Datasets:

roseking
/

openAI-math-dataset

	import Mathlib
	import OAI.Combinatorics.KServer.Minimization
	import OAI.Combinatorics.KServer.Simplex

	namespace OAI

	noncomputable section
	open scoped BigOperators Topology
	open Finset Set Filter MeasureTheory

	/-! The exact source primitives installed in the actual affine minimizer
	recursion. The coefficients use raw core flags; fine caps are deliberately
	absent. The zero-potential cases do not evaluate the marked denominator. -/
	namespace KServer.SimplexFamilies
	open scoped BigOperators
	open Finset Set MeasureTheory
	open KServer.SimplexAnalytic KServer.AdaptiveMinimization
	variable {J R : Type*} [Fintype J] [DecidableEq J] [Fintype R]

	structure Data (J R : Type*) [Fintype J] where
	active : Finset J
	flags : J → R → Bool
	F : J → ℝ → ℝ
	eta : J → ℝ
	base : J → ℝ
	zeroPotential : Bool
	measurable : ∀ i, Measurable (F i)
	positive : ∀ i y, 0 ≤ y → 0 < F i y
	integrable : ∀ i, IntervalIntegrable (singularKernel (F i)) volume 0 1
	base_mem : ∀ i, base i ∈ Set.Icc 0 1

	omit [DecidableEq J] [Fintype R] in
	def K (d : Data J R) (a : J → ℝ) (i : J) : ℝ := ∫ s in d.base i..a i, kernel (d.F i) s
	omit [DecidableEq J] [Fintype R] in
	def Z (d : Data J R) (a : J → ℝ) (i : J) : ℝ := ∫ s in d.base i..a i, singularKernel (d.F i) s

	def core (d : Data J R) (p : J × R → ℝ) (i : J) : ℝ := ∑ r, if d.flags i r then p (i,r) else 0

	def potential (C ell : ℝ) (d : Data J R) (B a : J → ℝ) : ℝ :=
	if d.zeroPotential then 0 else ∑ i, primitive (d.F i) C ell (d.eta i) (∑ j, B j) (B i) (d.base i) (a i)

	def coefficient (C ell : ℝ) (d : Data J R) (a : J → ℝ) (ir : J × R) : ℝ :=
	if d.zeroPotential then 0 else if d.flags ir.1 ir.2 then
	Cell((∑ i, K d a i)-(1+d.eta ir.1)*Z d a ir.1) else 0

	omit [DecidableEq J] [Fintype R] in
	lemma K_continuous (d : Data J R) (i : J) :
	ContinuousOn (fun a => K d a i) (ChangingDomains.simplex d.active) := by
	have hi := kernel_intervalIntegrable (d.measurable i) (d.positive i) (d.integrable i)
	have he : Set.uIcc (0:ℝ) 1=Set.Icc 0 1 := uIcc_of_le (by norm_num)
	have hc := intervalIntegral.continuousOn_primitive_interval' hi (he.symm ▸ d.base_mem i)
	rw [he] at hc
	exact hc.comp (continuous_apply i).continuousOn (fun a ha => ChangingDomains.simplex_coordinate ha i)

	omit [DecidableEq J] [Fintype R] in
	lemma Z_continuous (d : Data J R) (i : J) :
	ContinuousOn (fun a => Z d a i) (ChangingDomains.simplex d.active) := by
	have he : Set.uIcc (0:ℝ) 1=Set.Icc 0 1 := uIcc_of_le (by norm_num)
	have hc := intervalIntegral.continuousOn_primitive_interval' (d.integrable i) (he.symm ▸ d.base_mem i)
	rw [he] at hc
	exact hc.comp (continuous_apply i).continuousOn (fun a ha => ChangingDomains.simplex_coordinate ha i)

	omit [DecidableEq J] [Fintype R] in
	lemma coefficient_continuous (C ell : ℝ) (d : Data J R) (ir : J × R) :
	ContinuousOn (fun a => coefficient C ell d a ir) (ChangingDomains.simplex d.active) := by
	unfold coefficient
	split_ifs
	· exact continuousOn_const
	· exact ((continuousOn_finsetSum _ fun i _ => K_continuous d i).sub
	((Z_continuous d ir.1).const_mul _)).const_mul _
	· exact continuousOn_const

	def family (C ell : ℝ) : Family (Data J R) (J × R) J where
	domain d := ChangingDomains.simplex d.active
	compact d := ChangingDomains.simplex_compact d.active
	seed d := ⟨ChangingDomains.canonical d.active,ChangingDomains.canonical_mem d.active⟩
	offset _ _ := 0
	coefficient := coefficient C ell
	offset_continuous _ := continuousOn_const
	coefficient_continuous := coefficient_continuous C ell

	/-- Algebraic equality with the literal sum of the source's improper
	integrals. In particular the family does not substitute another potential. -/
	lemma value_eq (C ell : ℝ) (d : Data J R) (p : J × R → ℝ) (a : J → ℝ)
	(ha : a ∈ ChangingDomains.simplex d.active) :
	(family C ell).value d p a=potential C ell d (core d p) a := by
	by_cases hz : d.zeroPotential=true
	· simp [Family.value,family,coefficient,potential,hz]
	· have hi : ∀ i, primitive (d.F i) C ell (d.eta i) (∑ j, core d p j) (core d p i) (d.base i) (a i)=
	Cell((∑ j, core d p j)K d a i-(1+d.eta i)core d p i*Z d a i) := by
	intro i
	exact primitive_affine (d.measurable i) (d.positive i) (d.integrable i) C ell _ _ _
	(d.base_mem i) (ChangingDomains.simplex_coordinate ha i)
	simp only [Family.value,family,coefficient,potential,hz,Bool.false_eq_true,ite_false,zero_add]
	rw [Fintype.sum_prod_type]
	have hr (i : J) : (∑ r, (if d.flags i r=true then Cell((∑ j, K d a j)-(1+d.eta i)Z d a i) else 0)p (i,r))=
	Cell((∑ j, K d a j)-(1+d.eta i)Z d a i)core d p i := by
	rw [core,mul_sum]
	exact sum_congr rfl fun r _ => by split_ifs <;> simp_all
	simp_rw [hr,hi]
	have heL (i : J) : Cell((∑ j, K d a j)-(1+d.eta i)Z d a i)core d p i =
	(Cell(∑ j, K d a j))core d p i-Cell((1+d.eta i)core d p i*Z d a i) := by ring
	have heR (i : J) : Cell((∑ j, core d p j)K d a i-(1+d.eta i)core d p i*Z d a i) =
	(Cell(∑ j, core d p j))K d a i-Cell((1+d.eta i)core d p i*Z d a i) := by ring
	simp_rw [heL,heR]
	simp only [sum_sub_distrib,←mul_sum]
	ring

	/-- Uniform endpoint bound from the same dimension-free rank slopes; the
	finite number of coordinates may occur only in this additive endpoint. -/
	lemma value_bound (C ell : ℝ) (d : Data J R) (p : J × R → ℝ) (a : J → ℝ) {L : ℝ}
	(hc : ∀ ir, \|coefficient C ell d a ir\| ≤ L) (hp : ∀ ir, p ir ∈ Set.Icc 0 1) :
	\|(family C ell).value d p a\| ≤ L*Fintype.card (J × R) := by
	change \|0+∑ ir, coefficient C ell d a ir*p ir\| ≤ _
	rw [zero_add]
	calc _ ≤ ∑ ir, \|coefficient C ell d a ir*p ir\| := abs_sum_le_sum_abs _ _
	_ ≤ ∑ _ir : J × R, L := sum_le_sum fun ir _ => by
	rw [abs_mul,abs_of_nonneg (hp ir).1]
	exact (mul_le_mul_of_nonneg_left (hp ir).2 (abs_nonneg _)).trans (by simpa using hc ir)
	_ = _ := by simp [mul_comm]

	def regular (A : Finset J) (flags : J → R → Bool) (h eta : J → ℝ)
	(hh : ∀ i, 1 ≤ h i) : Data J R where
	active := A
	flags := flags
	F i := regularF (h i)
	eta := eta
	base _ := 0
	zeroPotential := false
	measurable i := by unfold regularF; fun_prop
	positive i y hy := (show 0 < h i by linarith [hh i]).trans_le (regularF_lower (by linarith [hh i]) hy)
	integrable i := regular_singular_intervalIntegrable (by linarith [hh i])
	base_mem _ := ⟨le_rfl,by norm_num⟩

	omit [DecidableEq J] [Fintype R] in
	lemma regular_coefficient_bound (A : Finset J) (flags : J → R → Bool) (h eta : J → ℝ)
	(hh : ∀ i, 1 ≤ h i) (heta : ∀ i, eta i ∈ Set.Icc 0 1)
	{C ell : ℝ} (hC : 0 ≤ C) (hell : 0 ≤ ell) {a : J → ℝ}
	(ha : a ∈ ChangingDomains.simplex A) (ir : J × R) :
	\|coefficient C ell (regular A flags h eta hh) a ir\| ≤ 3Cell := by
	have hai i := ChangingDomains.simplex_coordinate ha i
	have hb i := regular_coefficients (hh i) (hai i)
	have hs : (∑ i, K (regular A flags h eta hh) a i) ∈ Set.Icc 0 1 := by
	constructor
	· exact sum_nonneg fun i _ => (hb i).1.1
	· have hm := sum_le_sum fun i (_ : i∈Finset.univ) => (hb i).1.2
	change (∑ i, K (regular A flags h eta hh) a i) ≤ ∑ i, a i at hm
	rw [ha.2.1] at hm
	split_ifs at hm <;> linarith
	change \|if flags ir.1 ir.2=true then Cell((∑ i, K (regular A flags h eta hh) a i)-(1+eta ir.1)*Z (regular A flags h eta hh) a ir.1) else 0\| ≤ _
	split_ifs with hf
	· rw [abs_mul,abs_of_nonneg (mul_nonneg hC hell)]
	have hj : Z (regular A flags h eta hh) a ir.1 ∈ Set.Icc 0 1 := (hb ir.1).2
	have hprod := mul_le_mul (show 1+eta ir.1 ≤ 2 by linarith [(heta ir.1).2]) hj.2 hj.1 (by norm_num : (0:ℝ)≤2)
	have hprodn := mul_nonneg (show 0 ≤ 1+eta ir.1 by linarith [(heta ir.1).1]) hj.1
	have hab : \|(∑ i, K (regular A flags h eta hh) a i)-(1+eta ir.1)*Z (regular A flags h eta hh) a ir.1\| ≤ 3 := by
	rw [abs_le]
	constructor <;> linarith [hs.1,hs.2]
	nlinarith [mul_le_mul_of_nonneg_left hab (mul_nonneg hC hell)]
	· simp only [abs_zero]; positivity


	/-- The marked base point is exactly one; no value with `u=0` is accepted. -/
	def marked (A : Finset J) (flags : J → R → Bool) (o : J) (h eta : J → ℝ)
	(u : ℝ) (hu : 0 < u) (hu1 : u ≤ 1) (hh : ∀ i ≠ o, 1 ≤ h i) : Data J R where
	active := A
	flags := flags
	F i := if i=o then markedF u (1+Real.log (1/u)) else regularF (h i)
	eta := eta
	base i := if i=o then 1 else 0
	zeroPotential := false
	measurable i := by
	by_cases hi : i=o
	· simpa only [ite_eq_left hi] using (show Measurable (markedF u (1+Real.log (1/u))) by unfold markedF; fun_prop)
	· simpa only [ite_eq_right hi] using (show Measurable (regularF (h i)) by unfold regularF; fun_prop)
	positive i y hy := by
	split_ifs with hi
	· apply marked_positive hu _ y hy
	have := log_inverse_nonneg ⟨hu,hu1⟩
	linarith
	· exact (show 0 < h i by linarith [hh i hi]).trans_le
	(regularF_lower (by linarith [hh i hi]) hy)
	integrable i := by
	split_ifs with hi
	· apply marked_singular_intervalIntegrable hu
	have := log_inverse_nonneg ⟨hu,hu1⟩
	linarith
	· exact regular_singular_intervalIntegrable (by linarith [hh i hi])
	base_mem i := by split_ifs <;> constructor <;> norm_num

	omit [Fintype R] in
	lemma marked_coefficient_bound (A : Finset J) (flags : J → R → Bool) (o : J) (h eta : J → ℝ)
	{u C ell : ℝ} (hu : 0 < u) (hu1 : u ≤ 1) (hh : ∀ i ≠ o, 1 ≤ h i)
	(heta : ∀ i, eta i ∈ Set.Icc 0 1) (hC : 0 ≤ C) (hell : 0 ≤ ell) {a : J → ℝ}
	(ha : a ∈ ChangingDomains.simplex A) (ir : J × R) :
	\|coefficient C ell (marked A flags o h eta u hu hu1 hh) a ir\| ≤ 13Cell := by
	let d := marked A flags o h eta u hu hu1 hh
	have hai i := ChangingDomains.simplex_coordinate ha i
	have hK (i : J) : \|K d a i\| ≤ a i+if i=o then 4 else 0 := by
	by_cases hi : i=o
	· subst i
	simp only [K,d,marked,ite_true]
	rw [intervalIntegral.integral_symm,abs_neg,
	abs_of_nonneg (marked_coefficients hu hu1 rfl (hai o)).1.1]
	linarith [(marked_coefficients hu hu1 rfl (hai o)).1.2,ha.1 o]
	· simp only [K,d,marked,ite_eq_right hi,add_zero]
	rw [abs_of_nonneg (regular_coefficients (hh i hi) (hai i)).1.1]
	exact (regular_coefficients (hh i hi) (hai i)).1.2
	have hs : \|∑ i, K d a i\| ≤ 5 := by
	apply (abs_sum_le_sum_abs _ _).trans
	have ht := sum_le_sum fun i (_ : i∈Finset.univ) => hK i
	classical
	have he : (∑ i : J, if i=o then (4:ℝ) else 0)=4 := by simp
	rw [sum_add_distrib,he,ha.2.1] at ht
	split_ifs at ht <;> linarith
	have hZ (i : J) : \|Z d a i\| ≤ 4 := by
	by_cases hi : i=o
	· subst i
	simp only [Z,d,marked,ite_true]
	rw [intervalIntegral.integral_symm,abs_neg,
	abs_of_nonneg (marked_coefficients hu hu1 rfl (hai o)).2.1]
	exact (marked_coefficients hu hu1 rfl (hai o)).2.2
	· simp only [Z,d,marked,ite_eq_right hi]
	rw [abs_of_nonneg (regular_coefficients (hh i hi) (hai i)).2.1]
	linarith [(regular_coefficients (hh i hi) (hai i)).2.2]
	change \|if flags ir.1 ir.2=true then Cell((∑ i, K d a i)-(1+eta ir.1)*Z d a ir.1) else 0\| ≤ _
	split_ifs
	· rw [abs_mul,abs_of_nonneg (mul_nonneg hC hell)]
	have hprod : \|(1+eta ir.1)*Z d a ir.1\| ≤ 8 := by
	rw [abs_mul,abs_of_nonneg (show 0 ≤ 1+eta ir.1 by linarith [(heta ir.1).1])]
	have := mul_le_mul (show 1+eta ir.1 ≤ 2 by linarith [(heta ir.1).2])
	(hZ ir.1) (abs_nonneg _) (by norm_num : (0:ℝ)≤2)
	linarith
	have hab : \|(∑ i, K d a i)-(1+eta ir.1)*Z d a ir.1\| ≤ 13 :=
	(abs_sub _ _).trans (by linarith)
	nlinarith [mul_le_mul_of_nonneg_left hab (mul_nonneg hC hell)]
	· simp only [abs_zero]; positivity

	/-- The automatic empty/singleton rule is a genuine zero potential, not a
	singular marked potential evaluated at a zero side parameter. -/
	def automatic (A : Finset J) (flags : J → R → Bool) : Data J R :=
	{regular A flags (fun _ => 1) (fun _ => 0) (fun _ => le_rfl) with zeroPotential := true}

	lemma automatic_value (C ell : ℝ) (A : Finset J) (flags : J → R → Bool)
	(p : J × R → ℝ) (a : J → ℝ) : (family C ell).value (automatic A flags) p a=0 := by
	simp [Family.value,family,coefficient,automatic]

	end KServer.SimplexFamilies


	noncomputable section
	open scoped BigOperators Topology
	open Set Filter
	namespace KServer.SimplexTracker

	variable {ι : Type*} [Fintype ι]

	def simplex : Set (ι → ℝ) := {a \| (∀ i, 0 ≤ a i) ∧ ∑ i, a i = 1}

	def potential (f : ι → ℝ → ℝ) (a : ι → ℝ) : ℝ := ∑ i, f i (a i)

	def movement (a b : ι → ℝ) : ℝ := ∑ i, \|a i - b i\|

	def objective (f : ι → ℝ → ℝ) (S : ℝ) (a₀ a : ι → ℝ) : ℝ :=
	potential f a + S * movement a a₀

	lemma max_coordinate {a : ι → ℝ} (ha : a ∈ simplex) (i : ι) : a i ≤ 1 := by
	have h := Finset.single_le_sum (fun j _ => ha.1 j) (Finset.mem_univ i)
	rwa [ha.2] at h

	/-- Quantitative deficit from replacing alpha_i by alpha_i/(1+eta_i). -/
	lemma weighted_deficit {a h η : ι → ℝ} {ct ell ηmax : ℝ}
	(ha : a ∈ simplex) (hh : ∀ i, 0 ≤ h i) (hc : 0 < ct) (hl : 0 < ell)
	(hη : ∀ i, η i = ct * h i / ell) (hmax : ∀ i, η i ≤ ηmax) :
	∑ i, a i / (1 + η i) ≤
	1 - ct / (ell * (1 + ηmax)) * ∑ i, a i * h i := by
	by_cases he : IsEmpty ι
	· let := he
	have : (0 : ℝ) = 1 := by simpa using ha.2
	norm_num at this
	have : Nonempty ι := not_isEmpty_iff.mp he
	have hη0 : ∀ i, 0 ≤ η i := fun i => by rw [hη i]; exact div_nonneg (mul_nonneg hc.le (hh i)) hl.le
	have hmax0 : 0 ≤ ηmax := (hη0 (Classical.arbitrary ι)).trans (hmax _)
	have hi : ∀ i, a i / (1 + η i) ≤
	a i - ct / (ell * (1 + ηmax)) * (a i * h i) := by
	intro i
	have hden : 0 < 1 + η i := by linarith [hη0 i]
	have hdm : 0 < 1 + ηmax := by linarith
	have hle := mul_le_mul_of_nonneg_left (hmax i) (ha.1 i)
	have hhi : ell * η i = ct * h i := by rw [hη i]; field_simp
	apply (div_le_iff₀ hden).mpr
	apply (mul_le_mul_iff_right₀ (mul_pos hl hdm)).mp
	field_simp
	rw [← hhi]
	nlinarith [mul_nonneg (mul_nonneg (ha.1 i) hl.le)
	(mul_nonneg (hη0 i) (sub_nonneg.mpr (hmax i)))]
	have hs := Finset.sum_le_sum fun i (_ : i ∈ Finset.univ) => hi i
	simpa only [Finset.sum_sub_distrib, ← Finset.mul_sum, ha.2] using hs

