Surrounding Family In iff germ
surroundingFamilyIn_iff_germ
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 ≠...
Exact Lean statement
theorem surroundingFamilyIn_iff_germ {γ : E → ℝ → Loop F} :
SurroundingFamilyIn g b γ C Ω ↔
(∀ x, LoopFamilyGerm b x γ) ∧ ∀ x ∈ C, SurroundingFamilyGerm g Ω x γFormal artifact
Lean source
theorem surroundingFamilyIn_iff_germ {γ : E → ℝ → Loop F} : SurroundingFamilyIn g b γ C Ω ↔ (∀ x, LoopFamilyGerm b x γ) ∧ ∀ x ∈ C, SurroundingFamilyGerm g Ω x γ := by constructor · rintro ⟨⟨base, t₀, projI, family_surrounds, family_cont⟩, H⟩ exact ⟨fun x ↦ ⟨base x, t₀ x, projI x, fun t s ↦ family_cont.continuousAt⟩, fun x x_in ↦ ⟨family_surrounds x x_in, H x x_in⟩⟩ · rintro ⟨h, h'⟩ refine ⟨⟨fun x ↦ (h x).base, fun x ↦ (h x).t₀, fun x ↦ (h x).projI, fun x hx ↦ (h' x hx).Surrounds, ?_⟩, fun x hx ↦ (h' x hx).val_in'⟩ apply continuous_iff_continuousAt.mpr rintro ⟨x, t, s⟩ apply (h x).cont- Project
- Sphere eversion
- License
- Apache-2.0
- Commit
- ded8fda5e76b
- Source
- SphereEversion/Loops/Surrounding.lean:963-976
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Related declarations
Continuous of Path
Continuous.ofPath
Plain-language statement
Loop.ofPath is continuous, general version.
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.