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e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 | 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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