File size: 3,513 Bytes
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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.OnlineUniform
import OAI.Combinatorics.MatchingCount.Sampling.OnlineSample
namespace OAI
namespace MatchingFPRAS.OnlineSample
open PhysicalStream
abbrev LocalRandom (N k : ℕ) := Bits k × (Fin N → Bits k)
abbrev ScanRandom (N k : ℕ) := Bits k × LocalRandom N k
def localRep {N k : ℕ} (b : LocalRandom N k) : NumericLadder.Bits :=
(List.ofFn b.1,List.ofFn (fun i=>List.ofFn (b.2 i)))
def scanRep {N k : ℕ} (b : ScanRandom N k) : List Bool×NumericLadder.Bits :=
(List.ofFn b.1,localRep b.2)
lemma mean_words (k N : ℕ) (h : List (List Bool) → ℚ) :
(words (k,N)).mean h=qaverage (fun bs : Fin N → Bits k=>h (List.ofFn (fun i=>List.ofFn (bs i)))) := by
unfold words
rw [show List.replicate N k=List.ofFn (fun _ : Fin N=>k) by simp]
exact mean_traverse_ofFn word N (fun _=>k) (fun _ b=>List.ofFn b) (fun _=>mean_word_qaverage k) h
lemma mean_local (k N : ℕ) (h : NumericLadder.Bits → ℚ) :
(localWords (k,N)).mean h=qaverage (fun bs : LocalRandom N k=>h (localRep bs)) := by
simp only [localWords,Experiment.mean_bind,Experiment.mean_map,mean_words,mean_word_qaverage]
rw [qaverage_prod]
rfl
lemma mean_scan (k N : ℕ) (h : List Bool×NumericLadder.Bits → ℚ) :
(scan (k,N)).mean h=qaverage (fun bs : ScanRandom N k=>h (scanRep bs)) := by
simp only [scan,Experiment.mean_bind,Experiment.mean_map,mean_local,mean_word_qaverage]
rw [qaverage_prod]
rfl
lemma mean_scans (k N r : ℕ) (h : List (List Bool×NumericLadder.Bits) → ℚ) :
(scans ((k,N),r)).mean h=qaverage (fun bs : Fin r→ScanRandom N k=>h (List.ofFn (fun i=>scanRep (bs i)))) := by
unfold scans
rw [show List.replicate r (k,N)=List.ofFn (fun _ : Fin r=>(k,N)) by simp]
exact mean_traverse_ofFn scan r (fun _=>(k,N)) (fun _ b=>scanRep b) (fun _=>mean_scan k N) h
lemma mean_marks (km m : ℕ) (keys : Fin m→ℕ) (h : List (ℕ×List Bool) → ℚ) :
(marks (km,List.ofFn keys)).mean h=
qaverage (fun bs : Fin m→Bits km=>h (List.ofFn (fun i=>(keys i,List.ofFn (bs i))))) := by
simp only [marks,markRequests,List.map_ofFn]
apply mean_traverse_ofFn markWord m (fun i=>(keys i,km)) (fun i b=>(keys i,List.ofFn b))
intro i h
simp only [markWord,Experiment.mean_map,mean_word_qaverage]
abbrev Random (N k r m km : ℕ) := (Fin r→ScanRandom N k)×(Fin m→Bits km)
def sampleRep {N k r m km : ℕ} (keys : Fin m→ℕ) (bs : Random N k r m km) : NumericMarking.SampleBits :=
(List.ofFn (fun i=>scanRep (bs.1 i)),List.ofFn (fun i=>(keys i,List.ofFn (bs.2 i))))
lemma mean_sample (N k r m km : ℕ) (keys : Fin m→ℕ) (h : NumericMarking.SampleBits → ℚ) :
(sample (((k,N),r),(km,List.ofFn keys))).mean h=
qaverage (fun bs : Random N k r m km=>h (sampleRep keys bs)) := by
simp only [sample,Experiment.mean_bind,Experiment.mean_map,mean_marks,mean_scans]
rw [qaverage_prod]
rfl
lemma mean_many (S N k r m km : ℕ) (keys : Fin m→ℕ) (h : List NumericMarking.SampleBits → ℚ) :
(many (S,(((k,N),r),(km,List.ofFn keys)))).mean h=
qaverage (fun bs : Fin S→Random N k r m km=>h (List.ofFn (fun i=>sampleRep keys (bs i)))) := by
unfold many
rw [show List.replicate S (((k,N),r),(km,List.ofFn keys))=
List.ofFn (fun _ : Fin S=>(((k,N),r),(km,List.ofFn keys))) by simp]
exact mean_traverse_ofFn sample S (fun _=>(((k,N),r),(km,List.ofFn keys)))
(fun _ b=>sampleRep keys b) (fun _=>mean_sample N k r m km keys) h
end MatchingFPRAS.OnlineSample
end OAI
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