import OAI.Geometry.Immersion.ClosedSurface.PhaseMean namespace OAI noncomputable section open Set Complex Bundle Manifold open scoped ContDiff Matrix Topology Manifold BigOperators namespace ClosedSurfaceR4.PhaseMean open ClosedSurfaceR4.SmallModes ClosedSurfaceR4.RealModes ClosedSurfaceR4.RootMean open ClosedSurfaceR4.WeightedEstimates ClosedSurfaceR4.FiniteMean open Set structure LocalBounds (U V : Set Base) (s r ρ R : ℝ) (reference : Base → Tensor) (F : RField 4) (ψ : Base → ℝ) (Q : Base → Tensor →L[ℝ] ℝ) (χ e : Base → Base) : Prop where openU : IsOpen U smoothF : ContDiff ℝ ∞ F domain : RealModeDomain F V smoothPsi : ContDiffOn ℝ ∞ ψ V smoothQ : ContDiffOn ℝ ∞ Q V smoothChi : ContDiffOn ℝ ∞ χ U smoothInv : ContDiffOn ℝ ∞ e V chiInto : MapsTo χ U V invInto : MapsTo e V U smoothPullback : ContDiffOn ℝ ∞ (pullbackField χ) U margin : ∀ p ∈ V, ρ + ‖Q p‖ * r ≤ coefficient Q e reference p ∧ coefficient Q e reference p ≤ R - ‖Q p‖ * r structure Budgets (U V : Set Base) (s : ℝ) (F : RField 4) (ψ : Base → ℝ) (Q : Base → Tensor →L[ℝ] ℝ) (χ e : Base → Base) where inv : ℕ → ℝ chi : ℕ → ℝ psi : ℕ → ℝ forms : ℕ → ℝ pull : ℕ → ℝ normal : ℕ → ℝ mode : ℕ → ℝ inv_pos : ∀ m, 1 ≤ inv m chi_pos : ∀ m, 1 ≤ chi m psi_pos : ∀ m, 1 ≤ psi m forms_pos : ∀ m, 1 ≤ forms m pull_pos : ∀ m, 1 ≤ pull m normal_pos : ∀ m, 1 ≤ normal m mode_pos : ∀ m, 1 ≤ mode m inv_bound : ∀ m j, 1 ≤ j → j ≤ m → ∀ p ∈ V, ‖iteratedFDerivWithin ℝ j e V p‖ ≤ inv m chi_bound : ∀ m j, 1 ≤ j → j ≤ m → ∀ p ∈ U, ‖iteratedFDerivWithin ℝ j χ U p‖ ≤ chi m psi_bound : ∀ m, WeightedBound V s m (psi m) ψ forms_bound : ∀ m, WeightedBound V s m (forms m) Q pull_bound : ∀ m, WeightedBound U s m (pull m) (pullbackField χ) normal_bound : ∀ m, WeightedBound V s m (normal m) (freeNormal F) mode_bound : ∀ m, ReconstructionCoefficientBound (fun p => complexify (F p)) V s m (mode m) variable {U V : Set Base} {s r ρ R : ℝ} {reference : Base → Tensor} {F : RField 4} {ψ : Base → ℝ} {Q : Base → Tensor →L[ℝ] ℝ} {χ e : Base → Base} def inputFactor (d : Budgets U V s F ψ Q χ e) (m : ℕ) : ℝ := 2 ^ m * d.forms m * (m.factorial : ℝ) * d.inv m ^ m lemma inputFactor_nonneg (d : Budgets U V s F ψ Q χ e) (m : ℕ) : 0 ≤ inputFactor d m := by have hp := le_trans zero_le_one (d.forms_pos m) have hj := le_trans zero_le_one (d.inv_pos m) unfold inputFactor positivity lemma LocalBounds.amplitude_smooth (h : LocalBounds U V s r ρ R reference F ψ Q χ e) (hρ : 0 < ρ) {A : Base → Tensor} (hA : ContDiffOn ℝ ∞ A U) (hball : InTrialBall U reference r A) : ContDiffOn ℝ ∞ (phaseAmplitude ψ (coefficient Q e A)) V := by have hR := coefficient_trial_range h.invInto h.margin hball exact contDiffOn_phaseAmplitude h.smoothPsi (contDiffOn_coefficient h.smoothQ h.smoothInv h.invInto hA) (fun p hp => hρ.trans_le (hR hp).1) theorem LocalBounds.amplitude_bounds (h : LocalBounds U V s r ρ R reference F ψ Q χ e) (d : Budgets U V s F ψ Q χ e) (hs : 0 < s) (hs1 : s ≤ 1) (hρ : 0 < ρ) (n : ℕ) (C : ℝ) : ∃ A0 : ℝ, 1 ≤ A0 ∧ ∀ (A B : Base → Tensor) (D : ℝ), 0 ≤ C → 0 ≤ D → ContDiffOn ℝ ∞ A U → ContDiffOn ℝ ∞ B U → InTrialBall U reference r A → InTrialBall U reference r B → WeightedBound U s n C A → WeightedBound U s n C B → WeightedBound U s n D (A - B) → WeightedBound V s n A0 (phaseAmplitude ψ (coefficient Q e A)) ∧ WeightedBound V s n A0 (phaseAmplitude ψ (coefficient Q e B)) ∧ WeightedBound V s n (A0 * inputFactor d n * D) (fun p => phaseAmplitude ψ (coefficient Q e A) p - phaseAmplitude ψ (coefficient Q e B) p) := by let C' := max 1 (inputFactor d n * C) obtain ⟨A0, hA0, hb⟩ := compact_amplitude_bounds (R := R) h.domain.isOpen.uniqueDiffOn hρ (show 1 ≤ C' from le_max_left _ _) (le_trans zero_le_one (d.psi_pos n)) n refine ⟨A0, hA0, ?_⟩ intro A B D hC hD hA hB hballA hballB hbA hbB hbD have hcoeff (X : Base → Tensor) (hX : ContDiffOn ℝ ∞ X U) (hbX : WeightedBound U s n C X) : WeightedBound V s n C' (coefficient Q e X) := (weighted_coefficient h.openU.uniqueDiffOn h.domain.isOpen.uniqueDiffOn hs hs1 (d.inv_pos n) (le_trans zero_le_one (d.forms_pos n)) hC h.smoothQ h.smoothInv h.invInto hX (d.inv_bound n) (d.forms_bound n) hbX).mono_const (le_max_right _ _) have hdiff := weighted_coefficient_difference h.openU.uniqueDiffOn h.domain.isOpen.uniqueDiffOn hs hs1 (d.inv_pos n) (le_trans zero_le_one (d.forms_pos n)) hD h.smoothQ h.smoothInv h.invInto hA hB (d.inv_bound n) (d.forms_bound n) hbD have hu := contDiffOn_coefficient h.smoothQ h.smoothInv h.invInto hA have hv := contDiffOn_coefficient h.smoothQ h.smoothInv h.invInto hB have hh := hb s (inputFactor d n * D) ψ (coefficient Q e A) (coefficient Q e B) hs (mul_nonneg (inputFactor_nonneg d n) hD) h.smoothPsi hu hv (coefficient_trial_range h.invInto h.margin hballA) (coefficient_trial_range h.invInto h.margin hballB) (d.psi_bound n) (hcoeff A hA hbA) (hcoeff B hB hbB) hdiff refine ⟨hh.1, hh.2.1, ?_⟩ simpa only [mul_assoc] using hh.2.2 end ClosedSurfaceR4.PhaseMean namespace ClosedSurfaceR4.PhaseMean open ClosedSurfaceR4.SmallModes ClosedSurfaceR4.RealModes ClosedSurfaceR4.RootMean open ClosedSurfaceR4.WeightedEstimates ClosedSurfaceR4.FiniteMean open Set variable {U V : Set Base} {s r ρ R : ℝ} {reference : Base → Tensor} {F : RField 4} {ψ : Base → ℝ} {Q : Base → Tensor →L[ℝ] ℝ} {χ e : Base → Base} def outputFactor (d : Budgets U V s F ψ Q χ e) (m : ℕ) : ℝ := 2 ^ m * d.pull m * (m.factorial : ℝ) * d.chi m ^ m lemma outputFactor_nonneg (d : Budgets U V s F ψ Q χ e) (m : ℕ) : 0 ≤ outputFactor d m := by have hp := le_trans zero_le_one (d.pull_pos m) have hj := le_trans zero_le_one (d.chi_pos m) unfold outputFactor positivity lemma LocalBounds.transport_bound (h : LocalBounds U V s r ρ R reference F ψ Q χ e) (d : Budgets U V s F ψ Q χ e) (hs : 0 < s) (hs1 : s ≤ 1) {m : ℕ} {D : ℝ} {g : Base → Tensor} (hD : 0 ≤ D) (hg : ContDiffOn ℝ ∞ g V) (hb : WeightedBound V s m D g) : WeightedBound U s m (outputFactor d m * D) (fun p => pullbackField χ p (g (χ p))) := by have hc := hb.comp_coordinates h.openU.uniqueDiffOn h.domain.isOpen.uniqueDiffOn hs hs1 (d.chi_pos m) hD h.smoothChi hg h.chiInto (d.chi_bound m) have hchi := le_trans zero_le_one (d.chi_pos m) have hh := (d.pull_bound m).clm_apply h.openU.uniqueDiffOn hs.le (le_trans zero_le_one (d.pull_pos m)) (by positivity) h.smoothPullback (hg.comp h.smoothChi h.chiInto) hc convert hh using 1 <;> try rfl dsimp [outputFactor] ring lemma LocalBounds.mean_smooth (h : LocalBounds U V s r ρ R reference F ψ Q χ e) (hρ : 0 < ρ) {A : Base → Tensor} (hA : ContDiffOn ℝ ∞ A U) (hball : InTrialBall U reference r A) (δ : ℝ) (q : ℕ) (η : ℝ) : ContDiffOn ℝ ∞ (meanTerm δ s F ψ Q χ e q η A) U := by exact h.smoothPullback.clm_apply ((contDiffOn_meanTensor h.smoothF h.domain (h.amplitude_smooth hρ hA hball) δ (η * s) q).comp h.smoothChi h.chiInto) theorem LocalBounds.mean_estimates (h : LocalBounds U V s r ρ R reference F ψ Q χ e) (d : Budgets U V s F ψ Q χ e) (hs : 0 < s) (hs1 : s ≤ 1) (hρ : 0 < ρ) {δ : ℝ} (hδ : 0 < δ) (q m : ℕ) (C : ℝ) : ∃ β κ : ℝ, 1 ≤ β ∧ 1 ≤ κ ∧ ∀ (η : ℝ), 0 < η → η ≤ 1 → ∀ (A B : Base → Tensor) (D : ℝ), 0 ≤ C → 0 ≤ D → ContDiffOn ℝ ∞ A U → ContDiffOn ℝ ∞ B U → InTrialBall U reference r A → InTrialBall U reference r B → WeightedBound U s (m + (q + 2)) C A → WeightedBound U s (m + (q + 2)) C B → WeightedBound U s (m + (q + 2)) D (A - B) → WeightedBound U s m (η * β) (meanTerm δ s F ψ Q χ e q η A) ∧ WeightedBound U s m (κ * η * D) (meanTerm δ s F ψ Q χ e q η A - meanTerm δ s F ψ Q χ e q η B) := by let n := m + q + 2 obtain ⟨a, ha, hb⟩ := h.amplitude_bounds d hs hs1 hρ n C let Z := meanErrorConstant 4 m (d.mode n) q * (Real.sqrt 2 * 2 ^ n * d.normal n) ^ 2 * a ^ 2 have hz : 0 ≤ Z := mul_nonneg (mul_nonneg (meanErrorConstant_nonneg 4 m q (le_trans zero_le_one (d.mode_pos n))) (sq_nonneg _)) (sq_nonneg _) refine ⟨max 1 (outputFactor d m * Z), max 1 (outputFactor d m * (2 * Z * inputFactor d n)), le_max_left _ _, le_max_left _ _, ?_⟩ intro η hη hη1 A B D hC hD hA hB hballA hballB hbA hbB hbD have hn : m + (q + 2) = n := by dsimp [n]; omega rw [hn] at hbA hbB hbD obtain ⟨haA, haB, haD⟩ := hb A B D hC hD hA hB hballA hballB hbA hbB hbD have hsa := h.amplitude_smooth hρ hA hballA have hsb := h.amplitude_smooth hρ hB hballB have hτs : η * s ≤ s := by nlinarith have hcancel : η * s / s = η := mul_div_cancel_right₀ _ hs.ne' have hval := weighted_meanTensor h.smoothF h.domain hsa hδ (mul_pos hη hs) hs hτs hs1 (le_trans zero_le_one (d.mode_pos n)) (le_trans zero_le_one ha) (le_trans zero_le_one (d.normal_pos n)) q m (d.mode_bound n) haA (d.normal_bound n) have hval' : WeightedBound V s m (Z * η) (meanTensor δ (η * s) F (phaseAmplitude ψ (coefficient Q e A)) q) := by convert hval using 1 rw [hcancel] dsimp [Z, n] ring have hv := h.transport_bound d hs hs1 (mul_nonneg hz hη.le) (contDiffOn_meanTensor h.smoothF h.domain hsa δ (η * s) q) hval' have hdiff := weighted_meanTensor_difference h.smoothF h.domain hsa hsb hδ (mul_pos hη hs) hs hτs hs1 (le_trans zero_le_one (d.mode_pos n)) (le_trans zero_le_one ha) (mul_nonneg (mul_nonneg (le_trans zero_le_one ha) (inputFactor_nonneg d n)) hD) (le_trans zero_le_one (d.normal_pos n)) q m (d.mode_bound n) haA haB haD (d.normal_bound n) have hdiff' : WeightedBound V s m (2 * Z * inputFactor d n * D * η) (fun p => meanTensor δ (η * s) F (phaseAmplitude ψ (coefficient Q e A)) q p - meanTensor δ (η * s) F (phaseAmplitude ψ (coefficient Q e B)) q p) := by convert hdiff using 1 rw [hcancel] dsimp [Z, n] ring have hi := inputFactor_nonneg d n have hd := h.transport_bound d hs hs1 (by positivity) ((contDiffOn_meanTensor h.smoothF h.domain hsa δ (η * s) q).sub (contDiffOn_meanTensor h.smoothF h.domain hsb δ (η * s) q)) hdiff' constructor · apply hv.mono_const calc _ = η * (outputFactor d m * Z) := by ring _ ≤ _ := mul_le_mul_of_nonneg_left (le_max_right _ _) hη.le · have hbound : outputFactor d m * (2 * Z * inputFactor d n * D * η) ≤ max 1 (outputFactor d m * (2 * Z * inputFactor d n)) * η * D := by calc _ = (outputFactor d m * (2 * Z * inputFactor d n)) * (η * D) := by ring _ ≤ _ := by simpa only [mul_assoc] using mul_le_mul_of_nonneg_right (le_max_right 1 (outputFactor d m * (2 * Z * inputFactor d n))) (mul_nonneg hη.le hD) apply (hd.mono_const hbound).congr intro p hp exact (map_sub (pullbackField χ p) _ _).symm end ClosedSurfaceR4.PhaseMean namespace ClosedSurfaceR4.PhaseMean open ClosedSurfaceR4.SmallModes ClosedSurfaceR4.RealModes ClosedSurfaceR4.RootMean open ClosedSurfaceR4.WeightedEstimates ClosedSurfaceR4.FiniteMean open Set theorem actual_local_mean_bounds {U V : Set Base} {s r ρ R : ℝ} {reference : Base → Tensor} {F : RField 4} {ψ : Base → ℝ} {Q : Base → Tensor →L[ℝ] ℝ} {χ e : Base → Base} (h : LocalBounds U V s r ρ R reference F ψ Q χ e) (d : Budgets U V s F ψ Q χ e) (hs : 0 < s) (hs1 : s ≤ 1) (hρ : 0 < ρ) {δ : ℝ} (hδ : 0 < δ) (q : ℕ) : ∃ B K : ℕ → ℝ → ℝ, MeanBounds U s reference r (q + 2) (meanTerm δ s F ψ Q χ e q) B K := by choose β κ hβ hκ hb using fun m C => h.mean_estimates d hs hs1 hρ hδ q m C refine ⟨β, κ, ?_⟩ constructor · intro m C hC exact hβ m C · intro m C hC exact hκ m C · intro η hη hη1 A hA hball exact h.mean_smooth hρ hA hball δ q η · intro η hη hη1 m C A hC hA hball hAB have hzero : WeightedBound U s (m + (q + 2)) 0 (A - A) := by apply (weightedBound_zero U s (m + (q + 2)) (F := Tensor)).congr intro p hp simp exact (hb m C η hη hη1 A A 0 (le_trans zero_le_one hC) le_rfl hA hA hball hball hAB hAB hzero).1 · intro η hη hη1 m C D A B hC hD hA hB hballA hballB hbA hbB hbD exact (hb m C η hη hη1 A B D (le_trans zero_le_one hC) hD hA hB hballA hballB hbA hbB hbD).2 end ClosedSurfaceR4.PhaseMean end end OAI