# FixArgs.jl

- FixArgs.jl
- Motivation
- Introduction
- Symbolic computation and lazy evaluation
- Philosophy
- More examples
- Concerns
- API and internals

# Motivation

This package began as an exploration in generalizing `Base.Fix1`

and `Base.Fix2`

. These types represent particular forms of functions: `x -> f(v, x)`

and `x -> f(x, v)`

for a given `f`

and `v`

. We'll pick up that motivation shortly.

There is a secondary motivation for this package, aligned with the philosophy that choosing names is hard, and that a long name may be avoided when concepts are further broken down:

If you need underscores, it most likely means that you can work harder to find a better name, or perhaps that you are mixing together two concepts that should be separated.

There are a lot of great names within the Julia ecosystem, and meaning is important:

With multiple dispatch, just as the meaning of a function is important to get consistent, it's also important to get the meanings of the argument slots consistent when possible.

This package can help to re-use these names and put them together in new ways. Crucially, it relies on parametric types, the same facility that allows `Vector{Int64}`

and `Vector{Float32}`

to be given "orthogonal" names, instead of requiring names that bake together the concept of "vector" and "element type".

# Introduction

Let's illustrate `Fix1`

and `Fix2`

. We'll use the `string`

function in `Base`

, which concatenates the string representations of its arguments:

`julia> string("first ", "second")`

`"first second"`

Now, to construct and use the `Fix1`

and `Fix2`

types:

`julia> using Base: Fix1, Fix2`

`julia> f1 = Fix1(string, "one then ")`

`(::Base.Fix1{typeof(string), String}) (generic function with 1 method)`

`julia> f1("two")`

`"one then two"`

The function-call behavior of `Fix1(f, bind)`

is the same as `x -> f(bind, x)`

.

Similarly,

`julia> f2 = Fix2(string, " before two")`

`(::Base.Fix2{typeof(string), String}) (generic function with 1 method)`

`julia> f2("one")`

`"one before two"`

The function-call behavior of `Fix2(f, bind)`

is the same as `x -> f(x, bind)`

.

The key point of the `Fix1`

and `Fix2`

types is that methods can dispatch on

- the type of
`f`

- the type of
`bind`

- the position of
`bind`

within the function call

Dispatch is not tenable with anonymous functions. Let's illustrate while moving to a more practical example using `==`

instead of `string`

.

`julia> f1 = x -> x == 0`

`#1 (generic function with 1 method)`

`julia> f2 = Fix1(==, 0)`

`(::Base.Fix1{typeof(==), Int64}) (generic function with 1 method)`

Now define the "same" things again:

`julia> f3 = x -> x == 0`

`#3 (generic function with 1 method)`

`julia> f4 = Fix1(==, 0)`

`(::Base.Fix1{typeof(==), Int64}) (generic function with 1 method)`

The types of both the `Fix1`

values is the same:

`julia> typeof(f2) === typeof(f4)`

`true`

But each anonymous function definition introduces a new type with an opaque name:

`julia> typeof(f1), typeof(f3)`

`(Main.var"#1#2", Main.var"#3#4")`

A new anonymous function is always given a unique type, which allows methods to specialize on the specific anonymous function passed as an argument, but does not "permit" dispatch. To be more accurate, as far as dispatch is concerned, the type of anonymous functions is not special:

`julia> foo(::typeof(f1)) = "f1"`

`foo (generic function with 1 method)`

`julia> foo(::typeof(f3)) = "f3"`

`foo (generic function with 2 methods)`

`julia> foo(f1)`

`"f1"`

`julia> foo(f3)`

`"f3"`

But really we'd like to use a type that is less opaque and furthermore is "structural" in some ways, rather than purely "nominal".

## Examples of `Base.Fix2`

Where is it useful to dispatch on these special functions? Because `Base`

does not export and does not document these types, there aren't many methods in the Julia ecosystem.

