| import OAI.Geometry.Immersion.ClosedSurface.FiniteIncrement |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Complex Bundle Manifold |
| open scoped ContDiff Matrix Topology Manifold BigOperators |
|
|
| namespace ClosedSurfaceR4.RealModes |
| open ClosedSurfaceR4.SmallModes ClosedSurfaceR4.PhaseMean ClosedSurfaceR4.WeightedEstimates |
| open Set |
| open ClosedSurfaceR4.QuadraticMean (sumDisplacement displacement) |
|
|
| theorem constructed_quadratic_correction_explicit {ι : Type*} [Fintype ι] [DecidableEq ι] |
| {F : RField 4} (hF : ContDiff ℝ ∞ F) {φ ψ u : ι → Base → ℝ} {S : ι → Set Base} |
| (c : ∀ i, SupportedFreeChart F (φ i) (ψ i) (S i)) |
| (d : ∀ l : QuadraticLabel ι, SupportedSolveChart F (quadraticPhase φ l) (quadraticSupport S l)) |
| {τ s P : ℝ} (hτ : 0 < τ) (hs : 0 < s) (hτs : τ ≤ s) (hs1 : s ≤ 1) |
| (hP : 0 ≤ P) (hφ : ∀ i, ContDiff ℝ ∞ (φ i)) (q r m : ℕ) |
| (b : ∀ i, FreeBudget (c i) (u i) τ s q (m + r + 1 + 1)) |
| (g : ∀ l, ForcedGeometryBudget (d l) τ s r m) |
| (hpφ : ∀ i v, ‖v‖ ≤ 1 → WeightedBound univ s (m + r + 1) P (coordDeriv v (φ i))) : |
| let Q : ℝ := 4 * 2 ^ (m + r + 1) * ((1 + 2 ^ (m + r + 1) * P) * ∑ i, (b i).slowSize) ^ 2 |
| let B := (∑ l, (g l).sizeFactor) * Q |
| let R := (∑ l, (g l).errorFactor) * Q |
| 0 ≤ B ∧ 0 ≤ R ∧ ∀ δ : ℝ, 0 ≤ δ → |
| let Z := fun i => (c i).amplitude (u i) δ τ q |
| let V := fun p => ∑ l, (d l).solve τ (quadraticAmplitude τ φ Z l) r p |
| ContDiff ℝ ∞ V ∧ tsupport V ⊆ ⋃ i, S i ∧ |
| WeightedBound univ τ m (B * δ ^ 2) V ∧ |
| WeightedBound univ τ m (R * (τ / s) ^ (r + 1) * δ ^ 2) |
| (fun p => realLinearizedTensor F V p + nonzeroPhaseSum τ φ Z p) := by |
| classical |
| let Q : ℝ := 4 * 2 ^ (m + r + 1) * ((1 + 2 ^ (m + r + 1) * P) * ∑ i, (b i).slowSize) ^ 2 |
| have hQ : 0 ≤ Q := by dsimp [Q]; positivity |
| refine ⟨mul_nonneg (Finset.sum_nonneg fun l _ => (g l).sizeFactor_nonneg) hQ, |
| mul_nonneg (Finset.sum_nonneg fun l _ => (g l).errorFactor_nonneg) hQ, ?_⟩ |
| intro δ hδ |
| dsimp only |
| let Z := fun i => (c i).amplitude (u i) δ τ q |
| have hZ (i) := (c i).amplitude_smooth (b i).smoothAmplitude δ τ q |
| have hbA := weighted_free_quadraticAmplitude c hδ hτ hs hτs hs1 hP hφ b hpφ |
| have hA := quadraticAmplitude_smooth hφ (fun i => (hZ i).1) τ |
| let budget := fun l => (g l).toSolveBudget hs hs1 (mul_nonneg hQ (sq_nonneg δ)) (hA l) (hbA l) |
| obtain ⟨hsm, hsp, hsize, hres⟩ := finite_quadratic_cancellation hF hφ |
| (fun i => (hZ i).1) (fun i => (hZ i).2) d hτ hs hτs hs1 r m budget |
| refine ⟨hsm, hsp, ?_, ?_⟩ |
| · have he : (∑ l, (budget l).size) = ((∑ l, (g l).sizeFactor) * Q) * δ ^ 2 := by |
| simp only [budget, ForcedGeometryBudget.toSolveBudget_size, ← Finset.sum_mul] |
| ring |
| rwa [he] at hsize |
| · have he : (∑ l, (budget l).residual) = |
| ((∑ l, (g l).errorFactor) * Q) * (τ / s) ^ (r + 1) * δ ^ 2 := by |
