openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Machines /FiniteWordExecution.lean
| import OAI.Combinatorics.ContingencyTables.Machines.FairBits | |
| import OAI.Combinatorics.MatchingCount.Complexity.OnlineWordCost | |
| namespace OAI | |
| /-! # Executing a computed finite Boolean sampler | |
| The program reads a fresh word on the physical online machine, preserves | |
| the input, and runs the deterministic binary sampler on that word. | |
| -/ | |
| namespace ContingencyTables.FiniteWordExecution | |
| open MatchingFPRAS MatchingFPRAS.PhysicalStream Polynomial | |
| variable {A B : Type} [TreeTyped.Codec A] [TreeTyped.Codec B] | |
| def source (n : A → ℕ) (a : A) : Experiment (List Bool) := OnlineWord.prepend (n a) [] | |
| noncomputable def sourceRealizer {n : A → ℕ} (N : TreeTyped.Realizer n) : | |
| OnlineTree.Realizer (source n) := | |
| (OnlineTree.composition (OnlineTree.deterministic N) OnlineWord.naturalRealizer).congr | |
| (by intro a g; rfl) | |
| def wordMeasure (n : A → ℕ) (a : A) : ℕ := TreeTyped.weight a+n a | |
| theorem work_source {n : A → ℕ} {N : TreeTyped.Realizer n} | |
| (hN : TreeTyped.PolynomialTime N) : | |
| OnlineTree.WorkBound (sourceRealizer N) (wordMeasure n) := by | |
| unfold sourceRealizer | |
| apply OnlineTree.work_congr | |
| have hN' : OnlineTree.WorkBound (OnlineTree.deterministic N) (wordMeasure n) := | |
| OnlineTree.work_of_polynomial (OnlineTree.polynomial_deterministic hN) X (by | |
| intro a | |
| simp only [eval_X,wordMeasure] | |
| omega) | |
| apply OnlineTree.work_composition hN' OnlineWord.work_natural (5*X+1) | |
| intro a | |
| change n a+TreeTyped.weight (n a)≤(5*X+1).eval (wordMeasure n a) | |
| have hw := TreeTyped.weight_nat_numeric (n a) | |
| simp only [eval_add,eval_mul,eval_ofNat,eval_X,eval_one,wordMeasure] | |
| omega | |
| def experiment (n : A → ℕ) (f : A×List Bool → B) (a : A) : Experiment B := | |
| (source n a).map (fun bs => f (a,bs)) | |
| noncomputable def realizer {n : A → ℕ} {f : A×List Bool → B} | |
| (N : TreeTyped.Realizer n) (F : TreeTyped.Realizer f) : OnlineTree.Realizer (experiment n f) := | |
| (OnlineTree.composition | |
| (OnlineTree.pair (OnlineTree.deterministic TreeTyped.identity) (sourceRealizer N)) | |
| (OnlineTree.deterministic F)).congr (by | |
| intro a g | |
| simp only [Experiment.mean_bind,Experiment.mean,Experiment.mean_map,experiment,id_eq]) | |
| /-- Runtime is polynomial in the input's encoded size plus the number of | |
| fresh bits. -/ | |
| theorem work_realizer {n : A → ℕ} {f : A×List Bool → B} | |
| {N : TreeTyped.Realizer n} {F : TreeTyped.Realizer f} | |
| (hN : TreeTyped.PolynomialTime N) (hF : TreeTyped.PolynomialTime F) : | |
| OnlineTree.WorkBound (realizer N F) (wordMeasure n) := by | |
| unfold realizer | |
| apply OnlineTree.work_congr | |
| apply OnlineTree.work_then_polynomial _ (OnlineTree.polynomial_deterministic hF) | |
| have hid : OnlineTree.WorkBound (OnlineTree.deterministic (TreeTyped.identity (A := A))) | |
| (wordMeasure n) := | |
| OnlineTree.work_of_polynomial (OnlineTree.polynomial_deterministic TreeTyped.polynomial_identity) X (by | |
| intro a | |
| simp only [eval_X,wordMeasure] | |
| omega) | |
| apply OnlineTree.work_pair hid (work_source hN) X | |
| intro a | |
| simp only [eval_X,wordMeasure] | |
| omega | |
| omit [TreeTyped.Codec A] [TreeTyped.Codec B] in | |
| theorem experiment_mean (n : A → ℕ) (f : A×List Bool → B) (a : A) (g : B → ℚ) : | |
| (experiment n f a).mean g= | |
| average (fun bits : Bits (n a) => g (f (a,List.ofFn bits))) := by | |
| unfold experiment source | |
| rw [Experiment.mean_map,OnlineWord.mean_prepend_nil] | |
| end ContingencyTables.FiniteWordExecution | |
| end OAI | |