### A Pluto.jl notebook ###
# v0.19.40

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ a22d543c-6a50-11ed-002d-7f2fe5bd7276
using Plots, PlutoUI, LaTeXStrings, ShortCodes

# ╔═╡ 4de22c75-0d00-4838-8ba4-52bab4dab89e
md"""
# Discrete Fourier Transform

In this notebook, we compute the Discrete Fourier Transform (DFT) of a periodic function ``f(x): \mathbb{R} \rightarrow \mathbb{R}`` with a period ``L=2 \pi``, i.e., ``f(x+L) = f(x)``.

The discrete fourier transform with ``N`` equidistant points is given by
```math
\hat y_k = \frac{1}{N} \sum_{j=0}^{N-1} y_j e^{-i j k\frac{2\pi}{N}} ~~~~ (k=0, \ldots, N-1),
```
where ``y_j`` is the function at the sampling points ``y_j = f(x_j)`` with ``x_j = 2 \pi j/N``. We can rewrite the DFT as
```math
\hat y_k = \frac{1}{N} \sum_{j=0}^{N-1} y_j e^{-i k x_j} = \frac{1}{N} \left( \sum_{j=0}^{N-1} y_j \cos(k x_j) - i
\sum_{j=0}^{N-1} y_j \sin(k x_j)  \right) ~~~~ (k=0, \ldots, N-1).
```

The top-left graph presents the function ``f(x)``, which will undergo a Discrete Fourier Transform (DFT). If selected, this graph also includes the functions ``\cos(kx)`` and/or ``\sin(kx)`` that ``f(x)`` will be correlated with. The bottom-left graph displays the product of ``f(x)`` with ``\cos(kx)`` and/or ``\sin(kx)`` at the chosen sample points. On the top-right, the graph illustrates the real component of the DFT results, whereas the bottom-right graph visualizes the imaginary component of these results.
"""

# ╔═╡ 6e4a15eb-8c68-4837-81dd-caac7c81ba7e
md"""
##### How to add a new function
To add your own function (and show the corresponding DFT), you need to do the following things:
* Implement a new function of a single variable `x` (let's call it `myfun`).
* Add `myfun` to the dropdown menu above (under "Interpolation settings").
One example, for which this has already been done in this notebook, is `cos_plus_1`. To get started, simply copy `cos_plus_1`, rename it to something unique, and implement away 🙂
"""

# ╔═╡ 6b49c667-b16e-44f3-9f0d-2b8a62f5efda
md"""
## Auxiliary functions and internals
If you are only interested in *using* the interactive plotting features in this notebook, there is **no need to read any further**. The remainder of this document contains auxiliary functions for computing the Taylor polynomial. Furthermore, implementation details that are required for this workbook to execute correctly are included here.
"""

# ╔═╡ b8f81a3f-d773-4e67-b8d7-e1734b04abaa
md"""
### Load required packages
This noteboook requires some additional packages to work:
* Plots.jl is used for plotting the results
* PlutoUI.jl is used to create the interactive GUI elements
* LaTeXStrings.jl is used to make some mathematical notations look nicer
* ShortCodes.jl allows to embed references to papers etc. more easily
"""

# ╔═╡ 94c7aca0-1f23-4b2a-94ae-1aef90fed8d9
md"""
### Definitions for additional functions to interpolate
"""

# ╔═╡ 9e81ff03-1e93-4664-a030-32c593b2be0a
function _2cosx_2sin3x(x)
	return 2*cos(x) + 2*sin(3*x)
end

# ╔═╡ 889fb26d-338c-45f8-b621-640ad42c44ae
md"""
##### Fourier transform settings
Function to take DFT: ``f = ``$(@bind selected_function Select([cos, sin, _2cosx_2sin3x])) ``(\omega_0 (x - \phi))``

Frequency ``\omega_0``: $(@bind omega0 Slider(1:0.05:15, default=1, show_value=true))

Shift ``\phi``: $(@bind phi Slider(0:0.05:2 * pi, default=0, show_value=true))

Number of nodes ``N``: $(@bind N Slider(1:100, default=5, show_value=true))
"""

# ╔═╡ c03bb676-d1d5-48fc-b966-b680a3088d52
md"""
##### 
Current frequency ``k``: $(@bind k Slider(0:N-1, default=0, show_value=true))

