openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Selection /SelectionAuxiliaryConditional.lean
| import OAI.Combinatorics.BalancedRyser.Selection.SelectionAuxiliaryMeanBounds | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped BigOperators | |
| namespace Balanced.Selection.Process | |
| attribute [local instance] Classical.propDecidable | |
| variable {I V : Type*} [Fintype I] [Fintype V] | |
| theorem ae_domainTrack_conditional_drift_of_mass {p : Parameters} (hp : p.Admissible) | |
| {q t : ℕ} {G : Input I V} (hG : GraphBounds G q p.c p.C) | |
| (hq : 2 ≤ q) (hPq : 2 * p.Pstar ≤ q) | |
| (ht : t + 1 ≤ p.lastMainTime q (Fintype.card I)) | |
| (i : I) {ε : ℝ} (hε : 0 ≤ ε) | |
| (hmass : ∀ S : State I V, Precheck p q t G S → i ∈ S.remaining → | |
| domainOverflowMass p q G S i ≤ ε) : | |
| ∀ᵐ ω ∂snapshotPathLaw p q G, | |
| |(snapshotPathLaw p q G)[fun ω => | |
| domainTrack p q G i (t + 1) ω - domainTrack p q G i t ω | | |
| snapshotFiltration t] ω| ≤ (q : ℝ) * ε / p.smin := by | |
| have hbound : 0 ≤ (q : ℝ) * ε / p.smin := by | |
| exact div_nonneg (mul_nonneg (Nat.cast_nonneg q) hε) (Real.exp_pos _).le | |
| filter_upwards [condExp_domainTrack_increment p q t G i] with ω hω | |
| rw [hω] | |
| cases hstate : (ω t).1 with | |
| | none => simpa [domainTrackIncrement] using hbound | |
| | some S => | |
| by_cases hpre : Precheck p q t G S | |
| · rw [sum_drawStepWeight_of_precheck p q t G S hpre] | |
| by_cases hi : i ∈ S.remaining | |
| · exact (domainTrack_noise_drift_bound hp hG hq hPq ht S hpre i hi).trans | |
| (div_le_div_of_nonneg_right | |
| (mul_le_mul_of_nonneg_left (hmass S hpre hi) (Nat.cast_nonneg q)) | |
| (Real.exp_pos _).le) | |
| · simpa [domainTrackIncrement, hi] using hbound | |
| · simpa [Fintype.sum_option, drawStepWeight, hpre, domainTrackIncrement] | |
| using hbound | |
| theorem ae_degreeTrack_conditional_drift_of_mass {p : Parameters} (hp : p.Admissible) | |
| {q t : ℕ} {G : Input I V} (hG : GraphBounds G q p.c p.C) | |
| (hq : 2 ≤ q) (hPq : 2 * p.Pstar ≤ q) | |
| (ht : t + 1 ≤ p.lastMainTime q (Fintype.card I)) | |
| (v : V) {ε : ℝ} (hε : 0 ≤ ε) | |
| (hmass : ∀ S : State I V, Precheck p q t G S → | |
| S.remaining.card = Fintype.card I - t → v ∈ currentVertices G S → | |
| degreeCensorMass p q G S v ≤ ε) : | |
| ∀ᵐ ω ∂snapshotPathLaw p q G, | |
| (snapshotPathLaw p q G)[fun ω => | |
| degreeTrack p q G v (t + 1) ω - degreeTrack p q G v t ω | | |
| snapshotFiltration t] ω ≤ p.C * q * ε / (p.smin * p.a / 3) := by | |
| have hmin : 0 < p.smin * p.a / 3 := by | |
| have hs : 0 < p.smin := Real.exp_pos _ | |
| have ha := (Parameters.a_bounds hp).1 | |
| positivity | |
| have hcoeff : 0 ≤ p.C * q := mul_nonneg hp.2.1.le (Nat.cast_nonneg q) | |
| have hbound : 0 ≤ p.C * q * ε / (p.smin * p.a / 3) := | |
| div_nonneg (mul_nonneg hcoeff hε) hmin.le | |
| filter_upwards [condExp_degreeTrack_increment p q t G v, | |
| ae_snapshot_remaining_card p q G] with ω hω hcard | |
| rw [hω] | |
| cases hstate : (ω t).1 with | |
| | none => simpa [degreeTrackIncrement] using hbound | |
| | some S => | |
| by_cases hpre : Precheck p q t G S | |
| · rw [sum_drawStepWeight_of_precheck p q t G S hpre] | |
| by_cases hv : v ∈ currentVertices G S | |
