improved function partial application design

This replaces `Fix` (xref JuliaLang#54653) with `fix`. The usage is similar: use
`fix(i)(f, x)` instead of `Fix{i}(f, x)`.

Benefits:
* Improved type safety: creating an invalid type such as
  `Fix{:some_symbol}` or `Fix{-7}` is not possible.
* The design should be friendlier to future extensions. E.g., suppose
  that publicly-facing functionality for fixing a keyword (instead of
  positional) argument was desired, it could be achieved by adding a
  new method to `fix` taking a `Symbol`, instead of adding new public
  names.

Lots of changes are shared with PR JuliaLang#56425, if one of them gets merged
the other will be greatly simplified.

NEWS.md

@@ -87,7 +87,7 @@ New library functions
 * `waitany(tasks; throw=false)` and `waitall(tasks; failfast=false, throw=false)` which wait multiple tasks at once ([#53341]).
 * `uuid7()` creates an RFC 9652 compliant UUID with version 7 ([#54834]).
 * `insertdims(array; dims)` allows to insert singleton dimensions into an array which is the inverse operation to `dropdims`
-* The new `Fix` type is a generalization of `Fix1/Fix2` for fixing a single argument ([#54653]).
+* `Fix1`/`Fix2` are now generalized by `fix` ([#54653], [#56518]).
 
 New library features
 --------------------

base/Base_compiler.jl

@@ -213,6 +213,7 @@ include("error.jl")
 include("bool.jl")
 include("number.jl")
 include("int.jl")
+include("typedomainnumbers.jl")
 include("operators.jl")
 include("pointer.jl")
 include("refvalue.jl")

base/operators.jl

@@ -1153,55 +1153,149 @@ julia> filter(!isletter, str)
 !(f::Function) = (!) ∘ f
 !(f::ComposedFunction{typeof(!)}) = f.inner #allows !!f === f
 
+const _PositiveInteger = _TypeDomainNumbers.PositiveIntegers.PositiveInteger
+
+struct PartiallyAppliedFunction{Position <: _PositiveInteger, Func, Arg} <: Function
+    partially_applied_argument_position::Position
+    f::Func
+    x::Arg
+
+    function (::Type{PartiallyAppliedFunction{Position}})(func::Func, arg) where {Position <: _PositiveInteger, Func}
+        Pos = Position::DataType
+        pos = Pos.instance
+        new{Pos, _stable_typeof(func), _stable_typeof(arg)}(pos, func, arg)
+    end
+end
+
+function getproperty((@nospecialize v::PartiallyAppliedFunction), s::Symbol)
+    getfield(v, s)
+end  # avoid overspecialization
+
+function Base.show(
+    (@nospecialize io::Base.IO),
+    (@nospecialize unused::Type{PartiallyAppliedFunction{Position}}),
+) where {Position <: _PositiveInteger}
+    if Position isa DataType
+        print(io, "fix(")
+        show(io, Position.instance)
+        print(io, ')')
+    else
+        show(io, PartiallyAppliedFunction)
+        print(io, '{')
+        show(io, Position)
+        print(io, '}')
+    end
+end
+
+function Base.show(
+    (@nospecialize io::Base.IO),
+    (@nospecialize unused::Type{PartiallyAppliedFunction{Position, Func}}),
+) where {Position <: _PositiveInteger, Func}
+    show(io, PartiallyAppliedFunction{Position})
+    print(io, '{')
+    show(io, Func)
+    print(io, '}')
+end
+
+function Base.show(
+    (@nospecialize io::Base.IO),
+    (@nospecialize unused::Type{PartiallyAppliedFunction{Position, Func, Arg}}),
+) where {Position <: _PositiveInteger, Func, Arg}
+    show(io, PartiallyAppliedFunction{Position, Func})
+    print(io, '{')
+    show(io, Arg)
+    print(io, '}')
+end
+
+function Base.show((@nospecialize io::Base.IO), @nospecialize p::PartiallyAppliedFunction)
+    print(io, "fix(")
+    show(io, p.partially_applied_argument_position)
+    print(io, ")(")
+    show(io, p.f)
+    print(io, ", ")
+    show(io, p.x)
+    print(io, ')')
+end
+
+function _partially_applied_function_check(m::Int, nm1::Int)
+    if m < nm1
+        throw(ArgumentError(LazyString("expected at least ", nm1, " arguments to `fix(", nm1 + 1, ")`, but got ", m)))
+    end
+end
+
+function (partial::PartiallyAppliedFunction)(args::Vararg{Any,M}; kws...) where {M}
+    n = partial.partially_applied_argument_position
+    nm1 = _TypeDomainNumbers.PositiveIntegers.natural_predecessor(n)
+    _partially_applied_function_check(M, Int(nm1))
+    (args_left, args_right) = _TypeDomainNumberTupleUtils.split_tuple(args, nm1)
+    partial.f(args_left..., partial.x, args_right...; kws...)
+end
+
 """
-    Fix{N}(f, x)
+    fix(::Integer)::UnionAll
+
+Return a [`UnionAll`](@ref) type such that:
+* It's a constructor taking two arguments:
+    1. A function to be partially applied
+    2. An argument of the above function to be fixed
+* Its instances are partial applications of the function, with one positional argument fixed. The argument to `fix` is the one-based index of the position argument to be fixed.
 
-A type representing a partially-applied version of a function `f`, with the argument
-`x` fixed at position `N::Int`. In other words, `Fix{3}(f, x)` behaves similarly to
-`(y1, y2, y3...; kws...) -> f(y1, y2, x, y3...; kws...)`.
+For example, `fix(3)(f, x)` behaves similarly to `(y1, y2, y3...; kws...) -> f(y1, y2, x, y3...; kws...)`.
+
+See also: [`Fix1`](@ref), [`Fix2`](@ref).
 
