Youtube
/
Idris2-ep0
1
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Define the natural numbers

This commit is contained in:
Nathan McCarty 2023-06-29 20:53:15 -04:00
parent 7f40878575
commit 267fabf43e
Signed by: thatonelutenist
SSH Key Fingerprint: SHA256:hwQEcmak9E6sdU9bXc98RHw/Xd1AhpB5HZT7ZSVJkRM
1 changed files with 35 additions and 2 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

37
src/Naturals.idr
View File

@ -1,4 +1,37 @@
module Naturals
main : IO ()
main = putStrLn "Hello!"
%default total
data Natural : Type where
||| Zero is a natural number
Zero : Natural
||| The successor of a natural n is a natural
S : Natural -> Natural
one : Natural
one = S Zero
two : Natural
two = S (S Zero)
data Equality : Natural -> Natural -> Type where
||| A natural n is equal to itself
Reflexive : {n : Natural} -> Equality n n
oneEqualsOne : Equality (S Zero) (S Zero)
oneEqualsOne = Reflexive
zeroNotEqualOne : Equality (S Zero) Zero -> Void
zeroNotEqualOne x impossible
symmetric : (n, m : Natural) -> Equality n m -> Equality m n
symmetric n n Reflexive = Reflexive
transitive : (n, m, o : Natural) -> Equality n m -> Equality m o -> Equality n o
transitive n n n Reflexive Reflexive = Reflexive
injection : (n, m : Natural) -> Equality (S n) (S m) -> Equality n m
injection n n Reflexive = Reflexive
succNotZero : (n : Natural) -> Equality (S n) Zero -> Void
succNotZero _ Reflexive impossible