import Mathlib import OAI.Computability.DirectedFeedback.Machines.MachineSingleOrbitBridge namespace OAI section section section section section section section section section section section section section section section section section section section section section section section section section section section section section section section section section section section section section section section section section section section section namespace DFVSGames.Fourier.MatrixSubspaceCount open Module open DFVSGames.Integration.BinaryLinear (F2) section GeneralField variable {K V : Type*} [Field K] [AddCommGroup V] [Module K V] theorem endomorphism_range_surjective : Function.Surjective (fun f : V →ₗ[K] V => LinearMap.range f) := by intro S obtain ⟨T, hT⟩ := S.exists_isCompl exact ⟨S.projection T hT, Submodule.range_projection hT⟩ theorem card_subspaces_le_endomorphisms [Finite V] : Nat.card (Submodule K V) ≤ Nat.card (V →ₗ[K] V) := by let : Finite (V →ₗ[K] V) := Finite.of_injective (fun f : V →ₗ[K] V => (f : V → V)) DFunLike.coe_injective exact Nat.card_le_card_of_surjective _ endomorphism_range_surjective end GeneralField section BinaryField variable {C : Type*} [AddCommGroup C] [Module F2 C] [Module.Finite F2 C] private theorem finite_color_space_inline_MatrixSubspaceCount : Finite C := Finite.of_injective (Module.finBasis F2 C).equivFun (Module.finBasis F2 C).equivFun.injective private theorem finite_endomorphisms_inline_MatrixSubspaceCount : Finite (C →ₗ[F2] C) := by let : Finite C := finite_color_space_inline_MatrixSubspaceCount exact Finite.of_injective (fun f : C →ₗ[F2] C => (f : C → C)) DFunLike.coe_injective private theorem finite_subspaces_inline_MatrixSubspaceCount : Finite (Submodule F2 C) := by let : Finite (C →ₗ[F2] C) := finite_endomorphisms_inline_MatrixSubspaceCount exact Finite.of_surjective _ endomorphism_range_surjective theorem card_binary_subspaces_le : Nat.card (Submodule F2 C) ≤ 2 ^ (finrank F2 C * finrank F2 C) := by let : Finite C := finite_color_space_inline_MatrixSubspaceCount calc _ ≤ Nat.card (C →ₗ[F2] C) := card_subspaces_le_endomorphisms _ = _ := DFVSGames.Appendix.CompressionCount.natCard_linearMap theorem card_binary_subspaces_le_of_finrank_le (s : ℕ) (hs : finrank F2 C ≤ s) : Nat.card (Submodule F2 C) ≤ 2 ^ (s * s) := by exact card_binary_subspaces_le.trans (Nat.pow_le_pow_right (by decide) (Nat.mul_le_mul hs hs)) theorem card_admissible_subspaces_le (P : Submodule F2 C → Prop) : Nat.card {S : Submodule F2 C // P S} ≤ 2 ^ (finrank F2 C * finrank F2 C) := by let : Finite (Submodule F2 C) := finite_subspaces_inline_MatrixSubspaceCount exact (Nat.card_le_card_of_injective Subtype.val Subtype.val_injective).trans card_binary_subspaces_le theorem card_color_admissible_subspaces_le (P : Submodule F2 C → Prop) : Nat.card (C × {S : Submodule F2 C // P S}) ≤ Nat.card C * 2 ^ (finrank F2 C * finrank F2 C) := by rw [Nat.card_prod] exact Nat.mul_le_mul_left _ (card_admissible_subspaces_le P) theorem card_color_admissible_subspaces_le_of_finrank_le (P : Submodule F2 C → Prop) (s : ℕ) (hs : finrank F2 C ≤ s) : Nat.card (C × {S : Submodule F2 C // P S}) ≤ Nat.card C * 2 ^ (s * s) := by exact (card_color_admissible_subspaces_le P).trans (Nat.mul_le_mul_left _ (Nat.pow_le_pow_right (by decide) (Nat.mul_le_mul hs hs))) end BinaryField end DFVSGames.Fourier.MatrixSubspaceCount end section namespace DFVSGames.Fourier.MatrixEnergy