notebook.jl

### A Pluto.jl notebook ###
# v0.19.11

using Markdown
using InteractiveUtils

# ╔═╡ d5559dba-9fe4-11ec-3744-ebd1408e7dc4
using AbstractTrees, PlutoUI, HypertextLiteral, PlutoTest, LinearAlgebra, Plots

# ╔═╡ 4da7a594-aec5-4c8d-ab20-6913ed6ff8ac
TableOfContents(title="📚 Table of Contents", indent=true, depth=4, aside=true)

# ╔═╡ 1b768306-0e0e-414d-a501-5c62a2db720f
md"## Computations as Trees"

# ╔═╡ 6733267e-1251-49a7-86a0-ff45ba408205
md"### Tracking variables"

# ╔═╡ 17f9e2c0-aa70-4d4e-ba42-51782df43785
md"### Computational Tree (Forward Pass)"

# ╔═╡ 7299bfea-e23e-4993-bebd-9d8762427294
md"### Pretty Viz: Forward Pass + Reverse Mode Diff"

# ╔═╡ be693f92-de60-4b43-82b3-ee6cceea310a
md"""

$\frac{\partial}{\partial x} (2xy + (x-1)^2) = 2y + 2(x-1)$
$\frac{\partial}{\partial y} (2xy + (x-1)^2) = 2x$

If x = 3 and y = 5, then these evaluate to  14 and 6.
"""

# ╔═╡ 20571e1a-9687-4d42-98b3-b3bc0b207b3b
md"## Implementation in Code"

# ╔═╡ 5e02b2c0-10cf-4746-840d-7017648f89f8
md"""
Besides the mathematics of reverse mode differentiation, there are many interesting computer science issues regarding the implementation of reverse diff.  Probably the jury is still out which is best.

The implementation below is based on Julia's dispatch system.
This means we create a type `Tracked` for keeping track of our input variables and everything we'll need to calculate later.

The other main approach (not illustrated here) is source to source translation of code. In Julia the zygote package translates at the high level, while the increasingly popular Enzyme package translates at the lower level.
"""

# ╔═╡ d1312080-272c-4bd5-bcfe-bf47f8320961
md"### The Tracked Data Structure"

# ╔═╡ 99b6ab91-a022-449c-988c-0e5c5719c910
begin

	struct Tracked{T} <: Number
		val::T
		name::Symbol
		df
		deps::Vector{Tracked}
	end

	# the two argument version as in Tracked{Int}(3,:x)
	Tracked{T}(x, name=gensym()) where {T} = Tracked{T}(x, name, nothing, Tracked[])

	## The above is typical Julia, the below shows up often when you 
	## define a new number type and you want ordinary numbers to work together
	## with your newly defined numbers.
	## This is similar to when in mathematics, strictly speaking integers
	## are separate from the reals, but we all treat integers as embedded
	## in the reals.

	# Converting a Tracked variable, converts like the number
	Base.convert(T::Type{Tracked{S}}, x::Tracked) where {S} = T(convert(S, x.val), x.name, x.df, x.deps)

	# This tells Julia to convert any number added to a `Tracked` to a `Tracked` first
	Base.promote_rule(::Type{Tracked{S}}, ::Type{T}) where {S<:Number, T<:Number} = Tracked{promote_type(S, T)}

	# All this is just for nicer printing
	function Base.show(io::IO, x::Tracked)
		if x.df === nothing
			print(io, Base.isgensym(x.name) ? x.val : "$(x.val)=$(x.name)")
		else
			#print(io, "⋅Tracked(")
			print(io,"[")
			print(io, x.name)
			print(io,"]")
			#print(io, ", ")
			 print(io," → ")
			show(io, x.val)
			#print(io, ")")
		end
	end
	function Base.show(io::IO, x::Tracked{<:AbstractArray})
		io = IOContext(io, :compact=>true)
		print(io, Base.isgensym(x.name) ? x.val : string(x.name))
	end
	Base.show(io::IO, ::MIME"text/plain", x::Tracked) = print_tree(io, x)

	Base.:(==)(x::Tracked, y::Tracked) = x === y
end

# ╔═╡ 0b5e6560-81fd-4182-bba5-aca702fb3048
begin
   # The second argument of Tracked is a symbol only used for printing
   x = Tracked{Int}(3, :x)
   y = Tracked{Int}(5, :y)
   u = Tracked{Int}(2, :u)
end

# ╔═╡ 0d006e4d-72e3-42ab-beb3-e5304a4d9ba9
x

# ╔═╡ 8919faec-753d-41d0-b0a8-c84c025b2919
xfloat = convert( Tracked{Float64}, x)

# ╔═╡ be28f764-a415-46ed-a503-68e8d831f8a8
(aa,bb) = (promote(x,2.5)) # promotes x to be a float and also y to be a Tracked

# ╔═╡ a0c0f48d-e0b0-4f45-8e0e-d9ab5743958b
Dump(aa)

# ╔═╡ 814786be-a760-4023-a614-83e727681075
Dump(bb)

# ╔═╡ 2ba83769-89b4-4ba0-8726-9210f394b4e7
md"### Operations on Tracked"

# ╔═╡ 885cd51d-895f-4996-a23b-780498b5b810
md"""
All overloads will do the operation (e.g. sum `x` and `y`), but also remember the pullback map and input variables for the reverse pass.
"""

# ╔═╡ 4813e57f-557a-4179-afd2-7925687b3d35
## Simeon write your way too
# These operations are

# d(x/y)= (y dx - x dy)/y^2
# ∇(x/y) = [1/y; -x/y^2] # Notice that the Δ -> (?,?) is just the gradient

# ╔═╡ ea94bdf9-b0b1-42ad-ba44-aaba73ed1192
md"""
Let's look at our example from above, $z = x^2+y$.
We'll introduce an intermediate variable $u = x^2$ so we can write $z$ as $u + y$.

