### A Pluto.jl notebook ###
# v0.20.21

using Markdown
using InteractiveUtils

# ╔═╡ 1137268f-e9a3-4e23-9cce-40ebabd6d8aa
using Symbolics

# ╔═╡ 74c793d2-ac1a-453e-ba48-b6dec0bbb700
using Plots

# ╔═╡ 8dba72ec-c59d-469b-ab2b-5a10c632c92e
# use formatted printing to display answer in dollars and cents
using Printf

# ╔═╡ 3ea22dd8-0977-11f1-0d0f-0f0d8767c483
md"""
A maker of personal computers sells ``15\,000`` units per month of a certain
model.
The cost of manufacture is \$``600``/unit, and the wholesale
price is \$``850``.
During the last quarter the maker lowered the price \$``100`` in some markets,
and it resulted in ``40\%`` increase in sales.
The company has been advertising its
product nationwide at a cost of \$``4\,000`` per month.
The marketing consultant
claims that increasing the advertising budget by \$``2\,000``/month
would result
in a sales increase of ``400`` units/month. Management has agreed to consider
an increase in advertising budget to no more that \$``10\,000``/month.
"""

# ╔═╡ b23c356f-2d84-40e8-b1ac-7077cd850382
md"""Graph level curves of the objective function for profit ``P(x,y)``
where ``x`` is the wholesale price and ``y`` is the advertising budget
for ``x\in[600,1200]`` and ``y\in[1000,12000]``.
Overlay the advertising-budget constraint on the same plot.
"""

# ╔═╡ 96f7872a-5d2f-4a1e-a8b6-c81c37629f39
D(f,x)=expand_derivatives(Differential(x)(f))

# ╔═╡ c77fc2bd-1284-4dd3-9eda-7961485af953
@variables x,y

# ╔═╡ 1880a948-7822-4e0c-9c30-d88e10df9e5b
md"""
There are two natural ways to interpret how changing the price affects
the quantity sold.  The first way compounds the effects of the advertizing as
```math
   q(x,y)=\Big(15000+{400\over 2000}(y-4000)\Big)\Big(1+{40\over 10000}(850-x)\Big)
```
while the second way treats the increase in sales due to advertizing separate from the increase due to discounting the price as
```math
	q(x,y)=15000\Big(1+{40\over 10000}(850-x)\Big)+{400\over 2000}(y-4000).
```
Since first way leads to a quadratic equation when we later solve``\nabla P=\lambda\nabla g`` ignore that and continue with the second interpretation
of ``q`` which leads to a linear system.
"""

# ╔═╡ d95fa485-d71a-44a5-a1a5-bb398a75fc29
# This is the second option for q which is apparently the correct one
q(x,y)=15000*(1+40//10000*(850-x))+400//2000*(y-4000)

# ╔═╡ 8f7ff2e6-3485-445d-b49f-3684615d36f5
R(x,y)=x*q(x,y)

# ╔═╡ 88d9a4ec-b245-4f20-86f5-37af67709989
C(x,y)=600*q(x,y)+y

# ╔═╡ b42e6baf-7bc9-41be-a646-425ae9f0e468
P(x,y)=R(x,y)-C(x,y)

# ╔═╡ 4144ba5d-b228-4ddc-ab65-546be2540701
# Using display causes the output to wrap
display(P(x,y))

# ╔═╡ 47cd50ec-f726-4f32-befd-2d437543a9be
g(x,y)=y

# ╔═╡ f5591804-0b4b-410e-b40e-c0fdbd92d8f2
begin
	plt=contour(600:10:1200,1000:100:12000,g,levels=[10000]);
	contour!(plt,600:10:1200,1000:100:12000,P)
end
	

# ╔═╡ 18c35335-c0c6-4639-b248-92e7097b9215
md"""
Determine the price ``x`` and the advertising budget ``y`` that will
maximize profit subject to the constraint.  Model as a constrained
optimization problem and solve using the method of Lagrange multipliers.
What is ``P(x,y)`` at the optimal values of ``x`` and ``y``?
"""


# ╔═╡ 64d30209-6c03-444d-b1b2-7db4a7999356
dPdx=expand(D(P(x,y),x))

# ╔═╡ b7709e8a-ff1d-42cf-8d1e-2bdc0f026b19
dPdy=expand(D(P(x,y),y))

# ╔═╡ 5181f96f-d9d7-4cd1-902d-1e90c34e0d6f
dgdx=expand(D(g(x,y),x))

# ╔═╡ 4426b1bb-515d-4be6-9793-c35e21b11242
dgdy=expand(D(g(x,y),y))

# ╔═╡ 671180f6-94fb-4f15-bd03-a8b1534bd42d
@variables lambda

# ╔═╡ f38afc6b-6cfb-4568-b098-8c5135ad65d7
S1=Symbolics.solve_for(
	[dPdx~lambda*dgdx,dPdy~lambda*dgdy,g(x,y)~10000],[x,y,lambda]
)

