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theory Imperative_Loops
imports
"Refine_Imperative_HOL.Sepref_HOL_Bindings"
"Refine_Imperative_HOL.Pf_Mono_Prover"
"Refine_Imperative_HOL.Pf_Add"
begin
section \<open>Imperative Loops\<close>
text "An auxiliary while rule provided by Peter Lammich."
lemma heap_WHILET_rule:
assumes
"wf R"
"P \<Longrightarrow>\<^sub>A I s"
"\<And>s. <I s * true> bi s <\<lambda>r. I s * \<up>(r \<longleftrightarrow> b s)>\<^sub>t"
"\<And>s. b s \<Longrightarrow> <I s * true> f s <\<lambda>s'. I s' * \<up>((s', s) \<in> R)>\<^sub>t"
"\<And>s. \<not> b s \<Longrightarrow> I s \<Longrightarrow>\<^sub>A Q s"
shows "<P * true> heap_WHILET bi f s <Q>\<^sub>t"
proof -
have "<I s * true> heap_WHILET bi f s <\<lambda>s'. I s' * \<up>(\<not> b s')>\<^sub>t"
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)
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_star_mono assms(2) ent_refl)
apply clarsimp
apply (intro ent_star_mono assms(5) ent_refl)
.
qed
lemma heap_WHILET_rule':
assumes
"wf R"
"P \<Longrightarrow>\<^sub>A I s si * F"
"\<And>si s. <I s si * F> bi si <\<lambda>r. I s si * F * \<up>(r \<longleftrightarrow> b s)>\<^sub>t"
"\<And>si s. b s \<Longrightarrow> <I s si * F> f si <\<lambda>si'. \<exists>\<^sub>As'. I s' si' * F * \<up>((s', s) \<in> R)>\<^sub>t"
"\<And>si s. \<not> b s \<Longrightarrow> I s si * F \<Longrightarrow>\<^sub>A Q s si"
shows "<P> heap_WHILET bi f si <\<lambda>si. \<exists>\<^sub>As. Q s si>\<^sub>t"
proof -
have "<I s si * F> heap_WHILET bi f si <\<lambda>si'. \<exists>\<^sub>As'. I s' si' * F * \<up>(\<not> b s')>\<^sub>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
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_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 \<Longrightarrow>\<^sub>A I s"
"\<And>s. <I s * true> bi s <\<lambda>r. I s * \<up>(r \<longleftrightarrow> b s)>\<^sub>t"
"\<And>s. b s \<Longrightarrow> <I s * true> f s <\<lambda>s'. I s' * \<up>((s', s) \<in> R)>\<^sub>t"
"\<And>s. \<not> b s \<Longrightarrow> I s \<Longrightarrow>\<^sub>A Q s"
shows "<P> heap_WHILET bi f s <Q>\<^sub>t"
supply R = heap_WHILET_rule'[of R P "\<lambda>s si. \<up>(s = si) * I s" s _ true bi b f "\<lambda>s si.\<up>(s = si) * Q s * true"]
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 <\<lambda>s'. I s' * \<up>(\<not> b s')>\<^sub>t"
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)
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 clarsimp
apply(intro ent_star_mono assms(5) ent_refl)
.
qed
*)
end
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