| import OAI.Geometry.IsometricImmersion.Metric |
| import Mathlib.Analysis.SpecialFunctions.SmoothTransition |
| import Mathlib.Analysis.Calculus.Deriv.Slope |
| import Mathlib.Analysis.Calculus.ContDiff.Operations |
| import Mathlib.Tactic.FunProp |
| import Mathlib.Tactic.Linarith |
| import Mathlib.Tactic.NormNum |
| import Mathlib.Tactic.Ring |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.Model |
| open SmoothLocal.Geometry |
|
|
| def squareTransition (t : ℝ) : ℝ := Real.smoothTransition (4 * t - 1) |
|
|
| theorem squareTransition_contDiff : ContDiff ℝ ∞ squareTransition := |
| Real.smoothTransition.contDiff.comp ((contDiff_const.mul contDiff_id).sub contDiff_const) |
|
|
| theorem squareTransition_range (t : ℝ) : |
| 0 ≤ squareTransition t ∧ squareTransition t ≤ 1 := |
| ⟨Real.smoothTransition.nonneg _, Real.smoothTransition.le_one _⟩ |
|
|
| theorem squareTransition_zero {t : ℝ} (ht : t ≤ 1 / 4) : squareTransition t = 0 := |
| Real.smoothTransition.zero_of_nonpos (by linarith) |
|
|
| theorem squareTransition_one {t : ℝ} (ht : 1 / 2 ≤ t) : squareTransition t = 1 := |
| Real.smoothTransition.one_of_one_le (by linarith) |
|
|
| theorem squareTransition_monotone : Monotone squareTransition := by |
| intro x y hxy |
| exact Real.smoothTransition.monotone (by linarith) |
|
|
| theorem squareTransition_deriv_nonneg (t : ℝ) : 0 ≤ deriv squareTransition t := |
| squareTransition_monotone.deriv_nonneg |
|
|
| def modelProfile (y : ℝ) : ℝ := 1 - y ^ 2 * squareTransition (y ^ 2) |
|
|
| theorem modelProfile_contDiff : ContDiff ℝ ∞ modelProfile := |
| contDiff_const.sub ((contDiff_id.pow 2).mul |
| (squareTransition_contDiff.comp (contDiff_id.pow 2))) |
|
|
| theorem modelProfile_even : Function.Even modelProfile := by |
| intro y |
| simp [modelProfile] |
|
|
| theorem modelProfile_plateau {y : ℝ} (hy : y ∈ Icc (-(1 / 2 : ℝ)) (1 / 2)) : |
| modelProfile y = 1 := by |
| have hs : y ^ 2 ≤ (1 / 2 : ℝ) ^ 2 := sq_le_sq' hy.1 hy.2 |
| have hz := squareTransition_zero (t := y ^ 2) (by nlinarith) |
| simp [modelProfile, hz] |
| |
| theorem modelProfile_bounds (y : ℝ) : 1 - y ^ 2 ≤ modelProfile y ∧ modelProfile y ≤ 1 := by |
| have hh := squareTransition_range (y ^ 2) |
| have hprod := mul_le_mul_of_nonneg_left hh.2 (sq_nonneg y) |
| have hprod0 := mul_nonneg (sq_nonneg y) hh.1 |
| unfold modelProfile |
| constructor <;> nlinarith |
| |
| theorem modelProfile_pos_inside {y : ℝ} (hy : |y| < 1) : 0 < modelProfile y := by |
| have hs : y ^ 2 < 1 := (sq_lt_one_iff_abs_lt_one y).mpr hy |
| linarith [(modelProfile_bounds y).1] |
| |
| theorem modelProfile_eq_one_sub_sq {y : ℝ} (hy : 1 / 2 ≤ y ^ 2) : |
| modelProfile y = 1 - y ^ 2 := by |
