Datasets:

roseking
/

openAI-math-dataset

import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityRateLimit
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

	import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.CompatibilityRateLimit
	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