# ╔═╡ e3605cb8-3d6a-404e-9cf8-462d1e62fd27
x0=Float64(Symbolics.toexpr(S1[1]))

# ╔═╡ f416b7ac-dc58-4ba2-8099-5828d300a65e
y0=Float64(Symbolics.toexpr(S1[2]))

# ╔═╡ 7cafdb21-8e7a-47e1-a036-10f063ddb48e
lambda0=Float64(Symbolics.toexpr(S1[3]))

# ╔═╡ c199b852-123d-47ae-9a34-3dec455c8507
md"""
The profit is maximixed when the wholesale price is
``x =`` \$$(@sprintf("%.2f",x0))
and the advertising budget is ``y =`` \$$(@sprintf("%.2f",y0)).
In this case the
maximum profit is \$$(@sprintf("%.2f",P(x0,y0)))
"""

# ╔═╡ cee88d99-0e8e-4836-b116-878e0caadbb3
md"""
Determine the sensitivity of the decision variables (price and
advertising) to price elasticity (the ``40\%`` number).  In particular,
let ``\alpha`` be the price elasticity and compute
``S(x,\alpha)`` and ``S(y,\alpha)``.
"""

# ╔═╡ c6345306-38b3-4bda-a249-40a9a045ee68
@variables alpha

# ╔═╡ 97ff4118-2e2b-4558-b3b3-41bba7fd8051
md"""
In order to finish this modeling problem,
I define new functions that are the same as those in the previous
part, with an additional suffix and the parameter ``\alpha``.

In my answer to Quiz 2 I was able to write something like
```
	alpha=40.0
```
to find the maximum and then write
```
	@variables alpha
```
to dynamically change the meaning of alpha in the already defined
functions and then find the sensitivity.

With a Pluto worksheet the interface does not allow one to redefine
*anything* in the middle of the worksheet.
This allows the notebook interface to track data dependencies and update
the cells automatically based on the dependency relations between them.
"""

# ╔═╡ be2e8882-f840-4121-9f25-995d34f05248
qa(x,y)=15000*(1+alpha*1//10000*(850-x))+400//2000*(y-4000)

# ╔═╡ caa5e5b0-6cf6-4446-af89-6d2f0272e7b8
Ra(x,y)=x*qa(x,y)

# ╔═╡ 35eee9ff-b724-4218-b4c2-0bf17f60d876
Ca(x,y)=600*qa(x,y)+y

# ╔═╡ 478753d1-5c12-4ad3-aa35-4d0761ab87f6
Pa(x,y)=Ra(x,y)-Ca(x,y)

# ╔═╡ 36b30951-9912-4840-80d1-6f0e5cddecc7
dPadx=expand(D(Pa(x,y),x))

# ╔═╡ a8e97ccf-4c6a-4dd8-92bd-6f89098aa802
dPady=expand(D(P(x,y),y))

# ╔═╡ 17f32c72-a348-4140-9fd1-74639da047fd
md"""
Although it is possible to solve for ``x`` and ``y`` we
use implicit differentiation here to find the derivatives without
needing to explicity solve for ``x`` and ``y``.  Write
```math
	u=\partial P/\partial x-\lambda\partial g/\partial x,
\qquad
	v=\partial P/\partial x-\lambda\partial g/\partial x
\qquad\hbox{and}\qquad
	w=g(x,y)-10000
```
so differentiating with respect to $\alpha$ implies
```math
	{du\over d\alpha}={\partial u\over\partial\alpha}
	+{\partial u\over\partial x}{dx\over d\alpha}
	+{\partial u\over\partial y}{dy\over d\alpha}
	+{\partial u\over\partial\lambda}{d\lambda\over d\alpha}=0
```
```math
	{dv\over da}={\partial v\over\partial\alpha}
	+{\partial v\over\partial x}{dx\over d\alpha}
	+{\partial v\over\partial y}{dy\over d\alpha}
	+{\partial v\over\partial\lambda}{d\lambda\over d\alpha}=0
```
and
```math
	{dw\over da}={\partial w\over\partial\alpha}
	+{\partial w\over\partial x}{dx\over d\alpha}
	+{\partial w\over\partial y}{dy\over d\alpha}
	+{\partial w\over\partial\lambda}{d\lambda\over d\alpha}=0.
```
No matter how complicated the functional relationships for ``x``, ``y`` 
and ``\lambda`` actually are,
implicit differentiation always yields linear equations in the derivatives.

