### Total Functions

Functions are total if, for all inputs, they terminate with a well-typed result.

Idris cannot automatically determine if this is true for the general case (No program can due to the halting problem) However, in many cases Idris can determine a function is total by using heuristics like the following:

- All calls are themselves to total functions.
- If there is a recursive call then there must be a decreasing argument that converges toward a base case.

If we require a function to be total but Idris cannot automatically determine that, we may have to provide a proof.

Alternatively there is a notion of total for the dual of this for infinite data types, see below.

For example, Idris knows that this is total because the RHS always involve smaller numbers. Idris code here | ||| Recursive definition of Fibonacci. fibRec : Nat -> Nat fibRec Z = Z fibRec (S Z) = S Z fibRec (S (S k)) = fibRec (S k) + fibRec k |

x

||| Accumulator for fibItr. Parameters: ||| * nth Fibonacci number ||| * Fibonacci number before last ||| * last Fibonacci number fibAcc : Nat -> Nat -> Nat -> Nat fibAcc Z a _ = a fibAcc (S k) a b = fibAcc k b (a + b) ||| Iterative definition of Fibonacci. fibItr : Nat -> Nat fibItr n = fibAcc n 0 1 |

### Co Patterns

To be total it must produce a non-empty finite prefix of a well-typed infinite result in finite time.

discussion on forum here

fib : Stream Int fib = fib' 1 2 where fib' : Int -> Int -> Stream Int fib' p1 p2 = p1 :: fib' p2 (p1 + p2) |

Made up Agda-like syntax: | record Stream a where head : Stream a -> a tail : Stream a -> Stream a |