openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Recursion /ExistsFarSeparationPower.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Measures.AmbientGridCell3 | |
| namespace OAI | |
| /-! Recursive policies and quantitative depth estimates. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4FullFar | |
| open Set Metric | |
| lemma exists_far_separation_powerND (d : ℕ) : | |
| ∃ k : ℕ, 1000*(d+1:ℝ)<(2:ℝ)^k ∧ (2:ℝ)^k<4000*(d+1:ℝ) := by | |
| obtain ⟨n,hn,hn'⟩ := exists_nat_pow_near (x := 1000*(d+1:ℝ)) (y := (2:ℝ)) | |
| (by have hd : (0:ℝ)≤d := Nat.cast_nonneg d; linarith) (by norm_num) | |
| refine ⟨n+1,hn',?_⟩ | |
| rw [pow_succ] | |
| nlinarith | |
| def farSeparatedGridND {d : ℕ} (b : ℕ) (K : ℝ) (X Y : Fin d → ℤ) : Prop := | |
| K/(8*(2:ℝ)^b)≤dist (ambientGridCenter b X) (ambientGridCenter b Y) ∧ | |
| dist (ambientGridCenter b X) (ambientGridCenter b Y)≤4*K/(2:ℝ)^b | |
| theorem farSeparatedGridND_of_lengths {d b : ℕ} {K : ℝ} | |
| (hK : 16*(d+1:ℝ)≤K) {X Y : Fin d → ℤ} {x y : EuclideanSpace ℝ (Fin d)} | |
| (hx : x∈ambientGridCell b X) (hy : y∈ambientGridCell b Y) | |
| (hlo : K/(4*(2:ℝ)^b)≤dist x y) (hhi : dist x y≤K/(2:ℝ)^b) : | |
| farSeparatedGridND b K X Y := by | |
| have htp : 0<(2:ℝ)^b := by positivity | |
| have hcx := ambientGridCell_dist_le hx (ambientGridCenter_mem b X) | |
| have hcy := ambientGridCell_dist_le hy (ambientGridCenter_mem b Y) | |
| have hxy := dist_triangle4 x (ambientGridCenter b X) (ambientGridCenter b Y) y | |
| have hcc := dist_triangle4 (ambientGridCenter b X) x y (ambientGridCenter b Y) | |
| rw [dist_comm x (ambientGridCenter b X)] at hxy | |
| rw [dist_comm y (ambientGridCenter b Y)] at hcc | |
| have hscale := div_le_div_of_nonneg_right hK htp.le | |
| have hscale' : 16*((d+1:ℝ)/(2:ℝ)^b)≤K/(2:ℝ)^b := by simpa only [mul_div_assoc] using hscale | |
| have he₁ : K/(4*(2:ℝ)^b)=(K/(2:ℝ)^b)/4 := by ring | |
| have he₂ : K/(8*(2:ℝ)^b)=(K/(2:ℝ)^b)/8 := by ring | |
| have he₃ : 4*K/(2:ℝ)^b=4*(K/(2:ℝ)^b) := by ring | |
| dsimp [farSeparatedGridND] | |
| rw [he₂,he₃] | |
| rw [he₁] at hlo | |
| constructor <;> linarith | |
| theorem farSeparatedGridND_partner_card {d : ℕ} (b : ℕ) {K : ℝ} (hK : 0≤K) | |
| (X : Fin d → ℤ) (P : Finset (Fin d → ℤ)) | |
| (hP : ∀ Y∈P, farSeparatedGridND b K X Y) : | |
| (P.card:ℝ)≤(8*K+5)^d := by | |
| have hh := ambientGridCenter_ball_card b P (ambientGridCenter b X) | |
| (r := 4*K/(2:ℝ)^b) (by positivity) (fun Y hY => by | |
| rw [dist_comm]; exact (hP Y hY).2) | |
| convert hh using 1 | |
| congr 1 | |
| field_simp; ring | |
| theorem exists_far_common_depthND {D₁ D₂ K : ℝ} (hD₁ : 0<D₁) (hD₂ : 0<D₂) | |
| (hcomp : max D₁ D₂≤2*min D₁ D₂) (hsize : max D₁ D₂≤K) : | |
| ∃ b : ℕ, K/(4*(2:ℝ)^b)≤D₁ ∧ D₁≤K/(2:ℝ)^b ∧ | |
| K/(4*(2:ℝ)^b)≤D₂ ∧ D₂≤K/(2:ℝ)^b ∧ | |
| K/(2*(2:ℝ)^b)< max D₁ D₂ := by | |
| have hD : 0< max D₁ D₂ := (lt_min hD₁ hD₂).trans_le min_le_max | |
| have hK : 0<K := hD.trans_le hsize | |
| obtain ⟨b,hb,hb'⟩ := exists_nat_pow_near | |
| ((one_le_div hD).mpr hsize) (by norm_num : (1:ℝ)<2) | |
| have hp : 0<(2:ℝ)^b := by positivity | |
| have hu : max D₁ D₂≤K/(2:ℝ)^b := by | |
| apply (le_div_iff₀ hp).mpr | |
| have hh := (le_div_iff₀ hD).mp hb | |
| nlinarith | |
| have hl : K/(2*(2:ℝ)^b)< max D₁ D₂ := by | |
| apply (div_lt_iff₀ (by positivity : 0<2*(2:ℝ)^b)).mpr | |
| rw [pow_succ] at hb' | |
| have hh := (div_lt_iff₀ hD).mp hb' | |
| nlinarith | |
| have hm : K/(4*(2:ℝ)^b)≤ min D₁ D₂ := by | |
| have he : K/(2*(2:ℝ)^b)=2*(K/(4*(2:ℝ)^b)) := by ring | |
| rw [he] at hl | |
| linarith | |
| exact ⟨b,hm.trans (min_le_left _ _),(le_max_left _ _).trans hu, | |
| hm.trans (min_le_right _ _),(le_max_right _ _).trans hu,hl⟩ | |
| lemma far_common_depth_strictND {d a b : ℕ} {D K : ℝ} | |
| (hK : 16*(d+1:ℝ)≤K) (hl : K/(2*(2:ℝ)^b)<D) | |
| (hu : D≤(d+1:ℝ)/(2:ℝ)^a) : a<b := by | |
| by_contra h | |
| have hba : b≤a := by omega | |
| have hpow : (2:ℝ)^b≤(2:ℝ)^a := pow_le_pow_right₀ (by norm_num) hba | |
| have hKp : 0<K := by have hd : (0:ℝ)≤d := Nat.cast_nonneg d; linarith | |
| have hscale : K/(2*(2:ℝ)^a)≤K/(2*(2:ℝ)^b) := | |
| div_le_div_of_nonneg_left hKp.le (by positivity) (by gcongr) | |
| have hbad : K/(2*(2:ℝ)^a)<(d+1:ℝ)/(2:ℝ)^a := hscale.trans_lt (hl.trans_le hu) | |
| have he : K/(2*(2:ℝ)^a)=(K/2)/(2:ℝ)^a := by ring | |
| rw [he] at hbad | |
| have hh := (div_lt_div_iff_of_pos_right (by positivity : 0<(2:ℝ)^a)).mp hbad | |
| linarith | |
| end Attempt4FullFar | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |