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import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityControls
namespace OAI
/-! Joint tensor extraction, compatibility and entropy estimates. -/
noncomputable section
namespace MatrixMultiplication.JointCompatibilityCountProducer
open MatrixMultiplication.Foundation JointTypeCounts JointCompatibilityIncidence
open JointCompatibilityControls JointCompatibilityScaling JointCompatibilityRateLimit Filter
open scoped BigOperators Topology
attribute [local instance] Classical.propDecidable
private theorem exists_pos_le_family {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 A : C → Type*}
[∀ c, Fintype (Shape c)] [∀ c, DecidableEq (Shape c)]
[∀ c, Fintype (A c)] [∀ c, DecidableEq (A c)]
omit [DecidableEq C] [∀ c, DecidableEq (Shape c)] [∀ c, DecidableEq (A c)] in
private theorem exists_group_entropy_window (base : ∀ c, Shape c → ℕ)
(d : ∀ c, Shape c → Prop) (law : ∀ c, Shape c → A c → ℝ)
{ε : ℝ} (hε : 0 < ε) :
∃ δ : ℝ, 0 < δ ∧ ∀ (g : Σ c, Option (Shape c)) (q : A g.1 → ℝ),
(∀ a, |q a - groupCenter base d law g a| ≤ δ) →
finiteEntropy q ≤ groupCap base d law ε g := by
have hex : ∀ g : Σ c, Option (Shape c), ∃ δ : ℝ, 0 < δ ∧
∀ q : A g.1 → ℝ, (∀ a, |q a - groupCenter base d law g a| ≤ δ) →
|finiteEntropy q - finiteEntropy (groupCenter base d law g)| < ε :=
fun g => exists_entropy_window (groupCenter base d law g) hε
choose δ hδ hw using hex
obtain ⟨δ₀, hδ₀, hsmall⟩ := exists_pos_le_family δ hδ
refine ⟨δ₀, hδ₀, ?_⟩
intro g q hq
have hh := (abs_lt.mp (hw g q (fun a => (hq a).trans (hsmall g)))).2
change finiteEntropy q ≤ finiteEntropy (groupCenter base d law g) + ε
linarith
omit [Fintype C] [DecidableEq C] in
private theorem repeated_residual_width_le
{Pos : C → Type*} [∀ c, Fintype (Pos c)] [∀ c, DecidableEq (Pos c)]
(base : ∀ c, Shape c → ℕ) (d : ∀ c, Shape c → Prop)
(t : ℕ) (w : FixedClassWords Pos Shape (fun c u => t * base c u))
(χ η δ : ℝ) (hδ : 0 ≤ δ)
(hnum : ∀ c, η + (designatedBase base d c : ℝ) * χ ≤ δ) (c : C) :
((t : ℝ) * η + (∑ u : {u : Shape c // d c u}, ((t * base c u.val : ℕ) : ℝ)) * χ) /
Fintype.card {i : Pos c // ¬d c ((w c).val i)} ≤ δ := by
have hsum : (∑ u : {u : Shape c // d c u}, ((t * base c u.val : ℕ) : ℝ)) =
(t : ℝ) * designatedBase base d c := by
simp only [Nat.cast_mul, designatedBase, Nat.cast_sum, Finset.mul_sum]
rw [residual_card_repeated, hsum, Nat.cast_mul]
by_cases ht : t = 0
· simpa only [ht, Nat.cast_zero, zero_mul, add_zero, zero_add, zero_div] using hδ
by_cases hr : residualBase base d c = 0
· simpa only [hr, Nat.cast_zero, mul_zero, div_zero] using hδ
have htpos : (0 : ℝ) < t := Nat.cast_pos.mpr (Nat.pos_of_ne_zero ht)
have hrpos : (0 : ℝ) < residualBase base d c := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hr)
have hrone : (1 : ℝ) ≤ residualBase base d c := by exact_mod_cast Nat.one_le_iff_ne_zero.mpr hr
have hres : η + (designatedBase base d c : ℝ) * χ ≤ δ * residualBase base d c :=
(hnum c).trans (by simpa only [mul_one] using mul_le_mul_of_nonneg_left hrone hδ)