 !!! compat "Julia 1.12"
-    This general functionality requires at least Julia 1.12, while `Fix1` and `Fix2`
-    are available earlier.
+    Requires at least Julia 1.12 (`Fix1` and `Fix2` are available earlier, too).
 
 !!! note
-    When nesting multiple `Fix`, note that the `N` in `Fix{N}` is _relative_ to the current
+    When nesting multiple `fix`, note that the `n` in `fix(n)` is _relative_ to the current
     available arguments, rather than an absolute ordering on the target function. For example,
-    `Fix{1}(Fix{2}(f, 4), 4)` fixes the first and second arg, while `Fix{2}(Fix{1}(f, 4), 4)`
+    `fix(1)(fix(2)(f, 4), 4)` fixes the first and second arg, while `fix(2)(fix(1)(f, 4), 4)`
     fixes the first and third arg.
-"""
-struct Fix{N,F,T} <: Function
-    f::F
-    x::T
 
-    function Fix{N}(f::F, x) where {N,F}
-        if !(N isa Int)
-            throw(ArgumentError(LazyString("expected type parameter in `Fix` to be `Int`, but got `", N, "::", typeof(N), "`")))
-        elseif N < 1
-            throw(ArgumentError(LazyString("expected `N` in `Fix{N}` to be integer greater than 0, but got ", N)))
-        end
-        new{N,_stable_typeof(f),_stable_typeof(x)}(f, x)
-    end
-end
+### Examples
 
-function (f::Fix{N})(args::Vararg{Any,M}; kws...) where {N,M}
-    M < N-1 && throw(ArgumentError(LazyString("expected at least ", N-1, " arguments to `Fix{", N, "}`, but got ", M)))
-    return f.f(args[begin:begin+(N-2)]..., f.x, args[begin+(N-1):end]...; kws...)
-end
+```jldoctest
+julia> Base.fix(2)(-, 3)(7)
+4
 
-# Special cases for improved constant propagation
-(f::Fix{1})(arg; kws...) = f.f(f.x, arg; kws...)
-(f::Fix{2})(arg; kws...) = f.f(arg, f.x; kws...)
+julia> Base.fix(2) === Base.Fix2
+true
 
+julia> Base.fix(1)(Base.fix(2)(muladd, 3), 2)(5) === (x -> muladd(2, 3, x))(5)
+true
+```
 """
-Alias for `Fix{1}`. See [`Fix`](@ref Base.Fix).
+function fix(@nospecialize m::Integer)
+    n = Int(m)::Int
+    if n ≤ 0
+        throw(ArgumentError("the index of the partially applied argument must be positive"))
+    end
+    k = _TypeDomainNumbers.Utils.from_abs_int(n)
+    PartiallyAppliedFunction{typeof(k)}
+end
+
+"""
+    Fix1::UnionAll
+
+[`fix(1)`](@ref Base.fix).
 """
-const Fix1{F,T} = Fix{1,F,T}
+const Fix1 = fix(1)
 
 """
-Alias for `Fix{2}`. See [`Fix`](@ref Base.Fix).
+    Fix2::UnionAll
+
+[`fix(2)`](@ref Base.fix).
 """
-const Fix2{F,T} = Fix{2,F,T}
+const Fix2 = fix(2)
+
+# Special cases for improved constant propagation
+function (partial::Fix1)(x; kws...)
+    partial.f(partial.x, x; kws...)
+end
+function (partial::Fix2)(x; kws...)
+    partial.f(x, partial.x; kws...)
+end
 
 
 """

base/public.jl

@@ -14,7 +14,7 @@ public
     AsyncCondition,
     CodeUnits,
     Event,
-    Fix,
+    fix,
     Fix1,
     Fix2,
     Generator,

base/tuple.jl

@@ -1,5 +1,18 @@
 # This file is a part of Julia. License is MIT: https://julialang.org/license
 
+module _TupleTypeByLength
+    export Tuple1OrMore, Tuple2OrMore, Tuple32OrMore
+    const Tuple1OrMore = Tuple{Any, Vararg}
+    const Tuple2OrMore = Tuple{Any, Any, Vararg}
+    const Tuple32OrMore = Tuple{
+        Any, Any, Any, Any, Any, Any, Any, Any,
+        Any, Any, Any, Any, Any, Any, Any, Any,
+        Any, Any, Any, Any, Any, Any, Any, Any,
+        Any, Any, Any, Any, Any, Any, Any, Any,
+        Vararg{Any, N},
+    } where {N}
+end
+
 # Document NTuple here where we have everything needed for the doc system
 """
     NTuple{N, T}
@@ -358,11 +371,7 @@ map(f, t::Tuple{Any, Any})      = (@inline; (f(t[1]), f(t[2])))
 map(f, t::Tuple{Any, Any, Any}) = (@inline; (f(t[1]), f(t[2]), f(t[3])))
 map(f, t::Tuple)                = (@inline; (f(t[1]), map(f,tail(t))...))
 # stop inlining after some number of arguments to avoid code blowup
-const Any32{N} = Tuple{Any,Any,Any,Any,Any,Any,Any,Any,
-                       Any,Any,Any,Any,Any,Any,Any,Any,
-                       Any,Any,Any,Any,Any,Any,Any,Any,
-                       Any,Any,Any,Any,Any,Any,Any,Any,
-                       Vararg{Any,N}}
+const Any32 = _TupleTypeByLength.Tuple32OrMore
 const All32{T,N} = Tuple{T,T,T,T,T,T,T,T,
                          T,T,T,T,T,T,T,T,
                          T,T,T,T,T,T,T,T,