Quelle RepointProof.thy

Sprache: Isabelle

(*  Title:       BDD

    Author:      Veronika Ortner and Norbert Schirmer, 2004
    Maintainer:NorbertSchirmer,  norbert.schirmeratweb 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,  (at youroption any later version.

This library is distributed in the hope that it will          pa_def: "pa = (rep <propto>low) (rep )
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 ‹
impor b

(open) DistinctTreeProver.set_of tree.Node tree.Tip

(in Repoint_impl) Repoint_modifies:
shows "∀rp\notinrp"
{t. t may_only_modify_globals σ in [low,high]}"
apply (hoare_rule HoarePartial.ProcRec1)
apply (vcg spec=modifies)
done

low_high_exchange_dag:
pt_sa "\<forallpt set_of lt ⟶ p = lowa pt ∧"
pt_changed: "∀pt ∈ set_of lt. lowa pt = (rep ∝ id) (higha (rep p)))
higha pt = (rep ∝ high) pt"
rep_pt: "∀ lowa(rep p := pa)) (rep ∝
"∧q. Dag q (rep ∝
Dag q (rep ∝ lowa) (rep ∝
rep_pt
(induct rt)
case Tip thus ?case by simp

case (Node lrt q' rrt)
have "Dag q (rep ∝ high) (rep p) = (rep ∝
then obtain
q': "q = q'" and
q_notNull: "q ≠ Null" and
lrt: "Dag ((rep ∝ low) q) (rep ∝a simp add: null_comp_def)
rrt: "Dag ((rep ∝ high) q) (rep ∝ rrept_dag hahigha:
by auto
have rlowa_rlow: "((rep ∝ lowa) q) = ((rep ∝ id) (higha (rep p))) (rep ∝ high) rrept"
proof (cases "q ∈ set_of lt")
case True
note q_in_lt=this
with pt_changed have lowa_q: "lowa q = (rep ∝
by simp
thus "(rep ∝
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 ≠ Null"
show ?thesis
proof (cases "(rep ∝ 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 ∝ low) q ≠ Null"
with lrt lowa_q have "lowa q ∈ set_of lrt"
by auto
with Node.prems Node have "lowa q ∈with o_oanagadft
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 ∉ set_of lt"
with pt_same have "low q = lowa q"
by auto
is
by (simp add: null_comp_def)
qed
have rhigha_rhigh: "((rep ∝ higha) q) = ((rep ∝ high) q)"
roof (cases "q ∈")
case True
note q_in_lt=this
with pt_changed have higha_q: "higha q = (rep ∝ high) q"
by simp
thus ?thesis
proof (cases "high q = Null")
case True
with higha_q have "higha q = Null"
(si add: null_comp_def)
with True show ?thesis
by (simp add: null_comp_def)
next
assume hqnNull:: "higq \\noteq>Null"
show ?thesis
proof (cases "(rep ∝ 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 ∝ high) q ≠
withrrt have "hig q\in> set_of r"
by auto
with Node.prems Node have "higha q ∈propto> id) (higha (rep p)))
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 ∉
with pt_same have "high q = higha q"
by hav "∀ set_of rrept.
thus ?thesi
by (simp add: null_comp_def)
qed
with rrt have rhigha_mixed_dag:
java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
by simp
from lrt rlowa_rlow have rlowa_mixed_dag:
" Dag ((rep ∝
by simp
from Node.prems have lrt_rep_eq: " ∀pt∈set_of w rp_notin have "no ≠
by simp
from Node.prems have rrt_rep_eq: "∀pt∈set_of rrt. rep pt = pt"
by simp
from rlowa_mixed_dag lrt_rep_eq have lowa_lrt:
" Dag ((rep ∝ lowa) q) (rep ∝ lowa) (rep ∝<ropto )
apply -
apply (rule Node.hyps)
apply auto
done
from rhigha_mixed_dag rrt_rep_eq have higha_rrt:
" Dag ((rep ∝
apply -
apply (rule Node.hyps)
apply auto
done
with lowa_lrt q' q_notNull
show " Dag q (rep ∝
by simp

