9.autodiff.jl

### A Pluto.jl notebook ###
# v0.19.22

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

# ╔═╡ f2e3f609-33ff-4798-bf6b-f69ab5209519
using PlutoUI

# ╔═╡ 45bc0bb7-fcd1-4527-a51f-05ab7df03dc8
using Plots

# ╔═╡ 62dd6d53-7f55-4933-95ae-2350d1992cf6
using FiniteDifferences

# ╔═╡ 4647fb94-2b5d-467a-8952-5aba5da87527
using BenchmarkTools

# ╔═╡ f6cd50b4-ce0a-11ed-01cf-63021b4eb012
using ForwardDiff

# ╔═╡ 4d5d8755-7774-4364-aba4-cb69ed898809
using Enzyme

# ╔═╡ 838b51c0-3987-4fb4-b8b8-1799f724ecd6
md"""
```math
\newcommand{\comment}[1]{{\bf  \color{blue}{\text{◂~ #1}}}}
```
"""

# ╔═╡ 385dda1d-8cab-42ee-9242-132032d66b7f
html"""
<button onclick="present()"> present </button>
"""

# ╔═╡ 79921035-aafb-478c-ba36-e5655a4515dd
TableOfContents(depth=2)

# ╔═╡ 1dace657-b9ca-4ec2-81c9-209c823e9adb
md"# The four methods to differentiate a function"

# ╔═╡ f76cf1c5-3801-4c6d-ba10-195a69e63e45
md"""
“谁要你教，不是草头底下一个来回的回字么？”

孔乙己显出极高兴的样子，将两个指头的长指甲敲着柜台，点头说，“对呀对呀！……回字有四样写法，你知道么？”我愈不耐烦了，努着嘴走远。

孔乙己刚用指甲蘸了酒，想在柜上写字，见我毫不热心，便又叹一口气，显出极惋惜的样子。
"""

# ╔═╡ 86874020-ec91-44c1-83cc-483cb0cd9337
md"""
# The history of autodiff
"""

# ╔═╡ ada143c0-40a1-457f-b680-be9edb3e93c8
md"""
* 1964 ~ Robert Edwin Wengert, A simple automatic derivative evaluation program. ``\comment{first forward mode AD}``
* 1970 ~ Seppo Linnainmaa, Taylor expansion of the accumulated rounding error. ``\comment{first backward mode AD}``
* 1986 ~ Rumelhart, D. E., Hinton, G. E., and Williams, R. J., Learning representations by back-propagating errors.``\comment{bring AD to machine learning people.}``
* 1992 ~ Andreas Griewank, Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation. ``\comment{also known as optimal checkpointing.}``
* 2000s ~ The boom of tensor based AD frameworks for machine learning.
* 2018 ~ Re-inventing AD as differential programming ([wiki](https://en.wikipedia.org/wiki/Differentiable_programming).)
![](https://qph.fs.quoracdn.net/main-qimg-fb2f8470f2120eb49c8142b08d9c4132)
* 2020 ~ Moses, William and Churavy, Valentin, Instead of Rewriting Foreign Code for Machine Learning, Automatically Synthesize Fast Gradients ``\comment{AD on LLVM}``.
"""

# ╔═╡ 89915416-29bc-47c4-b11e-c70be1ef858b
md"# Differentiating the Bessel function"

# ╔═╡ 29415653-52fc-4210-9cca-551c6f5d4d2c
md"""
```math
    J_\nu(z) = \sum\limits_{n=0}^{\infty} \frac{(z/2)^\nu}{\Gamma(k+1)\Gamma(k+\nu+1)} (-z^2/4)^{n}
```
"""

# ╔═╡ 53c29044-94d3-4470-bacb-e72bd2401d1c
md"## Poorman's Bessel function"

# ╔═╡ 8da8ed7e-bfd2-4be0-a764-ad04566b9632
function poor_besselj(ν, z::T; atol=eps(T)) where T
    k = 0
    s = (z/2)^ν / factorial(ν)
    out = s
    while abs(s) > atol
        k += 1
        s *= (-1) / k / (k+ν) * (z/2)^2
        out += s
    end
    out
end

