openAI-math-dataset / lean /OAI /Probability /Ballisticity /Walk /FiniteOutwardRowApproximation.lean
| import Mathlib | |
| import OAI.Probability.Ballisticity.Entropy.FiniteLogStability | |
| namespace OAI | |
| section | |
| section | |
| open MeasureTheory ProbabilityTheory Filter Function | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| namespace DirectionalTransience | |
| lemma upperPairNeighborhood_mono {d : ℕ} (e : Direction d) (a : ℝ) | |
| (x : Lattice d × Lattice d) {N R : ℕ} (hNR : N ≤ R) : | |
| upperPairNeighborhood e a x N ⊆ upperPairNeighborhood e a x R := by | |
| intro y hy | |
| refine ⟨?_,hy.2⟩ | |
| have hball (z : Lattice d) : LatticeBall z N ⊆ LatticeBall z R := by | |
| intro w hw i | |
| exact (hw i).trans (by exact_mod_cast hNR) | |
| exact hy.1.elim (fun h => Or.inl (hball _ h)) (fun h => Or.inr (hball _ h)) | |
| lemma finite_outward_upper_row_approximation {d : ℕ} {κ : ℝ≥0} (hκ : 0 < κ) | |
| (e f : Direction d) (A : ℝ) (N : ℕ) {ε B : ℝ} (hε : 0 < ε) (hB : 0 ≤ B) : | |
| ∃ R : ℕ, ∀ (a : ℝ) (x : Lattice d × Lattice d), x ∈ PairAtHeight (realPosition (step e)) a → | |
| ∃ F : Environment d → ℝ, | |
| _ _ (rowSigma (upperPairNeighborhood e a x R)) _ F ∧ | |
| (∀ ω, 0 ≤ F ω ∧ F ω ≤ B) ∧ | |
| (∀ ω, (∀ y g, κ ≤ (ω y).1 g) → | |
| |min (finiteLogExcess A (fun n => outwardKernelMass e f (n+1) x) N ω) B-F ω| ≤ ε) := by | |
| have hloc (n : ℕ) := outwardLog_fixed_height_locality hκ e f (Nat.succ_pos n) hε | |
| choose r hr using hloc | |
| obtain ⟨R,hR⟩ := ((Finset.range (N+1)).image r).exists_le | |
| refine ⟨R,?_⟩ | |
| intro a x hx | |
| let q : ℕ → Environment d → ℝ := fun n ω => outwardKernelMass e f (n+1) x (upperPairPatch e a x (r n) ω) | |
| let F : Environment d → ℝ := fun ω => min (finiteLogExcess A q N ω) B | |
| refine ⟨F,?_,?_,?_⟩ | |
| · have hm : _ _ (rowSigma (upperPairNeighborhood e a x R)) _ (finiteLogExcess A q N) := by | |
| unfold finiteLogExcess | |
| apply Measurable.ennreal_toReal | |
| apply Measurable.iSup | |
| intro n | |
| have hnr : r n ≤ R := hR (r n) (Finset.mem_image.mpr ⟨n,Finset.mem_range.mpr n.isLt,rfl⟩) | |
| have hp := (upperPairPatch_measurable e a x hx (r n)).mono | |
| (rowSigma_mono (upperPairNeighborhood_mono e a x hnr)) le_rfl | |
| have hq := (measurable_outwardKernelMass e f (n+1) x).comp hp | |
| exact (hq.log.neg.sub_const _).ennreal_ofReal | |
| exact hm.min measurable_const | |
| · intro ω | |
| exact ⟨le_min ENNReal.toReal_nonneg hB,min_le_right _ _⟩ | |
| · intro ω hω | |
| have hd := finiteLogExcess_stability A (fun n => outwardKernelMass e f (n+1) x) q N ω hε.le | |
| (fun n hn => (hr n a x hx ω hω).le) | |
| have h₁ := (abs_le.mp hd).1 | |
| have h₂ := (abs_le.mp hd).2 | |
| apply abs_le.mpr | |
| constructor | |
| · have hh : min (finiteLogExcess A q N ω) B ≤ min (finiteLogExcess A (fun n => outwardKernelMass e f (n+1) x) N ω) B+ε := by | |
| have ht := min_le_min (show finiteLogExcess A q N ω ≤ finiteLogExcess A (fun n => outwardKernelMass e f (n+1) x) N ω+ε by linarith) (show B ≤ B+ε by linarith) | |
| simpa only [min_add_add_right] using ht | |
| exact by dsimp [F]; linarith | |
| · have hh : min (finiteLogExcess A (fun n => outwardKernelMass e f (n+1) x) N ω) B ≤ min (finiteLogExcess A q N ω) B+ε := by | |
| have ht := min_le_min (show finiteLogExcess A (fun n => outwardKernelMass e f (n+1) x) N ω ≤ finiteLogExcess A q N ω+ε by linarith) (show B ≤ B+ε by linarith) | |
| simpa only [min_add_add_right] using ht | |
| exact by dsimp [F]; linarith | |
| end DirectionalTransience | |
| end | |
| end | |
| end OAI | |