But these types are constructed with, for example, `==(3)`

or `in([1, 2, 3])`

. A type like these is useful as a predicate to pass to higher-order functions, e.g. `findfirst(==(3), some_array)`

to find the first element that equals `3`

. Brevity aside, these types are useful to define more efficient methods of generic higher-order functions. For example, take a specific method of the `findfirst`

function:

```
findfirst(p::Union{Fix2{typeof(isequal),Int},Fix2{typeof(==),Int}}, r::OneTo{Int}) =
1 <= p.x <= r.stop ? p.x : nothing
```

The fallback for `findfirst`

(triggered by e.g. `findfirst(x->x==3, 1:10)`

instead of `findfirst(==(3), 1:10)`

) would produce the same (correct) answer, but the method above will be quicker.

Dispatching on the *structure* of the predicate function enables a certain form of symbolic computation.

# Symbolic computation and lazy evaluation

This package provides a generalization of `Fix1`

and `Fix2`

in a few ways:

- A function of any positional arity can be used, and any number of its arguments can be bound, allowing the remaining arguments to be provided later.
- A function can have its keyword arguments bound.
- The function
`x -> f(x, b)`

is represented with types:

The third generalization is powerful, because it's effectively the lambda calculus.

It is worth considering first just `Call`

, which can serve the purpose of representing a delayed function call evaluation. If you prefer, you may also consider a thunk `() -> foo(1, 2)`

, which would be a `Lambda`

(with no arguments) *and* a `Call`

that does not mention any "free variables".

If laziness is all that is needed, then defining a Julia anonymous function will do the job. But this package allows an additional benefit since methods can dispatch on details of the lazy call.

In many domains, new types are introduced to represent this pattern.

`Base.Iterators`

`Base.Generator`

consists of two fields `f`

and `iter`

. This can be taken as a representation of `map(f, iter)`

:

`julia> using FixArgs`

`julia> gen = let f = string, iter = 1:10 @xquote map(f, iter) end`

`Call(Some(map), FrankenTuple((Some(string), Some(1:10)), NamedTuple()))`

It's certainly less nice to look at than `Base.Generator{UnitRange{Int64}, typeof(string)}(string, 1:10)`

. Better UX / ergonomics are be possible by defining a type alias:

`const MyGenerator{F, I} = FixArgs.Call{Some{typeof(map)}, FixArgs.FrankenTuples.FrankenTuple{Tuple{Some{F}, Some{I}, (), Tuple{}}}`

That is quite unsightly, and there are quite a few internals leaking out. We can use a macro instead:

`julia> const MyGenerator{F, I} = @xquoteT map(::F, ::I)`

`FixArgs.Call{Some{typeof(map)}, FrankenTuples.FrankenTuple{Tuple{Some{F}, Some{I}}, (), Tuple{}}} where {F, I}`

It should be made convenient to defining constructors and `show`

methods that correspond with the type alias.

To evaluate the call (i.e. "collect the iterator"):

`julia> xeval(gen)`

`10-element Vector{String}: "1" "2" "3" "4" "5" "6" "7" "8" "9" "10"`

This example is actually circular. The evaluation of the `map`

call is done in terms of `Generator`

! The definition:

`map(f, A) = collect(Generator(f,A))`

Breaking this circularity is possible by defining

```
function iterate(gen::(@xquoteT map(::F, ::I))) where F, I
f = FixArgs.xeval(gen.args[1]) # not the prettiest thing right now...
iter = FixArgs.xeval(gen.args[2])
# ...
end
```

and might also require a separation of the purposes of `collect`

and `map`

. See this issue.

Many types in `Base.Iterators`

can be seen as lazy calls of existing functions. For example `Base.Iterators.Filter(flt, itr)`

could be replaced with `@xquote filter(flt, itr)`

. `Base.Iterators.Filter`

would be an alias for `(@xquoteT filter(::F, ::I)) where {F, I}`

to enable the existing symbolic computation, e.g.:

`reverse(f::Filter) = Filter(f.flt, reverse(f.itr))`

`Base.Iterators.Flatten`

, which defines a convenience function

`flatten(itr) = Flatten(itr)`

*could* be written in terms of a function `flatten`

with no methods. However, it is perhaps better seen as `@xquote reduce(vcat, it)`

`Base.Rational`

What is `Rational`

but lazy division on integers?

```
julia> 1/9 * 3/2 # eager division
0.16666666666666666
```