open scoped BigOperators variable {Q : Type*} {I : Q → Type*} [Fintype Q] [∀ q, Fintype (I q)] noncomputable def energy (weight : Q → ℝ) (a : (q : Q) → I q → ℝ) : ℝ := ∑ q, weight q * ∑ i, a q i ^ 2 noncomputable def cutoffEnergy (weight : Q → ℝ) (keep : (q : Q) → I q → Prop) (a : (q : Q) → I q → ℝ) : ℝ := by classical exact ∑ q, weight q * ∑ i, if keep q i then a q i ^ 2 else 0 noncomputable def spectralEnergy (weight : Q → ℝ) (eigen a : (q : Q) → I q → ℝ) : ℝ := ∑ q, weight q * ∑ i, eigen q i * a q i ^ 2 theorem energy_nonneg (weight : Q → ℝ) (a : (q : Q) → I q → ℝ) (hw : ∀ q, 0 ≤ weight q) : 0 ≤ energy weight a := by exact Finset.sum_nonneg (fun q _ => mul_nonneg (hw q) (Finset.sum_nonneg (fun i _ => sq_nonneg (a q i)))) theorem cutoffEnergy_nonneg (weight : Q → ℝ) (keep : (q : Q) → I q → Prop) (a : (q : Q) → I q → ℝ) (hw : ∀ q, 0 ≤ weight q) : 0 ≤ cutoffEnergy weight keep a := by classical unfold cutoffEnergy apply Finset.sum_nonneg intro q _ apply mul_nonneg (hw q) apply Finset.sum_nonneg intro i _ split_ifs · exact sq_nonneg _ · exact le_rfl theorem cutoffEnergy_le_energy (weight : Q → ℝ) (keep : (q : Q) → I q → Prop) (a : (q : Q) → I q → ℝ) (hw : ∀ q, 0 ≤ weight q) : cutoffEnergy weight keep a ≤ energy weight a := by classical apply Finset.sum_le_sum intro q _ apply mul_le_mul_of_nonneg_left _ (hw q) apply Finset.sum_le_sum intro i _ split_ifs <;> nlinarith [sq_nonneg (a q i)] theorem energy_split (weight : Q → ℝ) (keep : (q : Q) → I q → Prop) (a : (q : Q) → I q → ℝ) : energy weight a = cutoffEnergy weight keep a + cutoffEnergy weight (fun q i => ¬ keep q i) a := by classical unfold energy cutoffEnergy rw [← Finset.sum_add_distrib] apply Finset.sum_congr rfl intro q _ rw [← mul_add, ← Finset.sum_add_distrib] congr 1 apply Finset.sum_congr rfl intro i _ by_cases h : keep q i <;> simp [h] theorem spectralEnergy_le_cutoff_add (weight : Q → ℝ) (keep : (q : Q) → I q → Prop) (eigen a : (q : Q) → I q → ℝ) (η : ℝ) (hw : ∀ q, 0 ≤ weight q) (hη : 0 ≤ η) (hone : ∀ q i, eigen q i ≤ 1) (hhigh : ∀ q i, ¬ keep q i → eigen q i ≤ η) : spectralEnergy weight eigen a ≤ cutoffEnergy weight keep a + η * energy weight a := by classical have hpoint (q : Q) (i : I q) : eigen q i * a q i ^ 2 ≤ (if keep q i then a q i ^ 2 else 0) + η * a q i ^ 2 := by by_cases hk : keep q i · have h := mul_le_mul_of_nonneg_right (hone q i) (sq_nonneg (a q i)) have hn := mul_nonneg hη (sq_nonneg (a q i)) simp only [hk, ite_true] nlinarith · simpa only [hk, ite_false, zero_add] using mul_le_mul_of_nonneg_right (hhigh q i hk) (sq_nonneg (a q i)) calc spectralEnergy weight eigen a ≤ ∑ q, weight q * ∑ i, ((if keep q i then a q i ^ 2 else 0) + η * a q i ^ 2) := by apply Finset.sum_le_sum intro q _ exact mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun i _ => hpoint q i)) (hw q) _ = cutoffEnergy weight keep a + η * energy weight a := by simp only [cutoffEnergy, energy, Finset.sum_add_distrib, ← Finset.mul_sum, mul_add] congr 1 rw [Finset.mul_sum] apply Finset.sum_congr rfl intro q _ ring theorem cutoffEnergy_gt_half (weight : Q → ℝ) (keep : (q : Q) → I q → Prop) (eigen a : (q : Q) → I q → ℝ) (δ : ℝ) (hw : ∀ q, 0 ≤ weight q) (hδ : 0 ≤ δ) (hone : ∀ q i, eigen q i ≤ 1) (hhigh : ∀ q i, ¬ keep q i → eigen q i ≤ δ / 2) (htotal : energy weight a = 1) (hvalue : δ < spectralEnergy weight eigen a) : δ / 2 < cutoffEnergy weight keep a := by have h := spectralEnergy_le_cutoff_add weight keep eigen a (δ / 2) hw (by linarith) hone hhigh rw [htotal] at h linarith theorem cutoffEnergy_le_twice_difference_add (weight : Q → ℝ) (keep : (q : Q) → I q → Prop) (H G : (q : Q) → I q → ℝ) (hw : ∀ q, 0 ≤ weight q) : cutoffEnergy weight keep H ≤ 2 * energy weight (fun q i => H q i - G q i) + 2 * cutoffEnergy weight keep G := by classical have hpoint (q : Q) (i : I q) : (if keep q i then H q i ^ 2 else 0) ≤ 2 * (H q i - G q i) ^ 2 + 2 * (if keep q i then G q i ^ 2 else 0) := by by_cases hk : keep q i · simp only [hk, ite_true] nlinarith [sq_nonneg (H q i - 2 * G q i)] · simp only [hk, ite_false, mul_zero, add_zero] nlinarith [sq_nonneg (H q i - G q i)] calc cutoffEnergy weight keep H ≤ ∑ q, weight q * ∑ i, (2 * (H q i - G q i) ^ 2 + 2 * (if keep q i then G q i ^ 2 else 0)) := by apply Finset.sum_le_sum intro q _ exact mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun i _ => hpoint q i)) (hw q) _ = 2 * energy weight (fun q i => H q i - G q i) + 2 * cutoffEnergy weight keep G := by simp only [energy, cutoffEnergy, Finset.sum_add_distrib, ← Finset.mul_sum, mul_add] congr 1 <;> rw [Finset.mul_sum] <;> apply Finset.sum_congr rfl <;> intro q _ <;> ring theorem deleted_energy_gt_eighth (weight : Q → ℝ) (keep : (q : Q) → I q → Prop) (H G : (q : Q) → I q → ℝ) (δ : ℝ) (hw : ∀ q, 0 ≤ weight q) (hlarge : δ / 2 < cutoffEnergy weight keep H) (hsmall : cutoffEnergy weight keep G ≤ δ / 8) : δ / 8 < energy weight (fun q i => H q i - G q i) := by have h := cutoffEnergy_le_twice_difference_add weight keep H G hw linarith theorem boolean_energy_eq_mass (weight : Q → ℝ) (H : (q : Q) → I q → ℝ) (hbool : ∀ q i, H q i = 0 ∨ H q i = 1) : energy weight H = ∑ q, weight q * ∑ i, H q i := by unfold energy apply Finset.sum_congr rfl intro q _ congr 1 apply Finset.sum_congr rfl intro i _ rcases hbool q i with h | h <;> simp [h] theorem weighted_union_bound {Ω J : Type*} [Fintype Ω] [Fintype J] (weight value : Ω → ℝ) (event : J → Ω → Prop) [∀ j, DecidablePred (event j)] (hw : ∀ ω, 0 ≤ weight ω) (hvalue : ∀ ω, value ω ≤ 1) (hcover : ∀ ω, 0 < value ω → ∃ j, event j ω) : (∑ ω, weight ω * value ω) ≤ ∑ j, ∑ ω, weight ω * (if event j ω then (1 : ℝ) else 0) := by classical rw [Finset.sum_comm] apply Finset.sum_le_sum intro ω _ rw [← Finset.mul_sum] apply mul_le_mul_of_nonneg_left _ (hw ω) by_cases hp : 0 < value ω · obtain ⟨j, hj⟩ := hcover ω hp have hs : (if event j ω then (1 : ℝ) else 0) ≤ ∑ k, if event k ω then (1 : ℝ) else 0 := by apply Finset.single_le_sum (s := Finset.univ) (f := fun k : J => if event k ω then (1 : ℝ) else 0) (a := j) · intro k _; split_ifs <;> norm_num · exact Finset.mem_univ j exact le_trans (hvalue ω) (by simpa [hj] using hs) · exact le_trans (le_of_not_gt hp) (Finset.sum_nonneg (by intro j _; split_ifs <;> norm_num)) theorem exists_fixed_event {J : Type*} [Fintype J] (mass : J → ℝ) (threshold budget : ℝ) (hcount : (Fintype.card J : ℝ) * threshold ≤ budget) (hmass : budget < ∑ j, mass j) : ∃ j, threshold < mass j := by classical by_contra h push Not at h have hs : (∑ j, mass j) ≤ (Fintype.card J : ℝ) * threshold := by calc (∑ j, mass j) ≤ ∑ _j : J, threshold := Finset.sum_le_sum (fun j _ => h j) _ = _ := by simp linarith theorem exists_fixed_event_scaled {J : Type*} [Fintype J] (mass : J → ℝ) (N : ℕ) (δ : ℝ) (hN : 0 < N) (hδ : 0 ≤ δ) (hcard : Fintype.card J ≤ N) (hmass : δ / 8 < ∑ j, mass j) : ∃ j, δ / (8 * (N : ℝ)) < mass j := by have hNreal : (0 : ℝ) < N := by exact_mod_cast