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37bcf52 | 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 | import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.BalancedBody
namespace OAI
/-! A computable time cutoff starts the machine-only periodic body. It changes
only during the final empty part of the preceding period, so the body is
exactly zero before one and exactly periodic from one onward. -/
noncomputable section
namespace ForcedComputation.BalancedBody
open ShearFlows Recorder.Planar PlanarHamiltonian Set
open scoped ContDiff
def activationStart (n : ℕ) : ℚ := 1 - 1 / (4 * ((n : ℚ) + 1))
def activationFinish (n : ℕ) : ℚ := 1 - 1 / (8 * ((n : ℚ) + 1))
theorem activation_intervals (n : ℕ) :
0 < activationStart n ∧ activationStart n < activationFinish n ∧ activationFinish n < 1 := by
have hn : (0 : ℚ) < n + 1 := by positivity
have h₄ : (0 : ℚ) < 4 * (n + 1) := by positivity
have h₈ : (0 : ℚ) < 8 * (n + 1) := by positivity
refine ⟨?_, ?_, ?_⟩
· unfold activationStart
have h : 1 / (4 * ((n : ℚ) + 1)) < 1 := (div_lt_one h₄).mpr (by nlinarith [Nat.cast_nonneg (α := ℚ) n])
linarith
· unfold activationStart activationFinish
have h : 1 / (8 * ((n : ℚ) + 1)) < 1 / (4 * ((n : ℚ) + 1)) :=
one_div_lt_one_div_of_lt h₄ (by nlinarith)
linarith
· unfold activationFinish
have h : (0 : ℚ) < 1 / (8 * (n + 1)) := by positivity
linarith
theorem finish_le_activationStart {n : ℕ} (i : Fin n) :
PlanarTiming.finish i ≤ activationStart n := by
have hn : (0 : ℚ) < (n : ℚ) + 1 := by positivity
have hi : (i.val : ℚ) + 1 ≤ n := by exact_mod_cast i.isLt
unfold PlanarTiming.finish activationStart
apply (le_sub_iff_add_le).mpr
apply (mul_le_mul_iff_left₀ (by positivity : (0 : ℚ) < 4 * (n + 1))).mp
field_simp
nlinarith
def bodyLength (M : Alternating.Machine) (hM : M.WellFormed) : ℕ :=
(actions (freshMachine M) (freshInput_valid (BalancedPlanar.referenceValid hM)).1).length
def activation (M : Alternating.Machine) (hM : M.WellFormed) : ℝ → ℝ :=
smoothRamp (activationStart (bodyLength M hM)) (activationFinish (bodyLength M hM))
def activated (M : Alternating.Machine) (hM : M.WellFormed) : Velocity :=
fun y => activation M hM y.1 • velocity M hM y
theorem raw_late_zero (M : Alternating.Machine) (hM : M.WellFormed) {t : ℝ}
(ht : t ∈ Icc (activationStart (bodyLength M hM) : ℝ) 1) (x : Space) :
raw M hM (t, x) = 0 := by
have hn := activation_intervals (bodyLength M hM)
have ht0 : 0 ≤ t := (Rat.cast_nonneg.mpr hn.1.le).trans ht.1
have hp : BalancedPlanar.bodyVelocity M hM t (horizontal x) = 0 := by
rw [show BalancedPlanar.bodyVelocity M hM =
PlanarHamiltonian.periodicVelocity
(normalizedPulse (BalancedPlanar.referenceInput M) (BalancedPlanar.referenceValid hM)) from rfl]
rw [periodicVelocity_on_unit
(normalizedPulse_valid (BalancedPlanar.referenceInput M) (BalancedPlanar.referenceValid hM))
(normalizedPulse_inUnit (BalancedPlanar.referenceInput M) (BalancedPlanar.referenceValid hM)) ⟨ht0, ht.2⟩,
processorVelocity_eq_sum]
apply Finset.sum_eq_zero
intro i _
have hfi : ((normalizedPulse (BalancedPlanar.referenceInput M)
(BalancedPlanar.referenceValid hM) i).finish : ℝ) ≤ t := by
exact (Rat.cast_le.mpr (finish_le_activationStart i)).trans ht.1
have hab := (normalizedPulse_valid (BalancedPlanar.referenceInput M)
(BalancedPlanar.referenceValid hM) i).2
rw [Pulse.velocity, smoothPulse_after (by exact_mod_cast hab) hfi, zero_smul]
simp only [raw, CompactLift.velocity, CompactLift.spatial, VelocityDetector.triangularLift,
VelocityDetector.horizontalLinear_eq, hp, map_zero, zero_smul, add_zero, smul_zero]
theorem velocity_late_zero (M : Alternating.Machine) (hM : M.WellFormed) {t : ℝ}
(ht : t ∈ Icc (activationStart (bodyLength M hM) : ℝ) 1) (x : Space) :
velocity M hM (t, x) = 0 := by
change BalancedStretch.stretch (raw M hM (t, BalancedStretch.stretch.symm x)) = 0
rw [raw_late_zero M hM ht, map_zero]
theorem activated_before (M : Alternating.Machine) (hM : M.WellFormed) {t : ℝ}
(ht : t ≤ 1) (x : Space) : activated M hM (t, x) = 0 := by
by_cases ha : t ≤ (activationStart (bodyLength M hM) : ℝ)
· rw [activated, activation, smoothRamp_before
(by exact_mod_cast (activation_intervals (bodyLength M hM)).2.1) ha, zero_smul]
· rw [activated, velocity_late_zero M hM ⟨(lt_of_not_ge ha).le, ht⟩, smul_zero]
theorem activated_tail (M : Alternating.Machine) (hM : M.WellFormed) {t : ℝ}
(ht : 1 ≤ t) (x : Space) : activated M hM (t, x) = velocity M hM (t, x) := by
rw [activated, activation, smoothRamp_after
(by exact_mod_cast (activation_intervals (bodyLength M hM)).2.1), one_smul]
have hfinish : (activationFinish (bodyLength M hM) : ℝ) ≤ 1 := by
exact_mod_cast (activation_intervals (bodyLength M hM)).2.2.le
exact hfinish.trans ht
theorem activated_smooth (M : Alternating.Machine) (hM : M.WellFormed) :
ContDiff ℝ ∞ (activated M hM) :=
((smoothRamp_smooth _ _).comp contDiff_fst).smul (velocity_smooth M hM)
theorem activated_support (M : Alternating.Machine) (hM : M.WellFormed) (t : ℝ) :
tsupport (fun x => activated M hM (t, x)) ⊆ support :=
(tsupport_smul_subset_right _ _).trans (velocity_support M hM t)
theorem activated_bounded (M : Alternating.Machine) (hM : M.WellFormed) :
BoundedMixedDerivatives (activated M hM) :=
boundedMixed_of_compact_tail (activated_smooth M hM) (velocity_smooth M hM)
support_compact (activated_support M hM) (velocity_support M hM) (velocity_periodic M hM)
(fun _ ht x => activated_before M hM (by linarith) x)
(fun _ ht x => activated_tail M hM ht.le x)
end ForcedComputation.BalancedBody
end
end OAI
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