Need to use Coq functional programming language to solve below question:

(a) Define the type Nat (capital N to distinguish it with inductive type nat). (3 points)

(b) Define expressions zero, one, two, and three that encode the corresponding numbers as elements

of Nat. (4 points)

Problem 3 (37 points). Alonzo Church has proposed an encoding for natural numbers in pure \-calculus, as follows: A natural number n can be encoded as a higher order function that receives a function f and an argument x, and applies f to x, n number of times, denoted by fx. For example, 04 If.lx.x 14 If..x.fx 24 f. dr.f (f) 34 f.dr.ff (fr)) n A X.X.r.f"c. Extending pure l-calculus with polymorphic data types, gives a polymorphic type equivalent for natural numbers according to this encoding: Nat 4 VX(X + X) + X X, where X is a type variable. Problem 3 (37 points). Alonzo Church has proposed an encoding for natural numbers in pure \-calculus, as follows: A natural number n can be encoded as a higher order function that receives a function f and an argument x, and applies f to x, n number of times, denoted by fx. For example, 04 If.lx.x 14 If..x.fx 24 f. dr.f (f) 34 f.dr.ff (fr)) n A X.X.r.f"c. Extending pure l-calculus with polymorphic data types, gives a polymorphic type equivalent for natural numbers according to this encoding: Nat 4 VX(X + X) + X X, where X is a type variable