`julia> using FixArgs`

`julia> (@xquote 1/9) * (@xquote 3/2)`

`ERROR: MethodError: no method matching *(::FixArgs.Call{Some{typeof(/)}, FrankenTuples.FrankenTuple{Tuple{Some{Int64}, Some{Int64}}, (), Tuple{}}}, ::FixArgs.Call{Some{typeof(/)}, FrankenTuples.FrankenTuple{Tuple{Some{Int64}, Some{Int64}}, (), Tuple{}}}) Closest candidates are: *(::Any, ::Any, !Matched::Any, !Matched::Any...) at /opt/hostedtoolcache/julia/1.7.3/x64/share/julia/base/operators.jl:655`

Of course, we have to do some more work.

```
using Base: divgcd
function Base.:*(
x::(@xquoteT ::T / ::T),
y::(@xquoteT ::T / ::T),
) where {T}
xn, yd = divgcd(something(x.args[1]), something(y.args[2]))
xd, yn = divgcd(something(x.args[2]), something(y.args[1]))
ret = @xquote $(xn * yn) / $(xd * yd) # TODO use `unsafe_rational` and `checked_mul`
ret
end
```

Now, try again:

`julia> q = (@xquote 1/9) * (@xquote 3/2)`

`Call(Some(/), FrankenTuple((Some(1), Some(6)), NamedTuple()))`

`julia> map(xeval, q.args) # make numerator and denominator plainly visible`

`2-element Vector{Int64}: 1 6`

compare with using `//`

to construct a `Base.Rational`

:

`julia> 1//9 * 3//2`

`1//6`

Finally, because we have encoded the relationship between this "new" rational type, and `/`

, we can do:

`julia> xeval(q)`

`0.16666666666666666`

We could define an alias:

`const MyRational{T} = @xquoteT(::T / ::T)`

`FixArgs.Call{Some{typeof(/)}, FrankenTuples.FrankenTuple{Tuple{Some{T}, Some{T}}, (), Tuple{}}} where T`

which would also enforce the same type for both the numerator and denominator, as is the case of `Base.Rational`

.

`julia> sizeof(MyRational{Int32})`

`8`

Occasionally, a user might find this to be a limitation, yet they would still like to use some of the generic algorithms that might apply.

The fields of `Base.Rational`

are `num`

and `den`

. They have to be named since there is nothing else that gives the fields any meaning at all. In our type, however, they can be distinguished by the role they play with respect to the `/`

function.

## Fixed-Point Numbers and "static" arguments

Replacing `Rational`

may be silly, but this approach comes with a benefit: it also generalizes fixed-point numbers. A fixed-point number is just a lazy division with a specified (i.e. "static") denominator.

If we have a large array of fixed-point numbers with the same denominator, we certainly do not want to store the denominator repeatedly. Furthermore, we want efficient arithmetic. There is no need to check for a common denominator (let alone find one) if the two denominators are known to be equal at code-generation time.

There are values that we can "bake in" (see `Base.isbitstype`

) into the type of `Call`

itself!

Here is an example that models `FixedPointNumbers.Fixed{Int8,7}`

from `FixedPointNumbers.jl`

. The macros use the notation `V::::S`

(quadruple colon) to mark an argument `V`

as "static". Also note the use of `$`

to escape subexpressions.

`julia> using FixArgs`

`julia> MyFixed{Int8, 128} === typeof(MyQ0f7(3))`

`true`

`julia> function Base.:+(a::MyFixed{N,D}, b::MyFixed{N,D})::MyFixed{N,D} where {N, D} n = something(a.args[1]) + something(b.args[1]) return (@xquote $(N(n)) / D::::S) end`

`julia> xeval(MyQ0f7(3) + MyQ0f7(2)) === 5/128`

`true`

`julia> sizeof(MyFixed{Int8, 128})`

`1`

`julia> sizeof(Int8)`

`1`

And the generated code appears to be equivalent between

```
using FixedPointNumbers
look_inside_1(x, y) = reinterpret(Fixed{Int8, 7}, Int8(x)) + reinterpret(Fixed{Int8, 7}, Int8(y))
```

and

`look_inside_2(x, y) = MyQ0f7(x) + MyQ0f7(y)`

## Pure-imaginary type and `Base.Complex`

Now that we can make some arguments static, we can introduce a meaningful example where the lazy call does not correspond to a valid eager call. You can define a type such that `xeval`

raises `MethodError`

and still represent the computation symbolically. You can define a type `A`

in terms of a function `f`

and a type `B`

even if it may not make sense to define a new method of `f`

on `B`

.

