## 1.1 - Product

```Ocaml
# let rec product = function
	| [] -> 0
	| x::t -> x * product t;;
```
## 1.2 - Count
```Ocaml
# let rec count x n =
	if x = [] then
		0
	else
		let e::t = x in
		if e = x then
			1 + count t n
		else
			count t n;;
(* Correction *)
# let rec count x = function
	| [] -> 0
	| e::t -> (if x = e then 1 else 0) + count x t;;
val count = 'a -> 'a list -> int = <fun>
```

## 1.3 - Search
```Ocaml
# let rec search x n =
	if x = [] then
		0
	else
		let e::t = x in
		if e = x then
			true
		else
			search t n;;
(* Correction *)
# let search x = function
	| [] -> false
	| e::t -> x = e || search x t
	| _::t -> search x t ;;
val search 'a -> 'a list -> bool = <fun>
```

## 1.4 - $n^{th}$
```Ocaml
# let nth l n =
	if n <= 0 then
		invalid_arg "n <= 0"
	else
		let rec nthrec = function 
			| ([],_) -> failwith "list too short"
			| (h::_,1)-> h
			| (_::t, n) -> nthrec(t, n-1)
		in
			ntrec(l, n);;
val nth 'a list -> int -> 'a = <fun>

```

## 1.5 - Maximum
```Ocaml
# let rec max_value list = match list with 
	| [] -> failwith "La liste est vide" 
	| [x] -> x 
	| hd :: tl -> 
		let max_tail = max_value tl in 
		if hd > max_tail then hd else max_tail;;

(* v2 *)
# let max_value list = 
	if list = [] then
		failwith "la list est vide"
	else 
		let rec mv = match list with
			| [x] -> x 
			| hd :: tl -> 
				let max_tail = max_value tl in 
				if hd > max_tail then hd else max_tail
		in mv list;;
```

## 1.6 - Bonus second
```Ocaml
# let rec second_smallest list =
  match list with
  | [] | [_] -> failwith "La liste ne contient pas au moins deux éléments distincts"
  | [x; y] -> if x < y then y else x
  | hd1 :: hd2 :: tl ->
    let min1, min2 =
      if hd1 < hd2 then (hd1, hd2) else (hd2, hd1)
    in
    let rec find_second_smallest rest =
      match rest with
      | [] -> min2
      | hd :: tl ->
        if hd < min1 then find_second_smallest (min1 :: tl)
        else if hd < min2 && hd > min1 then find_second_smallest (hd :: min1 :: tl)
        else find_second_smallest (min2 :: tl)
    in
    find_second_smallest tl;;
```

## Exercise 2.1
```
let arith_list n a r = 
	if n < 0 then
		invalid_arg "n must be superior to 0"
	else if n = 0 then
		[]
	if n = 1 then
		a::[]
	else let rec arl =
		|