Now solve these three linear equations in the
unknowns ``dx/d\alpha``, ``dy/d\alpha`` and ``d\lambda/d\alpha``.
"""

# ╔═╡ 758eb1bc-961f-4eb8-92e9-ca561a90844c
u=dPadx-lambda*dgdx

# ╔═╡ 66503583-be71-49b1-820f-8c25ee5c38d4
v=dPady-lambda*dgdy

# ╔═╡ 2342c27d-e511-4f76-b2a9-d6d91f5c755f
w=g(x,y)-10000

# ╔═╡ 6b9f0a29-b280-4e10-8b99-cd5e546d04d2
A=[D(u,x) D(u,y) D(u,lambda);
   D(v,x) D(v,y) D(v,lambda);
   D(w,x) D(w,y) D(w,lambda)]

# ╔═╡ 7718d1f9-3e0e-4476-a51e-175dafabcc18
b=-[D(u,alpha),D(v,alpha),D(w,alpha)]

# ╔═╡ 588b8f23-b70c-4586-8007-5c1a631b6332
X=A\b

# ╔═╡ 2ec6a879-d5e2-481d-b050-54028536b704
md"""
At this point ``X=[dx/d\alpha, dy/d\alpha, d\lambda/d\alpha]``.  Now compute the
sensitivities.
"""

# ╔═╡ 69b3bebb-d704-4ee0-add3-dc63a982075d
s0=[alpha=>40,x=>x0,y=>y0,lambda=>lambda0]

# ╔═╡ 3edc32b7-bebf-4f0e-8cb8-b23ac681a2f8
Sxa=substitute(alpha/x*X[1],s0)

# ╔═╡ e25e4014-152d-43c0-ab2b-fe7e1f9b810d
Sya=substitute(alpha/y*X[2],s0)

# ╔═╡ ce333e65-844e-47fc-99fd-054f9bf93f2d
md"""
It follows that
``S(x,\alpha)=`` $Sxa 
and ``S(y,\alpha)=`` $Sya.
"""

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uuid = "62fd8b95-f654-4bbd-a8a5-9c27f68ccd50"
version = "0.1.1"

[[deps.TermInterface]]
git-tree-sha1 = "d673e0aca9e46a2f63720201f55cc7b3e7169b16"
uuid = "8ea1fca8-c5ef-4a55-8b96-4e9afe9c9a3c"
version = "2.0.0"

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

[[deps.TranscodingStreams]]
git-tree-sha1 = "0c45878dcfdcfa8480052b6ab162cdd138781742"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
version = "0.11.3"

[[deps.URIs]]
git-tree-sha1 = "bef26fb046d031353ef97a82e3fdb6afe7f21b1a"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.6.1"

[[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.Unitful]]
deps = ["Dates", "LinearAlgebra", "Random"]
git-tree-sha1 = "6258d453843c466d84c17a58732dda5deeb8d3af"
uuid = "1986cc42-f94f-5a68-af5c-568840ba703d"
version = "1.24.0"

    [deps.Unitful.extensions]
    ConstructionBaseUnitfulExt = "ConstructionBase"
    ForwardDiffExt = "ForwardDiff"
    InverseFunctionsUnitfulExt = "InverseFunctions"
    PrintfExt = "Printf"

    [deps.Unitful.weakdeps]
    ConstructionBase = "187b0558-2788-49d3-abe0-74a17ed4e7c9"
    ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"
    Printf = "de0858da-6303-5e67-8744-51eddeeeb8d7"

[[deps.UnitfulLatexify]]
deps = ["LaTeXStrings", "Latexify", "Unitful"]
git-tree-sha1 = "af305cc62419f9bd61b6644d19170a4d258c7967"
uuid = "45397f5d-5981-4c77-b2b3-fc36d6e9b728"
version = "1.7.0"

[[deps.Unzip]]
git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.2.0"

[[deps.Vulkan_Loader_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Wayland_jll", "Xorg_libX11_jll", "Xorg_libXrandr_jll", "xkbcommon_jll"]
git-tree-sha1 = "2f0486047a07670caad3a81a075d2e518acc5c59"
uuid = "a44049a8-05dd-5a78-86c9-5fde0876e88c"
version = "1.3.243+0"

[[deps.Wayland_jll]]
deps = ["Artifacts", "EpollShim_jll", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll"]
git-tree-sha1 = "96478df35bbc2f3e1e791bc7a3d0eeee559e60e9"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.24.0+0"

[[deps.WeakCacheSets]]
git-tree-sha1 = "386050ae4353310d8ff9c228f83b1affca2f7f38"
uuid = "d30d5f5c-d141-4870-aa07-aabb0f5fe7d5"
version = "0.1.0"