Repoint_impl

"∀apply (ru x=rrept in exI)
∧)
🍋p :== CALL Repoint (🍋p)
{Dag 🍋p 🍋low 🍋high repappl rulerr)
(∀pt. pt \<          apply
(hoare_rule CallRec1_SamePost)
(vcg)
(rule conjI)
clariaappl (simp only: D)
2
(intro impI allI )
(simp add: null_comp_def) (clar s ad:id Dag_dag)
(rule conjI)
2
(clarsimp)
clarify
*)

lemma (in Repoint_impl for pb
shows
  "∀σ. Γ:"ighb" b fact
  🍋p :== PROC Repointhave a:"forallpt. pt ∉
  {∀.((Dag^bsupσ ) ^{) (^{<sigma\^>rep ^<sigma>^∝σ

  ∧pa   ptand higha pt = highbptby

  ⟶ Dag 🍋p 🍋fromeptbtbjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40

  \pt. pt ∉ set_of rept ⟶ ^<sigma>low pt = 🍋low pt ∧ ^<sigma>high pt = 🍋"

apply (oar HoarePartial.ProcR)

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 l hig p rep lowa h pa lowb hi pb rept

  assume p_nNull: "p ≠

  assume rp_nNull: " rep p ≠ ) lw hgbrpp=p) et

  assume rec_spec_lrept:

    "∀rept. Dag ((rep ∝

    ∧ (∀no              have "∀>set_of rept.

    ⟶ no = (rep \ <pr> low) no <an> (hig(rep p := pb)) no = (rep ∝

        (∀pt. pt ∉ set_of rept ⟶p

  assume rec_spec_rrept:

    "∀rept. Dag ((rep ∝

    ∧ (∀no∈no<>set_ofrept"

    ⟶ Dag pb lowb highb rept ∧

        (∀o= (r(rp \<>

  assume rept_dag: "Dag ((rep ∝ id) p) (rep ∝ high) no"

  assume rno_rept: "∀no∈no<>set_of")

  show " Dag (rep p) lowb (highb(rep p := pb)) rept ∧

    (∀pt. pt ∉ set_of reptwith lowb_highb_def pb_def ?thesis

  proof -

    from rp_nNull rept_dag p_nNull                     by simp

      rept_def: "rept = Node lrept (rep p) rrept"

      by auto

    with rept_dag p_nNull have lrept_dag:

      "Dag ((rep \<proptoassume set_of rrept"

      by simp

    from rept_def rept_dag p_nNull have rrept_dag:

      "Dag ((rep ∝prof(ass"<in set_of lrept")

      by simp

    from rno_rept rept_def have rno_lrept: "∀case True

      by auto

    from rno_rept rept_def have rno_rrept: "∀pntnleta_

      by auto

    have repoint_post_low:

      " Dag pa lowa highahaveab_nc_nolowano = lowbno<>higha  highb no

      (∀

    proof -

      from applyrule_tac allE

        by (simp add: id_trans)

      with  rec_spec_lrept rno_lrept show ?thesis

        apply -

        apply (erule_tac x=lrept in allE)

        apply (erule impEubgoal_tac <noteq> rep p")

        apply simp

        apply assumption

        done

    qed

    hence low_lowa: "(<forallpt set_of lrept <longrightarrow low pt= lowa pt high pt = higha pt)"

      by simp

    from lrept_dag repoint_post_low obtain

      pa_def: "pa = (rep ∝ low) (rep p)"

      lowa_higha_def: "(∀

      apply -

      apply (drule Dags_eq_hp_eq)

      apply auto

      done

    from rept_dag have lowa no  (rep <> low)no \>  no=(rep high) no"

      by (rule Dag_is_DAG)

    with rept_def have rp_notin_lrept: "rep p ∉ set_of lrept"

      by simp

    from rept_DAG rept_def have rp_notin_rrept: "rep p ∉ab_nc_no

      by simp

    have haveno<> low) no ∧ high

    proof -

      from low_lowa_nc

        by (auto simp add rp_notin_lrept show

      with rrept_dagapplysubgoal_tacnoteq rep p")

