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92416a2 | 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 | import OAI.Dynamics.EntropyCounterexample.Clock
import OAI.Dynamics.EntropyCounterexample.SphereCutoffs
import OAI.Dynamics.EntropyCounterexample.SphereDoubling
import OAI.Dynamics.EntropyCounterexample.PredictionDerivative
namespace OAI
noncomputable section
namespace Problem340.AmbientPrediction
abbrev Ambient := ℝ × SphereAmbient
def predictor (x : Ambient) : Ambient :=
(x.1 + Clock.increment x.1 (SphereCutoffs.clockGateAmbient x.2),
SphereDoubling.ambientMap x.2)
def cylinder : Set Ambient := Set.univ ×ˢ Metric.sphere (0 : SphereAmbient) 1
@[fun_prop]
theorem contDiff_predictor {n : ℕ∞} : ContDiff ℝ n predictor := by
exact (contDiff_fst.add (Clock.contDiff_increment contDiff_fst
(SphereCutoffs.contDiff_clockGateAmbient.comp contDiff_snd))).prodMk
((SphereDoubling.contDiff_ambientMap n).comp contDiff_snd)
theorem predictor_equivariant (x : Ambient) :
predictor (x + (100, 0)) = predictor x + (100, 0) := by
apply Prod.ext
· change x.1 + 100 + Clock.increment (x.1 + 100)
(SphereCutoffs.clockGateAmbient (x.2 + 0)) =
(x.1 + Clock.increment x.1 (SphereCutoffs.clockGateAmbient x.2)) + 100
rw [add_zero]
have hi := Clock.increment_periodic (SphereCutoffs.clockGateAmbient x.2) x.1
dsimp at hi
rw [hi]
ring
· simp [predictor]
theorem predictor_preserves_cylinder : Set.MapsTo predictor cylinder cylinder := by
intro x hx
exact ⟨Set.mem_univ _, SphereDoubling.ambientMap_mem_sphere ⟨x.2, hx.2⟩⟩
def clockPrediction (m : ℕ) (x : Ambient) : ℝ := (predictor^[m] x).1
@[fun_prop]
theorem contDiff_clockPrediction (m : ℕ) {n : ℕ∞} :
ContDiff ℝ n (clockPrediction m) :=
by
have hi : ContDiff ℝ n (predictor^[m]) := by
induction m with
| zero => simpa using (contDiff_id : ContDiff ℝ n (id : Ambient → Ambient))
| succ m ih =>
rw [Function.iterate_succ]
exact ih.comp contDiff_predictor
exact hi.fst
theorem exists_derivative_bound :
∃ B : ℝ, 1 ≤ B ∧ ∀ x ∈ cylinder, ‖fderiv ℝ predictor x‖ ≤ B := by
have hperiod (z : SphereAmbient) :
Function.Periodic (fun t => fderiv ℝ predictor (t, z)) 100 := by
intro t
have heq : fderiv ℝ predictor ((t, z) + (100, 0)) =
fderiv ℝ predictor (t, z) := by
calc
_ = fderiv ℝ (fun y => predictor (y + (100, 0))) (t, z) :=
(fderiv_comp_add_right (100, 0)).symm
_ = fderiv ℝ (fun y => predictor y + (100, 0)) (t, z) :=
congrArg (fun f => fderiv ℝ f (t, z)) (funext predictor_equivariant)
_ = fderiv ℝ predictor (t, z) := fderiv_add_const (100, 0)
simpa using heq
obtain ⟨C, hC⟩ :=
(isCompact_Icc.prod (isCompact_sphere (0 : SphereAmbient) 1)).exists_bound_of_continuousOn
((contDiff_predictor (n := 1)).continuous_fderiv one_ne_zero).continuousOn
refine ⟨max 1 C, le_max_left _ _, ?_⟩
intro x hx
obtain ⟨r, hr, heq⟩ := (hperiod x.2).exists_mem_Ico₀
(by norm_num : (0 : ℝ) < 100) x.1
rw [show fderiv ℝ predictor x = fderiv ℝ predictor (r, x.2) from heq]
exact (hC (r, x.2) ⟨⟨hr.1, hr.2.le⟩, hx.2⟩).trans (le_max_right _ _)
theorem exists_clockPrediction_derivative_bound :
∃ B : ℝ, 1 ≤ B ∧ ∀ m x, x ∈ cylinder →
‖fderiv ℝ (clockPrediction m) x‖ ≤ B ^ m := by
obtain ⟨B, hB, hbound⟩ := exists_derivative_bound
refine ⟨B, hB, ?_⟩
intro m x hx
exact Prediction.norm_fderiv_projection_iterate_le_on
(fun y _ => (contDiff_predictor (n := 1)).differentiable one_ne_zero y)
predictor_preserves_cylinder (by linarith) hbound
(ContinuousLinearMap.fst ℝ ℝ SphereAmbient)
(ContinuousLinearMap.norm_fst_le ℝ ℝ SphereAmbient) m x hx
theorem exists_gain :
∃ L B : ℝ, 2 < L ∧ 1 ≤ B ∧ B < L ∧ ∀ m x, x ∈ cylinder →
‖fderiv ℝ (clockPrediction m) x‖ ≤ B ^ m := by
obtain ⟨B, hB, hbound⟩ := exists_clockPrediction_derivative_bound
exact ⟨B + 3, B, by linarith, hB, by linarith, hbound⟩
theorem norm_fderiv_clockPrediction_chart_le
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{κ : E → Ambient} {x : E} {B C : ℝ}
(hB : 0 ≤ B)
(hbound : ∀ m y, y ∈ cylinder → ‖fderiv ℝ (clockPrediction m) y‖ ≤ B ^ m)
(hκ : DifferentiableAt ℝ κ x) (hx : κ x ∈ cylinder)
(hC : ‖fderiv ℝ κ x‖ ≤ C) (m : ℕ) :
‖fderiv ℝ ((clockPrediction m) ∘ κ) x‖ ≤ B ^ m * C := by
rw [fderiv_comp x ((contDiff_clockPrediction m (n := 1)).differentiable
one_ne_zero (κ x)) hκ]
exact (ContinuousLinearMap.opNorm_comp_le _ _).trans
(mul_le_mul (hbound m (κ x) hx) hC (norm_nonneg _) (pow_nonneg hB m))
end Problem340.AmbientPrediction
end
end OAI
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