We will first start with $1 \cdot dz$. The one here represents the fact that $\frac{\partial z}{\partial z} = 1$.

We then want to *pull* that *back* until we get something in terms of $dx$ and $dy$, so we then get the derivatives wrt $x$ and $y$

The pullback map for $+$ will first map $dz$ to $d(u + y) = du + dy$ and then we can use the pullback map for ^ to expand $du$ to $2x \cdot dx$.

Using that recursive method we end up with $dz = 2x \cdot dx + dy$, which is exactly what we'd get if we did this by hand. This tells us that $\frac{\partial z}{\partial x} = 2x$ and $\frac{\partial z}{\partial y} = y$.
"""

# ╔═╡ 13487e65-5e48-4a37-9bea-f262dd7b6d56
function Base.:+(x::Tracked, y::Tracked)
	Tracked(x.val + y.val, :+, Δ -> (Δ, Δ), Tracked[x, y])
end

# ╔═╡ c9d70049-4793-41db-9075-e9e73e926c1a
x+y

# ╔═╡ b0cc4665-eb45-48ea-9a33-5acf56d2a283
function Base.:-(x::Tracked, y::Tracked)
	Tracked(x.val - y.val, :-, Δ -> (Δ, -Δ), Tracked[x, y])
end

# ╔═╡ 73d638bf-30c1-4694-b3a8-4b29c5e3fa65
function Base.:*(x::Tracked, y::Tracked)
	Tracked(x.val * y.val, :*, Δ -> (Δ * y.val', x.val' * Δ), Tracked[x, y])
end

# ╔═╡ 95ab6c83-0757-40b3-a116-6d872a074f69
x*y

# ╔═╡ db4f871a-3ca1-4bd3-a2c5-da7d6f599281
(x*y).val

# ╔═╡ 29360c2a-c47b-48a3-b225-248eb3ec8a42
(x*y).name

# ╔═╡ b47754ac-cb17-4ea3-a97d-20371fd341a5
((x*y).df)(1) # For reference

# ╔═╡ bb75e790-a628-4821-b7b1-11cd34a73d38
(x*y).deps

# ╔═╡ ac097299-0a31-474c-ab26-a4fb24bb9046
function Base.:^(x::Tracked, n::Int)
	Tracked(x.val^n, Symbol("^$n"), Δ -> (Δ * n * x.val^(n-1),), Tracked[x,])
end

# ╔═╡ aa99b123-7164-4ed7-833f-b99b19118a0d
x^2+y

# ╔═╡ 0d2a3187-46f1-4705-bb3e-ecf74e34840d
z = (2x*y + (x-1)^2)

# ╔═╡ 61513f88-c126-43ab-9515-85a2c8a472bf
(x^2+y).deps

# ╔═╡ 2c3f3103-9ccb-446e-a289-87bbd0a1b97b
(x^2+y).deps[2]

# ╔═╡ 795c510e-6280-42d9-8820-290ca07be88d
((x^2+y).deps[2]).deps

# ╔═╡ e450329b-46ee-4f53-b6f0-891d6596938c
x^2+y

# ╔═╡ d48b9f23-b1b8-43f8-a05d-2234b69bf1f0
(2x*y + (x-1)^2)

# ╔═╡ 2141849b-675e-406c-8df4-34b2706507af
function Base.:/(x::Tracked, y::Tracked)
	Tracked(x.val / y.val, :/, Δ -> (Δ / y.val, -Δ * x.val / y.val^2), Tracked[x, y])
end

# ╔═╡ 8ab0f55d-a393-4a8a-a48c-9ced26033f57
function Base.sin(x::Tracked)
	Tracked(sin(x.val), :sin, Δ -> (Δ * cos(x.val),), Tracked[x,])
end

# ╔═╡ 4bfc2f7d-a5b0-44c7-8bb6-f1b834c1cc51
md"""
`Tracked` is a tree -- We just need to tell *AbstractTrees.jl* how to get the children for each node and we get tree printing and iteration over all nodes for free.
"""

# ╔═╡ 2188a663-5a85-4ce4-bc8d-20383481e59b
AbstractTrees.children(x::Tracked) = x.deps

# ╔═╡ ccafd0b9-95aa-4e58-8afc-26cd3ee61cc9
md"""
Straight away we get the primal result of our calculation:
"""

# ╔═╡ 81eb8a2d-a3a9-45af-a5a5-b96aefd48712
(2x*y + (x-1)^2).val # The result of `2x*y + (x-1)^2`

# ╔═╡ 01eacbd4-ef37-4524-aecc-2ef9a1044cf8
md"""
To also get the gradient, we'll use `PreOrderDFS` to traverse the tree we just created from the top down.
"""

# ╔═╡ e52aa672-69a9-419b-a992-e7a3d1364fb6
# `PreOrderDFS` traverses this tree from the top down
Text.(collect(PreOrderDFS(z)))

# ╔═╡ d5565717-239b-41ab-b311-d59b383130ed
md"""
Ok, let's create our function `grad` which will accumulate all intermediate gradients into a dictionary:
"""

