| import OAI.Probability.FirstPassage.EndpointCoarseGeometry |
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| namespace OAI |
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| namespace PlanarFPP |
| open Filter |
| open scoped Topology |
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| lemma endpoint_mesh_ratio_tendsto {ι : Type*} {l : Filter ι} (k m : ι → ℕ) |
| (hratio : Tendsto (fun i => (2 : ℝ)^(m i)/(2 : ℝ)^(k i)) l atTop) (J : ℕ) : |
| Tendsto (fun i => (gridMesh (m i) J : ℝ)/(2 : ℝ)^(k i)) l atTop := by |
| apply tendsto_atTop.2 |
| intro B |
| filter_upwards [hratio.eventually (eventually_ge_atTop (B*(2 : ℝ)^(4*J+1)))] with i hi |
| have hs : (0 : ℝ) < 2^(k i) := by positivity |
| have hc : (0 : ℝ) < 2^(4*J+1) := by positivity |
| have hh := (le_div_iff₀ hs).mp hi |
| have hm := gridMesh_scale_bound (m i) J |
| apply (le_div_iff₀ hs).mpr |
| nlinarith |
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| lemma endpoint_levels_eventually {ι : Type*} {l : Filter ι} (k m : ι → ℕ) |
| (hk : Tendsto k l atTop) |
| (hratio : Tendsto (fun i => (2 : ℝ)^(m i)/(2 : ℝ)^(k i)) l atTop) |
| (hm : ∀ᶠ i in l, m i ≤ 3*k i) (J : ℕ) : |
| ∀ᶠ i in l, J ≤ gridLastLevel (m i) ∧ gridLastLevel (m i) ≤ k i := by |
| filter_upwards [hratio.eventually (eventually_ge_atTop ((2 : ℝ)^(4*J+1))), |
| hk.eventually (eventually_ge_atTop 2),hm] with i hi hki hmi |
| have hs : (1 : ℝ) ≤ 2^(k i) := one_le_pow₀ (by norm_num) |
| have hpow : (2 : ℝ)^(4*J+1) ≤ (2 : ℝ)^(m i) := by |
| have hh := (le_div_iff₀ (by positivity : (0 : ℝ) < 2^(k i))).mp hi |
| have hc : (0 : ℝ) ≤ 2^(4*J+1) := by positivity |
| nlinarith |
| have hexp : 4*J+1 ≤ m i := by |
| by_contra hh |
| have hh' := pow_lt_pow_right₀ (by norm_num : (1 : ℝ) < 2) (show m i < 4*J+1 by omega) |
| exact (not_lt_of_ge hpow) hh' |
| unfold gridLastLevel |
| omega |
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| lemma endpoint_error_eventually {ι : Type*} {l : Filter ι} (k : ι → ℕ) |
| (hk : Tendsto k l atTop) (K a ε : ℝ) (hε : 0 < ε) (J : ℕ) |
| (hJ : 32*(K+200)*endpointErrorTail J ≤ ε/4) : |
| ∀ᶠ i in l, 2*endpointExtensionError K (k i) J+2*|a| ≤ ε*(2 : ℝ)^(k i)/2 := by |
| have hs := (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1 : ℝ) < 2)).comp hk |
| filter_upwards [hs.eventually (eventually_ge_atTop (8*|a| /ε))] with i hi |
| have hh := (div_le_iff₀ hε).mp hi |
| have hj := mul_le_mul_of_nonneg_right hJ (show 0 ≤ (2 : ℝ)^(k i) by positivity) |
| simp only [Function.comp_apply] at hh |
| unfold endpointExtensionError |
| nlinarith |
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| end PlanarFPP |
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| end OAI |
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