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59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 | import OAI.Geometry.IsometricImmersion.Energy.DirectedWeightJetCost
namespace OAI
noncomputable section
open scoped ContDiff
namespace SmoothLocal.Weighted
open SmoothLocal.Geometry
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
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]
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
def directedQuadraticCostConstant (D M lambda S MI : ℝ) : ℝ :=
(2+8*M)*directedWeightJetBudget D M lambda*Real.exp (lambda*S+MI)
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
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
def directedDefectCostConstant (D M lambda S MI c2 : ℝ) : ℝ :=
c2*D*Real.exp (lambda*S+MI) + directedQuadraticCostConstant D M lambda S MI
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)
end SmoothLocal.Weighted
end
end OAI
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