| simp [modelProfile, squareTransition_one hy] |
| |
| theorem modelProfile_neg_outside {y : ℝ} (hy : 1 < |y|) : modelProfile y < 0 := by |
| have hs : 1 < y ^ 2 := (one_lt_sq_iff_one_lt_abs y).mpr hy |
| rw [modelProfile_eq_one_sub_sq (by linarith : 1 / 2 ≤ y ^ 2)] |
| linarith |
| |
| theorem modelProfile_one : modelProfile 1 = 0 := by |
| rw [modelProfile_eq_one_sub_sq (by norm_num : (1 / 2 : ℝ) ≤ (1 : ℝ) ^ 2)] |
| norm_num |
| |
| theorem modelProfile_hasDerivAt (y : ℝ) : |
| HasDerivAt modelProfile |
| (-2 * y * squareTransition (y ^ 2) - 2 * y ^ 3 * deriv squareTransition (y ^ 2)) y := by |
| have hsq : HasDerivAt (fun t : ℝ => t ^ 2) (2 * y) y := by |
| convert (hasDerivAt_id y).pow 2 using 1 <;> first | rfl | simp |
| have hcut := ((squareTransition_contDiff.differentiable (by simp)) (y ^ 2)).hasDerivAt.comp y hsq |
| have hp := (hsq.mul hcut).const_sub 1 |
| convert hp using 1 <;> first | rfl | (dsimp only [Function.comp_def]; ring) |
| |
| theorem modelProfile_deriv (y : ℝ) : |
| deriv modelProfile y = |
| -2 * y * squareTransition (y ^ 2) - 2 * y ^ 3 * deriv squareTransition (y ^ 2) := |
| (modelProfile_hasDerivAt y).deriv |
| |
| theorem modelProfile_deriv_nonpos {y : ℝ} (hy : 0 ≤ y) : deriv modelProfile y ≤ 0 := by |
| rw [modelProfile_deriv] |
| have hfirst : 0 ≤ 2 * y * squareTransition (y ^ 2) := |
| mul_nonneg (mul_nonneg (by norm_num) hy) (squareTransition_range _).1 |
| have hsecond : 0 ≤ 2 * y ^ 3 * deriv squareTransition (y ^ 2) := |
| mul_nonneg (mul_nonneg (by norm_num) (pow_nonneg hy _)) (squareTransition_deriv_nonneg _) |
| linarith |
| |
| theorem modelProfile_deriv_one_neg : deriv modelProfile 1 < 0 := by |
| rw [modelProfile_deriv] |
| have hc := squareTransition_one (t := 1) (by norm_num) |
| have hd := squareTransition_deriv_nonneg 1 |
| norm_num only [one_pow] |
| rw [hc] |
| nlinarith only [hd] |
| |
| theorem modelProfile_spec : |
| ContDiff ℝ ∞ modelProfile ∧ Function.Even modelProfile ∧ |
| (∀ y ∈ Icc (-(1 / 2 : ℝ)) (1 / 2), modelProfile y = 1) ∧ |
| (∀ y ∈ Ici (0 : ℝ), deriv modelProfile y ≤ 0) ∧ |
| (∀ y ∈ Ioo (-1 : ℝ) 1, 0 < modelProfile y) ∧ |
| (∀ y : ℝ, 1 < |y| → modelProfile y < 0) ∧ |
| modelProfile 1 = 0 ∧ deriv modelProfile 1 < 0 := |
| ⟨modelProfile_contDiff, modelProfile_even, fun _ hy => modelProfile_plateau hy, |
| fun _ hy => modelProfile_deriv_nonpos hy, |
| fun _ hy => modelProfile_pos_inside (abs_lt.mpr hy), |
| fun _ hy => modelProfile_neg_outside hy, modelProfile_one, modelProfile_deriv_one_neg⟩ |
| |
| def modelCurvature (kappa : ℝ) (p : Coord) : ℝ := |
| kappa * ((p 0) ^ 2 - modelProfile (p 1)) |
| |
| theorem modelCurvature_contDiff (kappa : ℝ) : ContDiff ℝ ∞ (modelCurvature kappa) := |
