openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Representation /RepresentationBucketCapacity.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationBucketProbability | |
| namespace OAI | |
| /-! # Capacity of the actual designated parent buckets -/ | |
| noncomputable section | |
| namespace LowSpaceSubsetSum | |
| open scoped BigOperators | |
| private theorem hashBucket_mod_mul_low {α : Type*} (X : Finset α) (v : α → ℤ) | |
| (x : α) (p q : ℤ) : | |
| hashBucket (X.filter fun e => v e % p = v x % p) v (fun w => w % (p * q)) x = | |
| hashBucket X v (fun w => w % (p * q)) x := by | |
| classical | |
| ext e | |
| simp only [hashBucket, Finset.mem_filter] | |
| constructor | |
| · exact fun h => ⟨h.1.1, h.2⟩ | |
| · intro h | |
| exact ⟨⟨h.1, (show Int.ModEq (p * q) (v e) (v x) from h.2).of_dvd | |
| (show p ∣ p * q from ⟨q, rfl⟩)⟩, h.2⟩ | |
| namespace RepresentationSetup | |
| variable {n : ℕ} (R : RepresentationSetup n) | |
| /-- The low restriction removes no member of the full product-modulus bucket. -/ | |
| theorem chosenParentBucket_eq_hashBucket (h : R.PrimeHistory) (r : R.Residues h.2) | |
| (hn : 10000 ≤ n) | |
| (hlog : Real.logb 2 ((mixerSize n : ℝ) + 1) + 1 ≤ (n : ℝ) / 1000) | |
| (hr : R.ResidueSuccess (R.actualOffsets (R.allocatedChildAssignment hn hlog)) h r) | |
| (j : Fin 2) (q : ℕ) : | |
| R.chosenParentBucket h r hn hlog j ((prefixModulus h.2.val 0 : ℤ) * q) = | |
| hashBucket (R.chosenLowEntries h r hn hlog j) Entry.value | |
| (fun w => w % ((prefixModulus h.2.val 0 : ℤ) * q)) | |
| (RepresentationUniverse.entry (R.signedWeights h.1.1.1) | |
| (R.chosenRawParent h r hn hlog j)) := by | |
| classical | |
| have hw := R.chosenRawParent_weight h r hn hlog hr j | |
| ext e | |
| simp only [chosenParentBucket, chosenLowEntries, hashBucket, Finset.mem_filter] | |
| change _ ↔ (_ ∧ _) ∧ e.value % ((prefixModulus h.2.val 0 : ℤ) * q) = | |
| RepresentationUniverse.parentWeight (R.signedWeights h.1.1.1) (R.chosenRawParent h r hn hlog j) % | |
| ((prefixModulus h.2.val 0 : ℤ) * q) | |
| rw [hw] | |
| constructor | |
| · intro he | |
| exact ⟨⟨he.1, (show Int.ModEq ((prefixModulus h.2.val 0 : ℤ) * q) | |
| e.value (weight (R.signedWeights h.1.1.1) (R.chosenNamedParent h r hn hlog j)) | |
| from he.2).of_dvd ⟨q, rfl⟩⟩, he.2⟩ | |
| · exact fun he => ⟨he.1.1, he.2⟩ | |
| /-- The unrefined left bucket fits its actual fixed-capacity dictionary. -/ | |
| theorem chosenLeftBucket_fits (h : R.PrimeHistory) (r : R.Residues h.2) | |
| (hn : 10000 ≤ n) | |
| (hlog : Real.logb 2 ((mixerSize n : ℝ) + 1) + 1 ≤ (n : ℝ) / 1000) | |
| (hr : R.ResidueSuccess (R.actualOffsets (R.allocatedChildAssignment hn hlog)) h r) | |
| (q : ℕ) | |
| (hq : ((R.chosenOtherBucket h r hn hlog 0 q).card : ℝ) ≤ | |
| (compressionDictionaryCapacity n : ℝ) / 2) : | |
