import Mathlib import OAI.Computability.DirectedFeedback.Games.AffineWitnessNormalization namespace OAI namespace DFVSGames.Inverse.RowErasureMatrix open DFVSGames.Inverse.Shortcode open DFVSGames.Inverse.RowErasure open DFVSGames.Inverse.RowErasureDescriptions open scoped BigOperators noncomputable section theorem exists_replacement_of_budget {ell m r : ℕ} (B : Mat ell m → Prop) (α : ℝ) (hα : 0 < α) (hαone : α ≤ 1) (halphabet : 1 / (2 ^ ell : ℝ) ≤ α / 8) (hbudget : (2 : ℝ) ^ ((2 * r + 1) * (ell + m)) * Real.exp (-(α ^ 2 * (2 : ℝ) ^ ((ell - r) * (m - r))) / 32) < 1) : ∃ replacement : Mat ell m → Vector ell, ∀ d : Description ell m r, ((family ell m r).points d).Nonempty → (family ell m r).randomizedAgreement B replacement d < α / 4 := by let ρ := Real.exp (-(α ^ 2 * (2 : ℝ) ^ ((ell - r) * (m - r))) / 32) apply (family ell m r).exists_erasure_of_probability_bound B α ρ · intro d by_cases hne : ((family ell m r).points d).Nonempty · have htail := (family ell m r).randomizedAgreement_probability_le B α hα hαone (by simpa only [card_vector, Nat.cast_pow, Nat.cast_ofNat] using halphabet) d have hcardNat := d.toSlice.pow_dimension_le_card r le_rfl le_rfl hne have hcard : (2 : ℝ) ^ ((ell - r) * (m - r)) ≤ (((family ell m r).points d).card : ℝ) := by exact_mod_cast hcardNat apply htail.trans apply Real.exp_le_exp.mpr have hmul := mul_le_mul_of_nonneg_left hcard (sq_nonneg α) linarith · simpa [uniformMass, indicator, hne] using (Real.exp_pos (-(α ^ 2 * (2 : ℝ) ^ ((ell - r) * (m - r))) / 32)).le · simpa only [card_description, Nat.cast_pow, Nat.cast_ofNat] using hbudget theorem advice_mass_ge_of_inverse_and_budget {ell m r : ℕ} (f : Mat ell m → Vector ell) (η α : ℝ) (hη : 0 < η) (hα : 0 < α) (hαone : α ≤ 1) (halphabet : 1 / (2 ^ ell : ℝ) ≤ α / 8) (haccept : 4 * η ≤ Shortcode.equalityAcceptance f) (inverse : ∀ g : Mat ell m → Vector ell, η ≤ Shortcode.equalityAcceptance g → HasAffineSlice g α r) (hbudget : (2 : ℝ) ^ ((2 * r + 1) * (ell + m)) * Real.exp (-(α ^ 2 * (2 : ℝ) ^ ((ell - r) * (m - r))) / 32) < 1) : η / (2 ^ (ell * r) : ℝ) ≤ adviceMass ((family ell m r).goodAt f α) := by exact advice_mass_ge_of_inverse_and_erasure f η α hη hα haccept inverse (exists_replacement_of_budget ((family ell m r).erasedUnion f α) α hα hαone halphabet hbudget) end end DFVSGames.Inverse.RowErasureMatrix namespace DFVSGames.Inverse.RowErasure open Filter open scoped Topology theorem eventually_log_unionBound_neg (α : ℝ) (hα : 0 < α) (ell r : ℕ) (hr : r < ell) : ∀ᶠ m : ℕ in atTop, (((2 * r + 1) * (ell + m) : ℕ) : ℝ) * Real.log 2 - α ^ 2 * (2 : ℝ) ^ ((ell - r) * (m - r)) / 32 < 0 := by let C : ℝ := (2 * r + 1 : ℕ) let D : ℝ := (ell + r : ℕ) have hzero : Tendsto (fun n : ℕ => 1 / (2 : ℝ) ^ n) atTop (𝓝 0) := by simpa only [pow_zero] using (tendsto_pow_const_div_const_pow_of_one_lt 0 (show (1 : ℝ) < 2 by norm_num)) have hone : Tendsto (fun n : ℕ => (n : ℝ) / (2 : ℝ) ^ n) atTop (𝓝 0) := by simpa only [pow_one] using (tendsto_pow_const_div_const_pow_of_one_lt 1 (show (1 : ℝ) < 2 by norm_num)) have hlim' : Tendsto (fun n : ℕ => (C * Real.log 2) * (D * (1 / (2 : ℝ) ^ n) + (n : ℝ) / (2 : ℝ) ^ n)) atTop (𝓝 0) := by simpa only [mul_zero, add_zero] using ((hzero.const_mul D).add hone).const_mul (C * Real.log 2) have hlim : Tendsto (fun n : ℕ => C * (D + (n : ℝ)) * Real.log 2 / (2 : ℝ) ^ n) atTop (𝓝 0) := by convert hlim' using 1 ext n ring have hpos : 0 < α ^ 2 / 32 := by positivity have hsmall : ∀ᶠ n : ℕ in atTop, C * (D + (n : ℝ)) * Real.log 2 / (2 : ℝ) ^ n < α ^ 2 / 32 := hlim.eventually (gt_mem_nhds hpos) obtain ⟨N, hN⟩ := eventually_atTop.1 hsmall apply eventually_atTop.2 refine ⟨N + r, ?