	/-- The derivative contradiction in eq:alpha-derivative-contradiction,
	with coefficients general enough to include the marked denominator. -/
	lemma exists_negative_derivative {a h η F B γ : ι → ℝ}
	{ct ell ηmax Cα CF S : ℝ}
	(ha : a ∈ simplex) (hh : ∀ i, 0 ≤ h i) (hc : 0 < ct) (hl : 0 < ell)
	(hη : ∀ i, η i = ct * h i / ell) (hmax : ∀ i, η i ≤ ηmax)
	(hCa : 0 < Cα) (_hCF : 0 ≤ CF) (hS : 0 < S)
	(hB : ∑ i, B i = S) (hF : ∀ i, 0 ≤ F i)
	(hT : 0 < ∑ i, a i * h i)
	(hden : ∑ i, a i * F i ≤ CF * ∑ i, a i * h i)
	(hgap : 2 * CF * (1 + ηmax) < Cα * ct)
	(hidentity : ∀ i, a i * F i / (1 + η i) * γ i =
	Cα * ell * (S * (a i / (1 + η i)) - B i)) :
	∃ i, γ i < -2 * S := by
	have hη0 : ∀ i, 0 ≤ η i := fun i => by rw [hη i]; exact div_nonneg (mul_nonneg hc.le (hh i)) hl.le
	have hdef := weighted_deficit ha hh hc hl hη hmax
	by_contra hall
	push Not at hall
	have hlow : -2 * S * (CF * ∑ i, a i * h i) ≤
	∑ i, a i * F i / (1 + η i) * γ i := by
	calc
	_ ≤ -2 * S * ∑ i, a i * F i :=
	mul_le_mul_of_nonpos_left hden (by linarith)
	_ ≤ ∑ i, a i * F i / (1 + η i) * γ i := by
	rw [Finset.mul_sum]
	apply Finset.sum_le_sum
	intro i _
	have hp : 0 ≤ a i * F i := mul_nonneg (ha.1 i) (hF i)
	have hd : 0 < 1 + η i := by linarith [hη0 i]
	have hdiv : a i * F i / (1 + η i) ≤ a i * F i := by
	apply (div_le_iff₀ hd).mpr
	nlinarith [mul_nonneg hp (hη0 i)]
	have hmul := mul_le_mul_of_nonneg_left (hall i) (div_nonneg hp hd.le)
	nlinarith
	have heq : (∑ i, a i * F i / (1 + η i) * γ i) =
	Cα * ell * (S * ∑ i, a i / (1 + η i) - S) := by
	simp_rw [hidentity]
	rw [← Finset.mul_sum, Finset.sum_sub_distrib, ← Finset.mul_sum, hB]
	rw [heq] at hlow
	have hnon : Nonempty ι := by
	by_contra hn
	have : IsEmpty ι := not_nonempty_iff.mp hn
	simp at hT
	let := hnon
	have hmax0 : 0 ≤ ηmax := (hη0 (Classical.arbitrary ι)).trans (hmax _)
	have hdm : 0 < ell * (1 + ηmax) := by positivity
	have hup := mul_le_mul_of_nonneg_left hdef (show 0 ≤ Cα * ell * S by positivity)
	have hstrict := mul_pos (sub_pos.mpr hgap) (mul_pos hS hT)
	have hid : ell * (1 + ηmax) * (ct / (ell * (1 + ηmax))) = ct := by
	field_simp
	have hlow' := mul_le_mul_of_nonneg_right hlow (show 0 ≤ 1 + ηmax by linarith)
	have hup' := mul_le_mul_of_nonneg_right hup (show 0 ≤ 1 + ηmax by linarith)
	have hr : Cα * ell * S * (1 - ct / (ell * (1 + ηmax)) * ∑ i, a i * h i) *
	(1 + ηmax) = Cα * ell * S * (1 + ηmax) - Cα * ct * S * ∑ i, a i * h i := by
	field_simp
	rw [hr] at hup'
	nlinarith

	lemma simplex_isCompact : IsCompact (simplex (ι := ι)) := by
	have hc : IsClosed (simplex (ι := ι)) := by
	change IsClosed ({a : ι → ℝ \| ∀ i, 0 ≤ a i} ∩ {a : ι → ℝ \| ∑ i, a i = 1})
	apply IsClosed.inter
	· simp only [ofPred_forall]
	exact isClosed_iInter fun i => isClosed_le continuous_const (continuous_apply i)
	· exact isClosed_eq (continuous_finsetSum _ fun i _ => continuous_apply i) continuous_const
	apply (isCompact_pi_infinite fun _ : ι => isCompact_Icc (a := (0 : ℝ)) (b := 1)).of_isClosed_subset hc
	intro a ha i
	exact ⟨ha.1 i, max_coordinate ha i⟩

	lemma objective_continuousOn {f : ι → ℝ → ℝ} (hf : ∀ i, ContinuousOn (f i) (Icc 0 1))
	(S : ℝ) (a₀ : ι → ℝ) : ContinuousOn (objective f S a₀) simplex := by
	apply ContinuousOn.add
	· apply continuousOn_finsetSum _
	intro i _
	exact (hf i).comp (continuous_apply i).continuousOn
	(fun a ha => ⟨ha.1 i, max_coordinate ha i⟩)
	· apply ContinuousOn.const_mul
	apply continuousOn_finsetSum _
	intro i _
	exact ((continuous_apply i).sub continuous_const).abs.continuousOn

	lemma exists_minimizer {f : ι → ℝ → ℝ} {S : ℝ} {a₀ : ι → ℝ}
	(hf : ∀ i, ContinuousOn (f i) (Icc 0 1)) (ha₀ : a₀ ∈ simplex) :
	∃ a ∈ simplex, IsMinOn (objective f S a₀) simplex a :=
	simplex_isCompact.exists_isMinOn ⟨a₀, ha₀⟩ (objective_continuousOn hf S a₀)

	lemma minimizer_payment {f : ι → ℝ → ℝ} {S : ℝ} {a₀ a : ι → ℝ}
	(ha₀ : a₀ ∈ simplex) (hm : IsMinOn (objective f S a₀) simplex a) :
	S * movement a a₀ ≤ potential f a₀ - potential f a := by
	have h := hm ha₀
	change potential f a + S * movement a a₀ ≤ potential f a₀ + S * movement a₀ a₀ at h
	have hz : movement a₀ a₀ = 0 := by simp [movement]
	rw [hz, mul_zero, add_zero] at h
	linarith

	lemma sum_replace [DecidableEq ι] (f : ι → ℝ → ℝ) (w : ι → ℝ) (i : ι) (y : ℝ) :
	(∑ j, f j (Function.update w i y j)) = (∑ j, f j (w j)) - f i (w i) + f i y := by
	have he : (∑ j ∈ Finset.univ.erase i, f j (Function.update w i y j)) =
	∑ j ∈ Finset.univ.erase i, f j (w j) := by
	apply Finset.sum_congr rfl
	intro j hj
	simp [Function.update, (Finset.mem_erase.mp hj).1]
	have h₁ := Finset.sum_erase_add Finset.univ (fun j => f j (Function.update w i y j))
	(Finset.mem_univ i)
	have h₂ := Finset.sum_erase_add Finset.univ (fun j => f j (w j)) (Finset.mem_univ i)
	simp only [Function.update_self, he] at h₁
	linarith

	/-- Transfer epsilon from donor i to receiver j, exactly as in §04. -/
	def transfer [DecidableEq ι] (a : ι → ℝ) (i j : ι) (ε : ℝ) : ι → ℝ :=
	Function.update (Function.update a i (a i - ε)) j (a j + ε)

	lemma sum_transfer [DecidableEq ι] (f : ι → ℝ → ℝ) (a : ι → ℝ) {i j : ι}
	(hij : i ≠ j) (ε : ℝ) :
	(∑ l, f l (transfer a i j ε l)) =
	(∑ l, f l (a l)) - f i (a i) + f i (a i - ε) - f j (a j) + f j (a j + ε) := by
	simp only [transfer, sum_replace, Function.update_of_ne hij.symm]

	lemma transfer_mem_simplex [DecidableEq ι] {a : ι → ℝ} (ha : a ∈ simplex) {i j : ι}
	(hij : i ≠ j) {ε : ℝ} (hε : ε ∈ Set.Icc 0 (a i)) : transfer a i j ε ∈ simplex := by
	constructor
	· intro l
	by_cases hlj : l = j
	· subst l; simp only [transfer, Function.update_self]; linarith [ha.1 j, hε.1]
	by_cases hli : l = i
	· subst l; simp [transfer, Function.update_of_ne hij, sub_nonneg.mpr hε.2]
	· simpa [transfer, Function.update_of_ne hlj, Function.update_of_ne hli] using ha.1 l
	· rw [sum_transfer (fun _ x => x) a hij ε, ha.2]
	ring

	lemma transfer_min_comparison [DecidableEq ι] {f : ι → ℝ → ℝ} {S : ℝ} {a₀ a : ι → ℝ}
	(hS : 0 ≤ S) (ha : a ∈ simplex) (hm : IsMinOn (objective f S a₀) simplex a)
	{i j : ι} (hij : i ≠ j) {ε : ℝ} (hε : ε ∈ Set.Icc 0 (a i)) :
	f i (a i) + f j (a j) ≤ f i (a i - ε) + f j (a j + ε) + 2 * S * ε := by
	have h := hm (transfer_mem_simplex ha hij hε)
	change potential f a + S * movement a a₀ ≤ potential f (transfer a i j ε) +
	S * movement (transfer a i j ε) a₀ at h
	dsimp [potential, movement] at h
	rw [sum_transfer f a hij ε, sum_transfer (fun l x => \|x - a₀ l\|) a hij ε] at h
	have hi := abs_sub_le (a i - ε) (a i) (a₀ i)
	have hj := abs_sub_le (a j + ε) (a j) (a₀ j)
	have hei : \|a i - ε - a i\| = ε := by rw [show a i - ε - a i = -ε by ring, abs_neg, abs_of_nonneg hε.1]
	have hej : \|a j + ε - a j\| = ε := by rw [show a j + ε - a j = ε by ring, abs_of_nonneg hε.1]
	rw [hei] at hi
	rw [hej] at hj
	have hi' := mul_le_mul_of_nonneg_left hi hS
	have hj' := mul_le_mul_of_nonneg_left hj hS
	linarith

	lemma minimizer_derivative_gap [DecidableEq ι] {f : ι → ℝ → ℝ} {S : ℝ} {a₀ a : ι → ℝ}
	(hS : 0 ≤ S) (ha : a ∈ simplex) (hm : IsMinOn (objective f S a₀) simplex a)
	{i j : ι} (hij : i ≠ j) (hi : 0 < a i) {di dj : ℝ}
	(hdi : HasDerivAt (f i) di (a i)) (hdj : HasDerivAt (f j) dj (a j)) :
	di ≤ dj + 2 * S := by
	have hc : IsLocalMinOn (fun ε => f i (a i - ε) + f j (a j + ε) + 2 * S * ε)
	(Icc 0 (a i)) 0 := by
	apply IsMinOn.isLocalMinOn
	intro ε hε
	simpa using transfer_min_comparison hS ha hm hij hε
	have hd : HasDerivAt (fun ε => f i (a i - ε) + f j (a j + ε) + 2 * S * ε)
	(-di + dj + 2 * S) 0 := by
	have hi' : HasDerivAt (fun ε => f i (a i - ε)) (-di) 0 := by
	convert (hdi.comp_of_eq 0 ((hasDerivAt_id 0).const_sub (a i)) (by simp)) using 1 <;> first \| rfl \| simp
	have hj' : HasDerivAt (fun ε => f j (a j + ε)) dj 0 := by
	convert (hdj.comp_of_eq 0 ((hasDerivAt_id 0).const_add (a j)) (by simp)) using 1 <;> first \| rfl \| simp
	convert (hi'.add hj').add ((hasDerivAt_id 0).const_mul (2 * S)) using 1 <;> first \| rfl \| simp
	have ht : a i - 0 ∈ posTangentConeAt (Icc 0 (a i)) 0 :=
	sub_mem_posTangentConeAt_of_segment_subset
	((convex_Icc 0 (a i)).segment_subset ⟨le_rfl, hi.le⟩ ⟨hi.le, le_rfl⟩)
	have h := hc.hasFDerivWithinAt_nonneg hd.hasFDerivAt.hasFDerivWithinAt ht
	simp only [ContinuousLinearMap.toSpanSingleton_apply, smul_eq_mul, sub_zero] at h
	nlinarith

	/-- A singular inward slope cannot occur at a constrained global minimum:
	there is always a positive donor in the unit simplex. -/
	lemma minimizer_not_inward_atBot [DecidableEq ι] {f : ι → ℝ → ℝ} {S : ℝ} {a₀ a : ι → ℝ}
	(hS : 0 ≤ S) (ha : a ∈ simplex) (hm : IsMinOn (objective f S a₀) simplex a)
	(hderiv : ∀ i, 0 < a i → DifferentiableAt ℝ (f i) (a i))
	{j : ι} (hj : a j = 0) :
	¬ Tendsto (fun ε : ℝ => (f j ε - f j 0) / ε) (𝓝[>] 0) atBot := by
	intro hsing
	have hex : ∃ i, 0 < a i := by
	by_contra hn
	push Not at hn
	have hsum : (∑ i, a i) ≤ 0 := Finset.sum_nonpos fun i _ => hn i
	linarith [ha.2]
	obtain ⟨i, hi⟩ := hex
	have hij : i ≠ j := by intro he; subst i; linarith
	have hdi := (hderiv i hi).hasDerivAt
	have hd : HasDerivAt (fun ε : ℝ => f i (a i - ε)) (-deriv (f i) (a i)) 0 := by
	convert hdi.comp_of_eq 0 ((hasDerivAt_id 0).const_sub (a i)) (by simp) using 1
	all_goals first \| rfl \| simp
	have hdl : Tendsto (fun ε : ℝ => (f i (a i - ε) - f i (a i)) / ε)
	(𝓝[>] 0) (𝓝 (-deriv (f i) (a i))) := by
	convert hd.tendsto_slope_zero_right using 1
	simp [smul_eq_mul, div_eq_mul_inv, mul_comm]
	have hbot := tendsto_atBot_add_const_right (𝓝[>] (0 : ℝ)) (2 * S) (hdl.add_atBot hsing)
	have hev : ∀ᶠ ε : ℝ in 𝓝[>] 0,
	0 < ε ∧ ε ≤ a i ∧
	(f i (a i - ε) - f i (a i)) / ε + (f j ε - f j 0) / ε + 2 * S < 0 := by
	filter_upwards [eventually_mem_nhdsWithin,
	(eventually_le_nhds hi).filter_mono nhdsWithin_le_nhds,
	hbot.eventually (eventually_lt_atBot 0)] with ε hε hεi hneg
	exact ⟨hε, hεi, hneg⟩
	obtain ⟨ε, hε, hεi, hneg⟩ := hev.exists
	have h := transfer_min_comparison hS ha hm hij ⟨hε.le, hεi⟩
	rw [hj, zero_add] at h
	have hmul := mul_neg_of_neg_of_pos hneg hε
	have hid : ((f i (a i - ε) - f i (a i)) / ε + (f j ε - f j 0) / ε + 2 * S) * ε =
	f i (a i - ε) - f i (a i) + (f j ε - f j 0) + 2 * S * ε := by
	field_simp
	rw [hid] at hmul
	linarith

	/-- Complete minimizer feasibility reduction, using the genuine derivative and
	singular endpoint properties of the source primitives. -/
	lemma minimizer_feasible [DecidableEq ι]
	{f : ι → ℝ → ℝ} {a₀ a h η F B : ι → ℝ} {ct ell ηmax Cα CF S : ℝ}
	(ha : a ∈ simplex) (hm : IsMinOn (objective f S a₀) simplex a)
	(hh : ∀ i, 0 ≤ h i) (hc : 0 < ct) (hl : 0 < ell)
	(hη : ∀ i, η i = ct * h i / ell) (hmax : ∀ i, η i ≤ ηmax)
	(hCa : 0 < Cα) (hCF : 0 ≤ CF) (hS : 0 < S)
	(hB0 : ∀ i, 0 ≤ B i) (hB : ∑ i, B i = S)
	(hF : ∀ i, 0 ≤ F i) (hFp : ∀ i, 0 < a i → 0 < F i)
	(hT : 0 < ∑ i, a i * h i)
	(hden : ∑ i, a i * F i ≤ CF * ∑ i, a i * h i)
	(hgap : 2 * CF * (1 + ηmax) < Cα * ct)
	(hderiv : ∀ i, 0 < a i → HasDerivAt (f i)
	(Cα * ell * ((S - (1 + η i) * B i / a i) / F i)) (a i))
	(hsing : ∀ i, a i = 0 → 0 < B i →
	Tendsto (fun ε : ℝ => (f i ε - f i 0) / ε) (𝓝[>] 0) atBot) :
	∀ i, S * a i ≤ (1 + η i) * B i := by
	have hη0 : ∀ i, 0 ≤ η i := fun i => by
	rw [hη i]; exact div_nonneg (mul_nonneg hc.le (hh i)) hl.le
	have hz : ∀ i, a i = 0 → B i = 0 := by
	intro i hi
	by_contra hBi
	have hp : 0 < B i := lt_of_le_of_ne (hB0 i) (Ne.symm hBi)
	exact minimizer_not_inward_atBot hS.le ha hm
	(fun j hj => (hderiv j hj).differentiableAt) hi (hsing i hi hp)
	let γ : ι → ℝ := fun i => if 0 < a i then
	Cα * ell * ((S - (1 + η i) * B i / a i) / F i) else 0
	have hid : ∀ i, a i * F i / (1 + η i) * γ i =
	Cα * ell * (S * (a i / (1 + η i)) - B i) := by
	intro i
	by_cases hi : 0 < a i
	· have hηi : 1 + η i ≠ 0 := by linarith [hη0 i]
	dsimp [γ]; rw [ite_eq_left hi]
	field_simp [hi.ne', (hFp i hi).ne', hηi]
	· have hzero : a i = 0 := le_antisymm (le_of_not_gt hi) (ha.1 i)
	simp only [γ, ite_eq_right hi, hzero, hz i hzero, zero_mul, zero_div, mul_zero, sub_self]
	obtain ⟨j, hj⟩ := exists_negative_derivative ha hh hc hl hη hmax hCa hCF hS hB hF hT hden hgap hid
	have hjp : 0 < a j := by
	by_contra hjp
	simp only [γ, ite_eq_right hjp] at hj
	linarith
	intro i
	by_contra hbad
	have hbad' : (1 + η i) * B i < S * a i := lt_of_not_ge hbad
	have hi : 0 < a i := by
	have hp : 0 ≤ (1 + η i) * B i := mul_nonneg (by linarith [hη0 i]) (hB0 i)
	nlinarith
	have hγi : 0 < γ i := by
	dsimp [γ]; rw [ite_eq_left hi]
	apply mul_pos (mul_pos hCa hl)
	apply div_pos _ (hFp i hi)
	exact sub_pos.mpr ((div_lt_iff₀ hi).mpr hbad')
	have hij : i ≠ j := by intro he; subst i; linarith
	have h := minimizer_derivative_gap hS.le ha hm hij hi (hderiv i hi) (hderiv j hjp)
	have hei : γ i = Cα * ell * ((S - (1 + η i) * B i / a i) / F i) := ite_eq_left hi
	have hej : γ j = Cα * ell * ((S - (1 + η j) * B j / a j) / F j) := ite_eq_left hjp
	rw [← hei, ← hej] at h
	linarith

	end KServer.SimplexTracker


	namespace KServer.Allocation
	open Set Filter
	open scoped BigOperators Topology
	open SimplexTracker SimplexAnalytic

	variable {ι : Type*} [Fintype ι]

	lemma simplex_weight_pos {a h : ι → ℝ} (ha : a ∈ simplex) (hh : ∀ i, 0 < h i) :
	0 < ∑ i, a i * h i := by
	have hex : ∃ i, 0 < a i := by
	by_contra hn
	push Not at hn
	have : (∑ i, a i) ≤ 0 := Finset.sum_nonpos fun i _ => hn i
	linarith [ha.2]
	obtain ⟨i, hi⟩ := hex
	exact lt_of_lt_of_le (mul_pos hi (hh i))
	(Finset.single_le_sum (fun j _ => mul_nonneg (ha.1 j) (hh j).le) (Finset.mem_univ i))

	lemma log_inverse_nonneg_closed {a : ℝ} (ha : a ∈ Set.Icc 0 1) : 0 ≤ Real.log (1 / a) := by
	rcases eq_or_lt_of_le ha.1 with hz \| hp
	· subst a; simp
	· exact log_inverse_nonneg ⟨hp, ha.2⟩

	/-- The exact integral update, including compact existence, singular endpoint
	exclusion, feasibility and payment. Its analytic premises are proved above for
	both displayed denominator types; no convexity is presumed. -/
	lemma exists_simplex_update [DecidableEq ι]
	{F : ι → ℝ → ℝ} {a₀ h c B base : ι → ℝ} {ct ell ηmax Cα CF : ℝ}
	(ha₀ : a₀ ∈ simplex) (hh : ∀ i, 0 < h i) (hc : 0 < ct) (hl : 0 < ell)
	(hmax : ∀ i, ct * h i / ell ≤ ηmax)
	(hCa : 0 < Cα) (hCF : 0 ≤ CF)
	(hB0 : ∀ i, 0 ≤ B i) (hb : ∀ i, base i ∈ Set.Icc 0 1)
	(hFc : ∀ i, Continuous (F i)) (hci : ∀ i, 0 < c i)
	(hF : ∀ i y, 0 ≤ y → c i * (1 + y)^2 ≤ F i y)
	(hdecay : ∀ i, Tendsto (fun s : ℝ => s * F i (Real.log (1 / s))) (𝓝[>] 0) (𝓝 0))
	(hden : ∀ a ∈ simplex, ∑ i, a i * F i (Real.log (1 / a i)) ≤ CF * ∑ i, a i * h i)
	(hgap : 2 * CF * (1 + ηmax) < Cα * ct) :
	let S := ∑ i, B i
	let f := fun i => primitive (F i) Cα ell (ct * h i / ell) S (B i) (base i)
	∃ a ∈ simplex, (∀ i, S * a i ≤ (1 + ct * h i / ell) * B i) ∧
	S * movement a a₀ ≤ potential f a₀ - potential f a ∧
	IsMinOn (objective f S a₀) simplex a := by
	dsimp only
	let S := ∑ i, B i
	let η := fun i => ct * h i / ell
	let f := fun i => primitive (F i) Cα ell (η i) S (B i) (base i)
	have hη : ∀ i, 0 ≤ η i := fun i => div_nonneg (mul_nonneg hc.le (hh i).le) hl.le
	have hp : ∀ i y, 0 ≤ y → 0 < F i y := fun i y hy =>
	(show 0 < c i * (1 + y)^2 by have := hci i; positivity).trans_le (hF i y hy)
	have hint := fun i => singular_intervalIntegrable_of_quadratic (hFc i).measurable (hci i) (hF i)
	have hfc : ∀ i, ContinuousOn (f i) (Icc 0 1) := fun i =>
	primitive_continuousOn (hFc i).measurable (hp i) (hint i) Cα ell (η i) S (B i) (hb i)
	obtain ⟨a, ha, hm⟩ := exists_minimizer hfc ha₀ (S := S)
	refine ⟨a, ha, ?_, minimizer_payment ha₀ hm, hm⟩
	have hS0 : 0 ≤ S := Finset.sum_nonneg fun i _ => hB0 i
	rcases eq_or_lt_of_le hS0 with hS \| hS
	· intro i
	change S * a i ≤ (1 + η i) * B i
	rw [← hS, zero_mul]
	exact mul_nonneg (by linarith [hη i]) (hB0 i)
	have hF0 : ∀ i, 0 ≤ F i (Real.log (1 / a i)) := fun i =>
	(hp i _ (log_inverse_nonneg_closed ⟨ha.1 i, max_coordinate ha i⟩)).le
	apply minimizer_feasible ha hm (fun i => (hh i).le) hc hl (fun _ => rfl) hmax
	hCa hCF hS hB0 rfl hF0
	(fun i hi => hp i _ (log_inverse_nonneg ⟨hi, max_coordinate ha i⟩))
	(simplex_weight_pos ha hh) (hden a ha) hgap
	· intro i hi
	exact primitive_derivative (hFc i) (hp i) (hint i) Cα ell (η i) S (B i)
	(hb i) ⟨hi, max_coordinate ha i⟩
	· intro i _ hBi
	exact primitive_inward_atBot (hFc i) (hci i) (hF i) (hdecay i)
	hCa hl (hη i) hBi S (hb i)