| (R.chosenParentBucket h r hn hlog 0 ((prefixModulus h.2.val 0 : ℤ) * q)).card ≤ | |
| compressionDictionaryCapacity n := by | |
| rw [R.chosenParentBucket_eq_hashBucket h r hn hlog hr 0 q, hashBucket_card] | |
| have he := R.left_equalWeight_half_capacity h.1.1.1 h.1.1.2 h.1.2 | |
| (R.allocatedBackground h.1.1.1 h.1.1.2 hn hlog) (by omega) | |
| (R.chosenLowEntries h r hn hlog 0) (R.chosenLowEntries_subset h r hn hlog 0) | |
| (RepresentationUniverse.entry (R.signedWeights h.1.1.1) (R.chosenRawParent h r hn hlog 0)) | |
| have hs : ((sameWeightKeys (R.chosenLowEntries h r hn hlog 0) Entry.value | |
| (RepresentationUniverse.entry (R.signedWeights h.1.1.1) (R.chosenRawParent h r hn hlog 0))).card : ℝ) + | |
| (R.chosenOtherBucket h r hn hlog 0 q).card ≤ compressionDictionaryCapacity n := by | |
| linarith | |
| exact_mod_cast hs | |
| /-- A full right-mask prefix has one possible equal-weight key, while its | |
| unequal-weight keys are already counted in the unrefined collision bucket. -/ | |
| theorem chosenRightFullBucket_fits (h : R.PrimeHistory) (r : R.Residues h.2) | |
| (hn : 10000 ≤ n) | |
| (hlog : Real.logb 2 ((mixerSize n : ℝ) + 1) + 1 ≤ (n : ℝ) / 1000) | |
| (hr : R.ResidueSuccess (R.actualOffsets (R.allocatedChildAssignment hn hlog)) h r) | |
| (q : ℕ) (D : Finset (Fin n)) | |
| (hq : ((R.chosenOtherBucket h r hn hlog 1 q).card : ℝ) ≤ | |
| (compressionDictionaryCapacity n : ℝ) / 2) : | |
| ((R.chosenParentBucket h r hn hlog 1 ((prefixModulus h.2.val 0 : ℤ) * q)).filter | |
| (fun e => e.mask = D)).card ≤ compressionDictionaryCapacity n := by | |
| classical | |
| let X := R.chosenLowEntries h r hn hlog 1 | |
| let x := RepresentationUniverse.entry (R.signedWeights h.1.1.1) | |
| (R.chosenRawParent h r hn hlog 1) | |
| let Y := X.filter fun e => e.mask = D | |
| have heq : ((sameWeightKeys Y Entry.value x).card : ℝ) ≤ | |
| (compressionDictionaryCapacity n : ℝ) / 2 := by | |
| have hset : sameWeightKeys Y Entry.value x = X.filter (fun e => e.value = x.value ∧ e.mask = D) := by | |
| ext e | |
| simp only [sameWeightKeys, Y, Finset.mem_filter] | |
| tauto | |
| rw [hset] | |
| exact right_fullMask_half_capacity X x.value D (by omega) | |
| have hother : (otherCollisionKeys Y Entry.value | |
| (fun w => w % ((prefixModulus h.2.val 0 : ℤ) * q)) x).card ≤ | |
| (R.chosenOtherBucket h r hn hlog 1 q).card := by | |
| apply Finset.card_le_card | |
| intro e he | |
| simp only [otherCollisionKeys, Y, Finset.mem_filter] at he | |
| exact Finset.mem_filter.mpr ⟨Finset.mem_filter.mpr ⟨he.1.1.1, he.1.2⟩, he.2⟩ | |
| have hb : ((hashBucket Y Entry.value | |
| (fun w => w % ((prefixModulus h.2.val 0 : ℤ) * q)) x).card : ℝ) ≤ | |
| compressionDictionaryCapacity n := by | |
| rw [hashBucket_card, Nat.cast_add] | |