# ╔═╡ 99a3507b-ca03-429f-acde-e2d1ebb32054
function grad(f::Tracked)
	d = Dict{Any, Any}(f => 1)
	for x in PreOrderDFS(f) # recursively traverse all dependents
		x.df === nothing && continue # ignore untracked variables like constants
		dy = x.df(d[x]) # evaluate pullback
		for (yᵢ, dyᵢ) in zip(x.deps, dy)
			# store the gradient in d
			# if we have already stored a gradient for this variable, we need to add them
			d[yᵢ] = get(d, yᵢ, 0) .+ dyᵢ
		end
	end
	return d
end

# ╔═╡ d4e9b202-242e-4420-986b-12d2ab57af93
grad(f::Tracked, x::Tracked) = grad(f)[x]

# ╔═╡ 7fcccb65-4a7b-4527-97be-d25f481f6eaf
md"""
We can verify that it does the right thing:
"""

# ╔═╡ 18b1c55d-a6b5-44f6-b0b3-50bdb0aa9d96
w = x*y + x

# ╔═╡ 506d408e-dc2b-4e12-b917-286e3f4079a2
grad(w)

# ╔═╡ a7c8cb6a-6e17-4d8f-8958-fe3527c5b8e7
grad(w, x), grad(w, y)

# ╔═╡ e55dfad2-db50-459f-ab54-fa7637fc3638
md"""
## How can we visualize both the forward and the reverse pass?

We can further visualize each steps we just took. First we do the forwards calculation, where we also build up our tree, then we go down the tree in the opposite direction to accumulate our gradient.
"""

# ╔═╡ d9304d8d-9a34-46f2-908d-42d5cc5f5c5f
👽 = Tracked{Int}(12, :👽)

# ╔═╡ a6c2d3a2-326a-41dd-864d-aa3662466222
x,y

# ╔═╡ bcff2aa8-2387-44c7-a28f-39cd505a7adf
md"""
We can also visualize what Julia does in the forward pass on the code itself:
"""

# ╔═╡ 53ae4070-8818-4d21-8648-19df9319918a
md"""
## Let's write our own neural network!
"""

# ╔═╡ 8a70e835-dae6-4727-8204-95c87d5c23da
md"""
We need some more overloads
"""

# ╔═╡ 520f280d-c78e-433b-a0a2-8ef05b04f7cc
function Base.broadcasted(::typeof(tanh), x::Tracked)
	res = tanh.(x.val)
	Tracked(res, :tanh, Δ -> (Δ .* (1 .- res.^2),), Tracked[x,])
end

# ╔═╡ a5671f2f-48ca-4896-9428-147dc671d2b9
function Base.broadcasted(::typeof(+), x::Tracked{<:VecOrMat}, y::Tracked{<:Vector})
	Tracked(x.val .+ y.val, :.+, Δ -> (Δ, sum(Δ; dims=2)), Tracked[x, y])
end

# ╔═╡ e01114fa-a847-424a-931f-8f42e623109f
function LinearAlgebra.norm(x::Tracked)
	res = norm(x.val)
	Tracked(res, :norm, Δ -> (Δ/res .* x.val,), Tracked[x,])
end

# ╔═╡ e6ee1921-57aa-4194-b2eb-8e32fa4a6c44
macro t(s)
	:(Tracked{typeof($(esc(s)))}($(esc(s)), $(QuoteNode(s))))
end

# ╔═╡ 266bf2ed-d625-47be-8a87-21e12052b27f
md"""
Let's use three dense layers, all of the form

$x \mapsto \tanh.(W \cdot x + b)$
"""

# ╔═╡ 788903f0-a372-4265-bbf3-5b507a705100
NN(x, (W1, W2, W3, b1, b2, b3)) = W3 * tanh.(W2 * tanh.(W1 * @t(x) .+ b1) .+ b2) .+ b3

# ╔═╡ 337bc837-174f-4ed6-bee1-c9e80dc03b52
input = collect(range(0, 2π; length=100))'

# ╔═╡ 8f1ec8d7-89ab-4bc5-9eef-2b0a69019854
md"""
We'll try to approximate a sine curve:
"""

# ╔═╡ d2d21d41-9a4e-428b-b696-6878901849ba
ŷ = sin.(input) #.+ .2input .^ 2 .- cos.(5input)

# ╔═╡ 98807f0d-f8a5-4fe0-9983-2b95290347d9
md"""
### Tada! 🎉

(It's even reasonably fast for neural nets)
"""

# ╔═╡ 85098c31-04a1-421c-b3f6-25fec9360588
md"### What if we use a (leaky) rectified linear unit (ReLU) as activation?"

# ╔═╡ 7dca2c78-6faf-450a-8096-1c5b9f7f1295
leaky_relu(x) = x > 0 ? x : .1x

# ╔═╡ b2622dc4-2fd3-4d80-8efd-a6a985a6b2e9
function Base.broadcasted(::typeof(leaky_relu), x::Tracked)
	Tracked(leaky_relu.(x.val), :tanh, Δ -> (map((x, Δ) -> x > 0 ? Δ : .1Δ, x.val, Δ),), Tracked[x,])
end