hN have hcardreal : (Fintype.card J : ℝ) ≤ N := by exact_mod_cast hcard have hthreshold : 0 ≤ δ / (8 * (N : ℝ)) := div_nonneg hδ (mul_nonneg (by norm_num) hNreal.le) have hcancel : (N : ℝ) * (δ / (8 * (N : ℝ))) = δ / 8 := by field_simp [ne_of_gt hNreal] apply exists_fixed_event mass (δ / (8 * (N : ℝ))) (δ / 8) _ hmass rw [← hcancel] exact mul_le_mul_of_nonneg_right hcardreal hthreshold section LinearMaps open MatrixCharacters MatrixFourier variable {E F : Type*} variable [AddCommGroup E] [Module F2 E] [AddCommGroup F] [Module F2 F] variable [FiniteDimensional F2 E] [FiniteDimensional F2 F] variable [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] noncomputable def linearRankOn (C : Submodule F2 F) (S : F →ₗ[F2] E) : ℕ := Module.finrank F2 (LinearMap.range (S.comp C.subtype)) omit [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] in theorem linearRankOn_top (S : F →ₗ[F2] E) : linearRankOn (⊤ : Submodule F2 F) S = Module.finrank F2 (LinearMap.range S) := by have hrange : LinearMap.range (S.comp (⊤ : Submodule F2 F).subtype) = LinearMap.range S := by ext x constructor · rintro ⟨v, hv⟩ exact ⟨v.1, hv⟩ · rintro ⟨v, hv⟩ exact ⟨⟨v, by simp⟩, hv⟩ exact congrArg (fun P : Submodule F2 E => Module.finrank F2 P) hrange omit [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)] in theorem linearRankOn_range {V : Type*} [AddCommGroup V] [Module F2 V] (S : F →ₗ[F2] E) (ι : V →ₗ[F2] F) : linearRankOn (LinearMap.range ι) S = Module.finrank F2 (LinearMap.range (S.comp ι)) := by have hrange : LinearMap.range (S.comp (LinearMap.range ι).subtype) = LinearMap.range (S.comp ι) := by ext x constructor · rintro ⟨v, hv⟩ obtain ⟨u, hu⟩ := v.property refine ⟨u, ?_⟩ change S (ι u) = x rw [hu] exact hv · rintro ⟨u, hu⟩ exact ⟨⟨ι u, ⟨u, rfl⟩⟩, hu⟩ exact congrArg (fun P : Submodule F2 E => Module.finrank F2 P) hrange noncomputable def rankProjection (C : Submodule F2 F) (r : ℕ) (f : (E →ₗ[F2] F) → ℝ) : (E →ₗ[F2] F) → ℝ := by classical exact ∑ S : F →ₗ[F2] E, (if linearRankOn C S ≤ r then linearCoeff f S else 0) • (fun X => (linearTraceCharacter S X).re) omit [FiniteDimensional F2 E] [FiniteDimensional F2 F] [Fintype (F →ₗ[F2] E)] in theorem linearCoeff_sub (f g : (E →ₗ[F2] F) → ℝ) (S : F →ₗ[F2] E) : linearCoeff (fun X => f X - g X) S = linearCoeff f S - linearCoeff g S := by simp only [linearCoeff, sub_mul, Finset.expect_sub_distrib] theorem linearCoeff_rankProjection (C : Submodule F2 F) (r : ℕ) (f : (E →ₗ[F2] F) → ℝ) (S : F →ₗ[F2] E) : linearCoeff (rankProjection C r f) S = if linearRankOn C S ≤ r then linearCoeff f S else 0 := by classical unfold rankProjection rw [linearCoeff_sum] simp only [linearCoeff_smul, linearCoeff_character, mul_ite, mul_one, mul_zero] simp theorem rankProjection_parseval (C : Submodule F2 F) (r : ℕ) (f : (E →ₗ[F2] F) → ℝ) : (𝔼 X, rankProjection C r f X ^ 2) = ∑ S : F →ₗ[F2] E, if linearRankOn C S ≤ r then linearCoeff f S ^ 2 else 0 := by classical rw [← linear_parseval] apply Finset.sum_congr rfl intro S _ rw [linearCoeff_rankProjection] split_ifs <;> simp theorem rankProjection_contraction (C : Submodule F2 F) (r : ℕ) (f : (E →ₗ[F2] F) → ℝ) : (𝔼 X, rankProjection C r f X ^ 2) ≤ 𝔼 X, f X ^ 2 := by classical rw [rankProjection_parseval, ← linear_parseval] apply Finset.sum_le_sum intro S _ split_ifs · exact le_rfl · exact sq_nonneg _ theorem rankProjection_orthogonal (C : Submodule F2 F) (r : ℕ) (f : (E →ₗ[F2] F) → ℝ) : (𝔼 X, rankProjection C