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theory Defs
imports Main
begin
type_synonym intervals = "(nat*nat) list"
fun inv' :: "nat \<Rightarrow> intervals \<Rightarrow> bool" where
"inv' x [] \<longleftrightarrow> True"
| "inv' x ((a,b)#is) \<longleftrightarrow> (x\<le>a \<and> a\<le>b \<and> inv' (Suc (Suc b)) is)"
definition inv where "inv \<equiv> inv' 0"
fun set_of :: "intervals => nat set"
where
"set_of [] = {}"
| "set_of ((a,b)#is) = {a..b} \<union> set_of is"
end
theory Submission
imports Defs
begin
lemma inv'_mono: "inv' n is \<Longrightarrow> m\<le>n \<Longrightarrow> inv' m is"
by (induction m "is" rule: inv'.induct) auto
fun addi :: "nat \<Rightarrow> nat \<Rightarrow> intervals \<Rightarrow> intervals" where
"addi _ = undefined"
lemma addi_correct:
assumes "inv is" "i\<le>j"
shows "inv (addi i j is)" "set_of (addi i j is) = {i..j} \<union> (set_of is)"
sorry
end
theory Check
imports Submission
begin
lemma addi_correct:
assumes "inv is" "i\<le>j"
shows "inv (addi i j is)" "set_of (addi i j is) = {i..j} \<union> (set_of is)"
by (rule addi_correct[OF assms])+
end