### A Pluto.jl notebook ###
# v0.17.3

using Markdown
using InteractiveUtils

# ╔═╡ 9dc62b95-1bc9-4df7-aa14-a23adf0638a1
using OrdinaryDiffEq

# ╔═╡ 2a02b5fb-1c15-438b-ae32-1d5221e20748
using TaylorIntegration

# ╔═╡ 178f0e34-f8ed-4c75-8664-987d88fb5d02
using LinearAlgebra

# ╔═╡ ad2764df-1d51-47d9-972f-10b881578a94
using Plots

# ╔═╡ bce95956-cf2e-4140-9dcf-99abdadea5d1
function TMM_odesys!(primes, dependent, params, independent)
    γ = params[1]
    θ = independent
    Mr = dependent[1]
    Mθ = dependent[2]
    gm1o2 = (γ - 1) / 2
    Mθsq = Mθ * Mθ
    Mθsqm1 = Mθsq - 1
    Mθcotθ = Mθ * cot(θ)
    primes[1] = ((Mθ * ((gm1o2 * ((Mr * Mr) + (Mr * Mθcotθ))) + Mθsq - 1)) / Mθsqm1)
    primes[2] = (((((gm1o2 * Mθsq) + 1) * (Mr + Mθcotθ)) - (Mr * Mθsqm1)) / Mθsqm1)
    return nothing
end

# ╔═╡ 2f9ad515-6f5d-494f-a812-baf840ded39f
function solptdiff(u0, u1)
  diff = u0 - u1
  sqrt(dot(diff, diff))
end

# ╔═╡ 51c943cf-bff7-4cd5-80ee-3cf28cfce8a1
cf_test_prob = ODEProblem(TMM_odesys!, [2.5, 0.0], (deg2rad(0.1), 0.4), [1.4])

# ╔═╡ 97cfdb65-0c81-4075-8071-34423098fe54
Vern7sol = solve(cf_test_prob, Vern7(), abstol = 1e-12, reltol = 0.0)

# ╔═╡ b938f909-3e63-4d71-b7d2-76298aff23b9
TIsol = solve(cf_test_prob, TaylorMethod(30), abstol = 1e-13, parse_eqs = false)

# ╔═╡ 3947f2c3-3d59-41c6-a670-54af58114fa1
solptdiff(Vern7sol.u[end], TIsol.u[end])

# ╔═╡ 8a5d9b14-9c2c-4014-a9d5-e18df027cc9e
matching_t_sol = taylorinteg(TMM_odesys!, [2.5, 0.0], Vern7sol.t, 30, 1e-13, [1.4], parse_eqs = false)

# ╔═╡ d99167c4-90ff-4d5a-be6a-5eb64cc6a91f
function mat2vecvec(m)
  nr = size(m, 1)
  v = Vector{Vector{typeof(m[1,1])}}(undef, nr)
  for i = 1:nr
    v[i] = m[i, :]
  end
  return v
end

# ╔═╡ e775aee2-2e98-4573-9db7-93b1b7fb01bc
matching_t_sol_vv = mat2vecvec(matching_t_sol);

# ╔═╡ eacca520-9af6-4b33-bd51-bab98887da00
diffs = solptdiff.(Vern7sol.u, matching_t_sol_vv)

# ╔═╡ 975ace45-7b3d-428d-8025-4cd752ff81bb
plot(Vern7sol.t[2:end], diffs[2:end], yscale = :log10, legend_position = false)

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
TaylorIntegration = "92b13dbe-c966-51a2-8445-caca9f8a7d42"

[compat]
OrdinaryDiffEq = "~5.71.0"
Plots = "~1.25.2"
TaylorIntegration = "~0.8.9"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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[[HTTP]]
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[[HarfBuzz_jll]]
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[[HostCPUFeatures]]
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[[Hwloc]]
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[[Hwloc_jll]]
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[[IfElse]]
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[[Inflate]]
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[[IniFile]]
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[[InteractiveUtils]]
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[[InverseFunctions]]
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[[IrrationalConstants]]
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[[IterativeSolvers]]
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[[IteratorInterfaceExtensions]]
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[[JLLWrappers]]
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[[JSON]]
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[[JpegTurbo_jll]]
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[[LAME_jll]]
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[[LZO_jll]]
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[[LaTeXStrings]]
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[[LabelledArrays]]
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[[Latexify]]
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[[LayoutPointers]]
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[[LibCURL]]
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[[LibCURL_jll]]
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[[LibGit2]]
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[[LibSSH2_jll]]
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[[Libdl]]
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[[Libffi_jll]]
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[[Libgcrypt_jll]]
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[[Libglvnd_jll]]
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[[Libgpg_error_jll]]
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[[Libiconv_jll]]
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[[Libmount_jll]]
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[[Libtiff_jll]]
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[[Libuuid_jll]]
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[[LineSearches]]
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[[LogExpFunctions]]
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[[Logging]]
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[[LoopVectorization]]
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[[MacroTools]]
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[[Markdown]]
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[[MbedTLS]]
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[[MbedTLS_jll]]
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[[Measures]]
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[[Missings]]
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[[Mmap]]
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[[MozillaCACerts_jll]]
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[[MuladdMacro]]
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[[NLSolversBase]]
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[[NLsolve]]
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[[NaNMath]]
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[[NetworkOptions]]
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[[NonlinearSolve]]
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[[OffsetArrays]]
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[[Ogg_jll]]
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[[OpenLibm_jll]]
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[[OpenSSL_jll]]
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[[OpenSpecFun_jll]]
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[[Opus_jll]]
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[[OrderedCollections]]
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[[OrdinaryDiffEq]]
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[[PCRE_jll]]
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[[PDMats]]
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[[Parameters]]
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[[Parsers]]
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[[Pixman_jll]]
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[[Pkg]]
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[[PlotThemes]]
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[[PlotUtils]]
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[[Plots]]
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[[Polyester]]
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[[PolyesterWeave]]
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[[PreallocationTools]]
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[[Preferences]]
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[[Printf]]
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[[Qt5Base_jll]]
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[[QuadGK]]
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[[REPL]]
deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"

