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### Definitions File

### Template File

### Check File

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

theory Submission imports Defs begin fun erase :: "level ⇒ com ⇒ com" where "erase _ _ = undefined" theorem erase_correct: "⟦ (c,s) ⇒ s'; (erase l c,t) ⇒ t'; 0 ⊢ c; s = t (< l) ⟧ ⟹ s' = t' (< l)" sorry text ‹ In the theorem above we assumed that both @{term"(c,s)"} and @{term "(erase l c,t)"} terminate. How about the following two properties: › lemma "⟦ (c,s) ⇒ s'; 0 ⊢ c; s = t (< l) ⟧ ⟹ ∃t'. (erase l c,t) ⇒ t' ∧ s' = t' (< l)" oops lemma "⟦ (erase l c,s) ⇒ s'; 0 ⊢ c; s = t (< l) ⟧ ⟹ ∃t'. (c,t) ⇒ t'" oops text ‹Give an informal justification or a counterexample for each property!› theorem well_initialized_commands: assumes "D A c B" assumes "s1 = s2 on A" assumes "(c,s1) ⇒ s1'" shows "∃s2'. (c,s2) ⇒ s2' ∧ s1'=s2' on B" sorry end

theory Check imports Submission begin theorem erase_correct: "⟦ (c,s) ⇒ s'; (erase l c,t) ⇒ t'; 0 ⊢ c; s = t (< l) ⟧ ⟹ s' = t' (< l)" by (rule Submission.erase_correct) theorem well_initialized_commands: assumes "D A c B" assumes "s1 = s2 on A" assumes "(c,s1) ⇒ s1'" shows "∃s2'. (c,s2) ⇒ s2' ∧ s1'=s2' on B" using assms by (rule Submission.well_initialized_commands) end

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