| simp only [budget, ForcedGeometryBudget.toSolveBudget_residual, ← Finset.sum_mul] |
| ring |
| rwa [he] at hres |
|
|
|
|
| theorem constructed_finite_increment_explicit {ι : Type*} [Fintype ι] [DecidableEq ι] |
| {F : RField 4} (hF : ContDiff ℝ ∞ F) {φ ψ u : ι → Base → ℝ} {S : ι → Set Base} |
| (c : ∀ i, SupportedFreeChart F (φ i) (ψ i) (S i)) |
| (d : ∀ l : QuadraticLabel ι, SupportedSolveChart F (quadraticPhase φ l) (quadraticSupport S l)) |
| {τ s P : ℝ} (hτ : 0 < τ) (hs : 0 < s) (hτs : τ ≤ s) (hs1 : s ≤ 1) |
| (hP : 0 ≤ P) (hφ : ∀ i, ContDiff ℝ ∞ (φ i)) (q m : ℕ) |
| (b : ∀ i, FreeBudget (c i) (u i) τ s q (m + 1 + q + 1 + 1)) |
| (g : ∀ l, ForcedGeometryBudget (d l) τ s q (m + 1)) |
| (hpφ : ∀ i v, ‖v‖ ≤ 1 → WeightedBound univ s (m + 1 + q + 1) P (coordDeriv v (φ i))) : |
| let low := fun i => (b i).mono_order (r := m + 1) (by omega) |
| let A : ℝ := ∑ i, (low i).size |
| let L : ℝ := ∑ i, (low i).errorFactor |
| let Q : ℝ := 4 * 2 ^ (m + 1 + q + 1) * ((1 + 2 ^ (m + 1 + q + 1) * P) * ∑ i, (b i).slowSize) ^ 2 |
| let B := (∑ l, (g l).sizeFactor) * Q |
| let R := (∑ l, (g l).errorFactor) * Q |
| let C := A + B |
| let E := L + R |
| let T := 4 * 2 ^ m * (2 * A * B + B ^ 2) |
| 0 ≤ C ∧ 0 ≤ E ∧ 0 ≤ T ∧ ∀ δ : ℝ, 0 ≤ δ → δ ≤ τ → |
| let Z := fun i => (c i).amplitude (u i) δ τ q |
| let X := sumDisplacement τ φ Z |
| let V := fun p => ∑ l, (d l).solve τ (quadraticAmplitude τ φ Z l) q p |
| let U := fun p => X p + V p |
| ContDiff ℝ ∞ U ∧ tsupport U ⊆ ⋃ i, S i ∧ |
| WeightedBound univ τ m (C * (δ * τ)) U ∧ |
| WeightedBound univ τ m (E * δ * (τ / s) ^ (q + 1) + T * (δ ^ 3 / τ)) |
| (fun p => realMetricTensor (fun x => F x + U x) p - realMetricTensor F p - zeroPhaseSum τ φ Z p) := by |
| classical |
| let low := fun i => (b i).mono_order (r := m + 1) (by omega) |
| let A : ℝ := ∑ i, (low i).size |
| let L : ℝ := ∑ i, (low i).errorFactor |
| have hA : 0 ≤ A := Finset.sum_nonneg fun i _ => (low i).size_nonneg |
| have hL : 0 ≤ L := Finset.sum_nonneg fun i _ => (low i).errorFactor_nonneg |
| let Q : ℝ := 4 * 2 ^ (m + 1 + q + 1) * ((1 + 2 ^ (m + 1 + q + 1) * P) * ∑ i, (b i).slowSize) ^ 2 |
| let B := (∑ l, (g l).sizeFactor) * Q |
| let R := (∑ l, (g l).errorFactor) * Q |
| have hh := constructed_quadratic_correction_explicit hF c d hτ hs hτs hs1 |
| hP hφ q q (m + 1) b g hpφ |
| obtain ⟨hB, hR, hv⟩ := hh |
| refine ⟨add_nonneg hA hB, |
| add_nonneg hL hR, by positivity, ?_⟩ |
| intro δ hδ hδτ |
| dsimp only |
| let Z := fun i => (c i).amplitude (u i) δ τ q |
| let X := sumDisplacement τ φ Z |
| let V := fun p => ∑ l, (d l).solve τ (quadraticAmplitude τ φ Z l) q p |
| obtain ⟨hZ, hX, hXsp, hXb, hXr⟩ := finite_free_displacement hF c hδ hτ hs hτs hs1 q (m + 1) low |
| obtain ⟨hV, hVsp, hVb, hVr⟩ := hv δ hδ |
| have hXS : ContDiffOn ℝ ∞ X univ := hX.contDiffOn |
| have hVS : ContDiffOn ℝ ∞ V univ := hV.contDiffOn |
| have hXb' : WeightedBound univ τ (m + 1) (A * δ * τ) X := by |