	/-- Instantiation on a domain without a dominant child, with the manuscript's
	regular denominators and base point zero, at all simplex points including zero. -/
	lemma regular_simplex_update [DecidableEq ι]
	{a₀ h B : ι → ℝ} {ct ell ηmax Cα Cexp : ℝ}
	(ha₀ : a₀ ∈ simplex) (hh : ∀ i, 1 ≤ h i) (hc : 0 < ct) (hl : 0 < ell)
	(hmax : ∀ i, ct * h i / ell ≤ ηmax) (hCa : 0 < Cα)
	(hB0 : ∀ i, 0 ≤ B i) (hCexp : 0 ≤ Cexp)
	(htail : ∑ i, Real.exp (-h i) ≤ Cexp)
	(hgap : 2 * (25 + 32 * Cexp) * (1 + ηmax) < Cα * ct) :
	let S := ∑ i, B i
	let f := fun i => primitive (regularF (h i)) Cα ell (ct * h i / ell) S (B i) 0
	∃ a ∈ simplex, (∀ i, S * a i ≤ (1 + ct * h i / ell) * B i) ∧
	S * movement a a₀ ≤ potential f a₀ - potential f a ∧
	IsMinOn (objective f S a₀) simplex a := by
	apply exists_simplex_update (c := fun i => 1 / h i) (CF := 25 + 32 * Cexp)
	ha₀ (fun i => by linarith [hh i]) hc hl hmax hCa (by positivity) hB0
	(fun _ => ⟨le_rfl, by norm_num⟩)
	(fun i => by unfold regularF; fun_prop)
	(fun i => by have := hh i; positivity)
	(fun i _ hy => regular_quadratic_lower (hh i) hy)
	(fun i => regular_decay (h i))
	· intro a ha
	exact regular_denominator_sum ha.1 ha.2 hh htail
	· exact hgap

	/-- Marked-domain instantiation; the base point is one exactly on the marked
	coordinate, preserving the source's cancellation under parameter changes. -/
	lemma marked_simplex_update [DecidableEq ι]
	{a₀ h B : ι → ℝ} {ct ell ηmax Cα Cexp c u H : ℝ} (o : ι)
	(ha₀ : a₀ ∈ simplex) (hu : 0 < u) (hu1 : u ≤ 1)
	(hH : H = 1 + Real.log (1 / u)) (ho : h o = u)
	(hh : ∀ i ≠ o, 1 ≤ h i) (hc : 0 < ct) (hl : 0 < ell)
	(hmax : ∀ i, ct * h i / ell ≤ ηmax) (hCa : 0 < Cα)
	(hB0 : ∀ i, 0 ≤ B i) (hCexp : 0 ≤ Cexp)
	(hsideC : 0 < c) (hsideC1 : c ≤ 1) (hside : ∀ i ≠ o, c * H ≤ h i)
	(htail : ∑ i ∈ Finset.univ.erase o, Real.exp (-h i) ≤ Cexp * u)
	(hgap : 2 * ((25 * c + 2 + 32 * Cexp) / c) * (1 + ηmax) < Cα * ct) :
	let S := ∑ i, B i
	let f := fun i => primitive (if i = o then markedF u H else regularF (h i))
	Cα ell (ct * h i / ell) S (B i) (if i = o then 1 else 0)
	∃ a ∈ simplex, (∀ i, S * a i ≤ (1 + ct * h i / ell) * B i) ∧
	S * movement a a₀ ≤ potential f a₀ - potential f a ∧
	IsMinOn (objective f S a₀) simplex a := by
	have hH1 : 1 ≤ H := by
	rw [hH]; linarith [log_inverse_nonneg ⟨hu, hu1⟩]
	have hHp : 0 < H := by linarith
	have hhp : ∀ i, 0 < h i := by
	intro i
	by_cases hi : i = o
	· simpa [hi, ho] using hu
	· linarith [hh i hi]
	obtain ⟨cm, hcm, hFm⟩ := marked_lower_constant hu hHp
	apply exists_simplex_update (c := fun i => if i = o then cm else 1 / h i)
	(CF := (25 * c + 2 + 32 * Cexp) / c) ha₀ hhp hc hl hmax hCa
	(by positivity) hB0
	· intro i; split_ifs <;> norm_num
	· intro i
	split_ifs
	· unfold markedF; fun_prop
	· unfold regularF; fun_prop
	· intro i
	split_ifs
	· exact hcm
	· exact one_div_pos.mpr (hhp i)
	· intro i y hy
	by_cases hi : i = o
	· simpa [hi] using hFm y hy
	· simpa [hi] using regular_quadratic_lower (hh i hi) hy
	· intro i
	by_cases hi : i = o
	· simpa [hi] using marked_decay u H
	· simpa [hi] using regular_decay (h i)
	· intro a ha
	have hd := marked_denominator_sum o ha.1 ha.2 hu.le (hu1.trans hH1)
	hsideC hsideC1 hh hside hCexp htail
	rw [← Finset.add_sum_erase _ _ (Finset.mem_univ o)]
	simp only [ite_true]
	have hs : (∑ i ∈ Finset.univ.erase o,
	a i * (if i = o then markedF u H else regularF (h i)) (Real.log (1 / a i))) =
	∑ i ∈ Finset.univ.erase o, a i * regularF (h i) (Real.log (1 / a i)) := by
	apply Finset.sum_congr rfl
	intro i hi
	rw [ite_eq_right (Finset.mem_erase.mp hi).1]
	rw [hs]
	have ht : (∑ i, a i * h i) = a o * u + ∑ i ∈ Finset.univ.erase o, a i * h i := by
	rw [← Finset.add_sum_erase _ _ (Finset.mem_univ o), ho]
	rw [ht]
	exact hd
	· exact hgap

	end KServer.Allocation


	namespace KServer.SupportedAllocation
	open scoped BigOperators
	open Finset
	variable {J : Type*} [Fintype J] [DecidableEq J]

	def restrict (A : Finset J) (a : J → ℝ) : A → ℝ := fun i => a i

	def extend (A : Finset J) (a : A → ℝ) : J → ℝ := fun i => if hi : i∈A then a ⟨i,hi⟩ else 0

	omit [Fintype J] in
	lemma extend_apply_mem (A : Finset J) (a : A → ℝ) (i : A) : extend A a i=a i := by
	simp [extend,i.property]

	omit [Fintype J] in
	lemma extend_apply_not_mem (A : Finset J) (a : A → ℝ) {i : J} (hi : i∉A) : extend A a i=0 := by
	simp [extend,hi]

	omit [Fintype J] in
	lemma restrict_extend (A : Finset J) (a : A → ℝ) : restrict A (extend A a)=a := by
	funext i; exact extend_apply_mem A a i

	omit [Fintype J] in
	lemma extend_restrict (A : Finset J) (a : J → ℝ) (ha : ∀ i∉A, a i=0) :
	extend A (restrict A a)=a := by
	funext i
	by_cases hi : i∈A
	· simp [extend,restrict,hi]
	· simp [extend,hi,ha i hi]

	omit [DecidableEq J] in
	lemma sum_restrict (A : Finset J) (a : J → ℝ) (ha : ∀ i∉A, a i=0) :
	(∑ i : A, a i)=∑ i, a i := by
	rw [←Finset.sum_subtype A (fun _ => Iff.rfl)]
	exact Finset.sum_subset (subset_univ A) fun i _ hi => ha i hi

	lemma sum_extend (A : Finset J) (a : A → ℝ) : (∑ i, extend A a i)=∑ i, a i := by
	have he := sum_restrict A (extend A a) (fun i hi => extend_apply_not_mem A a hi)
	simpa only [extend_apply_mem] using he.symm

	omit [DecidableEq J] in
	lemma restrict_simplex {A : Finset J} (hA : A.Nonempty) {a : J → ℝ}
	(ha : a ∈ ChangingDomains.simplex A) : restrict A a ∈ SimplexTracker.simplex := by
	refine ⟨fun i => ha.1 i,?_⟩
	change (∑ i : A, a i)=1
	rw [sum_restrict A a ha.2.2,ha.2.1,ite_eq_left hA]

	lemma extend_simplex {A : Finset J} (hA : A.Nonempty) {a : A → ℝ}
	(ha : a ∈ SimplexTracker.simplex) : extend A a ∈ ChangingDomains.simplex A := by
	refine ⟨?_,?_,fun i hi => extend_apply_not_mem A a hi⟩
	· intro i
	unfold extend
	split_ifs with hi
	· exact ha.1 ⟨i,hi⟩
	· exact le_rfl
	· rw [sum_extend,ite_eq_left hA]; exact ha.2

	omit [DecidableEq J] in
	lemma movement_restrict {A : Finset J} {a b : J → ℝ}
	(ha : ∀ i∉A, a i=0) (hb : ∀ i∉A, b i=0) :
	SimplexTracker.movement (restrict A a) (restrict A b)=ChangingDomains.variation a b := by
	apply sum_restrict A (fun i => \|a i-b i\|)
	intro i hi; simp [ha i hi,hb i hi]

	lemma movement_extend {A : Finset J} (a b : A → ℝ) :
	ChangingDomains.variation (extend A a) (extend A b)=SimplexTracker.movement a b := by
	have h := movement_restrict (fun i hi => extend_apply_not_mem A a hi)
	(fun i hi => extend_apply_not_mem A b hi)
	simpa only [restrict_extend] using h.symm

	omit [DecidableEq J] in
	lemma potential_restrict {A : Finset J} {f : J → ℝ → ℝ} {a : J → ℝ}
	(ha : ∀ i∉A, a i=0) (hf : ∀ i∉A, f i 0=0) :
	SimplexTracker.potential (fun i : A => f i) (restrict A a)=∑ i, f i (a i) := by
	apply sum_restrict A (fun i => f i (a i))
	intro i hi; rw [ha i hi,hf i hi]

	/-- Restriction is exact, including both the movement penalty and the literal
	source potentials. No dummy inactive coordinate enters the denominator bound. -/
	lemma restrict_isMin {A : Finset J} (hA : A.Nonempty) {f : J → ℝ → ℝ}
	{S : ℝ} {a a₀ : J → ℝ} (ha : a ∈ ChangingDomains.simplex A)
	(ha₀ : a₀ ∈ ChangingDomains.simplex A) (hf : ∀ i∉A, f i 0=0)
	(hm : IsMinOn (fun b => (∑ i, f i (b i))+S*ChangingDomains.variation b a₀)
	(ChangingDomains.simplex A) a) :
	IsMinOn (SimplexTracker.objective (fun i : A => f i) S (restrict A a₀))
	SimplexTracker.simplex (restrict A a) := by
	intro b hb
	have he := extend_simplex hA hb
	have hh := hm he
	change SimplexTracker.potential (fun i : A => f i) (restrict A a)+
	S*SimplexTracker.movement (restrict A a) (restrict A a₀) ≤
	SimplexTracker.potential (fun i : A => f i) b+S*SimplexTracker.movement b (restrict A a₀)
	rw [potential_restrict ha.2.2 hf,movement_restrict ha.2.2 ha₀.2.2]
	have hp := potential_restrict he.2.2 hf
	have hmove := movement_restrict he.2.2 ha₀.2.2
	rw [restrict_extend] at hp hmove
	rw [hp,hmove]
	exact hh

	end KServer.SupportedAllocation

	namespace KServer.Allocation
	open Set Filter
	open scoped BigOperators Topology
	open SimplexTracker SimplexAnalytic
	variable {ι : Type*} [Fintype ι] [DecidableEq ι]

	/-- The feasibility argument holds for every attained minimizer, in particular
	for the fixed noncomputable choice in the causal recurrence. -/
	lemma every_simplex_minimizer_feasible
	{F : ι → ℝ → ℝ} {a₀ a h c B base : ι → ℝ} {ct ell ηmax Cα CF : ℝ}
	(ha : a ∈ simplex) (hh : ∀ i, 0 < h i) (hc : 0 < ct) (hl : 0 < ell)
	(hmax : ∀ i, ct*h i/ell ≤ ηmax) (hCa : 0 < Cα) (hCF : 0 ≤ CF)
	(hB0 : ∀ i, 0 ≤ B i) (hb : ∀ i, base i ∈ Set.Icc 0 1)
	(hFc : ∀ i, Continuous (F i)) (hci : ∀ i, 0 < c i)
	(hF : ∀ i y, 0 ≤ y → c i*(1+y)^2 ≤ F i y)
	(hdecay : ∀ i, Tendsto (fun s : ℝ => s*F i (Real.log (1/s))) (𝓝[>] 0) (𝓝 0))
	(hden : ∀ b ∈ simplex, ∑ i, b iF i (Real.log (1/b i)) ≤ CF∑ i, b i*h i)
	(hgap : 2CF(1+ηmax) < Cα*ct)
	(hm : IsMinOn (objective (fun i => primitive (F i) Cα ell (ct*h i/ell)
	(∑ j, B j) (B i) (base i)) (∑ i, B i) a₀) simplex a) :
	∀ i, (∑ j, B j)a i ≤ (1+cth i/ell)*B i := by
	let S := ∑ i, B i
	let η := fun i => ct*h i/ell
	have hη : ∀ i, 0 ≤ η i := fun i => div_nonneg (mul_nonneg hc.le (hh i).le) hl.le
	have hp : ∀ i y, 0 ≤ y → 0 < F i y := fun i y hy =>
	(show 0 < c i*(1+y)^2 by have := hci i; positivity).trans_le (hF i y hy)
	have hint := fun i => singular_intervalIntegrable_of_quadratic (hFc i).measurable (hci i) (hF i)
	have hS0 : 0 ≤ S := Finset.sum_nonneg fun i _ => hB0 i
	rcases eq_or_lt_of_le hS0 with hS \| hS
	· intro i
	change Sa i ≤ (1+η i)B i
	rw [←hS,zero_mul]
	exact mul_nonneg (by linarith [hη i]) (hB0 i)
	have hF0 : ∀ i, 0 ≤ F i (Real.log (1/a i)) := fun i =>
	(hp i _ (log_inverse_nonneg_closed ⟨ha.1 i,max_coordinate ha i⟩)).le
	apply minimizer_feasible ha hm (fun i => (hh i).le) hc hl (fun _ => rfl) hmax
	hCa hCF hS hB0 rfl hF0
	(fun i hi => hp i _ (log_inverse_nonneg ⟨hi,max_coordinate ha i⟩))
	(simplex_weight_pos ha hh) (hden a ha) hgap
	· intro i hi
	exact primitive_derivative (hFc i) (hp i) (hint i) Cα ell (η i) S (B i)
	(hb i) ⟨hi,max_coordinate ha i⟩
	· intro i _ hBi
	exact primitive_inward_atBot (hFc i) (hci i) (hF i) (hdecay i)
	hCa hl (hη i) hBi S (hb i)

	end KServer.Allocation

	namespace KServer.Allocation
	open Set Filter
	open scoped BigOperators Topology
	open SimplexTracker SimplexAnalytic
	variable {ι : Type*} [Fintype ι] [DecidableEq ι]

	lemma every_regular_minimizer_feasible
	{a₀ a h B : ι → ℝ} {ct ell ηmax Cα Cexp : ℝ}
	(ha : a ∈ simplex) (hh : ∀ i, 1≤h i) (hc : 0<ct) (hl : 0<ell)
	(hmax : ∀ i, ct*h i/ell≤ηmax) (hCa : 0<Cα) (hB0 : ∀ i, 0≤B i)
	(hCexp : 0≤Cexp) (htail : ∑ i, Real.exp (-h i)≤Cexp)
	(hgap : 2(25+32Cexp)(1+ηmax)<Cαct)
	(hm : IsMinOn (objective (fun i => primitive (regularF (h i)) Cα ell (ct*h i/ell)
	(∑ j, B j) (B i) 0) (∑ i, B i) a₀) simplex a) :
	∀ i, (∑ j, B j)a i≤(1+cth i/ell)*B i := by
	apply every_simplex_minimizer_feasible (c := fun i => 1/h i) (CF := 25+32*Cexp)
	ha (fun i => by linarith [hh i]) hc hl hmax hCa (by positivity) hB0
	(fun _ => ⟨le_rfl,by norm_num⟩)
	(fun i => by unfold regularF; fun_prop)
	(fun i => by have := hh i; positivity)
	(fun i _ hy => regular_quadratic_lower (hh i) hy)
	(fun i => regular_decay (h i))
	· intro b hb; exact regular_denominator_sum hb.1 hb.2 hh htail
	· exact hgap
	· exact hm

	end KServer.Allocation

	namespace KServer.SupportedAllocation
	open scoped BigOperators
	open SimplexTracker SimplexAnalytic
	variable {J : Type*} [Fintype J] [DecidableEq J]

	/-- Uniform feasibility on the changing active domain. Its exp-tail hypothesis
	is a sum over active children only, exactly as in the source. -/
	lemma regular_supported_minimizer_feasible {A : Finset J} (hA : A.Nonempty)
	{a₀ a h B : J → ℝ} {ct ell ηmax Cα Cexp : ℝ}
	(ha : a ∈ ChangingDomains.simplex A) (ha₀ : a₀ ∈ ChangingDomains.simplex A)
	(hh : ∀ i∈A, 1≤h i) (hc : 0<ct) (hl : 0<ell)
	(hmax : ∀ i∈A, ct*h i/ell≤ηmax) (hCa : 0<Cα) (hB0 : ∀ i, 0≤B i)
	(hBout : ∀ i∉A, B i=0) (hCexp : 0≤Cexp) (htail : ∑ i∈A, Real.exp (-h i)≤Cexp)
	(hgap : 2(25+32Cexp)(1+ηmax)<Cαct)
	(hm : IsMinOn (fun b => (∑ i, primitive (regularF (h i)) Cα ell (ct*h i/ell)
	(∑ j, B j) (B i) 0 (b i))+(∑ i, B i)*ChangingDomains.variation b a₀)
	(ChangingDomains.simplex A) a) :
	∀ i, (∑ j, B j)a i≤(1+cth i/ell)*B i := by
	have hsum : (∑ j : A, B j)=∑ j, B j := sum_restrict A B hBout
	have htail' : (∑ i : A, Real.exp (-h i))≤Cexp := by
	rw [←Finset.sum_subtype A (fun _ => Iff.rfl) (fun i => Real.exp (-h i))]
	exact htail
	have hm' := restrict_isMin hA ha ha₀ (fun i _ => by simp [primitive]) hm
	have he := Allocation.every_regular_minimizer_feasible
	(a₀ := restrict A a₀) (a := restrict A a) (h := fun i : A => h i) (B := fun i : A => B i)
	(restrict_simplex hA ha) (fun i => hh i i.property) hc hl (fun i => hmax i i.property)
	hCa (fun i => hB0 i) hCexp htail' hgap
	have hh' : ∀ i : A, (∑ j, B j)a i≤(1+cth i/ell)*B i := by
	have hm'' : IsMinOn (objective (fun i : A => primitive (regularF (h i)) Cα ell (ct*h i/ell)
	(∑ j : A, B j) (B i) 0) (∑ i : A, B i) (restrict A a₀))
	simplex (restrict A a) := by simpa only [hsum] using hm'
	simpa only [hsum,restrict] using he hm''
	intro i
	by_cases hi : i∈A
	· exact hh' ⟨i,hi⟩
	· simp [ha.2.2 i hi,hBout i hi]

	end KServer.SupportedAllocation

	namespace KServer.Allocation
	open Set Filter
	open scoped BigOperators Topology
	open SimplexTracker SimplexAnalytic
	variable {ι : Type*} [Fintype ι] [DecidableEq ι]

	lemma every_marked_minimizer_feasible
	{a₀ a h B : ι → ℝ} {ct ell ηmax Cα Cexp c u H : ℝ} (o : ι)
	(ha : a ∈ simplex) (hu : 0 < u) (hu1 : u ≤ 1)
	(hH : H = 1 + Real.log (1 / u)) (ho : h o = u)
	(hh : ∀ i ≠ o, 1 ≤ h i) (hc : 0 < ct) (hl : 0 < ell)
	(hmax : ∀ i, ct * h i / ell ≤ ηmax) (hCa : 0 < Cα)
	(hB0 : ∀ i, 0 ≤ B i) (hCexp : 0 ≤ Cexp)
	(hsideC : 0 < c) (hsideC1 : c ≤ 1) (hside : ∀ i ≠ o, c * H ≤ h i)
	(htail : ∑ i ∈ Finset.univ.erase o, Real.exp (-h i) ≤ Cexp * u)
	(hgap : 2 * ((25 * c + 2 + 32 * Cexp) / c) * (1 + ηmax) < Cα * ct)
	(hm : IsMinOn (objective (fun i => primitive (if i = o then markedF u H else regularF (h i))
	Cα ell (ct*h i/ell) (∑ j, B j) (B i) (if i=o then 1 else 0))
	(∑ j, B j) a₀) simplex a) :
	∀ i, (∑ j, B j)a i ≤ (1+cth i/ell)*B i := by
	have hH1 : 1 ≤ H := by
	rw [hH]; linarith [log_inverse_nonneg ⟨hu, hu1⟩]
	have hHp : 0 < H := by linarith
	have hhp : ∀ i, 0 < h i := by
	intro i
	by_cases hi : i = o
	· simpa [hi, ho] using hu
	· linarith [hh i hi]
	obtain ⟨cm, hcm, hFm⟩ := marked_lower_constant hu hHp
	apply every_simplex_minimizer_feasible
	(F := fun i => if i=o then markedF u H else regularF (h i))
	(base := fun i => if i=o then 1 else 0)
	(c := fun i => if i = o then cm else 1 / h i)
	(CF := (25 * c + 2 + 32 * Cexp) / c) ha hhp hc hl hmax hCa
	(by positivity) hB0
	· intro i; split_ifs <;> norm_num
	· intro i
	split_ifs
	· unfold markedF; fun_prop
	· unfold regularF; fun_prop
	· intro i
	split_ifs
	· exact hcm
	· exact one_div_pos.mpr (hhp i)
	· intro i y hy
	by_cases hi : i = o
	· simpa [hi] using hFm y hy
	· simpa [hi] using regular_quadratic_lower (hh i hi) hy
	· intro i
	by_cases hi : i = o
	· simpa [hi] using marked_decay u H
	· simpa [hi] using regular_decay (h i)
	· intro a ha
	have hd := marked_denominator_sum o ha.1 ha.2 hu.le (hu1.trans hH1)
	hsideC hsideC1 hh hside hCexp htail
	rw [← Finset.add_sum_erase _ _ (Finset.mem_univ o)]
	simp only [ite_true]
	have hs : (∑ i ∈ Finset.univ.erase o,
	a i * (if i = o then markedF u H else regularF (h i)) (Real.log (1 / a i))) =
	∑ i ∈ Finset.univ.erase o, a i * regularF (h i) (Real.log (1 / a i)) := by
	apply Finset.sum_congr rfl
	intro i hi
	rw [ite_eq_right (Finset.mem_erase.mp hi).1]
	rw [hs]
	have ht : (∑ i, a i * h i) = a o * u + ∑ i ∈ Finset.univ.erase o, a i * h i := by
	rw [← Finset.add_sum_erase _ _ (Finset.mem_univ o), ho]
	rw [ht]
	exact hd
	· exact hgap

