| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Algorithm.OutputLengthBound |
| import OAI.Combinatorics.MatchingCount.Algorithm.TreeWeights |
| import OAI.Combinatorics.MatchingCount.Machines.OnlineUniform |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.PhysicalStream |
| lemma qaverage_const {A : Type} [Fintype A] [Nonempty A] (q : ℚ) : |
| qaverage (fun _ : A=>q)=q := by |
| simp only [qaverage,Finset.sum_const,Finset.card_univ,nsmul_eq_mul] |
| have h : (Fintype.card A:ℚ)≠0 := by exact_mod_cast Fintype.card_ne_zero |
| field_simp |
| end MatchingFPRAS.PhysicalStream |
|
|
| namespace MatchingFPRAS.OnlineTrial |
| open PhysicalStream TreeTyped TreeFraction ExecutableSource ExecutableScaling Transport UnitRefresh Enlargement |
|
|
| attribute [local instance] NumberedHierarchy.labelOrder |
| attribute [local instance] Transport.sym2Order Transport.bagEdgeOrder Transport.bagEdgeFintype |
| attribute [local instance] UnitRefresh.Computable.allowedDec Transport.prodOrder Transport.sumOrder |
| variable {G : GraphInput} [Nontrivial (Fin G.n)] |
|
|
| lemma mean_drawAppend (p : Trial.Parameters G) {j : ℕ} (hj : j≤p.K) |
| (L : FiniteTrial.Layer p j) {a : ℕ} (f : (Fin a→Bool)→L.Ω) |
| (h : (L.append p hj).Ω→ℚ) : |
| qaverage (fun b=>h (L.drawAppend p hj f b))= |
| qaverage (fun c=>qaverage (fun b=>h ⟨f c,(L.state (f c)).drawStage p hj b⟩)) := by |
| change qaverage ((fun c : (Fin a→Bool)×(Fin (FiniteTrial.stageBits p)→Bool)=> |
| h ⟨f c.1,(L.state (f c.1)).drawStage p hj c.2⟩) ∘ pairBitsEquiv a (FiniteTrial.stageBits p))=_ |
| rw [qaverage_equiv,qaverage_prod] |
|
|
| lemma mean_layer_succ (p : Trial.Parameters G) {j : ℕ} (hj : j+1≤p.K) |
| (h : (FiniteTrial.layers p (j+1) hj).Ω→ℚ) : |
| qaverage (fun b=>h (FiniteTrial.drawLayer p (j+1) hj b))= |
| qaverage (fun a=>qaverage (fun b=>h |
| ⟨FiniteTrial.drawLayer p j (by omega) a, |
| ((FiniteTrial.layers p j (by omega)).state (FiniteTrial.drawLayer p j (by omega) a)).drawStage p (by omega) b⟩)) := by |
| change qaverage ((fun b=>h ((FiniteTrial.layers p j (by omega)).drawAppend p (by omega) |
| (FiniteTrial.drawLayer p j (by omega)) b)) ∘ FiniteTrial.successorBitsEquiv j (FiniteTrial.stageBits p))=_ |
| rw [qaverage_equiv] |
| exact mean_drawAppend p (by omega) (FiniteTrial.layers p j (by omega)) |
| (FiniteTrial.drawLayer p j (by omega)) h |
|
|
| noncomputable def product {j : ℕ} (p : Trial.Parameters G) (L : FiniteTrial.Layer p j) (x : L.Ω) : ℚ := |
| (Z (completeGraph G.n):ℚ)*∏ i,L.ratios x i |
| omit [Nontrivial (Fin G.n)] in |
| lemma product_nonneg [Nontrivial (Fin G.n)] {j : ℕ} (p : Trial.Parameters G) (L : FiniteTrial.Layer p j) (x : L.Ω) : |
| 0≤product p L x := mul_nonneg (Nat.cast_nonneg _) (Finset.prod_nonneg (fun i _=>L.ratios_nonneg x i)) |
| lemma product_append {j : ℕ} (p : Trial.Parameters G) (hj : j≤p.K) (L : FiniteTrial.Layer p j) |
| (x : (L.append p hj).Ω) : |
| product p (L.append p hj) x = product p L x.1 * |
| (L.state x.1).ratioEstimate (RationalSource.sourceMarkedEquiv (L.state x.1).activity p.K ∘ x.2) := by |
| unfold product FiniteTrial.Layer.append |
| dsimp only |
| rw [Fin.prod_snoc,mul_assoc] |
|
|
| end MatchingFPRAS.OnlineTrial |
|
|
| end OAI |
|
|
