```
pub trait Step: Clone + PartialOrd<Self> {
fn steps_between(start: &Self, end: &Self) -> Option<usize>;
fn forward_checked(start: Self, count: usize) -> Option<Self>;
fn backward_checked(start: Self, count: usize) -> Option<Self>;
fn forward(start: Self, count: usize) -> Self { ... }
unsafe fn forward_unchecked(start: Self, count: usize) -> Self { ... }
fn backward(start: Self, count: usize) -> Self { ... }
unsafe fn backward_unchecked(start: Self, count: usize) -> Self { ... }
}
```

## Expand description

Objects that have a notion of *successor* and *predecessor* operations.

The *successor* operation moves towards values that compare greater.
The *predecessor* operation moves towards values that compare lesser.

## Required Methods

Returns the number of *successor* steps required to get from `start`

to `end`

.

Returns `None`

if the number of steps would overflow `usize`

(or is infinite, or if `end`

would never be reached).

##### Invariants

For any `a`

, `b`

, and `n`

:

`steps_between(&a, &b) == Some(n)`

if and only if`Step::forward_checked(&a, n) == Some(b)`

`steps_between(&a, &b) == Some(n)`

if and only if`Step::backward_checked(&b, n) == Some(a)`

`steps_between(&a, &b) == Some(n)`

only if`a <= b`

- Corollary:
`steps_between(&a, &b) == Some(0)`

if and only if`a == b`

- Note that
`a <= b`

does*not*imply`steps_between(&a, &b) != None`

; this is the case when it would require more than`usize::MAX`

steps to get to`b`

- Corollary:
`steps_between(&a, &b) == None`

if`a > b`

#### fn forward_checked(start: Self, count: usize) -> Option<Self>

#### fn forward_checked(start: Self, count: usize) -> Option<Self>

Returns the value that would be obtained by taking the *successor*
of `self`

`count`

times.

If this would overflow the range of values supported by `Self`

, returns `None`

.

##### Invariants

For any `a`

, `n`

, and `m`

:

`Step::forward_checked(a, n).and_then(|x| Step::forward_checked(x, m)) == Step::forward_checked(a, m).and_then(|x| Step::forward_checked(x, n))`

For any `a`

, `n`

, and `m`

where `n + m`

does not overflow:

`Step::forward_checked(a, n).and_then(|x| Step::forward_checked(x, m)) == Step::forward_checked(a, n + m)`

For any `a`

and `n`

:

`Step::forward_checked(a, n) == (0..n).try_fold(a, |x, _| Step::forward_checked(&x, 1))`

- Corollary:
`Step::forward_checked(&a, 0) == Some(a)`

- Corollary:

#### fn backward_checked(start: Self, count: usize) -> Option<Self>

#### fn backward_checked(start: Self, count: usize) -> Option<Self>

Returns the value that would be obtained by taking the *predecessor*
of `self`

`count`

times.

If this would overflow the range of values supported by `Self`

, returns `None`

.

##### Invariants

For any `a`

, `n`

, and `m`

:

`Step::backward_checked(a, n).and_then(|x| Step::backward_checked(x, m)) == n.checked_add(m).and_then(|x| Step::backward_checked(a, x))`

`Step::backward_checked(a, n).and_then(|x| Step::backward_checked(x, m)) == try { Step::backward_checked(a, n.checked_add(m)?) }`

For any `a`

and `n`

:

`Step::backward_checked(a, n) == (0..n).try_fold(a, |x, _| Step::backward_checked(&x, 1))`

- Corollary:
`Step::backward_checked(&a, 0) == Some(a)`

- Corollary:

## Provided Methods

Returns the value that would be obtained by taking the *successor*
of `self`

`count`

times.

If this would overflow the range of values supported by `Self`

,
this function is allowed to panic, wrap, or saturate.
The suggested behavior is to panic when debug assertions are enabled,
and to wrap or saturate otherwise.

Unsafe code should not rely on the correctness of behavior after overflow.

##### Invariants

For any `a`

, `n`

, and `m`

, where no overflow occurs:

`Step::forward(Step::forward(a, n), m) == Step::forward(a, n + m)`

For any `a`

and `n`

, where no overflow occurs:

`Step::forward_checked(a, n) == Some(Step::forward(a, n))`

`Step::forward(a, n) == (0..n).fold(a, |x, _| Step::forward(x, 1))`

- Corollary:
`Step::forward(a, 0) == a`

- Corollary:
`Step::forward(a, n) >= a`

`Step::backward(Step::forward(a, n), n) == a`

#### unsafe fn forward_unchecked(start: Self, count: usize) -> Self

#### unsafe fn forward_unchecked(start: Self, count: usize) -> Self

Returns the value that would be obtained by taking the *successor*
of `self`

`count`

times.

##### Safety

It is undefined behavior for this operation to overflow the
range of values supported by `Self`

. If you cannot guarantee that this
will not overflow, use `forward`

or `forward_checked`

instead.

##### Invariants

For any `a`

:

- if there exists
`b`

such that`b > a`

, it is safe to call`Step::forward_unchecked(a, 1)`

- if there exists
`b`

,`n`

such that`steps_between(&a, &b) == Some(n)`

, it is safe to call`Step::forward_unchecked(a, m)`

for any`m <= n`

.

For any `a`

and `n`

, where no overflow occurs:

`Step::forward_unchecked(a, n)`

is equivalent to`Step::forward(a, n)`

Returns the value that would be obtained by taking the *predecessor*
of `self`

`count`

times.

If this would overflow the range of values supported by `Self`

,
this function is allowed to panic, wrap, or saturate.
The suggested behavior is to panic when debug assertions are enabled,
and to wrap or saturate otherwise.

Unsafe code should not rely on the correctness of behavior after overflow.

##### Invariants

For any `a`

, `n`

, and `m`

, where no overflow occurs:

`Step::backward(Step::backward(a, n), m) == Step::backward(a, n + m)`

For any `a`

and `n`

, where no overflow occurs:

`Step::backward_checked(a, n) == Some(Step::backward(a, n))`

`Step::backward(a, n) == (0..n).fold(a, |x, _| Step::backward(x, 1))`

- Corollary:
`Step::backward(a, 0) == a`

- Corollary:
`Step::backward(a, n) <= a`

`Step::forward(Step::backward(a, n), n) == a`

#### unsafe fn backward_unchecked(start: Self, count: usize) -> Self

#### unsafe fn backward_unchecked(start: Self, count: usize) -> Self

Returns the value that would be obtained by taking the *predecessor*
of `self`

`count`

times.

##### Safety

It is undefined behavior for this operation to overflow the
range of values supported by `Self`

. If you cannot guarantee that this
will not overflow, use `backward`

or `backward_checked`

instead.

##### Invariants

For any `a`

:

- if there exists
`b`

such that`b < a`

, it is safe to call`Step::backward_unchecked(a, 1)`

- if there exists
`b`

,`n`

such that`steps_between(&b, &a) == Some(n)`

, it is safe to call`Step::backward_unchecked(a, m)`

for any`m <= n`

.

For any `a`

and `n`

, where no overflow occurs:

`Step::backward_unchecked(a, n)`

is equivalent to`Step::backward(a, n)`