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theory Defs imports "HOL-IMP.Small_Step" "HOL-IMP.Live_True" begin fun bury :: "com \<Rightarrow> vname set \<Rightarrow> com" where "bury SKIP X = SKIP" | "bury (x ::= a) X = (if x \<in> X then x ::= a else SKIP)" | "bury (c\<^sub>1;; c\<^sub>2) X = (bury c\<^sub>1 (L c\<^sub>2 X);; bury c\<^sub>2 X)" | "bury (IF b THEN c\<^sub>1 ELSE c\<^sub>2) X = IF b THEN bury c\<^sub>1 X ELSE bury c\<^sub>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: "\<lbrakk> mono f; mono g; lfp f \<subseteq> U; lfp g \<subseteq> U; \<And>X. X \<subseteq> U \<Longrightarrow> f X = g X \<rbrakk> \<Longrightarrow> lfp f = lfp g" sorry lemmas [simp] = L.simps(5) lemmas L_mono2 = L_mono[unfolded mono_def] theorem L_bury[simp]: "X \<subseteq> Y \<Longrightarrow> L (bury c Y) X = L c X" proof(induction c arbitrary: X Y) sorry qed theorem bury_bury: "X \<subseteq> Y \<Longrightarrow> bury (bury c Y) X = bury c X" sorry corollary "bury (bury c X) X = bury c X" end
theory Check imports Submission begin theorem lfp_eq: "\<lbrakk> mono f; mono g; lfp f \<subseteq> U; lfp g \<subseteq> U; \<And>X. X \<subseteq> U \<Longrightarrow> f X = g X \<rbrakk> \<Longrightarrow> lfp f = lfp g" by (rule Submission.lfp_eq) theorem L_bury: "X \<subseteq> Y \<Longrightarrow> L (bury c Y) X = L c X" by (rule Submission.L_bury) theorem bury_bury: "X \<subseteq> Y \<Longrightarrow> bury (bury c Y) X = bury c X" by (rule Submission.bury_bury) end