# ╔═╡ 0911a08a-4290-455b-9b26-0bf2862296da
plot(leaky_relu; label="leaky relu")

# ╔═╡ 0b094198-cf44-41d7-a8dc-fd8fd0716bb4
md"""
---
### Helper Functions
"""

# ╔═╡ d82adc20-4c8c-4f2c-9839-d03ad7e7f581
begin
	struct EX
		x::Any
		function EX(ex)
			if Meta.isexpr(ex, :call) && ex.args[1] === :+ && length(ex.args) > 3
				new(Expr(:call, :+, Expr(:call, :+, ex.args[2:end-1]...), ex.args[end]))
			else
				new(ex)
			end
		end
	end
	show_tree(ex::Expr) = show_tree(EX(ex))
	function Base.show(io::IO, ex::EX)
		Base.show_unquoted(io, Meta.isexpr(ex.x, :call) ? ex.x.args[1] : ex.x)
		if Meta.isexpr(ex.x, :call) && ex.x.args[1] === :^
			print(io, ex.x.args[3])
		end
	end
	function AbstractTrees.children(ex::EX)
		if Meta.isexpr(ex.x, :call)
			ex.x.args[1] === :^ ? [EX(ex.x.args[2])] : EX.(ex.x.args[2:end])
		else
			EX[]
		end
	end
	Base.:(==)(ex1::EX, ex2::EX) = ex1.x == ex2.x
	Base.hash(ex::EX, i::UInt) = hash(ex.x, i)
end

# ╔═╡ 96286b65-1a22-4458-a399-46579248cce4
begin
	_rand(s...) = randn(s...) / √prod(s)
	W1, W2, W3 = _rand(32, 1), _rand(32, 32), _rand(1, 32)
	b1, b2, b3 = _rand(32), _rand(32), _rand(1)
	params = [@t(W1), @t(W2), @t(W3), @t(b1), @t(b2), @t(b3)]
	p = @animate for i in 1:10000
		loss = norm(NN(input, params) - @t(ŷ))
		∇ = grad(loss)
		for p in params
			p.val .-= 1e-3 .* ∇[p]
		end
		plot(input', [NN(input, params).val; ŷ]'; label=["prediction" "training data"])
	end every 200
	gif(p; fps=5)
end

# ╔═╡ 49333dc7-b809-4bb1-b285-20a12d8b9209
let ŷ_t = @t(ŷ)

	loss = norm(NN(input, params) - ŷ_t)
	∇ = grad(loss)
	params′ = deepcopy(params)
	for (p, p′) in zip(params, params′)
		p′.val .-= 1e-3 .* ∇[p]
	end

	plot(input', [NN(input, params).val; ŷ]'; label=["prediction" "training data"])
	quiver!(input[1:4:end], NN(input, params).val[1:4:end]; quiver=(zero(input[1:4:end]), 3(NN(input, params′).val .- NN(input, params).val)[1:4:end]))
end

# ╔═╡ 86fa378b-815d-4c3d-9121-1338ee54f30f
let
	NN(x, (W1, W2, W3, b1, b2, b3)) = W3 * leaky_relu.(W2 * leaky_relu.(W1 * @t(x) .+ b1) .+ b2) .+ b3

	_rand(s...) = randn(s...) / √prod(s)
	W1, W2, W3 = _rand(32, 1), _rand(32, 32), _rand(1, 32)
	b1, b2, b3 = _rand(32), _rand(32), _rand(1)
	params = [@t(W1), @t(W2), @t(W3), @t(b1), @t(b2), @t(b3)]
	p = @animate for i in 1:10000
		loss = norm(NN(input, params) - @t(ŷ))
		∇ = grad(loss)
		for p in params
			p.val .-= 1e-3 .* ∇[p]
		end
		plot(input', [NN(input, params).val; ŷ]'; label=["prediction" "training data"])
	end every 200
	gif(p; fps=5)
end

# ╔═╡ 8110f306-a7bb-43a2-bb36-6182c59b4b2e
begin
	struct TTREE
		x
		d::Dict{Any, Any}
	end
	function Base.show(io::IO, x::TTREE)
		show(io, x.x)
		print(io, " ")
		show(io, MIME("text/html"), get(x.d, x.x, @htl("")))
	end
	AbstractTrees.children(x::TTREE) = (TTREE(i, x.d) for i in children(x.x))
end

# ╔═╡ 1f1b384a-6588-45a5-9dd3-6de3face8bfb
function ad_steps(x::Expr; color_fwd="red", color_bwd="green", font_size=".8em")
	x = EX(x)
	repr(x) = sprint(show, x; context=:compact=>true)
	span_fwd = @htl "<span style='color: $color_fwd; font-size: $font_size'>"
	span_bwd = @htl "<span style='color: $color_bwd; font-size: $font_size'>"

    d1 = Dict(
		let e = eval(i.x)
			i => @htl "&ensp;$(span_fwd)$(repr(e isa Tracked ?  e.val : e))</span>"
		end
		for i in PostOrderDFS(x) if isempty(children(i))
	)

	res = accumulate(Iterators.filter(x -> !isempty(children(x)), PostOrderDFS(x)); init=d1) do d,i
		d = copy(d)
		e = eval(i.x)
		d[i] = @htl "&ensp;$(span_fwd)$(repr(e isa Tracked ?  e.val : e))</span>" 
		d
	end

	pushfirst!(res, d1)

	f = eval(x.x)
	d = Dict{Any, Any}(f => 1)
	let d1 = copy(res[end])
		d1[x] = @htl "$(d1[x])&ensp;$(span_bwd)1</span>"
		push!(res, d1)
	end
	for (x, e) in zip(PreOrderDFS.((f, x))...)
		x.df === nothing && continue
		dy = x.df(d[x])
		for (yᵢ, dyᵢ, e) in zip(x.deps, dy, children(e))
			d1 = copy(res[end])
			if haskey(d, yᵢ)
				d1[e] = @htl "$(get(d1, e, ""))$(span_bwd) + $(repr(dyᵢ))</span>"
			else
				d1[e] = @htl "$(get(d1, e, ""))&ensp;$(span_bwd)$(repr(dyᵢ))</span>"
			end
			push!(res, d1)
			d[yᵢ] = get(d, yᵢ, 0) + dyᵢ
		end
	end
	res
end