| · exact (degreeTrack_noise_drift_bound hp hG hq hPq ht S hpre | |
| (hcard t S hstate) v hv).trans | |
| (div_le_div_of_nonneg_right | |
| (mul_le_mul_of_nonneg_left (hmass S hpre (hcard t S hstate) hv) hcoeff) hmin.le) | |
| · simpa [degreeTrackIncrement, hv] using hbound | |
| · simpa [Fintype.sum_option, drawStepWeight, hpre, degreeTrackIncrement] | |
| using hbound | |
| theorem ae_domainTrack_accumulated_drift_of_mass {p : Parameters} (hp : p.Admissible) | |
| {q T t : ℕ} {G : Input I V} (hG : GraphBounds G q p.c p.C) | |
| (hq : 2 ≤ q) (hPq : 2 * p.Pstar ≤ q) | |
| (hT : T ≤ p.lastMainTime q (Fintype.card I)) (ht : t ≤ T) | |
| (i : I) {ε : ℝ} (hε : 0 ≤ ε) | |
| (hmass : ∀ j < T, ∀ S : State I V, Precheck p q j G S → i ∈ S.remaining → | |
| domainOverflowMass p q G S i ≤ ε) : | |
| ∀ᵐ ω ∂snapshotPathLaw p q G, | |
| |∑ j ∈ Finset.range t, (snapshotPathLaw p q G)[fun ω => | |
| domainTrack p q G i (j + 1) ω - domainTrack p q G i j ω | | |
| snapshotFiltration j] ω| ≤ (t : ℝ) * ((q : ℝ) * ε / p.smin) := by | |
| have hstep : ∀ j ∈ Finset.range t, ∀ᵐ ω ∂snapshotPathLaw p q G, | |
| |(snapshotPathLaw p q G)[fun ω => | |
| domainTrack p q G i (j + 1) ω - domainTrack p q G i j ω | | |
| snapshotFiltration j] ω| ≤ (q : ℝ) * ε / p.smin := by | |
| intro j hj | |
| have hjt := Finset.mem_range.mp hj | |
| exact ae_domainTrack_conditional_drift_of_mass hp hG hq hPq (by omega) | |
| i hε (hmass j (by omega)) | |
| filter_upwards [ae_all_iff.2 (fun j => ae_all_iff.2 (hstep j))] with ω hω | |
| calc | |
| _ ≤ ∑ j ∈ Finset.range t, |(snapshotPathLaw p q G)[fun ω => | |
| domainTrack p q G i (j + 1) ω - domainTrack p q G i j ω | | |
| snapshotFiltration j] ω| := Finset.abs_sum_le_sum_abs _ _ | |
| _ ≤ ∑ _j ∈ Finset.range t, (q : ℝ) * ε / p.smin := | |
| Finset.sum_le_sum hω | |
| _ = _ := by simp | |
| theorem ae_degreeTrack_accumulated_drift_of_mass {p : Parameters} (hp : p.Admissible) | |
| {q T t : ℕ} {G : Input I V} (hG : GraphBounds G q p.c p.C) | |
| (hq : 2 ≤ q) (hPq : 2 * p.Pstar ≤ q) | |
| (hT : T ≤ p.lastMainTime q (Fintype.card I)) (ht : t ≤ T) | |
| (v : V) {ε : ℝ} (hε : 0 ≤ ε) | |
| (hmass : ∀ j < T, ∀ S : State I V, Precheck p q j G S → | |
| S.remaining.card = Fintype.card I - j → | |
| v ∈ currentVertices G S → degreeCensorMass p q G S v ≤ ε) : | |
| ∀ᵐ ω ∂snapshotPathLaw p q G, | |
| (∑ j ∈ Finset.range t, (snapshotPathLaw p q G)[fun ω => | |
| degreeTrack p q G v (j + 1) ω - degreeTrack p q G v j ω | | |
| snapshotFiltration j] ω) ≤ | |
| (t : ℝ) * (p.C * q * ε / (p.smin * p.a / 3)) := by | |
| have hstep : ∀ j ∈ Finset.range t, ∀ᵐ ω ∂snapshotPathLaw p q G, | |
| (snapshotPathLaw p q G)[fun ω => | |
| degreeTrack p q G v (j + 1) ω - degreeTrack p q G v j ω | | |
| snapshotFiltration j] ω ≤ p.C * q * ε / (p.smin * p.a / 3) := by | |
| intro j hj | |
| have hjt := Finset.mem_range.mp hj | |
| exact ae_degreeTrack_conditional_drift_of_mass hp hG hq hPq (by omega) | |
| v hε (hmass j (by omega)) | |
| filter_upwards [ae_all_iff.2 (fun j => ae_all_iff.2 (hstep j))] with ω hω | |
| calc | |
| _ ≤ ∑ _j ∈ Finset.range t, p.C * q * ε / (p.smin * p.a / 3) := | |
| Finset.sum_le_sum hω | |
| _ = _ := by simp | |
| end Balanced.Selection.Process | |
| end | |
| end OAI | |