### Odds and Evens

Due by Thu, 24 Jan 2019

Please e-mail your assignments to siddhartha.gadgil@gmail.com with "LTS2019" in the subject. The submissions will be added to the repository the day after the due date, and no late submissions will be accepted.

All problems should be solved in an Idris file which should be in a folder including the files Intro.idr and Evens.idr from the course. If your user name is myusername@iisc.ac.in, begin the idris file with the following.

module Myusername.Odds

import Evens

import Intro


1. Define an inductive type family IsOdd : Nat -> Type such that IsOdd n is inhabited if and only if n is odd.
2. Show that $3$ is odd.
3. Show that $2$ is not odd.
4. Show that if n: Nat is odd, then its successor S n is even.
5. Show that if n: Nat is even, then its successor S n is odd.
6. Show that every number n : Nat is either even or odd.
7. Show that no number n: Nat is both even and odd.
8. Show that if n: Nat and m: Nat are odd, then add n m is even (here add is from the intro).
9. Show that if n: Nat is odd, then there exists m: Nat such that n = S (double m).