| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Algorithm.RationalObservables |
| import OAI.Combinatorics.MatchingCount.Algorithm.RationalScale |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.FiniteTrial |
| open Finset Enlargement Transport |
|
|
| variable {V : Type*} [LinearOrder V] |
| def sampleAverage {Ω : Type*} {S : ℕ} (f : Ω → ℚ) (x : Fin S → Ω) : ℚ := |
| (∑ i,f (x i))/(S:ℚ) |
| lemma sampleAverage_cast {Ω : Type*} {S : ℕ} (f : Ω → ℚ) (x : Fin S → Ω) : |
| (sampleAverage f x:ℝ)=Enlargement.sampleAverage (fun a=>(f a:ℝ)) x := by |
| simp only [sampleAverage,Enlargement.sampleAverage,Rat.cast_div,Rat.cast_sum,Rat.cast_natCast] |
| def probeMatrix (z : LogicalEdge V → ℚ) (i j : V) : ℚ := |
| if hij : i=j then 0 else z (pairEdge i j hij) |
| lemma probeMatrix_cast (z : LogicalEdge V → ℚ) (i j : V) : |
| (probeMatrix z i j:ℝ)=Enlargement.probeMatrix (fun e=>(z e:ℝ)) i j := by |
| simp only [probeMatrix,Enlargement.probeMatrix] |
| split_ifs <;> norm_cast |
| lemma probeMatrix_symm (z : LogicalEdge V → ℚ) (i j : V) : probeMatrix z i j=probeMatrix z j i := by |
| apply Rat.cast_injective (α:=ℝ) |
| rw [probeMatrix_cast,probeMatrix_cast,Enlargement.probeMatrix_symm] |
| def estimateMatrix (D : ℕ) (dbar : ℚ) (zbar : LogicalEdge V → ℚ) (i j : V) : ℚ := |
| Counting.upperEstimate D dbar (probeMatrix zbar i j) |
| lemma estimateMatrix_cast (D : ℕ) (dbar : ℚ) (zbar : LogicalEdge V → ℚ) (i j : V) : |
| (estimateMatrix D dbar zbar i j:ℝ)=Enlargement.estimateMatrix D dbar (fun e=>(zbar e:ℝ)) i j := by |
| rw [estimateMatrix,Counting.upperEstimate_cast,probeMatrix_cast] |
| rfl |
| lemma estimateMatrix_symm (D : ℕ) (dbar : ℚ) (zbar : LogicalEdge V → ℚ) (i j : V) : |
| estimateMatrix D dbar zbar i j=estimateMatrix D dbar zbar j i := by |
| rw [estimateMatrix,estimateMatrix,probeMatrix_symm] |
| lemma estimateMatrix_bounds (D : ℕ) (dbar : ℚ) (zbar : LogicalEdge V → ℚ) (i j : V) : |
| Counting.alpha≤estimateMatrix D dbar zbar i j ∧ estimateMatrix D dbar zbar i j≤3 := by |
| unfold estimateMatrix Counting.upperEstimate |
| split_ifs |
| · exact ⟨le_max_left _ _,max_le (by norm_num [Counting.alpha]) (min_le_left _ _)⟩ |
| · norm_num [Counting.alpha] |
|
|
| variable {G : GraphInput} {j K S : ℕ} [Nontrivial (Fin G.n)] |
| attribute [local instance] NumberedHierarchy.labelOrder sym2Order bagEdgeOrder bagEdgeFintype |
| attribute [local instance] UnitRefresh.Computable.allowedDec prodOrder sumOrder |
| noncomputable local instance (w : Fin G.n → Fin G.n → ℚ) (K : ℕ) : |
| DecidableEq (SourceLabel (fun x y=>(w x y:ℝ)) K) := Classical.decEq _ |
| noncomputable local instance (w : Fin G.n → Fin G.n → ℚ) (K : ℕ) : |
| Fintype (Transport.BagEdge (sourceBags (fun x y=>(w x y:ℝ)) K)) := Fintype.ofFinite _ |
|
|
| def sourceD (K : ℕ) : ℕ := |
| 100*Fintype.card (SourceVertex (Fin G.n) K)^2*(100000000*(G.n+1)^4) |
| omit [Nontrivial (Fin G.n)] in |
| lemma sourceD_eq (K : ℕ) : sourceD (G:=G) K=Enlargement.sourceD (V:=Fin G.n) K := by |
| simp only [sourceD,Enlargement.sourceD,Fintype.card_fin] |
|
|
| def ScaleState.matrix (s : ScaleState G j) (x : Fin S → RationalSource.Marked s.activity K) : |
| Fin G.n → Fin G.n → ℚ := |
| estimateMatrix (sourceD (G:=G) K) |
| (sampleAverage (RationalSource.observableD s.activity s.nextActivity K) x) |
| (fun e=>sampleAverage (RationalSource.observableProbe s.activity s.nextActivity K e) x) |
| lemma ScaleState.matrix_bounds (s : ScaleState G j) (x : Fin S → RationalSource.Marked s.activity K) (a b : Fin G.n) : |
| Counting.alpha ≤ s.matrix x a b ∧ s.matrix x a b≤3 := estimateMatrix_bounds _ _ _ _ _ |
| lemma ScaleState.matrix_symm (s : ScaleState G j) (x : Fin S → RationalSource.Marked s.activity K) (a b : Fin G.n) : |
| s.matrix x a b=s.matrix x b a := estimateMatrix_symm _ _ _ _ _ |
| def ScaleState.updateVector (hn : 2≤G.n) (s : ScaleState G j) |
| (x : Fin S → RationalSource.Marked s.activity K) : Fin G.n → ℚ := |
| Counting.scalingTrace hn (s.matrix x) G.n |
| lemma ScaleState.updateVector_bounds (hn : 2≤G.n) (s : ScaleState G j) |