_⟩ intro m hm have hrm : r ≤ m := by omega have hn : N ≤ m - r := by omega have hmcast : (m : ℝ) = (r : ℝ) + ((m - r : ℕ) : ℝ) := by exact_mod_cast (show m = r + (m - r) by omega) have hcast : (((2 * r + 1) * (ell + m) : ℕ) : ℝ) = C * (D + ((m - r : ℕ) : ℝ)) := by dsimp [C, D] push_cast rw [hmcast] ring have hlinear := (div_lt_iff₀ (show (0 : ℝ) < 2 ^ (m - r) by positivity)).1 (hN (m - r) hn) have hexponent : m - r ≤ (ell - r) * (m - r) := by have h := Nat.mul_le_mul_right (m - r) (show 1 ≤ ell - r by omega) simpa only [one_mul] using h have hpower : (2 : ℝ) ^ (m - r) ≤ 2 ^ ((ell - r) * (m - r)) := pow_le_pow_right₀ (by norm_num) hexponent have hproduct := mul_le_mul_of_nonneg_left hpower hpos.le rw [hcast] nlinarith theorem eventually_unionBound_lt_one (α : ℝ) (hα : 0 < α) (ell r : ℕ) (hr : r < ell) : ∀ᶠ m : ℕ in atTop, (2 : ℝ) ^ ((2 * r + 1) * (ell + m)) * Real.exp (-(α ^ 2) * (2 : ℝ) ^ ((ell - r) * (m - r)) / 32) < 1 := by filter_upwards [eventually_log_unionBound_neg α hα ell r hr] with m hm have hpositive : (0 : ℝ) < 2 ^ ((2 * r + 1) * (ell + m)) := by positivity calc (2 : ℝ) ^ ((2 * r + 1) * (ell + m)) * Real.exp (-(α ^ 2) * (2 : ℝ) ^ ((ell - r) * (m - r)) / 32) = Real.exp ((((2 * r + 1) * (ell + m) : ℕ) : ℝ) * Real.log 2 - α ^ 2 * (2 : ℝ) ^ ((ell - r) * (m - r)) / 32) := by rw [← Real.exp_log hpositive, ← Real.exp_add, Real.log_pow] congr 1 ring _ < 1 := Real.exp_lt_one_iff.mpr hm theorem exists_unionBound_threshold (α : ℝ) (hα : 0 < α) (ell r : ℕ) (hr : r < ell) : ∃ m₀ : ℕ, ∀ m : ℕ, m₀ ≤ m → r ≤ m ∧ (2 : ℝ) ^ ((2 * r + 1) * (ell + m)) * Real.exp (-(α ^ 2) * (2 : ℝ) ^ ((ell - r) * (m - r)) / 32) < 1 := by exact eventually_atTop.1 ((eventually_ge_atTop r).and (eventually_unionBound_lt_one α hα ell r hr)) end DFVSGames.Inverse.RowErasure namespace DFVSGames.Inverse.RowErasureMatrix open DFVSGames.Inverse.Shortcode open DFVSGames.Inverse.RowErasure open DFVSGames.Inverse.RowErasureDescriptions open scoped BigOperators noncomputable section theorem exists_erasure_threshold (α : ℝ) (hα : 0 < α) (hαone : α ≤ 1) (ell r : ℕ) (hr : r < ell) (halphabet : 1 / (2 ^ ell : ℝ) ≤ α / 8) : ∃ m₀ : ℕ, ∀ m : ℕ, m₀ ≤ m → ∀ B : Mat ell m → Prop, ∃ replacement : Mat ell m → Vector ell, ∀ d : Description ell m r, ((family ell m r).points d).Nonempty → (family ell m r).randomizedAgreement B replacement d < α / 4 := by obtain ⟨m₀, hm₀⟩ := exists_unionBound_threshold α hα ell r hr refine ⟨m₀, ?_⟩ intro m hm B apply exists_replacement_of_budget B α hα hαone halphabet simpa only [neg_mul] using (hm₀ m hm).2 theorem exists_fixed_context_advice_threshold (η α : ℝ) (hη : 0 < η) (hα : 0 < α) (hαone : α ≤ 1) (ell r : ℕ) (hr : r < ell) (halphabet : 1 / (2 ^ ell : ℝ) ≤ α / 8) : ∃ m₀ : ℕ, ∀ m : ℕ, m₀ ≤ m → (∀ g : Mat ell m → Vector ell, η ≤ Shortcode.equalityAcceptance g → HasAffineSlice g α r) → ∀ f : Mat ell m → Vector ell, 4 * η ≤ Shortcode.equalityAcceptance f → η / (2 ^ (ell * r) : ℝ) ≤ adviceMass ((family ell m r).goodAt f α) := by obtain ⟨m₀, hm₀⟩ := exists_erasure_threshold α hα hαone ell r hr halphabet refine ⟨m₀, ?_⟩ intro m hm inverse f haccept exact advice_mass_ge_of_inverse_and_erasure f η α hη hα haccept inverse (hm₀ m hm ((family ell m r).erasedUnion f α)) theorem exists_contextual_advice_threshold (η α : ℝ) (hη : 0 < η) (hα : 0 < α) (hαone : α ≤ 1) (ell r : ℕ) (hr : r < ell) (halphabet : 1 / (2 ^ ell : ℝ) ≤ α / 8) : ∃ m₀ : ℕ, ∀ m : ℕ, m₀ ≤ m → (∀ g : Mat ell m → Vector ell, η ≤ Shortcode.equalityAcceptance g → HasAffineSlice g α r) → ∀ (Q : Type) [Fintype Q] (f : Q → Mat ell m → Vector ell), 10 * η ≤ uniformMass (fun q => 4 * η ≤ Shortcode.equalityAcceptance (f q)) → 10 * η ^ 2 / (2 ^ (ell * r) : ℝ) ≤ Finset.univ.expect (fun q => adviceMass ((family ell m r).goodAt (f q) α)) := by obtain ⟨m₀, hm₀⟩ := exists_erasure_threshold α hα hαone ell r hr halphabet refine ⟨m₀, ?