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Update docs (#164)
* fix link in index.md * copy docs from README.md * update index.md * simplify README * fix mod (LaTeX)
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‎README.md

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@@ -7,94 +7,22 @@ Interval Sets for Julia
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[![Coverage](https://codecov.io/gh/JuliaMath/IntervalSets.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaMath/IntervalSets.jl)
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[![Aqua QA](https://raw.githubusercontent.com/JuliaTesting/Aqua.jl/master/badge.svg)](https://github.com/JuliaTesting/Aqua.jl)
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This package represents intervals of an ordered set. For an interval
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spanning from `a` to `b`, all values `x` that lie between `a` and `b`
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are defined as being members of the interval.
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Currently this package defines one concrete type, `Interval`.
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These define the set spanning from `a` to `b`, meaning the
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interval is defined as the set `{x}` satisfying `a ≤ x ≤ b`. This is
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sometimes written `[a,b]` (mathematics syntax, not Julia syntax) or
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`a..b`.
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Optionally, `Interval{L,R}` can represent open and half-open intervals. The type
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parameters `L` and `R` correspond to the left and right endpoint respectively.
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The notation `ClosedInterval` is short for `Interval{:closed,:closed}`, while `OpenInterval` is short for `Interval{:open,:open}`. For example, the interval `Interval{:open,:closed}` corresponds to the set `{x}` satisfying `a < x ≤ b`.
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## Usage
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You can construct `ClosedInterval`s in a variety of ways:
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## Quick start
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```julia
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julia> using IntervalSets
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julia> ClosedInterval{Float64}(1,3)
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julia> i1 = 1.0 .. 3.0
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1.0 .. 3.0
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julia> 0.5..2.5
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0.5 .. 2.5
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julia> 1.5±1
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0.5 .. 2.5
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```
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Similarly, you can construct `OpenInterval`s and `Interval{:open,:closed}`s, and `Interval{:closed,:open}`:
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```julia
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julia> OpenInterval{Float64}(1,3)
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1.0 .. 3.0 (open)
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julia> OpenInterval(0.5..2.5)
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0.5 .. 2.5 (open)
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julia> Interval{:open,:closed}(1,3)
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1 .. 3 (open-closed)
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```
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The `±` operator may be typed as `\pm<TAB>` (using Julia's LaTeX
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syntax tab-completion).
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Intervals also support the expected set operations:
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```julia
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julia> 1.75 ∈ 1.5±1 # \in<TAB>; can also use `in`
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true
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julia> 0 ∈ 1.5±1
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false
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julia> 1 ∈ OpenInterval(0..1)
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false
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julia> intersect(1..5, 3..7) # can also use `a ∩ b`, where the symbol is \cap<TAB>
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3 .. 5
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julia> i2 = OpenInterval(0..4)
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0 .. 4 (open)
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julia> isempty(intersect(1..5, 10..11))
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julia> i1 ⊆ i2
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true
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julia> (0.25..5) ∪ (3..7.4) # \cup<TAB>; can also use union()
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0.25 .. 7.4
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julia> isclosedset(0.5..2.0)
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true
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julia> isopenset(OpenInterval(0.5..2.5))
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true
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julia> isleftopen(2..3)
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julia> i2 ⊆ i1
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false
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```
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When computing the union, the result must also be an interval:
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```julia
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julia> (0.25..5) ∪ (6..7.4)
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ERROR: ArgumentError: Cannot construct union of disjoint sets.
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Stacktrace:
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[1] union(d1::ClosedInterval{Float64}, d2::ClosedInterval{Float64})
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@ IntervalSets ~/.julia/dev/IntervalSets/src/interval.jl:127
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[2] top-level scope
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@ REPL[2]:1
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```
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### Importing the .. operator
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To import the `..` operator, use `import IntervalSets: (..)`. The parantheses are necessary to avoid parsing issues.
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Please refer to the [documentation](https://JuliaMath.github.io/IntervalSets.jl/stable) for comprehensive guides and examples.

