import Mathlib
import OAI.Analysis.BiholderTransport.Model

namespace OAI

section

section

noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff

namespace WeakMTWTransport
section ModelMixedConorm
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 tangentFiniteModelMixed (x : M) :
    FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
  inferInstanceAs (FiniteDimensional ℝ (Model n))
end ModelMixedConorm
end WeakMTWTransport

end

end

section

noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff

namespace WeakMTWTransport
section UniformMixedConorm
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 tangentFiniteUniformConorm (x : M) :
    FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
  inferInstanceAs (FiniteDimensional ℝ (Model n))
end UniformMixedConorm
end WeakMTWTransport

end

end

section

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

namespace WeakMTWTransport
section UniformRadialProfile
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 tangentFiniteUniformProfile (x : M) :
    FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
  inferInstanceAs (FiniteDimensional ℝ (Model n))
end UniformRadialProfile
end WeakMTWTransport

end

end

section

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

namespace WeakMTWTransport
section AffineMTW
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 tangentFiniteAffine (x : M) :
    FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
  inferInstanceAs (FiniteDimensional ℝ (Model n))
end AffineMTW
end WeakMTWTransport

end

end

section

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

namespace WeakMTWTransport
section UniformJoinSpectrum
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 tangentFiniteJoinBound (x : M) :
    FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
  inferInstanceAs (FiniteDimensional ℝ (Model n))
end UniformJoinSpectrum
end WeakMTWTransport

end

end

section

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

namespace WeakMTWTransport
section UniformAffineProfile
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 tangentFiniteAffineProfile (x : M) :
    FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
  inferInstanceAs (FiniteDimensional ℝ (Model n))
end UniformAffineProfile
end WeakMTWTransport

end

end

section

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

namespace WeakMTWTransport
section AffineSpectrumComparison
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 tangentFiniteAffineSpectrum (x : M) :
    FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
  inferInstanceAs (FiniteDimensional ℝ (Model n))
end AffineSpectrumComparison
end WeakMTWTransport

end

end

section

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

namespace WeakMTWTransport
section FixedJoinDeterminant
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 tangentFiniteJoinDet (x : M) :
    FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) :=
  inferInstanceAs (FiniteDimensional ℝ (Model n))
end FixedJoinDeterminant
end WeakMTWTransport

end

end

section

noncomputable section
open Module

namespace WeakMTWTransport
section NormDetBound
variable {E F : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [NormedAddCommGroup F] [InnerProductSpace ℝ F]
  [FiniteDimensional ℝ F]

lemma norm_eigenvector_image_eq_singularValue (A : E →ₗ[ℝ] F)
    {n : ℕ} (hn : finrank ℝ E=n) (i : Fin n) :
    ‖A (A.isSymmetric_adjoint_comp_self.eigenvectorBasis hn i)‖=A.singularValues i := by
  let v := A.isSymmetric_adjoint_comp_self.eigenvectorBasis hn i
  have hv : ‖v‖=1 := (A.isSymmetric_adjoint_comp_self.eigenvectorBasis hn).norm_eq_one i
  have he : (A.adjoint.comp A) v=A.isSymmetric_adjoint_comp_self.eigenvalues hn i • v :=
    A.isSymmetric_adjoint_comp_self.apply_eigenvectorBasis hn i
  have hsq : ‖A v‖^2=A.isSymmetric_adjoint_comp_self.eigenvalues hn i := by
    rw [←real_inner_self_eq_norm_sq,←A.adjoint_inner_left v (A v)]
    change inner ℝ ((A.adjoint.comp A) v) v=_
    rw [he,real_inner_smul_left,real_inner_self_eq_norm_sq,hv]
    ring
  apply (sq_eq_sq₀ (norm_nonneg _) (A.singularValues_nonneg i)).mp
  rw [hsq,A.sq_singularValues_fin hn i]
end NormDetBound
end WeakMTWTransport

end

end

end

end OAI