| import OAI.Geometry.IsometricImmersion.Immersions.ActualShearHeight |
| import OAI.Geometry.IsometricImmersion.Calculus.CoordinateJetReverseNorm |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set |
| open scoped ContDiff BigOperators |
|
|
| namespace SmoothLocal.Pulse |
| open SmoothLocal.Geometry |
|
|
| def inverseShearCLM (q0 : ℝ) : Coord →L[ℝ] Coord := |
| LinearMap.toContinuousLinearMap |
| { toFun := inverseShearCoordinates q0 |
| map_add' := by |
| intro left right |
| ext index |
| fin_cases index |
| · rfl |
| · dsimp [inverseShearCoordinates] |
| ring |
| map_smul' := by |
| intro scalar point |
| ext index |
| fin_cases index |
| · rfl |
| · dsimp [inverseShearCoordinates] |
| ring } |
|
|
| @[simp] theorem inverseShearCLM_apply (q0 : ℝ) (p : Coord) : |
| inverseShearCLM q0 p = inverseShearCoordinates q0 p := rfl |
|
|
| theorem inverseShearCLM_norm_le (q0 : ℝ) : ‖inverseShearCLM q0‖ ≤ 1 + |q0| := by |
| apply ContinuousLinearMap.opNorm_le_bound _ (by positivity) |
| intro v |
| apply (pi_norm_le_iff_of_nonneg (by positivity : 0 ≤ (1 + |q0|) * ‖v‖)).mpr |
| intro i |
| fin_cases i |
| · change ‖v 0‖ ≤ _ |
| exact (norm_le_pi_norm v 0).trans (by nlinarith [mul_nonneg (abs_nonneg q0) (norm_nonneg v)]) |
| · change ‖v 1 - q0 * v 0‖ ≤ _ |
| calc |
| _ ≤ ‖v 1‖ + ‖q0 * v 0‖ := norm_sub_le _ _ |
| _ = ‖v 1‖ + |q0| * ‖v 0‖ := by rw [norm_mul, Real.norm_eq_abs q0] |
| _ ≤ ‖v‖ + |q0| * ‖v‖ := add_le_add (norm_le_pi_norm v 1) |
| (mul_le_mul_of_nonneg_left (norm_le_pi_norm v 0) (abs_nonneg q0)) |
| _ = _ := by ring |
|
|
| theorem local_linear_full_jet_norm_bound {f : Coord → ℝ} {U : Set Coord} |
| (hf : ContDiffOn ℝ ∞ f U) (hU : IsOpen U) (L : Coord →L[ℝ] Coord) |
| {p : Coord} (hp : L p ∈ U) (n : ℕ) : |
| ‖iteratedFDeriv ℝ n (f ∘ L) p‖ ≤ ‖iteratedFDeriv ℝ n f (L p)‖ * ‖L‖^n := by |
| have hpre := hU.preimage L.continuous |
| have heq := L.iteratedFDerivWithin_comp_right hf hU.uniqueDiffOn hpre.uniqueDiffOn hp |
| (i := n) (WithTop.coe_le_coe.mpr le_top) |
| rw [iteratedFDerivWithin_of_isOpen n hpre hp, |
| iteratedFDerivWithin_of_isOpen n hU hp] at heq |
| rw [heq] |
| simpa only [Finset.prod_const, Finset.card_univ, Fintype.card_fin] using |
| ContinuousMultilinearMap.norm_compContinuousLinearMap_le |
| (iteratedFDeriv ℝ n f (L p)) (fun _ : Fin n => L) |
|
|
| theorem sheared_full_jet_bound_of_coordinateBound {z : Coord → ℝ} {U S : Set Coord} |
| {N n : ℕ} {B : ℝ} (hz : ContDiffOn ℝ ∞ z U) (hU : IsOpen U) |
| (hSU : S ⊆ U) (hbound : CoordinateBound z S N B) (hB : 0 ≤ B) |
| (q0 : ℝ) (hn : n ≤ N) {p : Coord} (hp : inverseShearCoordinates q0 p ∈ S) : |
| ‖iteratedFDeriv ℝ n (heightInShearCoordinates z q0) p‖ ≤ |
| (2 * (1 + |q0|))^N * B := by |
| have hlocal := local_linear_full_jet_norm_bound hz hU (inverseShearCLM q0) (hSU hp) n |
| have hbase := hbound.full_jet_norm_bound hz hU hSU hB hn hp |
| have hscale : 1 ≤ 2 * (1 + |q0|) := by nlinarith [abs_nonneg q0] |
| calc |
| _ ≤ ‖iteratedFDeriv ℝ n z (inverseShearCoordinates q0 p)‖ * ‖inverseShearCLM q0‖^n := hlocal |
| _ ≤ ((2 : ℝ)^n * B) * (1 + |q0|)^n := mul_le_mul hbase |
| (pow_le_pow_left₀ (norm_nonneg _) (inverseShearCLM_norm_le q0) n) |
| (pow_nonneg (norm_nonneg _) n) (mul_nonneg (by positivity) hB) |
| _ = (2 * (1 + |q0|))^n * B := by rw [mul_pow]; ring |
| _ ≤ (2 * (1 + |q0|))^N * B := |
| mul_le_mul_of_nonneg_right (pow_le_pow_right₀ hscale hn) hB |
|
|
| theorem CoordinateBound.inverse_shear {z : Coord → ℝ} {U S : Set Coord} |
| {N : ℕ} {B : ℝ} (hbound : CoordinateBound z S N B) |
| (hz : ContDiffOn ℝ ∞ z U) (hU : IsOpen U) (hSU : S ⊆ U) (hB : 0 ≤ B) (q0 : ℝ) : |
| CoordinateBound (heightInShearCoordinates z q0) (inverseShearCoordinates q0 ⁻¹' S) |
| N ((2 * (1 + |q0|))^N * B) := by |
| intro ds hds p hp |
| apply (norm_iteratedCoordPartial_le_jet (heightInShearCoordinates_contDiffOn hz q0) |
| (hU.preimage (inverseShearCoordinates_contDiff q0).continuous) ds (hSU hp)).trans |
| exact sheared_full_jet_bound_of_coordinateBound hz hU hSU hbound hB q0 hds hp |
|
|
| end SmoothLocal.Pulse |
|
|
| end |
|
|
| end OAI |
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|
