| import OAI.Geometry.IsometricImmersion.Darboux.QInitialEnergy |
| import OAI.Geometry.IsometricImmersion.Energy.LocalQResidualL2 |
| import OAI.Geometry.IsometricImmersion.Metrics.MetricHorizontalRequests |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set |
| open scoped ContDiff NNReal |
|
|
| namespace SmoothLocal.HighEquation |
| open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.ODE SmoothLocal.Hyperbolic |
|
|
| def QLowBounds (g : MetricField) (z : Coord → ℝ) (S : Set Coord) |
| (s0 M speed : ℝ) : Prop := |
| ∀ p ∈ S, s0 ≤ heightQCoefficient g z 5 p ∧ |
| |heightQCoefficient g z 2 p| ≤ M ∧ |heightQCoefficient g z 3 p| ≤ M ∧ |
| |heightQCoefficient g z 5 p| ≤ M ∧ |
| |coordPartial 0 (heightQCoefficient g z 4) p| ≤ M ∧ |
| |coordPartial 0 (heightQCoefficient g z 5) p| ≤ M ∧ |
| |coordPartial 1 (heightQCoefficient g z 5) p| ≤ M ∧ |
| |heightQCoefficient g z 4 p| + Real.sqrt (heightQCoefficient g z 5 p) ≤ speed |
|
|
| theorem QLowBounds.mono_set {g : MetricField} {z : Coord → ℝ} {S T : Set Coord} |
| {s0 M speed : ℝ} (h : QLowBounds g z S s0 M speed) (hTS : T ⊆ S) : |
| QLowBounds g z T s0 M speed := fun p hp => h p (hTS hp) |
|
|
| theorem QLowBounds.high_coefficients |
| {g : MetricField} {z : Coord → ℝ} {S : Set Coord} {s0 M speed : ℝ} |
| (h : QLowBounds g z S s0 M speed) (hM : 0 ≤ M) (ell : ℕ) {p : Coord} (hp : p ∈ S) : |
| |qHighBTheta g z ell p| ≤ M * (ell + 1) ∧ |
| |qHighBXi g z ell p| ≤ M * (ell + 1) ∧ |
| |coordPartial 0 (heightQCoefficient g z 4) p| ≤ M * (ell + 1) ∧ |
| |coordPartial 0 (heightQCoefficient g z 5) p| ≤ M * (ell + 1) ∧ |
| |coordPartial 1 (heightQCoefficient g z 5) p| ≤ M * (ell + 1) := by |
| obtain ⟨_, h2, h3, _, h4x, h5x, h5t, _⟩ := h p hp |
| have he : 0 ≤ (ell : ℝ) := Nat.cast_nonneg ell |
| have hB : M ≤ M * (ell + 1) := by nlinarith |
| have hbt : |qHighBTheta g z ell p| ≤ M * (ell + 1) := by |
| unfold qHighBTheta |
| calc |
| _ ≤ |heightQCoefficient g z 3 p| + |(ell : ℝ) * coordPartial 0 (heightQCoefficient g z 4) p| := abs_add_le _ _ |
| _ ≤ M + (ell : ℝ) * M := by |
| rw [abs_mul, abs_of_nonneg he] |
| exact add_le_add h3 (mul_le_mul_of_nonneg_left h4x he) |
| _ = _ := by ring |
| have hbx : |qHighBXi g z ell p| ≤ M * (ell + 1) := by |
| unfold qHighBXi |
| calc |
| _ ≤ |heightQCoefficient g z 2 p| + |(ell : ℝ) * coordPartial 0 (heightQCoefficient g z 5) p| := abs_add_le _ _ |
| _ ≤ M + (ell : ℝ) * M := by |
| rw [abs_mul, abs_of_nonneg he] |
| exact add_le_add h2 (mul_le_mul_of_nonneg_left h5x he) |
| _ = _ := by ring |
| exact ⟨hbt, hbx, h4x.trans hB, h5x.trans hB, h5t.trans hB⟩ |
|
|
| theorem shrinkingSlab_mono_top {xl xr a b t speed : ℝ} (ht : t ≤ b) : |
| shrinkingSlab xl xr a t speed ⊆ shrinkingSlab xl xr a b speed := by |
| intro p hp |
| exact ⟨⟨hp.1.1, hp.1.2.trans ht⟩, hp.2⟩ |
|
|
| theorem SpatialSliceL2Bound.mono_order_budget |
| {ι : Type*} {F : ι → Coord → ℝ} {theta left right : ℝ} |
| {N n : ℕ} {H H' : ℝ≥0} |
| (h : SpatialSliceL2Bound F theta left right N H) (hn : n ≤ N) (hH : H ≤ H') : |
| SpatialSliceL2Bound F theta left right n H' := by |
| intro i k hk |
| exact (h i k (hk.trans hn)).trans (by exact_mod_cast hH) |
|
|
| end SmoothLocal.HighEquation |
|
|
| end |
|
|
| end OAI |
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|
