openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Dense /DenseBoundedRejection.lean
| import OAI.Combinatorics.ContingencyTables.Dense.DenseBoundedTrial | |
| import OAI.Combinatorics.ContingencyTables.Dense.DenseIdealRejection | |
| import OAI.Combinatorics.ContingencyTables.Transport.RejectionLaw | |
| namespace OAI | |
| /-! # The bounded dense sampler's actual finite output law -/ | |
| namespace ContingencyTables.GridBoundary | |
| open scoped BigOperators Classical | |
| open ResidualMixture FirstSuccess | |
| noncomputable section | |
| variable {I J : Type*} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J] | |
| variable {q : ℕ} | |
| def trialSuccess (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ) | |
| (e : I × J ≃ Fin q) (htotal : ∑ i, R i = ∑ j, P j) | |
| (t : DenseTrial R P d q) : Option (Table R P) := | |
| if ht : TrialAccepted R P d e t then some (trialTable R P d e htotal t ht) else none | |
| omit [DecidableEq I] [DecidableEq J] in | |
| theorem trialSuccess_some_iff (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ) | |
| (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j) | |
| (e : I × J ≃ Fin q) (htotal : ∑ i, R i = ∑ j, P j) | |
| (t : DenseTrial R P d q) (X : Table R P) : | |
| trialSuccess R P d e htotal t = some X ↔ t = tableTrial R P d hd hR hP e X := by | |
| constructor | |
| · intro h | |
| unfold trialSuccess at h | |
| split_ifs at h with ht | |
| · have he := Option.some.inj h | |
| rw [← he, tableTrial_trialTable] | |
| · rintro rfl | |
| simp only [trialSuccess, dite_eq_left (tableTrial_accepted R P d hd hR hP e X), | |
| trialTable_tableTrial] | |
| theorem trialSuccess_ideal_mass (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ) | |
| (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j) | |
| (e : I × J ≃ Fin q) (htotal : ∑ i, R i = ∑ j, P j) (X : Table R P) : | |
| successMass (rationalTrialLaw R P d hd hR hP e) (trialSuccess R P d e htotal) X = | |
| rationalTableMass R P d hd hR hP e X := by | |
| unfold successMass | |
| simp_rw [trialSuccess_some_iff R P d hd hR hP e htotal] | |
| simp only [Finset.sum_ite_eq', Finset.mem_univ, ite_true] | |
| rfl | |
| theorem trialSuccess_ideal_failure (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ) | |
| (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j) | |
| (e : I × J ≃ Fin q) (htotal : ∑ i, R i = ∑ j, P j) : | |
| failureMass (rationalTrialLaw R P d hd hR hP e) (trialSuccess R P d e htotal) = | |
| 1 - ∑ X, rationalTableMass R P d hd hR hP e X := by | |
| have h := success_failure_total (rationalTrialLaw R P d hd hR hP e) (trialSuccess R P d e htotal) | |
| simp_rw [trialSuccess_ideal_mass R P d hd hR hP e htotal] at h | |
| linarith | |
| def denseBoundedLaw (R : Option I → ℕ) (P : Option J → ℕ) (d : ℕ) | |
| (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j) | |
| (hfree : Fintype.card (I × J) ≤ d) (e : I × J ≃ Fin q) | |
| (htotal : ∑ i, R i = ∑ j, P j) (h : ℕ) | |
| (a : Grid (I := Fin q) (4 * d ^ 8)) (fallback : Table R P) : RationalLaw (Table R P) := | |
| retryLaw (boundedTrialLaw R P d hd hR hP hfree e h a) | |
| (trialSuccess R P d e htotal) fallback (d ^ 4 * (h + 1)) | |
