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f61e547 | 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 | import OAI.Probability.FirstPassage.EndpointCoarseGeometry
namespace OAI
namespace PlanarFPP
open Filter
open scoped Topology
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
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
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
end PlanarFPP
end OAI
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