Our site saves small pieces of text information (cookies) on your device in order to deliver better content and for statistical purposes. You can disable the usage of cookies by changing the settings of your browser. By browsing our website without changing the browser settings you grant us permission to store that information on your device.

# Homework 7_1

This is the task corresponding to homework 7_1.

## Resources

### Definitions File

```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```

### Template File

```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```

### Check File

```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```

Terms and Conditions