r f X * (f X - rankProjection C r f X)) = 0 := by classical rw [← linear_parseval_inner] apply Finset.sum_eq_zero intro S _ rw [linearCoeff_sub, linearCoeff_rankProjection] split_ifs <;> simp theorem linear_energy_parseval (weight : Q → ℝ) (f : Q → (E →ₗ[F2] F) → ℝ) : energy weight (fun q S => linearCoeff (f q) S) = ∑ q, weight q * (𝔼 X, f q X ^ 2) := by simp only [energy, linear_parseval] theorem linear_difference_energy_parseval (weight : Q → ℝ) (H G : Q → (E →ₗ[F2] F) → ℝ) : energy weight (fun q S => linearCoeff (H q) S - linearCoeff (G q) S) = ∑ q, weight q * (𝔼 X, (H q X - G q X) ^ 2) := by simp only [← linearCoeff_sub, linear_energy_parseval] theorem linear_deleted_energy_gt_eighth (weight : Q → ℝ) (C : Submodule F2 F) (r : ℕ) (H G : Q → (E →ₗ[F2] F) → ℝ) (δ : ℝ) (hw : ∀ q, 0 ≤ weight q) (hlarge : δ / 2 < cutoffEnergy weight (fun _ S => linearRankOn C S ≤ r) (fun q S => linearCoeff (H q) S)) (hsmall : cutoffEnergy weight (fun _ S => linearRankOn C S ≤ r) (fun q S => linearCoeff (G q) S) ≤ δ / 8) : δ / 8 < ∑ q, weight q * (𝔼 X, (H q X - G q X) ^ 2) := by rw [← linear_difference_energy_parseval] exact deleted_energy_gt_eighth weight (fun _ S => linearRankOn C S ≤ r) (fun q S => linearCoeff (H q) S) (fun q S => linearCoeff (G q) S) δ hw hlarge hsmall noncomputable def colorIndicator {X Y : Type*} (label : X → Y) (y : Y) : X → ℝ := by classical exact fun x => if label x = y then 1 else 0 theorem colorIndicator_partition {X Y : Type*} [Fintype Y] (label : X → Y) (x : X) : ∑ y, colorIndicator label y x ^ 2 = 1 := by classical simp [colorIndicator, eq_comm] theorem colorIndicator_parseval {Y : Type*} [Fintype Y] (label : (E →ₗ[F2] F) → Y) : (∑ y, ∑ S : F →ₗ[F2] E, linearCoeff (colorIndicator label y) S ^ 2) = 1 := by simp_rw [linear_parseval] rw [← Finset.expect_sum_comm] simp_rw [colorIndicator_partition] exact Fintype.expect_const 1 noncomputable def labelingCoeff {Y : Type*} (label : Q → (E →ₗ[F2] F) → Y) (qy : Q × Y) (S : F →ₗ[F2] E) : ℝ := linearCoeff (colorIndicator (label qy.1) qy.2) S theorem labelingCoeff_energy_one {Y : Type*} [Fintype Y] (weight : Q → ℝ) (label : Q → (E →ₗ[F2] F) → Y) (hnorm : ∑ q, weight q = 1) : energy (fun qy : Q × Y => weight qy.1) (labelingCoeff label) = 1 := by classical calc energy (fun qy : Q × Y => weight qy.1) (labelingCoeff label) = ∑ q, weight q * (∑ y, ∑ S : F →ₗ[F2] E, linearCoeff (colorIndicator (label q) y) S ^ 2) := by simp only [energy, labelingCoeff, Fintype.sum_prod_type, Finset.mul_sum] _ = ∑ q, weight q := by simp only [colorIndicator_parseval, mul_one] _ = 1 := hnorm theorem labeling_lowRank_energy {Y : Type*} [Fintype Y] (weight : Q → ℝ) (label : Q → (E →ₗ[F2] F) → Y) (C : Submodule F2 F) (r : ℕ) (eigen : Q → (F →ₗ[F2] E) → ℝ) (δ : ℝ) (hw : ∀ q, 0 ≤ weight q) (hnorm : ∑ q, weight q = 1) (hδ : 0 ≤ δ) (hone : ∀ q S, eigen q S ≤ 1) (hhigh : ∀ q S, r < linearRankOn C S → eigen q S ≤ δ / 2) (hvalue : δ < spectralEnergy (fun qy : Q × Y => weight qy.1) (fun qy S => eigen qy.1 S) (labelingCoeff label)) : δ / 2 < cutoffEnergy (fun qy : Q × Y => weight qy.1) (fun _ S => linearRankOn C S ≤ r) (labelingCoeff label) := by apply cutoffEnergy_gt_half _ _ _ _ δ · intro qy; exact hw qy.1 · exact hδ · intro qy S; exact hone qy.1 S · intro qy S hS; exact hhigh qy.1 S (Nat.lt_of_not_ge hS) · exact labelingCoeff_energy_one weight label hnorm · exact hvalue end LinearMaps section DependentLinearMaps open MatrixCharacters MatrixFourier