# ╔═╡ c28f22cc-71b7-4339-8c81-9d76d2244c94
md"""
In each step, the state transfer can be described as $(k_i, s_i, out_i) \rightarrow (k_{i+1}, s_{k+1}, out_{i+1})$.
"""

# ╔═╡ ff53a172-634f-438f-a04a-1481ac20f140
let
	x = 0.0:0.01:10
	plt = plot([], []; label="", xlabel="x", ylabel="y")
	for i=0:5
		yi = poor_besselj.(i, x)
		plot!(plt, x, yi; label="J(ν=$i)", lw=2)
	end
	plt
end

# ╔═╡ c763e489-a6a9-42ac-8da1-a2ba48a0f1a8
@bind select_gradient_method Select(["Manual", "Forward", "Backward", "FiniteDiff"])

# ╔═╡ 5e782b60-3721-40d2-ae61-69edb4fc50f0
let
	x = 0.0:0.01:10
	plt = plot([], []; label="", xlabel="x", ylabel="y")
	for i=0:3
		yi = poor_besselj.(i, x)
		if select_gradient_method == "Forward"
			gi = [autodiff(Forward, poor_besselj, i, Enzyme.Duplicated(xi, 1.0))[1] for xi in x]
		elseif select_gradient_method == "Manual"
			gi = ((i == 0 ? -poor_besselj.(i+1, x) : poor_besselj.(i-1, x)) - poor_besselj.(i+1, x)) ./ 2
		elseif select_gradient_method == "Backward"
			gi = [autodiff(Reverse, poor_besselj, i, Enzyme.Active(xi))[1] for xi in x]
		elseif select_gradient_method == "FiniteDiff"
			gi = [autodiff(Reverse, poor_besselj, i, Enzyme.Active(xi))[1] for xi in x]
		end
		plot!(plt, x, yi; label="J(ν=$i)", lw=2, color=i)
		plot!(plt, x, gi; label="g(ν=$i)", lw=2, color=i, ls=:dash)
	end
	plt
end

# ╔═╡ 1d2a3579-b9c4-4ffb-819c-2689d3f1b83c
md"""
# Finite difference

First order forward Difference
```math
\frac{\partial f}{\partial x} \approx \frac{f(x+\Delta) - f(x)}{\Delta}
```


First order backward Difference
```math
\frac{\partial f}{\partial x} \approx \frac{f(x) - f(x-\Delta)}{\Delta}
```


First order central Difference
```math
\frac{\partial f}{\partial x} \approx \frac{f(x+\Delta) - f(x-\Delta)}{2\Delta}
```

Table of finite difference coefficient: [wiki page](https://en.wikipedia.org/wiki/Finite_difference_coefficient).

"""

# ╔═╡ 742504f5-8ae3-4388-8f05-f0a256815cbc
md"""
## Example: central finite difference to the 4th order
1. Check the table

|  -2  | -1  | 0 | 1 | 2 |
| --- | --- | --- | --- | --- |
| 1/12 | −2/3 | 0 | 2/3 | −1/12 |

2. Apply the fomula
```math
\frac{\partial f}{\partial x} \approx \frac{f(x-2\Delta) - 8f(x-\Delta) + 8f(x+\Delta) - f(x+2\Delta)}{12\Delta}
```
"""

# ╔═╡ 1fbdd9c5-fb52-4bcc-9a6a-156ab8bfe4f6
md"""
```math
\left(\begin{matrix}
f(x-2\Delta)\\f(x-\Delta)\\f(x)\\f(x+\Delta)\\f(x+2\Delta)
\end{matrix}\right) \approx \left(\begin{matrix}
1 & (-2)^1 & (-2)^{2} & (-2)^3 &  & (-2)^4\\
1 & (-1)^1 & (-1)^{2} & (-1)^3 &  & (-1)^4\\
1 & 0 & 0 & 0 &  & 0\\
1 & (1)^1 & (1)^{2} & (1)^3 &  & (1)^4\\
1 & (2)^1 & (2)^{2} & (2)^3 &  & (2)^4
\end{matrix}\right)\left(\begin{matrix}
f(x)\\f'(x)\Delta\\f''(x)\Delta^2/2\\f'''(x)\Delta^3/6\\f''''(x)\Delta^4/24
\end{matrix}\right)
```

