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theory Defs imports "HOL-IMP.Def_Init_Small" begin hide_const D consts AV :: "com \<Rightarrow> vname set" consts D :: "vname set \<Rightarrow> com \<Rightarrow> bool" end
theory Submission
imports Defs
begin
fun AV :: "com \<Rightarrow> vname set" where
"AV _ = undefined"
fun D :: "vname set \<Rightarrow> com \<Rightarrow> bool" where
"D _ = undefined"
theorem %invisible D_progress:
assumes "c \<noteq> SKIP"
shows "D (dom s) c \<Longrightarrow> \<exists> cs'. (c,s) \<rightarrow> cs'"
using assms
proof (induction c arbitrary: s)
case Assign thus ?case by auto (metis aval_Some small_step.Assign)
next
case (If b c1 c2)
then obtain bv where "bval b s = Some bv" by (auto dest!: bval_Some)
then show ?case
by(cases bv) (auto intro: small_step.IfTrue small_step.IfFalse)
qed (fastforce intro: small_step.intros)+
lemma %invisible D_incr: "(c,s) \<rightarrow> (c',s') \<Longrightarrow> dom s \<union> AV c \<subseteq> dom s' \<union> AV c'"
by (induction rule: small_step_induct) auto
lemma D_mono: "A \<subseteq> A' \<Longrightarrow> D A c \<Longrightarrow> D A' c"
sorry
theorem D_preservation: "(c,s) \<rightarrow> (c',s') \<Longrightarrow> D (dom s) c \<Longrightarrow> D (dom s') c'"
sorry
theorem D_sound: "(c,s) \<rightarrow>* (c',s') \<Longrightarrow> c' \<noteq> SKIP \<Longrightarrow> D (dom s) c \<Longrightarrow> \<exists>cs''. (c',s') \<rightarrow> cs''"
sorry
end
theory Check imports Submission begin lemma D_mono: "A \<subseteq> A' \<Longrightarrow> D A c \<Longrightarrow> D A' c" by (rule Submission.D_mono) theorem D_preservation: "(c,s) \<rightarrow> (c',s') \<Longrightarrow> D (dom s) c \<Longrightarrow> D (dom s') c'" by (rule Submission.D_preservation) theorem D_sound: "(c,s) \<rightarrow>* (c',s') \<Longrightarrow> c' \<noteq> SKIP \<Longrightarrow> D (dom s) c \<Longrightarrow> \<exists>cs''. (c',s') \<rightarrow> cs''" by (rule Submission.D_sound) end