| have ho : ((otherCollisionKeys Y Entry.value | |
| (fun w => w % ((prefixModulus h.2.val 0 : ℤ) * q)) x).card : ℝ) ≤ | |
| (compressionDictionaryCapacity n : ℝ) / 2 := | |
| le_trans (by exact_mod_cast hother) hq | |
| linarith | |
| have hset : (R.chosenParentBucket h r hn hlog 1 | |
| ((prefixModulus h.2.val 0 : ℤ) * q)).filter (fun e => e.mask = D) = | |
| hashBucket Y Entry.value (fun w => w % ((prefixModulus h.2.val 0 : ℤ) * q)) x := by | |
| rw [R.chosenParentBucket_eq_hashBucket h r hn hlog hr 1 q] | |
| ext e | |
| simp only [hashBucket, Y, X, x, Finset.mem_filter] | |
| tauto | |
| rw [hset] | |
| exact_mod_cast hb | |
| /-- Both actual bucket capacities hold for at least nine tenths of the extra | |
| prime choices, with no independence assumption between the two counts. -/ | |
| theorem chosenBuckets_probability (h : R.PrimeHistory) (r : R.Residues h.2) | |
| (hn : 20000 ≤ n) | |
| (hlog : Real.logb 2 ((mixerSize n : ℝ) + 1) + 1 ≤ (n : ℝ) / 1000) | |
| (hr : R.ResidueSuccess (R.actualOffsets (R.allocatedChildAssignment (by omega) hlog)) h r) | |
| (hgood : R.AllocatedResourceGood hn hlog ⟨h, r⟩) | |
| (hs : mixerSpreadTest (R.signedWeights h.1.1.1) (R.orderedMixers h.1.1.1) n) | |
| (hscale : Real.logb 2 (inputScale R.input : ℝ) ≤ (n : ℝ) / 1000000000) | |
| (D : Finset (Fin n)) : | |
| (9 / 10 : ℝ) ≤ | |
| ∑ q ∈ (primeRange (R.extraPrimeParameter h.1.1.1)).filter (fun q : ℕ => | |
| (R.chosenParentBucket h r (by omega) hlog 0 ((prefixModulus h.2.val 0 : ℤ) * q)).card ≤ | |
| compressionDictionaryCapacity n ∧ | |
| ((R.chosenParentBucket h r (by omega) hlog 1 ((prefixModulus h.2.val 0 : ℤ) * q)).filter | |
| (fun e => e.mask = D)).card ≤ compressionDictionaryCapacity n), | |
| primeProbability (R.extraPrimeParameter h.1.1.1) q := by | |
| have hp := R.chosenOtherBuckets_probability h r hn hlog hr hgood | |
| have hloss := R.representation_bucket_loss h.1.1.1 h.1.1.2 hn hs hscale | |
| have hsmall : (9 / 10 : ℝ) ≤ | |
| ∑ q ∈ (primeRange (R.extraPrimeParameter h.1.1.1)).filter (fun q : ℕ => | |
| ((R.chosenOtherBucket h r (by omega) hlog 0 q).card : ℝ) ≤ | |
| (compressionDictionaryCapacity n : ℝ) / 2 ∧ | |
| ((R.chosenOtherBucket h r (by omega) hlog 1 q).card : ℝ) ≤ | |
| (compressionDictionaryCapacity n : ℝ) / 2), | |
| primeProbability (R.extraPrimeParameter h.1.1.1) q := by linarith | |
| apply hsmall.trans | |
| apply Finset.sum_le_sum_of_subset_of_nonneg | |
| · intro q hq | |
| obtain ⟨hmem, hleft, hright⟩ := Finset.mem_filter.mp hq | |
| exact Finset.mem_filter.mpr ⟨hmem, | |
| R.chosenLeftBucket_fits h r (by omega) hlog hr q hleft, | |
| R.chosenRightFullBucket_fits h r (by omega) hlog hr q D hright⟩ | |
| · intro _ _ _ | |
| unfold primeProbability | |
| positivity | |
| /-- The extra-prime draw is uniform on the concrete prime range. -/ | |
