| import OAI.Geometry.IsometricImmersion.Energy.DirectedFluxCost |
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| namespace OAI |
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| noncomputable section |
| open scoped ContDiff |
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| namespace SmoothLocal.Weighted |
| open SmoothLocal.Geometry |
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| def directedWeightJetBudget (D M lambda : ℝ) : ℝ := 16 + 3*D + 2*D*(M+lambda) |
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| theorem directedWeightJetBudget_bounds {D M lambda : ℝ} |
| (hD : 0 ≤ D) (hM : 0 ≤ M) (hlambda : 0 ≤ lambda) : |
| 0 ≤ directedWeightJetBudget D M lambda ∧ |
| D ≤ directedWeightJetBudget D M lambda ∧ |
| 2*D ≤ directedWeightJetBudget D M lambda ∧ |
| D*M ≤ directedWeightJetBudget D M lambda ∧ |
| 8+D*(M+lambda) ≤ directedWeightJetBudget D M lambda ∧ |
| D+2*D*M ≤ directedWeightJetBudget D M lambda ∧ |
| 16+2*D*(M+lambda) ≤ directedWeightJetBudget D M lambda := by |
| have hDM := mul_nonneg hD hM |
| have hDl := mul_nonneg hD hlambda |
| unfold directedWeightJetBudget |
| constructor |
| · positivity |
| constructor |
| · nlinarith |
| constructor |
| · nlinarith |
| constructor |
| · nlinarith |
| constructor |
| · nlinarith |
| constructor <;> nlinarith |
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| theorem directedWeight_value_and_partials_bound |
| {I : Coord → ℝ} {p : Coord} {b lambda D S MI M : ℝ} |
| (hId : DifferentiableAt ℝ I p) (hlambda : 0 ≤ lambda) (hM : 0 ≤ M) |
| (hd : 0 ≤ edgeDistance b p) (hdD : edgeDistance b p ≤ D) |
| (hs : |p 1| ≤ S) (hIv : |I p| ≤ MI) |
| (hIi : ∀ i : Fin 2, |coordPartial i I p| ≤ M) : |
| |directedWeight b lambda I p| ≤ D*(Real.exp (lambda*S+MI)*edgeDistance b p^7) ∧ |
| |coordPartial 0 (directedWeight b lambda I) p| ≤ |
| (D*M)*(Real.exp (lambda*S+MI)*edgeDistance b p^7) ∧ |
| |coordPartial 1 (directedWeight b lambda I) p| ≤ |
| (8+D*(M+lambda))*(Real.exp (lambda*S+MI)*edgeDistance b p^7) := by |
| have hD : 0 ≤ D := hd.trans hdD |
| have hW : 0 ≤ directedWeight b lambda I p := by unfold directedWeight; positivity |
| have hE : 0 ≤ Real.exp (lambda*S+MI)*edgeDistance b p^7 := by positivity |
| have hw : |directedWeight b lambda I p| ≤ D*(Real.exp (lambda*S+MI)*edgeDistance b p^7) := by |
| rw [abs_of_nonneg hW] |
| convert directedWeight_le_seven hlambda hd hdD hs hIv using 1 |
| ring |
| have hphase : weightPhase lambda I p ≤ lambda*S+MI := by |
| have hs' : -p 1 ≤ S := by linarith [(abs_le.mp hs).1] |
| have hm := mul_le_mul_of_nonneg_left hs' hlambda |
| unfold weightPhase |
| linarith [(abs_le.mp hIv).2] |
| refine ⟨hw, ?_, ?_⟩ |
| · rw [directedWeight_partial_t b lambda hId, abs_mul] |
| have hh := mul_le_mul hw (hIi 0) (abs_nonneg _) (mul_nonneg hD hE) |
| convert hh using 1 |
| ring |
| · rw [directedWeight_partial_s b lambda hId] |
| have hIs : |coordPartial 1 I p - lambda| ≤ M+lambda := by |
| calc |
| _ ≤ |coordPartial 1 I p| + |lambda| := abs_sub _ _ |
| _ ≤ M+lambda := by rw [abs_of_nonneg hlambda]; exact add_le_add (hIi 1) le_rfl |
| have hlead : |-8*edgeDistance b p^7*Real.exp (weightPhase lambda I p)| ≤ |
| 8*(Real.exp (lambda*S+MI)*edgeDistance b p^7) := by |
| have hh := mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr hphase) |
| (show 0 ≤ 8*edgeDistance b p^7 by positivity) |
| calc |
| _ = 8*edgeDistance b p^7*Real.exp (weightPhase lambda I p) := by |
| rw [abs_mul, abs_mul, abs_of_nonneg (pow_nonneg hd 7), |
| abs_of_pos (Real.exp_pos _)] |
| norm_num |
| _ ≤ 8*edgeDistance b p^7*Real.exp (lambda*S+MI) := hh |
| _ = _ := by ring |
| have hrest : |directedWeight b lambda I p*(coordPartial 1 I p-lambda)| ≤ |
| (D*(M+lambda))*(Real.exp (lambda*S+MI)*edgeDistance b p^7) := by |
| rw [abs_mul] |
| calc |
| _ ≤ (D*(Real.exp (lambda*S+MI)*edgeDistance b p^7))*(M+lambda) := |
| (mul_le_mul_of_nonneg_left hIs (abs_nonneg _)).trans |
