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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
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