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import OAI.LinearAlgebra.MatrixFields.Construction.JointCompatibilityScaling
namespace OAI
namespace MatrixAllFields
open scoped BigOperators Topology Polynomial
section
noncomputable section
namespace MatrixMultiplication.JointCompatibilityRateLimit
open MatrixMultiplication.Foundation JointTypeCounts JointCompatibilityIncidence
open JointCompatibilityEntropyBound JointCompatibilityScaling Filter
open scoped BigOperators Topology
attribute [local instance] Classical.propDecidable
private theorem exists_pos_le_family_inline_MatrixMultiplication_JointCompatibilityRateLimit {K : Type*} [Fintype K]
(δ : K → ℝ) (hδ : ∀ k, 0 < δ k) :
∃ ε : ℝ, 0 < ε ∧ ∀ k, ε ≤ δ k := by
classical
have aux : ∀ s : Finset K, ∃ ε : ℝ, 0 < ε ∧ ∀ k ∈ s, ε ≤ δ k := by
intro s
induction s using Finset.induction_on with
| empty => exact ⟨1, zero_lt_one, by simp⟩
| @insert k s hk ih =>
obtain ⟨ε, hε, hb⟩ := ih
refine ⟨min (δ k) ε, lt_min (hδ k) hε, ?_⟩
intro j hj
rcases Finset.mem_insert.mp hj with rfl | hj
· exact min_le_left _ _
· exact (min_le_right _ _).trans (hb j hj)
obtain ⟨ε, hε, hb⟩ := aux Finset.univ
exact ⟨ε, hε, fun k => hb k (Finset.mem_univ k)⟩
variable {C : Type*} [Fintype C] [DecidableEq C]
{Shape Pair Left Right : C → Type*}
[∀ c, Fintype (Shape c)] [∀ c, DecidableEq (Shape c)]
[∀ c, Fintype (Pair c)] [∀ c, DecidableEq (Pair c)]
[∀ c, Fintype (Left c)] [∀ c, DecidableEq (Left c)]
[∀ c, Fintype (Right c)] [∀ c, DecidableEq (Right c)]
def totalBase (base : ∀ c, Shape c → ℕ) : ℕ := ∑ c, baseSize base c
def pairReferenceEntropy (base : ∀ c, Shape c → ℕ) (q : ∀ c, Pair c → ℝ) : ℝ :=
∑ c, (baseSize base c : ℝ) * finiteEntropy (q c)
def projectedReferenceEntropy (base : ∀ c, Shape c → ℕ)
(d : ∀ c, Shape c → Prop) (H : (Σ c, Option (Shape c)) → ℝ) : ℝ :=
∑ g : Σ c, Option (Shape c), (baseGroupSize base d g : ℝ) * H g
def compatibilityRate (base : ∀ c, Shape c → ℕ) (q : ∀ c, Pair c → ℝ)
(designatedLeft designatedRight : ∀ c, Shape c → Prop)
(HL HR : (Σ c, Option (Shape c)) → ℝ) : ℝ :=
baseEntropy base - pairReferenceEntropy base q +
projectedReferenceEntropy base designatedLeft HL +
projectedReferenceEntropy base designatedRight HR
omit [DecidableEq C] in
theorem projectedReferenceEntropy_add_const (base : ∀ c, Shape c → ℕ)
(d : ∀ c, Shape c → Prop) (H : (Σ c, Option (Shape c)) → ℝ) (δ : ℝ) :
projectedReferenceEntropy base d (fun g => H g + δ) =
projectedReferenceEntropy base d H + (totalBase base : ℝ) * δ := by
simp only [projectedReferenceEntropy, mul_add, Finset.sum_add_distrib,
← Finset.sum_mul, ← Nat.cast_sum, sum_baseGroupSize, totalBase, baseTotal]
omit [DecidableEq C] [∀ c, DecidableEq (Pair c)] in
theorem compatibilityRate_add_caps (base : ∀ c, Shape c → ℕ)
(q : ∀ c, Pair c → ℝ) (designatedLeft designatedRight : ∀ c, Shape c → Prop)
(HL HR : (Σ c, Option (Shape c)) → ℝ) (δ : ℝ) :
compatibilityRate base q designatedLeft designatedRight
(fun g => HL g + δ) (fun g => HR g + δ) =
compatibilityRate base q designatedLeft designatedRight HL HR +
2 * (totalBase base : ℝ) * δ := by
simp only [compatibilityRate, projectedReferenceEntropy_add_const]
ring
def PairWindow (base : ∀ c, Shape c → ℕ) (q : ∀ c, Pair c → ℝ)
(Pos : C → Type*) [∀ c, Fintype (Pos c)]
