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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 57 58 59 60 61 62 63 | import OAI.Probability.FirstPassage.EndpointTail
import OAI.Probability.FirstPassage.EndpointGeometry
namespace OAI
namespace PlanarFPP
open Set Finset
lemma gridMesh_le_scale (m j : ℕ) : (gridMesh m j : ℝ) ≤ (2 : ℝ)^m := by
simp only [gridMesh,Int.cast_pow,Int.cast_ofNat]
apply pow_le_pow_right₀ (by norm_num)
unfold gridMeshExponent
omega
lemma grid_endpoint_height (K : ℝ) (hK : 0 ≤ K) (k j : ℕ) (hj : j ≤ k) :
(K+200)*(j : ℝ)^2*gridWidth k j ≤ 2*(K+200)*(2 : ℝ)^k := by
have h := mul_le_mul_of_nonneg_left (endpointErrorTerm_le_two j)
(show 0 ≤ (K+200)*(2 : ℝ)^k by positivity)
rw [mul_assoc,gridWidth_endpointError k j hj]
nlinarith
noncomputable def endpointExtensionError (K : ℝ) (k J : ℕ) : ℝ :=
16*(K+200)*(2 : ℝ)^k*endpointErrorTail J
lemma endpoint_extension_cost_sum (K : ℝ) (hK : 0 ≤ K) (k m J : ℕ)
(hlast : gridLastLevel m ≤ k) (hJ : J ≤ gridLastLevel m) :
(∑ i ∈ range (gridLastLevel m-J+1),16*((K+200)*((J+i : ℕ) : ℝ)^2*gridWidth k (J+i))) ≤
endpointExtensionError K k J := by
have heq : (∑ i ∈ range (gridLastLevel m-J+1),16*((K+200)*((J+i : ℕ) : ℝ)^2*gridWidth k (J+i))) =
16*(K+200)*(2 : ℝ)^k*(∑ i ∈ range (gridLastLevel m-J+1),endpointErrorTerm (J+i)) := by
rw [mul_sum]
apply sum_congr rfl
intro i hi
have hi' := mem_range.mp hi
have hj : J+i ≤ k := by omega
have hh := gridWidth_endpointError k (J+i) hj
calc
_ = 16*(K+200)*(((J+i : ℕ) : ℝ)^2*gridWidth k (J+i)) := by ring
_ = _ := by rw [hh]; ring
rw [heq]
exact mul_le_mul_of_nonneg_left (endpointError_sum_le_tail J _ ) (by positivity)
lemma endpoint_localization (b A R C : ℝ) (_hC : 0 ≤ C) (k m : ℕ) (p z : Vertex) (d : ℤ)
(hd : (d : ℝ) ≤ (2 : ℝ)^m)
(hp : |(p.1 : ℝ)| ≤ A*(2 : ℝ)^m) (hr : |transverse b p| ≤ R*(2 : ℝ)^k)
(hx : p.1-3*d ≤ z.1 ∧ z.1 ≤ p.1+3*d)
(hz : |transverse b z-transverse b p| ≤ 2*C*(2 : ℝ)^k) :
|(z.1 : ℝ)| ≤ (A+4)*(2 : ℝ)^m ∧
|transverse b z| ≤ (R+2*C)*(2 : ℝ)^k := by
have hx' : (p.1 : ℝ)-3*(d : ℝ) ≤ (z.1 : ℝ) ∧
(z.1 : ℝ) ≤ (p.1 : ℝ)+3*(d : ℝ) := by exact_mod_cast hx
have hp' := abs_le.mp hp
refine ⟨abs_le.mpr ⟨?_,?_⟩,?_⟩
· linarith [pow_pos (by norm_num : (0 : ℝ)<2) m]
· linarith [pow_pos (by norm_num : (0 : ℝ)<2) m]
· have hh := abs_add_le (transverse b z-transverse b p) (transverse b p)
rw [sub_add_cancel] at hh
linarith
end PlanarFPP
end OAI
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