| contDiff_const.mul (((contDiff_apply ℝ ℝ 0).pow 2).sub |
| (modelProfile_contDiff.comp (contDiff_apply ℝ ℝ 1))) |
| |
| theorem modelCurvature_central_square {kappa : ℝ} (hkappa : 0 < kappa) {p : Coord} |
| (hp : ∀ i : Fin 2, p i ∈ Icc (-(1 / 5 : ℝ)) (1 / 5)) : |
| modelCurvature kappa p < -kappa / 2 := by |
| have hplateau : modelProfile (p 1) = 1 := modelProfile_plateau |
| ⟨by linarith [(hp 1).1], by linarith [(hp 1).2]⟩ |
| have hx : (p 0) ^ 2 ≤ (1 / 5 : ℝ) ^ 2 := sq_le_sq' (hp 0).1 (hp 0).2 |
| have hmul := mul_le_mul_of_nonneg_left hx hkappa.le |
| unfold modelCurvature |
| rw [hplateau] |
| nlinarith |
|
|
| theorem modelCurvature_pos_lateral {kappa : ℝ} (hkappa : 0 < kappa) {p : Coord} |
| (hp : 1 < |p 0|) : 0 < modelCurvature kappa p := by |
| have hx : 1 < (p 0) ^ 2 := (one_lt_sq_iff_one_lt_abs _).mpr hp |
| apply mul_pos hkappa |
| linarith [(modelProfile_bounds (p 1)).2] |
|
|
| theorem modelCurvature_lateral_band_lower {kappa : ℝ} (hkappa : 0 < kappa) {p : Coord} |
| (hp : 3 / 2 < |p 0|) : 5 / 4 * kappa < modelCurvature kappa p := by |
| have hx : (3 / 2 : ℝ) ^ 2 < (p 0) ^ 2 := sq_lt_sq.mpr (by |
| rw [abs_of_pos (by norm_num : (0 : ℝ) < 3 / 2)] |
| exact hp) |
| have hmul := mul_lt_mul_of_pos_left hx hkappa |
| have hh := mul_le_mul_of_nonneg_left (modelProfile_bounds (p 1)).2 hkappa.le |
| unfold modelCurvature |
| nlinarith |
|
|
| theorem modelCurvature_pos_transverse {kappa : ℝ} (hkappa : 0 < kappa) {p : Coord} |
| (hp : 1 < |p 1|) : 0 < modelCurvature kappa p := by |
| have hh := modelProfile_neg_outside hp |
| apply mul_pos hkappa |
| linarith [sq_nonneg (p 0)] |
|
|
| theorem modelCurvature_pos_lower_flow_band {kappa y s t : ℝ} (hkappa : 0 < kappa) |
| (hys : |y - s| ≤ 1 / 50) (hs : s < -3 / 2) : |
| 0 < modelCurvature kappa ![t, y] := by |
| have hy : y < -1 := by linarith [(abs_le.mp hys).2] |
| apply modelCurvature_pos_transverse hkappa |
| simp only [Matrix.cons_val_one, Matrix.cons_val_zero] |
| rw [abs_of_neg (by linarith : y < 0)] |
| linarith |
|
|
| theorem modelCurvature_abs_le_model_square {kappa : ℝ} (hkappa : 0 ≤ kappa) {p : Coord} |
| (hp : ∀ i : Fin 2, p i ∈ Icc (-3 : ℝ) 3) : |
| |modelCurvature kappa p| ≤ 17 * kappa := by |
| have hx : (p 0) ^ 2 ≤ (3 : ℝ) ^ 2 := sq_le_sq' (hp 0).1 (hp 0).2 |
| have hy : (p 1) ^ 2 ≤ (3 : ℝ) ^ 2 := sq_le_sq' (hp 1).1 (hp 1).2 |
| have hh := modelProfile_bounds (p 1) |
| have habs : |(p 0) ^ 2 - modelProfile (p 1)| ≤ 17 := by |
| apply abs_le.mpr |
| constructor <;> nlinarith [sq_nonneg (p 0)] |
| unfold modelCurvature |
| rw [abs_mul, abs_of_nonneg hkappa] |
| simpa only [mul_comm] using mul_le_mul_of_nonneg_left habs hkappa |
|
|
| end SmoothLocal.Model |
|
|
| end |
|
|
| end OAI |
|
|