Let the finite difference coefficients be $\vec \alpha^T = (\alpha_{-2}, \alpha_{-1}, \alpha_{0}, \alpha_{1}, \alpha_{2})$, we want $\alpha^T \vec f= f'(x)\Delta +O(\Delta^5)$, where $\vec f=A \vec g$ is the vector on the left side. $\vec \alpha$ can be solved by $A^T \backslash (0, 1, 0, 0, 0)^T$
"""

# ╔═╡ cee92def-01e4-4199-a964-b81412a0563f
let
	b = [0.0, 1, 0, 0, 0]
	A = [i^j for i=-2:2, j=0:4]
	A' \ b
end

# ╔═╡ b2d72996-27a0-467e-8c2d-20afcc28b24c
[i^j for i=-2:2, j=0:4]

# ╔═╡ 84182832-c653-49df-aa55-a78181974612
central_fdm(5, 1)(x->poor_besselj(2, x), 0.5)

# ╔═╡ 78e9ebd0-92eb-4fdc-b440-56b4fdaf7a60
@benchmark central_fdm(5, 1)(y->poor_besselj(2, y), x) setup=(x=0.5)

# ╔═╡ 54f754d5-6cb4-47ef-9b19-aa8c47c14b36
md"# Forward mode automatic differentiation"

# ╔═╡ f25a2d8e-5cd5-4df7-bd39-2830d6e2dea6
md"""
Forward mode AD attaches a infitesimal number $\epsilon$ to a variable, when applying a function $f$, it does the following transformation
```math
\begin{align}
    f(x+g \epsilon) = f(x) + f'(x) g\epsilon + \mathcal{O}(\epsilon^2)
\end{align}
```

The higher order infinitesimal is ignored. 

**In the program**, we can define a *dual number* with two fields, just like a complex number
```
f((x, g)) = (f(x), f'(x)*g)
```
"""

# ╔═╡ 4b3a4f50-4339-43e5-9bc1-473151c1f436
res = sin(ForwardDiff.Dual(π/4, 2.0))

# ╔═╡ 9359eddc-bfbb-428d-8197-00d1e9cd8eeb
res === ForwardDiff.Dual(sin(π/4), cos(π/4)*2.0)

# ╔═╡ bbadece5-e92a-42f5-998c-6149c067b130
md"
We can apply this transformation consecutively, it reflects the chain rule.
```math
\begin{align}
\frac{\partial \vec y_{i+1}}{\partial x} &= \boxed{\frac{\partial \vec y_{i+1}}{\partial \vec y_i}}\frac{\partial \vec y_i}{\partial x}\\
&\text{local Jacobian}
\end{align}
```
"

# ╔═╡ 2ec9461e-c7a3-43f5-9fef-fb52980fa196
md"""
**Example:** Computing two gradients $\frac{\partial z\sin x}{\partial x}$ and $\frac{\partial \sin^2x}{\partial x}$ at one sweep
"""

# ╔═╡ d2222941-f31b-4411-aac8-6de450aa2fbe
autodiff(Forward, poor_besselj, 2, Duplicated(0.5, 1.0))[1]

# ╔═╡ 5db5eef5-d217-4a61-9230-62ada8b7606f
@benchmark autodiff(Forward, poor_besselj, 2, Duplicated(x, 1.0))[1] setup=(x=0.5)

# ╔═╡ 0956c11a-63d7-4ab1-a99b-9056208759ac
md"""
**What if we want to compute gradients for multiple inputs?**

The computing time grows **linearly** as the number of variables that we want to differentiate. But does not grow significantly with the number of outputs.
"""

