| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Algorithm.OnlineLadderSchedule |
| import OAI.Combinatorics.MatchingCount.Complexity.OnlineRequestCost |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.TreeTyped |
| open NumericBounds |
|
|
| variable {A B C : Type} [Codec A] [Codec B] [Codec C] |
| lemma work_comp_bound {f : A→B} {g : B→C} {F : Realizer f} {G : Realizer g} |
| {m : A→ℕ} {n : B→ℕ} (hF : WorkBound F m) (hG : WorkBound G n) |
| (h : Bound (n ∘ f) m) : WorkBound (composition F G) m := by |
| obtain ⟨p,hp⟩:=h |
| exact work_composition hF hG p hp |
| lemma work_lift_bound {f : A→B} {F : Realizer f} {m : A→ℕ} |
| (hF : PolynomialTime F) (h : Bound weight m) : WorkBound F m := by |
| obtain ⟨p,hp⟩:=h |
| exact work_of_polynomial hF p hp |
| lemma work_remeasure_bound {f : A→B} {F : Realizer f} {m n : A→ℕ} |
| (hF : WorkBound F m) (h : Bound m n) : WorkBound F n := by |
| obtain ⟨p,hp⟩:=h |
| exact work_remeasure hF p hp |
| end MatchingFPRAS.TreeTyped |
|
|
| namespace MatchingFPRAS.OnlineSchedule |
| open TreeTyped TreeFraction Polynomial NumericBounds ExecutableSource Transport |
|
|
| def vertexMeasure (x : ℕ×ℕ) : ℕ := x.1+x.2+TreeTyped.weight x |
| lemma work_vertexNumber : WorkBound vertexNumberRealizer vertexMeasure := by |
| have lift {A : Type} [Codec A] {f : (ℕ×ℕ)→A} {F : Realizer f} (hF : PolynomialTime F) : |
| WorkBound F vertexMeasure := work_of_polynomial hF X (fun x=>by simp [vertexMeasure]) |
| unfold vertexNumberRealizer |
| apply work_then_polynomial _ polynomial_listLength |
| apply work_then_polynomial _ polynomial_vertices |
| apply work_pair _ _ X (fun x=>by simp [vertexMeasure]) |
| · exact work_composition (lift polynomial_first) range_numeric_work X (by |
| intro x;simp only [Polynomial.eval_X,id_eq];unfold vertexMeasure;omega) |
| · apply work_fuelAfter (lift (polynomial_addNat |
| (polynomial_mulNat (polynomial_constant _) polynomial_second) (polynomial_constant _))) (2*X+2) |
| intro x |
| simp only [Polynomial.eval_add,Polynomial.eval_mul,Polynomial.eval_ofNat,Polynomial.eval_X] |
| change 2*x.2+2 ≤ 2*vertexMeasure x+2 |
| unfold vertexMeasure;omega |
|
|
| def ladderMeasure (x : LadderInput) : ℕ := x.1.1+x.1.2+x.2+TreeTyped.weight x |
| lemma work_ladderSearchInput : WorkBound ladderSearchInputRealizer ladderMeasure := by |
| have lift {A : Type} [Codec A] {f : LadderInput→A} {F : Realizer f} (hF : PolynomialTime F) : |
| WorkBound F ladderMeasure := work_of_polynomial hF X (fun x=>by simp [ladderMeasure]) |
| unfold ladderSearchInputRealizer |
| apply work_pair _ _ X (fun x=>by simp [ladderMeasure]) |
| · apply work_then_polynomial _ polynomial_ladderPowerPlan |
| apply work_pair (lift polynomial_first) _ X (fun x=>by simp [ladderMeasure]) |
| apply work_fuelAfter (lift polynomial_second) X |
| intro x;simp only [Polynomial.eval_X];unfold ladderMeasure;omega |
| · apply work_fuelAfter (lift polynomial_tierBound) (X^2*(X+2*X+2)) |
| intro x |
| have hn : x.1.1 ≤ ladderMeasure x := by unfold ladderMeasure;omega |
| have hd : x.1.2 ≤ ladderMeasure x := by unfold ladderMeasure;omega |
| have hk : x.2 ≤ ladderMeasure x := by unfold ladderMeasure;omega |
| have hb:=NumericBounds.clog_le x.1.2 |
| simp only [Polynomial.eval_mul,Polynomial.eval_pow,Polynomial.eval_add,Polynomial.eval_X,Polynomial.eval_ofNat] |
| change tierBound x.1.1 x.1.2 x.2 ≤ _ |
| unfold tierBound |
| gcongr; omega |
| lemma work_ladderSchedule : WorkBound ladderScheduleRealizer ladderMeasure := |
| work_then_polynomial work_ladderSearchInput polynomial_findPower |
| end MatchingFPRAS.OnlineSchedule |
|
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| end OAI |
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