(pairCounts : ∀ c, Pair c → ℕ) (η : ℝ) : Prop :=
∀ c, 0 < baseSize base c → ∀ a,
|(pairCounts c a : ℝ) / Fintype.card (Pos c) - q c a| ≤ η
omit [DecidableEq C] [∀ c, DecidableEq (Pair c)] in
theorem exists_pair_entropy_window (q : ∀ c, Pair c → ℝ)
{κ ηMax : ℝ} (hκ : 0 < κ) (hηMax : 0 < ηMax) :
∃ η : ℝ, 0 < η ∧ η ≤ ηMax ∧ ∀ c (r : Pair c → ℝ),
(∀ a, |r a - q c a| ≤ η) → finiteEntropy (q c) - κ ≤ finiteEntropy r := by
have hex : ∀ c, ∃ δ : ℝ, 0 < δ ∧ ∀ r : Pair c → ℝ,
(∀ a, |r a - q c a| ≤ δ) → |finiteEntropy r - finiteEntropy (q c)| < κ :=
fun c => exists_entropy_window (q c) hκ
choose δ hδ hw using hex
obtain ⟨η₀, hη₀, hsmall⟩ := exists_pos_le_family_inline_MatrixMultiplication_JointCompatibilityRateLimit δ hδ
refine ⟨min ηMax η₀, lt_min hηMax hη₀, min_le_left _ _, ?_⟩
intro c r hr
have h := (abs_lt.mp (hw c r (fun a =>
(hr a).trans ((min_le_right _ _).trans (hsmall c))))).1
linarith
omit [Fintype C] [DecidableEq C] in
theorem repeated_positions_card
(Pos : C → Type*) [∀ c, Fintype (Pos c)] [∀ c, DecidableEq (Pos c)]
(base : ∀ c, Shape c → ℕ) (t : ℕ)
(w : FixedClassWords Pos Shape (fun c u => t * base c u)) (c : C) :
Fintype.card (Pos c) = t * baseSize base c := by
rw [fixedClassWords_size Pos Shape _ w c]
simp only [baseSize, Finset.mul_sum]
omit [DecidableEq C] in
theorem repeated_positions_card_le
(Pos : C → Type*) [∀ c, Fintype (Pos c)] [∀ c, DecidableEq (Pos c)]
(base : ∀ c, Shape c → ℕ) (t : ℕ)
(w : FixedClassWords Pos Shape (fun c u => t * base c u)) (c : C) :
Fintype.card (Pos c) ≤ t * totalBase base := by
rw [repeated_positions_card Pos base t w c]
exact Nat.mul_le_mul_left _
(Finset.single_le_sum (fun _ _ => Nat.zero_le _) (Finset.mem_univ c))
omit [DecidableEq C] [∀ c, DecidableEq (Pair c)] in
theorem pair_classEntropy_lower
(Pos : C → Type*) [∀ c, Fintype (Pos c)] [∀ c, DecidableEq (Pos c)]
(base : ∀ c, Shape c → ℕ) (q : ∀ c, Pair c → ℝ) (t : ℕ)
(pairCounts : ∀ c, Pair c → ℕ)
(w : FixedClassWords Pos Shape (fun c u => t * base c u))
{η κ : ℝ}
(hwindow : ∀ c (r : Pair c → ℝ), (∀ a, |r a - q c a| ≤ η) →
finiteEntropy (q c) - κ ≤ finiteEntropy r)
(hpair : PairWindow base q Pos pairCounts η) :
(t : ℝ) * (pairReferenceEntropy base q - (totalBase base : ℝ) * κ) ≤
classEntropy Pos Pair pairCounts := by
have hc (c : C) :
(t : ℝ) * (baseSize base c : ℝ) * (finiteEntropy (q c) - κ) ≤
(Fintype.card (Pos c) : ℝ) *
finiteEntropy (fun a => (pairCounts c a : ℝ) / Fintype.card (Pos c)) := by
by_cases hz : baseSize base c = 0
· simp only [repeated_positions_card Pos base t w c, hz, mul_zero,
Nat.cast_zero, zero_mul, le_refl]
· have he := hwindow c (fun a => (pairCounts c a : ℝ) / Fintype.card (Pos c))
(hpair c (Nat.pos_of_ne_zero hz))
have hh := mul_le_mul_of_nonneg_left he
(mul_nonneg (Nat.cast_nonneg t) (Nat.cast_nonneg (baseSize base c)))
simpa only [repeated_positions_card Pos base t w c, Nat.cast_mul] using hh
calc
(t : ℝ) * (pairReferenceEntropy base q - (totalBase base : ℝ) * κ) =
∑ c, (t : ℝ) * (baseSize base c : ℝ) * (finiteEntropy (q c) - κ) := by
simp only [pairReferenceEntropy, totalBase, Nat.cast_sum, Finset.mul_sum,
Finset.sum_mul, mul_sub, Finset.sum_sub_distrib, mul_assoc]
_ ≤ classEntropy Pos Pair pairCounts := Finset.sum_le_sum fun c _ => hc c
def errorCoefficient (Shape Pair Left Right : C → Type*)
