| import OAI.Geometry.IsometricImmersion.Energy.DirectedWeightJetCost |
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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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| theorem multiplierProduct_partial_abs_bound |
| {A m : Coord → ℝ} {p : Coord} {M V : ℝ} |
| (hA : DifferentiableAt ℝ A p) (hm : DifferentiableAt ℝ m p) |
| (hM : 0 ≤ M) (hV : 0 ≤ V) (hAv : |A p| ≤ M) (hmv : |m p| ≤ V) |
| (hAi : ∀ i : Fin 2, |coordPartial i A p| ≤ M) |
| (hmi : ∀ i : Fin 2, |coordPartial i m p| ≤ V) (i : Fin 2) : |
| |coordPartial i (fun q => A q*m q) p| ≤ 2*M*V := by |
| rw [HessianCalculus.coordPartial_mul_at hA hm i] |
| calc |
| _ ≤ |coordPartial i A p*m p| + |A p*coordPartial i m p| := abs_add_le _ _ |
| _ ≤ M*V + M*V := by |
| simp only [abs_mul] |
| exact add_le_add (mul_le_mul (hAi i) hmv (abs_nonneg _) hM) |
| ((mul_le_mul_of_nonneg_left (hmi i) (abs_nonneg _)).trans |
| (mul_le_mul_of_nonneg_right hAv hV)) |
| _ = _ := by ring |
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| theorem multiplierCoefficients_abs_bound |
| {A B C m n : Coord → ℝ} {p : Coord} {M V : ℝ} |
| (hA : DifferentiableAt ℝ A p) (hm : DifferentiableAt ℝ m p) (hn : DifferentiableAt ℝ n p) |
| (hM : 0 ≤ M) (hV : 0 ≤ V) |
| (hAv : |A p| ≤ M) (hBv : |B p| ≤ M) (hCv : |C p| ≤ M) |
| (hAi : ∀ i : Fin 2, |coordPartial i A p| ≤ M) |
| (hmv : |m p| ≤ V) (hnv : |n p| ≤ V) |
| (hmi : ∀ i : Fin 2, |coordPartial i m p| ≤ V) |
| (hni : ∀ i : Fin 2, |coordPartial i n p| ≤ V) : |
| |multiplierT B m n p| ≤ (1+M)*V ∧ |
| |multiplierS A C m n p| ≤ (3*M)*V ∧ |
| |multiplierJ A B C m n p| ≤ (1+4*M)*V := by |
| have hAm := multiplierProduct_partial_abs_bound hA hm hM hV hAv hmv hAi hmi |
| have hAn := multiplierProduct_partial_abs_bound hA hn hM hV hAv hnv hAi hni |
| have hprod {a b : ℝ} (ha : |a| ≤ M) (hb : |b| ≤ V) : |a*b| ≤ M*V := by |
| rw [abs_mul] |
| exact mul_le_mul ha hb (abs_nonneg _) hM |
| have hbn := hprod hBv hnv |
| have hbm := hprod hBv hmv |
| have hcm := hprod hCv hmv |
| have hcn := hprod hCv hnv |
| refine ⟨?_, ?_, ?_⟩ |
| · unfold multiplierT |
| have hdiff : |coordPartial 1 m p - coordPartial 0 n p| ≤ 2*V := |
| (abs_sub _ _).trans (by linarith [hmi 1, hni 0]) |
| have hhalf : |(coordPartial 1 m p-coordPartial 0 n p)/2| ≤ V := by |
| rw [abs_div] |
| norm_num |
| linarith |
| exact (abs_add_le _ _).trans (by nlinarith only [hhalf, hbn]) |
| · unfold multiplierS |
| have h1 : |-coordPartial 1 (fun q => A q*m q) p/2| ≤ M*V := by |
| rw [abs_div, abs_neg] |
| norm_num |
| linarith [hAm 1] |
| have h2 : |coordPartial 0 (fun q => A q*n q) p/2| ≤ M*V := by |
| rw [abs_div] |
| norm_num |
| linarith [hAn 0] |
| calc |
| _ ≤ |-coordPartial 1 (fun q => A q*m q) p/2 + |
| coordPartial 0 (fun q => A q*n q) p/2| + |C p*m p| := abs_add_le _ _ |
