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# Homework 08.1

This is the task corresponding to the first part of homework 8.

## Resources

### Definitions File

```theory Defs
imports "HOL-IMP.Small_Step" "HOL-IMP.Live_True"
begin

fun bury :: "com ⇒ vname set ⇒ com" where
"bury SKIP X = SKIP" |
"bury (x ::= a) X = (if x ∈ X then x ::= a else SKIP)" |
"bury (c⇩1;; c⇩2) X = (bury c⇩1 (L c⇩2 X);; bury c⇩2 X)" |
"bury (IF b THEN c⇩1 ELSE c⇩2) X = IF b THEN bury c⇩1 X ELSE bury c⇩2 X" |
"bury (WHILE b DO c) X = WHILE b DO bury c (L (WHILE b DO c) X)"

end```

### Template File

```theory Submission
imports Defs
begin

theorem lfp_eq: "⟦ mono f; mono g; lfp f ⊆ U; lfp g ⊆ U;
⋀X. X ⊆ U ⟹ f X = g X ⟧ ⟹ lfp f = lfp g"
sorry

lemmas [simp] = L.simps(5)
lemmas L_mono2 = L_mono[unfolded mono_def]

theorem L_bury[simp]: "X ⊆ Y ⟹ L (bury c Y) X = L c X"
proof(induction c arbitrary: X Y)
case SKIP
then show ?case sorry
next
case (Assign x1 x2)
then show ?case sorry
next
case (Seq c1 c2)
then show ?case sorry
next
case (If x1 c1 c2)
then show ?case sorry
next
case (While x1 c)
then show ?case sorry
qed

theorem bury_bury: "X ⊆ Y ⟹ bury (bury c Y) X = bury c X"
sorry

corollary "bury (bury c X) X = bury c X"

end```

### Check File

```theory Check
imports Submission
begin

theorem lfp_eq: "⟦ mono f; mono g; lfp f ⊆ U; lfp g ⊆ U;
⋀X. X ⊆ U ⟹ f X = g X ⟧ ⟹ lfp f = lfp g"
by (rule Submission.lfp_eq)

theorem L_bury: "X ⊆ Y ⟹ L (bury c Y) X = L c X"
by (rule Submission.L_bury)

theorem bury_bury: "X ⊆ Y ⟹ bury (bury c Y) X = bury c X"
by (rule Submission.bury_bury)

end```

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