(*  Title:       BDD

    Author:      Veronika Ortner and Norbert Schirmer, 2004
    Maintainer:  Norbert Schirmer,  norbert.schirmer at web de
    License:     LGPL
*)

(*  
ShareRepProof.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 ‹Proof of Procedure ShareRep›
theory ShareRepProof imports ProcedureSpecs Simpl.HeapList begin

lemma (in ShareRep_impl) ShareRep_modifies:
  shows "∀σ. Γ⊨{σ} PROC ShareRep (🚫nodeslist, 🚫p)
             {t. t may_only_modify_globals σ in [rep]}"
  apply (hoare_rule HoarePartial.ProcRec1)
  apply (vcg spec=modifies)
  done


lemma hd_filter_cons: 
"∧ i. [ P (xs ! i) p; i < length xs; ∀ no ∈ set (take i xs). ¬ P no p; ∀ a b. P a b = P b a]
  ==> xs ! i = hd (filter (P p) xs)"
apply (induct xs)
apply simp
apply (case_tac "P a p")
apply simp
apply (case_tac i)
apply simp
apply simp
apply (case_tac i)
apply simp
apply auto
done

lemma (in ShareRep_impl) ShareRep_spec_total:
shows 
  "∀σ ns. Γ,Θ⊨_t
  {σ. List 🚫nodeslist 🚫next ns ∧
     (∀no ∈ set ns. no ≠ Null ∧
       ((no→🚫low = Null) = (no→🚫high = Null)) ∧
       (isLeaf_pt 🚫p 🚫low 🚫high ⟶ isLeaf_pt no 🚫low 🚫high) ∧
       no→🚫var = 🚫p→🚫var) ∧
       🚫p ∈ set ns}
  PROC ShareRep (🚫nodeslist, 🚫p)
  { (^<sigma>p → 🚫rep = hd (filter (λ sn. repNodes_eq sn ^<sigma>p ^<sigma>low ^<sigma>high ^<sigma>rep) ns)) ∧
    (∀pt. pt ≠ ^<sigma>p ⟶ pt→^<sigma>rep = pt→🚫rep) ∧
    (^<sigma>p→🚫rep→^<sigma>var = ^<sigma>p → ^<sigma>var)}"
apply (hoare_rule HoareTotal.ProcNoRec1)
apply (hoare_rule anno=  
  "IF (isLeaf_pt 🚫p 🚫low 🚫high)
   THEN 🚫p → 🚫rep :== 🚫nodeslist
   ELSE
     WHILE (🚫nodeslist ≠ Null)
     INV {∃prx sfx. List 🚫nodeslist 🚫next sfx ∧ ns=prx@sfx ∧
           ¬ isLeaf_pt 🚫p 🚫low 🚫high ∧
           (∀no ∈ set ns. no ≠ Null ∧
             ((no→^<sigma>low = Null) = (no→^<sigma>high = Null)) ∧
             (isLeaf_pt ^<sigma>p ^<sigma>low ^<sigma>high ⟶ isLeaf_pt no ^<sigma>low ^<sigma>high) ∧
             no→^<sigma>var = ^<sigma>p→^<sigma>var) ∧
        ^<sigma>p ∈ set ns ∧
        ((∃pt ∈ set prx. repNodes_eq pt ^<sigma>p ^<sigma>low ^<sigma>high ^<sigma>rep)
         ⟶ 🚫rep ^<sigma>p = hd (filter (λ sn. repNodes_eq sn ^<sigma>p ^<sigma>low ^<sigma>high ^<sigma>rep) prx) ∧
             (∀pt. pt ≠ ^<sigma>p ⟶ pt→^<sigma>rep = pt→🚫rep)) ∧
        ((∀pt ∈ set prx. ¬ repNodes_eq pt ^<sigma>p ^<sigma>low ^<sigma>high ^<sigma>rep) ⟶ ^<sigma>rep = 🚫rep) ∧
        (🚫nodeslist ≠ Null ⟶
           (∀pt ∈ set prx. ¬ repNodes_eq pt ^<sigma>p ^<sigma>low ^<sigma>high ^<sigma>rep)) ∧
        (🚫p = ^<sigma>p ∧ 🚫high = ^<sigma>high ∧ 🚫low = ^<sigma>low)}
     VAR MEASURE (length (list 🚫nodeslist 🚫next))
     DO
       IF (repNodes_eq 🚫nodeslist 🚫p 🚫low 🚫high 🚫rep)
       THEN 🚫p→🚫rep :== 🚫nodeslist;; 🚫nodeslist :== Null
       ELSE 🚫nodeslist :== 🚫nodeslist→🚫next
       FI
     OD
  FI" in HoareTotal.annotateI)
apply vcg
using  [[simp_depth_limit = 2]]
apply   (rule conjI)
apply    clarify
apply    (simp (no_asm_use))
prefer 2
apply    clarify
apply    (rule_tac x="[]" in exI)
apply    (rule_tac x=ns in exI)
apply    (simp (no_asm_use))
prefer 2
apply   clarify
apply   (rule conjI)
apply    clarify
apply    (rule conjI)
apply     (clarsimp simp add: List_list) (* solving termination contraint *)
apply    (simp (no_asm_use))
apply    (rule conjI)
apply    assumption
prefer 2
apply    clarify
apply    (simp (no_asm_use))
