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The directory layout of a Rust project should look like this (source)

.
├── Cargo.lock
├── Cargo.toml
├── benches
│ └── large-input.rs
├── examples
│ └── simple.rs
├── src
│ ├── bin
│ │ └── another_executable.rs
│ ├── lib.rs
│ └── main.rs
└── tests
 └── some-integration-tests.rs

What is the file simple.rs under examples? How do I execute it? How should the file look like?

Examples are useful in library crates to show how the crate is used.

An example can be an executable with a main method or a library; it can either be in a single file examples/example-name.rs or consist of several files in a subdirectory examples/example-name/, with the main method in main.rs. To compile a library example you need to specify its crate type in Cargo.toml:

[[example]]
name = "example-name"
crate-type = ["lib"]

Examples are compiled by cargo test to ensure that they are up to date with the crate. You can run a specific executable example by

cargo run --example <example-name>

and selectively build any example with

cargo build --example <example-name>

This is documented in the Cargo Reference.