Plot cos: $(@bind plot_cos Select(["no", "yes"]))

Plot sin: $(@bind plot_sin Select(["no", "yes"]))
"""

# ╔═╡ 8156b2cc-bd11-4714-82c5-00f97eceec8c
md"""
### Definitions for functions to compute node locations
The following function compute the node locations with an equidistant distribution.
"""

# ╔═╡ e560e608-ac7a-427d-9e2f-715ad01283da
"""
    equidistant_nodes(a, b, N)

Compute ``N`` equidistant node locations (including end points) on the interval
``[a, b]``.
"""
function equidistant_nodes(N)

	# Initialize nodes to zero and compute spacing `h`
	nodes = zeros(N)
	h = 2 * pi / N

	# Compute node locations
	for i in 1:N
		nodes[i] = (i - 1) * h
	end

	return nodes
end

# ╔═╡ ac216c60-8288-465b-ab2b-2ba3e7701d02
begin
	using FFTW
	# Auxiliary variables
	n = N - 1
	x_min = 0.0
	x_max = 2 * pi
	xvalues = range(x_min, x_max, 1001)
	
	# Compute exact function values
	exact_values = selected_function.(omega0 * (xvalues .- phi))
	y_min = min(-2, minimum(exact_values) - 0.1)
	y_max = max(2, maximum(exact_values) + 0.1) 
	
	# Compute nodes
	nodes = equidistant_nodes(N)
	yvalues = selected_function.(omega0 * (nodes .- phi))

	# Compute Fourier frequencies
	fourier_real_cont = cos.(k * xvalues)
	fourier_imag_cont = sin.(k * xvalues)
	fourier_real_disc = cos.(k * nodes)
	fourier_imag_disc = sin.(k * nodes)
	
	# Plot canvas 1 and set some default options
	p1 = plot(
		xlims=(x_min - 0.1, x_max),
		ylims=(y_min, y_max),
		legend=:bottomright,
		legendfontsize=10,
		tickfontsize=10,
		gridlinewidth=1.5,
		title=L"\textrm{Function \, Values}",
	)
	
	# Plot exact function
	label_exact = L"f = %$(nameof(selected_function))$\,(%$(omega0) x)$"
    plot!(p1, xvalues, exact_values, label=label_exact, linewidth=3, linecolor=:blue)

	# Plot markers at node locations
	scatter!(p1, nodes, yvalues, label=nothing, markersize=5, markercolor=:white, markershape = :rect, markerstrokecolor=:blue, markerstrokewidth = 2) #"nodes"

	# Plot Fourier functions
	if plot_cos == "yes"
		plot!(p1, xvalues, fourier_real_cont, label=L"\cos(k x)", linewidth=3, linestyle = :dash, linecolor = :green)
		scatter!(p1, nodes, fourier_real_disc, label=nothing, markersize=5, markercolor=:green, markershape = :diamond)
	end
	if plot_sin == "yes"
		plot!(p1, xvalues, fourier_imag_cont, label=L"\sin(k x)", linewidth=3, linestyle = :dash, linecolor = :red)
		scatter!(p1, nodes, fourier_imag_disc, label=nothing, markersize=5, markercolor=:red)
	end

	real_part = (fourier_real_disc .* yvalues)
	imag_part = (fourier_imag_disc .* yvalues)


	# Plot canvas 4 and set some default options
	p4 = plot(
		xlims=(x_min - 0.1, x_max),
		ylims=(y_min, y_max),
		legend=:bottomright,
		legendfontsize=10,
		tickfontsize=10,
		gridlinewidth=1.5,
		title=L"f \cdot e^{i\,k\,x}",
	)
	
	if plot_cos == "yes"
		for i in 1:N
			plot!(p4, [nodes[i], nodes[i]], [0, real_part[i]], label=nothing, linecolor=:green, linewidth = 3) #, markersize=5
		end
		scatter!(p4, nodes, real_part, label=L"y_j \cos(k x_j)", markershape = :diamond, markercolor=:green)
	end
	if plot_sin == "yes"
		for i in 1:N
			plot!(p4, [nodes[i], nodes[i]], [0, imag_part[i]], label=nothing, linecolor=:red, linewidth = 3, markercolor=:red) #, markersize=5
		end
		scatter!(p4, nodes, imag_part, label=L"y_j \sin(k x_j)", markercolor=:red)
	end
	
