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# FDS Week 7 Homework

Week 7 homework.

## Resources

Download Files

### 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 Template
imports Defs
begin

fun del :: "nat \<Rightarrow> intervals \<Rightarrow> intervals" where
"del _ _ = undefined"

lemma del_correct_1:
"inv is \<Longrightarrow> inv (del x is)"
sorry

lemma del_correct_2:
"inv is \<Longrightarrow> set_of (del x is) = (set_of is) - {x}"
sorry

fun addi :: "nat \<Rightarrow> nat \<Rightarrow> intervals \<Rightarrow> intervals" where
"addi _ _ _ = undefined"

lemma addi_correct_1:
"inv is \<Longrightarrow> i\<le>j \<Longrightarrow> inv (addi i j is)"
sorry

lemma addi_correct_2:
"inv is \<Longrightarrow> i\<le>j \<Longrightarrow> set_of (addi i j is) = {i..j} \<union> (set_of is)"
sorry

end
```

### Check File

```theory Check
imports Template
begin

lemma del_correct_1:
"Defs.inv is \<Longrightarrow> Defs.inv (del x is)"
by(rule del_correct_1)

lemma del_correct_2:
"Defs.inv is \<Longrightarrow> Defs.set_of (del x is) = (Defs.set_of is) - {x}"
by(rule del_correct_2)

lemma addi_correct_1:
"Defs.inv is \<Longrightarrow> i\<le>j \<Longrightarrow> Defs.inv (addi i j is)"
by(rule addi_correct_1)

lemma addi_correct_2:
"Defs.inv is \<Longrightarrow> i\<le>j \<Longrightarrow> Defs.set_of (addi i j is) = {i..j} \<union> (Defs.set_of is)"
by(rule addi_correct_2)

end
```

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