- **Today**:
	- Proofs by rule (natural) induction
	- Simultaneous inductive definitions

**Definition:** We call a set of inference rules an **inductive definition** of a judgement if the rules are **exhaustive** e.e.


- If a judgement holds, it can be inferred from the rules, and
- If a judgement can be inferred, it holds

**Announcement:** 4 weeks for assignment after the christmas week. You probably only need 2 weeks.

If $s_1N s_2 L \implies s_1s_2 L$, $\epsilon L$, $s L\implies (s)N$ 

Assume $s_1 L$ and $s_2L$ and we know that $s_1 N s_2 L \implies s_1s_2 L$

# Syntax

- So far
	- Revision of inference rules, natural (rule) induction
	- First steps in Haskell
	- Simple grammars specified using inference rules
- The inference rule for SExpr defined the **concrete syntax** of a simple language, including precedence and associativity

- The concrete syntax of a language is designed with the human user in mind.