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caec230 | 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 | import OAI.MathematicalPhysics.RapidForcing.CurlDerivatives
import OAI.MathematicalPhysics.RapidForcing.JointSmoothness
namespace OAI
open scoped BigOperators ENNReal Topology
open Set MeasureTheory
namespace RapidForcing
lemma curl_affine_cross (a c x : Space) :
curl (fun y => (1 / 2 : ℝ) • cross a (y - c)) x = a := by
let v : Space → Space := fun y => (1 / 2 : ℝ) • cross a (y - c)
have hv : DifferentiableAt ℝ v x := by
have h : ContDiff ℝ 1 v := by
apply (contDiff_piLp 2).mpr
intro i
fin_cases i <;> dsimp [v, cross, vec] <;> fun_prop
exact h.differentiable (by norm_num) x
have hf (i j : Fin 3) (b : Space) :
fderiv ℝ (fun y : Space => (1 / 2 : ℝ) *
(a i * (y j - c j) - a j * (y i - c i))) x b =
(1 / 2 : ℝ) * (a i * b j - a j * b i) := by
have h := (((((EuclideanSpace.proj j : Space →L[ℝ] ℝ).hasFDerivAt (x := x)).sub_const (c j)).const_mul (a i)).sub
((((EuclideanSpace.proj i : Space →L[ℝ] ℝ).hasFDerivAt (x := x)).sub_const (c i)).const_mul (a j))).const_mul (1 / 2 : ℝ)
have hf := congrArg (fun L : Space →L[ℝ] ℝ => L b) h.fderiv
simpa [PiLp.proj_apply, Pi.sub_apply] using hf
change curl v x = a
ext k
fin_cases k
· change (fderiv ℝ v x (basis 1)) 2 - (fderiv ℝ v x (basis 2)) 1 = a 0
rw [← fderiv_coordinate hv 2, ← fderiv_coordinate hv 1]
change fderiv ℝ (fun y : Space => (1 / 2 : ℝ) *
(a 0 * (y 1 - c 1) - a 1 * (y 0 - c 0))) x (basis 1) -
fderiv ℝ (fun y : Space => (1 / 2 : ℝ) *
(a 2 * (y 0 - c 0) - a 0 * (y 2 - c 2))) x (basis 2) = _
rw [hf, hf]
simp [basis]
ring
· change (fderiv ℝ v x (basis 2)) 0 - (fderiv ℝ v x (basis 0)) 2 = a 1
rw [← fderiv_coordinate hv 0, ← fderiv_coordinate hv 2]
change fderiv ℝ (fun y : Space => (1 / 2 : ℝ) *
(a 1 * (y 2 - c 2) - a 2 * (y 1 - c 1))) x (basis 2) -
fderiv ℝ (fun y : Space => (1 / 2 : ℝ) *
(a 0 * (y 1 - c 1) - a 1 * (y 0 - c 0))) x (basis 0) = _
rw [hf, hf]
simp [basis]
ring
· change (fderiv ℝ v x (basis 0)) 1 - (fderiv ℝ v x (basis 1)) 0 = a 2
rw [← fderiv_coordinate hv 1, ← fderiv_coordinate hv 0]
change fderiv ℝ (fun y : Space => (1 / 2 : ℝ) *
(a 2 * (y 0 - c 0) - a 0 * (y 2 - c 2))) x (basis 0) -
fderiv ℝ (fun y : Space => (1 / 2 : ℝ) *
(a 1 * (y 2 - c 2) - a 2 * (y 1 - c 1))) x (basis 1) = _
rw [hf, hf]
simp [basis]
ring
lemma curl_congr_of_eventuallyEq {v w : Space → Space} {x : Space}
(h : v =ᶠ[𝓝 x] w) : curl v x = curl w x := by
unfold curl
rw [h.fderiv_eq]
lemma zeta_scaled_eventually_one (δ : ℝ) (c : Space) :
∀ᶠ y in 𝓝 c, ζ (δ⁻¹ • (y - c)) = 1 := by
have ho : IsOpen {y : Space | ‖δ⁻¹ • (y - c)‖ < (1 / 16 : ℝ)} :=
isOpen_lt (by fun_prop) continuous_const
have he : ∀ᶠ y in 𝓝 c, ‖δ⁻¹ • (y - c)‖ < (1 / 16 : ℝ) :=
ho.mem_nhds (by simp)
filter_upwards [he] with y hy
exact zeta_one (fun i => (PiLp.norm_apply_le (δ⁻¹ • (y - c)) i).trans hy.le)
