| import OAI.Dynamics.StandardMap.BridgeLinear |
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| namespace OAI |
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| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
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| open MeasureTheory Set Filter Metric |
| open scoped Topology ENNReal |
| namespace StandardMapEntropy |
| lemma geometric_Ico_bound (q : ℝ) (hq : 0 ≤ q) (hqhalf : q ≤ 1/2) (l B : ℕ) : |
| ∑ i ∈ Finset.Ico l B, q^i ≤ 2*q^l := by |
| rw [Finset.sum_Ico_eq_sum_range] |
| simp_rw [pow_add,mul_comm (q^l)] |
| rw [← Finset.sum_mul] |
| have hs : ∑ i ∈ Finset.range (B-l), q^i ≤ 2 := by |
| calc |
| _ ≤ ∑ i ∈ Finset.range (B-l), (1/2:ℝ)^i := by gcongr |
| _ ≤ 2 := sum_geometric_two_le (B-l) |
| exact mul_le_mul_of_nonneg_right hs (pow_nonneg hq _) |
| lemma stable_error_sum (M c : ℝ) (D : ℕ → ℝ) (l B : ℕ) (hM : 1 < M) |
| (_hc : 0 < c) (hq : M ^ (-2*c) ≤ 1/2) (_hl : 1 ≤ l) (_hlB : l ≤ B) |
| (hD : ∀ i, l ≤ i → i ≤ B → M^(c*(i:ℝ)) ≤ D i) : |
| ∑ i ∈ Finset.Ico l B, M/(D i*D (i+1)) ≤ 2*M^(1-c-2*c*(l:ℝ)) := by |
| have hpos : 0 < M := by linarith |
| have hqpos : 0 ≤ M ^ (-2*c) := Real.rpow_nonneg hpos.le _ |
| have hpoint (i : ℕ) (hi : i ∈ Finset.Ico l B) : |
| M/(D i*D (i+1)) ≤ M^(1-c)*(M^(-2*c))^i := by |
| obtain ⟨hli,hiB⟩:=Finset.mem_Ico.mp hi |
| have hDi := hD i hli (by omega) |
| have hDi' := hD (i+1) (by omega) (by omega) |
| have hi0 : 0 < D i := (Real.rpow_pos_of_pos hpos _).trans_le hDi |
| have hi1 : 0 < D (i+1) := (Real.rpow_pos_of_pos hpos _).trans_le hDi' |
| calc |
| _ ≤ M/(M^(c*(i:ℝ))*M^(c*((i+1:ℕ):ℝ))) := by |
| apply div_le_div_of_nonneg_left hpos.le (by positivity) |
| exact mul_le_mul hDi hDi' (Real.rpow_nonneg hpos.le _) hi0.le |
| _ = _ := by |
| conv_lhs => lhs; rw [← Real.rpow_one M] |
| rw [← Real.rpow_add hpos,← Real.rpow_sub hpos,← Real.rpow_mul_natCast,← Real.rpow_add hpos] |
| · congr 1 |
| push_cast |
| ring |
| · exact hpos.le |
| calc |
| _ ≤ ∑ i ∈ Finset.Ico l B, M^(1-c)*(M^(-2*c))^i := Finset.sum_le_sum hpoint |
| _ = M^(1-c)*∑ i ∈ Finset.Ico l B, (M^(-2*c))^i := by rw [Finset.mul_sum] |
| _ ≤ M^(1-c)*(2*(M^(-2*c))^l) := |
| mul_le_mul_of_nonneg_left (geometric_Ico_bound _ hqpos hq l B) (Real.rpow_nonneg hpos.le _) |
| _ = _ := by |
| rw [← Real.rpow_mul_natCast hpos.le] |
| have hh:=Real.rpow_add hpos (1-c) ((-2*c)*(l:ℝ)) |
| rw [show 1-c-2*c*(l:ℝ)=1-c+(-2*c)*(l:ℝ) by ring,hh] |
| ring |
| lemma stable_prefix_power_bound (M : ℝ) (D : ℕ → ℝ) (l B : ℕ) |
| (hM : 1 < M) (hq : M^(-(49/25:ℝ)) ≤ 1/2) (hl : 1 ≤ l) (hlB : l ≤ B) |
| (hD : ∀ i, l ≤ i → i ≤ B → M^((49/50:ℝ)*(i:ℝ)) ≤ D i) |
| (hu : D l ≤ M^(l:ℝ)) : |
| (D l)⁻¹+D l*∑ i ∈ Finset.Ico l B, M/(D i*D (i+1)) ≤ 3*M^(-(47/50:ℝ)*(l:ℝ)) := by |
| have hp : 0 < M := by linarith |
| have hlR : (1:ℝ) ≤ l := by exact_mod_cast hl |
| have hsum:=stable_error_sum M (49/50) D l B hM (by norm_num) (by convert hq using 1 ; norm_num) hl hlB hD |
| have hDl :=hD l le_rfl hlB |
| have hDpos : 0 < D l := (Real.rpow_pos_of_pos hp _).trans_le hDl |
| have hinv : (D l)⁻¹ ≤ M^(-(49/50:ℝ)*(l:ℝ)) := by |
| rw [neg_mul,Real.rpow_neg hp.le] |
| exact inv_le_inv₀ hDpos (Real.rpow_pos_of_pos hp _) |>.mpr hDl |
| have hi : M^(-(49/50:ℝ)*(l:ℝ)) ≤ M^(-(47/50:ℝ)*(l:ℝ)) := |
| Real.rpow_le_rpow_of_exponent_le hM.le (by nlinarith) |
| have hmul : D l*(∑ i ∈ Finset.Ico l B, M/(D i*D (i+1))) ≤ 2*M^(-(47/50:ℝ)*(l:ℝ)) := by |
| calc |
| _ ≤ M^(l:ℝ)*(2*M^(1-49/50-2*(49/50)*(l:ℝ))) := |
| mul_le_mul hu hsum (Finset.sum_nonneg fun i hi => by |
| have hi':=Finset.mem_Ico.mp hi |
| have h0 := (Real.rpow_pos_of_pos hp _).trans_le (hD i hi'.1 (by omega)) |
| have h1 := (Real.rpow_pos_of_pos hp _).trans_le (hD (i+1) (by omega) (by omega)) |
| positivity) (Real.rpow_nonneg hp.le _) |
| _ = 2*M^((l:ℝ)+(1-49/50-2*(49/50)*(l:ℝ))) := by rw [Real.rpow_add hp]; ring |
| _ ≤ _ := mul_le_mul_of_nonneg_left (Real.rpow_le_rpow_of_exponent_le hM.le (by linarith)) (by norm_num) |
| nlinarith |
| end StandardMapEntropy |
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| end OAI |
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