| import OAI.Geometry.IsometricImmersion.Darboux.SixVariableDarboux |
| import OAI.Geometry.IsometricImmersion.Calculus.QuotientLowBounds |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped ContDiff Topology BigOperators Matrix |
| open Set |
|
|
| namespace SmoothLocal.HighEquation |
| open SmoothLocal.Geometry |
|
|
| def stateSegment (w0 w1 : DarbouxState) (sigma : ℝ) : DarbouxState := |
| (1 - sigma) • w0 + sigma • w1 |
|
|
| def heightJetSegment (z0 z : Coord → ℝ) (sigma : ℝ) (p : Coord) : DarbouxState := |
| stateSegment (solutionJet z0 p) (solutionJet z p) sigma |
|
|
| theorem statePoint_segment {w0 w1 : DarbouxState} |
| (hp : statePoint w0 = statePoint w1) (sigma : ℝ) : |
| statePoint (stateSegment w0 w1 sigma) = statePoint w0 := by |
| have h0 : w0 0 = w1 0 := congrFun hp 0 |
| have h1 : w0 1 = w1 1 := congrFun hp 1 |
| ext i |
| fin_cases i <;> simp [statePoint, stateSegment, Pi.add_apply, Pi.smul_apply, |
| smul_eq_mul, ← h0, ← h1] <;> ring |
|
|
| theorem statePoint_heightJetSegment (z0 z : Coord → ℝ) (sigma : ℝ) (p : Coord) : |
| statePoint (heightJetSegment z0 z sigma p) = p := by |
| rw [heightJetSegment, statePoint_segment (by simp only [statePoint_solutionJet]), |
| statePoint_solutionJet] |
|
|
| theorem stateDenominator_segment (g : MetricField) {w0 w1 : DarbouxState} |
| (hp : statePoint w0 = statePoint w1) (sigma : ℝ) : |
| stateDenominator g (stateSegment w0 w1 sigma) = |
| (1 - sigma) * stateDenominator g w0 + sigma * stateDenominator g w1 := by |
| unfold stateDenominator |
| rw [statePoint_segment hp, ← hp] |
| simp only [jetYY, jetConnection, stateGradient, stateSegment, Pi.add_apply, |
| Pi.smul_apply, smul_eq_mul, Fin.sum_univ_two, Matrix.cons_val_zero, |
| Matrix.cons_val_one] |
| ring |
|
|
| theorem stateMixed_segment (g : MetricField) {w0 w1 : DarbouxState} |
| (hp : statePoint w0 = statePoint w1) (sigma : ℝ) : |
| stateMixed g (stateSegment w0 w1 sigma) = |
| (1 - sigma) * stateMixed g w0 + sigma * stateMixed g w1 := by |
| unfold stateMixed |
| rw [statePoint_segment hp, ← hp] |
| simp only [jetMixed, jetConnection, stateGradient, stateSegment, Pi.add_apply, |
| Pi.smul_apply, smul_eq_mul, Fin.sum_univ_two, Matrix.cons_val_zero, |
| Matrix.cons_val_one] |
| ring |
|
|
| theorem stateDenominator_heightJetSegment (g : MetricField) (z0 z : Coord → ℝ) |
| (sigma : ℝ) (p : Coord) : |
| stateDenominator g (heightJetSegment z0 z sigma p) = |
| (1 - sigma) * covHessian g z0 p 1 1 + sigma * covHessian g z p 1 1 := by |
| rw [heightJetSegment, stateDenominator_segment (g := g) |
| (by simp only [statePoint_solutionJet]), stateDenominator_solutionJet, |
| stateDenominator_solutionJet] |
|
|
| theorem stateDenominator_difference (g : MetricField) {w0 w1 : DarbouxState} |
| (hp : statePoint w0 = statePoint w1) : |
| stateDenominator g w1 - stateDenominator g w0 = |
| (w1 5 - w0 5) - christoffel g 0 1 1 (statePoint w0) * (w1 2 - w0 2) - |
| christoffel g 1 1 1 (statePoint w0) * (w1 3 - w0 3) := by |
| simp only [stateDenominator, ← hp, jetYY, jetConnection, stateGradient, |
| Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one] |
| ring |
|
|
| theorem stateMixed_difference (g : MetricField) {w0 w1 : DarbouxState} |
| (hp : statePoint w0 = statePoint w1) : |
| stateMixed g w1 - stateMixed g w0 = |
| (w1 4 - w0 4) - christoffel g 0 0 1 (statePoint w0) * (w1 2 - w0 2) - |
