52 lines
1.8 KiB
Markdown
52 lines
1.8 KiB
Markdown
# Expressions
|
|
|
|
Lean is based on dependent type theory, and is very similar to the one
|
|
used in the [Boole](https://github.com/avigad/boole) and
|
|
[Coq](http://coq.inria.fr/) systems. In contrast to Coq, Lean is
|
|
classical.
|
|
|
|
In Lean, we have the following kind of expressions: _constants_,
|
|
,_function applications_, _(heterogeneous) equality_, _local variables_,
|
|
_lambdas_, _dependent function spaces_ (aka _Pis_), _let expressions_,
|
|
and _Types_.
|
|
|
|
## Constants
|
|
|
|
Constants are essentially references to variable declarations, definitions, axioms and theorems in the
|
|
environment. In the following example, we use the command `variables` to declare `x` and `y` as integers.
|
|
The `check` command displays the type of the given expression. The `x` and `y` in the `check` command
|
|
are constants. They reference the objects declared using the command `variables`.
|
|
|
|
```lean
|
|
variables x y : Nat
|
|
check x + y
|
|
```
|
|
|
|
In the following example, we define the constant `s` as the sum of `x` and `y` using the `definition` command.
|
|
The `eval` command evaluates (normalizes) the expression `s + 1`. In this example, `eval` will just expand
|
|
the definition of `s`, and return `x + y + 1`.
|
|
|
|
```lean
|
|
definition s := x + y
|
|
eval s + 1
|
|
```
|
|
|
|
## Function applications
|
|
|
|
In Lean, the expression `f t` is a function application, where `f` is a function that is applied to `t`.
|
|
In the following example, we define the function `max`. The `eval` command evaluates the application `double 3`,
|
|
and returns the value `6`.
|
|
|
|
```lean
|
|
import tactic -- load basic tactics such as 'simp'
|
|
definition double (x : Nat) : Nat := x + x
|
|
eval double 3
|
|
```
|
|
|
|
In the following command, we define the function `inc`, and evaluate some expressions using `inc` and `max`.
|
|
|
|
```lean
|
|
definition inc (x : Nat) : Nat := x + 1
|
|
eval inc (inc (inc 2))
|
|
eval double (inc 3)
|
|
```
|