        "Dag ((rep ∝

        apply

      thm low_high_exchange_dag

      with low_lowa_nc lowa_higha_def rno_rrept

        "Dag ((rep ∝ set_of lrept"

        apply -

        apply (rule low_high_exchange_dag no_in_rept no_notin_rrept : "no = rep p"

        apply auto

        done

      have "Dag ((rep ∝ id) (highawi rp_notin low_lowa_nc

        Dag ((rep ∝ high no = higha no"

      proof -

        have "∀

          (rep ∝

        proof

          fix no

          assume no_in_rrept: "no ∈ set_of rrept"

          with rp_notin_rrept have "no ≠ rep p"

            by blast

          thus "(rep ∝ low) no = lowb<>high no = highb no"

            (rep ∝ higha) no = (rep ∝ higha) no"

            by (simp add: null_comp_def)

        qedd

        thus ?thesis

          by (rule heaps_eq_Dag_eq

      qed

      with

        by simp

    qed

    with rec_spec_rrept rno_rrept have repoint_rrept: "Dag pb lowb highb rrept ∧

      (∀pt. pt ∉ set_of rrept ⟶

      (lowa(rep p := pa)) pt = lowb pt ∧ higha pt = highb pt)"

      apply -

      apply (erule_tac x=rrept in allE)

      apply (erule impE (rep p) (rep ∝ low) (rep ∝

      apply simp

      apply assumption

      done

    then have ab_nc: "(∀pt. pt ∉ set_of rrept ⟶ apply(rue hehep_qDg_q)

      (lowa(rep p := pa)) pt = lowb pt ∧

      by simp

    from rointret retdgotain

      pb_def: "pb = ((rep ∝

      lowb_highb_def: "(∀ no ∈

      apply -

      apply (drule Dags_eq_hp_eq)

      apply auto

      done

    have rept_end_dag: " Dag (rep p) lowb (highb(rephigh  highbrep p : pb pt)"

    proof -

      have "∀ noproof

      proof

        fix no

        assume no_in_rept: " no ∈ set_of rept"

        showlowb = (rep <propto  = ( 🚫

        proof (cases "no ∈ set_of rrept")

          case True

          with lowb_highb_def pb_def show ?thesis

            by simp

        next

          assume no_notin_rrept: "pt ≠

          show ?thesis

          proof (cases "no ∈

            case True

            with no_notin_rrept rp_notin_lrept ab_nc

            have ab_nc_no:withow_lowa_nc

              apply -

              apply (erule_tac x=no in allE)



              apply simp

              apply (subgoal_tac "no ≠

              apply simp

              apply blast

              done

            from lowa_higha_def True have

              "lowa

              by auto

            with ab_nc_no have "lowb no = (rep ∝

              by simp

            with rp_notin_lrept True show ?thesis

              apply (subgoal_tac "no ≠ rep p")

              apply simp

              apply blast

              done

          next

            assume nod

            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:

              qed

              by auto

            from rp_notin_rrept no_rp ab_nc have

              "(lowa(rep p := pa)) no = lowb no ∧ higha no = highb no"

              by auto

            with a_nc pa_def no_rp have "(rep ∝ low) no = lowb no ∧ 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 ∝ low) (rep ∝ 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 "(∀pt. pt ∉ set_of rept ⟶ low pt = lowb pt ∧ high pt = (highb(rep p := pb)) pt)"

    proof (intro allI impI)

      fix pt

      assume pt_notin_rept: "pt ∉ set_of rept"

      with rept_def obtain

        pt_notin_lrept: "pt ∉ set_of lrept" and

        pt_notin_rrept: "pt ∉ set_of rrept" and

        pt_neq_rp: "pt ≠ rep p"

        by simp

      with low_lowa_nc ab_nc show "low pt = lowb pt ∧ 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