	· exact hm

	end KServer.Allocation

	namespace KServer.SupportedAllocation
	open SimplexTracker SimplexAnalytic
	variable {J : Type*} [Fintype J] [DecidableEq J]

	/-- Feasibility for the actual changing marked domain; inactive coordinates
	are zero instead of being assigned a spurious positive denominator weight. -/
	lemma marked_supported_minimizer_feasible {A : Finset J} {o : J} (hoA : o ∈ A)
	{a₀ a h B : J → ℝ} {ct ell ηmax Cα Cexp c u H : ℝ}
	(ha : a ∈ ChangingDomains.simplex A) (ha₀ : a₀ ∈ ChangingDomains.simplex A)
	(hu : 0 < u) (hu1 : u ≤ 1) (hH : H = 1 + Real.log (1 / u)) (ho : h o = u)
	(hh : ∀ i∈A, i ≠ o → 1 ≤ h i) (hc : 0<ct) (hl : 0<ell)
	(hmax : ∀ i∈A, ct*h i/ell ≤ ηmax) (hCa : 0<Cα)
	(hB0 : ∀ i, 0≤B i) (hBout : ∀ i∉A, B i=0) (hCexp : 0≤Cexp)
	(hsideC : 0<c) (hsideC1 : c≤1) (hside : ∀ i∈A, i≠o → c*H≤h i)
	(htail : ∑ i∈A.erase o, Real.exp (-h i) ≤ Cexp*u)
	(hgap : 2((25c+2+32Cexp)/c)(1+ηmax)<Cα*ct)
	(hm : IsMinOn (fun b => (∑ i, primitive (if i=o then markedF u H else regularF (h i))
	Cα ell (ct*h i/ell) (∑ j, B j) (B i) (if i=o then 1 else 0) (b i))+
	(∑ i, B i)*ChangingDomains.variation b a₀) (ChangingDomains.simplex A) a) :
	∀ i, (∑ j, B j)a i ≤ (1+cth i/ell)*B i := by
	have hA : A.Nonempty := ⟨o,hoA⟩
	let o' : A := ⟨o,hoA⟩
	have hsum : (∑ j : A, B j)=∑ j, B j := sum_restrict A B hBout
	have hcon (i : A) : (i=o') ↔ ((i : J)=o) := Subtype.ext_iff
	have hm' := restrict_isMin hA ha ha₀
	(fun i hi => by simp [ite_eq_right (show i≠o by intro he; exact hi (he ▸ hoA)), primitive]) hm
	have htail' : (∑ i∈Finset.univ.erase o', Real.exp (-h (i : J))) ≤ Cexp*u := by
	have he : (∑ i∈Finset.univ.erase o', Real.exp (-h (i : J))) =
	∑ i∈A.erase o, Real.exp (-h i) := by
	apply Finset.sum_bij (fun (i : A) (_ : i∈Finset.univ.erase o') => (i : J))
	· intro i hi
	exact Finset.mem_erase.mpr ⟨fun he => (Finset.mem_erase.mp hi).1
	(Subtype.ext he), i.property⟩
	· intro i _ j _ hij
	exact Subtype.ext hij
	· intro i hi
	refine ⟨⟨i,(Finset.mem_erase.mp hi).2⟩, ?_,rfl⟩
	exact Finset.mem_erase.mpr ⟨fun he => (Finset.mem_erase.mp hi).1
	(congrArg Subtype.val he),Finset.mem_univ _⟩
	· intro i _; rfl
	rw [he]; exact htail
	have hm'' : IsMinOn (objective (fun i : A => primitive
	(if i=o' then markedF u H else regularF (h i)) Cα ell (ct*h i/ell)
	(∑ j : A, B j) (B i) (if i=o' then 1 else 0))
	(∑ i : A, B i) (restrict A a₀)) simplex (restrict A a) := by
	simpa only [hcon,hsum] using hm'
	have he := Allocation.every_marked_minimizer_feasible o'
	(restrict_simplex hA ha) hu hu1 hH ho
	(fun i hi => hh i i.property (fun he => hi (Subtype.ext he))) hc hl
	(fun i => hmax i i.property) hCa (fun i => hB0 i) hCexp hsideC hsideC1
	(fun i hi => hside i i.property (fun he => hi (Subtype.ext he))) htail' hgap hm''
	intro i
	by_cases hi : i∈A
	· simpa only [hsum,restrict] using he ⟨i,hi⟩
	· simp [ha.2.2 i hi,hBout i hi]
	end KServer.SupportedAllocation

	/-! The literal two-stage update of §04: prepare with the new core flags on
	common coordinates, minimize on the old simplex, transpose at exactly zero
	cost, then minimize on the new simplex. The bounds below calculate its edit
	charge from the actual flags and held parameters, not from an assumed drift
	or policy bound. -/
	namespace KServer.AdaptiveAllocation
	open Finset SimplexAnalytic SimplexFamilies AdaptiveMinimization
	variable {J R : Type*} [Fintype J] [DecidableEq J] [Fintype R]

	lemma update_isMin (C ell : ℝ) (d : Data J R) (p : J × R → ℝ) (c : ℝ) (a : J → ℝ) :
	IsMinOn (fun b => potential C ell d (core d p) b+c*ChangingDomains.variation b a)
	(ChangingDomains.simplex d.active) ((family C ell).update d p c a) := by
	have hm := ((family C ell).exists_update d p c a).choose_spec
	intro b hb
	have hm' := hm.2 hb
	change (family C ell).value d p ((family C ell).update d p c a)+c*movement ((family C ell).update d p c a) a ≤
	(family C ell).value d p b+c*movement b a at hm'
	rw [value_eq C ell d p ((family C ell).update d p c a)
	((family C ell).update_mem d p c a), value_eq C ell d p b hb] at hm'
	exact hm'

	lemma regular_update_feasible {A : Finset J} (hA : A.Nonempty)
	(f : J → R → Bool) (h : J → ℝ) (hh : ∀ i, 1≤h i)
	{C ell ct ηmax Cexp : ℝ} (hC : 0<C) (hl : 0<ell) (hc : 0<ct)
	(hmax : ∀ i∈A, ct*h i/ell≤ηmax) (hCexp : 0≤Cexp)
	(htail : ∑ i∈A, Real.exp (-h i)≤Cexp)
	(hgap : 2(25+32Cexp)(1+ηmax)<Cct)
	(p : J × R → ℝ) (hp : ∀ ir, 0≤p ir) (hf : ∀ i∉A, ∀ r, f i r=false)
	(a : J → ℝ) (ha : a∈ChangingDomains.simplex A) :
	let d := regular A f h (fun i => ct*h i/ell) hh
	let B := core d p
	let a' := (family C ell).update d p (∑ i, B i) a
	∀ i, (∑ j, B j)a' i≤(1+cth i/ell)*B i := by
	intro d B a'
	have hB : ∀ i, 0≤B i := by
	intro i
	exact ChangingDomains.core_nonneg (fun i r => hp (i,r)) i
	have hBz : ∀ i∉A, B i=0 := by
	intro i hi
	simp [B,core,d,regular,hf i hi]
	apply SupportedAllocation.regular_supported_minimizer_feasible hA
	((family C ell).update_mem d p (∑ i, B i) a) ha
	(fun i _ => hh i) hc hl hmax hC hB hBz hCexp htail hgap
	exact update_isMin C ell d p (∑ i, B i) a

	omit [DecidableEq J] in
	lemma core_eq (A : Finset J) (f : J → R → Bool) (h eta : J → ℝ)
	(hh : ∀ i, 1≤h i) (p : J × R → ℝ) :
	core (regular A f h eta hh) p = ChangingDomains.core f (fun i r => p (i,r)) := rfl

	/-- The prepared input is concrete and uses only new flags on the common
	old/new active domain, as in §04 (not intersection of old and new flags). -/
	def preparedFlags (A B : Finset J) (g : J → R → Bool) : J → R → Bool :=
	ChangingDomains.syntheticFlags A B g

	def preparedInput (A B : Finset J) (g : J → R → Bool) (p : J × R → ℝ) : J → ℝ :=
	ChangingDomains.core (preparedFlags A B g) (fun i r => p (i,r))

	omit [Fintype J] [Fintype R] in
	lemma prepared_flags_outside (A B : Finset J) (g : J → R → Bool) :
	∀ i∉A, ∀ r, preparedFlags A B g i r=false := by
	intro i hi r
	simp [preparedFlags,ChangingDomains.syntheticFlags,hi]

	omit [Fintype J] in
	lemma prepared_input_outside (A B : Finset J) (g : J → R → Bool) (p : J × R → ℝ) :
	∀ i∉B, preparedInput A B g p i=0 := by
	intro i hi
	simp [preparedInput,preparedFlags,ChangingDomains.core,ChangingDomains.syntheticFlags,hi]

	omit [DecidableEq J] [Fintype R] in
	lemma regular_potential_zero (A : Finset J) (f : J → R → Bool) (h eta : J → ℝ)
	(hh : ∀ i, 1≤h i) (C ell : ℝ) (a : J → ℝ) :
	potential C ell (regular A f h eta hh) (fun _ => 0) a=0 := by
	simp [potential,regular,primitive]

	omit [DecidableEq J] in
	lemma nonneg_total_zero {B : J → ℝ} (hB : ∀ i, 0≤B i) (hS : (∑ i, B i)=0) :
	B=(fun _ => 0) := by
	funext i
	exact le_antisymm (by simpa [hS] using (single_le_sum (fun j _ => hB j) (mem_univ i))) (hB i)

	omit [Fintype R] in
	lemma regular_transpose_value (A B : Finset J) (f : J → R → Bool) (h eta : J → ℝ)
	(hh : ∀ i, 1≤h i) (C ell : ℝ) (v a : J → ℝ) (hv : ∀ i, 0≤v i) :
	potential C ell (regular A f h eta hh) v
	(ChangingDomains.transposeSimplex B (∑ i, v i) a)=
	potential C ell (regular A f h eta hh) v a := by
	by_cases hS : (∑ i, v i)=0
	· rw [nonneg_total_zero hv hS]
	simp only [regular_potential_zero]
	· simp [ChangingDomains.transposeSimplex,hS]

	end KServer.AdaptiveAllocation

	namespace KServer.AdaptiveAllocation
	open Finset SimplexAnalytic SimplexFamilies AdaptiveMinimization
	variable {J R : Type*} [Fintype J] [DecidableEq J] [Fintype R]

	omit [DecidableEq J] [Fintype R] in
	lemma regular_input_bound (A : Finset J) (hA : A.Nonempty) (f : J → R → Bool)
	(h eta : J → ℝ) (hh : ∀ i, 1≤h i) (he : ∀ i, eta i∈Set.Icc 0 1)
	{C ell : ℝ} (hC : 0≤C) (hl : 0≤ell) {B B' a : J → ℝ}
	(ha : a∈ChangingDomains.simplex A) :
	\|potential C ell (regular A f h eta hh) B a-
	potential C ell (regular A f h eta hh) B' a\| ≤
	3Cell*ChangingDomains.variation B B' := by
	have hs : (∑ i, a i)=1 := by simpa only [ite_eq_left hA] using ha.2.1
	exact regular_input_slopes ha.1 hs hh (fun i => (he i).1) (fun i => (he i).2) hC hl

	omit [DecidableEq J] [Fintype R] in
	lemma regular_parameter_jump (A A' : Finset J) (f g : J → R → Bool)
	(h h' eta eta' : J → ℝ) (hh : ∀ i, 1≤h i) (hh' : ∀ i, 1≤h' i)
	(he : ∀ i, eta i∈Set.Icc 0 1) (he' : ∀ i, eta' i∈Set.Icc 0 1)
	(heq : ∀ i, h i=h' i → eta i=eta' i)
	{C ell : ℝ} (hC : 0≤C) (hl : 0≤ell) {B a : J → ℝ}
	(hB : ∀ i, 0≤B i) (ha : ∀ i, a i∈Set.Icc 0 1)
	(hcap : ∀ i, (∑ j, B j)a i≤(1+eta i)B i) :
	\|potential C ell (regular A' g h' eta' hh') B a-
	potential C ell (regular A f h eta hh) B a\| ≤
	8Cell*(∑ i, if h i=h' i then 0 else B i) := by
	have hS : 0≤∑ i, B i := sum_nonneg fun i _ => hB i
	have hi (i : J) :
	\|primitive (regularF (h' i)) C ell (eta' i) (∑ j, B j) (B i) 0 (a i)-
	primitive (regularF (h i)) C ell (eta i) (∑ j, B j) (B i) 0 (a i)\| ≤
	8Cell*(if h i=h' i then 0 else B i) := by
	by_cases hhx : h i=h' i
	· simp [hhx,heq i hhx]
	· rw [ite_eq_right hhx]
	have ho := regular_component_bound (hh i) (ha i) hC hl (he i).1 hS (hB i)
	have hn := regular_component_bound (hh' i) (ha i) hC hl (he' i).1 hS (hB i)
	have hep := mul_le_mul_of_nonneg_right (he i).2 (hB i)
	have he'p := mul_le_mul_of_nonneg_right (he' i).2 (hB i)
	have hc' := hcap i
	have hlo : (∑ j, B j)a i+(1+eta i)B i≤4*B i := by nlinarith
	have hln : (∑ j, B j)a i+(1+eta' i)B i≤4*B i := by nlinarith
	have hco := mul_le_mul_of_nonneg_left hlo (mul_nonneg hC hl)
	have hcn := mul_le_mul_of_nonneg_left hln (mul_nonneg hC hl)
	have hab := abs_sub
	(primitive (regularF (h' i)) C ell (eta' i) (∑ j, B j) (B i) 0 (a i))
	(primitive (regularF (h i)) C ell (eta i) (∑ j, B j) (B i) 0 (a i))
	nlinarith
	change \|(∑ i, primitive (regularF (h' i)) C ell (eta' i) (∑ j, B j) (B i) 0 (a i))-
	∑ i, primitive (regularF (h i)) C ell (eta i) (∑ j, B j) (B i) 0 (a i)\| ≤ _
	rw [←sum_sub_distrib,mul_sum]
	exact (abs_sum_le_sum_abs _ _).trans (sum_le_sum fun i _ => hi i)

	omit [DecidableEq J] in
	lemma total_variation_bound (B B' : J → ℝ) :
	\|(∑ i, B i)-(∑ i, B' i)\|≤ChangingDomains.variation B B' := by
	rw [←sum_sub_distrib]
	exact abs_sum_le_sum_abs _ _

	omit [Fintype J] in
	lemma transpose_cap {B : Finset J} {S : ℝ} {a v gamma : J → ℝ}
	(hv : ∀ i, 0≤v i) (hg : ∀ i, 0≤gamma i)
	(hcap : ∀ i, Sa i≤gamma iv i) :
	∀ i, S(ChangingDomains.transposeSimplex B S a) i≤gamma iv i := by
	intro i
	by_cases hs : S=0
	· simp only [hs,zero_mul]
	exact mul_nonneg (hg i) (hv i)
	· simpa [ChangingDomains.transposeSimplex,hs] using hcap i

	/-- This is the prepared minimizer from the manuscript; no drift inequality
	is an input to its definition. -/
	def regularPrepare (C ell ct : ℝ) (A B : Finset J) (g : J → R → Bool)
	(h : J → ℝ) (hh : ∀ i, 1≤h i) (p : J × R → ℝ) (a : J → ℝ) : J → ℝ :=
	(family C ell).update (regular A (preparedFlags A B g) h (fun i => ct*h i/ell) hh)
	p (∑ i, preparedInput A B g p i) a

	/-- Delete and insert zero coordinates following the prepared minimizing
	update, with a canonical simplex state only when its weight is exactly zero. -/
	def regularTransport (C ell ct : ℝ) (A B : Finset J) (g : J → R → Bool)
	(h : J → ℝ) (hh : ∀ i, 1≤h i) (p : J × R → ℝ) (a : J → ℝ) : J → ℝ :=
	ChangingDomains.transposeSimplex B (∑ i, preparedInput A B g p i)
	(regularPrepare C ell ct A B g h hh p a)

	lemma regular_transport_mem {A B : Finset J} (hA : A.Nonempty) (hB : B.Nonempty)
	(g : J → R → Bool) (h : J → ℝ) (hh : ∀ i, 1≤h i)
	{C ell ct ηmax Cexp : ℝ} (hC : 0<C) (hl : 0<ell) (hc : 0<ct)
	(hmax : ∀ i∈A, ct*h i/ell≤ηmax) (hCexp : 0≤Cexp)
	(htail : ∑ i∈A, Real.exp (-h i)≤Cexp)
	(hgap : 2(25+32Cexp)(1+ηmax)<Cct)
	(p : J × R → ℝ) (hp : ∀ ir, 0≤p ir) (a : J → ℝ)
	(ha : a∈ChangingDomains.simplex A) :
	regularTransport C ell ct A B g h hh p a∈ChangingDomains.simplex B := by
	have hp' : ∀ i r, 0≤p (i,r) := fun i r => hp (i,r)
	have hcap := regular_update_feasible hA (preparedFlags A B g) h hh hC hl hc hmax
	hCexp htail hgap p hp (prepared_flags_outside A B g) a ha
	exact ChangingDomains.transposeSimplex_mem (v := preparedInput A B g p)
	hA hB ((family C ell).update_mem _ _ _ _)
	(sum_nonneg fun i _ => ChangingDomains.core_nonneg hp' i) hcap
	(prepared_input_outside A B g p)

	end KServer.AdaptiveAllocation

	namespace KServer.AdaptiveAllocation
	open Finset SimplexAnalytic SimplexFamilies AdaptiveMinimization
	variable {J R : Type*} [Fintype J] [DecidableEq J] [Fintype R]

	lemma literal_update_payment (C ell : ℝ) (d : Data J R) (p : J × R → ℝ) (c : ℝ)
	{a : J → ℝ} (ha : a∈ChangingDomains.simplex d.active) :
	c*ChangingDomains.variation ((family C ell).update d p c a) a ≤
	potential C ell d (core d p) a-
	potential C ell d (core d p) ((family C ell).update d p c a) := by
	have hm := update_isMin C ell d p c a ha
	change potential C ell d (core d p) ((family C ell).update d p c a)+
	c*ChangingDomains.variation ((family C ell).update d p c a) a ≤
	potential C ell d (core d p) a+c*ChangingDomains.variation a a at hm
	rw [ChangingDomains.variation_self,mul_zero,add_zero] at hm
	linarith

	/-- The final update after genuine preparation and transport. -/
	def regularStep (C ell ct : ℝ) (A B : Finset J) (g : J → R → Bool)
	(h h' : J → ℝ) (hh : ∀ i, 1≤h i) (hh' : ∀ i, 1≤h' i)
	(p : J × R → ℝ) (a : J → ℝ) : J → ℝ :=
	(family C ell).update (regular B g h' (fun i => ct*h' i/ell) hh') p
	(∑ i, ChangingDomains.core g (fun i r => p (i,r)) i)
	(regularTransport C ell ct A B g h hh p a)