| _ ≤ (|-coordPartial 1 (fun q => A q*m q) p/2| + |
| |coordPartial 0 (fun q => A q*n q) p/2|) + |C p*m p| := |
| add_le_add (abs_add_le _ _) le_rfl |
| _ ≤ _ := by nlinarith only [h1, h2, hcm] |
| · unfold multiplierJ |
| calc |
| _ ≤ |-coordPartial 0 m p + B p*m p - coordPartial 1 (fun q => A q*n q) p| + |
| |C p*n p| := abs_add_le _ _ |
| _ ≤ (|-coordPartial 0 m p+B p*m p| + |coordPartial 1 (fun q => A q*n q) p|) + |
| |C p*n p| := add_le_add (abs_sub _ _) le_rfl |
| _ ≤ ((|-coordPartial 0 m p|+|B p*m p|) + |coordPartial 1 (fun q => A q*n q) p|) + |
| |C p*n p| := add_le_add (add_le_add (abs_add_le _ _) le_rfl) le_rfl |
| _ ≤ _ := by rw [abs_neg]; nlinarith only [hmi 0, hbm, hAn 1, hcn] |
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| theorem multiplierQuadratic_abs_bound |
| {A B C m n u : Coord → ℝ} {p : Coord} {M V : ℝ} |
| (hA : DifferentiableAt ℝ A p) (hm : DifferentiableAt ℝ m p) (hn : DifferentiableAt ℝ n p) |
| (hM : 0 ≤ M) (hV : 0 ≤ V) |
| (hAv : |A p| ≤ M) (hBv : |B p| ≤ M) (hCv : |C p| ≤ M) |
| (hAi : ∀ i : Fin 2, |coordPartial i A p| ≤ M) |
| (hmv : |m p| ≤ V) (hnv : |n p| ≤ V) |
| (hmi : ∀ i : Fin 2, |coordPartial i m p| ≤ V) |
| (hni : ∀ i : Fin 2, |coordPartial i n p| ≤ V) : |
| |multiplierQuadratic A B C m n u p| ≤ |
| (2+8*M)*V*((coordPartial 0 u p)^2+(coordPartial 1 u p)^2) := by |
| obtain ⟨hT,hS,hJ⟩ := multiplierCoefficients_abs_bound hA hm hn hM hV hAv hBv hCv hAi hmv hnv hmi hni |
| have hh := scalarQuadratic_abs_bound (x := coordPartial 0 u p) (y := coordPartial 1 u p) hT hJ hS |
| have heq : multiplierQuadratic A B C m n u p = |
| multiplierT B m n p*(coordPartial 0 u p)^2 + |
| multiplierJ A B C m n p*(coordPartial 0 u p*coordPartial 1 u p) + |
| multiplierS A C m n p*(coordPartial 1 u p)^2 := by unfold multiplierQuadratic; ring |
| rw [heq] |
| convert hh using 1 |
| ring |
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| def directedQuadraticCostConstant (D M lambda S MI : ℝ) : ℝ := |
| (2+8*M)*directedWeightJetBudget D M lambda*Real.exp (lambda*S+MI) |
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| theorem directedQuadratic_abs_le_seven |
| {A B C I u : Coord → ℝ} {p : Coord} {b lambda epsilon D S MI M : ℝ} |
| (hA : DifferentiableAt ℝ A p) (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) |
| (hAv : |A p| ≤ M) (hBv : |B p| ≤ M) (hCv : |C p| ≤ M) |
| (hAi : ∀ i : Fin 2, |coordPartial i A p| ≤ M) : |
| |multiplierQuadratic A B C (directedM b lambda I) (directedN b lambda epsilon I) u p| ≤ |
| directedQuadraticCostConstant D M lambda S MI * weightedGradient b u p := by |
| obtain ⟨hmv,hnv,hmi,hni⟩ := directedMultiplier_firstJet_abs_le_seven |
| hId hlambda hM heps heps1 ht hd hdD hs hIv hIi |
| have hD : 0 ≤ D := hd.trans hdD |
| have hN := (directedWeightJetBudget_bounds hD hM hlambda).1 |