  "∀σ rept. Γ⊨ {σ. Dag ((🍋rep ∝ id) 🍋p) (🍋rep ∝ 🍋low) (🍋rep ∝ 🍋high) rept

  ∧ (∀ no ∈ set_of rept. 🍋rep no = no) }

  🍋p :== PROC Repoint (🍋p)

  {Dag 🍋p 🍋low 🍋high rept ∧

  (∀pt. pt ∉ set_of rept ⟶ ^<sigma>low pt = 🍋low pt ∧ ^<sigma>high pt = 🍋high pt)}"

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 ∝ id) p) (rep ∝ low) (rep ∝ high) rept"

  assume rno_rept: "∀no∈set_of rept. rep no = no"

  assume p_nNull: "p ≠ Null"

  assume rp_nNull: " rep p ≠ Null"

  show "∃lrept.

             Dag ((rep ∝ id) (low (rep p))) (rep ∝ low) (rep ∝ high) lrept ∧

             (∀no∈set_of lrept. rep no = no) ∧

             (∀lowa higha pa.

                 Dag pa lowa higha lrept ∧

                 (∀pt. pt ∉ set_of lrept ⟶

                       low pt = lowa pt ∧ high pt = higha pt) ⟶

                 (∃rrept.

                     Dag ((rep ∝ id) (higha (rep p))) (rep ∝ lowa(rep p := pa))

                      (rep ∝ higha) rrept ∧

                     (∀no∈set_of rrept. rep no = no) ∧

                     (∀lowb highb pb.

                         Dag pb lowb highb rrept ∧

                         (∀pt. pt ∉ set_of rrept ⟶

                               (lowa(rep p := pa)) pt = lowb pt ∧

                               higha pt = highb pt) ⟶

                         Dag (rep p) lowb (highb(rep p := pb)) rept ∧

                         (∀pt. pt ∉ set_of rept ⟶

                               low pt = lowb pt ∧

                               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 ∝ low) (rep p)) (rep ∝ low) (rep ∝ high) lrept"

      by simp

    from rept_def rept_dag p_nNull have rrept_dag:

      "Dag ((rep ∝ high) (rep p)) (rep ∝ low) (rep ∝ high) rrept"

      by simp

    from rno_rept rept_def have rno_lrept: "∀ no ∈ set_of lrept. rep no = no"

      by auto

    from rno_rept rept_def have rno_rrept: "∀ no ∈ 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:

          "∀pt. pt ∉ set_of lrept ⟶ low pt = lowa pt ∧ high pt = higha pt" by fact

        from lrept_dag lrepta obtain

          pa_def: "pa = (rep ∝ low) (rep p)" and

          lowa_higha_def: "∀no ∈ set_of lrept.

          lowa no = (rep ∝ low) no ∧ higha no = (rep ∝ 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 ∉ set_of lrept"

          by simp

        from rept_DAG rept_def have rp_notin_rrept: "rep p ∉ set_of rrept"

          by simp

        have rrepta: "Dag ((rep ∝ id) (higha (rep p)))

                         (rep ∝ lowa(rep p := pa)) (rep ∝ higha) rrept"

        proof -

          from low_lowa_nc rp_notin_lrept

          have "(rep ∝ high) (rep p) = (rep ∝ higha) (rep p)"

            by (auto simp add: null_comp_def)

          with rrept_dag have higha_mixed_rrept:

            "Dag ((rep ∝ id) (higha (rep p))) (rep ∝ low) (rep ∝ 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 ∝ id) (higha (rep p))) (rep ∝ lowa) (rep ∝ higha) rrept"

            apply -

            apply (rule low_high_exchange_dag)

            apply auto

            done

          have "Dag ((rep ∝ id) (higha (rep p))) (rep ∝ lowa) (rep ∝ higha) rrept =

                Dag ((rep ∝ id) (higha (rep p)))

                        (rep ∝ lowa(rep p := pa)) (rep ∝ higha) rrept"

          proof -

            have "∀no ∈ set_of rrept.

