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

This is the task corresponding to homework 7.

## Resources

### Definitions File

```theory Defs
imports "HOL-IMP.Sec_Type_Expr" "HOL-IMP.Def_Init_Small"
begin

inductive sec_type :: "nat \<Rightarrow> com \<Rightarrow> bool" ("(_/ \<turnstile> _)" [0,0] 50) where
Skip: "l \<turnstile> SKIP" |
Assign: "\<lbrakk> sec x \<ge> sec a;  sec x \<ge> l \<rbrakk> \<Longrightarrow> l \<turnstile> x ::= a" |
Seq: "\<lbrakk> l \<turnstile> c\<^sub>1;  l \<turnstile> c\<^sub>2 \<rbrakk> \<Longrightarrow> l \<turnstile> c\<^sub>1;;c\<^sub>2" |
If: "\<lbrakk> max (sec b) l \<turnstile> c\<^sub>1;  max (sec b) l \<turnstile> c\<^sub>2 \<rbrakk> \<Longrightarrow> l \<turnstile> IF b THEN c\<^sub>1 ELSE c\<^sub>2" |
While: "max (sec b) l \<turnstile> c \<Longrightarrow> l \<turnstile> WHILE b DO c"

inductive_cases [elim!]: "l \<turnstile> x ::= a"  "l \<turnstile> c\<^sub>1;;c\<^sub>2"  "l \<turnstile> IF b THEN c\<^sub>1 ELSE c\<^sub>2"  "l \<turnstile> WHILE b DO c"

lemma anti_mono: "\<lbrakk> l \<turnstile> c;  l' \<le> l \<rbrakk> \<Longrightarrow> l' \<turnstile> c"
apply(induction arbitrary: l' rule: sec_type.induct)
apply (metis sec_type.intros(1))
apply (metis le_trans sec_type.intros(2))
apply (metis sec_type.intros(3))
apply (metis If le_refl sup_mono sup_nat_def)
apply (metis While le_refl sup_mono sup_nat_def)
done

inductive sec_type2 :: "com \<Rightarrow> level \<Rightarrow> bool" ("(\<turnstile> _ : _)" [0,0] 50) where
Skip2: "\<turnstile> SKIP : l" |
Assign2: "sec x \<ge> sec a \<Longrightarrow> \<turnstile> x ::= a : sec x" |
Seq2: "\<lbrakk> \<turnstile> c\<^sub>1 : l\<^sub>1;  \<turnstile> c\<^sub>2 : l\<^sub>2 \<rbrakk> \<Longrightarrow> \<turnstile> c\<^sub>1;;c\<^sub>2 : min l\<^sub>1 l\<^sub>2 " |
If2: "\<lbrakk> sec b \<le> min l\<^sub>1 l\<^sub>2;  \<turnstile> c\<^sub>1 : l\<^sub>1;  \<turnstile> c\<^sub>2 : l\<^sub>2 \<rbrakk> \<Longrightarrow> \<turnstile> IF b THEN c\<^sub>1 ELSE c\<^sub>2 : min l\<^sub>1 l\<^sub>2" |
While2:  "\<lbrakk> sec b \<le> l;  \<turnstile> c : l \<rbrakk> \<Longrightarrow> \<turnstile> WHILE b DO c : l"

end```

### Template File

```theory Submission
imports Defs
begin

theorem bottom_up_impl_top_down: "\<turnstile> c : l \<Longrightarrow> l \<turnstile> c"
sorry

theorem top_down_impl_bottom_up: "l \<turnstile> c \<Longrightarrow> \<exists> l' \<ge> l. \<turnstile> c : l'"
sorry

end```

### Check File

```theory Check
imports Submission
begin

theorem bottom_up_impl_top_down: "\<turnstile> c : l \<Longrightarrow> l \<turnstile> c"
by (rule Submission.bottom_up_impl_top_down)

theorem top_down_impl_bottom_up: "l \<turnstile> c \<Longrightarrow> \<exists> l' \<ge> l. \<turnstile> c : l'"
by (rule Submission.top_down_impl_bottom_up)

end```

Terms and Conditions