| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Sampling.OnlineSample |
| import OAI.Combinatorics.MatchingCount.Complexity.OnlineWordCost |
| import OAI.Combinatorics.MatchingCount.Complexity.OnlineTraverseCost |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.OnlineSample |
| open TreeTyped PhysicalStream Polynomial |
|
|
| variable {A B : Type} [Codec A] [Codec B] |
| lemma work_repeatInput : TreeTyped.WorkBound (repeatInputRealizer (A:=A)) (fun x=>x.1+weight x) := by |
| unfold repeatInputRealizer |
| apply TreeTyped.work_congr |
| apply TreeTyped.work_then_polynomial _ (polynomial_mapWith polynomial_first) |
| apply TreeTyped.work_pair _ _ X (by intro x; simp only [Polynomial.eval_X];omega) |
| · exact TreeTyped.work_of_polynomial polynomial_second X (fun x=>by simp only [Polynomial.eval_X];omega) |
| · apply TreeTyped.work_composition |
| (TreeTyped.work_of_polynomial polynomial_first X (fun x=>by simp only [Polynomial.eval_X];omega)) |
| range_numeric_work X |
| intro x |
| simp only [Polynomial.eval_X,id_eq] |
| omega |
|
|
| def mass (m : A→ℕ) (as : List A) : ℕ := weight as+(as.map m).sum |
| lemma weight_le_mass (m : A→ℕ) (as : List A) : weight as ≤ mass m as := Nat.le_add_right _ _ |
| lemma mass_mem (m : A→ℕ) (as : List A) (a : A) (ha : a∈as) : m a ≤ mass m as := |
| by |
| have h : m a ≤ (as.map m).sum := by |
| induction as with |
| | nil => simp at ha |
| | cons b bs ih => |
| simp only [List.mem_cons] at ha |
| simp only [List.map_cons,List.sum_cons] |
| rcases ha with rfl | ha |
| · omega |
| · have := ih ha;omega |
| unfold mass;omega |
| lemma weight_replicate (n : ℕ) (a : A) : weight (List.replicate n a)=n*(1+weight a)+1 := by |
| induction n with |
| | zero => simp |
| | succ n ih => rw [List.replicate_succ,weight_cons,ih,Nat.succ_mul];omega |
| lemma mass_replicate (m : A→ℕ) (n : ℕ) (a : A) : |
| mass m (List.replicate n a)=n*(1+weight a+m a)+1 := by |
| simp only [mass,weight_replicate,List.map_replicate,List.sum_replicate,Nat.nsmul_eq_mul] |
| ring |
| lemma work_traverse_mass {f : A→Experiment B} {F : OnlineTree.Realizer f} {m : A→ℕ} |
| (hF : OnlineTree.WorkBound F m) : OnlineTree.WorkBound (OnlineTree.traverse F) (mass m) := |
| OnlineTree.work_traverse hF X X (by intro as;simpa using weight_le_mass m as) |
| (by intro as a ha;simpa using mass_mem m as a ha) |
|
|
| def repeatMeasure (m : A→ℕ) (x : ℕ×A) : ℕ := x.1+weight x+m x.2 |
| lemma work_repeat {f : A→Experiment B} {F : OnlineTree.Realizer f} {m : A→ℕ} |
| (hF : OnlineTree.WorkBound F m) : OnlineTree.WorkBound |
| (OnlineTree.composition (OnlineTree.deterministic repeatInputRealizer) (OnlineTree.traverse F)) |
| (repeatMeasure m) := by |
| apply OnlineTree.work_composition |
| (OnlineTree.work_deterministic (TreeTyped.work_remeasure work_repeatInput X |
| (fun x=>by simp only [Polynomial.eval_X,repeatMeasure];omega))) (work_traverse_mass hF) (X*(X+1)+1) |
| intro x |
| change mass m (List.replicate x.1 x.2) ≤ _ |
| rw [mass_replicate] |
| simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_X,Polynomial.eval_one] |
| have hx : x.1 ≤ repeatMeasure m x := by unfold repeatMeasure;omega |
| have hw : weight x.2+m x.2 ≤ repeatMeasure m x := by |
| have := weight_snd x;unfold repeatMeasure;omega |
| nlinarith |
|
|