                      (rep ∝ lowa) no = (rep ∝ lowa(rep p := pa)) no ∧

                      (rep ∝ higha) no = (rep ∝ higha) no"

            proof

              fix no

              assume no_in_rrept: "no ∈ set_of rrept"

              with rp_notin_rrept have "no ≠ rep p"

                by blast

              thus "(rep ∝ lowa) no = (rep ∝ lowa(rep p := pa)) no ∧

                (rep ∝ higha) no = (rep ∝ 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: "∀pt. pt ∉ set_of rrept ⟶

                            (lowa(rep p := pa)) pt = lowb pt ∧ higha pt = highb pt" by fact

            from rreptb rrept_dag obtain

              pb_def: "pb = ((rep ∝ high) (rep p))" and

              lowb_highb_def: "∀no ∈ set_of rrept.

                                  lowb no = (rep ∝ low) no ∧ highb no = (rep ∝ 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 "∀no ∈ set_of rept.

                    lowb no = (rep ∝ low) no ∧ (highb(rep p := pb)) no = (rep ∝ high) no"

              proof

                fix no

                assume no_in_rept: " no ∈ set_of rept"

                show "lowb no = (rep ∝ low) no ∧

                      (highb(rep p := pb)) no = (rep ∝ high) no"

                proof (cases "no ∈ set_of rrept")

                  case True

                  with lowb_highb_def pb_def show ?thesis

                    by simp

                next

                  assume no_notin_rrept: " no ∉ set_of rrept"

                  show ?thesis

                  proof (cases "no ∈ set_of lrept")

                    case True

                    with no_notin_rrept rp_notin_lrept ab_nc

                    have ab_nc_no: "lowa no = lowb no ∧ higha no = highb no"

                      apply -

                      apply (erule_tac x=no in allE)

                      apply (erule impE)

                      apply simp

                      apply (subgoal_tac "no ≠ rep p")

                      apply simp

                      apply blast

                      done

                    from lowa_higha_def True have

                      "lowa no = (rep ∝ low) no ∧ higha no = (rep ∝ high) no"

                      by auto

                    with ab_nc_no

                    have "lowb no = (rep ∝ low) no ∧ highb no =(rep ∝ high) no"

                      by simp

                    with rp_notin_lrept True show ?thesis

                      apply (subgoal_tac "no ≠ rep p")

                      apply simp

                      apply blast

                      done

                  next

                    assume no_notin_lrept: " no ∉ 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 ∧ high no = higha no"

                      by auto

                    from rp_notin_rrept no_rp ab_nc

                    have "(lowa(rep p := pa)) no = lowb no ∧ higha no = highb no"

                      by auto

                    with a_nc pa_def no_rp

                    have "(rep ∝ low) no = lowb no ∧ 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 ∝ low) (rep ∝ high) rept"

                apply -

                apply (rule heaps_eq_Dag_eq)

                apply auto

                done

              with rept_dag p_nNull show ?thesis

                by simp

            qed

            have "(∀pt. pt ∉ set_of rept ⟶ low pt = lowb pt ∧

                        high pt = (highb(rep p := pb)) pt)"

            proof (intro allI impI)

              fix pt

              assume pt_notin_rept: "pt ∉ set_of rept"

              with rept_def obtain

                pt_notin_lrept: "pt ∉ set_of lrept" and

                pt_notin_rrept: "pt ∉ set_of rrept" and

                pt_neq_rp: "pt ≠ rep p"

                by simp

              with low_lowa_nc ab_nc

              show "low pt = lowb pt ∧ 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

  "∀σ rept. Γ⊨_t {σ. Dag ((🍋rep ∝ id) 🍋p) (🍋rep ∝ 🍋low) (🍋rep ∝ 🍋high) rept

  ∧ (∀ no ∈ set_of rept. 🍋rep no = no) }

  🍋p :== PROC Repoint (🍋p)