Here is an over-the-top example:

`julia> using FixArgs`

`julia> struct ImaginaryUnit end # if we want to be really cute, can do `@xquote sqrt((-1)::::S)``

`julia> const Imaginary{T} = @xquoteT ::T * ::ImaginaryUnit`

`FixArgs.Call{Some{typeof(*)}, FrankenTuples.FrankenTuple{Tuple{Some{T}, Some{Main.ImaginaryUnit}}, (), Tuple{}}} where T`

`julia> Imaginary(x) = @xquote x * $(ImaginaryUnit()) # note escaping`

`FixArgs.Call{Some{typeof(*)}, FrankenTuples.FrankenTuple{Tuple{Some{T}, Some{Main.ImaginaryUnit}}, (), Tuple{}}} where T`

note that if we assume we have no `Base.Complex`

or anything like it, we don't have a way to further evaluate:

`julia> xeval(Imaginary(3))`

`ERROR: MethodError: no method matching *(::Int64, ::Main.ImaginaryUnit) Closest candidates are: *(::Any, ::Any, !Matched::Any, !Matched::Any...) at /opt/hostedtoolcache/julia/1.7.3/x64/share/julia/base/operators.jl:655 *(::T, !Matched::T) where T<:Union{Int128, Int16, Int32, Int64, Int8, UInt128, UInt16, UInt32, UInt64, UInt8} at /opt/hostedtoolcache/julia/1.7.3/x64/share/julia/base/int.jl:88 *(::Union{Int16, Int32, Int64, Int8}, !Matched::BigInt) at /opt/hostedtoolcache/julia/1.7.3/x64/share/julia/base/gmp.jl:542 ...`

We represented pure imaginary numbers as lazy multiplication of numbers and a singleton type `ImaginaryUnit`

, and it is basically as if we had defined

```
struct Imaginary{T}
_::T
end
```

Let's just go ahead and represent complex numbers too:

`julia> # const MyComplex{R, I} = @xquoteT ::R + (::I * ::ImaginaryUnit) # TODO this macro doesn't work MyComplex(r, i) = @xquote r + i * $(ImaginaryUnit())`

`MyComplex (generic function with 1 method)`

Note this monster of a type has the same size as `Base.Complex`

:

`julia> sizeof(Complex(1, 2))`

`16`

`julia> sizeof(MyComplex(1, 2))`

`16`

and the same layout too:

`julia> reinterpret(Int64, [Complex(1, 2)])`

`2-element reinterpret(Int64, ::Vector{Complex{Int64}}): 1 2`

`julia> reinterpret(Int64, [MyComplex(1, 2)])`

`2-element reinterpret(Int64, ::Vector{FixArgs.Call{Some{typeof(+)}, FrankenTuples.FrankenTuple{Tuple{Some{Int64}, FixArgs.Call{Some{typeof(*)}, FrankenTuples.FrankenTuple{Tuple{Some{Int64}, Some{Main.ImaginaryUnit}}, (), Tuple{}}}}, (), Tuple{}}}}): 1 2`

Of course, there are many different types that would all be mathematically equivalent by swapping the arguments to `+`

or `*`

. Note that swapping the arguments to `+`

would give a different memory layout.

## Faster set operations by deferring computation

Suppose we have some generic function to return bounds on set-like objects.

```
"""Produce the unique best bound, in the sense that `x ∈ input` implies `x ∈ result`"""
function bounding#=(result_type, input)=# end
```

`Main.bounding`

The objects may be shapes in space, and `result_type`

could correspond to important categories of bounding volumes. To keep things simple, let us deal with sets of integers as represented by e.g. `Vector`

and `UnitRange`

.

