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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 | import OAI.Dynamics.StandardMap.NonlinearDirichlet
namespace OAI
open MeasureTheory Set
open scoped ENNReal BigOperators
open MeasureTheory Set Filter Metric
open scoped Topology ENNReal
namespace StandardMapEntropy
lemma weighted_green_exponent (l p : ℕ) :
(4/5:ℝ)*((l:ℝ)+1)+(min l p:ℕ)-(9/10)*((max l p:ℕ)+1)+1-
(8/5)*((p:ℝ)+1) ≤ -(1/10)-(3/5)*((p:ℝ)+1) := by
have hl : (0:ℝ) ≤ l := Nat.cast_nonneg _
have hp : (0:ℝ) ≤ p := Nat.cast_nonneg _
rcases le_total l p with h|h
· rw [min_eq_left h,max_eq_right h]
have hh : (l:ℝ) ≤ p := by exact_mod_cast h
linarith
· rw [min_eq_right h,max_eq_left h]
linarith
lemma finite_green_weight_sum (M : ℝ) (hM : 1 < M) (hq : M^(-(3/5:ℝ)) ≤ 1/2) (n : ℕ) :
∑ i : Fin n, M^(-(1/10:ℝ)-(3/5)*((i:ℝ)+1)) ≤ 2*M^(-(7/10:ℝ)) := by
have hp : 0 < M := by linarith
have he (i : Fin n) : M^(-(1/10:ℝ)-(3/5)*((i:ℝ)+1)) =
M^(-(7/10:ℝ))*(M^(-(3/5:ℝ)))^(i:ℕ) := by
rw [← Real.rpow_mul_natCast hp.le,← Real.rpow_add hp]
congr 1
ring
simp_rw [he]
rw [← Finset.mul_sum,Fin.sum_univ_eq_sum_range]
have hs : ∑ i ∈ Finset.range n, (M^(-(3/5:ℝ)))^i ≤ 2 := by
calc
_ ≤ ∑ i ∈ Finset.range n, (1/2:ℝ)^i := by gcongr
_ ≤ 2 := sum_geometric_two_le n
nlinarith [Real.rpow_pos_of_pos hp (-(7/10:ℝ))]
lemma green_weighted_rows (M C R : ℝ) (n : ℕ) (t U : ℕ → ℝ)
(hM : 1 < M) (hC : 0 ≤ C) (hR : 0 ≤ R) (hq : M^(-(3/5:ℝ)) ≤ 1/2)
(ht : ∀ p, 1 ≤ p → p ≤ n → |t p| ≤ M^((p:ℝ)-1))
(hU : ∀ p, 1 ≤ p → p ≤ n → |U p| ≤ C*M^(-(9/10:ℝ)*(p:ℝ))) :
∀ i : Fin n,
∑ j : Fin n,
|M^((4/5:ℝ)*((i:ℝ)+1))*(t (min (i:ℕ) (j:ℕ)+1)*U (max (i:ℕ) (j:ℕ)+1))| *
((2*Real.pi*M)*R/M^((4/5:ℝ)*((j:ℝ)+1)))/M^((4/5:ℝ)*((j:ℝ)+1)) ≤
(4*Real.pi*C*R)*M^(-(7/10:ℝ)) := by
have hp : 0 < M := by linarith
intro i
calc
_ ≤ ∑ j : Fin n, (2*Real.pi*C*R)*M^(-(1/10:ℝ)-(3/5)*((j:ℝ)+1)) := by
apply Finset.sum_le_sum
intro j hj
have hmin : min (i:ℕ) (j:ℕ)+1 ≤ n := by have:=min_le_left (i:ℕ) (j:ℕ); omega
have hmax : max (i:ℕ) (j:ℕ)+1 ≤ n := by omega
have ht' := ht _ (by omega) hmin
have hU' := hU _ (by omega) hmax
rw [abs_mul,abs_of_pos (Real.rpow_pos_of_pos hp _),abs_mul]
calc
_ ≤ M^((4/5:ℝ)*((i:ℝ)+1))*(M^((min (i:ℕ) (j:ℕ):ℝ))*(C*M^(-(9/10:ℝ)*((max (i:ℕ) (j:ℕ):ℝ)+1)))) *
((2*Real.pi*M)*R/M^((4/5:ℝ)*((j:ℝ)+1)))/M^((4/5:ℝ)*((j:ℝ)+1)) := by
gcongr
· simpa only [Nat.cast_add,Nat.cast_one,add_sub_cancel_right,Nat.cast_min] using ht'
· simpa only [Nat.cast_add,Nat.cast_one,Nat.cast_max] using hU'
_ = (2*Real.pi*C*R)*M^((4/5:ℝ)*((i:ℝ)+1)+(min (i:ℕ) (j:ℕ):ℝ)-
(9/10)*((max (i:ℕ) (j:ℕ):ℝ)+1)+1-(8/5)*((j:ℝ)+1)) := by
rw [show (4/5:ℝ)*((i:ℝ)+1)+(min (i:ℕ) (j:ℕ):ℝ)-
(9/10)*((max (i:ℕ) (j:ℕ):ℝ)+1)+1-(8/5)*((j:ℝ)+1) =
((4/5:ℝ)*((i:ℝ)+1)+(min (i:ℕ) (j:ℕ):ℝ)+
(-(9/10)*((max (i:ℕ) (j:ℕ):ℝ)+1))+1)-
((4/5)*((j:ℝ)+1)+(4/5)*((j:ℝ)+1)) by ring]
rw [Real.rpow_sub hp]
simp only [Real.rpow_add hp,Real.rpow_one]
ring
_ ≤ _ := mul_le_mul_of_nonneg_left
(Real.rpow_le_rpow_of_exponent_le hM.le (by simpa only [Nat.cast_min,Nat.cast_max] using weighted_green_exponent i j)) (by positivity)
_ = (2*Real.pi*C*R)*∑ j : Fin n, M^(-(1/10:ℝ)-(3/5)*((j:ℝ)+1)) := by rw [Finset.mul_sum]
_ ≤ (2*Real.pi*C*R)*(2*M^(-(7/10:ℝ))) :=
mul_le_mul_of_nonneg_left (finite_green_weight_sum M hM hq n) (by positivity)
_ = _ := by ring
end StandardMapEntropy
end OAI
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