| have hV : 0 ≤ directedWeightJetBudget D M lambda*Real.exp (lambda*S+MI)*edgeDistance b p^7 := by |
| positivity |
| have hh := multiplierQuadratic_abs_bound (u := u) hA |
| (directedM_differentiableAt b lambda hId) (directedN_differentiableAt b lambda epsilon hId) |
| hM hV hAv hBv hCv hAi hmv hnv hmi hni |
| convert hh using 1 |
| unfold directedQuadraticCostConstant weightedGradient |
| ring |
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| theorem nonnegative_cutoff_defect_bound {chi a q : ℝ} |
| (hchi : 0 ≤ chi ∧ chi ≤ 1) (ha : 0 ≤ a) : chi*max 0 (a-q) ≤ a+|q| := by |
| have hmax : max 0 (a-q) ≤ a+|q| := by |
| apply max_le |
| · exact add_nonneg ha (abs_nonneg q) |
| · linarith [neg_le_abs q] |
| have hh := mul_le_mul hchi.2 hmax (le_max_left (0 : ℝ) (a-q)) (by norm_num : (0 : ℝ) ≤ 1) |
| simpa only [one_mul] using hh |
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| def directedDefectCostConstant (D M lambda S MI c2 : ℝ) : ℝ := |
| c2*D*Real.exp (lambda*S+MI) + directedQuadraticCostConstant D M lambda S MI |
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| theorem directedCoercivityDefect_le_seven |
| {A B C chi I u : Coord → ℝ} {p : Coord} {b lambda epsilon D S MI M c2 : ℝ} |
| (hA : DifferentiableAt ℝ A p) (hId : DifferentiableAt ℝ I p) |
| (hlambda : 0 ≤ lambda) (hM : 0 ≤ M) (hc2 : 0 ≤ c2) |
| (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) |
| (hAv : |A p| ≤ M) (hBv : |B p| ≤ M) (hCv : |C p| ≤ M) |
| (hAi : ∀ i : Fin 2, |coordPartial i A p| ≤ M) (hchi : 0 ≤ chi p ∧ chi p ≤ 1) : |
| directedCoercivityDefect A B C chi I u b lambda epsilon c2 p ≤ |
| directedDefectCostConstant D M lambda S MI c2 * weightedGradient b u p := by |
| have hQ := directedQuadratic_abs_le_seven (u := u) hA hId hlambda hM heps heps1 ht hd hdD |
| hs hIv hIi hAv hBv hCv hAi |
| have hW := directedWeight_le_seven hlambda hd hdD hs hIv |
| have hW0 : 0 ≤ directedWeight b lambda I p := by unfold directedWeight; positivity |
| have hE : 0 ≤ (coordPartial 0 u p)^2+(coordPartial 1 u p)^2 := |
| add_nonneg (sq_nonneg _) (sq_nonneg _) |
| have hfirst := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hW hc2) hE |
| have hbase := nonnegative_cutoff_defect_bound |
| (q := multiplierQuadratic A B C (directedM b lambda I) (directedN b lambda epsilon I) u p) |
| hchi (mul_nonneg (mul_nonneg hc2 hW0) hE) |
| change directedCoercivityDefect A B C chi I u b lambda epsilon c2 p ≤ _ at hbase |
| exact hbase.trans (by |
| have hh := add_le_add hfirst hQ |
| convert hh using 1 |
| unfold directedDefectCostConstant weightedGradient |
| ring) |
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| end SmoothLocal.Weighted |
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| end |
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| end OAI |
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