apply (div_le_iff₀ (mul_pos htpos hrpos)).mpr
calc
(t : ℝ) * η + ((t : ℝ) * designatedBase base d c) * χ =
(t : ℝ) * (η + (designatedBase base d c : ℝ) * χ) := by ring
_ ≤ (t : ℝ) * (δ * residualBase base d c) :=
mul_le_mul_of_nonneg_left hres htpos.le
_ = δ * ((t : ℝ) * residualBase base d c) := by ring
omit [DecidableEq C] in
theorem exists_uniform_sideEntropyControl_all_positions
(Pair : C → Type*) [∀ c, Fintype (Pair c)] [∀ c, DecidableEq (Pair c)]
(base : ∀ c, Shape c → ℕ) (d : ∀ c, Shape c → Prop)
(law : ∀ c, Shape c → A c → ℝ) (part : ∀ c, Pair c → A c)
(h0 : ∀ c u a, 0 ≤ law c u a) (h1 : ∀ c u a, law c u a ≤ 1)
{ε χMax ηMax : ℝ} (hε : 0 < ε) (hχMax : 0 < χMax) (hηMax : 0 < ηMax) :
∃ χ η : ℝ, 0 < χ ∧ χ ≤ χMax ∧ 0 < η ∧ η ≤ ηMax ∧
∀ (t : ℕ) (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)),
(∀ c a, |((∑ s : {s : Pair c // part c s = a}, pairCounts c s.val : ℕ) : ℝ) -
(t : ℝ) * totalCenterCount base law c a| ≤ (t : ℝ) * η) →
Nonempty (SideEntropyControl Pos Shape Pair A (fun c u => t * base c u)
pairCounts part d law χ (groupCap base d law ε) w) := by
obtain ⟨δ, hδ, hwindow⟩ := exists_group_entropy_window base d law hε
let D : ℕ := ∑ c, designatedBase base d c
have hD : (0 : ℝ) ≤ D := Nat.cast_nonneg _
have hden : 0 < 2 * ((D : ℝ) + 1) := by positivity
let χ₀ : ℝ := δ / (2 * ((D : ℝ) + 1))
have hχ₀ : 0 < χ₀ := div_pos hδ hden
have heq : (2 * ((D : ℝ) + 1)) * χ₀ = δ := mul_div_cancel₀ _ hden.ne'
have hχδ : χ₀ ≤ δ := by nlinarith [mul_nonneg hD hχ₀.le]
have hDχ : (D : ℝ) * χ₀ ≤ δ / 2 := by nlinarith
let χ := min χMax χ₀
let η := min ηMax (δ / 2)
have hχ : 0 < χ := lt_min hχMax hχ₀
have hη : 0 < η := lt_min hηMax (half_pos hδ)
have hχδ' : χ ≤ δ := (min_le_right _ _).trans hχδ
have hnum (c : C) : η + (designatedBase base d c : ℝ) * χ ≤ δ := by
have hc : (designatedBase base d c : ℝ) ≤ D := by
exact_mod_cast (Finset.single_le_sum (fun _ _ => Nat.zero_le _)
(Finset.mem_univ c) : designatedBase base d c ≤ ∑ c, designatedBase base d c)
have hm : (designatedBase base d c : ℝ) * χ ≤ (D : ℝ) * χ₀ :=
mul_le_mul hc (min_le_right _ _) hχ.le hD
have he : η ≤ δ / 2 := min_le_right _ _
linarith
refine ⟨χ, η, hχ, min_le_left _ _, hη, min_le_left _ _, ?_⟩
intro t Pos _ _ pairCounts w haggr
refine ⟨{
residualLaw := residualCenter base d law
totalTarget := fun c a => (t : ℝ) * totalCenterCount base law c a
totalError := fun _ => (t : ℝ) * η
aggregate_window := haggr
mixture_balance := repeated_mixture_balance base d law t w
nonneg := groupCap_nonneg base d law h0 h1 hε.le
designated_entropy := ?_
residual_entropy := ?_ }⟩
· intro c u _ q hq
exact hwindow ⟨c, some u⟩ q (fun a => (hq a).trans hχδ')
· intro c q hq
exact hwindow ⟨c, none⟩ q (fun a => (hq a).trans
(repeated_residual_width_le base d t w χ η δ hδ.le hnum c))
section PairMarginals
variable {Pair : C → Type*}
[∀ c, Fintype (Pair c)] [∀ c, DecidableEq (Pair c)]
def aggregateCoefficient (base : ∀ c, Shape c → ℕ)
(Pair : C → Type*) [∀ c, Fintype (Pair c)] : ℕ :=
∑ c, baseSize base c * Fintype.card (Pair c)