# ╔═╡ 1a154bb7-93a3-4973-8908-788db77ac294
s2 = @htl """
<link rel="stylesheet" href="https://fperucic.github.io/treant-js/Treant.css"/>
<style>
.Treant > .node {
	padding: 5px; border: 2px solid #484848; border-radius: 8px;
	box-sizing: unset;
	min-width: fit-content;
	font-size: 1.6em;
}
.Treant > .node > span {
	vertical-align: middle;
}

.Treant .collapse-switch { width: 100%; height: 100%; border: none; }
.Treant .node.collapsed { background-color: var(--main-bg-color); }
.Treant .node.collapsed .collapse-switch { background: none;}
</style>

<script src="https://fperucic.github.io/treant-js/vendor/jquery.min.js"></script>
<script src="https://fperucic.github.io/treant-js/vendor/jquery.easing.js"></script>
<script src="https://fperucic.github.io/treant-js/vendor/raphael.js"></script>
<script src="https://fperucic.github.io/treant-js/Treant.js"></script>
"""

# ╔═╡ 6b1fb808-e993-4c2b-b81b-6710f8206de7
function to_json(x)
	d = Dict{Symbol, Any}(
		:innerHTML => sprint(AbstractTrees.printnode, x),
		:children => Any[to_json(c) for c in children(x)],
		#:collapsed => !isempty(children(x)),
	)
end

# ╔═╡ 437285d4-ec53-4bb7-9966-fcfb5352e205
function show_tree(x; height=400)
	id = gensym()
	@htl """
	$s2
	<div id="$id" style="width:100%; height: $(height)px"> </div>
	<script>
	var simple_chart_config = {
		chart: {
			container: "#$id",

			//animateOnInit: true,

			node: {
				collapsable: true,
			},

			nodeAlign: "BOTTOM",

			connectors: {
				type: "straight",
				style: {
					stroke: getComputedStyle(document.documentElement).getPropertyValue('--cm-editor-text-color')
				}
			},
			animation: {
				nodeAnimation: "easeOutBounce",
				nodeSpeed: 500,
				connectorsAnimation: "bounce",
				connectorsSpeed: 500
			},
		},

		nodeStructure: $(to_json(x))
	};
	var my_chart = new Treant(simple_chart_config);
	</script>
	"""
end

# ╔═╡ 86aa821b-a373-4814-953e-535f3a33c002
show_tree(norm(NN(input, params) - @t(ŷ)); height = 1000)

# ╔═╡ 094bf7ee-7c9b-458c-98dc-521768831654
function show_steps(steps; height=400)
	id1, id2 = gensym(), gensym()
	@htl """
	$s2
	<div id="$id1" style="width:100%; height: $(height)px"> </div>
	<span style="color: red; font-size: 1.5em; margin-bottom: 1em">Move me!</span>
	<input id="$id2" type="range" min="0" max="$(length(steps)-1)" value="0" style="width: 100%">
	<script>
	var steps = $(to_json.(steps));
	var simple_chart_config = {
		chart: {
			container: "#$id1",

			//animateOnInit: true,

			node: {
				collapsable: true,
			},

			nodeAlign: "BOTTOM",

			connectors: {
				type: "straight",
				style: {
					stroke: getComputedStyle(document.documentElement).getPropertyValue('--cm-editor-text-color')
				}
			},
			animation: {
				nodeAnimation: "easeOutBounce",
				nodeSpeed: 500,
				connectorsAnimation: "bounce",
				connectorsSpeed: 500
			},
		},
		nodeStructure: steps[0]
	};
	var my_chart = new Treant(simple_chart_config, null, \$);
	var slider = document.getElementById("$id2");
	slider.addEventListener('input', function(){
		simple_chart_config.nodeStructure = steps[this.value];
		document.getElementById("$id1").innerHTML = "";
		my_chart = new Treant(simple_chart_config, null, \$);
	});
	</script>
	"""
end

# ╔═╡ faf69c6f-b179-4ca2-aa22-641e9560e17b
macro ad_steps(ex)
	quote
		ex = $(QuoteNode(ex))
		steps = ad_steps(ex)
		show_steps([TTREE(EX(ex), step) for step in steps])
	end
end

# ╔═╡ 27b39d7d-fc08-4ccc-aea4-b64f8a4f5726
#@ad_steps 3y*x + 2(x-1)*x
@ad_steps (2x*y + (x-1)^2)  # type the steps for the macro

# ╔═╡ ec723b90-1c8b-4a87-8a86-fe17d6e85f22
@ad_steps (x+y)

# ╔═╡ 0d186a1b-eb89-4be8-a237-e5a9608f4a30
@ad_steps (x*y)

# ╔═╡ db3389c0-cb00-404c-81a8-8c4ccdab22f4
@ad_steps ((x*y)*u)

# ╔═╡ 8b486bd8-25a1-4895-99c8-0541c9e3c8b0
@ad_steps (x/y)

# ╔═╡ 2eb6af9c-9ac7-4b64-93fb-c5a2f037d303
@ad_steps y^2

# ╔═╡ f6ce8448-d9ce-4453-9e47-dc6443d50f55
s1 = html"""
<style>
p-frame-viewer {
	display: inline-flex;
	flex-direction: column;
}
p-frames,
p-frame-controls {
	display: inline-flex;
}
p-frame-controls {
	margin-top: 20px;
}
line-like {
	font-size: 30px;
}
"""