	# Compute FFT (FFTW does not divide by N)
	freq = fft(yvalues) / N
	
	# Plot canvas 2 and set some default options
	y_min = min(-1, -maximum(abs.(real(freq)))) - 0.1
	y_max = max(1, maximum(abs.(real(freq)))) + 0.1
	p2 = plot(
		xlims=(0, N),
		ylims=(y_min, y_max),
		legend=:bottomright,
		legendfontsize=10,
		tickfontsize=10,
		gridlinewidth=1.5,
		title=L"\textrm{Re}\, (\hat{y}_k) = \frac{1}{N}\sum_j y_j \cos(k x_j)",
	)

	# Plot markers at node locations on the x-axis
	scatter!(p2, 0:k, real(freq[1:k + 1]), label=nothing, markersize=5, markercolor=:red)
	plot!(p2, [(N/2), (N/2)], [y_min, y_max], linecolor = :blue, label = nothing)
	annotate!(p2, (N/2), y_min + 0.25, text(L"\textrm{Wavelength} = 2 \Delta x", :blue, :left, 12))

	# Plot canvas 2 and set some default options
	y_min = min(-1, -maximum(abs.(imag(freq)))) - 0.1
	y_max = max(1, maximum(abs.(imag(freq)))) + 0.1
	p3 = plot(
		xlims=(0, N),
		ylims=(y_min, y_max),
		legend=:bottomright,
		legendfontsize=10,
		tickfontsize=10,
		gridlinewidth=1.5,
		title=L"\textrm{Im}\, (\hat{y}_k) = -\frac{1}{N}\sum_j y_j \sin(k x_j)",
	)
	# Plot markers at node locations on the x-axis
	scatter!(p3, 0:k, imag(freq[1:k + 1]), label=nothing, markersize=5, markercolor=:red)
	plot!(p3, [(N/2), (N/2)], [y_min, y_max], linecolor = :blue, label = nothing)
	annotate!(p3, (N/2), y_min + 0.25, text(L"\textrm{Wavelength} = 2 \Delta x", :blue, :left, 12))

	# Create joint plot
	plot(p1, p2, p4, p3, layout = (2, 2), size=(1200, 450))
end

# ╔═╡ 4d297775-84a7-42e3-ac72-792338351354
md"""
### Definitions of some helper functions
"""

# ╔═╡ bf205421-a919-4090-acb1-9b2f2555b056
md"""
### Notebook style adjustments
Make notebook wider to show plots side by side:
"""

# ╔═╡ 282da2f2-c0da-4310-9bc4-54b5addc6b11
html"""
<style>
	main {
		margin: 0 auto;
		max-width: 2000px;
    	padding-left: max(160px, 10%);
    	padding-right: max(160px, 10%);
	}
</style>
"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
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LaTeXStrings = "~1.3.0"
Plots = "~1.36.3"
PlutoUI = "~0.7.48"
ShortCodes = "~0.3.5"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
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[[deps.Random]]
deps = ["SHA", "Serialization"]
uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[deps.RecipesBase]]
deps = ["SnoopPrecompile"]
git-tree-sha1 = "d12e612bba40d189cead6ff857ddb67bd2e6a387"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
version = "1.3.1"

[[deps.RecipesPipeline]]
deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase", "SnoopPrecompile"]
git-tree-sha1 = "e974477be88cb5e3040009f3767611bc6357846f"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.6.11"

[[deps.Reexport]]
git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "1.2.2"

[[deps.RelocatableFolders]]
deps = ["SHA", "Scratch"]
git-tree-sha1 = "90bc7a7c96410424509e4263e277e43250c05691"
uuid = "05181044-ff0b-4ac5-8273-598c1e38db00"
version = "1.0.0"

[[deps.Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "838a3a4188e2ded87a4f9f184b4b0d78a1e91cb7"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.3.0"

[[deps.SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"
version = "0.7.0"

[[deps.Scratch]]
deps = ["Dates"]
git-tree-sha1 = "f94f779c94e58bf9ea243e77a37e16d9de9126bd"
uuid = "6c6a2e73-6563-6170-7368-637461726353"
version = "1.1.1"

[[deps.Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[deps.ShortCodes]]
deps = ["Base64", "CodecZlib", "HTTP", "JSON3", "Memoize", "UUIDs"]
git-tree-sha1 = "95479a28f0bb4ad37ec7c7ece7fdbfc400c643e0"
uuid = "f62ebe17-55c5-4640-972f-b59c0dd11ccf"
version = "0.3.5"