[[deps.XZ_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "fee71455b0aaa3440dfdd54a9a36ccef829be7d4"
uuid = "ffd25f8a-64ca-5728-b0f7-c24cf3aae800"
version = "5.8.1+0"

[[deps.Xorg_libICE_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "a3ea76ee3f4facd7a64684f9af25310825ee3668"
uuid = "f67eecfb-183a-506d-b269-f58e52b52d7c"
version = "1.1.2+0"

[[deps.Xorg_libSM_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libICE_jll"]
git-tree-sha1 = "9c7ad99c629a44f81e7799eb05ec2746abb5d588"
uuid = "c834827a-8449-5923-a945-d239c165b7dd"
version = "1.2.6+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "b5899b25d17bf1889d25906fb9deed5da0c15b3b"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.8.12+0"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "aa1261ebbac3ccc8d16558ae6799524c450ed16b"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.13+0"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "6c74ca84bbabc18c4547014765d194ff0b4dc9da"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.4+0"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "52858d64353db33a56e13c341d7bf44cd0d7b309"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.6+0"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "a4c0ee07ad36bf8bbce1c3bb52d21fb1e0b987fb"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.7+0"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "9caba99d38404b285db8801d5c45ef4f4f425a6d"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "6.0.1+0"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "a376af5c7ae60d29825164db40787f15c80c7c54"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.8.3+0"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll"]
git-tree-sha1 = "a5bc75478d323358a90dc36766f3c99ba7feb024"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.6+0"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "aff463c82a773cb86061bce8d53a0d976854923e"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.5+0"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "7ed9347888fac59a618302ee38216dd0379c480d"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.12+0"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXau_jll", "Xorg_libXdmcp_jll"]
git-tree-sha1 = "bfcaf7ec088eaba362093393fe11aa141fa15422"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.17.1+0"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "e3150c7400c41e207012b41659591f083f3ef795"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.3+0"

[[deps.Xorg_xcb_util_cursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_jll", "Xorg_xcb_util_renderutil_jll"]
git-tree-sha1 = "c5bf2dad6a03dfef57ea0a170a1fe493601603f2"
uuid = "e920d4aa-a673-5f3a-b3d7-f755a4d47c43"
version = "0.1.5+0"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "f4fc02e384b74418679983a97385644b67e1263b"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.1+0"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll"]
git-tree-sha1 = "68da27247e7d8d8dafd1fcf0c3654ad6506f5f97"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.1+0"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "44ec54b0e2acd408b0fb361e1e9244c60c9c3dd4"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.1+0"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "5b0263b6d080716a02544c55fdff2c8d7f9a16a0"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.10+0"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "f233c83cad1fa0e70b7771e0e21b061a116f2763"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.2+0"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "801a858fc9fb90c11ffddee1801bb06a738bda9b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.7+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "00af7ebdc563c9217ecc67776d1bbf037dbcebf4"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.44.0+0"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "a63799ff68005991f9d9491b6e95bd3478d783cb"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.6.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+1"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "446b23e73536f84e8037f5dce465e92275f6a308"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.7+1"

[[deps.eudev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "c3b0e6196d50eab0c5ed34021aaa0bb463489510"
uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
version = "3.2.14+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "b6a34e0e0960190ac2a4363a1bd003504772d631"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.61.1+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "4bba74fa59ab0755167ad24f98800fe5d727175b"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.12.1+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "125eedcb0a4a0bba65b657251ce1d27c8714e9d6"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.17.4+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.11.0+0"

[[deps.libdecor_jll]]
deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"]
git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f"
uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f"
version = "0.2.2+0"

[[deps.libevdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "56d643b57b188d30cccc25e331d416d3d358e557"
uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
version = "1.13.4+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "646634dd19587a56ee2f1199563ec056c5f228df"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.4+0"

[[deps.libinput_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "eudev_jll", "libevdev_jll", "mtdev_jll"]
git-tree-sha1 = "91d05d7f4a9f67205bd6cf395e488009fe85b499"
uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
version = "1.28.1+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "07b6a107d926093898e82b3b1db657ebe33134ec"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.50+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll"]
git-tree-sha1 = "11e1772e7f3cc987e9d3de991dd4f6b2602663a5"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.8+0"

[[deps.mtdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "b4d631fd51f2e9cdd93724ae25b2efc198b059b1"
uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
version = "1.1.7+0"

[[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+2"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "14cc7083fc6dff3cc44f2bc435ee96d06ed79aa7"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "10164.0.1+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e7b67590c14d487e734dcb925924c5dc43ec85f3"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "4.1.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "fbf139bce07a534df0e699dbb5f5cc9346f95cc1"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.9.2+0"
"""

# ╔═╡ Cell order:
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