| import Mathlib |
| import OAI.Combinatorics.KServer.OutputRules |
| import OAI.Combinatorics.KServer.Trees |
|
|
| namespace OAI |
|
|
| /-! Conditional ranks of literal complete-prefix cells, connected to the |
| allocation's filtered process and compatible lower envelopes. No local |
| allocation or competitiveness assertion is an input. -/ |
| noncomputable section |
| open scoped BigOperators |
| open Finset |
| namespace KServer.HierarchicalRanks |
| attribute [local instance] Classical.propDecidable Classical.decEq |
| open RankTracking PosteriorRanks HierarchyCounts |
| variable {Ω A Y : Type} [Fintype Ω] [Fintype A] {k d : ℕ} |
| variable {w : Ω → ℝ} (hw : ∀ ω,0≤w ω) |
| variable (H : ℕ → Ω → ℕ) |
| variable (hr : ∀ t ω ρ,H (t+1) ω=H (t+1) ρ → H t ω=H t ρ) |
| variable (maps : ℕ → Ω → ℕ → Y → A) (q : ℕ → Ω → Fin k → Y) |
|
|
| def family (word : Fin d → A) (a : Fin k) : FilteredRanks w := |
| rankProcess hw H hr (fun t ω=>count (maps t ω) word (q t ω)) a.val |
|
|
| def lower (word : Fin d → A) (z : ℝ) (t : ℕ) (ω : Ω) : ℝ := |
| ∑ a : Fin k, RankFunctions.rank z ((family hw H hr maps q word a).p t ω) |
|
|
| def size (word : Fin d → A) (t : ℕ) (ω : Ω) : ℝ := |
| CoarseProcess.size (family hw H hr maps q word) t ω |
|
|
| omit [Fintype A] in |
| lemma family_value (word : Fin d → A) (a : Fin k) (t : ℕ) (ω : Ω) : |
| (family hw H hr maps q word a).p t ω= |
| RankOrder.countRank (conditional w (H t) ω) |
| (fun ρ=>count (maps t ρ) word (q t ρ)) a.val := |
| (conditional_rank w (H t) ω _ _).symm |
|
|
| omit [Fintype A] in |
| lemma lower_eq (word : Fin d → A) (z : ℝ) (t : ℕ) (ω : Ω) : |
| lower hw H hr maps q word z t ω=CompatibleBounds.L k (conditional w (H t) ω) |
| (fun ρ=>count (maps t ρ) word (q t ρ)) z := by |
| simp only [lower,CompatibleBounds.L,family_value] |
|
|
| omit [Fintype A] in |
| lemma size_eq (word : Fin d → A) (t : ℕ) (ω : Ω) : |
| size hw H hr maps q word t ω=CompatibleBounds.M k (conditional w (H t) ω) |
| (fun ρ=>count (maps t ρ) word (q t ρ)) := by |
| simp only [size,CoarseProcess.size,CompatibleBounds.M,family_value] |
|
|
| lemma lower_superadditive (word : Fin d → A) {z : ℝ} |
| (hz : z∈Set.Icc 0 (1/100:ℝ)) (t : ℕ) (ω : Ω) (hω : 0<w ω) : |
| (∑ a : A,lower hw H hr maps q (Fin.snoc word a) z t ω) ≤ lower hw H hr maps q word z t ω := by |
| simp_rw [lower_eq] |
| exact (HierarchyCounts.compatible w hw (H t) ω hω (maps t) word (q t) hz).1 |
|
|
| lemma size_superadditive (word : Fin d → A) (t : ℕ) (ω : Ω) (hω : 0<w ω) : |
| (∑ a : A,size hw H hr maps q (Fin.snoc word a) t ω) ≤ size hw H hr maps q word t ω := by |
| simp_rw [size_eq] |
| exact (HierarchyCounts.compatible w hw (H t) ω hω (maps t) word (q t) |
| (z:=0) (by norm_num)).2.1 |
|
|
| omit [Fintype A] in |
| lemma family_root (word : Fin 0 → A) (a : Fin k) (t : ℕ) (ω : Ω) : |
| (family hw H hr maps q word a).p t ω=0 := by |
| change posterior w (H t) (PosteriorRanks.rank _ a.val) ω=0 |
| have he : PosteriorRanks.rank (fun ρ=>count (maps t ρ) word (q t ρ)) a.val=fun _=>0 := by |