# ╔═╡ 38cc8a4d-0107-4dee-b0f0-76959ad9cc92
md"""
# Reverse mode automatic differentiation
"""

# ╔═╡ 17e21230-2299-4dda-87ef-40e629d66213
md"On the other side, the back-propagation can differentiate **many inputs** with respect to a **single output** efficiently"

# ╔═╡ 38ca5970-78ed-495b-9d69-74a6a22ec027
md"""
```math
\begin{align}
    \frac{\partial \mathcal{L}}{\partial \vec y_i} = \frac{\partial \mathcal{L}}{\partial \vec y_{i+1}}&\boxed{\frac{\partial \vec y_{i+1}}{\partial \vec y_i}}\\
&\text{local jacobian?}
\end{align}
```
"""

# ╔═╡ 18065ade-1e2d-4b37-a075-8de966de306d
autodiff(Reverse, poor_besselj, 2, Enzyme.Active(0.5))[1]

# ╔═╡ 4a4d5c19-a784-4285-8ea0-f2203cd80eb3
@benchmark autodiff(Reverse, poor_besselj, 2, Enzyme.Active(x))[1] setup=(x=0.5)

# ╔═╡ 298873a4-ba02-43c3-9507-ce99a00f7245
md"### How to visit local Jacobians in the reversed order? "

# ╔═╡ 62484b76-033b-455a-bf51-d1f57adbc8a0
md"""
Caching intermediate results in a stack!
"""

# ╔═╡ 52fa22a0-98ae-4d32-884d-b40cdb9cff62
md"""
# Rule based autodiff
"""

# ╔═╡ 721bfffc-b76f-4d0d-8ed6-be0c327eef59
md"""
The backward rule of the Bessel function is
```math
\begin{align}
&J'_{\nu}(z) =  \frac{J_{\nu-1}(z) - J_{\nu+1}(z) }2\\
&J'_{0}(z) =  - J_{1}(z)
\end{align}
```
"""

# ╔═╡ 0aaa152e-29cf-43f4-8a07-ad7dd4a5f304
0.5 * (poor_besselj(1, 0.5) - poor_besselj(3, 0.5))

# ╔═╡ ee1f777e-c8fb-4139-aae8-503ed4fd2391
@benchmark 0.5 * (poor_besselj(1, x) - poor_besselj(3, x)) setup=(x=0.5)

# ╔═╡ 88199efc-975c-4971-8481-3319209bf139
md"""
# Deriving the backward rule of matrix multiplication
"""

# ╔═╡ 3d1bfe11-a863-4a63-9bb7-0a052dc04d25
md"Please check [blog](https://giggleliu.github.io/posts/2019-04-02-einsumbp/)"

# ╔═╡ 0e1de67c-592a-408d-849f-984dd47664cd
md"""
## Rule based or not?
"""

# ╔═╡ 164c3f50-cb0c-4f74-8580-303dedbc712b
html"""
<table>
<tr>
<th width=200></th>
<th width=300>rule based</th>
<th width=300>differential programming</th>
</tr>
<tr style="vertical-align:top">
<td>meaning</td>
<td>defining backward rules manully for functions on tensors</td>
<td>defining backward rules on a limited set of basic scalar operations, and generate gradient code using source code transformation</td>
</tr>
<tr style="vertical-align:top">
<td>pros and cons</td>
<td>
<ol>
<li style="color:green">Good tensor performance</li>
<li style="color:green">Mature machine learning ecosystem</li>
<li style="color:red">Need to define backward rules manually</li>
</ol>
</td>
<td>
<ol>
<li style="color:green">Reasonalbe scalar performance</li>
<li style="color:red">hard to utilize BLAS</li>
</ol>
</td>
<td>
</td>
</tr>
<tr style="vertical-align:top">
<td>packages</td>
<td>Jax<br>PyTorch</td>
<td><a href="http://tapenade.inria.fr:8080/tapenade/">Tapenade</a><br>
<a href="http://www.met.reading.ac.uk/clouds/adept/">Adept</a><br>
<a href="https://github.com/EnzymeAD/Enzyme">Enzyme</a>
</td>
</tr>
</table>
"""