`julia> bounding(::Type{UnitRange}, v::Vector{<:Integer}) = UnitRange(extrema(v)...)`

`bounding (generic function with 1 method)`

`julia> bounding(UnitRange, [1, 3, 5])`

`1:5`

Consider the following computation:

`julia> eager = bounding(UnitRange, union(1:3, 5:7))`

`1:7`

It might be worth deferring that `union`

call. It produces a representation with a size linear in the number of elements, whereas a deferred computation is representable in constant size.

```
using FixArgs
function bounding(
::Type{UnitRange},
_union::(@xquoteT union(::UnitRange{T}, ::UnitRange{T}))
) where T <: Integer
(a, b) = something.(_union.args)
UnitRange(min(minimum(a), minimum(b)), max(maximum(a), maximum(b)))
end
```

`bounding (generic function with 2 methods)`

Now to use our specialized method for bounding unions of `UnitRanges`

, we simply defer one part of the previous computation:

`julia> lazy = bounding(UnitRange, @xquote union($(1:3),$(5:7)))`

`1:7`

`julia> eager == lazy`

`true`

## Ergonomics and Syntax

Ideally, this package would support exactly the syntax that Julia supports. However, it is a bit challenging not being able to hook into lowering exactly.

- https://docs.julialang.org/en/v1/devdocs/ast/#Bracketed-forms
- https://github.com/JuliaLang/julia/pull/32201

Anonymous function syntax is still under discussion:

There are packages that define anonymous function syntax, and it would be nice to ensure that they compose with e.g. `@xquote`

.

# Philosophy

## Names

(see also nominative and structural type systems)

The Julia ecosystem goes to great lengths to find the right generic functions and to ensure that all methods defined on generic functions are semantically compatible. This effort enables generic programming and interoperability. Much care goes into naming these functions and deciding their generic meaning, and it makes sense to reuse those names when possible, instead of creating a new name.

One possible advantage of using compositional names for types is that two packages can declare methods on the same type without knowing about each other, as long as both packages know about the constituents of the composition. There may be situations in which using a compositional name is a better strategy than trying to establish a "FooBase" package.

It is certainly possible to go overboard, however.

## Type Piracy

The Julia manual discusses not overloading methods on container types defined by others (hashed).

Arguably `Call`

can be seen as a container type, but the whole purpose of it is for methods to be defined on it. As far as type piracy is concerned, e.g. `@xquoteT ::Int64 / ::Int64`

is equivalent to using `Rational{Int64}`

since in both cases, all types are owned by `Base`

(assuming `Call`

, `Lambda`

, etc. is not owned by your package (module) either). In this example, the types are `typeof(/)`

and `Int64`

.

For a function owned by your package, e.g. `my_foo`

, by all means define `my_foo(::(@xquoteT ::Int64 / ::Int64))`

.

For a function not owned your package, e.g. `Base.:*`

, to avoid type piracy, `*`

should be defined by a package that owns at least one of the constituent types, e.g.

`Base.:*(::(@xquote MyFunction(::NotMyType))`

`Base.:*(::(@xquote NotMyFunction(::MyType))`

`Base.:*(::(@xquote MyFunction(::MyType))`

If it is controversial what `f(::(@xquoteT func(::A, ::B)`

should mean, then it is likely worth defining a new type instead of using this package.

If there are multiple functions that apply to the fields of a struct, as in:

```
struct MyType{T1, T2}
arg1::T1
arg2::T2
end
f1(t::MyType) = *(t.arg1, t.arg2)
f2(t::MyType) = +(t.arg1, t.arg2)
```

Then it also seems like type would also not be a good candidate to be replaced with a `Call`

. Which of `*`

or `+`

would one choose?

# More examples

## Types of geometry primitives from scratch

What is better?

```
struct Disc{R, C}
radius::R
center::C
end
```

or

```
using LinearAlgebra: norm
Disc(r, c) = @xquote x -> norm(x - c) ≤ r
```

Honestly, I don't know. I imagine there is a trade-off. But consider a situation where you'd like to distinguish between the disc with boundary, disc without boundary, and just the boundary (a circle). Those distinctions are as simple as `≤`

, `<`

, and `==`

. You might more accurately call the struct above `DiscWithBoundary`

, and introduce `DiscWithoutBoundary`

and `Circle`

. Or instead you might use a convention that all the regions are closed sets and introduce wrapper types `WithoutBoundary`

and `OnlyBoundary`

. In both cases, there's a lot of names to choose...