| funext ρ |
| simp only [PosteriorRanks.rank,count_zero] |
| exact ite_eq_right (by omega) |
| rw [he] |
| simp [posterior,numerator] |
|
|
| omit [Fintype A] in |
| lemma lower_root (word : Fin 0 → A) (z : ℝ) (t : ℕ) (ω : Ω) : |
| lower hw H hr maps q word z t ω=k := by |
| simp only [lower,family_root,RankFunctions.rank_zero] |
| simp |
|
|
| omit [Fintype A] in |
| lemma size_root (word : Fin 0 → A) (t : ℕ) (ω : Ω) : |
| size hw H hr maps q word t ω=k := by |
| simp only [size,CoarseProcess.size,family_root] |
| norm_num [RankFunctions.sizeRank] |
|
|
| omit [Fintype A] in |
| lemma lower_nonneg (word : Fin d → A) {z : ℝ} (hz : z∈Set.Icc 0 (1/100:ℝ)) |
| (t : ℕ) (ω : Ω) : 0≤lower hw H hr maps q word z t ω := |
| sum_nonneg fun _ _=>RankFunctions.rank_nonneg hz _ |
|
|
| omit [Fintype A] in |
| lemma lower_size (word : Fin d → A) {z : ℝ} (hz : z∈Set.Icc 0 (1/100:ℝ)) |
| (t : ℕ) (ω : Ω) : lower hw H hr maps q word z t ω ≤36*size hw H hr maps q word t ω := by |
| unfold lower size CoarseProcess.size |
| rw [mul_sum] |
| exact sum_le_sum fun a _=>CompatibleBounds.rank_size_domination hz |
| ((family hw H hr maps q word a).range t ω).1 |
|
|
| omit [Fintype A] in |
| lemma size_bound (word : Fin d → A) (t : ℕ) (ω : Ω) : size hw H hr maps q word t ω ≤ k := by |
| calc _ ≤ ∑ _a : Fin k,(1:ℝ) := by |
| unfold size CoarseProcess.size |
| apply sum_le_sum |
| intro a _ |
| have h:=(family hw H hr maps q word a).range t ω |
| exact (RankFunctions.sizeRank_le_complement h).trans (by linarith [h.1]) |
| _ = _ := by simp |
|
|
| omit [Fintype A] in |
| / |
| has a member in the current observed cell. This is a fiber, not an independence |
| assertion and does not condition on any rounding outcomes. -/ |
| lemma first_rank_zero {k : ℕ} (hk : 0<k) (q : ℕ → Ω → Fin k → Y) |
| (word : Fin d → A) (t : ℕ) (ω : Ω) |
| (hs : ∀ ρ,H t ρ=H t ω → ∃ l,hit (maps t ρ) word (q t ρ l)) : |
| (family hw H hr maps q word ⟨0,hk⟩).p t ω=0 := by |
| change posterior w (H t) (PosteriorRanks.rank _ 0) ω=0 |
| unfold posterior numerator |
| have hn : (∑ ρ∈fiber (H t) (H t ω),w ρ*PosteriorRanks.rank |
| (fun z=>count (maps t z) word (q t z)) 0 ρ)=0 := by |
| apply sum_eq_zero |
| intro ρ hρ |
| obtain ⟨l,hl⟩:=hs ρ (mem_filter.mp hρ).2 |
| have hc : 0<count (maps t ρ) word (q t ρ) := card_pos.mpr ⟨l,by simp [hl]⟩ |
| simp [PosteriorRanks.rank,not_le.mpr hc] |
| rw [hn,zero_div] |
|
|
| omit [Fintype A] in |
| lemma served_lower {k : ℕ} (hk : 0<k) (q : ℕ → Ω → Fin k → Y) |
| (word : Fin d → A) {z : ℝ} (hz : z∈Set.Icc 0 (1/100:ℝ)) (t : ℕ) (ω : Ω) |
| (hs : ∀ ρ,H t ρ=H t ω → ∃ l,hit (maps t ρ) word (q t ρ l)) : |
| 1≤lower hw H hr maps q word z t ω := by |
| have h:=single_le_sum (s:=univ) |
| (fun a _=>RankFunctions.rank_nonneg hz ((family hw H hr maps q word a).p t ω)) |
| (mem_univ (⟨0,hk⟩:Fin k)) |
| rw [first_rank_zero hw H hr maps hk q word t ω hs,RankFunctions.rank_zero] at h |
| exact h |
|
|
| end KServer.HierarchicalRanks |
