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# Homework 1

This is the task corresponding to homework 1.

## Resources

### Definitions File

```theory Defs
imports Main
begin

text \<open>Definitions and lemmas from the tutorial\<close>

fun snoc :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list" where
"snoc [] x = [x]" |
"snoc (y # ys) x = y # (snoc ys x)"

fun reverse :: "'a list \<Rightarrow> 'a list" where
"reverse [] = []" |
"reverse (x # xs) = snoc (reverse xs) x"

lemma reverse_snoc: "reverse (snoc xs y) = y # reverse xs"
by (induct xs) auto

theorem reverse_reverse: "reverse (reverse xs) = xs"
by (induct xs) (auto simp add: reverse_snoc)

end```

### Template File

```theory Submission
imports Defs
begin

fun repeat :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
"repeat _ = undefined"

value "repeat 5 (0::nat) = [0, 0, 0, 0, 0]"
value "repeat 3 (1::nat) = [1, 1, 1]"

theorem rep_len: "length (repeat n a) = n"
sorry

theorem rep_rev: "reverse (repeat n a) = repeat n a"
sorry

end```

### Check File

```theory Check
imports Submission
begin

theorem rep_len: "length (repeat n a) = n"
by (rule Submission.rep_len)

theorem rep_rev: "reverse (repeat n a) = repeat n a"
by (rule Submission.rep_rev)

end```

### Definitions File

```theory Defs
imports Main
begin

declare [[names_short]]

fun snoc :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list" where
"snoc [] x = [x]" |
"snoc (y # ys) x = y # (snoc ys x)"

fun reverse :: "'a list \<Rightarrow> 'a list" where
"reverse [] = []" |
"reverse (x # xs) = snoc (reverse xs) x"

consts double :: "'a list \<Rightarrow> 'a list"

end```

### Template File

```theory Submission
imports Defs
begin

fun double :: "'a list \<Rightarrow> 'a list"  where
"double _ = undefined"

value "double [1,2,(3::nat)] = [1,1,2,2,3,3]"

theorem double_len: "length (double xs) = 2 * length xs"
sorry

theorem reverse_double: "reverse (double xs) = double (reverse xs)"
sorry

theorem rev_double: "rev (double xs) = double (rev xs)"
sorry

end```

### Check File

```theory Check
imports Submission
begin

theorem double_len: "length (double xs) = 2 * length xs"
by (rule Submission.double_len)

theorem reverse_double: "reverse (double xs) = double (reverse xs)"
by (rule Submission.reverse_double)

theorem rev_double: "rev (double xs) = double (rev xs)"
by (rule Submission.rev_double)

end```

Terms and Conditions