theorem movingCurl_at_center (c : ℝ → Space) (δ σ : ℝ) :
movingCurl c δ σ (c σ) = deriv c σ := by
unfold movingCurl
calc
_ = curl (fun y => (1 / 2 : ℝ) • cross (deriv c σ) (y - c σ)) (c σ) := by
apply curl_congr_of_eventuallyEq
filter_upwards [zeta_scaled_eventually_one δ (c σ)] with y hy
rw [hy, one_smul]
_ = _ := curl_affine_cross _ _ _
lemma other_addressCurl_zero (M : Machine) (w : M.Input) (n m m' : ℕ)
(hne : m' ≠ m) (σ : ℝ) :
movingCurl (addressPath M (M.scaleData w) n m') ((M.scaleData w).δ n) σ
(addressPath M (M.scaleData w) n m σ) = 0 := by
apply image_eq_zero_of_notMem_tsupport
intro hx
have h := abs_le.mp (addressCurl_tsupport_subset_box M w n m' σ hx 1)
have h' := abs_le.mp (addressPath_bound M w n m σ 1)
have hb := (M.scaleData w).b_lt_delta_div_eight n
have hδ := (M.scaleData w).delta_pos n
simp only [addressCenter, vec_one_coord] at h h'
rcases lt_or_gt_of_ne hne with hm | hm
· have hg : (m' : ℝ) + 1 ≤ (m : ℝ) := by exact_mod_cast hm
nlinarith
· have hg : (m : ℝ) + 1 ≤ (m' : ℝ) := by exact_mod_cast hm
nlinarith
lemma addressedVelocity_on_step (M : Machine) (w : M.Input) (n : ℕ) {t : ℝ}
(ht : t ∈ Icc (1 + (n : ℝ)) (2 + (n : ℝ))) :
addressedVelocity M w t = stepVelocity M w n t := by
rw [addressedVelocity_eq_finite M w (n + 2) (by push_cast; linarith [ht.2])]
rw [loadingVelocity_right_collar M w (by have := Nat.cast_nonneg (α := ℝ) n; linarith [ht.1]),
zero_add]
apply Finset.sum_eq_single n
· intro j _ hj
rcases lt_or_gt_of_ne hj with hj | hj
· apply stepVelocity_right_collar
have : (j : ℝ) + 1 ≤ (n : ℝ) := by exact_mod_cast hj
linarith [ht.1]
· apply stepVelocity_left_collar
have : (n : ℝ) + 1 ≤ (j : ℝ) := by exact_mod_cast hj
linarith [ht.2]
· intro hn
exact (hn (Finset.mem_range.mpr (by omega))).elim
theorem addressedVelocity_at_addressPath (M : Machine) (w : M.Input) (n m : ℕ)
(hm : m ≤ (M.scaleData w).D n) {σ : ℝ} (hσ : σ ∈ Icc (0 : ℝ) 1) :
addressedVelocity M w (1 + (n : ℝ) + σ) (addressPath M (M.scaleData w) n m σ) =
deriv (addressPath M (M.scaleData w) n m) σ := by
rw [addressedVelocity_on_step M w n (by constructor <;> linarith [hσ.1, hσ.2])]
unfold stepVelocity
rw [show 1 + (n : ℝ) + σ - 1 - (n : ℝ) = σ by ring]
calc
_ = movingCurl (addressPath M (M.scaleData w) n m) ((M.scaleData w).δ n) σ
(addressPath M (M.scaleData w) n m σ) := by
apply Finset.sum_eq_single m
· intro m' _ hne
exact other_addressCurl_zero M w n m m' hne σ
· intro hnm
exact (hnm (Finset.mem_range.mpr (by omega))).elim
_ = _ := movingCurl_at_center _ _ _
theorem addressedVelocity_at_loadingPath (M : Machine) (w : M.Input) {t : ℝ} (ht : t ≤ 1) :
let c := fun s => θ s • vec (-1) ((M.scaleData w).δ 0 * (M.initialDigit w : ℝ)) 0
addressedVelocity M w t (c t) = deriv c t := by
dsimp
unfold addressedVelocity
rw [ite_eq_left ht]
exact movingCurl_at_center _ _ _
end RapidForcing
end OAI
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