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# Homework 05.1

This is the task corresponding to the first part of homework 5.

## Resources

### Definitions File

```theory Defs
imports "HOL-IMP.Small_Step" "HOL-IMP.BExp" "HOL-IMP.Star"
begin

end```

### Template File

```theory Submission
imports Defs
begin

fun small :: "com * state ⇒ (com * state) option" where
"small _ = undefined"

theorem small_small_step_equiv: "(c,s) → (c',s') ⟷ small (c,s) = Some (c',s')"
sorry

fun smalls :: "nat ⇒ com * state ⇒ (com * state) option" where
"smalls _ _ = undefined"

theorem smalls_small_steps_equiv:
"(∃s'. (c,s) →* (c',s')) ⟷ (
if c' = SKIP then
(∃n. smalls n (c, s) = None)
else
(∃n s'. smalls n (c, s) = Some (c', s'))
)"
sorry

end```

### Check File

```theory Check
imports Submission
begin

theorem small_small_step_equiv: "(c,s) → (c',s') ⟷ small (c,s) = Some (c',s')"
by (rule Submission.small_small_step_equiv)

theorem smalls_small_steps_equiv:
"(∃s'. (c,s) →* (c',s')) ⟷ (
if c' = SKIP then
(∃n. smalls n (c, s) = None)
else
(∃n s'. smalls n (c, s) = Some (c', s'))
)"
by (rule Submission.smalls_small_steps_equiv)

end```

Terms and Conditions