epicours/Algo/Courses/Chapter 7 - High Order (exercises).md

Ex 1.2

# let sum n = 
	if n < 0 then
		invalid_arg "n<0"
	else
		let rec sumrc n  =
			if n=0 then
				0
			else
				n + sum(n-1)
		in sumrc n;;
val sum : int -> int = <fun>

(*Correction*)
let sigma f n = 
	if n<0 then
		invalid_arg "n<0"
	else
		let rec sig = function
			| 0 -> f0
			| n -> fn + sig (n-1)
		in sig n;;
val sigma: (int -> int) -> int -> int = <fun>

Ex 2.1

# let rec map f = function
	| [] -> []
	|e::t -> f e::map f t ;;
val map : ('a -> 'b) -> 'a list -> 'b list = <fun>

Ex 2.2

# let rec for_all p = function
	| [] -> true
	| e::t -> p e && for_all p t;;
val for_all: ('a -> bool) -> 'a list -> bool = <fun>

Ex 2.3

# let rec exists p = function
	| [] -> true
	| e::t -> p e || for_all p t;;
val exists: ('a -> bool) -> 'a list -> bool = <fun>

Ex 2.4

# let rec find p = function
	| [] -> true
	| e::t -> if p e then e else find p t;;
val exists: (bool -> bool) -> bool list -> bool = <fun>

Ex 2.5

# let rec filter p : function
	| [] -> []
	| e::t -> if p e then
		e::filter p t
		else filter p t;;
val filter: ('a -> bool) -> 'a list -> 'a 

Ex 2.6

let rec partition p l =
	match l with 
		| [] -> ([], [])
		| h==t> let (l1, l2) = partition p t in
			if p h then (h::l1,l2)
			else (l1, h::l2);;
val partition : ('a -> bool) -> 'a list -> 'a list * 'a list = <fun>

Ex 2.7

# let less2 p k l1 l2 =
	if k < 0 then invalid_arg "k needs to be positive" else
		let rec less p k l1 l2 acc = match (l1, l2) with 
			| ([],[]) -> acc < k
			| (e::t, e::t2) -> if p e e2 then less p k t t2 (acc + 1)
				else less p k t t2 acc
		in less p k l1 l2 0;;
val less2 : ('a -> 'b -> bool) -> int -> 'a list -> 'b list -> bool = <fun>

Ex 3.1