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# Homework 02

This is the task corresponding to homework 2.

## Resources

Download Files

### Definitions File

```theory Defs
imports "HOL-IMP.AExp" "HOL-IMP.BExp"
begin

datatype aexp = N int | V vname | Plus aexp aexp | Mult int aexp

fun aval :: "aexp ⇒ state ⇒ val" where
"aval (N n) s = n" |
"aval (V x) s = s x" |
"aval (Plus a⇩1 a⇩2) s = aval a⇩1 s + aval a⇩2 s" |
"aval (Mult i a) s = i * aval a s"

end```

### Template File

```theory Submission
imports Defs
begin

fun rlenc :: "'a ⇒ nat ⇒ 'a list ⇒ ('a × nat) list" where
"rlenc _ = undefined"

value "replicate (3::nat) (1::nat) = [1,1,1]"

theorem test1:
‹rlenc 0 0 ([1,3,3,8] :: int list) = [(0,0),(1,1),(3,2),(8,1)]›
by eval
theorem test2:
‹rlenc 1 0 ([3,4,5] :: int list) = [(1,0),(3,1),(4,1),(5,1)]›
by eval

fun rldec :: "('a × nat) list ⇒ 'a list" where
"rldec _ = undefined"

theorem enc_dec: "rldec (rlenc a 0 l) = l"
sorry

lemmas [simp] = algebra_simps

fun normal :: "aexp ⇒ bool" where
"normal _ = undefined"

fun normalize :: "aexp ⇒ aexp" where
"normalize _ = undefined"

theorem semantics_unchanged: "aval (normalize a) s = aval a s"
sorry

theorem normalize_normalizes: "normal (normalize a)"
sorry

end```

### Check File

```theory Check
imports Submission
begin

theorem test1:
‹rlenc 0 0 ([1,3,3,8] :: int list) = [(0,0),(1,1),(3,2),(8,1)]›
by (rule Submission.test1)

theorem test2:
‹rlenc 1 0 ([3,4,5] :: int list) = [(1,0),(3,1),(4,1),(5,1)]›
by (rule Submission.test2)

theorem enc_dec: "rldec (rlenc a 0 l) = l"
by (rule Submission.enc_dec)

theorem semantics_unchanged: "aval (normalize a) s = aval a s"
by (rule Submission.semantics_unchanged)

theorem normalize_normalizes: "normal (normalize a)"
by (rule Submission.normalize_normalizes)

end```

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