# ╔═╡ 9a141034-17cb-4d85-a5a2-4724a38dd269
macro visual_debug(expr)
	s1
	quote
		$(esc(:(PlutoTest.@eval_step_by_step($expr)))) .|> PlutoTest.SlottedDisplay |> PlutoTest.frames |> PlutoTest.with_slotted_css
	end
end

# ╔═╡ 79f71f9d-b491-4a2c-85a4-29ae8da4f312
@visual_debug 3y + 2(x-1)

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
AbstractTrees = "1520ce14-60c1-5f80-bbc7-55ef81b5835c"
HypertextLiteral = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoTest = "cb4044da-4d16-4ffa-a6a3-8cad7f73ebdc"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"

[compat]
AbstractTrees = "~0.4.2"
HypertextLiteral = "~0.9.4"
Plots = "~1.31.7"
PlutoTest = "~0.2.2"
PlutoUI = "~0.7.39"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised

julia_version = "1.8.0"
manifest_format = "2.0"
project_hash = "eceb496e896aa6e7e79fc43936b5726532da0604"

[[deps.AbstractPlutoDingetjes]]
deps = ["Pkg"]
git-tree-sha1 = "8eaf9f1b4921132a4cff3f36a1d9ba923b14a481"
uuid = "6e696c72-6542-2067-7265-42206c756150"
version = "1.1.4"

[[deps.AbstractTrees]]
git-tree-sha1 = "5c0b629df8a5566a06f5fef5100b53ea56e465a0"
uuid = "1520ce14-60c1-5f80-bbc7-55ef81b5835c"
version = "0.4.2"

[[deps.Adapt]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "195c5505521008abea5aee4f96930717958eac6f"
uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
version = "3.4.0"

[[deps.ArgTools]]
uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f"
version = "1.1.1"

[[deps.Artifacts]]
uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33"

[[deps.Base64]]
uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f"

[[deps.Bzip2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "19a35467a82e236ff51bc17a3a44b69ef35185a2"
uuid = "6e34b625-4abd-537c-b88f-471c36dfa7a0"
version = "1.0.8+0"

[[deps.Cairo_jll]]
deps = ["Artifacts", "Bzip2_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "JLLWrappers", "LZO_jll", "Libdl", "Pixman_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll", "Zlib_jll", "libpng_jll"]
git-tree-sha1 = "4b859a208b2397a7a623a03449e4636bdb17bcf2"
uuid = "83423d85-b0ee-5818-9007-b63ccbeb887a"
version = "1.16.1+1"

[[deps.ChainRulesCore]]
deps = ["Compat", "LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "80ca332f6dcb2508adba68f22f551adb2d00a624"
uuid = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
version = "1.15.3"

[[deps.ChangesOfVariables]]
deps = ["ChainRulesCore", "LinearAlgebra", "Test"]
git-tree-sha1 = "38f7a08f19d8810338d4f5085211c7dfa5d5bdd8"
uuid = "9e997f8a-9a97-42d5-a9f1-ce6bfc15e2c0"
version = "0.1.4"

[[deps.CodecZlib]]
deps = ["TranscodingStreams", "Zlib_jll"]
git-tree-sha1 = "ded953804d019afa9a3f98981d99b33e3db7b6da"
uuid = "944b1d66-785c-5afd-91f1-9de20f533193"
version = "0.7.0"

[[deps.ColorSchemes]]
deps = ["ColorTypes", "ColorVectorSpace", "Colors", "FixedPointNumbers", "Random"]
git-tree-sha1 = "1fd869cc3875b57347f7027521f561cf46d1fcd8"
uuid = "35d6a980-a343-548e-a6ea-1d62b119f2f4"
version = "3.19.0"

[[deps.ColorTypes]]
deps = ["FixedPointNumbers", "Random"]
git-tree-sha1 = "eb7f0f8307f71fac7c606984ea5fb2817275d6e4"
uuid = "3da002f7-5984-5a60-b8a6-cbb66c0b333f"
version = "0.11.4"

[[deps.ColorVectorSpace]]
deps = ["ColorTypes", "FixedPointNumbers", "LinearAlgebra", "SpecialFunctions", "Statistics", "TensorCore"]
git-tree-sha1 = "d08c20eef1f2cbc6e60fd3612ac4340b89fea322"
uuid = "c3611d14-8923-5661-9e6a-0046d554d3a4"
version = "0.9.9"

[[deps.Colors]]
deps = ["ColorTypes", "FixedPointNumbers", "Reexport"]
git-tree-sha1 = "417b0ed7b8b838aa6ca0a87aadf1bb9eb111ce40"
uuid = "5ae59095-9a9b-59fe-a467-6f913c188581"
version = "0.12.8"

[[deps.Compat]]
deps = ["Dates", "LinearAlgebra", "UUIDs"]
git-tree-sha1 = "5856d3031cdb1f3b2b6340dfdc66b6d9a149a374"
uuid = "34da2185-b29b-5c13-b0c7-acf172513d20"
version = "4.2.0"

[[deps.CompilerSupportLibraries_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "e66e0078-7015-5450-92f7-15fbd957f2ae"
version = "0.5.2+0"