apply    (rule conjI)
apply    (clarsimp simp add: List_list) (* solving termination constraint *)
apply    (simp only: List_not_Null simp_thms triv_forall_equality)
apply    clarify
apply    (simp only: triv_forall_equality)
apply    (rename_tac sfx)
apply    (rule_tac x="prx@[nodeslist]" in exI)
apply    (rule_tac x="sfx" in exI)
apply    (rule conjI)
apply     assumption
apply    (rule conjI)
apply     simp
prefer 4
apply   (elim exE conjE)
apply   (simp (no_asm_use))
apply   hypsubst
using  [[simp_depth_limit = 100]]
proof -
  (* IF-THEN to postcondition *)
  fix ns var low high rep "next" p nodeslist
  assume ns: "List nodeslist next ns"
  assume no_prop:  "∀no∈set ns.
           no ≠ Null ∧
           (low no = Null) = (high no = Null) ∧
           (isLeaf_pt p low high ⟶ isLeaf_pt no low high) ∧ var no = var p"
  assume p_in_ns: "p ∈ set ns" 
  assume p_Leaf: "isLeaf_pt p low high"
  show "nodeslist = hd [sn←ns . repNodes_eq sn p low high rep] ∧
        var nodeslist = var p"
  proof -
    from p_in_ns no_prop have p_not_Null: "p≠Null"
      using [[simp_depth_limit=2]]
      by auto
    from p_in_ns have "ns ≠ []"
      by (cases ns) auto
    with ns obtain ns' where ns': "ns = nodeslist#ns'" 
      by (cases "nodeslist=Null") auto
    with no_prop p_Leaf obtain 
      "isLeaf_pt nodeslist low high" and
      var_eq: "var nodeslist = var p" and
      "nodeslist≠Null"
      using [[simp_depth_limit=2]]
      by auto
    with p_not_Null p_Leaf have "repNodes_eq nodeslist p low high rep"
      by (simp add: repNodes_eq_def isLeaf_pt_def null_comp_def)
    with ns' var_eq
    show ?thesis
      by simp
  qed
next
  (* From invariant to postcondition *)
  fix var::"ref→nat" and low high rep repa p prx sfx "next"
  assume sfx: "List Null next sfx"
  assume p_in_ns: "p ∈ set (prx @ sfx)"
  assume no_props: "∀no∈set (prx @ sfx).
           no ≠ Null ∧
           (low no = Null) = (high no = Null) ∧
           (isLeaf_pt p low high ⟶ isLeaf_pt no low high) ∧ var no = var p"
  assume match_prx: "(∃pt∈set prx. repNodes_eq pt p low high rep) ⟶
                       repa p = hd [sn←prx . repNodes_eq sn p low high rep] ∧
                      (∀pt. pt ≠ p ⟶ rep pt = repa pt)"
  show "repa p = hd [sn←prx @ sfx . repNodes_eq sn p low high rep] ∧
          (∀pt. pt ≠ p ⟶ rep pt = repa pt) ∧ var (repa p) = var p"
  proof -
    from sfx
    have sfx_Nil: "sfx=[]"
      by simp
    with p_in_ns have ex_match: "(∃pt∈set prx. repNodes_eq pt p low high rep)"
      apply -
      apply (rule_tac x=p in bexI)
      apply  (simp add: repNodes_eq_def)
      apply simp
      done
    hence not_empty: "[sn←prx . repNodes_eq sn p low high rep] ≠ []"
      apply -
      apply (erule bexE)
      apply (rule filter_not_empty)
      apply auto
      done
    from ex_match match_prx obtain
      found: "repa p = hd [sn←prx . repNodes_eq sn p low high rep]" and
      unmodif: "∀pt. pt ≠ p ⟶ rep pt = repa pt"
      by blast
    from hd_filter_in_list [OF not_empty] found
    have "repa p ∈ set prx"
      by simp
    with no_props
    have "var (repa p) = var p"
      using [[simp_depth_limit=2]]
      by simp
    with found unmodif sfx_Nil
    show ?thesis
      by simp
  qed
next
  (* Invariant to invariant; ELSE part *)
  fix var low high p repa "next" nodeslist prx sfx
  assume nodeslist_not_Null: "nodeslist ≠ Null" 
