep \<propto> low) (rep p)) (rep \<propto> low) (rep \<propto> high) lrept" | |
| by simp | |
| from rept_def rept_dag p_nNull have rrept_dag: | |
| "Dag ((rep \<propto> high) (rep p)) (rep \<propto> low) (rep \<propto> high) rrept" | |
| by simp | |
| from rno_rept rept_def have rno_lrept: "\<forall> no \<in> set_of lrept. rep no = no" | |
| by auto | |
| from rno_rept rept_def have rno_rrept: "\<forall> no \<in> set_of rrept. rep no = no" | |
| by auto | |
| show ?thesis | |
| apply (rule_tac x=lrept in exI) | |
| apply (rule conjI) | |
| apply (simp add: id_trans lrept_dag) | |
| apply (rule conjI) | |
| apply (rule rno_lrept) | |
| apply clarify | |
| subgoal premises prems for lowa higha pa | |
| proof - | |
| have lrepta: "Dag pa lowa higha lrept" by fact | |
| have low_lowa_nc: | |
| "\<forall>pt. pt \<notin> set_of lrept \<longrightarrow> low pt = lowa pt \<and> high pt = higha pt" by fact | |
| from lrept_dag lrepta obtain | |
| pa_def: "pa = (rep \<propto> low) (rep p)" and | |
| lowa_higha_def: "\<forall>no \<in> set_of lrept. | |
| lowa no = (rep \<propto> low) no \<and> higha no = (rep \<propto> high) no" | |
| apply - | |
| apply (drule Dags_eq_hp_eq) | |
| apply auto | |
| done | |
| from rept_dag have rept_DAG: "DAG rept" | |
| by (rule Dag_is_DAG) | |
| with rept_def have rp_notin_lrept: "rep p \<notin> set_of lrept" | |
| by simp | |
| from rept_DAG rept_def have rp_notin_rrept: "rep p \<notin> set_of rrept" | |
| by simp | |
| have rrepta: "Dag ((rep \<propto> id) (higha (rep p))) | |
| (rep \<propto> lowa(rep p := pa)) (rep \<propto> higha) rrept" | |
| proof - | |
| from low_lowa_nc rp_notin_lrept | |
| have "(rep \<propto> high) (rep p) = (rep \<propto> higha) (rep p)" | |
| by (auto simp add: null_comp_def) | |
| with rrept_dag have higha_mixed_rrept: | |
| "Dag ((rep \<propto> id) (higha (rep p))) (rep \<propto> low) (rep \<propto> high) rrept" | |
| by (simp add: id_trans) | |
| thm low_high_exchange_dag | |
| with low_lowa_nc lowa_higha_def rno_rrept | |
| have lowa_higha_rrept: | |
| "Dag ((rep \<propto> id) (higha (rep p))) (rep \<propto> lowa) (rep \<propto> higha) rrept" | |
| apply - | |
| apply (rule low_high_exchange_dag) | |
| apply auto | |
| done | |
| have "Dag ((rep \<propto> id) (higha (rep p))) (rep \<propto> lowa) (rep \<propto> higha) rrept = | |
| Dag ((rep \<propto> id) (higha (rep p))) | |
| (rep \<propto> lowa(rep p := pa)) (rep \<propto> higha) rrept" | |
| proof - | |
| have "\<forall>no \<in> set_of rrept. | |
| (rep \<propto> lowa) no = (rep \<propto> lowa(rep p := pa)) no \<and> | |
| (rep \<propto> higha) no = (rep \<propto> higha) no" | |
| proof | |
| fix no | |
| assume no_in_rrept: "no \<in> set_of rrept" | |
| with rp_notin_rrept have "no \<noteq> rep p" | |
| by blast | |
| thus "(rep \<propto> lowa) no = (rep \<propto> lowa(rep p := pa)) no \<and> | |
| (rep \<propto> higha) no = (rep \<propto> higha) no" | |
| by (simp add: null_comp_def) | |
| qed | |
| thus ?thesis | |
| by (rule heaps_eq_Dag_eq) | |
| qed | |
| with lowa_higha_rrept show ?thesis | |
| by simp | |
| qed | |
| show ?thesis | |
| apply (rule_tac x=rrept in exI) | |
| apply (rule conjI) | |
| apply (rule rrepta) | |
| apply (rule conjI) | |
| apply (rule rno_rrept) | |
| apply clarify | |
| subgoal premises prems for lowb highb pb | |
| proof - | |
