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# Homework 1

This is the task corresponding to homework 1.

## Resources

### Definitions File

```theory Defs
imports Main
begin

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 listsum :: "int list \<Rightarrow> int"

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

end```

### Template File

```theory Submission
imports Defs
begin

fun listsum :: "int list \<Rightarrow> int"  where
"listsum _ = undefined"

value "listsum [1,2,3] = 6"
value "listsum [] = 0"
value "listsum [1,-2,3] = 2"

lemma listsum_filter_z: "listsum (filter (\<lambda>x. x\<noteq>0) l) = listsum l"
sorry

lemma listsum_rev: "listsum (rev xs) = listsum xs"
sorry

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

value "flatten [[1,2,3],[2]] = [1,2,3,2::int]"
value "flatten [[1,2,3],[],[2]] = [1,2,3,2::int]"

lemma sum_flatten: "listsum (flatten xs) = listsum (map listsum xs)"
sorry

end```

### Check File

```theory Check
imports Submission
begin

lemma listsum_filter_z: "listsum (filter (\<lambda>x. x\<noteq>0) l) = listsum l"
by (rule Submission.listsum_filter_z)

lemma listsum_rev: "listsum (rev xs) = listsum xs"
by (rule Submission.listsum_rev)

lemma sum_flatten: "listsum (flatten xs) = listsum (map listsum xs)"
by (rule Submission.sum_flatten)

end```

Terms and Conditions