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# Week 8 bonus 1

Week 8 bonus 1

## Resources

### Definitions File

```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```

### Template File

```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

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```

### Check File

```theory Check
imports Submission
begin