| convert hXb using 1 |
| first | rfl | ring |
| have hc := weighted_cubic_remainder isOpen_univ hτ hδ hδτ hA hB hXS hVS hXb' hVb |
| have hXS0 := contDiffOn_realLinearizedTensor isOpen_univ hF.contDiffOn hXS |
| have hVS0 := (contDiffOn_realLinearizedTensor isOpen_univ hF.contDiffOn hVS).add |
| (contDiff_nonzeroPhaseSum hφ (fun i => (hZ i).1) τ).contDiffOn |
| have hCS0 := (contDiffOn_realLinearizedTensor isOpen_univ hXS hVS).add |
| (contDiffOn_realMetricTensor isOpen_univ hVS) |
| have hη : 0 ≤ (τ / s) ^ (q + 1) := by positivity |
| have hτ1 := hτs.trans hs1 |
| have hδ1 := hδτ.trans hτ1 |
| have hXR : WeightedBound univ τ m (L * δ * (τ / s) ^ (q + 1)) (realLinearizedTensor F X) := by |
| apply (hXr.mono_order (Nat.le_succ m)).mono_const |
| have he : (∑ i, (low i).residual) * (δ * τ) = L * δ * (τ / s) ^ (q + 1) * τ := by |
| simp only [FreeBudget.residual_eq, ← Finset.sum_mul, L] |
| ring |
| rw [he] |
| exact mul_le_of_le_one_right (by positivity) hτ1 |
| have hVR : WeightedBound univ τ m (R * δ * (τ / s) ^ (q + 1)) |
| (fun p => realLinearizedTensor F V p + nonzeroPhaseSum τ φ Z p) := by |
| apply (hVr.mono_order (Nat.le_succ m)).mono_const |
| have hd2 : δ ^ 2 ≤ δ := by nlinarith |
| calc |
| R * (τ / s) ^ (q + 1) * δ ^ 2 ≤ R * (τ / s) ^ (q + 1) * δ := |
| mul_le_mul_of_nonneg_left hd2 (mul_nonneg hR hη) |
| _ = _ := by ring |
| refine ⟨hX.add hV, ?_, ?_, ?_⟩ |
| · intro p hp |
| by_contra hn |
| have hx : p ∉ tsupport X := fun hh => hn (hXsp hh) |
| have hy : p ∉ tsupport V := fun hh => hn (hVsp hh) |
| have he : (fun p => X p + V p) =ᶠ[nhds p] fun _ => 0 := by |
| filter_upwards [notMem_tsupport_iff_eventuallyEq.mp hx, |
| notMem_tsupport_iff_eventuallyEq.mp hy] with x hx hy |
| simp [hx, hy] |
| exact (notMem_tsupport_iff_eventuallyEq.mpr he) hp |
| · have hb := (hXb.mono_order (Nat.le_succ m)).add isOpen_univ.uniqueDiffOn hτ.le hXS hVS |
| (hVb.mono_order (Nat.le_succ m)) |
| apply hb.mono_const |
| change A * (δ * τ) + B * δ ^ 2 ≤ (A + B) * (δ * τ) |
| nlinarith [mul_le_mul_of_nonneg_left hδτ (mul_nonneg hB hδ)] |
| · have hb := ((hXR.add isOpen_univ.uniqueDiffOn hτ.le hXS0 hVS0 hVR).add |
| isOpen_univ.uniqueDiffOn hτ.le (hXS0.add hVS0) hCS0 hc) |
| have he : L * δ * (τ / s) ^ (q + 1) + R * δ * (τ / s) ^ (q + 1) + |
| (4 : ℝ) * 2 ^ m * (2 * A * B + B ^ 2) * (δ ^ 3 / τ) = |
| (L + R) * δ * (τ / s) ^ (q + 1) + |
| 4 * 2 ^ m * (2 * A * B + B ^ 2) * (δ ^ 3 / τ) := by ring |
| norm_num only [Nat.cast_ofNat] at hb |
| rw [he] at hb |
| apply hb.congr |
| intro p hp |
| have hh := finite_phase_metric_identity (hF.differentiable (by simp) p) |
| (hV.differentiable (by simp) p) (fun i => (hφ i).differentiable (by simp) p) |
| (fun i => (hZ i).1.differentiable (by simp) p) τ (zeroPhaseSum τ φ Z) |
| simpa only [Z, X, V, sub_self, add_zero] using hh |
|
|
|
|
| end ClosedSurfaceR4.RealModes |
|
|
| end |
|
|
| end OAI |
|
|