	lemma regular_step_payment {A B : Finset J} (hA : A.Nonempty) (hB : B.Nonempty)
	(f g : J → R → Bool) (h h' : J → ℝ)
	(hh : ∀ i, 1≤h i) (hh' : ∀ i, 1≤h' i)
	{C ell ct ηmax Cexp : ℝ} (hC : 0<C) (hl : 0<ell) (hc : 0<ct)
	(he : ∀ i, cth i/ell∈Set.Icc 0 1) (he' : ∀ i, cth' i/ell∈Set.Icc 0 1)
	(hmax : ∀ i∈A, ct*h i/ell≤ηmax) (hCexp : 0≤Cexp)
	(htail : ∑ i∈A, Real.exp (-h i)≤Cexp)
	(hgap : 2(25+32Cexp)(1+ηmax)<Cct)
	(p : J × R → ℝ) (hp : ∀ ir, p ir∈Set.Icc 0 1)
	(hf : ∀ i∉A, ∀ r, f i r=false) (hg : ∀ i∉B, ∀ r, g i r=false)
	(a : J → ℝ) (ha : a∈ChangingDomains.simplex A) :
	let d₀ := regular A f h (fun i => ct*h i/ell) hh
	let d₃ := regular B g h' (fun i => ct*h' i/ell) hh'
	let B₀ := core d₀ p
	let B₃ := core d₃ p
	let B₁ := preparedInput A B g p
	let a₃ := regularStep C ell ct A B g h h' hh hh' p a
	(∑ i, B₃ i)*ChangingDomains.variation a₃ a ≤
	potential C ell d₀ B₀ a-potential C ell d₃ B₃ a₃+
	(6Cell+4)*ChangingDomains.changedFlags f g+
	8Cell*(∑ i, if h i=h' i then 0 else B₁ i) := by
	intro d₀ d₃ B₀ B₃ B₁ a₃
	let d₁ := regular A (preparedFlags A B g) h (fun i => ct*h i/ell) hh
	let a₁ := regularPrepare C ell ct A B g h hh p a
	let a₂ := regularTransport C ell ct A B g h hh p a
	have hp₀ : ∀ ir, 0≤p ir := fun ir => (hp ir).1
	have hp₀' : ∀ i r, 0≤p (i,r) := fun i r => (hp (i,r)).1
	have hB₁ : ∀ i, 0≤B₁ i := fun i => ChangingDomains.core_nonneg hp₀' i
	have hS₁ : 0≤∑ i, B₁ i := sum_nonneg fun i _ => hB₁ i
	have ha₁ : a₁∈ChangingDomains.simplex A := (family C ell).update_mem _ _ _ _
	have ha₂ : a₂∈ChangingDomains.simplex B :=
	regular_transport_mem hA hB g h hh hC hl hc hmax hCexp htail hgap p hp₀ a ha
	have hcap₁ : ∀ i, (∑ j, B₁ j)a₁ i≤(1+cth i/ell)*B₁ i :=
	regular_update_feasible hA (preparedFlags A B g) h hh hC hl hc hmax hCexp htail hgap
	p hp₀ (prepared_flags_outside A B g) a ha
	have hcap₂ : ∀ i, (∑ j, B₁ j)a₂ i≤(1+cth i/ell)*B₁ i :=
	transpose_cap hB₁ (fun i => by linarith [(he i).1]) hcap₁
	have hin := ChangingDomains.synthetic_errors (fun i r => hp (i,r)) hf hg
	have herr₀ : ChangingDomains.variation B₀ B₁≤ChangingDomains.changedFlags f g := hin.1
	have herr₃ : ChangingDomains.variation B₃ B₁≤ChangingDomains.changedFlags f g := hin.2
	have hST : (∑ i, B₁ i)≤∑ i, B₃ i :=
	sub_nonneg.mp (ChangingDomains.synthetic_total (A := A) (B := B) (g := g) hp₀')
	have hδ : (∑ i, B₃ i)-(∑ i, B₁ i)≤ChangingDomains.changedFlags f g :=
	(le_abs_self _).trans ((total_variation_bound B₃ B₁).trans herr₃)
	have him₀ := regular_input_bound A hA f h (fun i => ct*h i/ell) hh he hC.le hl.le
	(B := B₀) (B' := B₁) ha
	have him₃ := regular_input_bound B hB g h' (fun i => ct*h' i/ell) hh' he' hC.le hl.le
	(B := B₃) (B' := B₁) ha₂
	have ht := regular_parameter_jump A B f g h h' (fun i => ct*h i/ell)
	(fun i => ct*h' i/ell) hh hh' he he' (fun i hi => by rw [hi]) hC.le hl.le
	hB₁ (fun i => ChangingDomains.simplex_coordinate ha₂ i) hcap₂
	have hpm₁ : (∑ i, B₁ i)*ChangingDomains.variation a₁ a ≤
	potential C ell d₀ B₁ a-potential C ell d₀ B₁ a₁ :=
	literal_update_payment C ell d₁ p (∑ i, B₁ i) ha
	have hpm₃ : (∑ i, B₃ i)*ChangingDomains.variation a₃ a₂ ≤
	potential C ell d₃ B₃ a₂-potential C ell d₃ B₃ a₃ :=
	literal_update_payment C ell d₃ p (∑ i, B₃ i) ha₂
	have htrans : potential C ell d₀ B₁ a₂=potential C ell d₀ B₁ a₁ :=
	regular_transpose_value A B f h (fun i => ct*h i/ell) hh C ell B₁ a₁ hB₁
	have hweight := ChangingDomains.prepared_weight_comparison ha ha₁ ha₂ hS₁ hST
	(ChangingDomains.transposeSimplex_payment B (∑ i, B₁ i) a₁) (a₃ := a₃)
	have hinput₀ := mul_le_mul_of_nonneg_left herr₀ (by positivity : 0≤3Cell)
	have hinput₃ := mul_le_mul_of_nonneg_left herr₃ (by positivity : 0≤3Cell)
	have he₀ := (neg_le_of_abs_le him₀)
	have he₃ := (le_of_abs_le him₃)
	have hep := le_of_abs_le ht
	change _ ≤ 8Cell*(∑ i, if h i=h' i then 0 else B₁ i) at hep
	change -(3Cell*ChangingDomains.variation B₀ B₁) ≤
	potential C ell d₀ B₀ a-potential C ell d₀ B₁ a at he₀
	change potential C ell d₃ B₃ a₂-potential C ell d₃ B₁ a₂ ≤
	3Cell*ChangingDomains.variation B₃ B₁ at he₃
	change potential C ell d₃ B₁ a₂-potential C ell d₀ B₁ a₂ ≤ _ at hep
	nlinarith

	end KServer.AdaptiveAllocation

	namespace KServer.AdaptiveAllocation
	open Finset SimplexAnalytic SimplexFamilies AdaptiveMinimization
	variable {J R : Type*} [Fintype J] [DecidableEq J] [Fintype R]

	lemma marked_update_feasible {A : Finset J} {o : J} (hoA : o∈A)
	(f : J → R → Bool) (h : J → ℝ) {u : ℝ} (hu : 0<u) (hu1 : u≤1)
	(ho : h o=u) (hh : ∀ i≠o, 1≤h i)
	{C ell ct ηmax Cexp c : ℝ} (hC : 0<C) (hl : 0<ell) (hc : 0<ct)
	(hmax : ∀ i∈A, ct*h i/ell≤ηmax) (hCexp : 0≤Cexp)
	(hsideC : 0<c) (hsideC1 : c≤1)
	(hside : ∀ i∈A, i≠o → c*(1+Real.log (1/u))≤h i)
	(htail : ∑ i∈A.erase o, Real.exp (-h i)≤Cexp*u)
	(hgap : 2((25c+2+32Cexp)/c)(1+ηmax)<C*ct)
	(p : J × R → ℝ) (hp : ∀ ir, 0≤p ir) (hf : ∀ i∉A, ∀ r, f i r=false)
	(a : J → ℝ) (ha : a∈ChangingDomains.simplex A) :
	let d := marked A f o h (fun i => ct*h i/ell) u hu hu1 hh
	let B := core d p
	let a' := (family C ell).update d p (∑ i, B i) a
	∀ i, (∑ j, B j)a' i≤(1+cth i/ell)*B i := by
	intro d B a'
	have hB : ∀ i, 0≤B i := fun i => ChangingDomains.core_nonneg (fun i r => hp (i,r)) i
	have hBz : ∀ i∉A, B i=0 := by
	intro i hi
	simp [B,core,d,marked,hf i hi]
	apply SupportedAllocation.marked_supported_minimizer_feasible hoA
	((family C ell).update_mem d p (∑ i, B i) a) ha hu hu1 rfl ho
	(fun i _ hi => hh i hi) hc hl hmax hC hB hBz hCexp hsideC hsideC1 hside htail hgap
	exact update_isMin C ell d p (∑ i, B i) a

	omit [Fintype R] in
	lemma marked_input_bound (A : Finset J) (hA : A.Nonempty) (f : J → R → Bool)
	(o : J) (h eta : J → ℝ) {u : ℝ} (hu : 0<u) (hu1 : u≤1)
	(hh : ∀ i≠o, 1≤h i) (he : ∀ i, eta i∈Set.Icc 0 1)
	{C ell : ℝ} (hC : 0≤C) (hl : 0≤ell) {B B' a : J → ℝ}
	(ha : a∈ChangingDomains.simplex A) :
	\|potential C ell (marked A f o h eta u hu hu1 hh) B a-
	potential C ell (marked A f o h eta u hu hu1 hh) B' a\| ≤
	13Cell*ChangingDomains.variation B B' := by
	have hs : (∑ i, a i)=1 := by simpa only [ite_eq_left hA] using ha.2.1
	exact marked_input_slopes o ha.1 hs hh hu hu1 rfl (fun i => (he i).1)
	(fun i => (he i).2) hC hl

	omit [Fintype R] in
	lemma marked_potential_zero (A : Finset J) (f : J → R → Bool) (o : J)
	(h eta : J → ℝ) {u : ℝ} (hu : 0<u) (hu1 : u≤1) (hh : ∀ i≠o, 1≤h i)
	(C ell : ℝ) (a : J → ℝ) :
	potential C ell (marked A f o h eta u hu hu1 hh) (fun _ => 0) a=0 := by
	simp [potential,marked,primitive]

	omit [Fintype R] in
	lemma marked_transpose_value (A B : Finset J) (f : J → R → Bool) (o : J)
	(h eta : J → ℝ) {u : ℝ} (hu : 0<u) (hu1 : u≤1) (hh : ∀ i≠o, 1≤h i)
	(C ell : ℝ) (v a : J → ℝ) (hv : ∀ i, 0≤v i) :
	potential C ell (marked A f o h eta u hu hu1 hh) v
	(ChangingDomains.transposeSimplex B (∑ i, v i) a)=
	potential C ell (marked A f o h eta u hu hu1 hh) v a := by
	by_cases hS : (∑ i, v i)=0
	· rw [nonneg_total_zero hv hS]
	simp only [marked_potential_zero]
	· simp [ChangingDomains.transposeSimplex,hS]

	lemma simplex_side_weight {A : Finset J} (hA : A.Nonempty) {a B eta : J → ℝ}
	(ha : a∈ChangingDomains.simplex A) (hB : ∀ i, 0≤B i)
	(he : ∀ i, eta i≤1) (hcap : ∀ i, (∑ j, B j)a i≤(1+eta i)B i) (o : J) :
	(∑ j, B j)(1-a o)≤2(∑ i∈Finset.univ.erase o, B i) := by
	have hs : (∑ i, a i)=1 := by simpa only [ite_eq_left hA] using ha.2.1
	have hx : 1-a o=∑ i∈Finset.univ.erase o, a i := by
	have ht := Finset.add_sum_erase Finset.univ (fun i => a i) (Finset.mem_univ o)
	rw [hs] at ht
	linarith
	rw [hx,mul_sum,mul_sum]
	exact sum_le_sum fun i _ => (hcap i).trans
	(mul_le_mul_of_nonneg_right (by linarith [he i]) (hB i))

	omit [Fintype R] in
	/-- Marked parameter edits charge the dominant component only when `u`
	changes, and charge it to the side total, not the dominant mass. -/
	lemma marked_parameter_jump (A A' : Finset J) (hA' : A'.Nonempty)
	(f g : J → R → Bool) (o : J) (h h' eta eta' : J → ℝ)
	{u u' : ℝ} (hu : 0<u) (hu1 : u≤1) (hu' : 0<u') (hu1' : u'≤1)
	(hh : ∀ i≠o, 1≤h i) (hh' : ∀ i≠o, 1≤h' i)
	(he : ∀ i, eta i∈Set.Icc 0 1) (he' : ∀ i, eta' i∈Set.Icc 0 1)
	(heq : ∀ i, h i=h' i → eta i=eta' i)
	(heqo : u=u' → eta o=eta' o) (heo : eta o≤u) (heo' : eta' o≤u')
	{C ell : ℝ} (hC : 0≤C) (hl : 0≤ell) {B a : J → ℝ}
	(hB : ∀ i, 0≤B i) (ha : a∈ChangingDomains.simplex A')
	(hcap : ∀ i, (∑ j, B j)a i≤(1+eta i)B i) :
	\|potential C ell (marked A' g o h' eta' u' hu' hu1' hh') B a-
	potential C ell (marked A f o h eta u hu hu1 hh) B a\| ≤
	8Cell*(∑ i∈Finset.univ.erase o, if h i=h' i then 0 else B i)+
	24Cell*(if u=u' then 0 else
	(∑ i∈Finset.univ.erase o, B i)+(u+u')*(∑ i, B i)) := by
	let d₀ := marked A f o h eta u hu hu1 hh
	let d₁ := marked A' g o h' eta' u' hu' hu1' hh'
	let term := fun i => primitive (d₁.F i) C ell (eta' i) (∑ j, B j) (B i) (d₁.base i) (a i)-
	primitive (d₀.F i) C ell (eta i) (∑ j, B j) (B i) (d₀.base i) (a i)
	have hS : 0≤∑ i, B i := sum_nonneg fun i _ => hB i
	have hBs : 0≤∑ i∈Finset.univ.erase o, B i := sum_nonneg fun i _ => hB i
	have hai := fun i => ChangingDomains.simplex_coordinate ha i
	have hside := simplex_side_weight hA' ha hB (fun i => (he i).2) hcap o
	have hSpr : (∑ j, B j)=B o+∑ i∈Finset.univ.erase o, B i :=
	(Finset.add_sum_erase Finset.univ B (Finset.mem_univ o)).symm
	have hregular (i : J) (hi : i∈Finset.univ.erase o) :
	\|term i\|≤8Cell*(if h i=h' i then 0 else B i) := by
	have hio : i≠o := (Finset.mem_erase.mp hi).1
	change \|primitive (if i=o then markedF u' (1+Real.log (1/u')) else regularF (h' i))
	C ell (eta' i) (∑ j, B j) (B i) (if i=o then 1 else 0) (a i)-
	primitive (if i=o then markedF u (1+Real.log (1/u)) else regularF (h i))
	C ell (eta i) (∑ j, B j) (B i) (if i=o then 1 else 0) (a i)\|≤_
	simp only [ite_eq_right hio]
	by_cases hx : h i=h' i
	· simp [hx,heq i hx]
	· rw [ite_eq_right hx]
	have ho := regular_component_bound (hh i hio) (hai i) hC hl (he i).1 hS (hB i)
	have hn := regular_component_bound (hh' i hio) (hai i) hC hl (he' i).1 hS (hB i)
	have hep := mul_le_mul_of_nonneg_right (he i).2 (hB i)
	have he'p := mul_le_mul_of_nonneg_right (he' i).2 (hB i)
	have hlo : (∑ j, B j)a i+(1+eta i)B i≤4*B i := by nlinarith [hcap i]
	have hln : (∑ j, B j)a i+(1+eta' i)B i≤4*B i := by nlinarith [hcap i]
	have hco := mul_le_mul_of_nonneg_left hlo (mul_nonneg hC hl)
	have hcn := mul_le_mul_of_nonneg_left hln (mul_nonneg hC hl)
	have hab := abs_sub
	(primitive (regularF (h' i)) C ell (eta' i) (∑ j, B j) (B i) 0 (a i))
	(primitive (regularF (h i)) C ell (eta i) (∑ j, B j) (B i) 0 (a i))
	nlinarith
	have hmarked : \|term o\|≤24Cell*(if u=u' then 0 else
	(∑ i∈Finset.univ.erase o, B i)+(u+u')*(∑ i, B i)) := by
	change \|primitive (if o=o then markedF u' (1+Real.log (1/u')) else regularF (h' o))
	C ell (eta' o) (∑ j, B j) (B o) (if o=o then 1 else 0) (a o)-
	primitive (if o=o then markedF u (1+Real.log (1/u)) else regularF (h o))
	C ell (eta o) (∑ j, B j) (B o) (if o=o then 1 else 0) (a o)\|≤_
	simp only [ite_true]
	by_cases hx : u=u'
	· simp [hx,heqo hx]
	· rw [ite_eq_right hx]
	have ho := marked_component_bound hu hu1 rfl (hai o) hC hl (he o).1 heo (hB o) hBs hSpr
	have hn := marked_component_bound hu' hu1' rfl (hai o) hC hl (he' o).1 heo' (hB o) hBs hSpr
	have hab := abs_sub
	(primitive (markedF u' (1+Real.log (1/u'))) C ell (eta' o) (∑ j, B j) (B o) 1 (a o))
	(primitive (markedF u (1+Real.log (1/u))) C ell (eta o) (∑ j, B j) (B o) 1 (a o))
	have hs := mul_le_mul_of_nonneg_left hside (by positivity : 0≤8Cell)
	have hpos : 0≤Cell(u+u')*(∑ j, B j) := by positivity
	nlinarith
	change \|(∑ i, primitive (d₁.F i) C ell (eta' i) (∑ j, B j) (B i) (d₁.base i) (a i))-
	∑ i, primitive (d₀.F i) C ell (eta i) (∑ j, B j) (B i) (d₀.base i) (a i)\|≤_
	rw [←sum_sub_distrib]
	change \|∑ i, term i\|≤_
	have ht := (abs_sum_le_sum_abs term Finset.univ)
	rw [←Finset.add_sum_erase Finset.univ (fun i => \|term i\|) (Finset.mem_univ o)] at ht
	have hr := sum_le_sum hregular
	rw [←mul_sum] at hr
	linarith

	end KServer.AdaptiveAllocation

	namespace KServer.AdaptiveAllocation
	open Finset SimplexAnalytic SimplexFamilies AdaptiveMinimization
	variable {J R : Type*} [Fintype J] [DecidableEq J] [Fintype R]

	def markedPrepare (C ell ct : ℝ) (A B : Finset J) (g : J → R → Bool)
	(o : J) (h : J → ℝ) (u : ℝ) (hu : 0<u) (hu1 : u≤1)
	(hh : ∀ i≠o, 1≤h i) (p : J × R → ℝ) (a : J → ℝ) : J → ℝ :=
	(family C ell).update
	(marked A (preparedFlags A B g) o h (fun i => ct*h i/ell) u hu hu1 hh)
	p (∑ i, preparedInput A B g p i) a

	def markedTransport (C ell ct : ℝ) (A B : Finset J) (g : J → R → Bool)
	(o : J) (h : J → ℝ) (u : ℝ) (hu : 0<u) (hu1 : u≤1)
	(hh : ∀ i≠o, 1≤h i) (p : J × R → ℝ) (a : J → ℝ) : J → ℝ :=
	ChangingDomains.transposeSimplex B (∑ i, preparedInput A B g p i)
	(markedPrepare C ell ct A B g o h u hu hu1 hh p a)

	lemma marked_transport_mem {A B : Finset J} {o : J} (hoA : o∈A) (hB : B.Nonempty)
	(g : J → R → Bool) (h : J → ℝ) {u : ℝ} (hu : 0<u) (hu1 : u≤1)
	(ho : h o=u) (hh : ∀ i≠o, 1≤h i)
	{C ell ct ηmax Cexp c : ℝ} (hC : 0<C) (hl : 0<ell) (hc : 0<ct)
	(hmax : ∀ i∈A, ct*h i/ell≤ηmax) (hCexp : 0≤Cexp)
	(hsideC : 0<c) (hsideC1 : c≤1)
	(hside : ∀ i∈A, i≠o → c*(1+Real.log (1/u))≤h i)
	(htail : ∑ i∈A.erase o, Real.exp (-h i)≤Cexp*u)
	(hgap : 2((25c+2+32Cexp)/c)(1+ηmax)<C*ct)
	(p : J × R → ℝ) (hp : ∀ ir, 0≤p ir) (a : J → ℝ)
	(ha : a∈ChangingDomains.simplex A) :
	markedTransport C ell ct A B g o h u hu hu1 hh p a∈ChangingDomains.simplex B := by
	have hp' : ∀ i r, 0≤p (i,r) := fun i r => hp (i,r)
	have hcap := marked_update_feasible hoA (preparedFlags A B g) h hu hu1 ho hh
	hC hl hc hmax hCexp hsideC hsideC1 hside htail hgap p hp
	(prepared_flags_outside A B g) a ha
	exact ChangingDomains.transposeSimplex_mem (v := preparedInput A B g p)
	⟨o,hoA⟩ hB ((family C ell).update_mem _ _ _ _)
	(sum_nonneg fun i _ => ChangingDomains.core_nonneg hp' i) hcap
	(prepared_input_outside A B g p)

	def markedStep (C ell ct : ℝ) (A B : Finset J) (g : J → R → Bool)
	(o : J) (h h' : J → ℝ) (u u' : ℝ) (hu : 0<u) (hu1 : u≤1)
	(hu' : 0<u') (hu1' : u'≤1) (hh : ∀ i≠o, 1≤h i) (hh' : ∀ i≠o, 1≤h' i)
	(p : J × R → ℝ) (a : J → ℝ) : J → ℝ :=
	(family C ell).update (marked B g o h' (fun i => ct*h' i/ell) u' hu' hu1' hh') p
	(∑ i, ChangingDomains.core g (fun i r => p (i,r)) i)
	(markedTransport C ell ct A B g o h u hu hu1 hh p a)

