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### Definitions File

### Template File

### Check 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

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" by (simp add: bury_bury) end

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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