Simplify.jl

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Algebraic simplification in Julia

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# Simplify.jl

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**Simplify.jl** implements methods for symbolic algebraic simplification in the Julia language.

## Examples

Normalization involves determining the unique normal form of an expression ("simplest" equivalent expression) through repeated application of rules. *Simplify.jl* will use its [internal set of algebraic rules](./src/rules.jl) by default, which includes trigonometry, logarithms, differentiation (based on [DiffRules.jl](https://github.com/JuliaDiff/DiffRules.jl)), and more.
```julia
julia> @syms x y b θ;

julia> normalize(@term(1 / (sin(-θ) / cos(-θ))))
@term(1 / (-(sin(θ)) / cos(θ)))

julia> normalize(@term(log(b, 1 / (b^abs(x^2)))))
@term(log(b, 1 / b ^ abs(x ^ 2)))

julia> normalize(@term(diff(sin(2x) - log(x+y), x)))
@term(1 * -(inv(x + y) * (1 * diff(y, x) + 1 * one(x))) + 1 * cos(2x) * (2 * one(x) + x * 0))

julia> normalize(@term(!x & x | (y & (y | true))))
@term(!x & x | (y | true) & y)

julia> normalize(@term(y^(6 - 3log(x, x^2))))
@term(y ^ (-(6 * log(x, x)) + 6))
```

In many cases, it is useful to specify entirely custom rules by passing a Term Rewriting System as the second argument to `normalize`. This may be done either by manually constructing a `Rules` object or by using the `RULES` strategy for `@term`.
```julia
julia> @syms f g h;
       @vars x y;

julia> normalize(@term(f(x, f(y, y))), @term RULES [
          f(x, x) => 1
          f(x, 1) => x
       ])
@term(x)

julia> normalize(@term(f(g(f(1), h()))), Rules(
          @term(f(x)) => @term(x),
          @term(h())  => @term(3),
       ))
@term(g(1, 3))

julia> using Simplify: EvalRule

julia> normalize(@term(f(g(f(1), h()))), Rules(
          @term(f(x)) => @term(x),
          @term(h())  => @term(3),
          EvalRule(g, (a, b) -> 2a + b)
       ))
@term(5)
```

Variables may contain information about their domain, which may result in more specific normalizations.
```julia
julia> using SpecialSets

julia> @syms x y z;

julia> ctx = [get_context(); Image(y, GreaterThan(3)); Image(z, Even ∩ LessThan(0))];

julia> with_context(ctx) do
           normalize(@term(abs(x)))
       end
@term(abs(x))

julia> with_context(ctx) do
           normalize(@term(abs(y)))
       end
@term(y)

julia> with_context(ctx) do
           normalize(@term(abs(z)))
       end
@term(-z)
```

```julia
julia> ctx = [get_context(); Image(x, TypeSet(Int)); Image(y, TypeSet(Int))];

julia> with_context(ctx) do
           normalize(@term(diff(sin(2x) - log(x + y), x)))
       end
@term(cos(2x) * 2 + -(inv(x + y) * (diff(y, x) + 1)))
```

## Acknowledgements
- [*Term Rewriting and All That*](https://www21.in.tum.de/~nipkow/TRaAT/)
- [@MasonProtter](https://github.com/MasonProtter)
- [@Yijia-Chen](https://github.com/Yijia-Chen)

Simplify.jl

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