OCaml :

Sorted lists

Consider the functions for ordered lists from the ordered list.ml le in the code-examples direc-

tory of the public class repository:

let rec place e l = match l with

| [ ] -> [e]

| x::xs -> if e < x then e::x::xs

else x :: (place e xs)

let rec is_elem e l = match l with

| [ ] -> false

| x::xs -> e = x || (e > x && is_elem e xs)

let rec sorted l = match l with

| [ ] -> true

| x::[] -> true

| x1::x2::xs -> x1 <= x2 && sorted (x2::xs)

Using induction, show that

sorted l => sorted (place e l)

Your proof must explicitly and clearly indicate the base case you prove, the inductive case you prove and what the inductive hypothesis provides in the proof.

Each step in your proof must be accompanied by a justication describing why that step could be taken.