Eventually exists surrounding Pts approx Surrounding Points At
SmoothSurroundingFamily.eventually_exists_surroundingPts_approxSurroundingPointsAt
Plain-language statement
The key property from which it should be easy to construct localCenteringDensity, localCenteringDensityNhd etc below.
Exact Lean statement
theorem eventually_exists_surroundingPts_approxSurroundingPointsAt :
∀ᶠ z : E × ℕ in 𝓝 x ×ˢ atTop,
∃ w, SurroundingPts (g z.1) (γ.approxSurroundingPointsAt x z.1 z.2) wFormal artifact
Lean source
theorem eventually_exists_surroundingPts_approxSurroundingPointsAt : ∀ᶠ z : E × ℕ in 𝓝 x ×ˢ atTop, ∃ w, SurroundingPts (g z.1) (γ.approxSurroundingPointsAt x z.1 z.2) w := by let a : ι → E × ℕ → F := fun i z ↦ γ.approxSurroundingPointsAt x z.1 z.2 i suffices ∀ i, Tendsto (a i) (𝓝 x ×ˢ atTop) (𝓝 (γ.surroundingPointsAt x i)) by have hg : Tendsto (fun z : E × ℕ ↦ g z.fst) (𝓝 x ×ˢ atTop) (𝓝 (g x)) := Tendsto.comp γ.smooth_surrounded.continuous.continuousAt tendsto_fst exact eventually_surroundingPts_of_tendsto_of_tendsto' ⟨_, γ.surroundPtsPointsWeightsAt x⟩ this hg intro i let t := γ.surroundingParametersAt x i change Tendsto (fun z : E × ℕ ↦ (γ z.1).mollify z.2 t) (𝓝 x ×ˢ atTop) (𝓝 (γ x t)) exact Loop.tendsto_mollify_apply γ γ.smooth.continuous x t- Project
- Sphere eversion
- License
- Apache-2.0
- Commit
- ded8fda5e76b
- Source
- SphereEversion/Loops/Reparametrization.lean:147-159
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Continuous of Path
Continuous.ofPath
Plain-language statement
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Source project: Sphere eversion
Person-level attribution pending.
Injective update iff
DualPair.injective_update_iff
Project documentation
Map a dual pair under a linear equivalence. -/ @[simps] def map (p : DualPair E) (L : E ≃L[ℝ] E') : DualPair E' := ⟨p.π ∘L ↑L.symm, L p.v, (congr_arg p.π <| L.symm_apply_apply p.v).trans p.pairing⟩ theorem update_comp_left (p : DualPair E) (ψ' : F →L[ℝ] G) (φ : E →L[ℝ] F) (w : F) : p.update (ψ' ∘L φ) (ψ' w) = ψ' ∘L p.update φ w := by ext1 u simp only [upd...
Source project: Sphere eversion
Person-level attribution pending.
Extend loops
extend_loops
Project documentation
A more precise version of sfHomotopy_in. -/ theorem sfHomotopy_in' {ι} (h₀ : SurroundingFamily g b γ₀ U) (h₁ : SurroundingFamily g b γ₁ U) (τ : ι → ℝ) (x : ι → E) (i : ι) {V : Set E} (hx : x i ∈ V) {t : ℝ} (ht : t ∈ I) {s : ℝ} (h_in₀ : ∀ i, x i ∈ V → ∀ t ∈ I, ∀ (s : ℝ), τ i ≠ 1 → (x i, γ₀ (x i) t s) ∈ Ω) (h_in₁ : ∀ i, x i ∈ V → ∀ t ∈ I, ∀ (s : ℝ), τ i ≠...
Source project: Sphere eversion
Person-level attribution pending.