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 C*ell*((∑ 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)= C*ell*((∑ 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 C*ell*((∑ j, K d a j)-(1+d.eta i)*Z d a i) else 0)*p (i,r))= C*ell*((∑ 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) : C*ell*((∑ j, K d a j)-(1+d.eta i)*Z d a i)*core d p i = (C*ell*(∑ j, K d a j))*core d p i-C*ell*((1+d.eta i)*core d p i*Z d a i) := by ring have heR (i : J) : C*ell*((∑ j, core d p j)*K d a i-(1+d.eta i)*core d p i*Z d a i) = (C*ell*(∑ j, core d p j))*K d a i-C*ell*((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| ≤ 3*C*ell := 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 C*ell*((∑ 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| ≤ 13*C*ell := 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 C*ell*((∑ 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 i*F i (Real.log (1/b i)) ≤ CF*∑ i, b i*h i) (hgap : 2*CF*(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+ct*h 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 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) 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 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+ct*h 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 (∑ 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+ct*h 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+ct*h 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+ct*h 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 (∑ 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+ct*h 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 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+ct*h 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| ≤ 3*C*ell*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| ≤ 8*C*ell*(∑ 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)| ≤ 8*C*ell*(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, S*a i≤gamma i*v i) : ∀ i, S*(ChangingDomains.transposeSimplex B S a) i≤gamma i*v 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 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 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+ (6*C*ell+4)*ChangingDomains.changedFlags f g+ 8*C*ell*(∑ 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+ct*h 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+ct*h 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≤3*C*ell) have hinput₃ := mul_le_mul_of_nonneg_left herr₃ (by positivity : 0≤3*C*ell) have he₀ := (neg_le_of_abs_le him₀) have he₃ := (le_of_abs_le him₃) have hep := le_of_abs_le ht change _ ≤ 8*C*ell*(∑ i, if h i=h' i then 0 else B₁ i) at hep change -(3*C*ell*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 ≤ 3*C*ell*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 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+ct*h 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 (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 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 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 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|≤8*C*ell*(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|≤24*C*ell*(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≤8*C*ell) have hpos : 0≤C*ell*(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 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 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 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 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+ (26*C*ell+4)*ChangingDomains.changedFlags f g+ 8*C*ell*(∑ i∈Finset.univ.erase o, if h i=h' i then 0 else B₁ i)+ 24*C*ell*(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+ct*h 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+ct*h 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≤13*C*ell) have hinput₃ := mul_le_mul_of_nonneg_left herr₃ (by positivity : 0≤13*C*ell) have he₀ := neg_le_of_abs_le him₀ have he₃ := le_of_abs_le him₃ have hep := le_of_abs_le ht change -(13*C*ell*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 ≤ 13*C*ell*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 potential (8*M) b θ z B D v+D*ChangingDomains.variation v a₀) (ChangingDomains.sideDomain A W) a) : (∀ i, D*a i≤(b+θ i)*B i) ∧ ((∑ i, B i)≤2*D → ∀ i, b*B i-θ i*z i*D≤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 -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 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*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)*ell*W*|D-D'|+ c*M*(Real.log (1+W)+2)/cw*ell*ChangingDomains.variation B B' := by have hp : 0 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*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+c*M*(Real.log (1+W)+2)/cw)*ell* (∑ i, if h i=h' i then 0 else B i) := by have hp : 0 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+c*M*(Real.log (1+W)+2)/cw)*ell*B 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*(D*a i)+ c*M*(Real.log (1+W)+2)/cw*ell*B i := he _ ≤ c*(1/cw+1)*ell*(M*B i)+ c*M*(Real.log (1+W)+2)/cw*ell*B 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 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