| (mul_le_mul_of_nonneg_right hw (add_nonneg hM hlambda)) |
| _ = _ := by ring |
| exact (abs_add_le _ _).trans (by |
| have hh := add_le_add hlead hrest |
| convert hh using 1 |
| ring) |
|
|
| theorem directedMultiplier_firstJet_abs_le_seven |
| {I : Coord → ℝ} {p : Coord} {b lambda epsilon D S MI M : ℝ} |
| (hId : DifferentiableAt ℝ I p) (hlambda : 0 ≤ lambda) (hM : 0 ≤ M) |
| (heps : 0 ≤ epsilon) (heps1 : epsilon ≤ 1) (ht : |p 0| ≤ 2) |
| (hd : 0 ≤ edgeDistance b p) (hdD : edgeDistance b p ≤ D) |
| (hs : |p 1| ≤ S) (hIv : |I p| ≤ MI) |
| (hIi : ∀ i : Fin 2, |coordPartial i I p| ≤ M) : |
| let V := directedWeightJetBudget D M lambda * Real.exp (lambda*S+MI)*edgeDistance b p^7 |
| |directedM b lambda I p| ≤ V ∧ |directedN b lambda epsilon I p| ≤ V ∧ |
| (∀ i : Fin 2, |coordPartial i (directedM b lambda I) p| ≤ V) ∧ |
| (∀ i : Fin 2, |coordPartial i (directedN b lambda epsilon I) p| ≤ V) := by |
| let E := Real.exp (lambda*S+MI)*edgeDistance b p^7 |
| have hE : 0 ≤ E := by dsimp [E]; positivity |
| have hD : 0 ≤ D := hd.trans hdD |
| have het : |epsilon*p 0| ≤ (2 : ℝ) := by |
| rw [abs_mul, abs_of_nonneg heps] |
| exact (mul_le_mul_of_nonneg_left ht heps).trans (by linarith) |
| have heabs : |epsilon| ≤ 1 := by rwa [abs_of_nonneg heps] |
| obtain ⟨hw, hwt, hws⟩ := directedWeight_value_and_partials_bound hId hlambda hM hd hdD hs hIv hIi |
| obtain ⟨hN, hND, hN2D, hNDM, hNs, hNt, hNss⟩ := directedWeightJetBudget_bounds hD hM hlambda |
| change |directedWeight b lambda I p| ≤ D*E at hw |
| change |coordPartial 0 (directedWeight b lambda I) p| ≤ (D*M)*E at hwt |
| change |coordPartial 1 (directedWeight b lambda I) p| ≤ (8+D*(M+lambda))*E at hws |
| have hlift {a x : ℝ} (hx : |x| ≤ a*E) (ha : a ≤ directedWeightJetBudget D M lambda) : |
| |x| ≤ directedWeightJetBudget D M lambda*Real.exp (lambda*S+MI)*edgeDistance b p^7 := by |
| have hh := hx.trans (mul_le_mul_of_nonneg_right ha hE) |
| convert hh using 1 |
| dsimp only [E] |
| ring |
| dsimp only |
| refine ⟨?_, ?_, ?_, ?_⟩ |
| · unfold directedM |
| rw [abs_neg] |
| exact hlift hw hND |
| · have hh : |directedN b lambda epsilon I p| ≤ (2*D)*E := by |
| unfold directedN |
| rw [abs_mul] |
| convert mul_le_mul het hw (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 2) using 1 |
| ring |
| exact hlift hh hN2D |
| · intro i |
| rw [directedM_partial b lambda i, abs_neg] |
| fin_cases i |
| · simpa only [Fin.mk_zero] using hlift hwt hNDM |
| · simpa only [Fin.mk_one] using hlift hws hNs |
| · intro i |
| rw [directedN_partial b lambda epsilon hId i] |
| fin_cases i |
| · simp only [Fin.mk_zero, Fin.isValue, ↓reduceIte, mul_one] |
| have hfirst : |epsilon*directedWeight b lambda I p| ≤ D*E := by |
| rw [abs_mul] |
| have hh := mul_le_mul heabs hw (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 1) |
| simpa only [one_mul] using hh |
| have hsecond : |epsilon*p 0*coordPartial 0 (directedWeight b lambda I) p| ≤ (2*D*M)*E := by |
| rw [abs_mul] |
| convert mul_le_mul het hwt (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 2) using 1 |
| ring |
| have hh : |epsilon*directedWeight b lambda I p + |
| epsilon*p 0*coordPartial 0 (directedWeight b lambda I) p| ≤ (D+2*D*M)*E := by |
| exact (abs_add_le _ _).trans (by |
| convert add_le_add hfirst hsecond using 1 |
| ring) |
| exact hlift hh hNt |
| · simp only [Fin.mk_one, Fin.isValue, show (0 : Fin 2) ≠ 1 by decide, ↓reduceIte, |
| mul_zero, zero_mul, zero_add] |
| have hh : |epsilon*p 0*coordPartial 1 (directedWeight b lambda I) p| ≤ |
| (16+2*D*(M+lambda))*E := by |
| rw [abs_mul] |
| convert mul_le_mul het hws (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 2) using 1 |
| ring |
| exact hlift hh hNss |
|
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| end SmoothLocal.Weighted |
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| end |
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| end OAI |
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