| 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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|
