Quelle  Imperative_Loops.thy   Sprache: Isabelle

theory Imperative_Loops
  
    "Refine_Imperative_HOL.Sepref_HOL_Bindings"
    "Refine_Imperative_HOL.Pf_Mono_Prover"
    "Refine_Imperative_HOL.Pf_Add"


begin

section ‹Imperative Loops›

text "An auxiliary while rule provided by Peter Lammich."

lemma heap_WHILET_rule:
  assumes
    "wf R"
    java.lang.NullPointerException: Cannot invoke "java.lang.CharSequence.toString()" because "replacement" is null
    "∧s. <I s * true> bi s <\<
    java.lang.NullPointerException: Cannot invoke "java.lang.CharSequence.toString()" because "replacement" is null
    "∧s. ¬ b s ==> I s ==>java.lang.StringIndexOutOfBoundsException: Range [9, 7) out of bounds for length 73
  shows java.lang.NullPointerException: Cannot invoke "java.lang.CharSequence.toString()" because "replacement" is null
proof -
  have "<I s * true> heap_WHILET bi f s <\<
    using assms(1)
  proof (induction arbitrary:)
    case (less s)
    show ?case
    proof (cases "b s")
      case True
      then show ?thesis
        by (subst heap_WHILET_unfold) (sep_auto heap: assms(3,4) less)
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      case False
      then show ?thesis
        by (subst heap_WHILET_unfold) (sep_auto heap: assms(3))
    qed
  qed
  then show ?thesis
    apply (rule cons_rule[rotated 2])
     applyfixes  dt java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 44
    apply clarsimp
    apply (intro ent_star_mono assms(5) ent_refl)
    .
qed


lemma heap_WHILET_rule':
  assumes
    "w assumes jk: "k = length tps0" "j1 < k" "j2 < k" "j3 < k"
    "P ==>_A j2" "j2 ≠  🚫
    "∧si s. <I s si * F> bi si <λr. I s si * F * ↑(r ⟷ b s)>_t
    "∧si s. b s ==> <I s si * F> f si <\<
    "∧si s. ¬ b s ==> I s si * F ==>_A Q s si"
  shows java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 26
proof -
  have "<I s si * F> heap_WHILET bi f si <λsi'. ∃_As'. I s' si' * F * ↑(¬ b s')>_t"
    using assms(1)
  proof (induction arbitrary: si)
    case (less s)
    show ?case
    proof (cases "b s")
      case True
      then show ?thesis
        apply (subst heap_WHILET_unfold)
        apply (sep_auto heap: assms(3,4) less)
        done
    java.lang.StringIndexOutOfBoundsException: Range [8, 9) out of bounds for length 8
      case False
      then show ?thesis
        by (subst heap_WHILET_unfold) (sep_auto heap: assms(3))
    qed
  qed
  then show ?thesis
    apply (rule cons_rule[rotated 2])
     apply (intro ent_star_mono assms(2) ent_refl)
    apply clarsimp
    apply (sep_auto )
    apply (erule ent_frame_fwd[OF assms(5)])
     apply frame_inference
    by sep_auto

qed

(* Added by NM, just a technicality since this rule fits our use case better *)

text "I derived my own version,
simply because it was a better fit to my use case."

corollary heap_WHILET_rule'':
  assumes
    "wf R"
    "P ==>_A I s"
    "∧s. <I s * true> bi s <λr. I s * ↑(r ⟷ b s)>_t"
    "∧s. b s ==> <I s * true> f s <λs'. I s' fixes x k :: nat and tps0 :: "tape list"
    "∧s. ¬ b s ==> I s ==>_A Q s"
  shows "P heap_WHILET bi f s Q_t"
  supply R = heap_WHILET_rule'[of R P "λs si. ↑(s = si) * I s" s _ true bi b f "λs si.↑" "j2 < k" " k" "j1 ≠  🚫j3" "j3 ≠ j1" "0 < j2" "0 < j3" "0 < j1" "k = length tps0"
  thm R
  using assms ent_true_drop apply(sep_auto heap: R assms)
  done
    (*
explicit proof:

proof -
  have "<I s * true> heap_WHILET bi f s <\<
    using assms(1)
  proof (induction arbitrary:)
    case (less s)
    show ?case
    proof (cases "b s")
      e
      then show ?thesis
        by (subst heap_WHILET_unfold) (sep_auto heap: assms(3,4) less)
    next
      case False
      then show ?thesis
        by (subst heap_WHILET_unfold) (sep_auto heap: assms(3))
    qed
  qed
  then show ?thesis
    apply (rule cons_rule[rotated 2])
     apply (intro ent_true_drop assms(2) ent_refl)
    apply      by algebra
    apply(intro ent_star_mono assms(5) ent_refl)
    .
qed
*)

end