lean/OAI/Geometry/Cannon/Modulus/IndicesPreimageImage.lean

import Mathlib
import OAI.Geometry.Cannon.Modulus.Weights

namespace OAI

section
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noncomputable section
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open Set Topology
open Set Topology Filter Metric unitInterval
open scoped InnerProductSpace
open Set unitInterval
open Set Metric Topology
open Set Topology unitInterval

namespace CannonCells

open CannonDuality

variable {X Y ι : Type*} [Fintype ι]

lemma indices_preimage_image (e : X → Y) (C : ι → Set Y) (A : Set X) :
    indices (fun i => e ⁻¹' C i) A = indices C (e '' A) := by
  ext i
  rw [mem_indices,mem_indices]
  constructor
  · rintro ⟨x,hx,hc⟩
    exact ⟨e x,⟨x,hx,rfl⟩,hc⟩
  · rintro ⟨_,⟨x,hx,rfl⟩,hc⟩
    exact ⟨x,hx,hc⟩

lemma incidence_preimage_image (e : X → Y) (C : ι → Set Y) (A : Set X) :
    incidence (fun i => e ⁻¹' C i) A = incidence C (e '' A) := by
  unfold incidence
  rw [indices_preimage_image]

lemma familyVectors_preimage_image (e : X → Y) (C : ι → Set Y) (F : Set (Set X)) :
    familyVectors (fun i => e ⁻¹' C i) F =
      familyVectors C {B | ∃ A ∈ F, B = e '' A} := by
  ext v
  rw [mem_familyVectors,mem_familyVectors]
  constructor
  · rintro ⟨A,hA,rfl⟩
    exact ⟨e '' A,⟨A,hA,rfl⟩,(incidence_preimage_image e C A).symm⟩
  · rintro ⟨B,⟨A,hA,rfl⟩,rfl⟩
    exact ⟨A,hA,incidence_preimage_image e C A⟩

lemma modulus_preimage_image (e : X → Y) (C : ι → Set Y) (F : Set (Set X)) :
    modulus (fun i => e ⁻¹' C i) F = modulus C {B | ∃ A ∈ F, B = e '' A} := by
  unfold modulus
  rw [familyVectors_preimage_image]

end CannonCells

end
end
end
end
end
end
end
end
end

end OAI