_⟩ intro m hm inverse Q _ f hcontext have hlocal : ∀ q : Q, 4 * η ≤ Shortcode.equalityAcceptance (f q) → η ≤ uniformMass (goodUnion ((family ell m r).goodAt (f q) α)) := by intro q hq exact (family ell m r).union_mass_ge_of_inverse_and_erasure left right (f q) η α hη hα (by simpa only [acceptance_eq] using hq) left_uniform right_uniform (described_inverse α η inverse) (hm₀ m hm ((family ell m r).erasedUnion (f q) α)) have h := advice_average_ge_ten_eta_sq_div_card (fun q => 4 * η ≤ Shortcode.equalityAcceptance (f q)) (fun q => (family ell m r).goodAt (f q) α) η hη.le hcontext hlocal simpa only [card_rowMap, Nat.cast_pow, Nat.cast_ofNat] using h end end DFVSGames.Inverse.RowErasureMatrix namespace DFVSGames.Decoder.ActualGoodRows open Integration.BinaryLinear Reduction ActualSource open Inverse Inverse.RowErasure open Inverse.Shortcode (Mat HasAffineSlice) open scoped BigOperators Classical noncomputable section local instance homFintype {D F : Type*} [AddCommGroup D] [Module F2 D] [AddCommGroup F] [Module F2 F] [Fintype D] [Fintype F] : Fintype (D →ₗ[F2] F) := Fintype.ofInjective (fun M : D →ₗ[F2] F => (M : D → F)) DFunLike.coe_injective def table (S : Source) (k s d : ℕ) (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s)) (U : ActualGame.Question S k) (T : ActualHomogeneous.E k →ₗ[F2] Vector d) (M : Mat s (1 + 2 * k)) : Vector s := TableKeysGame.unfolded S k s d labeling (U, ((MatrixCoordinates.mapEquiv k s).symm M).prod T) theorem table_acceptance (S : Source) (k s d : ℕ) (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s)) (U : ActualGame.Question S k) (T : ActualHomogeneous.E k →ₗ[F2] Vector d) : Shortcode.equalityAcceptance (table S k s d labeling U T) = ActualSpectral.fiberAcceptance S k s d labeling U T := by unfold table rw [← MatrixCoordinates.shortcodeAcceptance_eq k s (fun M : ActualHomogeneous.E k →ₗ[F2] Vector s => TableKeysGame.unfolded S k s d labeling (U, M.prod T))] unfold ActualSpectral.fiberAcceptance SurrogateFibers.shortcodeAcceptance apply Finset.expect_congr (by ext M; simp only [Finset.mem_univ]) intro M _ apply Finset.expect_congr (by ext a; simp only [Finset.mem_univ]) intro a _ apply Finset.expect_congr (by ext l; simp only [Finset.mem_univ]) intro l _ rfl def adviceMass (S : Source) (k s d r : ℕ) (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s)) (α : ℝ) : ℝ := 𝔼 U : ActualGame.Question S k, 𝔼 T : ActualHomogeneous.E k →ₗ[F2] Vector d, RowErasure.adviceMass ((RowErasureMatrix.family s (1 + 2 * k) r).goodAt (table S k s d labeling U T) α) def goodRowMass (rstar s r : ℕ) : ℝ := 10 * ActualSpectral.eta rstar ^ 2 / (2 : ℝ) ^ (s * r) theorem goodRowMass_pos (rstar s r : ℕ) : 0 < goodRowMass rstar s r := by unfold goodRowMass have h := ActualSpectral.eta_pos rstar positivity theorem exists_actual_advice_threshold (rstar : ℕ) (α : ℝ) (hα : 0 < α) (hαone : α ≤ 1) (s r : ℕ) (hr : r < s) (halphabet : 1 / (2 : ℝ) ^ s ≤ α / 8) : ∃ k₀ : ℕ, ∀ k : ℕ, k₀ ≤ k → (∀ f : Mat s (1 + 2 * k) → Vector s, ActualSpectral.eta rstar ≤ Shortcode.equalityAcceptance f → HasAffineSlice f α r) → ∀ (S : Source) (d : ℕ) (g : SplitGadget s d) (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s)), (∀ L : Ambient s d →ₗ[F2] ActualHomogeneous.E k, rstar ≤ Module.finrank F2 (L.comp (alphabetEmbedding s d)).range → Integration.SplitGadget.kernelProbability g L ≤ 7 / 8) → (99 : ℚ) / 100 ≤ TableKeysGame.acceptanceProbability S k g labeling → goodRowMass rstar s r ≤ adviceMass S k s d r labeling α := by obtain ⟨m₀, hm₀⟩ := RowErasureMatrix.exists_contextual_advice_threshold (ActualSpectral.eta rstar) α (ActualSpectral.eta_pos rstar) hα hαone s r hr halphabet refine ⟨m₀, ?