# 15-150: Principles of Functional Programming

# Lecture 3: Recursion and Induction

Today we wrote some recursive functions for integers and lists.

We proved correctness of two implementations of the `power`
function. We used standard/mathematical induction for one
implementation and strong/complete induction for a different
implementation of that same function.

We also proved totality of
our implementation of the `length` function, using structural
induction over lists.

### Key Concepts

- Recursive functions
- Proofs of correctness
- Standard (mathematical) induction ---
Mathematical induction can be useful when an integer
variable is reduced by 1 in the recursive part of a function.
- Strong induction --- Strong induction can be useful when an integer
variable is reduced by more than 1.
- Structural induction --- Structural induction can be useful for recursion over datatypes more general than integers.
- Correspondence between recursive function
clauses and proof by induction.

Here is an introduction to
lists.

(There will be additional notes about structural
induction over lists and trees next time.)