# ╔═╡ b648761c-1fa7-4e70-b0d6-654c51b2935f
md"""
# Obtaining Hessian
"""

# ╔═╡ 2fac953b-7a04-4fc1-a54d-a2062b590b20
md"""
Hessian is the Jacobian of the gradient. We can use **forward over backward**.
"""

# ╔═╡ 18ae2723-f947-4ff6-864e-2dd4a0bb32f9
md"""
# Optimal checkpointing, towards solving the memory wall problem
"""

# ╔═╡ 02ca5e2f-1705-4de8-bd89-0b4b69e1fee7
md"""
# Game: Pass the ball

In each step, if you have the ball, you pick one of the following actions
1. raise your hand, and pass the ball to the next,
2. pass the ball to the next without raising your hand,
2. only if you are the last one in the queue, you can left the queue and pass the ball to those raising hands.
Otherwise, you may
1. put down your hand, or
2. do nothing.

**Goal: We require the number of raised hands being at most $m$ at the same time, please empty the queue while minimizing the number of ball passings.**

### The connection to checkpointing
* A person: a computing state $s_k$,
* The queue: a linear program $s_1, s_2, \ldots, s_n$,
* Passing ball: program running forward $s_{k}\rightarrow s_{k+1}$,
* Left queue: the gradient $g_k$ being computed,
* Rasing hand: create a checkpoint in the main memory,
* put down the hand: deallocate a checkpoint.
"""

# ╔═╡ 2f715b1a-2e32-4068-86fd-912d735b307a
md"# Homeworks
1. Given the binomial function $\eta(\tau, \delta) = \frac{(\tau + \delta)!}{\tau!\delta!}$, show that the following statement is true.
```math
\eta(\tau,\delta) = \sum_{k=0}^\delta \eta(\tau-1,k)
```
2. Given the following program to compute the $l_2$-norm of a vector $x\in R^n$.
```julia
function poorman_norm(x::Vector{<:Real})
	nm2 = zero(real(eltype(x)))
	for i=1:length(x)
		nm2 += abs2(x[i])
	end
	ret = sqrt(nm2)
	return ret
end
```

In the program, the `abs2` and `sqrt` functions can be treated as primitive functions, which means they should not be further decomposed as more elementary functions.

### Tasks
1. Rewrite the program (on paper or with code) to implement the forward mode autodiff, where you can use the notation $\dot y \equiv (\frac{\partial y}{\partial x_i}, \frac{\partial y}{\partial x_2},\ldots \frac{\partial y}{\partial x_n})$ to denote a derivative.

**Example**
To compute the gradient of
```julia
function f(x, y)
	a = 2 * x
	b = sin(a)
	c = cos(y)
    return b + c
end
```

The forward autodiff rewritten program is
```julia
function f_forward((x, ̇̇x), (y, ̇y))
	(a, ̇a) = (2 * x, 2 * ̇x)
	(b, ̇b) = (sin(a), cos(a) .* ̇a)
	(c, ̇c) = （cos(y), -sin(y) .* ̇y）
    return (b + c, ̇b + ̇c)
end
```
2. Rewrite the program (on paper or with code) to implement the reverse mode autodiff, where you can use the notation $\overline y \equiv \frac{\partial \mathcal{L}}{\partial y}$ to denote an adjoint, $y \rightarrow T$ to denote pushing a variable to the global stack, and $y \leftarrow T$ to denote poping a variable from the global stack. In your submission, both the forward pass and backward pass should be included.
3. Estimate how many intermediate states is cached in your reverse mode autodiff program?

### Reference
* Griewank A, Walther A. Evaluating derivatives: principles and techniques of algorithmic differentiation[M]. Society for industrial and applied mathematics, 2008.
"

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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"]
git-tree-sha1 = "c6edfe154ad7b313c01aceca188c05c835c67360"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.4+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", "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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# ╠═18065ade-1e2d-4b37-a075-8de966de306d
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# ╟─b648761c-1fa7-4e70-b0d6-654c51b2935f
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# ╟─2f715b1a-2e32-4068-86fd-912d735b307a
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# ╟─00000000-0000-0000-0000-000000000002