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import OAI.Geometry.Cannon.Modulus.Weights
namespace OAI
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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
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