|
|
| end |
|
|
|
|
| /-! Actual top-down fractional quotas on all complete-prefix labels. The |
| parent budget is obtained from conditional count superadditivity, not posited |
| as an allocation oracle. Fixed unused labels remain present. -/ |
| noncomputable section |
| open scoped BigOperators |
| open Finset |
| namespace KServer.HierarchicalQuota |
| attribute [local instance] Classical.propDecidable Classical.decEq |
| open RankTracking ActualStar |
| variable {Ω A Y:Type} [Fintype Ω] [Fintype A] {k:ℕ} {w:Ω→ℝ} |
| variable (hw:∀ ω,0≤w ω) (H:ℕ→Ω→ℕ) |
| variable (hr:∀ t ω ρ,H (t+1) ω=H (t+1) ρ→H t ω=H t ρ) |
| variable (maps:ℕ→Ω→ℕ→Y→A) (q:ℕ→Ω→Fin k→Y) |
|
|
| def children {d:ℕ} (word:Fin d→A):A→Fin k→FilteredRanks w:= |
| fun a=>HierarchicalRanks.family hw H hr maps q (Fin.snoc word a) |
| def parent {d:ℕ} (word:Fin d→A):Fin k→FilteredRanks w:= |
| HierarchicalRanks.family hw H hr maps q word |
|
|
| def quota (hk:1≤(k:ℝ)): (d:ℕ)→(Fin d→A)→ℕ→Ω→ℝ |
| | 0,_ ,_,_ => k |
| | d+1,word,t,ω=>ActualOutput.out hk (children hw H hr maps q (Fin.init word)) |
| (parent hw H hr maps q (Fin.init word)) t ω |
| (quota hk d (Fin.init word) t ω) (word (Fin.last d)) |
|
|
| lemma quota_snoc (hk:1≤(k:ℝ)) {d:ℕ} (word:Fin d→A) (a:A) (t:ℕ) (ω:Ω): |
| quota hw H hr maps q hk (d+1) (Fin.snoc word a) t ω= |
| ActualOutput.out hk (children hw H hr maps q word) (parent hw H hr maps q word) |
| t ω (quota hw H hr maps q hk d word t ω) a:=by |
| simp only [quota,Fin.init_snoc,Fin.snoc_last] |
|
|
| lemma sizes {d:ℕ} (word:Fin d→A) (hwpos:∀ ω,0<w ω) (t:ℕ) (ω:Ω): |
| (∑ a,CoarseProcess.size (children hw H hr maps q word a) t ω)≤ |
| CoarseProcess.size (parent hw H hr maps q word) t ω:= |
| HierarchicalRanks.size_superadditive hw H hr maps q word t ω (hwpos ω) |
| omit [Fintype A] in |
| lemma size_bound {d:ℕ} (word:Fin d→A) (t:ℕ) (ω:Ω): |
| CoarseProcess.size (parent hw H hr maps q word) t ω≤k:= |
| HierarchicalRanks.size_bound hw H hr maps q word t ω |
|
|
| lemma parent_budget (hk:1≤(k:ℝ)) {d:ℕ} (word:Fin d→A) (hwpos:∀ ω,0<w ω) |
| (t:ℕ) (ω:Ω) (x:ℝ) (hx:ActualOutput.lower k (parent hw H hr maps q word) t ω≤x): |
| (∑ a,(ActualOutput.n (children hw H hr maps q word) t ω a- |
| β k (parent hw H hr maps q word) t ω*ActualOutput.B (children hw H hr maps q word) t ω a+ |
| ActualOutput.f k (children hw H hr maps q word) (parent hw H hr maps q word) t ω a))≤x:=by |
| have hz:=(ActualCaps.ranges hk (parent hw H hr maps q word) t ω (size_bound hw H hr maps q word t ω)).1 |
| have hs:=HierarchicalRanks.lower_superadditive hw H hr maps q word hz t ω (hwpos ω) |
| have heq : (∑ a,(ActualOutput.n (children hw H hr maps q word) t ω a- |
| β k (parent hw H hr maps q word) t ω*ActualOutput.B (children hw H hr maps q word) t ω a+ |
| ActualOutput.f k (children hw H hr maps q word) (parent hw H hr maps q word) t ω a))= |
| ∑ a,HierarchicalRanks.lower hw H hr maps q (Fin.snoc word a) |
| (ActualCaps.z k (parent hw H hr maps q word) t ω) t ω := by |
| apply sum_congr rfl |