variable {J : Type*} [Fintype J] {E : J → Type*} {F : Type*} [∀ j, AddCommGroup (E j)] [∀ j, Module F2 (E j)] [AddCommGroup F] [Module F2 F] [∀ j, FiniteDimensional F2 (E j)] [FiniteDimensional F2 F] [∀ j, Fintype (E j →ₗ[F2] F)] [∀ j, Fintype (F →ₗ[F2] E j)] theorem dependent_linear_energy_parseval (weight : J → ℝ) (H : (j : J) → (E j →ₗ[F2] F) → ℝ) : energy weight (fun j S => linearCoeff (H j) S) = ∑ j, weight j * (𝔼 X, H j X ^ 2) := by unfold energy apply Finset.sum_congr rfl intro j _ rw [linear_parseval] omit [∀ j, FiniteDimensional F2 (E j)] [FiniteDimensional F2 F] in theorem dependent_remainder_cutoff_le_eighth (weight : J → ℝ) (C : Submodule F2 F) (r : ℕ) (G : (j : J) → (E j →ₗ[F2] F) → ℝ) (κ δ : ℝ) (hw : ∀ j, 0 ≤ weight j) (hκ : 0 ≤ κ) (hκδ : κ ≤ δ / 8) (hlevel : ∀ j, (∑ S : F →ₗ[F2] E j, if linearRankOn C S ≤ r then linearCoeff (G j) S ^ 2 else 0) ≤ κ * (𝔼 X, G j X ^ 2)) (hmass : (∑ j, weight j * (𝔼 X, G j X ^ 2)) ≤ 1) : cutoffEnergy weight (fun _ S => linearRankOn C S ≤ r) (fun j S => linearCoeff (G j) S) ≤ δ / 8 := by classical have hs : cutoffEnergy weight (fun _ S => linearRankOn C S ≤ r) (fun j S => linearCoeff (G j) S) ≤ κ * (∑ j, weight j * (𝔼 X, G j X ^ 2)) := by calc _ ≤ ∑ j, weight j * (κ * (𝔼 X, G j X ^ 2)) := by apply Finset.sum_le_sum intro j _ apply mul_le_mul_of_nonneg_left _ (hw j) simpa using hlevel j _ = _ := by rw [Finset.mul_sum] apply Finset.sum_congr rfl intro j _ ring have hm := mul_le_mul_of_nonneg_left hmass hκ linarith theorem dependent_deleted_energy_from_level (weight : J → ℝ) (C : Submodule F2 F) (r : ℕ) (H G : (j : J) → (E j →ₗ[F2] F) → ℝ) (κ δ : ℝ) (hw : ∀ j, 0 ≤ weight j) (hκ : 0 ≤ κ) (hκδ : κ ≤ δ / 8) (hlarge : δ / 2 < cutoffEnergy weight (fun _ S => linearRankOn C S ≤ r) (fun j S => linearCoeff (H j) S)) (hlevel : ∀ j, (∑ S : F →ₗ[F2] E j, if linearRankOn C S ≤ r then linearCoeff (G j) S ^ 2 else 0) ≤ κ * (𝔼 X, G j X ^ 2)) (hmass : (∑ j, weight j * (𝔼 X, G j X ^ 2)) ≤ 1) : δ / 8 < ∑ j, weight j * (𝔼 X, (H j X - G j X) ^ 2) := by have hsmall := dependent_remainder_cutoff_le_eighth weight C r G κ δ hw hκ hκδ hlevel hmass have h := deleted_energy_gt_eighth weight (fun _ S => linearRankOn C S ≤ r) (fun j S => linearCoeff (H j) S) (fun j S => linearCoeff (G j) S) δ hw hlarge hsmall simpa only [← linearCoeff_sub, energy, linear_parseval, Pi.sub_apply] using h end DependentLinearMaps def boolReal (b : Bool) : ℝ := if b then 1 else 0 @[simp] theorem boolReal_sq (b : Bool) : boolReal b ^ 2 = boolReal b := by cases b <;> norm_num [boolReal] theorem boolReal_le_one (b : Bool) : boolReal b ≤ 1 := by cases b <;> norm_num [boolReal] noncomputable def labelBool {X Y : Type*} (label : X → Y) (y : Y) : X → Bool := by classical exact fun x => decide (label x = y) theorem labelBool_real {X Y : Type*} (label : X → Y) (y : Y) (x : X) : boolReal (labelBool label y x) = colorIndicator label y x := by classical simp [boolReal, labelBool, colorIndicator] theorem removed_boolReal {X Row Witness : Type*} (S : Deletion.RestrictionSystem X Row Witness) (H : X → Bool) (p d : ℕ) (x : X) : boolReal (Deletion.removed S H p d x) = boolReal (H x) - boolReal (Deletion.remainder S H p d x) := by classical by_cases hd : Deletion.DeletedSet S H p d x · simp [Deletion.removed, Deletion.remainder, hd, boolReal] · simp [Deletion.removed, Deletion.remainder, hd, boolReal] theorem removed_difference_sq {X Row Witness : Type*} (S : Deletion.RestrictionSystem