## A way to avoid naming a new method

Computing the square norm of a vector can usually be done more efficiently than computing the norm and then computing the square.

`julia> using FixArgs`

`julia> using LinearAlgebra: norm, norm_sqr`

`julia> vec = [1, 1]`

`2-element Vector{Int64}: 1 1`

instead of writing

`julia> norm_sqr(vec)`

`2.0000000000000004`

maybe you would rather write

`julia> xeval(@xquote norm(vec)^2::::S)`

`2.0000000000000004`

Well, perhaps better with an alternative macro that applies `::::S`

to any `isbitstype`

literals, and also does `xeval`

.

`@a_good_short_name norm(vec)^2`

Whatever owns `norm`

can add a new method to `xeval`

to detect this form and use the more efficient method.

`xeval(c::(@xquoteT norm(::T)^2::::S)) where T = norm_sqr(c.args[1].args[1])`

This is different from having a package define its own macro. This would be one macro that can be hooked into by defining new methods on `xeval`

.

## Alternative to `Base.literal_pow`

`julia> Meta.@lower x^2`

`:($(Expr(:thunk, CodeInfo( @ none within `top-level scope` 1 ─ %1 = Core.apply_type(Base.Val, 2) │ %2 = (%1)() │ %3 = Base.literal_pow(^, x, %2) └── return %3 ))))`

Could instead lower to roughly

```
%1 = @xquote x^(2::::S) # i.e. something roughly like `Call(^, x, Val(2))`
%2 = xeval(%1)
```

`xeval`

could call `Base.literal_pow`

for backwards compatibility:

`julia> using FixArgs`

`julia> function FixArgs.xeval(x::(@xquoteT (::B)^(E::::S))) where {B, E} println("I was called") base = xeval(x.args[1]) Base.literal_pow(^, base, Val(E)) end`

`julia> let x = 3 xeval(@xquote x^(2::::S)) end`

`I was called 9`

but going forward, types that wish to hook into this functionality would define a method on `xeval`

directly.

## Related Patterns and Possible Applications:

Base.Broadcast.Broadcasted is like `Call`

with extra information `Style`

and `axes::Axes`

. `materialize`

is like `xeval`

. There's more information in the manual.

Base.Generator, as already discussed.

### partial application with keyword arguments

`isapprox(y; kwargs...) = x -> isapprox(x, y; kwargs...)`

probably would use `Base.Fix2`

if it also supported keyword arguments. All other docstrings with "Create a function" do.

### array-of-struct and struct-of-array representations

```
soa = (a=[1,2,3], b=[10, 20, 30])
aos_eager = map(NamedTuple{(:a, :b)} ∘ tuple, soa.a, soa.b)
```

```
3-element Vector{NamedTuple{(:a, :b), Tuple{Int64, Int64}}}:
(a = 1, b = 10)
(a = 2, b = 20)
(a = 3, b = 30)
```

### non-standard evaluation

### nominal vs structural fields

https://github.com/JuliaLang/julia/pull/37517/files Instead of field names `inner`

and `outer`

, the arguments can be distinguished by the role they play with respect to the function `∘`

.

What if `∘`

itself is defined as `∘(a, b) = @xquote a ∘ b`

? Then define function without methods: `function ∘ end`

# Concerns

The poor compiler...

The poor user trying to understand the error message...

Method ambiguities

Unifying too many things can lead to too much coupling.

# API and internals

`FixArgs.ArgPos`

— Type`FixArgs.Arity`

— TypeRepresent the arity of a `Lambda`

.

Currently, only represents a fixed number of positional arguments, but may be generalized to include optional and keyword arguments.

`P`

is 0, 1, 2, ... `KW`

is always `NoKeywordArguments`

, and may be extended in the future.