omit [DecidableEq C] [∀ c, Fintype (A c)] in
theorem aggregate_error_of_pairWindow
(base : ∀ c, Shape c → ℕ) (q : ∀ c, Pair c → ℝ)
(law : ∀ c, Shape c → A c → ℝ) (part : ∀ c, Pair c → A c)
(hbalance : ∀ c a, (baseSize base c : ℝ) *
(∑ b : {b : Pair c // part c b = a}, q c b.val) = totalCenterCount base law c a)
(t : ℕ) (ht : 0 < t) (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) {η : ℝ} (hη : 0 ≤ η)
(hp : PairWindow base q Pos pairCounts η) (c : C) (a : A c) :
|((∑ b : {b : Pair c // part c b = a}, pairCounts c b.val : ℕ) : ℝ) -
(t : ℝ) * totalCenterCount base law c a| ≤
(t : ℝ) * (aggregateCoefficient base Pair : ℝ) * η := by
by_cases hz : baseSize base c = 0
· have hsum : (∑ b, pairCounts c b) = 0 := by
rw [← JointCompatibilityEntropyBound.fixedClassWords_size Pos Pair pairCounts z c,
repeated_positions_card Pos base t w c, hz, Nat.mul_zero]
have hcount (b : Pair c) : pairCounts c b = 0 :=
Nat.eq_zero_of_le_zero ((Finset.single_le_sum (fun _ _ => Nat.zero_le _)
(Finset.mem_univ b)).trans_eq hsum)
have hcenter : totalCenterCount base law c a = 0 := by
rw [← hbalance c a, hz, Nat.cast_zero, zero_mul]
simpa only [hcount, Finset.sum_const_zero, Nat.cast_zero, hcenter, mul_zero,
sub_self, abs_zero] using
mul_nonneg (mul_nonneg (Nat.cast_nonneg t)
(Nat.cast_nonneg (aggregateCoefficient base Pair))) hη
have hb : 0 < baseSize base c := Nat.pos_of_ne_zero hz
have hn : (0 : ℝ) < Fintype.card (Pos c) := by
rw [repeated_positions_card Pos base t w c, Nat.cast_mul]
exact mul_pos (Nat.cast_pos.mpr ht) (Nat.cast_pos.mpr hb)
have hcoord (b : Pair c) :
|(pairCounts c b : ℝ) - (t : ℝ) * (baseSize base c : ℝ) * q c b| ≤
(t : ℝ) * (baseSize base c : ℝ) * η := by
have heq : (Fintype.card (Pos c) : ℝ) *
((pairCounts c b : ℝ) / Fintype.card (Pos c) - q c b) =
(pairCounts c b : ℝ) - (Fintype.card (Pos c) : ℝ) * q c b := by
rw [mul_sub, mul_div_cancel₀ _ hn.ne']
have h := calc
|(pairCounts c b : ℝ) - (Fintype.card (Pos c) : ℝ) * q c b| =
|(Fintype.card (Pos c) : ℝ) *
((pairCounts c b : ℝ) / Fintype.card (Pos c) - q c b)| :=
congrArg abs heq.symm
_ = (Fintype.card (Pos c) : ℝ) *
|(pairCounts c b : ℝ) / Fintype.card (Pos c) - q c b| := by
rw [abs_mul, abs_of_pos hn]
_ ≤ (Fintype.card (Pos c) : ℝ) * η :=
mul_le_mul_of_nonneg_left (hp c hb b) hn.le
simpa only [repeated_positions_card Pos base t w c, Nat.cast_mul] using h
have hm := marginal_coordinate_error
(fun b : {b : Pair c // part c b = a} => (pairCounts c b.val : ℝ))
(fun b => (t : ℝ) * (baseSize base c : ℝ) * q c b.val)
((t : ℝ) * (baseSize base c : ℝ) * η) (fun b => hcoord b.val)
have hsum : (∑ b : {b : Pair c // part c b = a},
(t : ℝ) * (baseSize base c : ℝ) * q c b.val) =
(t : ℝ) * totalCenterCount base law c a := by
rw [← Finset.mul_sum, mul_assoc, hbalance c a]
have hm' :
|((∑ b : {b : Pair c // part c b = a}, pairCounts c b.val : ℕ) : ℝ) -
(t : ℝ) * totalCenterCount base law c a| ≤
(Fintype.card {b : Pair c // part c b = a} : ℝ) *
((t : ℝ) * (baseSize base c : ℝ) * η) := by
simpa only [Nat.cast_sum, hsum] using hm
