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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 | 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
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