[[deps.Contour]]
git-tree-sha1 = "d05d9e7b7aedff4e5b51a029dced05cfb6125781"
uuid = "d38c429a-6771-53c6-b99e-75d170b6e991"
version = "0.6.2"

[[deps.DataAPI]]
git-tree-sha1 = "fb5f5316dd3fd4c5e7c30a24d50643b73e37cd40"
uuid = "9a962f9c-6df0-11e9-0e5d-c546b8b5ee8a"
version = "1.10.0"

[[deps.DataStructures]]
deps = ["Compat", "InteractiveUtils", "OrderedCollections"]
git-tree-sha1 = "d1fff3a548102f48987a52a2e0d114fa97d730f0"
uuid = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8"
version = "0.18.13"

[[deps.DataValueInterfaces]]
git-tree-sha1 = "bfc1187b79289637fa0ef6d4436ebdfe6905cbd6"
uuid = "e2d170a0-9d28-54be-80f0-106bbe20a464"
version = "1.0.0"

[[deps.Dates]]
deps = ["Printf"]
uuid = "ade2ca70-3891-5945-98fb-dc099432e06a"

[[deps.DelimitedFiles]]
deps = ["Mmap"]
uuid = "8bb1440f-4735-579b-a4ab-409b98df4dab"

[[deps.DocStringExtensions]]
deps = ["LibGit2"]
git-tree-sha1 = "5158c2b41018c5f7eb1470d558127ac274eca0c9"
uuid = "ffbed154-4ef7-542d-bbb7-c09d3a79fcae"
version = "0.9.1"

[[deps.Downloads]]
deps = ["ArgTools", "FileWatching", "LibCURL", "NetworkOptions"]
uuid = "f43a241f-c20a-4ad4-852c-f6b1247861c6"
version = "1.6.0"

[[deps.EarCut_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3f3a2501fa7236e9b911e0f7a588c657e822bb6d"
uuid = "5ae413db-bbd1-5e63-b57d-d24a61df00f5"
version = "2.2.3+0"

[[deps.Expat_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "bad72f730e9e91c08d9427d5e8db95478a3c323d"
uuid = "2e619515-83b5-522b-bb60-26c02a35a201"
version = "2.4.8+0"

[[deps.Extents]]
git-tree-sha1 = "5e1e4c53fa39afe63a7d356e30452249365fba99"
uuid = "411431e0-e8b7-467b-b5e0-f676ba4f2910"
version = "0.1.1"

[[deps.FFMPEG]]
deps = ["FFMPEG_jll"]
git-tree-sha1 = "b57e3acbe22f8484b4b5ff66a7499717fe1a9cc8"
uuid = "c87230d0-a227-11e9-1b43-d7ebe4e7570a"
version = "0.4.1"

[[deps.FFMPEG_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "JLLWrappers", "LAME_jll", "Libdl", "Ogg_jll", "OpenSSL_jll", "Opus_jll", "Pkg", "Zlib_jll", "libaom_jll", "libass_jll", "libfdk_aac_jll", "libvorbis_jll", "x264_jll", "x265_jll"]
git-tree-sha1 = "ccd479984c7838684b3ac204b716c89955c76623"
uuid = "b22a6f82-2f65-5046-a5b2-351ab43fb4e5"
version = "4.4.2+0"

[[deps.FileWatching]]
uuid = "7b1f6079-737a-58dc-b8bc-7a2ca5c1b5ee"

[[deps.FixedPointNumbers]]
deps = ["Statistics"]
git-tree-sha1 = "335bfdceacc84c5cdf16aadc768aa5ddfc5383cc"
uuid = "53c48c17-4a7d-5ca2-90c5-79b7896eea93"
version = "0.8.4"

[[deps.Fontconfig_jll]]
deps = ["Artifacts", "Bzip2_jll", "Expat_jll", "FreeType2_jll", "JLLWrappers", "Libdl", "Libuuid_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "21efd19106a55620a188615da6d3d06cd7f6ee03"
uuid = "a3f928ae-7b40-5064-980b-68af3947d34b"
version = "2.13.93+0"

[[deps.Formatting]]
deps = ["Printf"]
git-tree-sha1 = "8339d61043228fdd3eb658d86c926cb282ae72a8"
uuid = "59287772-0a20-5a39-b81b-1366585eb4c0"
version = "0.4.2"

[[deps.FreeType2_jll]]
deps = ["Artifacts", "Bzip2_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "87eb71354d8ec1a96d4a7636bd57a7347dde3ef9"
uuid = "d7e528f0-a631-5988-bf34-fe36492bcfd7"
version = "2.10.4+0"

[[deps.FriBidi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "aa31987c2ba8704e23c6c8ba8a4f769d5d7e4f91"
uuid = "559328eb-81f9-559d-9380-de523a88c83c"
version = "1.0.10+0"

[[deps.GLFW_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pkg", "Xorg_libXcursor_jll", "Xorg_libXi_jll", "Xorg_libXinerama_jll", "Xorg_libXrandr_jll"]
git-tree-sha1 = "d972031d28c8c8d9d7b41a536ad7bb0c2579caca"
uuid = "0656b61e-2033-5cc2-a64a-77c0f6c09b89"
version = "3.3.8+0"

[[deps.GR]]
deps = ["Base64", "DelimitedFiles", "GR_jll", "HTTP", "JSON", "Libdl", "LinearAlgebra", "Pkg", "Printf", "Random", "RelocatableFolders", "Serialization", "Sockets", "Test", "UUIDs"]
git-tree-sha1 = "cf0a9940f250dc3cb6cc6c6821b4bf8a4286cf9c"
uuid = "28b8d3ca-fb5f-59d9-8090-bfdbd6d07a71"
version = "0.66.2"

