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### Definitions File

### Template File

### Check File

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

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

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

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