In an `Inductive`

type definition, each constructor can take zero or more arguments. When applied to pattern matching, each constructor could be destructed back into parts.

Generalizing the definition of pairs, we notice that a list of numbers is either an empty list or a pair of a number and another list, much like a natural number is either zero or the successor of another natural number.

We could tell Coq how to parenthesize expressions involving multiple uses of binary operators using `Notation`

. For example, this statement…

```
Notation "[ x ; .. ; y ]" := (cons x .. (cons y nil) ..).
```

…gives an expression like `1`

`::`

`2`

`::`

`3`

`::`

`nil`

a clear meaning.

To prove a proposition $P(l)$ that holds for all lists $l$, we could reason as follows:

- First, show $P(l)$ is true if $l$ is
`nil`

. - Then show that $P(l)$ is true when $l$ is a pair of some number $n$ and some smaller list $l’$, assuming $P(l’)$ is true.

This strategy works because larger lists can always be broken down into smaller ones, eventually reaching `nil`

.

We could use `Search`

command to search for theorems, using wildcards like `(_`

`+`

`_`

`=`

`_`

`+`

`_)`

, or using `(?x`

`+`

`?y`

`=`

`?y`

`+`

`?x)`

if we want a more precise search.

Sometimes, an operation is not valid, such as taking the first element out of an empty list; in such cases, we could create a new data-type “wrapping around” the original type `T`

, with two constructors `Some`

`T`

and `None`

, together with a function for unwrapping.

Coq doesn’t have a built-in boolean type, any inductive type defined with exactly two constructors are considered a boolean; the guard is considered true if it evaluates to the first constructor and false if it evaluates to the second.

Partial maps are constructed similar to lists, with an “empty” constructor and the other constructor taking an id, a value, and the rest of the partial map (also a partial map). Updating a partial map can be achieved by simply adding a record to the front; the new record shadows the original one.