[[Random]]
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uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[RecipesBase]]
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[[RecipesPipeline]]
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[[RecursiveArrayTools]]
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[[RecursiveFactorization]]
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[[Reexport]]
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[[Requires]]
deps = ["UUIDs"]
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[[Rmath]]
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[[Rmath_jll]]
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[[SHA]]
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[[SIMDDualNumbers]]
deps = ["ForwardDiff", "IfElse", "SLEEFPirates", "VectorizationBase"]
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[[SIMDTypes]]
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[[SLEEFPirates]]
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git-tree-sha1 = "1410aad1c6b35862573c01b96cd1f6dbe3979994"
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[[SciMLBase]]
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git-tree-sha1 = "6f3c31990f63ac67ff6a485ccf77c5d8f7257547"
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[[Scratch]]
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[[Serialization]]
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[[Setfield]]
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git-tree-sha1 = "0afd9e6c623e379f593da01f20590bacc26d1d14"
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[[SharedArrays]]
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uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383"

[[Showoff]]
deps = ["Dates", "Grisu"]
git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de"
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[[SimpleTraits]]
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[[Sockets]]
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[[SortingAlgorithms]]
deps = ["DataStructures"]
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[[SparseArrays]]
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[[SparseDiffTools]]
deps = ["Adapt", "ArrayInterface", "Compat", "DataStructures", "FiniteDiff", "ForwardDiff", "Graphs", "LinearAlgebra", "Requires", "SparseArrays", "StaticArrays", "VertexSafeGraphs"]
git-tree-sha1 = "f33a0f6721b270cdf417f0c986e93c973e5913c8"
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[[SpecialFunctions]]
deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "e08890d19787ec25029113e88c34ec20cac1c91e"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
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[[Static]]
deps = ["IfElse"]
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[[StaticArrays]]
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[[Statistics]]
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uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[StatsAPI]]
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uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
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[[StatsBase]]
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git-tree-sha1 = "2bb0cb32026a66037360606510fca5984ccc6b75"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
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[[StatsFuns]]
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git-tree-sha1 = "bedb3e17cc1d94ce0e6e66d3afa47157978ba404"
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[[StrideArraysCore]]
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[[StructArrays]]
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git-tree-sha1 = "2ce41e0d042c60ecd131e9fb7154a3bfadbf50d3"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
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[[SuiteSparse]]
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uuid = "4607b0f0-06f3-5cda-b6b1-a6196a1729e9"

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

[[TableTraits]]
deps = ["IteratorInterfaceExtensions"]
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uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c"
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[[Tables]]
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git-tree-sha1 = "bb1064c9a84c52e277f1096cf41434b675cd368b"
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[[Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"

[[TaylorIntegration]]
deps = ["DiffEqBase", "Espresso", "InteractiveUtils", "LinearAlgebra", "Markdown", "OrdinaryDiffEq", "RecursiveArrayTools", "Reexport", "Requires", "StaticArrays", "TaylorSeries"]
git-tree-sha1 = "b7af537b6fefba170e3cc36de26d0c2da3124ffd"
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[[TaylorSeries]]
deps = ["LinearAlgebra", "Markdown", "Requires", "SparseArrays"]
git-tree-sha1 = "e59006471450151223b59921bc6582a687d61c23"
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[[Test]]
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uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[ThreadingUtilities]]
deps = ["ManualMemory"]
git-tree-sha1 = "03013c6ae7f1824131b2ae2fc1d49793b51e8394"
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[[TreeViews]]
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[[TriangularSolve]]
deps = ["CloseOpenIntervals", "IfElse", "LayoutPointers", "LinearAlgebra", "LoopVectorization", "Polyester", "Static", "VectorizationBase"]
git-tree-sha1 = "ec9a310324dd2c546c07f33a599ded9c1d00a420"
uuid = "d5829a12-d9aa-46ab-831f-fb7c9ab06edf"
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[[URIs]]
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[[UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[UnPack]]
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[[Unicode]]
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[[UnicodeFun]]
deps = ["REPL"]
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[[VectorizationBase]]
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[[VertexSafeGraphs]]
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[[Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
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[[Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "66d72dc6fcc86352f01676e8f0f698562e60510f"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
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[[XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
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[[XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
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[[Xorg_libX11_jll]]
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git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
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[[Xorg_libXau_jll]]
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git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
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[[Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
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[[Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
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[[Xorg_libXext_jll]]
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[[Xorg_libXfixes_jll]]
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[[Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
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[[Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
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[[Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
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[[Xorg_libXrender_jll]]
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uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
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[[Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
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[[Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
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version = "1.13.0+3"

[[Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
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[[Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
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[[Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
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[[Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
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[[Xorg_xcb_util_renderutil_jll]]
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git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
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[[Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
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[[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"

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

[[Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
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[[Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+0"

[[ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.2"

[[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"

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

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

[[libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+0"

[[nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

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

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

[[xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

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