```

## Number validations: floats, greater than and less than

Number validations also work for `float` values.

Here's where it becomes important to be able to declare <abbr title="greater than"><code>gt</code></abbr> and not just <abbr title="greater than or equal"><code>ge</code></abbr>. As with it you can require, for example, that a value must be greater than `0`, even if it is less than `1`.

また、数値のバリデーションを宣言することもできます:

* `gt`: より大きい（`g`reater `t`han）
* `ge`: 以上（`g`reater than or `e`qual）
* `lt`: より小さい（`l`ess `t`han）
* `le`: 以下（`l`ess than or `e`qual）

!!! info "情報"
    `Query`、`Path`などは後に共通の`Param`クラスのサブクラスを見ることになります。（使う必要はありません）

    そして、それらすべては、これまで見てきた追加のバリデーションとメタデータと同じパラメータを共有しています。

    assert response.status_code == 422
    assert response.json() == IsDict(
        {
            "detail": [
                {
                    "type": "greater_than",
                    "loc": ["body", "item", "price"],
                    "msg": "Input should be greater than 0",
                    "input": -3.0,
                    "ctx": {"gt": 0.0},
                }
            ]
        }
    ) | IsDict(

    assert response.json() == IsDict(
        {
            "detail": [
                {
                    "type": "greater_than",
                    "loc": ["body", 0, "age"],
                    "msg": "Input should be greater than 0",
                    "input": -1.0,
                    "ctx": {"gt": 0},
                }
            ]
        }
    ) | IsDict(

You send it a token and it returns an authenticated user.

This provider might be charging you per request, and calling it might take some extra time than if you had a fixed mock user for tests.

You probably want to test the external provider once, but not necessarily call it for every test that runs.

You can define those cases using a list in the type declaration.

You will receive all the values from the duplicate header as a Python `list`.

For example, to declare a header of `X-Token` that can appear more than once, you can write:

=== "Python 3.10+"

    ```Python hl_lines="9"
    {!> ../../../docs_src/header_params/tutorial003_an_py310.py!}
    ```

=== "Python 3.9+"

    ```Python hl_lines="9"

```Python hl_lines="8"
{!../../../docs_src/events/tutorial001.py!}
```

In this case, the `startup` event handler function will initialize the items "database" (just a `dict`) with some values.

You can add more than one event handler function.

And your application won't start receiving requests until all the `startup` event handlers have completed.

### `shutdown` event

# Separate OpenAPI Schemas for Input and Output or Not

When using **Pydantic v2**, the generated OpenAPI is a bit more exact and **correct** than before. 😎

In fact, in some cases, it will even have **two JSON Schemas** in OpenAPI for the same Pydantic model, for input and output, depending on if they have **default values**.

Let's see how that works and how to change it if you need to do that.

## Pydantic Models for Input and Output

    ```Python hl_lines="19"
    {!> ../../../docs_src/dependencies/tutorial004.py!}
    ```

...and **FastAPI** will know what to do.

!!! tip
    If that seems more confusing than helpful, disregard it, you don't *need* it.

# SQL (Relational) Databases

!!! info
    These docs are about to be updated. 🎉

    The current version assumes Pydantic v1, and SQLAlchemy versions less than 2.0.

    The new docs will include Pydantic v2 and will use <a href="https://sqlmodel.tiangolo.com/" class="external-link" target="_blank">SQLModel</a> (which is also based on SQLAlchemy) once it is updated to use Pydantic v2 as well.

**FastAPI** doesn't require you to use a SQL (relational) database.