_⟩ intro k hk hinverse S d g labeling hkernel haccept let Q := ActualGame.Question S k × (ActualHomogeneous.E k →ₗ[F2] Vector d) let f : Q → Mat s (1 + 2 * k) → Vector s := fun q => table S k s d labeling q.1 q.2 have hcontext : 10 * ActualSpectral.eta rstar ≤ uniformMass (fun q : Q => 4 * ActualSpectral.eta rstar ≤ Shortcode.equalityAcceptance (f q)) := by have h := ActualSpectral.actual_good_fiber_mass S k g labeling rstar hkernel haccept dsimp only [Q, f] unfold uniformMass rw [ActualSpectral.expect_prod] simpa only [indicator, table_acceptance] using h have h := hm₀ (1 + 2 * k) (by omega) hinverse Q f hcontext dsimp only [Q, f] at h rw [ActualSpectral.expect_prod] at h simpa only [goodRowMass, adviceMass] using h end end DFVSGames.Decoder.ActualGoodRows namespace DFVSGames.Decoder.ConstantSelection open DFVSGames.Foundations.Information open SparseLaw noncomputable section def visibleMass (α g₀ : ℝ) (ell r : ℕ) : ℝ := α * g₀ / (8 * (2 : ℝ) ^ (ell * r)) def conditionalMass (α : ℝ) (ell r : ℕ) : ℝ := (α / 8) ^ 2 / (2 : ℝ) ^ (ell * r) / (2 : ℝ) ^ r def decodingMass (α g₀ : ℝ) (ell r : ℕ) : ℝ := visibleMass α g₀ ell r * conditionalMass α ell r def variationBudget (α g₀ : ℝ) (ell r : ℕ) : ℝ := visibleMass α g₀ ell r / (1 + α / 4) theorem visibleMass_pos {α g₀ : ℝ} (hα : 0 < α) (hg₀ : 0 < g₀) (ell r : ℕ) : 0 < visibleMass α g₀ ell r := by unfold visibleMass positivity theorem conditionalMass_pos {α : ℝ} (hα : 0 < α) (ell r : ℕ) : 0 < conditionalMass α ell r := by unfold conditionalMass positivity theorem decodingMass_pos {α g₀ : ℝ} (hα : 0 < α) (hg₀ : 0 < g₀) (ell r : ℕ) : 0 < decodingMass α g₀ ell r := mul_pos (visibleMass_pos hα hg₀ ell r) (conditionalMass_pos hα ell r) theorem variationBudget_pos {α g₀ : ℝ} (hα : 0 < α) (hg₀ : 0 < g₀) (ell r : ℕ) : 0 < variationBudget α g₀ ell r := by exact div_pos (visibleMass_pos hα hg₀ ell r) (by positivity) theorem variationBudget_suffices {α g₀ ε : ℝ} (hα : 0 < α) (ell r : ℕ) (hε : ε ≤ variationBudget α g₀ ell r) : (1 + α / 4) * ε ≤ α * g₀ * (1 / (2 : ℝ) ^ (ell * r)) / 8 := by have hd : 0 < 1 + α / 4 := by positivity have h := (le_div_iff₀ hd).mp hε calc (1 + α / 4) * ε = ε * (1 + α / 4) := mul_comm _ _ _ ≤ visibleMass α g₀ ell r := h _ = α * g₀ * (1 / (2 : ℝ) ^ (ell * r)) / 8 := by unfold visibleMass ring theorem cube_exponential_antitone {m n : ℕ} (hmn : m ≤ n) {N : ℝ} (hN : 0 < N) : Real.exp (-(n : ℝ) / (3600 * N)) ≤ Real.exp (-(m : ℝ) / (3600 * N)) := by apply Real.exp_le_exp.mpr apply div_le_div_of_nonneg_right _ (by positivity) exact neg_le_neg (by exact_mod_cast hmn) theorem exists_simultaneous_cube (V : Type*) [Fintype V] [Zero V] [DecidableEq V] {N ε γ : ℝ} (hN : 1 ≤ N) (hε : 0 < ε) (hγ : 0 < γ) (lower : ℕ) : ∃ m : ℕ, lower ≤ m ∧ 0 < m ∧ totalVariation (independentWeights (mixture (1 / (m : ℝ) ^ 2) (singletonPairWeights V)) (m ^ 3)) (uniformWeights (Fin (m ^ 3) → V × V)) ≤ ε ∧ (1 - ((1 / (m : ℝ) ^ 2) / N) / 3600) ^ (m ^ 3) < γ := by obtain ⟨m₀, hm₀, hexp⟩ := Clean.exists_cube_exponential_lt (show 0 < N by linarith) hγ obtain ⟨m, hlarge, hm, htv⟩ := exists_cubeSlope_totalVariation_le V ε hε (max lower m₀) have hbase : m₀ ≤ m := (le_max_right _ _).trans hlarge refine ⟨m, (le_max_left _ _).trans hlarge, hm, htv, ?