have hKnat : baseSize base c * Fintype.card {b : Pair c // part c b = a} ≤
aggregateCoefficient base Pair :=
(Nat.mul_le_mul_left _ (Fintype.card_subtype_le _)).trans
(Finset.single_le_sum
(f := fun d => baseSize base d * Fintype.card (Pair d))
(fun _ _ => Nat.zero_le _) (Finset.mem_univ c))
have hK : (baseSize base c : ℝ) *
(Fintype.card {b : Pair c // part c b = a} : ℝ) ≤
(aggregateCoefficient base Pair : ℝ) := by exact_mod_cast hKnat
have hmul := mul_le_mul_of_nonneg_left hK (mul_nonneg (Nat.cast_nonneg t) hη)
nlinarith
end PairMarginals
section CountProducer
variable {Pair Left Right : C → Type*}
[∀ c, Fintype (Pair c)] [∀ c, DecidableEq (Pair c)]
[∀ c, Fintype (Left c)] [∀ c, DecidableEq (Left c)]
[∀ c, Fintype (Right c)] [∀ c, DecidableEq (Right c)]
theorem exists_widths_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)))
(hL0 : ∀ c u a, 0 ≤ leftLaw c u a) (hL1 : ∀ c u a, leftLaw c u a ≤ 1)
(hR0 : ∀ c u a, 0 ≤ rightLaw c u a) (hR1 : ∀ c u a, rightLaw c u a ≤ 1)
(hleft : ∀ c a, (baseSize base c : ℝ) *
(∑ b : {b : Pair c // leftPart c b = a}, q c b.val) =
totalCenterCount base leftLaw c a)
(hright : ∀ c a, (baseSize base c : ℝ) *
(∑ b : {b : Pair c // rightPart c b = a}, q c b.val) =
totalCenterCount base rightLaw c a)
{ε χMax ηMax : ℝ} (hε : 0 < ε) (hχMax : 0 < χMax) (hηMax : 0 < ηMax) :
∃ χ η : ℝ, 0 < χ ∧ χ ≤ χMax ∧ 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)
(_ : FixedClassWords Pos Shape (fun c u => t * base c u))
(z : FixedClassWords Pos Pair pairCounts),
PairWindow base q Pos pairCounts η →
(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
(groupCap base designatedLeft leftLaw 0)
(groupCap base designatedRight rightLaw 0) + 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
(groupCap base designatedLeft leftLaw 0)
(groupCap base designatedRight rightLaw 0) + ε) := by
let δ : ℝ := ε / (4 * ((totalBase base : ℝ) + 1))
have hden : 0 < 4 * ((totalBase base : ℝ) + 1) := by positivity
have hδ : 0 < δ := div_pos hε hden
have hδeq : (4 * ((totalBase base : ℝ) + 1)) * δ = ε :=
mul_div_cancel₀ _ hden.ne'
have hδbound : 2 * (totalBase base : ℝ) * δ ≤ ε / 2 := by nlinarith
obtain ⟨χL, ηL, hχL, hχLMax, hηL, _, hCL⟩ :=
exists_uniform_sideEntropyControl_all_positions Pair base designatedLeft leftLaw leftPart
hL0 hL1 (ηMax := 1) hδ hχMax zero_lt_one
obtain ⟨χR, ηR, hχR, hχRMax, hηR, _, hCR⟩ :=
exists_uniform_sideEntropyControl_all_positions Pair base designatedRight rightLaw rightPart
hR0 hR1 (ηMax := 1) hδ hχMax zero_lt_one
let χ := min χL χR
have hχ : 0 < χ := lt_min hχL hχR
have hχMax' : χ ≤ χMax := (min_le_left _ _).trans hχLMax
let K : ℕ := aggregateCoefficient base Pair
have hKden : (0 : ℝ) < (K : ℝ) + 1 := by positivity
let ηUpper : ℝ := min ηMax (min ηL ηR / ((K : ℝ) + 1))
have hηUpper : 0 < ηUpper := lt_min hηMax (div_pos (lt_min hηL hηR) hKden)
obtain ⟨η, hη, hηUpper', hbound⟩ := exists_window_eventually_count_le
base q leftPart rightPart designatedLeft designatedRight leftLaw rightLaw hparts
(groupCap base designatedLeft leftLaw δ) (groupCap base designatedRight rightLaw δ)
