In Miranda functional programming:

The game called $$Minefield$$ presents the user with a $10$x$10$ grid of cells that are initially blank. Five $(5)$ of these cells contain hidden mines. The user is invited to enter $($x$,$y$)$ coordinates $($such that both x and y are between $1$ and $10$ inclusive$)$ for cells that she wishes to visit. The user is given one point for every cell that is visited that does not contain a mine. As soon as the user visits a cell that contains a mine, or visits a cell that has already been visited, the game is over and the program prints the user$$s score to the screen.

You are given the following type synonym definition for the game board:

$>$board $==[[$cell$]]$

$1.$Give an algebraic type definition for the type cell.

$2.$ Give the definition $($including its type$)$ of the function init$\_$board which takes as its argument a $($possibly infinite$)$ list of $($x$,$y$)$ coordinates representing the positions of the mines. Your function should generate a value of type board containing $95$ empty cells and $5$ cells containing mines in the appropriate positions.

$3.$ Give the definitions $($including types$)$ of the following two functions:

$$ usermove. This function takes two arguments: the board and a single $($x$,$y$)$ coordinate. It returns a two$-$tuple containing $($i$)$ a boolean according to whether the user has hit a mine or a previously visited cell, and $($ii$)$ a new board, suitably updated to indicate which cells have been visited.

$$ showboard. this function takes a single argument: the board. It returns a list of characters with newlines embedded appropriately in order to display the board on the screen. Previously$-$visited cells should be shown as the character $$X$$ and unvisited cells should be shown as the space character. Mines should be shown as spaces $($i$.$e$.$ hidden from the user$).$

$4.$ Assuming the existence of a function random which produces a potentially$-$infinite list of random $($x$,$y$)$ coordinates, give the definition for the main function for the above game. This function should provide a loop so that before each move the screen is cleared and the current state of the board is displayed on the screen. The user is then invited to enter a new $($x$,$y$)$ coordinate pair, and this is checked with the function usermove. Loop again if no mine $($or previously visited cell$)$ is hit: otherwise, terminate and print out the user$$s score.