| import OAI.Geometry.CylinderCovering.Radial.SweepDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.Nilpotent |
|
|
| namespace OAI |
|
|
| open Set |
|
|
| namespace CylinderCovering.RadialSweep |
|
|
| theorem velocity_derivWithin (D : Data) {t : ℝ} (ht : t ∈ Icc D.α D.β) : |
| derivWithin D.w (Icc D.α D.β) t = D.lam t • (D.a + t • D.b) := |
| (D.aligned t ht).derivWithin (uniqueDiffOn_Icc D.interval_nonempty t ht) |
|
|
| theorem velocity_contDiffOn (D : Data) : ContDiffOn ℝ 1 D.w (Icc D.α D.β) := by |
| apply (contDiffOn_one_iff_derivWithin (uniqueDiffOn_Icc D.interval_nonempty)).mpr |
| refine ⟨fun t ht => (D.aligned t ht).differentiableWithinAt, ?_⟩ |
| have hc : ContinuousOn (fun t => D.lam t • (D.a + t • D.b)) (Icc D.α D.β) := |
| D.scalar_continuous.smul (continuous_const.continuousOn.add |
| (continuous_id.continuousOn.smul continuous_const.continuousOn)) |
| exact hc.congr (fun t ht => velocity_derivWithin D ht) |
|
|
| theorem velocity_hasDerivAt (D : Data) {t : ℝ} (ht : t ∈ Ioo D.α D.β) : |
| HasDerivAt D.w (D.lam t • (D.a + t • D.b)) t := |
| (D.aligned t ⟨ht.1.le, ht.2.le⟩).hasDerivAt (Icc_mem_nhds ht.1 ht.2) |
|
|
| theorem velocity_contDiffOn_interior (D : Data) : |
| ContDiffOn ℝ 1 D.w (Ioo D.α D.β) := |
| (velocity_contDiffOn D).mono Ioo_subset_Icc_self |
|
|
| end CylinderCovering.RadialSweep |
|
|
| namespace RadialNilpotent |
|
|
| variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] |
|
|
| theorem isOpen_radialSector (e : (ℝ × ℝ) ≃L[ℝ] F) (α β : ℝ) : |
| IsOpen {z : F | 0 < (e.symm z).1 ∧ |
| (e.symm z).2 / (e.symm z).1 ∈ Ioo α β} := by |
| have heq : {z : F | 0 < (e.symm z).1 ∧ |
| (e.symm z).2 / (e.symm z).1 ∈ Ioo α β} = |
| {z : F | 0 < (e.symm z).1 ∧ |
| α * (e.symm z).1 < (e.symm z).2 ∧ |
| (e.symm z).2 < β * (e.symm z).1} := by |
| ext z |
| constructor |
| · rintro ⟨hz, hα, hβ⟩ |
| exact ⟨hz, (lt_div_iff₀ hz).mp hα, (div_lt_iff₀ hz).mp hβ⟩ |
| · rintro ⟨hz, hα, hβ⟩ |
| exact ⟨hz, (lt_div_iff₀ hz).mpr hα, (div_lt_iff₀ hz).mpr hβ⟩ |
| rw [heq] |
| have hu : Continuous (fun z : F => (e.symm z).1) := |
| continuous_fst.comp e.symm.continuous |
| have hv : Continuous (fun z : F => (e.symm z).2) := |
| continuous_snd.comp e.symm.continuous |
| exact (isOpen_lt continuous_const hu).inter |
| ((isOpen_lt (continuous_const.mul hu) hv).inter |
| (isOpen_lt hv (continuous_const.mul hu))) |
|
|
| theorem positive_radial_parameters_unique {a b : F} |
| (e : (ℝ × ℝ) ≃L[ℝ] F) |
| (he : ∀ u v : ℝ, e (u, v) = u • a + v • b) |
| {ρ σ t s : ℝ} (hρ : 0 < ρ) |
| (h : ρ • (a + t • b) = σ • (a + s • b)) : |
| ρ = σ ∧ t = s := by |
| have hc := congrArg e.symm h |
| rw [basis_coordinates_radial e he, basis_coordinates_radial e he] at hc |
| have hr : ρ = σ := congrArg Prod.fst hc |
| have ht : ρ * t = σ * s := congrArg Prod.snd hc |
| refine ⟨hr, ?_⟩ |
| rw [← hr] at ht |
| exact (mul_left_cancel₀ (ne_of_gt hρ)) ht |
|
|
| end RadialNilpotent |
|
|
| end OAI |
|
|
