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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 | import OAI.Geometry.IsometricImmersion.Caps.CapFlow
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.MeasureTheory.Integral.Bochner.Set
namespace OAI
noncomputable section
open Set Metric Filter Function MeasureTheory
open scoped ContDiff Topology Interval
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted
def verticalAverageDerivative (q : Coord → ℝ) (t y0 y1 : ℝ) : ℝ :=
∫ u in 0..1, coordPartial 1 q (coordinatePoint t (y0 + u * (y1 - y0)))
theorem verticalSegment_mem_modelSquare {t y0 y1 u : ℝ}
(ht : t ∈ Icc (-2 : ℝ) 2) (hy0 : y0 ∈ Icc (-3 : ℝ) 3)
(hy1 : y1 ∈ Icc (-3 : ℝ) 3) (hu : u ∈ Icc (0 : ℝ) 1) :
coordinatePoint t (y0 + u * (y1 - y0)) ∈ modelSquare := by
apply coordinatePoint_mem_modelSquare
⟨by linarith [ht.1], by linarith [ht.2]⟩
have hc := (convex_Icc (-3 : ℝ) 3) hy0 hy1
(show 0 ≤ 1 - u by linarith [hu.2]) hu.1 (show (1 - u) + u = 1 by ring)
have he : y0 + u * (y1 - y0) = (1 - u) * y0 + u * y1 := by ring
rw [he]
exact hc
theorem vertical_secant_identity {q : Coord → ℝ} {U : Set Coord}
(hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
{t y0 y1 : ℝ} (ht : t ∈ Icc (-2 : ℝ) 2)
(hy0 : y0 ∈ Icc (-3 : ℝ) 3) (hy1 : y1 ∈ Icc (-3 : ℝ) 3) :
q (coordinatePoint t y1) - q (coordinatePoint t y0) =
(y1 - y0) * verticalAverageDerivative q t y0 y1 := by
let z (u : ℝ) := y0 + u * (y1 - y0)
have hm (u : ℝ) (hu : u ∈ Icc (0 : ℝ) 1) : coordinatePoint t (z u) ∈ U :=
hSU (verticalSegment_mem_modelSquare ht hy0 hy1 hu)
have hc : ContinuousOn (fun u => coordPartial 1 q (coordinatePoint t (z u)))
(Icc (0 : ℝ) 1) :=
(partial_contDiffOn hq hU 1).continuousOn.comp
(by unfold coordinatePoint z; fun_prop) hm
have hd (u : ℝ) (hu : u ∈ uIcc (0 : ℝ) 1) :
HasDerivAt (fun v => q (coordinatePoint t (z v)))
(coordPartial 1 q (coordinatePoint t (z u)) * (y1 - y0)) u := by
have hu' : u ∈ Icc (0 : ℝ) 1 := by simpa using hu
have hqy := ((hq.contDiffAt (hU.mem_nhds (hm u hu'))).differentiableAt
(by simp)).hasFDerivAt.comp_hasDerivAt (z u) (coordinatePoint_hasDerivAt_y t (z u))
have hz : HasDerivAt z (y1 - y0) u := by
convert ((hasDerivAt_id u).mul_const (y1 - y0)).const_add y0 using 1 <;>
first | rfl | ring
convert hqy.comp u hz using 1 <;> rfl
have hi : IntervalIntegrable
(fun u => coordPartial 1 q (coordinatePoint t (z u))) volume 0 1 := by
apply ContinuousOn.intervalIntegrable
simpa using hc
have hint : IntervalIntegrable
(fun u => coordPartial 1 q (coordinatePoint t (z u)) * (y1 - y0)) volume 0 1 :=
hi.mul_const (y1 - y0)
have hFTC := intervalIntegral.integral_eq_sub_of_hasDerivAt hd hint
rw [intervalIntegral.integral_mul_const] at hFTC
have h0 : z 0 = y0 := by dsimp [z]; ring
have h1 : z 1 = y1 := by dsimp [z]; ring
rw [h0, h1] at hFTC
unfold verticalAverageDerivative
simpa only [z, mul_comm] using hFTC.symm
theorem verticalAverageDerivative_diagonal (q : Coord → ℝ) (t y : ℝ) :
verticalAverageDerivative q t y y = coordPartial 1 q (coordinatePoint t y) := by
simp [verticalAverageDerivative]
theorem flow_difference_hasDerivWithinAt {q : Coord → ℝ} {U : Set Coord}
(hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
{y0 y1 : ℝ → ℝ} {t : ℝ} (ht : t ∈ Icc (-2 : ℝ) 2)
(hy0 : y0 t ∈ Icc (-3 : ℝ) 3) (hy1 : y1 t ∈ Icc (-3 : ℝ) 3)
(hdy0 : HasDerivWithinAt y0 (-q (coordinatePoint t (y0 t))) (Icc (-2 : ℝ) 2) t)
(hdy1 : HasDerivWithinAt y1 (-q (coordinatePoint t (y1 t))) (Icc (-2 : ℝ) 2) t) :
HasDerivWithinAt (fun v => y1 v - y0 v)
(-verticalAverageDerivative q t (y0 t) (y1 t) * (y1 t - y0 t))
(Icc (-2 : ℝ) 2) t := by
have he := vertical_secant_identity hq hU hSU ht hy0 hy1
convert hdy1.sub hdy0 using 1
first | rfl | nlinarith only [he]
theorem scalar_linear_evolution_formula {a z : ℝ → ℝ} {R : ℝ} (hR : 0 < R)
(ha : ContinuousOn a (Ioo (-R) R))
(hz : ∀ t ∈ Ioo (-R) R, HasDerivAt z (-a t * z t) t)
{t : ℝ} (ht : t ∈ Ioo (-R) R) :
z t = z 0 * Real.exp (-(∫ r in 0..t, a r)) := by
let I (v : ℝ) := ∫ r in 0..v, a r
have hzero : (0 : ℝ) ∈ Ioo (-R) R := ⟨by linarith, hR⟩
have hI (v : ℝ) (hv : v ∈ Ioo (-R) R) : HasDerivAt I (a v) v := by
have hseg : uIcc 0 v ⊆ Ioo (-R) R := by
intro w hw
exact ⟨(lt_min hzero.1 hv.1).trans_le hw.1,
hw.2.trans_lt (max_lt hzero.2 hv.2)⟩
exact intervalIntegral.integral_hasDerivAt_right (ha.mono hseg).intervalIntegrable
(ContinuousOn.stronglyMeasurableAtFilter isOpen_Ioo ha v hv)
(ha.continuousAt (isOpen_Ioo.mem_nhds hv))
have hprod (v : ℝ) (hv : v ∈ Ioo (-R) R) :
HasDerivAt (fun w => z w * Real.exp (I w)) 0 v := by
convert (hz v hv).mul ((hI v hv).exp) using 1
first | rfl | ring
have heq : z t * Real.exp (I t) = z 0 := by
have hconst := isOpen_Ioo.is_const_of_deriv_eq_zero (convex_Ioo (-R) R).isPreconnected
(fun v hv => (hprod v hv).differentiableAt.differentiableWithinAt)
(fun v hv => (hprod v hv).deriv) ht hzero
simpa [I] using hconst
have he : Real.exp (I t) * Real.exp (-I t) = 1 := by
rw [← Real.exp_add]
simp
calc
z t = (z t * Real.exp (I t)) * Real.exp (-I t) := by rw [mul_assoc, he, mul_one]
_ = z 0 * Real.exp (-I t) := by rw [heq]
_ = _ := rfl
end SmoothLocal.Flow
end
end OAI
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