‎docs/src/index.md

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# IntervalSets.jl
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A Julia package implementing [IntervalSets](https://en.wikipedia.org/wiki/Quaternion).
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A Julia package implementing [interval sets](https://en.wikipedia.org/wiki/Interval_(mathematics)).
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[![Stable](https://img.shields.io/badge/docs-stable-blue.svg)](https://JuliaMath.github.io/IntervalSets.jl/stable)
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[![Dev](https://img.shields.io/badge/docs-dev-blue.svg)](https://JuliaMath.github.io/IntervalSets.jl/dev)
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[![Build Status](https://github.com/JuliaMath/IntervalSets.jl/workflows/CI/badge.svg)](https://github.com/JuliaMath/IntervalSets.jl/actions)
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[![Coverage](https://codecov.io/gh/JuliaMath/IntervalSets.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaMath/IntervalSets.jl)
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[![Aqua QA](https://raw.githubusercontent.com/JuliaTesting/Aqua.jl/master/badge.svg)](https://github.com/JuliaTesting/Aqua.jl)
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!!! note "Documentation"
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The documentation is still work in progress.
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```
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pkg> add IntervalSets
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```
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## Quick start
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```@repl
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using IntervalSets
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i1 = 1.0 .. 3.0
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i2 = OpenInterval(0..4)
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i1 ⊆ i2
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i2 ⊆ i1
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```
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Currently this package defines one concrete type, [`Interval`](@ref).
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These define the set spanning from `a` to `b`, meaning the interval is defined as the set ``\{x \ | \ a ≤ x ≤ b\}``.
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This is sometimes written ``[a,b]`` (mathematics syntax, not Julia syntax) or ``a..b``.
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Optionally, `Interval{L,R}` can represent open and half-open intervals.
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The type parameters `L` and `R` correspond to the left and right endpoint respectively.
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The notation [`ClosedInterval`](@ref) is short for `Interval{:closed,:closed}`,
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while [`OpenInterval`](@ref) is short for `Interval{:open,:open}`.
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For example, the interval `Interval{:open,:closed}` corresponds to the set ``\{x \ | \ a < x ≤ b\}``.
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## More examples
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```@setup more
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using IntervalSets
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```
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### Constructors
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```@repl more
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ClosedInterval{Float64}(1,3)
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OpenInterval{Float64}(1,3)
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Interval{:open, :closed}(1,3)
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0.5..2.5
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1.5 ± 1
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Interval{:open,:closed}(1,3)
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OpenInterval(0.5..2.5) # construct `OpenInterval` from `ClosedInterval`
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```
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The [`±`](@ref) operator may be typed as `\pm<TAB>` (using Julia's LaTeX syntax tab-completion).
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### Set operations
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```@repl
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1.75 ∈ 1.5±1 # \in<TAB>; can also use `in`
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0 ∈ 1.5±1
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1 ∈ OpenInterval(0..1)
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intersect(1..5, 3..7) # can also use `a ∩ b`, where the symbol is \cap<TAB>
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isempty(intersect(1..5, 10..11))
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(0.25..5) ∪ (3..7.4) # \cup<TAB>; can also use `union()`
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isclosedset(0.5..2.0)
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isopenset(OpenInterval(0.5..2.5))
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isleftopen(2..3)
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(0.25..5) ∪ (6..7.4) # union of interval must be an interval
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```
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### Importing the `..` operator
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To import the [`..`](@ref) operator, use `import IntervalSets: (..)`.
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The parantheses are necessary to avoid parsing issues.

‎src/IntervalSets.jl

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"""
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mod(x, i::AbstractInterval)
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Find `y` in the `i` interval such that ``x ≡ y (mod w)``, where `w = width(i)`.
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Find `y` in the `i` interval such that ``x ≡ y \\pmod w``, where `w = width(i)`.
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# Examples
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