import Mathlib
import OAI.Analysis.BiholderTransport.Model

namespace OAI

section
section
noncomputable section
open Set Filter Manifold Bundle Module
open scoped Topology ContDiff

namespace WeakMTWTransport
section FixedJoinSpectrum
variable {n : ℕ} {M : Type*} [MetricSpace M] [ChartedSpace (Model n) M]

local instance tangentFiniteSpectrum (x : M) :
    FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
  inferInstanceAs (FiniteDimensional ℝ (Model n))

lemma tangent_finrank (x : M) : finrank ℝ (TangentSpace 𝓘(ℝ,Model n) x)=n :=
  finrank_euclideanSpace_fin
end FixedJoinSpectrum
end WeakMTWTransport

end

end

section

noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff

namespace WeakMTWTransport
section EndpointJoinAction
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M]
  [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
  [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
  [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
    (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
  [IsRiemannianManifold 𝓘(ℝ,Model n) M]

local instance tangentFiniteEndpointJoin (x : M) :
    FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
  inferInstanceAs (FiniteDimensional ℝ (Model n))
end EndpointJoinAction
end WeakMTWTransport

end

end

end

end OAI