  {Dag 🍋p 🍋low 🍋high rept ∧

  (∀pt. pt ∉ set_of rept ⟶ ^<sigma>low pt = 🍋low pt ∧ ^<sigma>high pt = 🍋high pt)}"

apply (hoare_rule HoareTotal.ProcRec1

          [where r="measure (λ(s,p). size

                       (dag ((rep ∝ id) p) (rep ∝ low) (rep ∝ 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 ∝ id) p) (rep ∝ low) (rep ∝ high) rept"

  assume rno_rept: "∀no∈set_of rept. rep no = no"

  assume p_nNull: "p ≠ Null"

  assume rp_nNull: " rep p ≠ Null"

  show "∃lrept.

             Dag ((rep ∝ id) (low (rep p))) (rep ∝ low) (rep ∝ high) lrept ∧

             (∀no∈set_of lrept. rep no = no) ∧

             size (dag ((rep ∝ id) (low (rep p))) (rep ∝ low) (rep ∝ high))

             < size (dag ((rep ∝ id) p) (rep ∝ low) (rep ∝ high)) ∧

             (∀lowa higha pa.

                 Dag pa lowa higha lrept ∧

                 (∀pt. pt ∉ set_of lrept ⟶

                       low pt = lowa pt ∧ high pt = higha pt) ⟶

                 (∃rrept.

                     Dag ((rep ∝ id) (higha (rep p))) (rep ∝ lowa(rep p := pa))

                      (rep ∝ higha) rrept ∧

                     (∀no∈set_of rrept. rep no = no) ∧

                     size (dag ((rep ∝ id) (higha (rep p)))

                            (rep ∝ lowa(rep p := pa)) (rep ∝ higha))

                     < size (dag ((rep ∝ id) p) (rep ∝ low) (rep ∝ high)) ∧

                     (∀lowb highb pb.

                         Dag pb lowb highb rrept ∧

                         (∀pt. pt ∉ set_of rrept ⟶

                               (lowa(rep p := pa)) pt = lowb pt ∧

                               higha pt = highb pt) ⟶

                         Dag (rep p) lowb (highb(rep p := pb)) rept ∧

                         (∀pt. pt ∉ set_of rept ⟶

                               low pt = lowb pt ∧

                               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 ∝ low) (rep p)) (rep ∝ low) (rep ∝ high) lrept"

      by simp

    from rept_def rept_dag p_nNull have rrept_dag:

      "Dag ((rep ∝ high) (rep p)) (rep ∝ low) (rep ∝ high) rrept"

      by simp

    from rno_rept rept_def have rno_lrept: "∀ no ∈ set_of lrept. rep no = no"

      by auto

    from rno_rept rept_def have rno_rrept: "∀ no ∈ 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:

          "∀pt. pt ∉ set_of lrept ⟶ low pt = lowa pt ∧ high pt = higha pt" by fact

        from lrept_dag lrepta obtain

          pa_def: "pa = (rep ∝ low) (rep p)" and

          lowa_higha_def: "∀no ∈ set_of lrept.

          lowa no = (rep ∝ low) no ∧ higha no = (rep ∝ 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 ∉ set_of lrept"

          by simp

        from rept_DAG rept_def have rp_notin_rrept: "rep p ∉ set_of rrept"

          by simp

        have rrepta: "Dag ((rep ∝ id) (higha (rep p)))

                         (rep ∝ lowa(rep p := pa)) (rep ∝ higha) rrept"

        proof -

          from low_lowa_nc rp_notin_lrept

          have "(rep ∝ high) (rep p) = (rep ∝ higha) (rep p)"

            by (auto simp add: null_comp_def)

          with rrept_dag have higha_mixed_rrept:

            "Dag ((rep ∝ id) (higha (rep p))) (rep ∝ low) (rep ∝ 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 ∝ id) (higha (rep p))) (rep ∝ lowa) (rep ∝ higha) rrept"

            apply -

            apply (rule low_high_exchange_dag)

            apply auto

            done

          have "Dag ((rep ∝ id) (higha (rep p))) (rep ∝ lowa) (rep ∝ higha) rrept =

                Dag ((rep ∝ id) (higha (rep p)))

                        (rep ∝ lowa(rep p := pa)) (rep ∝ higha) rrept"

          proof -

            have "∀no ∈ set_of rrept.