[[deps.Showoff]]
deps = ["Dates", "Grisu"]
git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de"
uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f"
version = "1.0.3"

[[deps.SimpleBufferStream]]
git-tree-sha1 = "874e8867b33a00e784c8a7e4b60afe9e037b74e1"
uuid = "777ac1f9-54b0-4bf8-805c-2214025038e7"
version = "1.1.0"

[[deps.SnoopPrecompile]]
git-tree-sha1 = "f604441450a3c0569830946e5b33b78c928e1a85"
uuid = "66db9d55-30c0-4569-8b51-7e840670fc0c"
version = "1.0.1"

[[deps.Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[deps.SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "a4ada03f999bd01b3a25dcaa30b2d929fe537e00"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.1.0"

[[deps.SparseArrays]]
deps = ["Libdl", "LinearAlgebra", "Random", "Serialization", "SuiteSparse_jll"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[deps.SpecialFunctions]]
deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "d75bda01f8c31ebb72df80a46c88b25d1c79c56d"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "2.1.7"

[[deps.Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
version = "1.9.0"

[[deps.StatsAPI]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "f9af7f195fb13589dd2e2d57fdb401717d2eb1f6"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.5.0"

[[deps.StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "d1bf48bfcc554a3761a133fe3a9bb01488e06916"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.21"

[[deps.StructTypes]]
deps = ["Dates", "UUIDs"]
git-tree-sha1 = "ca4bccb03acf9faaf4137a9abc1881ed1841aa70"
uuid = "856f2bd8-1eba-4b0a-8007-ebc267875bd4"
version = "1.10.0"

[[deps.SuiteSparse_jll]]
deps = ["Artifacts", "Libdl", "Pkg", "libblastrampoline_jll"]
uuid = "bea87d4a-7f5b-5778-9afe-8cc45184846c"
version = "5.10.1+6"

[[deps.TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"
version = "1.0.3"

[[deps.Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"
version = "1.10.0"

[[deps.TensorCore]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "1feb45f88d133a655e001435632f019a9a1bcdb6"
uuid = "62fd8b95-f654-4bbd-a8a5-9c27f68ccd50"
version = "0.1.1"

[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[deps.TranscodingStreams]]
deps = ["Random", "Test"]
git-tree-sha1 = "8a75929dcd3c38611db2f8d08546decb514fcadf"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.9.9"

[[deps.Tricks]]
git-tree-sha1 = "6bac775f2d42a611cdfcd1fb217ee719630c4175"
uuid = "410a4b4d-49e4-4fbc-ab6d-cb71b17b3775"
version = "0.1.6"

[[deps.URIs]]
git-tree-sha1 = "e59ecc5a41b000fa94423a578d29290c7266fc10"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.4.0"

[[deps.UUIDs]]
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 = "ca0969166a028236229f63514992fc073799bb78"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.2.0"

[[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 = "1.2.13+0"

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

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.29.0+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"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+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 = "1.52.0+1"

[[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:
# ╟─4de22c75-0d00-4838-8ba4-52bab4dab89e
# ╟─ac216c60-8288-465b-ab2b-2ba3e7701d02
# ╟─889fb26d-338c-45f8-b621-640ad42c44ae
# ╟─c03bb676-d1d5-48fc-b966-b680a3088d52
# ╟─6e4a15eb-8c68-4837-81dd-caac7c81ba7e
# ╟─6b49c667-b16e-44f3-9f0d-2b8a62f5efda
# ╟─b8f81a3f-d773-4e67-b8d7-e1734b04abaa
# ╠═a22d543c-6a50-11ed-002d-7f2fe5bd7276
# ╟─94c7aca0-1f23-4b2a-94ae-1aef90fed8d9
# ╠═9e81ff03-1e93-4664-a030-32c593b2be0a
# ╟─8156b2cc-bd11-4714-82c5-00f97eceec8c
# ╟─e560e608-ac7a-427d-9e2f-715ad01283da
# ╟─4d297775-84a7-42e3-ac72-792338351354
# ╟─bf205421-a919-4090-acb1-9b2f2555b056
# ╠═282da2f2-c0da-4310-9bc4-54b5addc6b11
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002