	lemma marked_step_payment {A B : Finset J} {o : J} (hoA : o∈A) (hB : B.Nonempty)
	(f g : J → R → Bool) (h h' : J → ℝ) {u u' : ℝ}
	(hu : 0<u) (hu1 : u≤1) (hu' : 0<u') (hu1' : u'≤1)
	(ho : h o=u) (ho' : h' o=u') (hh : ∀ i≠o, 1≤h i) (hh' : ∀ i≠o, 1≤h' i)
	{C ell ct ηmax Cexp c : ℝ} (hC : 0<C) (hl : 0<ell) (hc : 0<ct)
	(he : ∀ i, cth i/ell∈Set.Icc 0 1) (he' : ∀ i, cth' i/ell∈Set.Icc 0 1)
	(hoeta : ctu/ell≤u) (hoeta' : ctu'/ell≤u')
	(hmax : ∀ i∈A, ct*h i/ell≤ηmax) (hCexp : 0≤Cexp)
	(hsideC : 0<c) (hsideC1 : c≤1)
	(hside : ∀ i∈A, i≠o → c*(1+Real.log (1/u))≤h i)
	(htail : ∑ i∈A.erase o, Real.exp (-h i)≤Cexp*u)
	(hgap : 2((25c+2+32Cexp)/c)(1+ηmax)<C*ct)
	(p : J × R → ℝ) (hp : ∀ ir, p ir∈Set.Icc 0 1)
	(hf : ∀ i∉A, ∀ r, f i r=false) (hg : ∀ i∉B, ∀ r, g i r=false)
	(a : J → ℝ) (ha : a∈ChangingDomains.simplex A) :
	let d₀ := marked A f o h (fun i => ct*h i/ell) u hu hu1 hh
	let d₃ := marked B g o h' (fun i => ct*h' i/ell) u' hu' hu1' hh'
	let B₀ := core d₀ p
	let B₃ := core d₃ p
	let B₁ := preparedInput A B g p
	let a₃ := markedStep C ell ct A B g o h h' u u' hu hu1 hu' hu1' hh hh' p a
	(∑ i, B₃ i)*ChangingDomains.variation a₃ a ≤
	potential C ell d₀ B₀ a-potential C ell d₃ B₃ a₃+
	(26Cell+4)*ChangingDomains.changedFlags f g+
	8Cell*(∑ i∈Finset.univ.erase o, if h i=h' i then 0 else B₁ i)+
	24Cell*(if u=u' then 0 else
	(∑ i∈Finset.univ.erase o, B₁ i)+(u+u')*(∑ i, B₁ i)) := by
	intro d₀ d₃ B₀ B₃ B₁ a₃
	let d₁ := marked A (preparedFlags A B g) o h (fun i => ct*h i/ell) u hu hu1 hh
	let a₁ := markedPrepare C ell ct A B g o h u hu hu1 hh p a
	let a₂ := markedTransport C ell ct A B g o h u hu hu1 hh p a
	have hA : A.Nonempty := ⟨o,hoA⟩
	have hp₀ : ∀ ir, 0≤p ir := fun ir => (hp ir).1
	have hp₀' : ∀ i r, 0≤p (i,r) := fun i r => (hp (i,r)).1
	have hB₁ : ∀ i, 0≤B₁ i := fun i => ChangingDomains.core_nonneg hp₀' i
	have hS₁ : 0≤∑ i, B₁ i := sum_nonneg fun i _ => hB₁ i
	have ha₁ : a₁∈ChangingDomains.simplex A := (family C ell).update_mem _ _ _ _
	have ha₂ : a₂∈ChangingDomains.simplex B :=
	marked_transport_mem hoA hB g h hu hu1 ho hh hC hl hc hmax hCexp
	hsideC hsideC1 hside htail hgap p hp₀ a ha
	have hcap₁ : ∀ i, (∑ j, B₁ j)a₁ i≤(1+cth i/ell)*B₁ i :=
	marked_update_feasible hoA (preparedFlags A B g) h hu hu1 ho hh hC hl hc hmax hCexp
	hsideC hsideC1 hside htail hgap p hp₀ (prepared_flags_outside A B g) a ha
	have hcap₂ : ∀ i, (∑ j, B₁ j)a₂ i≤(1+cth i/ell)*B₁ i :=
	transpose_cap hB₁ (fun i => by linarith [(he i).1]) hcap₁
	have hin := ChangingDomains.synthetic_errors (fun i r => hp (i,r)) hf hg
	have herr₀ : ChangingDomains.variation B₀ B₁≤ChangingDomains.changedFlags f g := hin.1
	have herr₃ : ChangingDomains.variation B₃ B₁≤ChangingDomains.changedFlags f g := hin.2
	have hST : (∑ i, B₁ i)≤∑ i, B₃ i :=
	sub_nonneg.mp (ChangingDomains.synthetic_total (A := A) (B := B) (g := g) hp₀')
	have hδ : (∑ i, B₃ i)-(∑ i, B₁ i)≤ChangingDomains.changedFlags f g :=
	(le_abs_self _).trans ((total_variation_bound B₃ B₁).trans herr₃)
	have him₀ := marked_input_bound A hA f o h (fun i => ct*h i/ell) hu hu1 hh he
	hC.le hl.le (B := B₀) (B' := B₁) ha
	have him₃ := marked_input_bound B hB g o h' (fun i => ct*h' i/ell) hu' hu1' hh' he'
	hC.le hl.le (B := B₃) (B' := B₁) ha₂
	have ht := marked_parameter_jump A B hB f g o h h' (fun i => ct*h i/ell)
	(fun i => ct*h' i/ell) hu hu1 hu' hu1' hh hh' he he'
	(fun i hi => by rw [hi]) (fun hi => by rw [ho,ho',hi])
	(by simpa [ho] using hoeta) (by simpa [ho'] using hoeta') hC.le hl.le hB₁ ha₂ hcap₂
	have hpm₁ : (∑ i, B₁ i)*ChangingDomains.variation a₁ a ≤
	potential C ell d₀ B₁ a-potential C ell d₀ B₁ a₁ :=
	literal_update_payment C ell d₁ p (∑ i, B₁ i) ha
	have hpm₃ : (∑ i, B₃ i)*ChangingDomains.variation a₃ a₂ ≤
	potential C ell d₃ B₃ a₂-potential C ell d₃ B₃ a₃ :=
	literal_update_payment C ell d₃ p (∑ i, B₃ i) ha₂
	have htrans : potential C ell d₀ B₁ a₂=potential C ell d₀ B₁ a₁ :=
	marked_transpose_value A B f o h (fun i => ct*h i/ell) hu hu1 hh C ell B₁ a₁ hB₁
	have hweight := ChangingDomains.prepared_weight_comparison ha ha₁ ha₂ hS₁ hST
	(ChangingDomains.transposeSimplex_payment B (∑ i, B₁ i) a₁) (a₃ := a₃)
	have hinput₀ := mul_le_mul_of_nonneg_left herr₀ (by positivity : 0≤13Cell)
	have hinput₃ := mul_le_mul_of_nonneg_left herr₃ (by positivity : 0≤13Cell)
	have he₀ := neg_le_of_abs_le him₀
	have he₃ := le_of_abs_le him₃
	have hep := le_of_abs_le ht
	change -(13Cell*ChangingDomains.variation B₀ B₁) ≤
	potential C ell d₀ B₀ a-potential C ell d₀ B₁ a at he₀
	change potential C ell d₃ B₃ a₂-potential C ell d₃ B₁ a₂ ≤
	13Cell*ChangingDomains.variation B₃ B₁ at he₃
	change potential C ell d₃ B₁ a₂-potential C ell d₀ B₁ a₂ ≤ _ at hep
	nlinarith

	end KServer.AdaptiveAllocation

	/-! Exact side primitive and compact minimizing update of §04,
	eq:side-primitive-explicit and lem:side-tracker. This is an allocation
	dependency of the absent main, not a separately counted main theorem. -/
	noncomputable section
	open scoped BigOperators
	open Set Filter
	namespace KServer.SideTracker

	/-- The explicit integral in eq:side-primitive-explicit. -/
	def primitive (c b θ z B D w : ℝ) : ℝ :=
	c * (D / θ * (w - z * Real.log (1 + w / z)) +
	D * z * Real.log (1 + w / z) / 2 -
	(b + θ / 2) * B * Real.log (1 + w / z) / θ)

	lemma primitive_derivative {c b θ z B D w : ℝ} (hθ : 0 < θ) (hz : 0 < z)
	(hw : 0 ≤ w) :
	HasDerivAt (primitive c b θ z B D)
	(c * (D * w - (b + θ / 2) * B + D * θ * z / 2) /
	(θ * (w + z))) w := by
	have hl : 0 < 1 + w / z := by positivity
	have hd := (((hasDerivAt_id w).div_const z).const_add 1).log hl.ne'
	have he : 1 / z / (1 + w / z) = 1 / (w + z) := by
	field_simp
	ring
	simp only [id_eq] at hd
	rw [he] at hd
	have H := ((((hasDerivAt_id w).sub (hd.const_mul z)).const_mul (D / θ)).add
	((hd.const_mul (D * z)).div_const 2)).sub
	((hd.const_mul ((b + θ / 2) * B)).div_const θ)
	have hwz : w + z ≠ 0 := by positivity
	have he' : c * (D * w - (b + θ / 2) * B + D * θ * z / 2) / (θ * (w + z)) =
	c * (D / θ * (1 - z * (1 / (w + z))) + D * z * (1 / (w + z)) / 2 -
	(b + θ / 2) * B * (1 / (w + z)) / θ) := by
	field_simp [hθ.ne', hwz]
	ring
	rw [he']
	exact H.const_mul c

	/-- One-sided necessary optimality, allowing a nondifferentiable movement cost. -/
	lemma derivative_le_of_left_comparison {f : ℝ → ℝ} {f' D x : ℝ}
	(hx : 0 < x) (hd : HasDerivAt f f' x)
	(hc : ∀ y ∈ Set.Icc 0 x, f x ≤ f y + D * (x - y)) : f' ≤ D := by
	have hm : IsLocalMinOn (fun y => f y - D * y) (Set.Icc 0 x) x := by
	apply IsMinOn.isLocalMinOn
	intro y hy
	have h := hc y hy
	change f x - D * x ≤ f y - D * y
	linarith
	have ht : 0 - x ∈ posTangentConeAt (Set.Icc 0 x) x :=
	sub_mem_posTangentConeAt_of_segment_subset
	((convex_Icc 0 x).segment_subset ⟨hx.le, le_rfl⟩ ⟨le_rfl, hx.le⟩)
	have h := hm.hasFDerivWithinAt_nonneg
	((hd.sub ((hasDerivAt_id x).const_mul D)).hasFDerivAt.hasFDerivWithinAt) ht
	simp only [ContinuousLinearMap.toSpanSingleton_apply, smul_eq_mul] at h
	nlinarith

	lemma neg_derivative_le_of_right_comparison {f : ℝ → ℝ} {f' D x ε : ℝ}
	(hε : 0 < ε) (hd : HasDerivAt f f' x)
	(hc : ∀ y ∈ Set.Icc x (x + ε), f x ≤ f y + D * (y - x)) : -D ≤ f' := by
	have hm : IsLocalMinOn (fun y => f y + D * y) (Set.Icc x (x + ε)) x := by
	apply IsMinOn.isLocalMinOn
	intro y hy
	have h := hc y hy
	change f x + D * x ≤ f y + D * y
	linarith
	have ht : (x + ε) - x ∈ posTangentConeAt (Set.Icc x (x + ε)) x :=
	sub_mem_posTangentConeAt_of_segment_subset
	((convex_Icc x (x + ε)).segment_subset ⟨le_rfl, by linarith⟩
	⟨by linarith, le_rfl⟩)
	have h := hm.hasFDerivWithinAt_nonneg
	((hd.add ((hasDerivAt_id x).const_mul D)).hasFDerivAt.hasFDerivWithinAt) ht
	simp only [ContinuousLinearMap.toSpanSingleton_apply, smul_eq_mul] at h
	nlinarith

	variable {ι : Type*} [Fintype ι]

	def domain (Cw : ℝ) : Set (ι → ℝ) := {w \| (∀ i, 0 ≤ w i) ∧ ∑ i, w i ≤ Cw}

	def potential (c b : ℝ) (θ z B : ι → ℝ) (D : ℝ) (w : ι → ℝ) : ℝ :=
	∑ i, primitive c b (θ i) (z i) (B i) D (w i)

	def movement (w v : ι → ℝ) : ℝ := ∑ i, \|w i - v i\|

	def objective (c b : ℝ) (θ z B : ι → ℝ) (D : ℝ) (w₀ w : ι → ℝ) : ℝ :=
	potential c b θ z B D w + D * movement w w₀

	lemma movement_self (w : ι → ℝ) : movement w w = 0 := by simp [movement]

	lemma domain_isCompact (Cw : ℝ) : IsCompact (domain (ι := ι) Cw) := by
	have hc : IsClosed (domain (ι := ι) Cw) := by
	change IsClosed ({w : ι → ℝ \| ∀ i, 0 ≤ w i} ∩ {w : ι → ℝ \| ∑ i, w i ≤ Cw})
	apply IsClosed.inter
	· simp only [ofPred_forall]
	exact isClosed_iInter fun i => isClosed_le continuous_const (continuous_apply i)
	· exact isClosed_le (continuous_finsetSum _ fun i _ => continuous_apply i) continuous_const
	apply (isCompact_pi_infinite fun _ : ι => isCompact_Icc (a := (0 : ℝ)) (b := Cw)).of_isClosed_subset hc
	intro w hw i
	exact ⟨hw.1 i, (Finset.single_le_sum (fun j _ => hw.1 j) (Finset.mem_univ i)).trans hw.2⟩

	lemma primitive_continuousOn {c b θ z B D : ℝ} (hθ : 0 < θ) (hz : 0 < z) :
	ContinuousOn (primitive c b θ z B D) (Ici 0) := by
	intro w hw
	exact (primitive_derivative hθ hz hw).continuousAt.continuousWithinAt

	lemma objective_continuousOn (c b D Cw : ℝ) (θ z B w₀ : ι → ℝ)
	(hθ : ∀ i, 0 < θ i) (hz : ∀ i, 0 < z i) :
	ContinuousOn (objective c b θ z B D w₀) (domain Cw) := by
	apply ContinuousOn.add
	· apply continuousOn_finsetSum _
	intro i _
	exact (primitive_continuousOn (hθ i) (hz i)).comp
	(continuous_apply i).continuousOn (fun _ hw => hw.1 i)
	· apply ContinuousOn.const_mul
	apply continuousOn_finsetSum _
	intro i _
	exact ((continuous_apply i).sub continuous_const).abs.continuousOn

	lemma exists_minimizer {c b D Cw : ℝ} {θ z B w₀ : ι → ℝ}
	(hθ : ∀ i, 0 < θ i) (hz : ∀ i, 0 < z i) (hw : w₀ ∈ domain Cw) :
	∃ w ∈ domain Cw, IsMinOn (objective c b θ z B D w₀) (domain Cw) w :=
	(domain_isCompact Cw).exists_isMinOn ⟨w₀, hw⟩
	(objective_continuousOn c b D Cw θ z B w₀ hθ hz)

	lemma minimizer_payment {c b D Cw : ℝ} {θ z B w₀ w : ι → ℝ}
	(hw₀ : w₀ ∈ domain Cw)
	(hm : IsMinOn (objective c b θ z B D w₀) (domain Cw) w) :
	D * movement w w₀ ≤ potential c b θ z B D w₀ - potential c b θ z B D w := by
	have h := hm hw₀
	change potential c b θ z B D w + D * movement w w₀ ≤
	potential c b θ z B D w₀ + D * movement w₀ w₀ at h
	rw [movement_self, mul_zero, add_zero] at h
	linarith

	lemma sum_replace [DecidableEq ι] (f : ι → ℝ → ℝ) (w : ι → ℝ) (i : ι) (y : ℝ) :
	(∑ j, f j (Function.update w i y j)) = (∑ j, f j (w j)) - f i (w i) + f i y := by
	have he : (∑ j ∈ Finset.univ.erase i, f j (Function.update w i y j)) =
	∑ j ∈ Finset.univ.erase i, f j (w j) := by
	apply Finset.sum_congr rfl
	intro j hj
	simp [Function.update, (Finset.mem_erase.mp hj).1]
	have h₁ := Finset.sum_erase_add Finset.univ (fun j => f j (Function.update w i y j))
	(Finset.mem_univ i)
	have h₂ := Finset.sum_erase_add Finset.univ (fun j => f j (w j)) (Finset.mem_univ i)
	simp only [Function.update_self, he] at h₁
	linarith

	lemma replace_mem_domain [DecidableEq ι] {Cw : ℝ} {w : ι → ℝ}
	(hw : w ∈ domain Cw) (i : ι) {y : ℝ} (hy : 0 ≤ y)
	(hs : y ≤ w i + (Cw - ∑ j, w j)) : Function.update w i y ∈ domain Cw := by
	constructor
	· intro j
	by_cases h : j = i
	· subst j; simpa using hy
	· simpa [Function.update, h] using hw.1 j
	· rw [sum_replace (fun _ x => x)]
	linarith

	lemma coordinate_min_comparison [DecidableEq ι] {c b D Cw : ℝ} {θ z B w₀ w : ι → ℝ}
	(hD : 0 ≤ D)
	(hm : IsMinOn (objective c b θ z B D w₀) (domain Cw) w)
	(i : ι) {y : ℝ} (hy : Function.update w i y ∈ domain Cw) :
	primitive c b (θ i) (z i) (B i) D (w i) ≤
	primitive c b (θ i) (z i) (B i) D y + D * \|y - w i\| := by
	have h := hm hy
	change potential c b θ z B D w + D * movement w w₀ ≤
	potential c b θ z B D (Function.update w i y) +
	D * movement (Function.update w i y) w₀ at h
	dsimp [potential, movement] at h
	rw [sum_replace (fun j x => primitive c b (θ j) (z j) (B j) D x),
	sum_replace (fun j x => \|x - w₀ j\|)] at h
	have ht := mul_le_mul_of_nonneg_left (abs_sub_le y (w i) (w₀ i)) hD
	linarith

	lemma derivative_upper_violation {M b θ z B D w : ℝ}
	(hb : 1 ≤ b) (hθ : 0 < θ) (hz : 0 < z) (hB : 0 ≤ B)
	(hD : 0 < D) (hw : 0 ≤ w) (hM : b + θ ≤ M)
	(hv : (b + θ) * B < D * w) :
	D < 8 * M * (D * w - (b + θ / 2) * B + D * θ * z / 2) / (θ * (w + z)) := by
	let N := D * w - (b + θ / 2) * B + D * θ * z / 2
	have hN : 0 < N := by
	dsimp [N]
	nlinarith [mul_nonneg hθ.le hB, mul_pos (mul_pos hD hθ) hz]
	have h1 : D * θ * w + D * θ * (b + θ) * z < 2 * (b + θ) * N := by
	have hh : 0 < (2 * b + θ) * (D * w - (b + θ) * B) :=
	mul_pos (by linarith) (sub_pos.mpr hv)
	dsimp [N]
	nlinarith
	have h2 : D * θ * (w + z) ≤ D * θ * w + D * θ * (b + θ) * z := by
	have hh := mul_le_mul_of_nonneg_left (show 1 ≤ b + θ by linarith)
	(mul_nonneg (mul_nonneg hD.le hθ.le) hz.le)
	nlinarith
	have h3 : 2 * (b + θ) * N ≤ 8 * M * N := by
	have hh := mul_le_mul_of_nonneg_right hM hN.le
	nlinarith [mul_pos (show 0 < M by linarith) hN]
	apply (lt_div_iff₀ (mul_pos hθ (by linarith : 0 < w + z))).mpr
	dsimp [N] at h2 h1 h3
	nlinarith [h2.trans_lt (h1.trans_le h3)]

	lemma derivative_lower_violation {M b θ z B D w : ℝ}
	(hb : 1 ≤ b) (hθ : 0 < θ) (hz : 0 < z)
	(hD : 0 < D) (hw : 0 ≤ w) (hM : b + θ ≤ M)
	(hv : D * w < b * B - θ * z * D) :
	8 * M * (D * w - (b + θ / 2) * B + D * θ * z / 2) / (θ * (w + z)) < -D := by
	let N := D * w - (b + θ / 2) * B + D * θ * z / 2
	have hB : 0 < B := by
	have : 0 < b * B := by nlinarith [mul_nonneg hD.le hw, mul_pos (mul_pos hθ hz) hD]
	exact (mul_pos_iff.mp this).resolve_right (by rintro ⟨h, _⟩; linarith) \|>.2
	have hN : N < 0 := by
	dsimp [N]
	nlinarith [mul_pos hθ hB, mul_pos (mul_pos hD hθ) hz]
	have h1 : 2 * b * N < -(D * θ * w + D * θ * b * z) := by
	have hh : 0 < (2 * b + θ) * (b * B - D * w - θ * z * D) :=
	mul_pos (by linarith) (by linarith)
	dsimp [N]
	nlinarith [mul_pos (mul_pos (mul_pos hD hθ) hθ) hz]
	have h2 : -(D * θ * w + D * θ * b * z) ≤ -(D * θ * (w + z)) := by
	have hh := mul_le_mul_of_nonneg_left hb (mul_nonneg (mul_nonneg hD.le hθ.le) hz.le)
	nlinarith
	have h3 : 8 * M * N ≤ 2 * b * N := by
	have hh := mul_le_mul_of_nonpos_right (show b ≤ M by linarith) hN.le
	nlinarith [mul_neg_of_pos_of_neg (show 0 < M by linarith) hN]
	apply (div_lt_iff₀ (mul_pos hθ (by linarith : 0 < w + z))).mpr
	dsimp [N] at h1 h3
	nlinarith [h3.trans_lt (h1.trans_le h2)]

	/-- Every minimizing update has the source's upper side feasibility. -/
	lemma minimizer_upper {M b D Cw : ℝ} {θ z B w₀ w : ι → ℝ}
	(hb : 1 ≤ b) (hθ : ∀ i, 0 < θ i) (hz : ∀ i, 0 < z i)
	(hB : ∀ i, 0 ≤ B i) (hD : 0 ≤ D) (hM : ∀ i, b + θ i ≤ M)
	(hw : w ∈ domain Cw)
	(hm : IsMinOn (objective (8 * M) b θ z B D w₀) (domain Cw) w) :
	∀ i, D * w i ≤ (b + θ i) * B i := by
	classical
	intro i
	by_cases hD0 : D = 0
	· rw [hD0, zero_mul]
	exact mul_nonneg (by linarith [hθ i]) (hB i)
	have hDp : 0 < D := lt_of_le_of_ne hD (Ne.symm hD0)
	by_contra h
	have hv := lt_of_not_ge h
	have hwi : 0 < w i := by
	nlinarith [mul_nonneg (show 0 ≤ b + θ i by linarith [hθ i]) (hB i)]
	have hc : ∀ y ∈ Set.Icc 0 (w i),
	primitive (8 * M) b (θ i) (z i) (B i) D (w i) ≤
	primitive (8 * M) b (θ i) (z i) (B i) D y + D * (w i - y) := by
	intro y hy
	have hh := coordinate_min_comparison hD hm i
	(replace_mem_domain hw i hy.1 (by linarith [hw.2, hy.2]))
	rw [abs_of_nonpos (sub_nonpos.mpr hy.2)] at hh
	linarith
	have hder := derivative_le_of_left_comparison hwi
	(primitive_derivative (c := 8 * M) (b := b) (B := B i) (D := D)
	(hθ i) (hz i) (hw.1 i)) hc
	have hvder := derivative_upper_violation hb (hθ i) (hz i) (hB i) hDp
	(hw.1 i) (hM i) hv
	linarith

	/-- The lower bound uses the slack proved from the upper bound, not a new
	assumption that the simplex constraint is inactive. -/
	lemma minimizer_lower {M b D Cw : ℝ} {θ z B w₀ w : ι → ℝ}
	(hb : 1 ≤ b) (hθ : ∀ i, 0 < θ i) (hz : ∀ i, 0 < z i)
	(hB : ∀ i, 0 ≤ B i) (hD : 0 ≤ D) (hM : ∀ i, b + θ i ≤ M)
	(hCw : 2 * M < Cw) (hS : ∑ i, B i ≤ 2 * D)
	(hw : w ∈ domain Cw)
	(hm : IsMinOn (objective (8 * M) b θ z B D w₀) (domain Cw) w) :
	∀ i, b * B i - θ i * z i * D ≤ D * w i := by
	classical
	intro i
	by_cases hD0 : D = 0
	· have hBi : B i = 0 := by
	have hh := Finset.single_le_sum (fun j _ => hB j) (Finset.mem_univ i)
	rw [hD0, mul_zero] at hS
	linarith [hB i]
	simp [hD0, hBi]
	have hDp : 0 < D := lt_of_le_of_ne hD (Ne.symm hD0)
	have hupper := minimizer_upper hb hθ hz hB hD hM hw hm
	have hs : ∑ j, w j ≤ 2 * M := by
	have hs₁ : D * ∑ j, w j ≤ M * ∑ j, B j := by
	simp only [Finset.mul_sum]
	exact Finset.sum_le_sum fun j _ => (hupper j).trans
	(mul_le_mul_of_nonneg_right (hM j) (hB j))
	have hMp : 0 ≤ M := by linarith [hM i, hθ i]
	have hh := mul_le_mul_of_nonneg_left hS hMp
	nlinarith
	have hgap : 0 < Cw - ∑ j, w j := by linarith
	have hc : ∀ y ∈ Set.Icc (w i) (w i + (Cw - ∑ j, w j)),
	primitive (8 * M) b (θ i) (z i) (B i) D (w i) ≤
	primitive (8 * M) b (θ i) (z i) (B i) D y + D * (y - w i) := by
	intro y hy
	have hh := coordinate_min_comparison hD hm i
	(replace_mem_domain hw i ((hw.1 i).trans hy.1) hy.2)
	rwa [abs_of_nonneg (sub_nonneg.mpr hy.1)] at hh
	have hder := neg_derivative_le_of_right_comparison hgap
	(primitive_derivative (c := 8 * M) (b := b) (B := B i) (D := D)
	(hθ i) (hz i) (hw.1 i)) hc
	by_contra h
	have hvder := derivative_lower_violation hb (hθ i) (hz i) hDp
	(hw.1 i) (hM i) (lt_of_not_ge h)
	linarith

