Research proof index

Loop.ofPath is continuous, general version.

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...

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 ≠...

The antipodal map on 𝕊^n ⊆ ℝ^{n+1} is an immersion.

Note: The conditions in this lemma are currently a bit weaker than the ones mentioned in the blueprint. TODO: use local_loops_def

Lemma lem:localisation_stability.