| have rreptb: "Dag pb lowb highb rrept" by fact | |
| have ab_nc: "\<forall>pt. pt \<notin> set_of rrept \<longrightarrow> | |
| (lowa(rep p := pa)) pt = lowb pt \<and> higha pt = highb pt" by fact | |
| from rreptb rrept_dag obtain | |
| pb_def: "pb = ((rep \<propto> high) (rep p))" and | |
| lowb_highb_def: "\<forall>no \<in> set_of rrept. | |
| lowb no = (rep \<propto> low) no \<and> highb no = (rep \<propto> high) no" | |
| apply - | |
| apply (drule Dags_eq_hp_eq) | |
| apply auto | |
| done | |
| have rept_end_dag: " Dag (rep p) lowb (highb(rep p := pb)) rept" | |
| proof - | |
| have "\<forall>no \<in> set_of rept. | |
| lowb no = (rep \<propto> low) no \<and> (highb(rep p := pb)) no = (rep \<propto> high) no" | |
| proof | |
| fix no | |
| assume no_in_rept: " no \<in> set_of rept" | |
| show "lowb no = (rep \<propto> low) no \<and> | |
| (highb(rep p := pb)) no = (rep \<propto> high) no" | |
| proof (cases "no \<in> set_of rrept") | |
| case True | |
| with lowb_highb_def pb_def show ?thesis | |
| by simp | |
| next | |
| assume no_notin_rrept: " no \<notin> set_of rrept" | |
| show ?thesis | |
| proof (cases "no \<in> set_of lrept") | |
| case True | |
| with no_notin_rrept rp_notin_lrept ab_nc | |
| have ab_nc_no: "lowa no = lowb no \<and> higha no = highb no" | |
| apply - | |
| apply (erule_tac x=no in allE) | |
| apply (erule impE) | |
| apply simp | |
| apply (subgoal_tac "no \<noteq> rep p") | |
| apply simp | |
| apply blast | |
| done | |
| from lowa_higha_def True have | |
| "lowa no = (rep \<propto> low) no \<and> higha no = (rep \<propto> high) no" | |
| by auto | |
| with ab_nc_no | |
| have "lowb no = (rep \<propto> low) no \<and> highb no =(rep \<propto> high) no" | |
| by simp | |
| with rp_notin_lrept True show ?thesis | |
| apply (subgoal_tac "no \<noteq> rep p") | |
| apply simp | |
| apply blast | |
| done | |
| next | |
| assume no_notin_lrept: " no \<notin> set_of lrept" | |
| with no_in_rept rept_def no_notin_rrept have no_rp: "no = rep p" | |
| by simp | |
| with rp_notin_lrept low_lowa_nc | |
| have a_nc: "low no = lowa no \<and> high no = higha no" | |
| by auto | |
| from rp_notin_rrept no_rp ab_nc | |
| have "(lowa(rep p := pa)) no = lowb no \<and> higha no = highb no" | |
| by auto | |
| with a_nc pa_def no_rp | |
| have "(rep \<propto> low) no = lowb no \<and> high no = highb no" | |
| by auto | |
| with pb_def no_rp show ?thesis | |
| by simp | |
| qed | |
| qed | |
| qed | |
| with rept_dag | |
| have "Dag (rep p) lowb (highb(rep p := pb)) rept = | |
| Dag (rep p) (rep \<propto> low) (rep \<propto> high) rept" | |
| apply - | |
| apply (rule heaps_eq_Dag_eq) | |
| apply auto | |
| done | |
| with rept_dag p_nNull show ?thesis | |
| by simp | |
| qed | |
| have "(\<forall>pt. pt \<notin> set_of rept \<longrightarrow> low pt = lowb pt \<and> | |
| high pt = (highb(rep p := pb)) pt)" | |
| proof (intro allI impI) | |
| fix pt | |
| assume pt_notin_rept: "pt \<notin> set_of rept" | |
| with rept_def obtain | |
| pt_notin_lrept: "pt \<notin> set_of lrept" and | |
| pt_notin_rrept: "pt \<notin> set_of rrept" and | |
| pt_neq_rp: "pt \<noteq> rep p" | |
| by simp | |
| with low_lowa_nc ab_nc | |
| show "low pt = lowb pt \<and> high pt = (highb(rep p := pb)) pt" | |
| by auto | |
| qed | |
| with rept_end_dag show ?thesis | |
| by simp | |
| qed | |
| done | |