	/-- Section 04, Lemma side-tracker, with the explicit sufficient multiplier
	C'_w = 8M. No rank assumption or positivity assumption on D is added. -/
	theorem side_update {M b D Cw : ℝ} {θ z B w₀ : ι → ℝ}
	(hb : 1 ≤ b) (hθ : ∀ i, 0 < θ i) (hz : ∀ i, 0 < z i)
	(hB : ∀ i, 0 ≤ B i) (hD : 0 ≤ D) (hM : ∀ i, b + θ i ≤ M)
	(hCw : 2 * M < Cw) (hw₀ : w₀ ∈ domain Cw) :
	∃ w ∈ domain Cw,
	IsMinOn (objective (8 * M) b θ z B D w₀) (domain Cw) w ∧
	(∀ i, D * w i ≤ (b + θ i) * B i) ∧
	((∑ i, B i) ≤ 2 * D → ∀ i, b * B i - θ i * z i * D ≤ D * w i) ∧
	D * movement w w₀ ≤ potential (8 * M) b θ z B D w₀ -
	potential (8 * M) b θ z B D w := by
	obtain ⟨w, hw, hm⟩ := exists_minimizer (c := 8 * M) (b := b) (B := B)
	(D := D) hθ hz hw₀
	exact ⟨w, hw, hm, minimizer_upper hb hθ hz hB hD hM hw hm,
	fun hS => minimizer_lower hb hθ hz hB hD hM hCw hS hw hm,
	minimizer_payment hw₀ hm⟩

	/-- The logarithmic formula is exactly the source's interval integral. -/
	lemma primitive_eq_integral {c b θ z B D w : ℝ} (hθ : 0 < θ) (hz : 0 < z)
	(hw : 0 ≤ w) :
	primitive c b θ z B D w =
	c * ∫ s in (0 : ℝ)..w,
	(D * s - (b + θ / 2) * B + D * θ * z / 2) / (θ * (s + z)) := by
	have hc : ContinuousOn (fun s : ℝ =>
	(D * s - (b + θ / 2) * B + D * θ * z / 2) / (θ * (s + z))) (Set.Icc 0 w) := by
	apply ContinuousOn.div
	· fun_prop
	· fun_prop
	· intro s hs
	exact mul_ne_zero hθ.ne' (ne_of_gt (by linarith [hs.1]))
	have hd : ∀ s ∈ uIcc (0 : ℝ) w,
	HasDerivAt (primitive 1 b θ z B D)
	((D * s - (b + θ / 2) * B + D * θ * z / 2) / (θ * (s + z))) s := by
	intro s hs
	rw [uIcc_of_le hw] at hs
	simpa only [one_mul] using primitive_derivative (c := (1 : ℝ))
	(b := b) (B := B) (D := D) hθ hz hs.1
	have h := intervalIntegral.integral_eq_sub_of_hasDerivAt hd
	((hc.mono (by rw [uIcc_of_le hw])).intervalIntegrable)
	simp [primitive] at h
	rw [h]
	rfl

	/-- Affinity in (B,D), used before every update decision in the all-step
	filtering telescope. In particular the held-D part is constant. -/
	lemma primitive_affine_inputs (c b θ z B₁ B₂ D₁ D₂ t w : ℝ) :
	primitive c b θ z ((1 - t) * B₁ + t * B₂) ((1 - t) * D₁ + t * D₂) w =
	(1 - t) * primitive c b θ z B₁ D₁ w + t * primitive c b θ z B₂ D₂ w := by
	unfold primitive
	ring

	end KServer.SideTracker

	namespace KServer.SideTracker
	open scoped BigOperators

	lemma log_factors {z w : ℝ} (hz : 0 < z) (hw : 0 ≤ w) :
	0 ≤ Real.log (1 + w / z) ∧
	0 ≤ z * Real.log (1 + w / z) ∧
	z * Real.log (1 + w / z) ≤ w := by
	have hq : 0 ≤ w / z := div_nonneg hw hz.le
	have hl : 0 ≤ Real.log (1 + w / z) := Real.log_nonneg (by linarith)
	refine ⟨hl, mul_nonneg hz.le hl, ?_⟩
	have hh := Real.log_le_sub_one_of_pos (show 0 < 1 + w / z by positivity)
	have hh' : Real.log (1 + w / z) ≤ w / z := by linarith
	have := mul_le_mul_of_nonneg_left hh' hz.le
	simpa [mul_div_cancel₀ _ hz.ne'] using this

	lemma side_log_bound {h W w : ℝ} (hh : 1 ≤ h) (hW : 0 ≤ W)
	(hw : w ∈ Set.Icc 0 W) :
	Real.log (1 + w / (Real.exp (1 - h) / h)) ≤
	(Real.log (1 + W) + 2) * h := by
	have hp : 0 < h := lt_of_lt_of_le zero_lt_one hh
	have hw0 : 0 ≤ w := hw.1
	have he : 1 ≤ Real.exp (h - 1) := Real.one_le_exp (by linarith)
	have hh' : h ≤ Real.exp (h - 1) := by
	have := Real.add_one_le_exp (h - 1)
	linarith
	have hE := Real.exp_pos (h - 1)
	have heq : 1 + w / (Real.exp (1 - h) / h) =
	1 + w * h * Real.exp (h - 1) := by
	have hn : 1 - h = -(h - 1) := by ring
	rw [hn, Real.exp_neg]
	field_simp
	rw [heq]
	have hmul : w * h * Real.exp (h - 1) ≤ W * Real.exp (h - 1) ^ 2 := by
	nlinarith [mul_le_mul_of_nonneg_right hw.2 hp.le,
	mul_le_mul_of_nonneg_left hh' hW,
	mul_le_mul_of_nonneg_right
	(mul_le_mul hw.2 hh' hp.le hW) hE.le]
	have hs : 1 ≤ Real.exp (h - 1) ^ 2 := by nlinarith
	have hbound : 1 + w * h * Real.exp (h - 1) ≤
	(1 + W) * Real.exp (2 * (h - 1)) := by
	rw [show 2 * (h - 1) = (h - 1) + (h - 1) by ring, Real.exp_add]
	nlinarith
	have hlog := Real.log_le_log (show 0 < 1 + w * h * Real.exp (h - 1) by positivity) hbound
	rw [Real.log_mul (by positivity) (Real.exp_ne_zero _), Real.log_exp] at hlog
	have hLW : 0 ≤ Real.log (1 + W) := Real.log_nonneg (by linarith)
	nlinarith

	/-- Coefficients of the actual primitive; all are nonnegative on the side
	domain. Their estimates use size-derived logarithmic cancellation. -/
	def Dcoef (c θ z w : ℝ) : ℝ :=
	c * ((w - z * Real.log (1 + w / z)) / θ + z * Real.log (1 + w / z) / 2)
	def Bcoef (c b θ z w : ℝ) : ℝ :=
	c * (b + θ / 2) * Real.log (1 + w / z) / θ

	lemma primitive_coefficients (c b θ z B D w : ℝ) :
	primitive c b θ z B D w = Dcoef c θ z w * D - Bcoef c b θ z w * B := by
	unfold primitive Dcoef Bcoef
	ring

	lemma coefficients_nonneg {c b θ z w : ℝ} (hc : 0 ≤ c) (hb : 0 ≤ b)
	(hθ : 0 < θ) (hz : 0 < z) (hw : 0 ≤ w) :
	0 ≤ Dcoef c θ z w ∧ 0 ≤ Bcoef c b θ z w := by
	obtain ⟨hL, hzL, hLw⟩ := log_factors hz hw
	constructor
	· exact mul_nonneg hc (add_nonneg (div_nonneg (sub_nonneg.mpr hLw) hθ.le)
	(by positivity))
	· exact div_nonneg (mul_nonneg (mul_nonneg hc (by positivity)) hL) hθ.le

	lemma Dcoef_bound {c θ z w cw ell h : ℝ} (hc : 0 ≤ c)
	(hz : 0 < z) (hw : 0 ≤ w) (hcw : 0 < cw) (hell : 1 ≤ ell)
	(hh : 1 ≤ h) (hθ : θ = cw * h / ell) :
	Dcoef c θ z w ≤ c * (1 / cw + 1) * ell * w := by
	have hellp : 0 < ell := lt_of_lt_of_le zero_lt_one hell
	have hp : 0 < h := lt_of_lt_of_le zero_lt_one hh
	have hθp : 0 < θ := by rw [hθ]; positivity
	obtain ⟨hL, hzL, hLw⟩ := log_factors hz hw
	have hi : 1 / θ ≤ ell / cw := by
	rw [div_le_div_iff₀ hθp hcw]
	rw [hθ]
	have he : ell * (cw * h / ell) = cw * h := by field_simp
	rw [he]
	nlinarith
	have ht : (w - z * Real.log (1 + w / z)) / θ ≤ w * (ell / cw) := by
	calc
	_ ≤ w / θ := div_le_div_of_nonneg_right (by linarith) hθp.le
	_ = w * (1 / θ) := by ring
	_ ≤ w * (ell / cw) := mul_le_mul_of_nonneg_left hi hw
	have hq : z * Real.log (1 + w / z) / 2 ≤ ell * w := by nlinarith
	unfold Dcoef
	calc
	_ ≤ c * (w * (ell / cw) + ell * w) :=
	mul_le_mul_of_nonneg_left (add_le_add ht hq) hc
	_ = c * (1 / cw + 1) * ell * w := by ring

	lemma Bcoef_bound {c b θ z w cw ell h K M : ℝ} (hc : 0 ≤ c)
	(hb : 0 ≤ b) (hz : 0 < z) (hw : 0 ≤ w) (hcw : 0 < cw)
	(hell : 0 < ell) (hh : 0 < h) (hθ : θ = cw * h / ell)
	(hBM : b + θ / 2 ≤ M)
	(hLK : Real.log (1 + w / z) ≤ K * h) :
	Bcoef c b θ z w ≤ c * M * K / cw * ell := by
	have hθp : 0 < θ := by rw [hθ]; positivity
	obtain ⟨hL, _, _⟩ := log_factors hz hw
	have hM : 0 ≤ M := (by positivity : 0 ≤ b + θ / 2).trans hBM
	unfold Bcoef
	calc
	_ ≤ c * M * (K * h) / θ := div_le_div_of_nonneg_right
	(mul_le_mul (mul_le_mul_of_nonneg_left hBM hc) hLK hL (mul_nonneg hc hM)) hθp.le
	_ = c * M * K / cw * ell := by rw [hθ]; field_simp

	lemma primitive_component_bound {c b θ z w cw ell h K M B D : ℝ}
	(hc : 0 ≤ c) (hb : 0 ≤ b) (hz : 0 < z) (hw : 0 ≤ w)
	(hcw : 0 < cw) (hell : 1 ≤ ell) (hh : 1 ≤ h)
	(hθ : θ = cw * h / ell) (hBM : b + θ / 2 ≤ M)
	(hLK : Real.log (1 + w / z) ≤ K * h) (hB : 0 ≤ B) (hD : 0 ≤ D) :
	\|primitive c b θ z B D w\| ≤
	c * (1 / cw + 1) * ell * (D * w) + c * M * K / cw * ell * B := by
	have he : 0 < ell := lt_of_lt_of_le zero_lt_one hell
	have hp : 0 < h := lt_of_lt_of_le zero_lt_one hh
	have hθp : 0 < θ := by rw [hθ]; positivity
	have hn := coefficients_nonneg hc hb hθp hz hw
	rw [primitive_coefficients]
	calc
	_ ≤ \|Dcoef c θ z w * D\| + \|Bcoef c b θ z w * B\| := abs_sub _ _
	_ = Dcoef c θ z w * D + Bcoef c b θ z w * B := by
	rw [abs_of_nonneg (mul_nonneg hn.1 hD), abs_of_nonneg (mul_nonneg hn.2 hB)]
	_ ≤ (c * (1 / cw + 1) * ell * w) * D + (c * M * K / cw * ell) * B :=
	add_le_add (mul_le_mul_of_nonneg_right (Dcoef_bound hc hz hw hcw hell hh hθ) hD)
	(mul_le_mul_of_nonneg_right (Bcoef_bound hc hb hz hw hcw he hp hθ hBM hLK) hB)
	_ = _ := by ring

	lemma side_z_identity {a A : ℝ} (ha : 0 < a) (hA : 0 < A) :
	a / (A * (1 + Real.log (A / a))) =
	Real.exp (1 - (1 + Real.log (A / a))) / (1 + Real.log (A / a)) := by
	rw [show 1 - (1 + Real.log (A / a)) = -Real.log (A / a) by ring,
	Real.exp_neg, Real.exp_log (div_pos hA ha)]
	field_simp

	lemma potential_input_bound {ι : Type*} [Fintype ι]
	{c b cw ell K M Cw : ℝ} {θ z h w : ι → ℝ}
	(hc : 0 ≤ c) (hb : 0 ≤ b) (hcw : 0 < cw) (hell : 1 ≤ ell)
	(hz : ∀ i, 0 < z i) (hh : ∀ i, 1 ≤ h i)
	(hθ : ∀ i, θ i = cw * h i / ell)
	(hBM : ∀ i, b + θ i / 2 ≤ M)
	(hLK : ∀ i, Real.log (1 + w i / z i) ≤ K * h i)
	(hw : w ∈ domain Cw) (B B' : ι → ℝ) (D D' : ℝ) :
	\|potential c b θ z B D w - potential c b θ z B' D' w\| ≤
	c * (1 / cw + 1) * ell * Cw * \|D - D'\| +
	c * M * K / cw * ell * (∑ i, \|B i - B' i\|) := by
	have he : 0 < ell := lt_of_lt_of_le zero_lt_one hell
	have hdc (i) := Dcoef_bound hc (hz i) (hw.1 i) hcw hell (hh i) (hθ i)
	have hbc (i) := Bcoef_bound hc hb (hz i) (hw.1 i) hcw he
	(lt_of_lt_of_le zero_lt_one (hh i)) (hθ i) (hBM i) (hLK i)
	have hnn (i) : 0 ≤ Dcoef c (θ i) (z i) (w i) ∧
	0 ≤ Bcoef c b (θ i) (z i) (w i) :=
	coefficients_nonneg hc hb (by rw [hθ i]; exact div_pos (mul_pos hcw (lt_of_lt_of_le zero_lt_one (hh i))) he) (hz i) (hw.1 i)
	have hdiff (i) : primitive c b (θ i) (z i) (B i) D (w i) -
	primitive c b (θ i) (z i) (B' i) D' (w i) =
	Dcoef c (θ i) (z i) (w i) * (D - D') -
	Bcoef c b (θ i) (z i) (w i) * (B i - B' i) := by
	rw [primitive_coefficients, primitive_coefficients]
	ring
	have hpoint (i) : \|primitive c b (θ i) (z i) (B i) D (w i) -
	primitive c b (θ i) (z i) (B' i) D' (w i)\| ≤
	c * (1 / cw + 1) * ell * w i * \|D - D'\| +
	c * M * K / cw * ell * \|B i - B' i\| := by
	rw [hdiff]
	calc
	_ ≤ \|Dcoef c (θ i) (z i) (w i) * (D - D')\| +
	\|Bcoef c b (θ i) (z i) (w i) * (B i - B' i)\| := abs_sub _ _
	_ = Dcoef c (θ i) (z i) (w i) * \|D - D'\| +
	Bcoef c b (θ i) (z i) (w i) * \|B i - B' i\| := by
	rw [abs_mul, abs_mul, abs_of_nonneg (hnn i).1, abs_of_nonneg (hnn i).2]
	_ ≤ _ := add_le_add (mul_le_mul_of_nonneg_right (hdc i) (abs_nonneg _))
	(mul_le_mul_of_nonneg_right (hbc i) (abs_nonneg _))
	unfold potential
	rw [← Finset.sum_sub_distrib]
	calc
	_ ≤ ∑ i, \|primitive c b (θ i) (z i) (B i) D (w i) -
	primitive c b (θ i) (z i) (B' i) D' (w i)\| := Finset.abs_sum_le_sum_abs _ _
	_ ≤ ∑ i, (c * (1 / cw + 1) * ell * w i * \|D - D'\| +
	c * M * K / cw * ell * \|B i - B' i\|) := Finset.sum_le_sum fun i _ => hpoint i
	_ = c * (1 / cw + 1) * ell * (∑ i, w i) * \|D - D'\| +
	c * M * K / cw * ell * (∑ i, \|B i - B' i\|) := by
	simp only [Finset.sum_add_distrib, ← Finset.sum_mul, ← Finset.mul_sum]
	_ ≤ _ := add_le_add (mul_le_mul_of_nonneg_right
	(mul_le_mul_of_nonneg_left hw.2 (by positivity)) (abs_nonneg _)) le_rfl

	end KServer.SideTracker

	namespace KServer.SideSupported
	open Finset
	variable {J : Type*} [Fintype J] [DecidableEq J]
	def restrict (A : Finset J) (a : J → ℝ) : A → ℝ := fun i => a i

	def extend (A : Finset J) (a : A → ℝ) : J → ℝ := fun i => if hi : i∈A then a ⟨i,hi⟩ else 0

	omit [Fintype J] in
	lemma extend_apply_mem (A : Finset J) (a : A → ℝ) (i : A) : extend A a i=a i := by
	simp [extend,i.property]

	omit [Fintype J] in
	lemma extend_apply_not_mem (A : Finset J) (a : A → ℝ) {i : J} (hi : i∉A) : extend A a i=0 := by
	simp [extend,hi]

	omit [Fintype J] in
	lemma restrict_extend (A : Finset J) (a : A → ℝ) : restrict A (extend A a)=a := by
	funext i; exact extend_apply_mem A a i

	omit [Fintype J] in
	lemma extend_restrict (A : Finset J) (a : J → ℝ) (ha : ∀ i∉A, a i=0) :
	extend A (restrict A a)=a := by
	funext i
	by_cases hi : i∈A
	· simp [extend,restrict,hi]
	· simp [extend,hi,ha i hi]

	omit [DecidableEq J] in
	lemma sum_restrict (A : Finset J) (a : J → ℝ) (ha : ∀ i∉A, a i=0) :
	(∑ i : A, a i)=∑ i, a i := by
	rw [←Finset.sum_subtype A (fun _ => Iff.rfl)]
	exact Finset.sum_subset (subset_univ A) fun i _ hi => ha i hi

	lemma sum_extend (A : Finset J) (a : A → ℝ) : (∑ i, extend A a i)=∑ i, a i := by
	have he := sum_restrict A (extend A a) (fun i hi => extend_apply_not_mem A a hi)
	simpa only [extend_apply_mem] using he.symm


	end KServer.SideSupported

	namespace KServer.SideSupported
	open Finset SideTracker
	variable {J : Type*} [Fintype J] [DecidableEq J]

	omit [DecidableEq J] in
	lemma restrict_mem {A : Finset J} {W : ℝ} {a : J → ℝ}
	(ha : a∈ChangingDomains.sideDomain A W) : restrict A a∈domain W := by
	refine ⟨fun i => ha.1 i, ?_⟩
	simpa only [restrict,sum_restrict A a ha.2.2] using ha.2.1

	lemma extend_mem {A : Finset J} {W : ℝ} {a : A → ℝ}
	(ha : a∈domain W) : extend A a∈ChangingDomains.sideDomain A W := by
	refine ⟨?_,?_,?_⟩
	· intro i
	simp only [extend]
	split_ifs with hi
	· exact ha.1 ⟨i,hi⟩
	· exact le_rfl
	· simpa only [sum_extend] using ha.2
	· exact fun i hi => extend_apply_not_mem A a hi

	omit [DecidableEq J] in
	lemma sideDomain_compact (A : Finset J) (W : ℝ) :
	IsCompact (ChangingDomains.sideDomain A W) := by
	have he : ChangingDomains.sideDomain A W=
	(domain W)∩{a : J → ℝ \| ∀ i∉A, a i=0} := by
	ext a
	simp only [ChangingDomains.sideDomain,domain,Set.mem_ofPred_eq,Set.mem_inter_iff]
	tauto
	rw [he]
	apply IsCompact.inter_right (domain_isCompact W)
	simp only [ofPred_forall]
	exact isClosed_iInter fun i => isClosed_iInter fun _ => isClosed_eq (continuous_apply i) continuous_const

	omit [DecidableEq J] in
	lemma movement_restrict {A : Finset J} {a b : J → ℝ}
	(ha : ∀ i∉A, a i=0) (hb : ∀ i∉A, b i=0) :
	movement (restrict A a) (restrict A b)=ChangingDomains.variation a b := by
	exact sum_restrict A (fun i => \|a i-b i\|) (fun i hi => by simp [ha i hi,hb i hi])

	omit [DecidableEq J] in
	lemma potential_restrict {A : Finset J} {c b D : ℝ} {θ z B a : J → ℝ}
	(ha : ∀ i∉A, a i=0) :
	potential c b (restrict A θ) (restrict A z) (restrict A B) D (restrict A a)=
	potential c b θ z B D a := by
	unfold potential restrict
	exact sum_restrict A (fun i => primitive c b (θ i) (z i) (B i) D (a i))
	(fun i hi => by simp [ha i hi,primitive])