_⟩ have hdecay := (Clean.cube_mask_rate_le_exp (Nat.succ_le_iff.mpr hm) hN).trans (cube_exponential_antitone hbase (by linarith : 0 < N)) have hbeta : (1 / (m : ℝ)) ^ 2 = 1 / (m : ℝ) ^ 2 := by ring rw [hbeta] at hdecay exact hdecay.trans_lt hexp theorem exists_matrix_cube (V : Type*) [Fintype V] [Zero V] [DecidableEq V] {α g₀ : ℝ} (hα : 0 < α) (hg₀ : 0 < g₀) (ell r dimW lower : ℕ) : ∃ m : ℕ, lower ≤ m ∧ 0 < m ∧ totalVariation (independentWeights (mixture (1 / (m : ℝ) ^ 2) (singletonPairWeights V)) (m ^ 3)) (uniformWeights (Fin (m ^ 3) → V × V)) ≤ variationBudget α g₀ ell r ∧ (1 - ((1 / (m : ℝ) ^ 2) / (2 : ℝ) ^ (dimW + r)) / 3600) ^ (m ^ 3) < decodingMass α g₀ ell r := by exact exists_simultaneous_cube V (one_le_pow₀ (by norm_num)) (variationBudget_pos hα hg₀ ell r) (decodingMass_pos hα hg₀ ell r) lower end end DFVSGames.Decoder.ConstantSelection namespace DFVSGames.Decoder.MatrixGapChoice open Integration.BinaryLinear Reduction ActualSource open Inverse.Shortcode (Mat HasAffineSlice InversePrinciple equalityAcceptance) noncomputable section structure InverseParameters (rstar : ℕ) where α : ℝ positive : 0 < α bounded : α ≤ 1 r : ℕ s : ℕ s_large : rstar ≤ s r_small : r < s trivial_small : 1 / (2 : ℝ) ^ (s - r) < α / 8 tupleThreshold : ℕ inverse : ∀ k : ℕ, tupleThreshold ≤ k → ∀ f : Mat s (1 + 2 * k) → Vector s, ActualSpectral.eta rstar ≤ equalityAcceptance f → HasAffineSlice f α r advice : ∀ k : ℕ, tupleThreshold ≤ k → ∀ (S : Source) (d : ℕ) (g : SplitGadget s d) (labeling : Fin (TableKeysGame.vertexCount S k s d) → Fin (2 ^ s)), (∀ L : Ambient s d →ₗ[F2] ActualHomogeneous.E k, rstar ≤ Module.finrank F2 (L.comp (alphabetEmbedding s d)).range → Integration.SplitGadget.kernelProbability g L ≤ 7 / 8) → (99 : ℚ) / 100 ≤ TableKeysGame.acceptanceProbability S k g labeling → ActualGoodRows.goodRowMass rstar s r ≤ ActualGoodRows.adviceMass S k s d r labeling α theorem eta_lt_one (rstar : ℕ) : ActualSpectral.eta rstar < 1 := by have hp : (1 : ℝ) ≤ 2 ^ rstar := one_le_pow₀ (by norm_num) have hi : 1 / (2 : ℝ) ^ rstar ≤ 1 := by exact (div_le_one (by positivity)).mpr hp unfold ActualSpectral.eta rw [← one_div] linarith theorem exists_inverse_parameters (hinverse : InversePrinciple) (rstar : ℕ) : Nonempty (InverseParameters rstar) := by obtain ⟨α, hα, hαone, r, _hr, s₀, hinv⟩ := hinverse (ActualSpectral.eta rstar) (ActualSpectral.eta_pos rstar) (eta_lt_one rstar) obtain ⟨s, hs, hrs, hsmall, _htiny⟩ := DimensionSelection.exists_logical_dimension hα (ActualSpectral.eta_pos rstar) r (max rstar s₀) have hsstar : rstar ≤ s := (le_max_left _ _).trans hs have hs₀ : s₀ ≤ s := (le_max_right _ _).trans hs obtain ⟨m₀, hm₀⟩ := hinv s hs₀ have halphabet : 1 / (2 : ℝ) ^ s ≤ α / 8 := by have hp : (2 : ℝ) ^ (s - r) ≤ 2 ^ s := pow_le_pow_right₀ (by norm_num) (Nat.sub_le s r) exact (one_div_le_one_div_of_le (by positivity) hp).trans hsmall.le obtain ⟨k₀, hk₀⟩ := ActualGoodRows.exists_actual_advice_threshold rstar α hα hαone s r hrs halphabet let threshold := max m₀ k₀ have hi (k : ℕ) (hk : threshold ≤ k) : ∀ f : Mat s (1 + 2 * k) → Vector s, ActualSpectral.eta rstar ≤ equalityAcceptance f → HasAffineSlice f α r := by apply hm₀ (1 + 2 * k) have hm : m₀ ≤ k := (le_max_left _ _).trans hk omega refine ⟨{ α := α positive := hα bounded := hαone r := r s := s s_large := hsstar r_small := hrs trivial_small := hsmall tupleThreshold := threshold inverse := hi advice := ?_ }⟩ intro k hk S d g labeling hkernel haccept exact hk₀ k ((le_max_right _ _).trans hk) (hi k hk) S d g labeling hkernel haccept theorem exists_cube {rstar : ℕ} (P : InverseParameters rstar) (d : ℕ) : ∃ m : ℕ, 0 < m ∧ P.tupleThreshold ≤ m ^ 3 ∧ 0 ≤ 1 / (m : ℝ) ^ 2 ∧ 1 / (m : ℝ) ^ 2 ≤ 1 ∧ Foundations.Information.totalVariation (SparseLaw.independentWeights (SparseLaw.mixture (1 / (m : ℝ) ^ 2) (SparseLaw.singletonPairWeights (Ambient P.s d))) (m ^ 3)) (SparseLaw.uniformWeights (Fin (m ^ 3) → Ambient P.s d × Ambient P.s d)) ≤ ConstantSelection.variationBudget P.