X Row Witness) (H : X → Bool) (p d : ℕ) (x : X) : (boolReal (H x) - boolReal (Deletion.remainder S H p d x)) ^ 2 = boolReal (Deletion.removed S H p d x) := by rw [← removed_boolReal, boolReal_sq] section QuestionProbabilities variable {Ω : Q → Type*} [∀ q, Fintype (Ω q)] variable {Y A : Type*} [Fintype Y] [Fintype A] noncomputable def questionEventMass (weight : Q → ℝ) (B : (q : Q) → Ω q → Prop) : ℝ := by classical exact ∑ q, weight q * (𝔼 X, if B q X then (1 : ℝ) else 0) theorem dependent_removed_union_bound (weight : Q → ℝ) (d : (q : Q) → Y → Ω q → ℝ) (B : (q : Q) → Y → A → Ω q → Prop) (hw : ∀ q, 0 ≤ weight q) (hd : ∀ q y X, d q y X ≤ 1) (hc : ∀ q y X, 0 < d q y X → ∃ a, B q y a X) : (∑ q, weight q * ∑ y, 𝔼 X, d q y X) ≤ ∑ ya : Y × A, questionEventMass weight (fun q X => B q ya.1 ya.2 X) := by classical have hp (q : Q) (y : Y) (X : Ω q) : d q y X ≤ ∑ a, if B q y a X then (1 : ℝ) else 0 := by by_cases hpos : 0 < d q y X · obtain ⟨a, ha⟩ := hc q y X hpos have hi := Finset.single_le_sum (s := Finset.univ) (f := fun b : A => if B q y b X then (1 : ℝ) else 0) (fun b _ => by split_ifs <;> norm_num) (Finset.mem_univ a) exact le_trans (hd q y X) (by simpa [ha] using hi) · exact le_trans (le_of_not_gt hpos) (Finset.sum_nonneg (by intro a _; split_ifs <;> norm_num)) calc _ ≤ ∑ q, weight q * ∑ y, 𝔼 X, ∑ a, if B q y a X then (1 : ℝ) else 0 := by apply Finset.sum_le_sum intro q _ apply mul_le_mul_of_nonneg_left _ (hw q) apply Finset.sum_le_sum intro y _ exact Finset.expect_le_expect (fun X _ => hp q y X) _ = _ := by simp only [questionEventMass, Finset.expect_sum_comm, Finset.mul_sum, Fintype.sum_prod_type] rw [Finset.sum_comm] apply Finset.sum_congr rfl intro y _ exact Finset.sum_comm theorem fixed_target_of_removed_mass (weight : Q → ℝ) (d : (q : Q) → Y → Ω q → ℝ) (B : (q : Q) → Y → A → Ω q → Prop) (hw : ∀ q, 0 ≤ weight q) (hd : ∀ q y X, d q y X ≤ 1) (hc : ∀ q y X, 0 < d q y X → ∃ a, B q y a X) (δ : ℝ) (N : ℕ) (hδ : 0 ≤ δ) (hN : 0 < N) (hcount : Fintype.card (Y × A) ≤ N) (hremoved : δ / 8 < ∑ q, weight q * ∑ y, 𝔼 X, d q y X) : ∃ y a, δ / (8 * (N : ℝ)) < questionEventMass weight (fun q X => B q y a X) := by have hu := dependent_removed_union_bound weight d B hw hd hc obtain ⟨⟨y, a⟩, hya⟩ := exists_fixed_event_scaled (fun ya : Y × A => questionEventMass weight (fun q X => B q ya.1 ya.2 X)) N δ hN hδ hcount (lt_of_lt_of_le hremoved hu) exact ⟨y, a, hya⟩ theorem boolean_subpartition_energy_le_one [∀ q, Nonempty (Ω q)] (weight : Q → ℝ) (label : (q : Q) → Ω q → Y) (G : (q : Q) → Y → Ω q → Bool) (hw : ∀ q, 0 ≤ weight q) (hnorm : ∑ q, weight q = 1) (hG : ∀ q y X, G q y X = true → label q X = y) : (∑ q, weight q * ∑ y, 𝔼 X, boolReal (G q y X) ^ 2) ≤ 1 := by classical have hp (q : Q) (X : Ω q) : (∑ y, boolReal (G q y X) ^ 2) ≤ 1 := by calc _ ≤ ∑ y, colorIndicator (label q) y X ^ 2 := by apply Finset.sum_le_sum intro y _ by_cases hg : G q y X = true · have hl := hG q y X hg simp [hg, boolReal, colorIndicator, hl] · simpa [boolReal, hg] using sq_nonneg (colorIndicator (label q) y X) _ = 1 := colorIndicator_partition (label q) X have hq (q : Q) : (∑ y, 𝔼 X, boolReal (G q y X) ^ 2) ≤ 1 := by rw [← Finset.expect_sum_comm] calc _ ≤ 𝔼 _X : Ω q, (1 : ℝ) := Finset.expect_le_expect (fun X _ => hp q X) _ = 1 := Fintype.expect_const 1 calc _ ≤ ∑ q, weight q * 1 := Finset.sum_le_sum (fun q _ => mul_le_mul_of_nonneg_left (hq q) (hw q)) _ = 1 := by simpa using hnorm theorem