[[deps.GR_jll]]
deps = ["Artifacts", "Bzip2_jll", "Cairo_jll", "FFMPEG_jll", "Fontconfig_jll", "GLFW_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pixman_jll", "Pkg", "Qt5Base_jll", "Zlib_jll", "libpng_jll"]
git-tree-sha1 = "2d908286d120c584abbe7621756c341707096ba4"
uuid = "d2c73de3-f751-5644-a686-071e5b155ba9"
version = "0.66.2+0"

[[deps.GeoInterface]]
deps = ["Extents"]
git-tree-sha1 = "fb28b5dc239d0174d7297310ef7b84a11804dfab"
uuid = "cf35fbd7-0cd7-5166-be24-54bfbe79505f"
version = "1.0.1"

[[deps.GeometryBasics]]
deps = ["EarCut_jll", "GeoInterface", "IterTools", "LinearAlgebra", "StaticArrays", "StructArrays", "Tables"]
git-tree-sha1 = "a7a97895780dab1085a97769316aa348830dc991"
uuid = "5c1252a2-5f33-56bf-86c9-59e7332b4326"
version = "0.4.3"

[[deps.Gettext_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "9b02998aba7bf074d14de89f9d37ca24a1a0b046"
uuid = "78b55507-aeef-58d4-861c-77aaff3498b1"
version = "0.21.0+0"

[[deps.Glib_jll]]
deps = ["Artifacts", "Gettext_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Libiconv_jll", "Libmount_jll", "PCRE_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "a32d672ac2c967f3deb8a81d828afc739c838a06"
uuid = "7746bdde-850d-59dc-9ae8-88ece973131d"
version = "2.68.3+2"

[[deps.Graphite2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "344bf40dcab1073aca04aa0df4fb092f920e4011"
uuid = "3b182d85-2403-5c21-9c21-1e1f0cc25472"
version = "1.3.14+0"

[[deps.Grisu]]
git-tree-sha1 = "53bb909d1151e57e2484c3d1b53e19552b887fb2"
uuid = "42e2da0e-8278-4e71-bc24-59509adca0fe"
version = "1.0.2"

[[deps.HTTP]]
deps = ["Base64", "CodecZlib", "Dates", "IniFile", "Logging", "LoggingExtras", "MbedTLS", "NetworkOptions", "Random", "SimpleBufferStream", "Sockets", "URIs", "UUIDs"]
git-tree-sha1 = "bf7fcabde6565fd8226d8a4c31bc8bb25fdd7f1d"
uuid = "cd3eb016-35fb-5094-929b-558a96fad6f3"
version = "1.3.0"

[[deps.HarfBuzz_jll]]
deps = ["Artifacts", "Cairo_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "Graphite2_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg"]
git-tree-sha1 = "129acf094d168394e80ee1dc4bc06ec835e510a3"
uuid = "2e76f6c2-a576-52d4-95c1-20adfe4de566"
version = "2.8.1+1"

[[deps.Hyperscript]]
deps = ["Test"]
git-tree-sha1 = "8d511d5b81240fc8e6802386302675bdf47737b9"
uuid = "47d2ed2b-36de-50cf-bf87-49c2cf4b8b91"
version = "0.0.4"

[[deps.HypertextLiteral]]
deps = ["Tricks"]
git-tree-sha1 = "c47c5fa4c5308f27ccaac35504858d8914e102f9"
uuid = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2"
version = "0.9.4"

[[deps.IOCapture]]
deps = ["Logging", "Random"]
git-tree-sha1 = "f7be53659ab06ddc986428d3a9dcc95f6fa6705a"
uuid = "b5f81e59-6552-4d32-b1f0-c071b021bf89"
version = "0.2.2"

[[deps.IniFile]]
git-tree-sha1 = "f550e6e32074c939295eb5ea6de31849ac2c9625"
uuid = "83e8ac13-25f8-5344-8a64-a9f2b223428f"
version = "0.5.1"

[[deps.InteractiveUtils]]
deps = ["Markdown"]
uuid = "b77e0a4c-d291-57a0-90e8-8db25a27a240"

[[deps.InverseFunctions]]
deps = ["Test"]
git-tree-sha1 = "b3364212fb5d870f724876ffcd34dd8ec6d98918"
uuid = "3587e190-3f89-42d0-90ee-14403ec27112"
version = "0.1.7"

[[deps.IrrationalConstants]]
git-tree-sha1 = "7fd44fd4ff43fc60815f8e764c0f352b83c49151"
uuid = "92d709cd-6900-40b7-9082-c6be49f344b6"
version = "0.1.1"

[[deps.IterTools]]
git-tree-sha1 = "fa6287a4469f5e048d763df38279ee729fbd44e5"
uuid = "c8e1da08-722c-5040-9ed9-7db0dc04731e"
version = "1.4.0"

[[deps.IteratorInterfaceExtensions]]
git-tree-sha1 = "a3f24677c21f5bbe9d2a714f95dcd58337fb2856"
uuid = "82899510-4779-5014-852e-03e436cf321d"
version = "1.0.0"

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deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unzip]]
git-tree-sha1 = "34db80951901073501137bdbc3d5a8e7bbd06670"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.1.2"

[[deps.Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "58443b63fb7e465a8a7210828c91c08b92132dff"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.14+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.12+3"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.1.1+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.48.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""

# ╔═╡ Cell order:
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