| qed | |
| done | |
| qed | |
| qed | |
| lemma (in Repoint_impl) Repoint_spec_total: | |
| shows | |
| "\<forall>\<sigma> rept. \<Gamma>\<turnstile>\<^sub>t \<lbrace>\<sigma>. Dag ((\<acute>rep \<propto> id) \<acute>p) (\<acute>rep \<propto> \<acute>low) (\<acute>rep \<propto> \<acute>high) rept | |
| \<and> (\<forall> no \<in> set_of rept. \<acute>rep no = no) \<rbrace> | |
| \<acute>p :== PROC Repoint (\<acute>p) | |
| \<lbrace>Dag \<acute>p \<acute>low \<acute>high rept \<and> | |
| (\<forall>pt. pt \<notin> set_of rept \<longrightarrow> \<^bsup>\<sigma>\<^esup>low pt = \<acute>low pt \<and> \<^bsup>\<sigma>\<^esup>high pt = \<acute>high pt)\<rbrace>" | |
| apply (hoare_rule HoareTotal.ProcRec1 | |
| [where r="measure (\<lambda>(s,p). size | |
| (dag ((\<^bsup>s\<^esup>rep \<propto> id) \<^bsup>s\<^esup>p) (\<^bsup>s\<^esup>rep \<propto> \<^bsup>s\<^esup>low) (\<^bsup>s\<^esup>rep \<propto> \<^bsup>s\<^esup>high)))"]) | |
| apply vcg | |
| apply (rule conjI) | |
| prefer 2 | |
| apply (clarsimp simp add: null_comp_def) | |
| apply clarify | |
| apply (rule conjI) | |
| prefer 2 | |
| apply clarsimp | |
| apply clarify | |
| proof - | |
| fix rept low high rep p | |
| assume rept_dag: "Dag ((rep \<propto> id) p) (rep \<propto> low) (rep \<propto> high) rept" | |
| assume rno_rept: "\<forall>no\<in>set_of rept. rep no = no" | |
| assume p_nNull: "p \<noteq> Null" | |
| assume rp_nNull: " rep p \<noteq> Null" | |
| show "\<exists>lrept. | |
| Dag ((rep \<propto> id) (low (rep p))) (rep \<propto> low) (rep \<propto> high) lrept \<and> | |
| (\<forall>no\<in>set_of lrept. rep no = no) \<and> | |
| size (dag ((rep \<propto> id) (low (rep p))) (rep \<propto> low) (rep \<propto> high)) | |
| < size (dag ((rep \<propto> id) p) (rep \<propto> low) (rep \<propto> high)) \<and> | |
| (\<forall>lowa higha pa. | |
| Dag pa lowa higha lrept \<and> | |
| (\<forall>pt. pt \<notin> set_of lrept \<longrightarrow> | |
| low pt = lowa pt \<and> high pt = higha pt) \<longrightarrow> | |
| (\<exists>rrept. | |
| Dag ((rep \<propto> id) (higha (rep p))) (rep \<propto> lowa(rep p := pa)) | |
| (rep \<propto> higha) rrept \<and> | |
| (\<forall>no\<in>set_of rrept. rep no = no) \<and> | |
| size (dag ((rep \<propto> id) (higha (rep p))) | |
| (rep \<propto> lowa(rep p := pa)) (rep \<propto> higha)) | |
| < size (dag ((rep \<propto> id) p) (rep \<propto> low) (rep \<propto> high)) \<and> | |
| (\<forall>lowb highb pb. | |
| Dag pb lowb highb rrept \<and> | |
| (\<forall>pt. pt \<notin> set_of rrept \<longrightarrow> | |
| (lowa(rep p := pa)) pt = lowb pt \<and> | |
| higha pt = highb pt) \<longrightarrow> | |
| Dag (rep p) lowb (highb(rep p := pb)) rept \<and> | |
| (\<forall>pt. pt \<notin> set_of rept \<longrightarrow> | |
| low pt = lowb pt \<and> | |
| high pt = (highb(rep p := pb)) pt))))" | |
| proof - | |
| from rp_nNull rept_dag p_nNull obtain lrept rrept where | |
| rept_def: "rept = Node lrept (rep p) rrept" | |
| by auto | |
| with rept_dag p_nNull have lrept_dag: | |
| "Dag ((rep \<propto> low) (rep p)) (rep \<propto> low) (rep \<propto> high) lrept" | |
| by simp | |
| from rept_def rept_dag p_nNull have rrept_dag: | |
| "Dag ((rep \<propto> high) (rep p)) (rep \<propto> low) (rep \<propto> high) rrept" | |
| by simp | |
| from rno_rept rept_def have rno_lrept: "\<forall> no \<in> set_of lrept. rep no = no" | |