	lemma restrict_isMin {A : Finset J} {c b D W : ℝ} {θ z B a a₀ : J → ℝ}
	(ha : a∈ChangingDomains.sideDomain A W) (ha₀ : a₀∈ChangingDomains.sideDomain A W)
	(hm : IsMinOn (fun v => potential c b θ z B D v+D*ChangingDomains.variation v a₀)
	(ChangingDomains.sideDomain A W) a) :
	IsMinOn (objective c b (restrict A θ) (restrict A z) (restrict A B) D (restrict A a₀))
	(domain W) (restrict A a) := by
	intro v hv
	have he := extend_mem hv
	have hh := hm he
	change potential c b (restrict A θ) (restrict A z) (restrict A B) D (restrict A a)+
	D*movement (restrict A a) (restrict A a₀) ≤
	potential c b (restrict A θ) (restrict A z) (restrict A B) D v+
	D*movement v (restrict A a₀)
	rw [potential_restrict ha.2.2,movement_restrict ha.2.2 ha₀.2.2]
	have hp := potential_restrict (c := c) (b := b) (D := D) (θ := θ) (z := z) (B := B) he.2.2
	have hmove := movement_restrict he.2.2 ha₀.2.2
	rw [restrict_extend] at hp hmove
	rw [hp,hmove]
	exact hh

	lemma supported_feasible {A : Finset J} {M b D W : ℝ} {θ z B a a₀ : J → ℝ}
	(ha : a∈ChangingDomains.sideDomain A W) (ha₀ : a₀∈ChangingDomains.sideDomain A W)
	(hb : 1≤b) (hθ : ∀ i, 0<θ i) (hz : ∀ i, 0<z i)
	(hB : ∀ i, 0≤B i) (hBz : ∀ i∉A, B i=0) (hD : 0≤D)
	(hM : ∀ i∈A, b+θ i≤M) (hW : 2*M<W)
	(hm : IsMinOn (fun v => potential (8M) b θ z B D v+DChangingDomains.variation v a₀)
	(ChangingDomains.sideDomain A W) a) :
	(∀ i, Da i≤(b+θ i)B i) ∧
	((∑ i, B i)≤2D → ∀ i, bB i-θ iz iD≤D*a i) := by
	have hm' := restrict_isMin ha ha₀ hm
	have hupper := minimizer_upper hb (fun i : A => hθ i) (fun i : A => hz i)
	(fun i : A => hB i) hD (fun i : A => hM i i.property) (restrict_mem ha) hm'
	constructor
	· intro i
	by_cases hi : i∈A
	· exact hupper ⟨i,hi⟩
	· simp [ha.2.2 i hi,hBz i hi]
	· intro hS
	have hS' : (∑ i : A, B i)≤2*D := by simpa only [sum_restrict A B hBz] using hS
	have hlower := minimizer_lower hb (fun i : A => hθ i) (fun i : A => hz i)
	(fun i : A => hB i) hD (fun i : A => hM i i.property) hW hS' (restrict_mem ha) hm'
	intro i
	by_cases hi : i∈A
	· exact hlower ⟨i,hi⟩
	· simp only [ha.2.2 i hi,hBz i hi,mul_zero,zero_sub]
	exact neg_nonpos.mpr (mul_nonneg (mul_nonneg (hθ i).le (hz i).le) hD)

	end KServer.SideSupported

	namespace KServer.SideFamilies
	open Finset SideTracker AdaptiveMinimization
	variable {J R : Type*} [Fintype J] [DecidableEq J] [Fintype R]

	structure Data (J R : Type*) where
	active : Finset J
	flags : J → R → Bool
	W : ℝ
	b : ℝ
	D : ℝ
	θ : J → ℝ
	z : J → ℝ
	W_nonneg : 0≤W
	θ_pos : ∀ i, 0<θ i
	z_pos : ∀ i, 0<z i

	def core (d : Data J R) (p : J × R → ℝ) (i : J) : ℝ :=
	∑ r, if d.flags i r then p (i,r) else 0

	def coefficient (c : ℝ) (d : Data J R) (a : J → ℝ) (ir : J × R) : ℝ :=
	if d.flags ir.1 ir.2 then -Bcoef c d.b (d.θ ir.1) (d.z ir.1) (a ir.1) else 0

	def offset (c : ℝ) (d : Data J R) (a : J → ℝ) : ℝ :=
	d.D * ∑ i, Dcoef c (d.θ i) (d.z i) (a i)

	lemma Dcoef_continuousOn {c θ z : ℝ} (hθ : 0<θ) (hz : 0<z) :
	ContinuousOn (Dcoef c θ z) (Set.Ici 0) := by
	have he : Dcoef c θ z=primitive c 0 θ z 0 1 := by
	funext w
	unfold Dcoef primitive
	ring
	rw [he]
	exact primitive_continuousOn hθ hz

	lemma Bcoef_continuousOn {c b θ z : ℝ} (hθ : 0<θ) (hz : 0<z) :
	ContinuousOn (Bcoef c b θ z) (Set.Ici 0) := by
	have he : Bcoef c b θ z=fun w => -primitive c b θ z 1 0 w := by
	funext w
	unfold Bcoef primitive
	ring
	rw [he]
	exact (primitive_continuousOn hθ hz).neg

	omit [DecidableEq J] [Fintype R] in
	lemma coefficient_continuous (c : ℝ) (d : Data J R) (ir : J × R) :
	ContinuousOn (fun a => coefficient c d a ir) (ChangingDomains.sideDomain d.active d.W) := by
	unfold coefficient
	split_ifs
	· exact ((Bcoef_continuousOn (d.θ_pos ir.1) (d.z_pos ir.1)).comp
	(continuous_apply ir.1).continuousOn (fun a ha => ha.1 ir.1)).neg
	· exact continuousOn_const

	omit [DecidableEq J] [Fintype R] in
	lemma offset_continuous (c : ℝ) (d : Data J R) :
	ContinuousOn (offset c d) (ChangingDomains.sideDomain d.active d.W) := by
	unfold offset
	apply ContinuousOn.const_mul
	exact continuousOn_finsetSum _ fun i _ =>
	(Dcoef_continuousOn (d.θ_pos i) (d.z_pos i)).comp
	(continuous_apply i).continuousOn (fun a ha => ha.1 i)

	def family (c : ℝ) : Family (Data J R) (J × R) J where
	domain d := ChangingDomains.sideDomain d.active d.W
	compact d := SideSupported.sideDomain_compact d.active d.W
	seed d := ⟨fun _ => 0,by
	refine ⟨fun _ => le_rfl,?_,fun _ _ => rfl⟩
	simpa using d.W_nonneg⟩
	offset := offset c
	coefficient := coefficient c
	offset_continuous := offset_continuous c
	coefficient_continuous := coefficient_continuous c

	omit [DecidableEq J] in
	lemma value_eq (c : ℝ) (d : Data J R) (p : J × R → ℝ) (a : J → ℝ) :
	(family c).value d p a=potential c d.b d.θ d.z (core d p) d.D a := by
	simp only [Family.value,family,offset,coefficient,potential]
	rw [Fintype.sum_prod_type]
	have hi (i : J) : (∑ r, (if d.flags i r then -Bcoef c d.b (d.θ i) (d.z i) (a i) else 0)*p (i,r))=
	-Bcoef c d.b (d.θ i) (d.z i) (a i)*core d p i := by
	rw [core,mul_sum]
	exact sum_congr rfl fun r _ => by split_ifs <;> simp_all
	simp_rw [hi,primitive_coefficients]
	rw [mul_sum,←sum_add_distrib]
	exact sum_congr rfl fun i _ => by ring

	lemma update_feasible (d : Data J R) {M : ℝ} (hb : 1≤d.b) (hD : 0≤d.D)
	(hM : ∀ i∈d.active, d.b+d.θ i≤M) (hW : 2*M<d.W)
	(p : J × R → ℝ) (hp : ∀ ir, 0≤p ir)
	(hf : ∀ i∉d.active, ∀ r, d.flags i r=false)
	(a : J → ℝ) (ha : a∈ChangingDomains.sideDomain d.active d.W) :
	let a' := (family (8*M)).update d p d.D a
	(∀ i, d.Da' i≤(d.b+d.θ i)core d p i) ∧
	((∑ i, core d p i)≤2*d.D → ∀ i,
	d.bcore d p i-d.θ id.z id.D≤d.Da' i) := by
	intro a'
	have hB : ∀ i, 0≤core d p i := by
	intro i
	unfold core
	apply sum_nonneg
	intro r _
	split_ifs
	· exact hp (i,r)
	· exact le_rfl
	have hBz : ∀ i∉d.active, core d p i=0 := by
	intro i hi
	simp [core,hf i hi]
	apply SideSupported.supported_feasible ((family (8*M)).update_mem _ _ _ _) ha hb
	d.θ_pos d.z_pos hB hBz hD hM hW
	intro v hv
	have hm := ((family (8*M)).exists_update d p d.D a).choose_spec.2 hv
	change (family (8M)).value d p a'+d.DAdaptiveMinimization.movement a' a ≤
	(family (8M)).value d p v+d.DAdaptiveMinimization.movement v a at hm
	simpa only [value_eq,AdaptiveMinimization.movement,ChangingDomains.variation, a', Set.mem_ofPred_eq] using hm

	omit [DecidableEq J] in
	lemma update_payment (c : ℝ) (d : Data J R) (p : J × R → ℝ) (a : J → ℝ)
	(ha : a∈ChangingDomains.sideDomain d.active d.W) :
	let a' := (family c).update d p d.D a
	d.D*ChangingDomains.variation a' a≤
	potential c d.b d.θ d.z (core d p) d.D a-
	potential c d.b d.θ d.z (core d p) d.D a' := by
	have hp := (family c).update_payment d p d.D ha
	simpa only [value_eq,AdaptiveMinimization.movement,ChangingDomains.variation] using hp

	end KServer.SideFamilies

	namespace KServer.SideFamilies
	open Finset SideTracker AdaptiveMinimization
	variable {J R : Type*} [Fintype J] [DecidableEq J] [Fintype R]

	omit [DecidableEq J] [Fintype R] in
	lemma domain_coordinate {d : Data J R} {a : J → ℝ}
	(ha : a∈ChangingDomains.sideDomain d.active d.W) (i : J) :
	a i∈Set.Icc 0 d.W :=
	⟨ha.1 i,(single_le_sum (fun j _ => ha.1 j) (mem_univ i)).trans ha.2.1⟩

	/-- The exact side regularizers obtained from the stored log size. -/
	def fromLog (A : Finset J) (f : J → R → Bool) (W b D cw ell : ℝ)
	(h : J → ℝ) (hW : 0≤W) (hcw : 0<cw) (hell : 0<ell) (hh : ∀ i, 1≤h i) : Data J R where
	active := A
	flags := f
	W := W
	b := b
	D := D
	θ i := cw*h i/ell
	z i := Real.exp (1-h i)/h i
	W_nonneg := hW
	θ_pos i := div_pos (mul_pos hcw (lt_of_lt_of_le zero_lt_one (hh i))) hell
	z_pos i := div_pos (Real.exp_pos _) (lt_of_lt_of_le zero_lt_one (hh i))

	omit [DecidableEq J] [Fintype R] in
	lemma fromLog_input_bound {A : Finset J} (_flags : J → R → Bool)
	{W b cw ell c M D D' : ℝ} (h : J → ℝ)
	(hW : 0≤W) (hcw : 0<cw) (hell : 1≤ell) (hh : ∀ i, 1≤h i)
	(hc : 0≤c) (hb : 0≤b) (hM : ∀ i, b+cw*h i/ell≤M)
	{a B B' : J → ℝ} (ha : a∈ChangingDomains.sideDomain A W) :
	\|potential c b (fun i => cw*h i/ell) (fun i => Real.exp (1-h i)/h i) B D a-
	potential c b (fun i => cw*h i/ell) (fun i => Real.exp (1-h i)/h i) B' D' a\|≤
	c(1/cw+1)ellW\|D-D'\|+
	cM(Real.log (1+W)+2)/cwellChangingDomains.variation B B' := by
	have hp : 0<ell := lt_of_lt_of_le zero_lt_one hell
	apply potential_input_bound hc hb hcw hell
	(fun i => div_pos (Real.exp_pos _) (lt_of_lt_of_le zero_lt_one (hh i))) hh
	(fun _ => rfl)
	· intro i
	have ht : 0≤cw*h i/ell := div_nonneg (mul_nonneg hcw.le (le_trans zero_le_one (hh i))) hp.le
	linarith [hM i]
	· intro i
	apply side_log_bound (hh i) hW
	exact ⟨ha.1 i,(single_le_sum (fun j _ => ha.1 j) (mem_univ i)).trans ha.2.1⟩
	· exact ⟨ha.1,ha.2.1⟩

	omit [DecidableEq J] in
	lemma parameter_jump {W b cw ell c M D : ℝ} {h h' B a : J → ℝ}
	(hW : 0≤W) (hcw : 0<cw) (hell : 1≤ell)
	(hh : ∀ i, 1≤h i) (hh' : ∀ i, 1≤h' i)
	(hc : 0≤c) (hb : 0≤b) (hD : 0≤D) (hB : ∀ i, 0≤B i)
	(hM : ∀ i, b+cwh i/ell≤M) (hM' : ∀ i, b+cwh' i/ell≤M)
	(ha : a∈domain W) (hcap : ∀ i, Da i≤MB i) :
	\|potential c b (fun i => cw*h' i/ell) (fun i => Real.exp (1-h' i)/h' i) B D a-
	potential c b (fun i => cw*h i/ell) (fun i => Real.exp (1-h i)/h i) B D a\|≤
	2(c(1/cw+1)M+cM(Real.log (1+W)+2)/cw)ell*
	(∑ i, if h i=h' i then 0 else B i) := by
	have hp : 0<ell := lt_of_lt_of_le zero_lt_one hell
	have hw (i : J) : a i∈Set.Icc 0 W :=
	⟨ha.1 i,(single_le_sum (fun j _ => ha.1 j) (mem_univ i)).trans ha.2⟩
	have hbound (q : J → ℝ) (hq : ∀ i, 1≤q i) (hMq : ∀ i, b+cw*q i/ell≤M) (i : J) :
	\|primitive c b (cw*q i/ell) (Real.exp (1-q i)/q i) (B i) D (a i)\|≤
	(c(1/cw+1)M+cM(Real.log (1+W)+2)/cw)ellB i := by
	have ht : 0≤cw*q i/ell := div_nonneg (mul_nonneg hcw.le (le_trans zero_le_one (hq i))) hp.le
	have he := primitive_component_bound hc hb
	(div_pos (Real.exp_pos _) (lt_of_lt_of_le zero_lt_one (hq i))) (ha.1 i)
	hcw hell (hq i) rfl (show b+(cw*q i/ell)/2≤M by linarith [hMq i])
	(side_log_bound (hq i) hW (hw i)) (hB i) hD
	have hcap' := mul_le_mul_of_nonneg_left (hcap i)
	(show 0≤c(1/cw+1)ell by positivity)
	calc
	_ ≤ c(1/cw+1)ell(Da i)+
	cM(Real.log (1+W)+2)/cwellB i := he
	_ ≤ c(1/cw+1)ell(MB i)+
	cM(Real.log (1+W)+2)/cwellB i := add_le_add hcap' le_rfl
	_ = _ := by ring
	unfold potential
	rw [←sum_sub_distrib,mul_sum]
	apply (abs_sum_le_sum_abs _ _).trans
	apply sum_le_sum
	intro i _
	by_cases hi : h i=h' i
	· simp [hi]
	· rw [ite_eq_right hi]
	have ho := hbound h hh hM i
	have hn := hbound h' hh' hM' i
	exact (abs_sub _ _).trans (by linarith)

	/-- Synthetic ranks use the new flags on the intersection of active sets. -/
	def prepareData (old new : Data J R) : Data J R :=
	{old with flags := ChangingDomains.syntheticFlags old.active new.active new.flags, D := new.D}

	def prepare (c : ℝ) (old new : Data J R) (p : J × R → ℝ) (a : J → ℝ) : J → ℝ :=
	(family c).update (prepareData old new) p new.D a

	def transport (c : ℝ) (old new : Data J R) (p : J × R → ℝ) (a : J → ℝ) : J → ℝ :=
	ChangingDomains.transposeSide new.active (prepare c old new p a)

	def step (c : ℝ) (old new : Data J R) (p : J × R → ℝ) (a : J → ℝ) : J → ℝ :=
	(family c).update new p new.D (transport c old new p a)

	lemma prepare_mem (c : ℝ) (old new : Data J R) (p : J × R → ℝ) (a : J → ℝ) :
	prepare c old new p a∈ChangingDomains.sideDomain old.active old.W :=
	(family c).update_mem _ _ _ _

	lemma transport_mem (c : ℝ) (old new : Data J R) (p : J × R → ℝ) (a : J → ℝ)
	(hW : old.W=new.W) :
	transport c old new p a∈ChangingDomains.sideDomain new.active new.W := by
	rw [←hW]
	exact ChangingDomains.transposeSide_mem (prepare_mem c old new p a)

	lemma step_mem (c : ℝ) (old new : Data J R) (p : J × R → ℝ) (a : J → ℝ) :
	step c old new p a∈ChangingDomains.sideDomain new.active new.W :=
	(family c).update_mem _ _ _ _

	omit [Fintype J] in
	lemma prepared_core_zero (old new : Data J R) (p : J × R → ℝ)
	{i : J} (hi : i∉old.active∩new.active) : core (prepareData old new) p i=0 := by
	simp [core,prepareData,ChangingDomains.syntheticFlags,hi]

	lemma transport_payment (old new : Data J R) {M : ℝ}
	(hb : 1≤old.b) (hD : 0≤new.D)
	(hM : ∀ i∈old.active, old.b+old.θ i≤M) (hW : 2*M<old.W)
	(p : J × R → ℝ) (hp : ∀ ir, 0≤p ir)
	(a : J → ℝ) (ha : a∈ChangingDomains.sideDomain old.active old.W) :
	new.DChangingDomains.variation (transport (8M) old new p a)
	(prepare (8*M) old new p a)=0 := by
	have hfeas := update_feasible (prepareData old new) hb hD hM hW p hp
	(fun i hi r => by
	change i∉old.active at hi
	simp [prepareData,ChangingDomains.syntheticFlags,hi]) a ha
	exact ChangingDomains.transposeSide_payment (prepare_mem (8*M) old new p a).1 hD
	hfeas.1 (fun i hi => prepared_core_zero old new p (by simp [hi]))

	lemma transport_value (c : ℝ) (old new : Data J R) (p : J × R → ℝ) (a : J → ℝ)
	(hzero : new.D*ChangingDomains.variation (transport c old new p a)
	(prepare c old new p a)=0) :
	(family c).value (prepareData old new) p (transport c old new p a)=
	(family c).value (prepareData old new) p (prepare c old new p a) := by
	rw [value_eq,value_eq]
	unfold potential
	apply sum_congr rfl
	intro i _
	by_cases hi : i∈new.active
	· simp [transport,ChangingDomains.transposeSide,hi]
	· have hB : core (prepareData old new) p i=0 :=
	prepared_core_zero old new p (by simp [hi])
	by_cases hD : new.D=0
	· rw [hB]
	change primitive c old.b (old.θ i) (old.z i) 0 new.D _ =
	primitive c old.b (old.θ i) (old.z i) 0 new.D _
	simp [hD,primitive]
	· have hvar : ChangingDomains.variation (transport c old new p a)
	(prepare c old new p a)=0 := (mul_eq_zero.mp hzero).resolve_left hD
	have hd := single_le_sum (s := univ)
	(f := fun j => \|transport c old new p a j-prepare c old new p a j\|)
	(fun j _ => abs_nonneg _) (mem_univ i)
	change \|transport c old new p a i-prepare c old new p a i\|≤
	ChangingDomains.variation (transport c old new p a) (prepare c old new p a) at hd
	rw [hvar] at hd
	have he := sub_eq_zero.mp (abs_nonpos_iff.mp hd)
	rw [he]

	/-- Local side edit payment for the actual two minimizers and deletion. The
	potential discrepancy terms will be bounded with the literal log parameters. -/
	lemma step_payment (old new : Data J R) {M : ℝ}
	(hb : 1≤old.b) (hD : 0≤new.D)
	(hM : ∀ i∈old.active, old.b+old.θ i≤M) (hW : 2*M<old.W)
	(p : J × R → ℝ) (hp : ∀ ir, 0≤p ir)
	(a : J → ℝ) (ha : a∈ChangingDomains.sideDomain old.active old.W)
	(hWW : old.W=new.W) :
	new.DChangingDomains.variation (step (8M) old new p a) a ≤
	(family (8M)).value old p a-(family (8M)).value new p (step (8*M) old new p a)+
	\|(family (8M)).value (prepareData old new) p a-(family (8M)).value old p a\|+
	\|(family (8M)).value new p (transport (8M) old new p a)-
	(family (8M)).value (prepareData old new) p (transport (8M) old new p a)\| := by
	have hz := transport_payment old new hb hD hM hW p hp a ha
	have he := transport_value (8*M) old new p a hz
	have h₁ := (family (8*M)).update_payment (prepareData old new) p new.D ha
	have h₂ := (family (8M)).update_payment new p new.D (transport_mem (8M) old new p a hWW)
	have hm₁ := ChangingDomains.variation_triangle (step (8*M) old new p a)
	(transport (8*M) old new p a) a
	have hm₂ := ChangingDomains.variation_triangle (transport (8*M) old new p a)
	(prepare (8*M) old new p a) a
	have hp₁ := mul_le_mul_of_nonneg_left hm₁ hD
	have hp₂ := mul_le_mul_of_nonneg_left hm₂ hD
	change new.DChangingDomains.variation (prepare (8M) old new p a) a ≤
	(family (8*M)).value (prepareData old new) p a-
	(family (8M)).value (prepareData old new) p (prepare (8M) old new p a) at h₁
	change new.DChangingDomains.variation (step (8M) old new p a)
	(transport (8*M) old new p a) ≤
	(family (8M)).value new p (transport (8M) old new p a)-
	(family (8M)).value new p (step (8M) old new p a) at h₂
	have hab₁ := le_abs_self ((family (8M)).value (prepareData old new) p a-(family (8M)).value old p a)
	have hab₂ := le_abs_self ((family (8M)).value new p (transport (8M) old new p a)-
	(family (8M)).value (prepareData old new) p (transport (8M) old new p a))
	nlinarith

	end KServer.SideFamilies

	end
	end
	end

	end OAI