openAI-math-dataset / lean /OAI /Combinatorics /ContingencyTables /Sampling /ExactPhysicalProgram.lean
| import OAI.Combinatorics.ContingencyTables.Sampling.ExactTypedProgram | |
| import OAI.Combinatorics.ContingencyTables.Sampling.PhysicalPartialSequence | |
| namespace OAI | |
| /-! # Physical execution of the parsed and printed exact sampler -/ | |
| namespace ContingencyTables.ExactPhysicalProgram | |
| open MatchingFPRAS MatchingFPRAS.PhysicalStream MatchingFPRAS.PhysicalPost | |
| open MatchingFPRAS.TreeTyped MatchingFPRAS.TreePost Turing | |
| open DensePrograms | |
| noncomputable def ofOnline (P : OnlineProc.Proc (TreeMachine.K TreeFunction.Reg)) : Program where | |
| State := OnlineStack.Q (TreeMachine.K TreeFunction.Reg) (fun _ => Bool) | |
| P.Label RegisterProc.Store | |
| table := liftBool (OnlineStack.table P.code P.coin) | |
| states := OnlineStack.states P.code | |
| supported := liftBool_supported _ _ | |
| (OnlineStack.table_supported P.code P.coin) | |
| entry := OnlineStack.initialState | |
| entry_mem := OnlineStack.initialState_mem P.code | |
| noncomputable def program : Program := ofOnline ExactTypedProgram.program | |
| attribute [local irreducible] ExactTypedProgram.program | |
| noncomputable def output (as : List Symbol) (o : Option MatrixCode) : Cfg Symbol program.State := | |
| upCfg (OnlineStack.cfg (ExactTypedProgram.output as .nil o)) | |
| lemma output_some (as : List Symbol) (X : MatrixCode) : | |
| output as (some X)=program.final (Tape.mk₁ (packed (GreedyFeasibleTable.printMatrix X))) := by | |
| change upCfg (OnlineStack.cfg | |
| (⟨none,none,inputStacks (GreedyFeasibleTable.printMatrix X)⟩ : | |
| OnlineTree.StoreCfg ExactTypedProgram.program))=_ | |
| exact OnlineTree.cfg_packed ExactTypedProgram.program (GreedyFeasibleTable.printMatrix X) none | |
| lemma output_none_live (as : List Symbol) : (output as none).state≠none := by | |
| change (some _ : Option program.State)≠none | |
| exact Option.some_ne_none _ | |
| /-- Every finite exact-sampling prefix runs on one fixed finite physical | |
| transition table, including its live failure endpoint. -/ | |
| theorem prefix_correct (as : List Symbol) (N : ℕ) : | |
| ∃ q,Implements program.table (output as) | |
| (program.initial (Tape.mk₁ (packed as))) q ∧ | |
| ∀ g : Option MatrixCode → ℚ, | |
| q.mean g=(ExactCoreProgram.prefixExperiment (ProfilePrograms.exactParsedMargins as) N).mean g := by | |
| obtain ⟨p,hp,hpm⟩ := ExactTypedProgram.prefix_correct as .nil N | |
| obtain ⟨q,hq,hqm,_⟩ := hp.compile (weight as) (inputStacks_bound as) | |
| have hu := hq.up | |
| have hi := OnlineTree.cfg_packed ExactTypedProgram.program as (some default) | |
| change upCfg (OnlineStack.cfg (OnlineTree.initial ExactTypedProgram.program (encode as) .nil))= | |
| program.initial (Tape.mk₁ (packed as)) at hi | |
| rw [hi] at hu | |
| exact ⟨q,hu,fun g => (hqm g).trans (hpm g)⟩ | |
| end ContingencyTables.ExactPhysicalProgram | |
| end OAI | |