| by auto | |
| from rno_rept rept_def have rno_rrept: "\<forall> no \<in> set_of rrept. rep no = no" | |
| by auto | |
| show ?thesis | |
| apply (rule_tac x=lrept in exI) | |
| apply (rule conjI) | |
| apply (simp add: id_trans lrept_dag) | |
| apply (rule conjI) | |
| apply (rule rno_lrept) | |
| apply (rule conjI) | |
| using rept_dag rept_def | |
| apply (simp only: Dag_dag) | |
| apply (clarsimp simp add: id_trans Dag_dag) | |
| apply clarify | |
| subgoal premises prems for lowa higha pa | |
| proof - | |
| have lrepta: "Dag pa lowa higha lrept" by fact | |
| have low_lowa_nc: | |
| "\<forall>pt. pt \<notin> set_of lrept \<longrightarrow> low pt = lowa pt \<and> high pt = higha pt" by fact | |
| from lrept_dag lrepta obtain | |
| pa_def: "pa = (rep \<propto> low) (rep p)" and | |
| lowa_higha_def: "\<forall>no \<in> set_of lrept. | |
| lowa no = (rep \<propto> low) no \<and> higha no = (rep \<propto> high) no" | |
| apply - | |
| apply (drule Dags_eq_hp_eq) | |
| apply auto | |
| done | |
| from rept_dag have rept_DAG: "DAG rept" | |
| by (rule Dag_is_DAG) | |
| with rept_def have rp_notin_lrept: "rep p \<notin> set_of lrept" | |
| by simp | |
| from rept_DAG rept_def have rp_notin_rrept: "rep p \<notin> set_of rrept" | |
| by simp | |
| have rrepta: "Dag ((rep \<propto> id) (higha (rep p))) | |
| (rep \<propto> lowa(rep p := pa)) (rep \<propto> higha) rrept" | |
| proof - | |
| from low_lowa_nc rp_notin_lrept | |
| have "(rep \<propto> high) (rep p) = (rep \<propto> higha) (rep p)" | |
| by (auto simp add: null_comp_def) | |
| with rrept_dag have higha_mixed_rrept: | |
| "Dag ((rep \<propto> id) (higha (rep p))) (rep \<propto> low) (rep \<propto> high) rrept" | |
| by (simp add: id_trans) | |
| thm low_high_exchange_dag | |
| with low_lowa_nc lowa_higha_def rno_rrept | |
| have lowa_higha_rrept: | |
| "Dag ((rep \<propto> id) (higha (rep p))) (rep \<propto> lowa) (rep \<propto> higha) rrept" | |
| apply - | |
| apply (rule low_high_exchange_dag) | |
| apply auto | |
| done | |
| have "Dag ((rep \<propto> id) (higha (rep p))) (rep \<propto> lowa) (rep \<propto> higha) rrept = | |
| Dag ((rep \<propto> id) (higha (rep p))) | |
| (rep \<propto> lowa(rep p := pa)) (rep \<propto> higha) rrept" | |
| proof - | |
| have "\<forall>no \<in> set_of rrept. | |
| (rep \<propto> lowa) no = (rep \<propto> lowa(rep p := pa)) no \<and> | |
| (rep \<propto> higha) no = (rep \<propto> higha) no" | |
| proof | |
| fix no | |
| assume no_in_rrept: "no \<in> set_of rrept" | |
| with rp_notin_rrept have "no \<noteq> rep p" | |
| by blast | |
| thus "(rep \<propto> lowa) no = (rep \<propto> lowa(rep p := pa)) no \<and> | |
| (rep \<propto> higha) no = (rep \<propto> higha) no" | |
| by (simp add: null_comp_def) | |
| qed | |
| thus ?thesis | |
| by (rule heaps_eq_Dag_eq) | |
| qed | |
| with lowa_higha_rrept show ?thesis | |
| by simp | |
| qed | |
| show ?thesis | |
| apply (rule_tac x=rrept in exI) | |
| apply (rule conjI) | |
| apply (rule rrepta) | |
| apply (rule conjI) | |
| apply (rule rno_rrept) | |
| apply (rule conjI) | |
| using rept_dag rept_def rrepta | |
| apply (simp only: Dag_dag) | |
| apply (clarsimp simp add: id_trans Dag_dag) | |
| apply clarify | |
| subgoal premises prems for lowb highb pb | |
| proof - | |
| have rreptb: "Dag pb lowb highb rrept" by fact | |
