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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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