α (ActualGoodRows.goodRowMass rstar P.s P.r) P.s P.r ∧ (1 - ((1 / (m : ℝ) ^ 2) / (2 : ℝ) ^ (d + P.r)) / 3600) ^ (m ^ 3) < ConstantSelection.decodingMass P.α (ActualGoodRows.goodRowMass rstar P.s P.r) P.s P.r := by obtain ⟨m, hlarge, hm, htv, hrate⟩ := ConstantSelection.exists_matrix_cube (Ambient P.s d) P.positive (ActualGoodRows.goodRowMass_pos rstar P.s P.r) P.s P.r d P.tupleThreshold have hmreal : (1 : ℝ) ≤ m := by exact_mod_cast (Nat.succ_le_iff.mpr hm) have hmsq : (1 : ℝ) ≤ (m : ℝ) ^ 2 := one_le_pow₀ hmreal refine ⟨m, hm, hlarge.trans (Nat.le_self_pow (by decide : (3 : ℕ) ≠ 0) m), by positivity, ?_, htv, hrate⟩ exact (div_le_one (by positivity)).mpr hmsq end end DFVSGames.Decoder.MatrixGapChoice namespace DFVSGames.Inverse.RowErasure noncomputable section variable {X Y A S D : Type*} def SliceFamily.chosenDescription (F : SliceFamily X Y A S D) (f : X → Y) (α : ℝ) (a : A) (s : S) (hgood : F.GoodAdvice f α a s) : D := Classical.choose hgood theorem SliceFamily.chosenDescription_spec (F : SliceFamily X Y A S D) (f : X → Y) (α : ℝ) (a : A) (s : S) (hgood : F.GoodAdvice f α a s) : F.rowMap (F.chosenDescription f α a s hgood) = a ∧ F.rowValue (F.chosenDescription f α a s hgood) = s ∧ (F.points (F.chosenDescription f α a s hgood)).Nonempty ∧ α / 2 ≤ F.agreement f (F.chosenDescription f α a s hgood) := Classical.choose_spec hgood theorem SliceFamily.chosenDescription_congr (F : SliceFamily X Y A S D) (f : X → Y) (α : ℝ) {a a' : A} {s s' : S} (hgood : F.GoodAdvice f α a s) (hgood' : F.GoodAdvice f α a' s') (ha : a = a') (hs : s = s') : F.chosenDescription f α a s hgood = F.chosenDescription f α a' s' hgood' := by subst a' subst s' rfl def SliceFamily.selectedDescription (F : SliceFamily X Y A S D) (f : X → Y) (α : ℝ) (a : A) (s : S) : Option D := by classical exact if hgood : F.GoodAdvice f α a s then some (F.chosenDescription f α a s hgood) else none theorem SliceFamily.selectedDescription_of_good (F : SliceFamily X Y A S D) (f : X → Y) (α : ℝ) (a : A) (s : S) (hgood : F.GoodAdvice f α a s) : F.selectedDescription f α a s = some (F.chosenDescription f α a s hgood) := by simp only [SliceFamily.selectedDescription, dite_eq_left hgood] theorem SliceFamily.good_of_selectedDescription (F : SliceFamily X Y A S D) (f : X → Y) (α : ℝ) (a : A) (s : S) {d : D} (hselected : F.selectedDescription f α a s = some d) : F.GoodAdvice f α a s := by classical by_contra hgood simp only [SliceFamily.selectedDescription, dite_eq_right hgood] at hselected cases hselected theorem SliceFamily.selectedDescription_spec (F : SliceFamily X Y A S D) (f : X → Y) (α : ℝ) (a : A) (s : S) {d : D} (hselected : F.selectedDescription f α a s = some d) : F.rowMap d = a ∧ F.rowValue d = s ∧ (F.points d).Nonempty ∧ α / 2 ≤ F.agreement f d := by have hgood := F.good_of_selectedDescription f α a s hselected rw [F.selectedDescription_of_good f α a s hgood] at hselected have hd := Option.some.inj hselected rw [← hd] exact F.chosenDescription_spec f α a s hgood theorem SliceFamily.selectedDescription_congr (F : SliceFamily X Y A S D) (f : X → Y) (α : ℝ) {a a' : A} {s s' : S} (ha : a = a') (hs : s = s') : F.selectedDescription f α a s = F.selectedDescription f α a' s' := by subst a' subst s' rfl def SliceFamily.selectedEvent (F : SliceFamily X Y A S D) (f : X → Y) (α : ℝ) (a : A) (x : X) : Prop := ∃ d, F.selectedDescription f α a (F.advice a x) = some d ∧ x ∈ F.points d def SliceFamily.selectedMatch (F : SliceFamily X Y A S D) (f : X → Y) (α : ℝ) (a : A) (x : X) : Prop := ∃ d, F.selectedDescription f α a (F.advice a x) = some d ∧ x ∈ F.points d ∧ f x = F.target d x theorem SliceFamily.selectedEvent_iff_of_advice (F : SliceFamily X Y A S D) (f : X → Y) (α : ℝ) (a : A) (s : S) (hgood : F.GoodAdvice f α a s) (x : X) (hx : F.advice a x = s) : F.selectedEvent f α a x ↔ x ∈ F.points (F.chosenDescription f α a s hgood) := by unfold SliceFamily.selectedEvent rw [hx, F.selectedDescription_of_good f α a s hgood] constructor · rintro ⟨d, hd, hmem⟩ have he := Option.some.inj hd simpa only [he] using hmem · intro hmem exact ⟨_, rfl, hmem⟩ theorem SliceFamily.selectedMatch_iff_of_advice (F : SliceFamily X Y A S D) (f : X → Y) (α : ℝ) (a : A) (s : S) (hgood : F.GoodAdvice f α a s) (x : X) (hx : F.advice a x = s) : F.selectedMatch f α a x ↔ x ∈ F.points (F.chosenDescription f α a s hgood) ∧ f x = F.target (F.chosenDescription f α a s hgood) x := by unfold SliceFamily.selectedMatch rw [hx, F.selectedDescription_of_good f α a s hgood] constructor · rintro ⟨d, hd, hmem, hmatch⟩ have he := Option.some.inj hd simpa only [he] using And.intro hmem hmatch · rintro ⟨hmem, hmatch⟩ exact ⟨_, rfl, hmem, hmatch⟩ theorem SliceFamily.selectedEvent_good (F : SliceFamily X Y A S D) (f : X → Y) (α : ℝ) (a : A) (x : X) (hx : F.selectedEvent f α a x) : F.goodAt f α a x := by obtain ⟨d, hd, _⟩ := hx exact F.good_of_selectedDescription f α a (F.advice a x) hd theorem SliceFamily.selectedMatch_event (F : SliceFamily X Y A S D) (f : X → Y) (α : ℝ) (a : A) (x : X) (hx : F.selectedMatch f α a x) : F.selectedEvent f α a x := by obtain ⟨d, hd, hmem, _⟩ := hx exact ⟨d, hd, hmem⟩ end end DFVSGames.Inverse.RowErasure namespace DFVSGames.Inverse.RowErasure.SelectionPartition open scoped BigOperators Classical noncomputable section variable {X S : Type*} [Fintype X] [Fintype S] def fiber (observe : X → S) (s : S) : Finset X := Finset.univ.filter fun x => observe x = s def event (observe : X → S) (good : S → Prop) (chosen : S → Finset X) (x : X) : Prop := good (observe x) ∧ x ∈ chosen (observe x) def matchEvent (observe : X → S) (good : S → Prop) (chosen : S → Finset X) (hit : S → X → Prop) (x : X) : Prop := event observe good chosen x ∧ hit (observe x) x theorem expect_le_expect_of_fiber_sum_le (observe : X → S) (f g : X → ℝ) (hle : ∀ s, (∑ x ∈ fiber observe s, f x) ≤ ∑ x ∈ fiber observe s, g x) : Finset.univ.expect f ≤ Finset.univ.expect g := by rw [Fintype.expect_eq_sum_div_card, Fintype.expect_eq_sum_div_card] apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _) rw [← Finset.sum_fiberwise Finset.univ observe f, ← Finset.sum_fiberwise Finset.univ observe g] exact Finset.sum_le_sum fun s _ => hle s omit [Fintype S] in theorem sum_event_mul_fiber (observe : X → S) (good : S → Prop) (chosen : S → Finset X) (support : ∀ s x, x ∈ chosen s → observe x = s) (s : S) (hgood : good s) (w : X → ℝ) : (∑ x ∈ fiber observe s, indicator (event observe good chosen x) * w x) = ∑ x ∈ chosen s, w x := by have hsub : chosen s ⊆ fiber observe s := by intro x hx exact Finset.mem_filter.mpr ⟨Finset.mem_univ x, support s x hx⟩ calc _ = ∑ x ∈ fiber observe s, if x ∈ chosen s then w x else 0 := by apply Finset.sum_congr rfl intro x hx have hobs : observe x = s := (Finset.mem_filter.mp hx).2 simp [event, indicator, hobs, hgood, ite_mul] _ = ∑ x ∈ chosen s, if x ∈ chosen s then w x else 0 := by exact (Finset.sum_subset hsub (by intro x _ hx simp only [ite_eq_right hx])).symm _ = ∑ x ∈ chosen s, w x := by apply Finset.sum_congr rfl intro x hx exact ite_eq_left hx omit [Fintype S] in theorem sum_event_fiber (observe : X → S) (good : S → Prop) (chosen : S → Finset X) (support : ∀ s x, x ∈ chosen s → observe x = s) (s : S) (hgood : good s) : (∑ x ∈ fiber observe s, indicator (event observe good chosen x)) = ((chosen s).card : ℝ) := by simpa using sum_event_mul_fiber observe good chosen support s hgood (fun _ => 1) omit [Fintype S] in theorem sum_match_fiber (observe : X → S) (good : S → Prop) (chosen : S → Finset X) (support : ∀ s x, x ∈ chosen s → observe x = s) (hit : S → X → Prop) (s : S) (hgood : good s) : (∑ x ∈ fiber observe s, indicator (matchEvent observe good chosen hit x)) = ∑ x ∈ chosen s, indicator (hit s x) := by calc _ = ∑ x ∈ fiber observe s, indicator (event observe good chosen x) * indicator (hit s x) := by apply Finset.sum_congr rfl intro x hx have hobs : observe x = s := (Finset.mem_filter.mp hx).2 by_cases hevent : event observe good chosen x <;> by_cases hhit : hit s x <;> simp [matchEvent, indicator, hevent, hobs, hhit] _ = ∑ x ∈ chosen s, indicator (hit s x) := sum_event_mul_fiber observe good chosen support s hgood (fun x => indicator (hit s x)) theorem mass_event_ge (observe : X → S) (good : S → Prop) (chosen : S → Finset X) (support : ∀ s x, x ∈ chosen s → observe x = s) (c : ℝ) (hsize : ∀ s, good s → c * ((fiber observe s).card : ℝ) ≤ ((chosen s).card : ℝ)) : c * uniformMass (fun x => good (observe x)) ≤ uniformMass (event observe good chosen) := by unfold uniformMass rw [Finset.mul_expect] apply expect_le_expect_of_fiber_sum_le observe intro s by_cases hgood : good s · rw [sum_event_fiber observe good chosen support s hgood] calc (∑ x ∈ fiber observe s, c * indicator (good (observe x))) = c * ((fiber observe s).card : ℝ) := by calc _ = ∑ _x ∈ fiber observe s, c := by apply Finset.sum_congr rfl intro x hx have hobs : observe x = s := (Finset.mem_filter.mp hx).2 simp [indicator, hobs, hgood] _ = _ := by simp [mul_comm] _ ≤ _ := hsize s hgood · have hz : ∀ x ∈ fiber observe s, indicator (good (observe x)) = 0 ∧ indicator (event observe good chosen x) = 0 := by intro x hx have hobs : observe x = s := (Finset.mem_filter.mp hx).2 simp [indicator, event, hobs, hgood] have hleft : (∑ x ∈ fiber observe s, c * indicator (good (observe x))) = 0 := Finset.sum_eq_zero fun x hx => by rw [(hz x hx).1, mul_zero] have hright : (∑ x ∈ fiber observe s, indicator (event observe good chosen x)) = 0 := Finset.sum_eq_zero fun x hx => (hz x hx).2 rw [hleft, hright] theorem mass_match_ge (observe : X → S) (good : S → Prop) (chosen : S → Finset X) (support : ∀ s x, x ∈ chosen s → observe x = s) (hit : S → X → Prop) (β : ℝ) (hmatch : ∀ s, good s → β * ((chosen s).card : ℝ) ≤ ∑ x ∈ chosen s, indicator (hit s x)) : β * uniformMass (event observe good chosen) ≤ uniformMass (matchEvent observe good chosen hit) := by unfold uniformMass rw [Finset.mul_expect] apply expect_le_expect_of_fiber_sum_le observe intro s by_cases hgood : good s · rw [← Finset.mul_sum, sum_event_fiber observe good chosen support s hgood, sum_match_fiber observe good chosen support hit s hgood] exact hmatch s hgood · have hz : ∀ x ∈ fiber observe s, indicator (event observe good chosen x) = 0 ∧ indicator (matchEvent observe good chosen hit x) = 0 := by intro x hx have hobs : observe x = s := (Finset.mem_filter.mp hx).2 simp [indicator, event, matchEvent, hobs, hgood] have hleft : (∑ x ∈ fiber observe s, β * indicator (event observe good chosen x)) = 0 := Finset.sum_eq_zero fun x hx => by rw [(hz x hx).1, mul_zero] have hright : (∑ x ∈ fiber observe s, indicator (matchEvent observe good chosen hit x)) = 0 := Finset.sum_eq_zero fun x hx => (hz x hx).2 rw [hleft, hright] end end DFVSGames.Inverse.RowErasure.SelectionPartition end OAI