| def pairMeasure (x : ℕ×ℕ) : ℕ := x.1+x.2+weight x |
| lemma work_firstWord : OnlineTree.WorkBound |
| (OnlineTree.composition (OnlineTree.deterministic (TreeTyped.first (A:=ℕ) (B:=ℕ))) OnlineWord.naturalRealizer) |
| pairMeasure := by |
| apply OnlineTree.work_composition |
| (OnlineTree.work_deterministic (TreeTyped.work_of_polynomial polynomial_first X |
| (fun x=>by simp [pairMeasure]))) OnlineWord.work_natural X |
| intro x |
| change x.1+weight x.1 ≤ _ |
| simp only [Polynomial.eval_X,pairMeasure] |
| have := weight_fst x |
| omega |
| lemma work_words : OnlineTree.WorkBound wordsRealizer pairMeasure := by |
| unfold wordsRealizer |
| apply OnlineTree.work_congr |
| apply OnlineTree.work_composition |
| (OnlineTree.work_deterministic (TreeTyped.work_composition |
| (TreeTyped.work_of_polynomial (polynomial_pair polynomial_second polynomial_first) X |
| (fun x=>by simp [pairMeasure])) work_repeatInput (2*X+1) (by |
| intro x;simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_X, |
| Polynomial.eval_one] |
| simp only [pairMeasure,weight_prod];have := weight_fst x;have := weight_snd x;omega))) |
| (work_traverse_mass OnlineWord.work_natural) (X*(2*X+1)+1) |
| intro x |
| change mass (fun n=>n+weight n) (List.replicate x.2 x.1) ≤ _ |
| rw [mass_replicate] |
| simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_X,Polynomial.eval_one,Polynomial.eval_ofNat] |
| have h1 : x.2 ≤ pairMeasure x := by unfold pairMeasure;omega |
| have h2 : weight x.1+x.1+weight x.1 ≤ 2*pairMeasure x := by have := weight_fst x;unfold pairMeasure;omega |
|
|
| have h := Nat.mul_le_mul h1 (show 1+weight x.1+(x.1+weight x.1) ≤ 2*pairMeasure x+1 by omega) |
| nlinarith |
| lemma work_local : OnlineTree.WorkBound localRealizer pairMeasure := by |
| unfold localRealizer |
| exact OnlineTree.work_pair work_firstWord work_words X (fun x=>by simp [pairMeasure]) |
| lemma work_scan : OnlineTree.WorkBound scanRealizer pairMeasure := by |
| unfold scanRealizer |
| exact OnlineTree.work_pair work_firstWord work_local X (fun x=>by simp [pairMeasure]) |
|
|
| def scansMeasure (x : (ℕ×ℕ)×ℕ) : ℕ := pairMeasure x.1+x.2+weight x |
| lemma work_scans : OnlineTree.WorkBound scansRealizer scansMeasure := by |
| unfold scansRealizer |
| apply OnlineTree.work_congr |
| apply OnlineTree.work_composition |
| (OnlineTree.work_deterministic (TreeTyped.work_composition |
| (TreeTyped.work_of_polynomial (polynomial_pair polynomial_second polynomial_first) X |
| (fun x=>by simp [scansMeasure])) work_repeatInput (2*X+1) (by |
| intro x |
| simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_X,Polynomial.eval_one] |
| simp only [scansMeasure,weight_prod] |
| have := weight_fst x;have := weight_snd x;omega))) |
| (work_traverse_mass work_scan) (X*(2*X+1)+1) |
| intro x |
| change mass pairMeasure (List.replicate x.2 x.1) ≤ _ |
| rw [mass_replicate] |
| simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_X,Polynomial.eval_one,Polynomial.eval_ofNat] |
| have h1 : x.2 ≤ scansMeasure x := by unfold scansMeasure;omega |
| have h2 : weight x.1+pairMeasure x.1 ≤ 2*scansMeasure x := by |
| have := weight_fst x;unfold scansMeasure;omega |
| have h := Nat.mul_le_mul h1 (show 1+weight x.1+pairMeasure x.1 ≤ 2*scansMeasure x+1 by omega) |
| omega |
|
|
