| import OAI.Combinatorics.ContingencyTables.Sampling.ExactPhysicalProgram |
| import OAI.Combinatorics.ContingencyTables.Sampling.ExactGateTyped |
|
|
| namespace OAI |
|
|
| /-! # The cheap exact-sampling prefix on the same physical program -/ |
|
|
| namespace ContingencyTables.ExactGatePhysical |
| open MatchingFPRAS MatchingFPRAS.PhysicalStream MatchingFPRAS.PhysicalPost |
| open MatchingFPRAS.TreeTyped MatchingFPRAS.TreePost Turing |
| open DensePrograms ProfilePrograms |
|
|
| attribute [local irreducible] ExactTypedProgram.program |
|
|
| noncomputable def output (as : List Symbol) (o : Option MatrixCode) : |
| Cfg Symbol ExactPhysicalProgram.program.State := |
| upCfg (OnlineStack.cfg (ExactGateTyped.output as .nil o)) |
|
|
| lemma output_some (as : List Symbol) (X : MatrixCode) : |
| output as (some X)=ExactPhysicalProgram.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 ExactPhysicalProgram.program.State)≠none |
| exact Option.some_ne_none _ |
|
|
| noncomputable def typedBudget (as : List Symbol) (B : ℕ) : ℕ := |
| exactParsedMarginsRealizer.cost as+ExactGatePrefix.budget (exactParsedMargins as)+B |
|
|
| noncomputable def budget (as : List Symbol) (B : ℕ) : ℕ := |
| typedBudget as B*OnlineStack.Executes.factor ExactTypedProgram.program.code |
| (weight as) (typedBudget as B) |
|
|
| theorem prefix_correct_cost (as : List Symbol) (B : ℕ) |
| (hB : (ExactGatePrefix.experiment (exactParsedMargins as)).All |
| (fun o => o.elim 0 GreedyFeasibleTable.printMatrixRealizer.cost≤B)) : |
| ∃ q,Implements ExactPhysicalProgram.program.table (output as) |
| (ExactPhysicalProgram.program.initial (Tape.mk₁ (packed as))) q ∧ |
| (∀ g : Option MatrixCode → ℚ, |
| q.mean g=(ExactGatePrefix.experiment (exactParsedMargins as)).mean g) ∧ |
| q.cost≤budget as B := by |
| obtain ⟨p,hp,hpm,hpc⟩ := ExactGateTyped.prefix_correct_cost as .nil B hB |
| obtain ⟨q,hq,hqm,hqc⟩ := 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))= |
| ExactPhysicalProgram.program.initial (Tape.mk₁ (packed as)) at hi |
| rw [hi] at hu |
| refine ⟨q,hu,(fun g => (hqm g).trans (hpm g)),?_⟩ |
| exact hqc.trans (Nat.mul_le_mul hpc (OnlineStack.Executes.factor_mono |
| (weight as) (weight as) p.cost (typedBudget as B) le_rfl hpc)) |
|
|
| end ContingencyTables.ExactGatePhysical |
|
|
| end OAI |
|
|