                      (rep ∝ lowa) no = (rep ∝ lowa(rep p := pa)) no ∧

                      (rep ∝ higha) no = (rep ∝ higha) no"

            proof

              fix no

              assume no_in_rrept: "no ∈ set_of rrept"

              with rp_notin_rrept have "no ≠ rep p"

                by blast

              thus "(rep ∝ lowa) no = (rep ∝ lowa(rep p := pa)) no ∧

                (rep ∝ higha) no = (rep ∝ 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: "∀pt. pt ∉ set_of rrept ⟶

                            (lowa(rep p := pa)) pt = lowb pt ∧ higha pt = highb pt" by fact

            from rreptb rrept_dag obtain

              pb_def: "pb = ((rep ∝ high) (rep p))" and

              lowb_highb_def: "∀no ∈ set_of rrept.

                                  lowb no = (rep ∝ low) no ∧ highb no = (rep ∝ 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 "∀no ∈ set_of rept.

                    lowb no = (rep ∝ low) no ∧ (highb(rep p := pb)) no = (rep ∝ high) no"

              proof

                fix no

                assume no_in_rept: " no ∈ set_of rept"

                show "lowb no = (rep ∝ low) no ∧

                      (highb(rep p := pb)) no = (rep ∝ high) no"

                proof (cases "no ∈ set_of rrept")

                  case True

                  with lowb_highb_def pb_def show ?thesis

                    by simp

                next

                  assume no_notin_rrept: " no ∉ set_of rrept"

                  show ?thesis

                  proof (cases "no ∈ set_of lrept")

                    case True

                    with no_notin_rrept rp_notin_lrept ab_nc

                    have ab_nc_no: "lowa no = lowb no ∧ higha no = highb no"

                      apply -

                      apply (erule_tac x=no in allE)

                      apply (erule impE)

                      apply simp

                      apply (subgoal_tac "no ≠ rep p")

                      apply simp

                      apply blast

                      done

                    from lowa_higha_def True have

                      "lowa no = (rep ∝ low) no ∧ higha no = (rep ∝ high) no"

                      by auto

                    with ab_nc_no

                    have "lowb no = (rep ∝ low) no ∧ highb no =(rep ∝ high) no"

                      by simp

                    with rp_notin_lrept True show ?thesis

                      apply (subgoal_tac "no ≠ rep p")

                      apply simp

                      apply blast

                      done

                  next

                    assume no_notin_lrept: " no ∉ 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 ∧ high no = higha no"

                      by auto

                    from rp_notin_rrept no_rp ab_nc

                    have "(lowa(rep p := pa)) no = lowb no ∧ higha no = highb no"

                      by auto

                    with a_nc pa_def no_rp

                    have "(rep ∝ low) no = lowb no ∧ 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 ∝ low) (rep ∝ high) rept"

                apply -

                apply (rule heaps_eq_Dag_eq)

                apply auto

                done

              with rept_dag p_nNull show ?thesis

                by simp

            qed

            have "(∀pt. pt ∉ set_of rept ⟶ low pt = lowb pt ∧

                        high pt = (highb(rep p := pb)) pt)"

            proof (intro allI impI)

              fix pt

              assume pt_notin_rept: "pt ∉ set_of rept"

              with rept_def obtain

                pt_notin_lrept: "pt ∉ set_of lrept" and

                pt_notin_rrept: "pt ∉ set_of rrept" and

                pt_neq_rp: "pt ≠ rep p"

                by simp

              with low_lowa_nc ab_nc

              show "low pt = lowb pt ∧ 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}}

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