Theory Counterc (Isabelle2005: October 2005)

# Theory Counterc

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theory Counterc
imports UNITY_Main
begin

    (*  Title:      HOL/UNITY/Counterc
        ID:         $Id: Counterc.thy,v 1.10 2005/06/17 14:13:10 haftmann Exp $
        Author:     Sidi O Ehmety, Cambridge University Computer Laboratory
        Copyright   2001  University of Cambridge

    A family of similar counters, version with a full use of "compatibility "

    From Charpentier and Chandy,
    Examples of Program Composition Illustrating the Use of Universal Properties
       In J. Rolim (editor), Parallel and Distributed Processing,
       Spriner LNCS 1586 (1999), pages 1215-1227.
    *)

    header{*A Family of Similar Counters: Version with Compatibility*}

    theory Counterc imports  UNITY_Main begin

    typedecl state
    arities state :: type

    consts
      C :: "state=>int"
      c :: "state=>nat=>int"

    consts
      sum  :: "[nat,state]=>int"
      sumj :: "[nat, nat, state]=>int"

    primrec (* sum I s = sigma_{i<I}. c s i *)
      "sum 0 s = 0"
      "sum (Suc i) s = (c s) i + sum i s"

    primrec
      "sumj 0 i s = 0"
      "sumj (Suc n) i s = (if n=i then sum n s else (c s) n + sumj n i s)"

    types command = "(state*state)set"

    constdefs
      a :: "nat=>command"
     "a i == {(s, s'). (c s') i = (c s) i + 1 & (C s') = (C s) + 1}"

      Component :: "nat => state program"
      "Component i == mk_total_program({s. C s = 0 & (c s) i = 0},
                                       {a i},
                                       \<Union>G ∈ preserves (%s. (c s) i). Acts G)"

    declare Component_def [THEN def_prg_Init, simp]
    declare Component_def [THEN def_prg_AllowedActs, simp]
    declare a_def [THEN def_act_simp, simp]

    (* Theorems about sum and sumj *)
    lemma sum_sumj_eq1 [rule_format]: "∀i. I<i--> (sum I s = sumj I i s)"
    by (induct_tac "I", auto)

    lemma sum_sumj_eq2 [rule_format]: "i<I --> sum I s  = c s i + sumj I i s"
    apply (induct_tac "I")
    apply (auto simp add: linorder_neq_iff sum_sumj_eq1)
    done

    lemma sum_ext [rule_format]:
         "(∀i. i<I --> c s' i = c s i) --> (sum I s' = sum I s)"
    by (induct_tac "I", auto)

    lemma sumj_ext [rule_format]:
         "(∀j. j<I & j≠i --> c s' j =  c s j) --> (sumj I i s' = sumj I i s)"
    apply (induct_tac "I", safe)
    apply (auto intro!: sum_ext)
    done

    lemma sum0 [rule_format]: "(∀i. i<I --> c s i = 0) -->  sum I s = 0"
    by (induct_tac "I", auto)

    (* Safety properties for Components *)

    lemma Component_ok_iff:
         "(Component i ok G) =
          (G ∈ preserves (%s. c s i) & Component i ∈ Allowed G)"
    apply (auto simp add: ok_iff_Allowed Component_def [THEN def_total_prg_Allowed])
    done
    declare Component_ok_iff [iff]
    declare OK_iff_ok [iff]
    declare preserves_def [simp]

    lemma p2: "Component i ∈ stable {s. C s = (c s) i + k}"
    by (simp add: Component_def, safety)

    lemma p3:
         "[| OK I Component; i∈I |]
          ==> Component i ∈ stable {s. ∀j∈I. j≠i --> c s j = c k j}"
    apply simp
    apply (unfold Component_def mk_total_program_def)
    apply (simp (no_asm_use) add: stable_def constrains_def)
    apply blast
    done

    lemma p2_p3_lemma1:
         "[| OK {i. i<I} Component; i<I |] ==>
          ∀k. Component i ∈ stable ({s. C s = c s i + sumj I i k} Int
                                    {s. ∀j∈{i. i<I}. j≠i --> c s j = c k j})"
    by (blast intro: stable_Int [OF p2 p3])

    lemma p2_p3_lemma2:
         "(∀k. F ∈ stable ({s. C s = (c s) i + sumj I i k} Int
                            {s. ∀j∈{i. i<I}. j≠i --> c s j = c k j}))
          ==> (F ∈ stable {s. C s = c s i + sumj I i s})"
    apply (simp add: constrains_def stable_def)
    apply (force intro!: sumj_ext)
    done

    lemma p2_p3:
         "[| OK {i. i<I} Component; i<I |]
          ==> Component i ∈ stable {s. C s = c s i + sumj I i s}"
    by (blast intro: p2_p3_lemma1 [THEN p2_p3_lemma2])

    (* Compositional correctness *)
    lemma safety:
         "[| 0<I; OK {i. i<I} Component |]
          ==> (\<Squnion>i∈{i. i<I}. (Component i)) ∈ invariant {s. C s = sum I s}"
    apply (simp (no_asm) add: invariant_def JN_stable sum_sumj_eq2)
    apply (auto intro!: sum0 p2_p3)
    done

    end

lemma sum_sumj_eq1:

      I < i ==> sum I s = sumj I i s

lemma sum_sumj_eq2:

      i < I ==> sum I s = c s i + sumj I i s

lemma sum_ext:

      (!!i. i < I ==> c s' i = c s i) ==> sum I s' = sum I s

lemma sumj_ext:

      (!!j. j < I ∧ j ≠ i ==> c s' j = c s j) ==> sumj I i s' = sumj I i s

lemma sum0:

      (!!i. i < I ==> c s i = 0) ==> sum I s = 0

lemma Component_ok_iff:

      Component i ok G = (G ∈ preserves (%s. c s i) ∧ Component i ∈ Allowed G)

lemma p2:

      Component i ∈ stable {s. C s = c s i + k}

lemma p3:

      [| OK I Component; i ∈ I |]
      ==> Component i ∈ stable {s. ∀j∈I. j ≠ i --> c s j = c k j}

lemma p2_p3_lemma1:

      [| OK {i. i < I} Component; i < I |]
      ==> ∀k. Component i
              ∈ stable
                 ({s. C s = c s i + sumj I i k} ∩
                  {s. ∀j∈{i. i < I}. j ≠ i --> c s j = c k j})

lemma p2_p3_lemma2:

      ∀k. F ∈ stable
               ({s. C s = c s i + sumj I i k} ∩
                {s. ∀j∈{i. i < I}. j ≠ i --> c s j = c k j})
      ==> F ∈ stable {s. C s = c s i + sumj I i s}

lemma p2_p3:

      [| OK {i. i < I} Component; i < I |]
      ==> Component i ∈ stable {s. C s = c s i + sumj I i s}

lemma safety:

      [| 0 < I; OK {i. i < I} Component |]
      ==> JOIN {i. i < I} Component ∈ invariant {s. C s = sum I s}