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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 | import OAI.Dynamics.EntropyCounterexample.GainProfile
import OAI.Dynamics.EntropyCounterexample.ClockPeriodization
namespace OAI
noncomputable section
namespace Problem340
def periodicGain (L : ℝ) (t : ℝ) : ℝ :=
ClockPeriodization.periodize 100 (fun x : ℝ × ℝ => gainProfile L x.1) (t, 0)
lemma gainProfile_nonzero_mem {L t : ℝ} (h : gainProfile L t ≠ 0) :
t ∈ Set.Ioo (6 : ℝ) 96 := by
by_contra ht
exact h (gainProfile_zero_outside ht)
lemma periodicGain_contDiff (L : ℝ) :
ContDiff ℝ (⊤ : ℕ∞) (periodicGain L) := by
apply (ClockPeriodization.contDiff_periodize (P := 100) (a := 6) (b := 96)
(by norm_num) ((gainProfile_contDiff L).comp contDiff_fst) ?_).comp
(contDiff_id.prodMk contDiff_const)
intro x hx
exact ⟨(gainProfile_nonzero_mem hx).1.le, (gainProfile_nonzero_mem hx).2.le⟩
lemma periodicGain_periodic (L : ℝ) : Function.Periodic (periodicGain L) 100 := by
intro t
exact ClockPeriodization.periodize_add_period 100
(fun x : ℝ × ℝ => gainProfile L x.1) t 0
lemma periodicGain_mem_Icc {L : ℝ} (hL : 0 ≤ L) (t : ℝ) :
periodicGain L t ∈ Set.Icc (0 : ℝ) L := by
apply ClockPeriodization.periodize_mem_Icc (a := 6) (b := 96)
(by norm_num) (by norm_num) hL
· intro x hx
exact gainProfile_nonzero_mem hx
· intro x
exact gainProfile_mem_Icc hL x.1
lemma periodicGain_eq {L t : ℝ} (ht : t ∈ Set.Icc (0 : ℝ) 100) :
periodicGain L t = gainProfile L t := by
apply ClockPeriodization.periodize_eq_on_fundamental (by norm_num) ?_ ht (0 : ℝ)
intro x hx
obtain ⟨hx0, hx1⟩ := gainProfile_nonzero_mem hx
constructor <;> linarith
lemma periodicGain_eq_gain (L : ℝ) {t : ℝ} (ht : t ∈ Set.Icc (7 : ℝ) 90) :
periodicGain L t = L := by
rw [periodicGain_eq (by constructor <;> linarith [ht.1, ht.2])]
exact gainProfile_eq_gain L ht
lemma periodicGain_eq_zero (L : ℝ) {t : ℝ} (ht : t ∈ Set.Icc (2 : ℝ) 5) :
periodicGain L t = 0 := by
rw [periodicGain_eq (by constructor <;> linarith [ht.1, ht.2])]
exact gainProfile_eq_zero L ht
theorem exists_periodic_gain {L : ℝ} (hL : 0 ≤ L) :
∃ a : ℝ → ℝ, ContDiff ℝ (⊤ : ℕ∞) a ∧ Function.Periodic a 100 ∧
(∀ t, a t ∈ Set.Icc (0 : ℝ) L) ∧
(∀ t ∈ Set.Icc (2 : ℝ) 5, a t = 0) ∧
(∀ t ∈ Set.Icc (7 : ℝ) 90, a t = L) :=
⟨periodicGain L, periodicGain_contDiff L, periodicGain_periodic L,
periodicGain_mem_Icc hL, fun _ ht => periodicGain_eq_zero L ht,
fun _ ht => periodicGain_eq_gain L ht⟩
section CircleLift
local instance : Fact ((0 : ℝ) < 100) := ⟨by norm_num⟩
lemma gainProfile_liftIco_coe_eq_periodicGain (L t : ℝ) :
AddCircle.liftIco (100 : ℝ) 0 (gainProfile L) (t : AddCircle (100 : ℝ)) =
periodicGain L t := by
let r := AddCircle.equivIco (100 : ℝ) 0 (t : AddCircle (100 : ℝ))
have hr : (r : ℝ) ∈ Set.Icc (0 : ℝ) 100 := by
simpa only [zero_add] using Set.Ico_subset_Icc_self r.property
have hrep : ((r : ℝ) : AddCircle (100 : ℝ)) = (t : AddCircle (100 : ℝ)) :=
AddCircle.coe_equivIco
have hp := periodicGain_periodic L
calc
AddCircle.liftIco (100 : ℝ) 0 (gainProfile L) (t : AddCircle (100 : ℝ)) =
gainProfile L r := rfl
_ = periodicGain L r := (periodicGain_eq hr).symm
_ = hp.lift ((r : ℝ) : AddCircle (100 : ℝ)) := (hp.lift_coe _).symm
_ = hp.lift (t : AddCircle (100 : ℝ)) := congrArg hp.lift hrep
_ = periodicGain L t := hp.lift_coe t
lemma gainProfile_liftIco_coe_contDiff (L : ℝ) :
ContDiff ℝ (⊤ : ℕ∞) (fun t : ℝ =>
AddCircle.liftIco (100 : ℝ) 0 (gainProfile L) (t : AddCircle (100 : ℝ))) := by
simpa only [gainProfile_liftIco_coe_eq_periodicGain] using periodicGain_contDiff L
end CircleLift
end Problem340
end
end OAI
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