remainder_subpartition_energy_le_one [∀ q, Nonempty (Ω q)] {Row Witness : Q → Type*} (S : (q : Q) → Deletion.RestrictionSystem (Ω q) (Row q) (Witness q)) (weight : Q → ℝ) (label : (q : Q) → Ω q → Y) (p d : ℕ) (hw : ∀ q, 0 ≤ weight q) (hnorm : ∑ q, weight q = 1) : (∑ q, weight q * ∑ y, 𝔼 X, boolReal (Deletion.remainder (S q) (labelBool (label q) y) p d X) ^ 2) ≤ 1 := by apply boolean_subpartition_energy_le_one weight label _ hw hnorm intro q y X hG have h := Deletion.remainder_le_original (S q) (labelBool (label q) y) p d X hG simpa [labelBool] using h end QuestionProbabilities section ActualBadRows open MatrixCharacters variable {C : Type*} [AddCommGroup C] [Module F2 C] [FiniteDimensional F2 C] [Fintype C] {E : Q → Type*} [∀ q, AddCommGroup (E q)] [∀ q, Module F2 (E q)] [∀ q, Finite (E q)] [∀ q, Fintype (E q →ₗ[F2] C)] theorem fixed_badRow_target_of_removed_mass (weight : Q → ℝ) (H : (q : Q) → C → (E q →ₗ[F2] C) → Bool) (r p d : ℕ) (δ : ℝ) (hw : ∀ q, 0 ≤ weight q) (hδ : 0 ≤ δ) (hremoved : δ / 8 < ∑ q, weight q * ∑ y, 𝔼 X, boolReal (Deletion.removed (MatrixRestrictions.system r) (H q y) p d X)) : ∃ (y : C) (C' : Submodule F2 C), Module.finrank F2 (C ⧸ C') ≤ r ∧ δ / (8 * (Nat.card C : ℝ) * (2 : ℝ) ^ (Module.finrank F2 C * Module.finrank F2 C)) < questionEventMass weight (fun q (X : E q →ₗ[F2] C) => Deletion.BadRow (MatrixRestrictions.system r) (H q y) p d ⟨C', C'.mkQ.comp X⟩) := by classical let : Finite (C →ₗ[F2] C) := Finite.of_injective (fun f : C →ₗ[F2] C => (f : C → C)) DFunLike.coe_injective let : Finite (Submodule F2 C) := Finite.of_surjective (fun f : C →ₗ[F2] C => LinearMap.range f) MatrixSubspaceCount.endomorphism_range_surjective let A := {C' : Submodule F2 C // Module.finrank F2 (C ⧸ C') ≤ r} let : Fintype A := Fintype.ofFinite A let N := Nat.card C * 2 ^ (Module.finrank F2 C * Module.finrank F2 C) let value (q : Q) (y : C) (X : E q →ₗ[F2] C) : ℝ := boolReal (Deletion.removed (MatrixRestrictions.system r) (H q y) p d X) let event (q : Q) (y : C) (a : A) (X : E q →ₗ[F2] C) : Prop := Deletion.BadRow (MatrixRestrictions.system r) (H q y) p d ⟨a.val, a.val.mkQ.comp X⟩ have hcover (q : Q) (y : C) (X : E q →ₗ[F2] C) (hpos : 0 < value q y X) : ∃ a : A, event q y a X := by have ht : Deletion.removed (MatrixRestrictions.system r) (H q y) p d X = true := by by_contra hn simp [value, boolReal, hn] at hpos have hdel : Deletion.DeletedSet (MatrixRestrictions.system r) (H q y) p d X := by by_contra hn simp [Deletion.removed, hn] at ht obtain ⟨C', hcodim, hbad⟩ := (MatrixRestrictions.deletedSet_iff_exists_admissible_target (H q y) r p d X).mp hdel exact ⟨⟨C', hcodim⟩, hbad⟩ have hcount : Fintype.card (C × A) ≤ N := by simpa only [A, N, Nat.card_eq_fintype_card] using (MatrixSubspaceCount.card_color_admissible_subspaces_le (fun C' : Submodule F2 C => Module.finrank F2 (C ⧸ C') ≤ r)) have hN : 0 < N := by simpa only [N, Nat.card_eq_fintype_card] using (Nat.mul_pos (Fintype.card_pos : 0 < Fintype.card C) (pow_pos (by norm_num : 0 < (2 : ℕ)) (Module.finrank F2 C * Module.finrank F2 C))) obtain ⟨y, a, ha⟩ := fixed_target_of_removed_mass weight value event hw (fun _ _ _ => boolReal_le_one _) hcover δ N hδ hN hcount hremoved refine ⟨y, a.val, a.property, ?_⟩ simpa only [N, event, Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat, mul_assoc] using ha end ActualBadRows end DFVSGames.Fourier.MatrixEnergy end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end end OAI