[∀ c, Fintype (Shape c)] [∀ c, Fintype (Pair c)]
[∀ c, Fintype (Left c)] [∀ c, Fintype (Right c)] : ℝ :=
classErrorCoefficient Shape + classErrorCoefficient Pair +
groupErrorCoefficient Shape Left + groupErrorCoefficient Shape Right
omit [DecidableEq C] in
theorem compatibilityExponent_le_rate_add
(Pos : C → Type*) [∀ c, Fintype (Pos c)] [∀ c, DecidableEq (Pos c)]
(base : ∀ c, Shape c → ℕ) (q : ∀ c, Pair c → ℝ)
(designatedLeft designatedRight : ∀ c, Shape c → Prop)
(HL HR : (Σ c, Option (Shape c)) → ℝ)
(t : ℕ) (ht : 0 < t) (pairCounts : ∀ c, Pair c → ℕ)
(w : FixedClassWords Pos Shape (fun c u => t * base c u))
{η κ : ℝ}
(hwindow : ∀ c (r : Pair c → ℝ), (∀ a, |r a - q c a| ≤ η) →
finiteEntropy (q c) - κ ≤ finiteEntropy r)
(hpair : PairWindow base q Pos pairCounts η) :
compatibilityExponent Pos Shape Pair Left Right (fun c u => t * base c u)
pairCounts designatedLeft designatedRight HL HR w ≤
(t : ℝ) * (compatibilityRate base q designatedLeft designatedRight HL HR +
(totalBase base : ℝ) * κ) +
errorCoefficient Shape Pair Left Right * typeErrorFactor (totalBase base) t := by
have hsize := repeated_positions_card Pos base t w
have hsizele := repeated_positions_card_le Pos base t w
have hs := classEntropy_repeated Pos base t ht hsize
have hl := groupedEntropy_repeated Pos base designatedLeft t HL w
have hr := groupedEntropy_repeated Pos base designatedRight t HR w
have hp := pair_classEntropy_lower Pos base q t pairCounts w hwindow hpair
have e₁ := classTypeError_le Pos Shape (totalBase base) t hsizele
have e₂ := classTypeError_le Pos Pair (totalBase base) t hsizele
have e₃ := groupedTypeError_le Pos Left (fun c u => t * base c u)
designatedLeft w (totalBase base) t hsizele
have e₄ := groupedTypeError_le Pos Right (fun c u => t * base c u)
designatedRight w (totalBase base) t hsizele
unfold compatibilityExponent
rw [hs, hl, hr]
dsimp only [compatibilityRate, projectedReferenceEntropy, baseGroupedEntropy,
errorCoefficient] at *
nlinarith
omit [DecidableEq C] in
theorem exists_window_eventually_exponent_le
(base : ∀ c, Shape c → ℕ) (q : ∀ c, Pair c → ℝ)
(designatedLeft designatedRight : ∀ c, Shape c → Prop)
(HL HR : (Σ c, Option (Shape c)) → ℝ)
{ε ηMax : ℝ} (hε : 0 < ε) (hηMax : 0 < ηMax) :
∃ η : ℝ, 0 < η ∧ η ≤ ηMax ∧
∀ᶠ t : ℕ in atTop,
∀ (Pos : C → Type*) [∀ c, Fintype (Pos c)] [∀ c, DecidableEq (Pos c)]
(pairCounts : ∀ c, Pair c → ℕ)
(w : FixedClassWords Pos Shape (fun c u => t * base c u)),
PairWindow base q Pos pairCounts η →
compatibilityExponent Pos Shape Pair Left Right (fun c u => t * base c u)
pairCounts designatedLeft designatedRight HL HR w ≤
(t : ℝ) * (compatibilityRate base q designatedLeft designatedRight HL HR + ε) := by
let κ : ℝ := ε / (2 * ((totalBase base : ℝ) + 1))
have hden : 0 < 2 * ((totalBase base : ℝ) + 1) := by positivity
have hκ : 0 < κ := div_pos hε hden
have hκeq : (2 * ((totalBase base : ℝ) + 1)) * κ = ε :=
mul_div_cancel₀ _ hden.ne'
have hκbound : (totalBase base : ℝ) * κ ≤ ε / 2 := by
nlinarith [(Nat.cast_nonneg (totalBase base) : (0 : ℝ) ≤ totalBase base)]
obtain ⟨η, hη, hηMax', hwindow⟩ := exists_pair_entropy_window q hκ hηMax
refine ⟨η, hη, hηMax', ?_⟩
have herr := tendsto_mul_typeErrorFactor_div
(errorCoefficient Shape Pair Left Right) (totalBase base)