| intro a _ |
| exact (TrackedCaps.decomposition (children hw H hr maps q word a) |
| (ActualCaps.z k (parent hw H hr maps q word) t ω) t ω).symm |
| rw [heq] |
| exact hs.trans hx |
|
|
| lemma quota_bounds (hk:1≤(k:ℝ)) (hwpos:∀ ω,0<w ω) (d:ℕ) (word:Fin d→A) (t:ℕ) (ω:Ω): |
| ActualOutput.lower k (parent hw H hr maps q word) t ω≤quota hw H hr maps q hk d word t ω ∧ |
| quota hw H hr maps q hk d word t ω≤72*CoarseProcess.size (parent hw H hr maps q word) t ω:=by |
| induction d with |
| | zero=> |
| have hl:=HierarchicalRanks.lower_root hw H hr maps q word (ActualCaps.z k (parent hw H hr maps q word) t ω) t ω |
| have hs:=HierarchicalRanks.size_root hw H hr maps q word t ω |
| change ActualOutput.lower k (parent hw H hr maps q word) t ω=(k:ℝ) at hl |
| change CoarseProcess.size (parent hw H hr maps q word) t ω=(k:ℝ) at hs |
| simp only [quota,hl,hs] |
| constructor |
| · rfl |
| · linarith |
| | succ d ih=> |
| have hp:=parent_budget hw H hr maps q hk (Fin.init word) hwpos t ω _ (ih (Fin.init word)).1 |
| have hf:=ActualOutput.out_feasible hk (children hw H hr maps q (Fin.init word)) |
| (parent hw H hr maps q (Fin.init word)) (sizes hw H hr maps q (Fin.init word) hwpos) |
| (size_bound hw H hr maps q (Fin.init word)) t ω _ hp |
| have hd:=ActualOutput.output_domination hk (children hw H hr maps q (Fin.init word)) |
| (parent hw H hr maps q (Fin.init word)) (sizes hw H hr maps q (Fin.init word) hwpos) |
| (size_bound hw H hr maps q (Fin.init word)) t ω _ hp (word (Fin.last d)) |
| have he:Fin.snoc (Fin.init word) (word (Fin.last d))=word:=Fin.snoc_init_self word |
| have hl := (hf.1 (word (Fin.last d))).1 |
| have hu := hd.2 |
| simp only [children,he] at hl hu |
| exact ⟨hl,hu⟩ |
|
|
| lemma children_budget (hk:1≤(k:ℝ)) (hwpos:∀ ω,0<w ω) {d:ℕ} (word:Fin d→A) (t:ℕ) (ω:Ω): |
| (∑ a,quota hw H hr maps q hk (d+1) (Fin.snoc word a) t ω)≤quota hw H hr maps q hk d word t ω:=by |
| simp_rw [quota_snoc] |
| exact (ActualOutput.out_feasible hk _ _ (sizes hw H hr maps q word hwpos) |
| (size_bound hw H hr maps q word) t ω _ (parent_budget hw H hr maps q hk word hwpos t ω _ |
| (quota_bounds hw H hr maps q hk hwpos d word t ω).1)).2 |
|
|
| lemma quota_nonneg (hk:1≤(k:ℝ)) (hwpos:∀ ω,0<w ω) {d:ℕ} (word:Fin d→A) (t:ℕ) (ω:Ω): |
| 0≤quota hw H hr maps q hk d word t ω:= |
| (ActualOutput.lower_nonneg hk _ t ω (size_bound hw H hr maps q word t ω)).trans |
| (quota_bounds hw H hr maps q hk hwpos d word t ω).1 |
|
|
| lemma served (hk:1≤(k:ℝ)) (hwpos:∀ ω,0<w ω) {d:ℕ} (word:Fin d→A) (t:ℕ) (ω:Ω) |
| (hs:∀ ρ,H t ρ=H t ω→∃ l,HierarchyCounts.hit (maps t ρ) word (q t ρ l)): |
| 1≤quota hw H hr maps q hk d word t ω:=by |
| have hkp:0<k:=by exact_mod_cast (show (0:ℝ)<k by linarith) |
| exact (HierarchicalRanks.served_lower hw H hr maps hkp q word |
| (ActualCaps.ranges hk _ t ω (size_bound hw H hr maps q word t ω)).1 t ω hs).trans |
| (quota_bounds hw H hr maps q hk hwpos d word t ω).1 |
| end KServer.HierarchicalQuota |
|
|
| end |
|
|
| end OAI |
|
|
