1. Tutorial 7

2. add_bin_list : bool list list -> bool list
Week 7 Exercise 1
Implement a function add_bin_list that adds a list of binary numbers. 
add_bin_list : bool list list -> bool list
example:
add_bin_list [ [true; true; false; true]; [false; false; true]; [true; false; true]] = [false; false; true; false; true; false; false]

3. Least significant bit (LSB) Most significant bit (MSB)
Week 7 Exercise 1
Implement a function add_bin_list that adds a list of binary numbers. 
add_bin_list : bool list list -> bool list
example:
add_bin_list [ [true; true; false; true]; [false; false; true]; [true; false; true]] = [false; false; true; false; true; false; false]
Least significant bit (LSB)
Most significant bit (MSB)

4. Least significant bit (LSB) Most significant bit (MSB)
Week 7 Exercise 1
1011
Implement a function add_bin_list that adds a list of binary numbers. 
add_bin_list : bool list list -> bool list
example:
add_bin_list [ [true; true; false; true]; [false; false; true]; [true; false; true]] = [false; false; true; false; true; false; false]
Least significant bit (LSB)
Most significant bit (MSB)

5. From Lecture let add_3 b1 b2 b3 = match b1, b2, b3 with
| true, true, true -> true, true
| true, true, false
| true, false, true
| false, true, true -> true, false
| true, false, false
| false, true, false
| false, false, true -> false, true
| false, false, false -> false, false;;
let rec add_bin' n1 n2 carry = match n1, n2 with
| [], [] -> [carry]
| [], n2 -> add_bin' [false] n2 carry
| n1, [] -> add_bin' n1 [false] carry
| d1 :: d1s, d2 :: d2s ->
let (carry, d) = add_3 d1 d2 carry in
d :: (add_bin' d1s d2s carry);;
let rec add_bin n1 n2 = add_bin' n1 n2 false;;

6. solution let add_bin_list = List.fold_left add_bin [false];;
val fold_left : ('a -> 'b -> 'a) -> 'a -> 'b list -> ’a
List.fold_left f a [b1; ...; bn] is f (... (f (f a b1) b2) ...) bn.

7. mul_bin_pow : bool list -> int -> bool list
Week 7 Exercise 2
Write a function mul_bin_pow n k which multiplies a binary number n by the number k-th power of 2. 
mul_bin_pow : bool list -> int -> bool list
For example: 
mul_bin_pow [true; true; false; true] 3 = [false; false; false; true; true; false; true]

8. solution Multiplying a number by the k-th power of its base
Shift the digits/bits k places to the left
Pad with zeros
Example: in decimal multiplying any number by 10
555 *10 =5550
The same with binary: multiplying any number by 2, shift bits to the left and add zero to the LSB
111 *(1000)=111000
2^3

9. solution let rec nfalse = function | 0 -> []
| n -> false :: (nfalse (n - 1));;
let mul_bin_pow n k = (nfalse n;;

10. mul_bin : bool list -> bool list -> bool list
Week 7 Problem
Implement the function mul_bin which multiplies two binary numbers. 
mul_bin : bool list -> bool list -> bool list
For example:
mul_bin [false; true; true] [true; false; true] = [false; true; true; true; true]

11. solution let rec bin_to_pows n = function | [] -> []
| true :: ns -> n :: (bin_to_pows (n + 1) ns)
| false :: ns -> (bin_to_pows (n + 1) ns);;
val bin_to_pows : int -> bool list -> int list = <fun>
let shift bs ps = List.map (fun k -> mul_bin_pow bs k) ps;;
val shift : bool list -> int list -> bool list list = <fun>
let rec mul_bin m n = add_bin_list (shift m (bin_to_pows 0 n));;
val mul_bin : bool list -> bool list -> bool list = <fun>