filter_upwards [(tendsto_order.1 herr).2 (ε / 2) (half_pos hε),
eventually_ge_atTop (1 : ℕ)] with t he ht
intro Pos _ _ pairCounts w hp
have htpos : 0 < t := lt_of_lt_of_le Nat.zero_lt_one ht
have htr : (0 : ℝ) < t := Nat.cast_pos.mpr htpos
have he' := (div_lt_iff₀ htr).mp he
have hκ' := mul_le_mul_of_nonneg_left hκbound htr.le
have hb := compatibilityExponent_le_rate_add (Left := Left) (Right := Right)
Pos base q designatedLeft designatedRight HL HR t htpos pairCounts w hwindow hp
nlinarith
theorem exists_window_eventually_count_le
(base : ∀ c, Shape c → ℕ) (q : ∀ c, Pair c → ℝ)
(leftPart : ∀ c, Pair c → Left c) (rightPart : ∀ c, Pair c → Right c)
(designatedLeft designatedRight : ∀ c, Shape c → Prop)
(leftLaw : ∀ c, Shape c → Left c → ℝ)
(rightLaw : ∀ c, Shape c → Right c → ℝ)
(hparts : ∀ c, Function.Injective (fun a : Pair c => (leftPart c a, rightPart c a)))
(HL HR : (Σ c, Option (Shape c)) → ℝ)
{ε ηMax : ℝ} (hε : 0 < ε) (hηMax : 0 < ηMax) :
∃ η : ℝ, 0 < η ∧ η ≤ ηMax ∧
∀ᶠ t : ℕ in atTop,
∀ (Pos : C → Type*) [∀ c, Fintype (Pos c)] [∀ c, DecidableEq (Pos c)]
(pairCounts : ∀ c, Pair c → ℕ)
(w : FixedClassWords Pos Shape (fun c u => t * base c u))
(z : FixedClassWords Pos Pair pairCounts) (χ : ℝ),
PairWindow base q Pos pairCounts η →
SideEntropyControl Pos Shape Pair Left (fun c u => t * base c u) pairCounts
leftPart designatedLeft leftLaw χ HL w →
SideEntropyControl Pos Shape Pair Right (fun c u => t * base c u) pairCounts
rightPart designatedRight rightLaw χ HR w →
(compatibleCandidateCount Pos Shape Pair Left Right (fun c u => t * base c u)
pairCounts leftPart rightPart designatedLeft designatedRight leftLaw rightLaw χ z : ℝ) ≤
Real.exp ((t : ℝ) *
(compatibilityRate base q designatedLeft designatedRight HL HR + ε)) ∧
(0 < compatibleCandidateCount Pos Shape Pair Left Right (fun c u => t * base c u)
pairCounts leftPart rightPart designatedLeft designatedRight leftLaw rightLaw χ z →
Real.log (compatibleCandidateCount Pos Shape Pair Left Right (fun c u => t * base c u)
pairCounts leftPart rightPart designatedLeft designatedRight leftLaw rightLaw χ z : ℝ) /
(t : ℝ) ≤ compatibilityRate base q designatedLeft designatedRight HL HR + ε) := by
obtain ⟨η, hη, hηMax', hbound⟩ := exists_window_eventually_exponent_le
(Left := Left) (Right := Right) base q designatedLeft designatedRight HL HR hε hηMax
refine ⟨η, hη, hηMax', ?_⟩
filter_upwards [hbound, eventually_ge_atTop (1 : ℕ)] with t hb ht
intro Pos _ _ pairCounts w z χ hp hL hR
have he := hb Pos pairCounts w hp
have hc := compatibleCandidateCount_le_exp Pos Shape Pair Left Right
(fun c u => t * base c u) pairCounts leftPart rightPart
designatedLeft designatedRight leftLaw rightLaw χ w z hparts HL HR hL hR
refine ⟨hc.trans (Real.exp_le_exp.mpr he), ?_⟩
intro hpos
have hl := log_compatibleCandidateCount_le Pos Shape Pair Left Right
(fun c u => t * base c u) pairCounts leftPart rightPart
designatedLeft designatedRight leftLaw rightLaw χ w z hparts HL HR hL hR hpos
have htr : (0 : ℝ) < t :=
Nat.cast_pos.mpr (lt_of_lt_of_le Nat.zero_lt_one ht)
exact (div_le_iff₀ htr).mpr (by simpa only [mul_comm] using hl.trans he)
end MatrixMultiplication.JointCompatibilityRateLimit
end
end
end MatrixAllFields
end OAI
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