(half_pos hε) hηUpper
have hηMax' : η ≤ ηMax := hηUpper'.trans (min_le_left _ _)
have hηscale : (K : ℝ) * η ≤ min ηL ηR := by
have hle : η ≤ min ηL ηR / ((K : ℝ) + 1) :=
hηUpper'.trans (min_le_right _ _)
have hmul := (le_div_iff₀ hKden).mp hle
nlinarith
have hrate :
compatibilityRate base q designatedLeft designatedRight
(groupCap base designatedLeft leftLaw δ) (groupCap base designatedRight rightLaw δ) +
ε / 2 ≤
compatibilityRate base q designatedLeft designatedRight
(groupCap base designatedLeft leftLaw 0) (groupCap base designatedRight rightLaw 0) + ε := by
have heq := compatibilityRate_add_caps base q designatedLeft designatedRight
(groupCap base designatedLeft leftLaw 0) (groupCap base designatedRight rightLaw 0) δ
have hcapL : (fun g => groupCap base designatedLeft leftLaw 0 g + δ) =
groupCap base designatedLeft leftLaw δ := by
funext g
simp only [groupCap, add_zero]
have hcapR : (fun g => groupCap base designatedRight rightLaw 0 g + δ) =
groupCap base designatedRight rightLaw δ := by
funext g
simp only [groupCap, add_zero]
rw [hcapL, hcapR] at heq
linarith only [heq, hδbound]
refine ⟨χ, η, hχ, hχMax', hη, hηMax', ?_⟩
filter_upwards [hbound, eventually_ge_atTop (1 : ℕ)] with t hb ht
intro Pos _ _ pairCounts w z hp
have htpos : 0 < t := lt_of_lt_of_le Nat.zero_lt_one ht
have haggrL : ∀ c a,
|((∑ b : {b : Pair c // leftPart c b = a}, pairCounts c b.val : ℕ) : ℝ) -
(t : ℝ) * totalCenterCount base leftLaw c a| ≤ (t : ℝ) * ηL := by
intro c a
have h := aggregate_error_of_pairWindow base q leftLaw leftPart hleft
t htpos Pos pairCounts w z hη.le hp c a
have hscale := mul_le_mul_of_nonneg_left
(hηscale.trans (min_le_left _ _)) (Nat.cast_nonneg t)
change _ ≤ (t : ℝ) * (K : ℝ) * η at h
nlinarith
have haggrR : ∀ c a,
|((∑ b : {b : Pair c // rightPart c b = a}, pairCounts c b.val : ℕ) : ℝ) -
(t : ℝ) * totalCenterCount base rightLaw c a| ≤ (t : ℝ) * ηR := by
intro c a
have h := aggregate_error_of_pairWindow base q rightLaw rightPart hright
t htpos Pos pairCounts w z hη.le hp c a
have hscale := mul_le_mul_of_nonneg_left
(hηscale.trans (min_le_right _ _)) (Nat.cast_nonneg t)
change _ ≤ (t : ℝ) * (K : ℝ) * η at h
nlinarith
obtain ⟨controlL⟩ := hCL t Pos pairCounts w haggrL
obtain ⟨controlR⟩ := hCR t Pos pairCounts w haggrR
have controlL' := control_mono_width (fun c u => t * base c u) pairCounts
leftPart designatedLeft leftLaw (groupCap base designatedLeft leftLaw δ) w
controlL (show χ ≤ χL from min_le_left _ _)
have controlR' := control_mono_width (fun c u => t * base c u) pairCounts
rightPart designatedRight rightLaw (groupCap base designatedRight rightLaw δ) w
controlR (show χ ≤ χR from min_le_right _ _)
have hcount := hb Pos pairCounts w z χ hp controlL' controlR'
refine ⟨hcount.1.trans (Real.exp_le_exp.mpr
(mul_le_mul_of_nonneg_left hrate (Nat.cast_nonneg t))), ?_⟩
intro hpos
exact (hcount.2 hpos).trans hrate
end CountProducer
end MatrixMultiplication.JointCompatibilityCountProducer
end
end OAI
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