lean2/doc/lean/expr.md
Leonardo de Moura 17eb2374ee doc(README): add link to tutorial in the main page
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
2014-02-02 19:14:02 -08:00

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# 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 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`.
We define the function `double`
```lean
import tactic -- load basic tactics such as 'simp'
definition double (x : Nat) : Nat := x + x
```
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
```