| have ab_nc: "\<forall>pt. pt \<notin> set_of rrept \<longrightarrow> | |
| (lowa(rep p := pa)) pt = lowb pt \<and> higha pt = highb pt" by fact | |
| from rreptb rrept_dag obtain | |
| pb_def: "pb = ((rep \<propto> high) (rep p))" and | |
| lowb_highb_def: "\<forall>no \<in> set_of rrept. | |
| lowb no = (rep \<propto> low) no \<and> highb no = (rep \<propto> high) no" | |
| apply - | |
| apply (drule Dags_eq_hp_eq) | |
| apply auto | |
| done | |
| have rept_end_dag: " Dag (rep p) lowb (highb(rep p := pb)) rept" | |
| proof - | |
| have "\<forall>no \<in> set_of rept. | |
| lowb no = (rep \<propto> low) no \<and> (highb(rep p := pb)) no = (rep \<propto> high) no" | |
| proof | |
| fix no | |
| assume no_in_rept: " no \<in> set_of rept" | |
| show "lowb no = (rep \<propto> low) no \<and> | |
| (highb(rep p := pb)) no = (rep \<propto> high) no" | |
| proof (cases "no \<in> set_of rrept") | |
| case True | |
| with lowb_highb_def pb_def show ?thesis | |
| by simp | |
| next | |
| assume no_notin_rrept: " no \<notin> set_of rrept" | |
| show ?thesis | |
| proof (cases "no \<in> set_of lrept") | |
| case True | |
| with no_notin_rrept rp_notin_lrept ab_nc | |
| have ab_nc_no: "lowa no = lowb no \<and> higha no = highb no" | |
| apply - | |
| apply (erule_tac x=no in allE) | |
| apply (erule impE) | |
| apply simp | |
| apply (subgoal_tac "no \<noteq> rep p") | |
| apply simp | |
| apply blast | |
| done | |
| from lowa_higha_def True have | |
| "lowa no = (rep \<propto> low) no \<and> higha no = (rep \<propto> high) no" | |
| by auto | |
| with ab_nc_no | |
| have "lowb no = (rep \<propto> low) no \<and> highb no =(rep \<propto> high) no" | |
| by simp | |
| with rp_notin_lrept True show ?thesis | |
| apply (subgoal_tac "no \<noteq> rep p") | |
| apply simp | |
| apply blast | |
| done | |
| next | |
| assume no_notin_lrept: " no \<notin> set_of lrept" | |
| with no_in_rept rept_def no_notin_rrept have no_rp: "no = rep p" | |
| by simp | |
| with rp_notin_lrept low_lowa_nc | |
| have a_nc: "low no = lowa no \<and> high no = higha no" | |
| by auto | |
| from rp_notin_rrept no_rp ab_nc | |
| have "(lowa(rep p := pa)) no = lowb no \<and> higha no = highb no" | |
| by auto | |
| with a_nc pa_def no_rp | |
| have "(rep \<propto> low) no = lowb no \<and> high no = highb no" | |
| by auto | |
| with pb_def no_rp show ?thesis | |
| by simp | |
| qed | |
| qed | |
| qed | |
| with rept_dag | |
| have "Dag (rep p) lowb (highb(rep p := pb)) rept = | |
| Dag (rep p) (rep \<propto> low) (rep \<propto> high) rept" | |
| apply - | |
| apply (rule heaps_eq_Dag_eq) | |
| apply auto | |
| done | |
| with rept_dag p_nNull show ?thesis | |
| by simp | |
| qed | |
| have "(\<forall>pt. pt \<notin> set_of rept \<longrightarrow> low pt = lowb pt \<and> | |
| high pt = (highb(rep p := pb)) pt)" | |
| proof (intro allI impI) | |
| fix pt | |
| assume pt_notin_rept: "pt \<notin> set_of rept" | |
| with rept_def obtain | |
| pt_notin_lrept: "pt \<notin> set_of lrept" and | |
| pt_notin_rrept: "pt \<notin> set_of rrept" and | |
| pt_neq_rp: "pt \<noteq> rep p" | |
| by simp | |
| with low_lowa_nc ab_nc | |
| show "low pt = lowb pt \<and> high pt = (highb(rep p := pb)) pt" | |
| by auto | |
| qed | |
| with rept_end_dag show ?thesis | |
| by simp | |
| qed | |
| done | |
| qed | |
| done | |
| qed | |
| qed | |
| end | |