| /-- The finite sampler always returns an actual table and is within the | |
| requested error of uniform, assuming finite-box interpolation and finite | |
| Cheeger inequalities. -/ | |
| theorem denseBoundedLaw_variation (R : Option I → ℕ) (P : Option J → ℕ) | |
| (d : ℕ) (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j) | |
| (hRmax : ∀ i, R (some i) ≤ R none) (hPmax : ∀ j, P (some j) ≤ P none) | |
| (hdimR : Fintype.card (Option I) ≤ d) (hdimP : Fintype.card (Option J) ≤ d) | |
| (hfree : Fintype.card (I × J) ≤ d) (htotal : ∑ i, R i = ∑ j, P j) | |
| (e : I × J ≃ Fin q) (hPL : PublishedInputs.FiniteBoxPrekopaLeindler q) | |
| (hCheeger : PublishedInputs.FiniteCheeger (Grid (I := Fin q) (4 * d ^ 8))) | |
| (h : ℕ) (a : Grid (I := Fin q) (4 * d ^ 8)) (fallback : Table R P) : | |
| let _ : Nonempty (Table R P) := ⟨fallback⟩ | |
| (variation (denseBoundedLaw R P d hd hR hP hfree e htotal h a fallback) | |
| (uniformLaw (α := Table R P)) : ℝ) ≤ (2 : ℝ) ^ (-(h : ℝ)) := by | |
| let : Nonempty (Table R P) := ⟨fallback⟩ | |
| let p := boundedTrialLaw R P d hd hR hP hfree e h a | |
| let q₀ := rationalTrialLaw R P d hd hR hP e | |
| let f := trialSuccess R P d e htotal | |
| let n := d ^ 4 * (h + 1) | |
| have hg : successMass q₀ f = rationalTableMass R P d hd hR hP e := | |
| funext (trialSuccess_ideal_mass R P d hd hR hP e htotal) | |
| have hf : failureMass q₀ f = 1 - ∑ X, rationalTableMass R P d hd hR hP e X := | |
| trialSuccess_ideal_failure R P d hd hR hP e htotal | |
| have hi : (variation (retryLaw q₀ f fallback n) (uniformLaw (α := Table R P)) : ℝ) ≤ | |
| (63 / 64 : ℝ) ^ n := by | |
| have hi₀ := dense_ideal_rejection_variation R P d hd hR hP hRmax hPmax hdimR hdimP | |
| hfree htotal e n fallback | |
| unfold variation | |
| change (((∑ X : Table R P, | |
| |(retryLaw q₀ f fallback n).mass X - uniformMass|) / 2 : ℚ) : ℝ) ≤ _ | |
| simp_rw [retryLaw_mass] | |
| rw [hg, hf] | |
| exact hi₀ | |
| have hv := retryLaw_variation p q₀ f fallback n | |
| have hvc : (variation (retryLaw p f fallback n) (retryLaw q₀ f fallback n) : ℝ) ≤ | |
| (n : ℝ) * (variation p q₀ : ℝ) := by exact_mod_cast hv | |
| have ht := boundedTrialLaw_variation R P d hd hR hP hRmax hPmax hdimR hdimP hfree htotal | |
| e hPL hCheeger h a | |
| have hm := mul_le_mul_of_nonneg_left ht (show (0 : ℝ) ≤ n by positivity) | |
| have htri := variation_triangle (retryLaw p f fallback n) (retryLaw q₀ f fallback n) | |
| (uniformLaw (α := Table R P)) | |
| have htric : (variation (retryLaw p f fallback n) (uniformLaw (α := Table R P)) : ℝ) ≤ | |
| (variation (retryLaw p f fallback n) (retryLaw q₀ f fallback n) : ℝ) + | |
| (variation (retryLaw q₀ f fallback n) (uniformLaw (α := Table R P)) : ℝ) := by exact_mod_cast htri | |
| change (variation (retryLaw p f fallback n) (uniformLaw (α := Table R P)) : ℝ) ≤ _ | |
| apply (htric.trans (add_le_add (hvc.trans hm) hi)).trans | |
| have he : (n : ℝ) * ((1 + (d : ℝ)) * (2 : ℝ) ^ (-(4 * ((h : ℝ) + d)))) = | |
| ((d ^ 4 * (h + 1) * (1 + d) : ℕ) : ℝ) * (2 : ℝ) ^ (-(4 * ((h : ℝ) + d))) := by | |
| dsimp [n] | |
| push_cast | |
| ring | |
| rw [he, add_comm] | |
| exact dense_total_error_budget d h hd | |
| end | |
| end ContingencyTables.GridBoundary | |
| end OAI | |