  assume p_no_Leaf: "¬ isLeaf_pt p low high"
  assume no_props: "∀no∈set prx ∪ set (nodeslist # sfx).
           no ≠ Null ∧ (low no = Null) = (high no = Null) ∧ var no = var p"
  assume p_in_ns: "p ∈ set prx ∨ p ∈ set (nodeslist # sfx)"
  assume match_prx: "(∃pt∈set prx. repNodes_eq pt p low high repa) ⟶
            repa p = hd [sn←prx . repNodes_eq sn p low high repa]"
  assume nomatch_prx: "∀pt∈set prx. ¬ repNodes_eq pt p low high repa"
  assume nomatch_nodeslist: "¬ repNodes_eq nodeslist p low high repa"
  assume sfx: "List (next nodeslist) next sfx"
  show "(∀no∈set prx ∪ set (nodeslist # sfx).
              no ≠ Null ∧ (low no = Null) = (high no = Null) ∧ var no = var p) ∧
        ((∃pt∈set (prx @ [nodeslist]). repNodes_eq pt p low high repa) ⟶
           repa p = hd [sn←prx @ [nodeslist] . repNodes_eq sn p low high repa]) ∧
        (next nodeslist ≠ Null ⟶
            (∀pt∈set (prx @ [nodeslist]). ¬ repNodes_eq pt p low high repa))"
  proof -
    from nomatch_prx nomatch_nodeslist
    have "((∃pt∈set (prx @ [nodeslist]). repNodes_eq pt p low high repa) ⟶
           repa p = hd [sn←prx @ [nodeslist] . repNodes_eq sn p low high repa])" 
      by auto
    moreover
    from nomatch_prx nomatch_nodeslist
    have "(next nodeslist ≠ Null ⟶
            (∀pt∈set (prx @ [nodeslist]). ¬ repNodes_eq pt p low high repa))"
      by auto
    ultimately show ?thesis
      using no_props
      by (intro conjI)
  qed
next
  (* Invariant to invariant: THEN part *)
  fix var low high p repa "next" nodeslist prx sfx
  assume nodeslist_not_Null: "nodeslist ≠ Null" 
  assume sfx: "List nodeslist next sfx" 
  assume p_not_Leaf: "¬ isLeaf_pt p low high"
  assume no_props: "∀no∈set prx ∪ set sfx.
           no ≠ Null ∧
           (low no = Null) = (high no = Null) ∧
           (isLeaf_pt p low high ⟶ isLeaf_pt no low high) ∧ var no = var p"
  assume p_in_ns: "p ∈ set prx ∨ p ∈ set sfx"
  assume match_prx: "(∃pt∈set prx. repNodes_eq pt p low high repa) ⟶
        repa p = hd [sn←prx . repNodes_eq sn p low high repa]"
  assume nomatch_prx: "∀pt∈set prx. ¬ repNodes_eq pt p low high repa"
  assume match: "repNodes_eq nodeslist p low high repa"
  show "(∀no∈set prx ∪ set sfx.
              no ≠ Null ∧
              (low no = Null) = (high no = Null) ∧
              (isLeaf_pt p low high ⟶ isLeaf_pt no low high) ∧ var no = var p) ∧
        (p ∈ set prx ∨ p ∈ set sfx) ∧
        ((∃pt∈set prx ∪ set sfx. repNodes_eq pt p low high repa) ⟶
           nodeslist =
           hd ([sn←prx . repNodes_eq sn p low high repa] @
               [sn←sfx . repNodes_eq sn p low high repa])) ∧
        ((∀pt∈set prx ∪ set sfx. ¬ repNodes_eq pt p low high repa) ⟶
           repa = repa(p := nodeslist))"
  proof -
    from nodeslist_not_Null sfx
    obtain sfx' where sfx': "sfx=nodeslist#sfx'"
      by (cases "nodeslist=Null") auto
    from nomatch_prx match sfx'
    have hd: "hd ([sn←prx . repNodes_eq sn p low high repa] @
               [sn←sfx . repNodes_eq sn p low high repa]) = nodeslist"
      by simp
    from match sfx'
    have triv: "((∀pt∈set prx ∪ set sfx. ¬ repNodes_eq pt p low high repa) ⟶
           repa = repa(p := nodeslist))" 
      by simp
    show ?thesis
      apply (rule conjI)
      apply (rule no_props)
      apply (intro conjI)
      apply   (rule p_in_ns)
      apply  (simp add: hd)
      apply (rule triv)
      done
  qed
qed

end