| (x : Fin S → RationalSource.Marked s.activity K) (i : Fin G.n) : |
| 0<s.updateVector hn x i ∧ s.updateVector hn x i≤2 := by |
| have h:=(Counting.scalingTrace_invariant hn (s.matrix x) (s.matrix_symm x) |
| (fun a b _=>s.matrix_bounds x a b) G.n).1 |
| exact ⟨h.pos i,h.upper i⟩ |
| def ScaleState.update (hn : 2≤G.n) (s : ScaleState G j) |
| (x : Fin S → RationalSource.Marked s.activity K) : ScaleState G (j+1) where |
| scale i:=s.scale i*s.updateVector hn x i |
| positive i:=mul_pos (s.positive i) (s.updateVector_bounds hn x i).1 |
| upper i:=by |
| rw [pow_succ] |
| exact mul_le_mul (s.upper i) (s.updateVector_bounds hn x i).2 (s.updateVector_bounds hn x i).1.le (by positivity) |
| lemma ScaleState.update_activity (hn : 2≤G.n) (s : ScaleState G j) |
| (x : Fin S → RationalSource.Marked s.activity K) : |
| (s.update hn x).activity=(fun a b=>s.nextActivity a b*s.updateVector hn x a*s.updateVector hn x b) := by |
| funext a b |
| simp only [ScaleState.activity,ScaleState.update,ScaleState.nextActivity] |
| ring |
|
|
| def ScaleState.ratioEstimate (s : ScaleState G j) (x : Fin S → RationalSource.Marked s.activity K) : ℚ := |
| Counting.ratioEstimate (sampleAverage (RationalSource.observableA s.activity K) x) |
| (sampleAverage (RationalSource.observableD s.activity s.nextActivity K) x) |
|
|
| lemma average_A (w : Fin G.n → Fin G.n → ℚ) (K : ℕ) |
| (x : Fin S → SourceMarked (fun a b=>(w a b:ℝ)) K) : |
| (sampleAverage (RationalSource.observableA w K) (RationalSource.sourceMarkedEquiv w K ∘ x):ℝ)= |
| Enlargement.sampleAverage (Enlargement.observableA (fun a b=>(w a b:ℝ)) K) x := by |
| simp only [sampleAverage,Enlargement.sampleAverage,Rat.cast_div,Rat.cast_sum,Rat.cast_natCast,Function.comp_apply, |
| RationalSource.observableA_cast] |
| lemma average_D (w w₁ : Fin G.n → Fin G.n → ℚ) (K : ℕ) |
| (x : Fin S → SourceMarked (fun a b=>(w a b:ℝ)) K) : |
| (sampleAverage (RationalSource.observableD w w₁ K) (RationalSource.sourceMarkedEquiv w K ∘ x):ℝ)= |
| Enlargement.sampleAverage (Enlargement.observableD (fun a b=>(w a b:ℝ)) (fun a b=>(w₁ a b:ℝ)) K) x := by |
| simp only [sampleAverage,Enlargement.sampleAverage,Rat.cast_div,Rat.cast_sum,Rat.cast_natCast,Function.comp_apply, |
| RationalSource.observableD_cast] |
| lemma average_Probe (w w₁ : Fin G.n → Fin G.n → ℚ) (K : ℕ) (e : LogicalEdge (Fin G.n)) |
| (x : Fin S → SourceMarked (fun a b=>(w a b:ℝ)) K) : |
| (sampleAverage (RationalSource.observableProbe w w₁ K e) (RationalSource.sourceMarkedEquiv w K ∘ x):ℝ)= |
| Enlargement.sampleAverage (Enlargement.observableProbe (fun a b=>(w a b:ℝ)) (fun a b=>(w₁ a b:ℝ)) K e) x := by |
| simp only [sampleAverage,Enlargement.sampleAverage,Rat.cast_div,Rat.cast_sum,Rat.cast_natCast,Function.comp_apply, |
| RationalSource.observableProbe_cast] |
| lemma ScaleState.matrix_cast (s : ScaleState G j) |
| (x : Fin S → SourceMarked (fun a b=>(s.activity a b:ℝ)) K) : |
| (fun a b=>(s.matrix (RationalSource.sourceMarkedEquiv s.activity K ∘ x) a b:ℝ))= |
| Enlargement.estimateMatrix (sourceD (G:=G) K) |
| (Enlargement.sampleAverage (Enlargement.observableD (fun a b=>(s.activity a b:ℝ)) |
| (fun a b=>(s.nextActivity a b:ℝ)) K) x) |
| (fun e=>Enlargement.sampleAverage (Enlargement.observableProbe (fun a b=>(s.activity a b:ℝ)) |
| (fun a b=>(s.nextActivity a b:ℝ)) K e) x) := by |
| funext a b |
| simp only [ScaleState.matrix,estimateMatrix_cast,average_D,average_Probe] |
| lemma ScaleState.ratioEstimate_cast (s : ScaleState G j) |
| (x : Fin S → SourceMarked (fun a b=>(s.activity a b:ℝ)) K) : |
| (s.ratioEstimate (RationalSource.sourceMarkedEquiv s.activity K ∘ x):ℝ)= |
| RealStage.ratioEstimate |
| (Enlargement.sampleAverage (Enlargement.observableA (fun a b=>(s.activity a b:ℝ)) K) x) |
| (Enlargement.sampleAverage (Enlargement.observableD (fun a b=>(s.activity a b:ℝ)) |
| (fun a b=>(s.nextActivity a b:ℝ)) K) x) := by |
| simp only [ScaleState.ratioEstimate,Counting.ratioEstimate_cast,average_A,average_D] |
| end MatchingFPRAS.FiniteTrial |
|
|
| end OAI |
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