| christoffel g 1 0 1 (statePoint w0) * (w1 3 - w0 3) := by |
| simp only [stateMixed, ← hp, jetMixed, jetConnection, stateGradient, |
| Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one] |
| ring |
|
|
| theorem three_term_jet_error {x y z c0 c1 B epsilon : ℝ} |
| (hB : 0 ≤ B) (heps : 0 ≤ epsilon) |
| (hx : |x| ≤ epsilon) (hy : |y| ≤ epsilon) (hz : |z| ≤ epsilon) |
| (h0 : |c0| ≤ B) (h1 : |c1| ≤ B) : |
| |x - c0 * y - c1 * z| ≤ (1 + 2 * B) * epsilon := by |
| have hy' : |c0 * y| ≤ B * epsilon := by |
| rw [abs_mul] |
| exact (mul_le_mul_of_nonneg_left hy (abs_nonneg _)).trans |
| (mul_le_mul_of_nonneg_right h0 heps) |
| have hz' : |c1 * z| ≤ B * epsilon := by |
| rw [abs_mul] |
| exact mul_le_mul h1 hz (abs_nonneg _) hB |
| calc |
| _ ≤ (|x| + |c0 * y|) + |c1 * z| := |
| (abs_sub _ _).trans (add_le_add (abs_sub _ _) le_rfl) |
| _ ≤ (epsilon + B * epsilon) + B * epsilon := add_le_add (add_le_add hx hy') hz' |
| _ = _ := by ring |
|
|
| theorem affine_segment_error {a b epsilon sigma : ℝ} |
| (hs : sigma ∈ Icc (0 : ℝ) 1) (heps : 0 ≤ epsilon) (hab : |b - a| ≤ epsilon) : |
| |((1 - sigma) * a + sigma * b) - a| ≤ epsilon := by |
| have heq : ((1 - sigma) * a + sigma * b) - a = sigma * (b - a) := by ring |
| rw [heq, abs_mul, abs_of_nonneg hs.1] |
| exact (mul_le_mul_of_nonneg_left hab hs.1).trans |
| (mul_le_of_le_one_left heps hs.2) |
|
|
| theorem stateDenominator_segment_error (g : MetricField) {w0 w1 : DarbouxState} |
| (hp : statePoint w0 = statePoint w1) {sigma B epsilon : ℝ} |
| (hs : sigma ∈ Icc (0 : ℝ) 1) (hB : 0 ≤ B) (heps : 0 ≤ epsilon) |
| (hΓ : ∀ r : Fin 2, |christoffel g r 1 1 (statePoint w0)| ≤ B) |
| (h2 : |w1 2 - w0 2| ≤ epsilon) (h3 : |w1 3 - w0 3| ≤ epsilon) |
| (h5 : |w1 5 - w0 5| ≤ epsilon) : |
| |stateDenominator g (stateSegment w0 w1 sigma) - stateDenominator g w0| ≤ |
| (1 + 2 * B) * epsilon := by |
| rw [stateDenominator_segment g hp] |
| apply affine_segment_error hs (mul_nonneg (by linarith) heps) |
| rw [stateDenominator_difference g hp] |
| exact three_term_jet_error hB heps h5 h2 h3 (hΓ 0) (hΓ 1) |
|
|
| theorem stateMixed_segment_error (g : MetricField) {w0 w1 : DarbouxState} |
| (hp : statePoint w0 = statePoint w1) {sigma B epsilon : ℝ} |
| (hs : sigma ∈ Icc (0 : ℝ) 1) (hB : 0 ≤ B) (heps : 0 ≤ epsilon) |
| (hΓ : ∀ r : Fin 2, |christoffel g r 0 1 (statePoint w0)| ≤ B) |
| (h2 : |w1 2 - w0 2| ≤ epsilon) (h3 : |w1 3 - w0 3| ≤ epsilon) |
| (h4 : |w1 4 - w0 4| ≤ epsilon) : |
| |stateMixed g (stateSegment w0 w1 sigma) - stateMixed g w0| ≤ |
| (1 + 2 * B) * epsilon := by |
| rw [stateMixed_segment g hp] |
| apply affine_segment_error hs (mul_nonneg (by linarith) heps) |
| rw [stateMixed_difference g hp] |
| exact three_term_jet_error hB heps h4 h2 h3 (hΓ 0) (hΓ 1) |
|
|
| theorem denominator_lower_of_error {a b nu : ℝ} |
| (hnu : nu ≤ |a|) (herr : |b - a| ≤ nu / 2) : nu / 2 ≤ |b| := by |
| have htriangle : |a| ≤ |b| + |b - a| := by |
| calc |
| |a| = |b - (b - a)| := by congr 1; ring |
| _ ≤ _ := abs_sub _ _ |
| linarith |
|
|
| theorem heightJetSegment_denominator_lower (g : MetricField) (z0 z : Coord → ℝ) |
| (p : Coord) {sigma B epsilon nu : ℝ} |
| (hs : sigma ∈ Icc (0 : ℝ) 1) (hB : 0 ≤ B) (heps : 0 ≤ epsilon) |
| (hΓ : ∀ r : Fin 2, |christoffel g r 1 1 p| ≤ B) |
| (hgrad : ∀ r : Fin 2, |coordPartial r z p - coordPartial r z0 p| ≤ epsilon) |