| abbrev ExtraPrimeSpace (z : Layout n) := ↥(primeRange (R.extraPrimeParameter z)) | |
| def extraPrimeLaw (z : Layout n) (hn : 100 ≤ n) : FiniteLaw (R.ExtraPrimeSpace z) := by | |
| let : Nonempty (R.ExtraPrimeSpace z) := | |
| ⟨⟨(primeRange_nonempty (R.extraPrimeParameter_range z hn).1).choose, | |
| (primeRange_nonempty (R.extraPrimeParameter_range z hn).1).choose_spec⟩⟩ | |
| exact FiniteLaw.uniform | |
| theorem extraPrimeLaw_probability (z : Layout n) (hn : 100 ≤ n) (P : ℕ → Prop) [DecidablePred P] : | |
| (R.extraPrimeLaw z hn).probability (fun q => P q.val) = | |
| ∑ q ∈ (primeRange (R.extraPrimeParameter z)).filter P, | |
| primeProbability (R.extraPrimeParameter z) q := by | |
| classical | |
| let : Nonempty (R.ExtraPrimeSpace z) := | |
| ⟨⟨(primeRange_nonempty (R.extraPrimeParameter_range z hn).1).choose, | |
| (primeRange_nonempty (R.extraPrimeParameter_range z hn).1).choose_spec⟩⟩ | |
| rw [extraPrimeLaw, FiniteLaw.uniform_probability, | |
| finiteUniformProbability_finset (primeRange (R.extraPrimeParameter z)) P] | |
| simp [primeProbability, div_eq_mul_inv, mul_comm] | |
| /-- Finite-kernel version of the two actual dictionary-capacity bounds. -/ | |
| theorem chosenBuckets_law_probability (h : R.PrimeHistory) (r : R.Residues h.2) | |
| (hn : 20000 ≤ n) | |
| (hlog : Real.logb 2 ((mixerSize n : ℝ) + 1) + 1 ≤ (n : ℝ) / 1000) | |
| (hr : R.ResidueSuccess (R.actualOffsets (R.allocatedChildAssignment (by omega) hlog)) h r) | |
| (hgood : R.AllocatedResourceGood hn hlog ⟨h, r⟩) | |
| (hs : mixerSpreadTest (R.signedWeights h.1.1.1) (R.orderedMixers h.1.1.1) n) | |
| (hscale : Real.logb 2 (inputScale R.input : ℝ) ≤ (n : ℝ) / 1000000000) | |
| (D : Finset (Fin n)) : | |
| (9 / 10 : ℝ) ≤ (R.extraPrimeLaw h.1.1.1 (by omega)).probability (fun q => | |
| (R.chosenParentBucket h r (by omega) hlog 0 ((prefixModulus h.2.val 0 : ℤ) * q.val)).card ≤ | |
| compressionDictionaryCapacity n ∧ | |
| ((R.chosenParentBucket h r (by omega) hlog 1 ((prefixModulus h.2.val 0 : ℤ) * q.val)).filter | |
| (fun e => e.mask = D)).card ≤ compressionDictionaryCapacity n) := by | |
| classical | |
| let P : ℕ → Prop := fun q => | |
| (R.chosenParentBucket h r (by omega) hlog 0 ((prefixModulus h.2.val 0 : ℤ) * q)).card ≤ | |
| compressionDictionaryCapacity n ∧ | |
| ((R.chosenParentBucket h r (by omega) hlog 1 ((prefixModulus h.2.val 0 : ℤ) * q)).filter | |
| (fun e => e.mask = D)).card ≤ compressionDictionaryCapacity n | |
| change (9 / 10 : ℝ) ≤ (R.extraPrimeLaw h.1.1.1 (by omega)).probability (fun q => P q.val) | |
| rw [R.extraPrimeLaw_probability h.1.1.1 (by omega) P] | |
| exact R.chosenBuckets_probability h r hn hlog hr hgood hs hscale D | |
| end RepresentationSetup | |
| end LowSpaceSubsetSum | |
| end | |
| end OAI | |