| def markMeasure (x : ℕ×ℕ) : ℕ := x.2+weight x |
| lemma work_markWord : OnlineTree.WorkBound markWordRealizer markMeasure := by |
| apply OnlineTree.work_mapSecond OnlineWord.work_natural X X |
| · intro x;simp only [Polynomial.eval_X,markMeasure];have := weight_snd x;omega |
| · intro x;simp only [Polynomial.eval_X,markMeasure];have := weight_fst x;omega |
| omit [Codec A] in |
| lemma sum_map_le (m : A→ℕ) (as : List A) (M : ℕ) (h : ∀ a∈as,m a ≤ M) : |
| (as.map m).sum ≤ as.length*M := by |
| induction as with |
| | nil => simp |
| | cons a as ih => |
| have h1 := h a (by simp) |
| have h2 := ih (fun a ha=>h a (by simp [ha])) |
| simp only [List.length_cons,List.map_cons,List.sum_cons,Nat.add_mul,Nat.one_mul] |
| omega |
| lemma mass_le (m : A→ℕ) (as : List A) (M : ℕ) (hw : weight as ≤ M) (h : ∀ a∈as,m a ≤ M) : |
| mass m as ≤ M*(M+1) := by |
| have h1 := sum_map_le m as M h |
| have h2 := Nat.mul_le_mul_right M ((list_length_le_weight as).trans hw) |
| unfold mass |
| nlinarith |
|
|
| def marksMeasure (x : ℕ×List ℕ) : ℕ := x.1+weight x |
| lemma work_marks : OnlineTree.WorkBound marksRealizer marksMeasure := by |
| have hp : TreeTyped.PolynomialTime markRequestsRealizer := polynomial_congr _ |
| (polynomial_mapWith (polynomial_pair polynomial_second polynomial_first)) |
| obtain ⟨p,hp'⟩ := hp |
| have hp : TreeTyped.PolynomialTime markRequestsRealizer := ⟨p,hp'⟩ |
| apply OnlineTree.work_composition |
| (OnlineTree.work_deterministic (TreeTyped.work_of_polynomial hp X (fun x=>by simp [marksMeasure]))) |
| (work_traverse_mass work_markWord) ((p+X)*((p+X)+1)) |
| intro x |
| simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_X,Polynomial.eval_one] |
| apply mass_le |
| · have hw : weight (markRequests x) ≤ p.eval (marksMeasure x) := |
| (Nat.le_add_left _ _).trans ((hp' x).trans (polynomial_eval_monotone p (by unfold marksMeasure;omega))) |
| omega |
| · intro a ha |
| have hw := weight_mem ha |
| have hb : weight (markRequests x) ≤ p.eval (marksMeasure x) := |
| (Nat.le_add_left _ _).trans ((hp' x).trans (polynomial_eval_monotone p (by unfold marksMeasure;omega))) |
| obtain ⟨i,hi,rfl⟩ := List.mem_map.mp ha |
| unfold marksMeasure at hb |
| unfold markMeasure marksMeasure |
| dsimp only |
| omega |
|
|
| def requestMeasure (x : Request) : ℕ := scansMeasure x.1+marksMeasure x.2+weight x |
| lemma work_sample : OnlineTree.WorkBound sampleRealizer requestMeasure := by |
| unfold sampleRealizer |
| apply OnlineTree.work_pair _ _ X (fun x=>by simp [requestMeasure]) |
| · apply OnlineTree.work_composition |
| (OnlineTree.work_deterministic (TreeTyped.work_of_polynomial polynomial_first X |
| (fun x=>by simp [requestMeasure]))) work_scans X |
| intro x |
| simp only [Experiment.All,Polynomial.eval_X] |
| unfold requestMeasure;omega |
| · apply OnlineTree.work_composition |
| (OnlineTree.work_deterministic (TreeTyped.work_of_polynomial polynomial_second X |
| (fun x=>by simp [requestMeasure]))) work_marks X |
| intro x |
| simp only [Experiment.All,Polynomial.eval_X] |
| unfold requestMeasure;omega |
| lemma work_many : OnlineTree.WorkBound manyRealizer (repeatMeasure requestMeasure) := work_repeat work_sample |
| end MatchingFPRAS.OnlineSample |
|
|
| end OAI |
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