| (hsecond : |coordPartial 1 (coordPartial 1 z) p - |
| coordPartial 1 (coordPartial 1 z0) p| ≤ epsilon) |
| (hbase : nu ≤ |covHessian g z0 p 1 1|) (hsmall : (1 + 2 * B) * epsilon ≤ nu / 2) : |
| nu / 2 ≤ |stateDenominator g (heightJetSegment z0 z sigma p)| := by |
| apply denominator_lower_of_error (a := covHessian g z0 p 1 1) hbase |
| have herr := stateDenominator_segment_error g |
| (w0 := solutionJet z0 p) (w1 := solutionJet z p) |
| (by simp only [statePoint_solutionJet]) hs hB heps |
| (by simpa only [statePoint_solutionJet] using hΓ) (hgrad 0) (hgrad 1) hsecond |
| simpa only [heightJetSegment, stateDenominator_solutionJet] using herr.trans hsmall |
|
|
| theorem heightJetSegment_mem_domain (g : MetricField) (z0 z : Coord → ℝ) |
| {U : Set Coord} {p : Coord} (hp : p ∈ U) {sigma nu : ℝ} (hnu : 0 < nu) |
| (hden : nu / 2 ≤ |stateDenominator g (heightJetSegment z0 z sigma p)|) : |
| heightJetSegment z0 z sigma p ∈ darbouxStateDomain g U := by |
| refine ⟨by simpa only [statePoint_heightJetSegment] using hp, ?_⟩ |
| intro hz |
| rw [hz, abs_zero] at hden |
| linarith |
| |
| theorem quotient_difference_bound {b h b0 h0 M epsilon c : ℝ} |
| (hc : 0 < c) (hM : 0 ≤ M) (heps : 0 ≤ epsilon) |
| (hh : c ≤ |h|) (hh0 : c ≤ |h0|) (hb0 : |b0| ≤ M) |
| (hb : |b - b0| ≤ epsilon) (hd : |h - h0| ≤ epsilon) : |
| |b / h - b0 / h0| ≤ epsilon / c + M * epsilon / c ^ 2 := by |
| have hne : h ≠ 0 := by intro hz; rw [hz, abs_zero] at hh; linarith |
| have h0ne : h0 ≠ 0 := by intro hz; rw [hz, abs_zero] at hh0; linarith |
| have heq : b / h - b0 / h0 = (b - b0) / h + b0 * (h0 - h) / (h * h0) := by |
| field_simp [hne, h0ne] |
| ring |
| rw [heq] |
| refine (abs_add_le _ _).trans (add_le_add ?_ ?_) |
| · exact LowQuotient.abs_div_bound heps hb hc hh |
| · refine LowQuotient.abs_div_bound (mul_nonneg hM heps) ?_ (sq_pos_of_pos hc) ?_ |
| · rw [abs_mul, abs_sub_comm h0 h] |
| exact mul_le_mul hb0 hd (abs_nonneg _) hM |
| · rw [abs_mul, pow_two] |
| exact mul_le_mul hh hh0 hc.le (abs_nonneg _) |
| |
| theorem stateSegment_quotient_error (g : MetricField) {w0 w1 : DarbouxState} |
| (hp : statePoint w0 = statePoint w1) {sigma B epsilon nu M : ℝ} |
| (hs : sigma ∈ Icc (0 : ℝ) 1) (hB : 0 ≤ B) (heps : 0 ≤ epsilon) |
| (hnu : 0 < nu) (hM : 0 ≤ M) |
| (hΓyy : ∀ r : Fin 2, |christoffel g r 1 1 (statePoint w0)| ≤ B) |
| (hΓxy : ∀ r : Fin 2, |christoffel g r 0 1 (statePoint w0)| ≤ B) |
| (h2 : |w1 2 - w0 2| ≤ epsilon) (h3 : |w1 3 - w0 3| ≤ epsilon) |
| (h4 : |w1 4 - w0 4| ≤ epsilon) (h5 : |w1 5 - w0 5| ≤ epsilon) |
| (hbase : nu ≤ |stateDenominator g w0|) (hsmall : (1 + 2 * B) * epsilon ≤ nu / 2) |
| (hmix : |stateMixed g w0| ≤ M) : |
| |stateMixed g (stateSegment w0 w1 sigma) / stateDenominator g (stateSegment w0 w1 sigma) - |
| stateMixed g w0 / stateDenominator g w0| ≤ |
| ((1 + 2 * B) * epsilon) / (nu / 2) + |
| M * ((1 + 2 * B) * epsilon) / (nu / 2) ^ 2 := by |
| have hden := stateDenominator_segment_error g hp hs hB heps hΓyy h2 h3 h5 |
| have hmixerr := stateMixed_segment_error g hp hs hB heps hΓxy h2 h3 h4 |
| have hlower := denominator_lower_of_error hbase (hden.trans hsmall) |
| exact quotient_difference_bound (half_pos hnu) hM |
| (mul_nonneg (by linarith) heps) hlower ((by linarith : nu / 2 ≤ nu).trans hbase) |
| hmix hmixerr hden |
| |
| end SmoothLocal.HighEquation |
| |
| end |
| |
| end OAI |
| |