`FixArgs.Call`

— TypeA call "f(args...)". `args`

may represent both positional and keyword arguments.

`FixArgs.Context`

— Typeterms are evaluated with respect to a `Context`

A `Context`

is an associations between bound variables and values, and they may be nested (via the `parent`

field).

`FixArgs.Lambda`

— TypeA lambda expression "args -> body"

`FixArgs.ParentScope`

— TypeNest `ArgPos`

in `ParentScope`

s to represent a reference to the formal parameters of a "parent" function. Forms a unary representation.

Related: [De Bruijn indices]https://en.wikipedia.org/wiki/De*Bruijn*index

`FixArgs.isexpr`

— Method```
isexpr(expr) -> Bool
isexpr(expr, head) -> Bool
```

Checks whether given value isa `Base.Expr`

and if further given `head`

, it also checks whether the `head`

matches `expr.head`

.

**Examples**

```
julia> using ExprParsers
julia> EP.isexpr(:(a = hi))
true
julia> EP.isexpr(12)
false
julia> EP.isexpr(:(f(a) = a), :(=))
true
julia> EP.isexpr(:(f(a) = a), :function)
false
```

`FixArgs.lc_expr`

— MethodConvert an expression (e.g. an `Expr`

) to a Lambda-Call value

`FixArgs.normalize_lambda_1_arg`

— Methodnormalize `:(x -> body)`

into `:((x,) -> body`

)

`FixArgs.postwalk`

— Method`postwalk(f, expr)`

Applies `f`

to each node in the given expression tree, returning the result. `f`

sees expressions *after* they have been transformed by the walk.

See also: `prewalk`

.

`FixArgs.prewalk`

— Method`prewalk(f, expr)`

Applies `f`

to each node in the given expression tree, returning the result. `f`

sees expressions *before* they have been transformed by the walk, and the walk will be applied to whatever `f`

returns.

This makes `prewalk`

somewhat prone to infinite loops; you probably want to try `postwalk`

first.

`FixArgs.relabel_args`

— Functionα-conversion in λ-calculus

`labeler(x)`

produces a `Symbol`

or similar from `x.referent_depth`

`x.antecedent_depth`

`x.arg_i`

`x.sym`

– name before relabeling

`x.referent_depth - x.antecedent_depth`

is number of `->`

s that are between the evaluation site and the definition site

`FixArgs.uneval`

— MethodGiven a value, produce an expression that when `eval`

'd produces the value.

e.g.

```
julia> eval(uneval(Expr(:my_call, :arg1, :arg2)))
:($(Expr(:my_call, :arg1, :arg2)))
julia> eval(eval(uneval(:(sqrt(9)))))
3.0
```

This function is used to return expressions from this package's macros. This is likely not a well-posed problem to begin with. Related issue.

Note the special case for `:(esc(x))`

.

`FixArgs.xapply`

— Functionapply a `Lambda`

expression to arguments.

`FixArgs.xeval`

— Methodevaluate a Lambda-Call expression

Currently only works on `Call`

expressions.

`FixArgs.@fix`

— MacroA convenience macro that implements the syntax of this PR

`@fix f(_, b)`

is the equivalent of `x -> f(x, b)`

`FixArgs.@quote_some`

— MacroThis macro is used to debug and introspect the escaping behavior of `@xquote`

```
julia> dump(let x = 9
@xquote sqrt(x)
end)
Expr
head: Symbol call
args: Array{Any}((2,))
1: sqrt (function of type typeof(sqrt))
2: Int64 9
```

`FixArgs.@xquote`

— Macro`FixArgs.@xquoteT`

— MacroThe types produced by this package are unwieldly. This macro permits a convenient syntax, e.g. `@xquoteT func(::Arg1Type, ::Arg2Type)`

to represent types.

```
let func = identity, arg = 1
typeof(@xquote func(arg)) == @xquoteT func(::typeof(arg))
end
# output
true
```

If an argument is "static", then it is part of the type, and the value is annotated as illustrated:

```
julia> @xquoteT string(123::::S)
FixArgs.Call{Some{typeof(string)},FrankenTuples.FrankenTuple{Tuple{Val{123}},(),Tuple{}}}
```