| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Machines.TreePostRealization |
| import OAI.Combinatorics.MatchingCount.Machines.PhysicalProgram |
| import OAI.Combinatorics.MatchingCount.Machines.OnlineTree |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.OnlineTree |
| open Turing TreeTyped TreeData TreeFunction TreeMachine TreePost PhysicalPost PhysicalStream |
|
|
| lemma input_height {A : Type} [Codec A] (a : A) : OnlineStack.height (inputStacks a)=weight a := by |
| apply le_antisymm |
| · exact Finset.sup_le (fun k _=>inputStacks_bound a k) |
| · have h:=OnlineStack.height_bound (inputStacks a) (Sum.inl (0 : Reg)) |
| simpa only [inputStacks,encodeEnv,env,ite_true,weight,TreeFunction.size] using h |
|
|
| lemma cfg_packed {A : Type} [Codec A] (P : OnlineProc.Proc (K Reg)) (a : A) (l : Option P.Label) : |
| upCfg (OnlineStack.cfg (⟨l,none,inputStacks a⟩ : StoreCfg P))= |
| ⟨l.map (fun u=>(Sum.inl (TM1to1.Λ'.normal (TM2to1.Λ'.normal u)),none)),Tape.mk₁ (packed a)⟩ := by |
| simp only [upCfg,OnlineStack.cfg,OnlineStack.stackCfg,input_height,CoinTM1.trCfg, |
| TM1to1.trCfg,TM1to1.trTape,TM1to1.trTape',Tape.mk₁,Tape.mk₂,Tape.mk'_left,Tape.mk'_right₀, |
| ListBlank.flatMap_mk,List.flatMap_nil,Option.map_map] |
| rw [Tape.map_mk'] |
| rfl |
|
|
| noncomputable def physicalProgram {A B : Type} [Codec A] [Codec B] |
| {f : A→Experiment B} (F : Realizer f) : Program where |
| State := OnlineStack.Q (K Reg) (fun _=>Bool) F.program.Label RegisterProc.Store |
| table := liftBool (OnlineStack.table F.program.code F.program.coin) |
| states := OnlineStack.states F.program.code |
| supported := liftBool_supported _ _ (OnlineStack.table_supported F.program.code F.program.coin) |
| entry := OnlineStack.initialState |
| entry_mem := OnlineStack.initialState_mem F.program.code |
|
|
| noncomputable def physicalCost {A B : Type} [Codec A] [Codec B] |
| {f : A→Experiment B} (F : Realizer f) (a : A) : ℕ := |
| F.cost a*OnlineStack.Executes.factor F.program.code (weight a) (F.cost a) |
|
|
| lemma physical_realizes {A B : Type} [Codec A] [Codec B] |
| {f : A→Experiment B} (F : Realizer f) (a : A) : |
| (physicalProgram F).Realizes (Tape.mk₁ (packed a)) |
| ((f a).map (fun b=>Tape.mk₁ (packed b))) (physicalCost F a) := by |
| obtain ⟨p,hp,hm,hc⟩:=F.correct a .nil |
| obtain ⟨q,hq,hqm,hqc⟩:=hp.compile (weight a) (inputStacks_bound a) |
| have he:=hq.up |
| have hi : upCfg (OnlineStack.cfg (initial F.program (encode a) .nil))= |
| (physicalProgram F).initial (Tape.mk₁ (packed a)) := cfg_packed F.program a (some default) |
| have ho (b : B) : upCfg (OnlineStack.cfg (final F.program (encode b) .nil))= |
| (physicalProgram F).final (Tape.mk₁ (packed b)) := cfg_packed F.program b none |
| rw [hi] at he |
| simp_rw [ho] at he |
| refine ⟨q.map (fun b=>Tape.mk₁ (packed b)),?_,?_,?_⟩ |
| · exact he.map_all _ _ (Experiment.all_true _ |>.mono (fun _ _=>rfl)) |
| · intro w |
| rw [Experiment.mean_map,hqm,hm,Experiment.mean_map] |
| · rw [Experiment.cost_map] |
| exact hqc.trans (Nat.mul_le_mul hc (OnlineStack.Executes.factor_mono _ _ _ _ le_rfl hc)) |
| end MatchingFPRAS.OnlineTree |
|
|
| end OAI |
|
|
