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# Homework 7

This is the task corresponding to homework 7.

## Resources

### Definitions File

```theory Defs
imports "HOL-IMP.Def_Init" "HOL-IMP.Big_Step" "HOL-IMP.Sec_Typing"
begin

consts erase :: "level \<Rightarrow> com \<Rightarrow> com"

end```

### Template File

```theory Submission
imports Defs
begin

fun erase :: "level \<Rightarrow> com \<Rightarrow> com"  where
"erase _ = undefined"

theorem erase_correct:
"\<lbrakk> (c,s) \<Rightarrow> s'; (erase l c,t) \<Rightarrow> t';  0 \<turnstile> c;  s = t (< l) \<rbrakk>
\<Longrightarrow> s' = t' (< l)"
sorry

prop "\<lbrakk> (c,s) \<Rightarrow> s';  0 \<turnstile> c;  s = t (< l) \<rbrakk>
\<Longrightarrow> \<exists>t'. (erase l c,t) \<Rightarrow> t' \<and> s' = t' (< l)"
prop "\<lbrakk> (erase l c,s) \<Rightarrow> s';  0 \<turnstile> c;  s = t (< l) \<rbrakk> \<Longrightarrow> \<exists>t'. (c,t) \<Rightarrow> t'"
theorem well_initialized_commands:
assumes "D A c B"
and "s = s' on A"
and "(c,s) \<Rightarrow> t"
and "(c,s') \<Rightarrow> t'"
shows "t=t' on B"
sorry

end```

### Check File

```theory Check
imports Submission
begin

theorem erase_correct: "\<lbrakk> (c,s) \<Rightarrow> s'; (erase l c,t) \<Rightarrow> t';  0 \<turnstile> c;  s = t (< l) \<rbrakk>
\<Longrightarrow> s' = t' (< l)"
by (rule Submission.erase_correct)

theorem well_initialized_commands: "(D A c B) \<Longrightarrow> (s = s' on A) \<Longrightarrow> ((c,s) \<Rightarrow> t) \<Longrightarrow> ((c,s') \<Rightarrow> t') \<Longrightarrow> t=t' on B"
by (rule Submission.well_initialized_commands)

end```

Terms and Conditions