| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Complexity.StackCompilerCost |
| import OAI.Combinatorics.MatchingCount.Graphs.SafeTrace |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.CoinStack |
| open Turing TM2to1 TM2to1.Λ' MachineCost StackCost SafeTrace |
|
|
| variable {K : Type} {Γ : K → Type} {Λ σ : Type} [DecidableEq K] |
|
|
| def safe (p : Λ → Prop) : Λ' K Γ Λ σ → Prop |
| | .normal l => p l |
| | .go _ _ _ | .ret _ => True |
|
|
| def coins (c : CoinTM1.Coins Λ) : CoinTM1.Coins (Λ' K Γ Λ σ) |
| | .normal l => (c l).map (fun f b=>.normal (f b)) |
| | .go _ _ _ | .ret _ => none |
|
|
| variable (M : Λ → TM2.Stmt Γ Λ σ) (p : Λ → Prop) |
| lemma go_path {k : K} (o : StAct K Γ σ k) (q : TM2.Stmt Γ Λ σ) (v : σ) |
| {S : List (Γ k)} {L : ListBlank (∀ k,Option (Γ k))} |
| (hL : L.map (proj k)=ListBlank.mk (S.map some).reverse) |
| (n : ℕ) (hn : n≤ S.length) : |
| Path (tr M) (safe p) n ⟨some (go k o q),v,Tape.mk' ∅ (addBottom L)⟩ |
| ⟨some (go k o q),v,(Tape.move .right)^[n] (Tape.mk' ∅ (addBottom L))⟩ := by |
| rw [←go_time M o q v hL n hn] |
| apply Path.of_run |
| intro i hi l hl |
| rw [go_time M o q v hL i (by omega)] at hl |
| cases hl |
| trivial |
|
|
| lemma return_path (q : TM2.Stmt Γ Λ σ) (v : σ) (L : ListBlank (∀ k,Option (Γ k))) (n : ℕ) : |
| Path (tr M) (safe p) n |
| ⟨some (ret q),v,(Tape.move .right)^[n] (Tape.mk' ∅ (addBottom L))⟩ |
| ⟨some (ret q),v,Tape.mk' ∅ (addBottom L)⟩ := by |
| induction n with |
| | zero => exact .refl _ |
| | succ n ih => |
| apply Path.cons (by intro l hl; cases hl; trivial) |
| have hs : advance (TM1.step (tr M)) |
| ⟨some (ret q),v,(Tape.move .right)^[n+1] (Tape.mk' ∅ (addBottom L))⟩= |
| ⟨some (ret q),v,(Tape.move .right)^[n] (Tape.mk' ∅ (addBottom L))⟩ := by |
| simp only [advance,TM1.step,tr,TM1.stepAux,Tape.move_right_n_head,Tape.mk'_nth_nat, |
| addBottom_nth_succ_fst,Bool.cond_false] |
| rw [Function.iterate_succ_apply',Tape.move_right_left] |
| rfl |
| rw [hs] |
| exact ih |
|
|
| theorem statement_path (q : TM2.Stmt Γ Λ σ) (v : σ) (S : ∀ k,List (Γ k)) |
| (L : ListBlank (∀ k,Option (Γ k))) (H : ℕ) (hS : StackBound S H) |
| (hL : ∀ k,L.map (proj k)=ListBlank.mk ((S k).map some).reverse) : |
| ∃ (t : ℕ) (b : TM1.Cfg (Γ' K Γ) (Λ' K Γ Λ σ) σ), |
| t≤ 2*accesses q*(H+accesses q+1) ∧ TrCfg (TM2.stepAux q v S) b ∧ |
| Path (tr M) (safe p) t |
| (TM1.stepAux (trNormal q) v (Tape.mk' ∅ (addBottom L))) b := by |
| induction q using stmtStRec generalizing v S L H with |
| | run k o q ih => |
| let v':=stVar v (S k) o |
| let S':=Function.update S k (stWrite v (S k) o) |
| have hS' : StackBound S' (H+1):=action_bound hS o v |
| obtain ⟨L',hL',hact⟩:=tr_respects_aux₂ |
| (q:=TM1.Stmt.goto fun _ _=>ret q) hL o |
| obtain ⟨t,b,ht,hb,he⟩:=ih v' S' L' (H+1) hS' hL' |
| have hgo:=go_path M p o q v (hL k) (S k).length le_rfl |
| have hs : advance (TM1.step (tr M)) |
| ⟨some (go k o q),v,(Tape.move .right)^[(S k).length] (Tape.mk' ∅ (addBottom L))⟩= |
| ⟨some (ret q),v',(Tape.move .right)^[(S' k).length] (Tape.mk' ∅ (addBottom L'))⟩ := by |
| simp only [advance,TM1.step] |
| rw [tr,TM1.stepAux,Tape.move_right_n_head,Tape.mk'_nth_nat,addBottom_nth_snd, |
| stk_nth_val _ (hL k),List.getElem?_eq_none (le_of_eq List.length_reverse)] |
| simp only [Option.isNone_none,Bool.cond_true] |
| rw [hact] |
| rfl |
| have hstep:=Path.single (M:=tr M) (safe:=safe p) |
| (⟨some (go k o q),v,(Tape.move .right)^[(S k).length] (Tape.mk' ∅ (addBottom L))⟩) |
| (by intro l hl; cases hl; trivial) |
| rw [hs] at hstep |
| have hret:=return_path M p q v' L' (S' k).length |
| have hlast:=Path.single (M:=tr M) (safe:=safe p) |
| (⟨some (ret q),v',Tape.mk' ∅ (addBottom L')⟩) |
| (by intro l hl; cases hl; trivial) |
| have hf : advance (TM1.step (tr M)) ⟨some (ret q),v',Tape.mk' ∅ (addBottom L')⟩= |
| TM1.stepAux (trNormal q) v' (Tape.mk' ∅ (addBottom L')) := by |
| simp only [advance,TM1.step,tr,TM1.stepAux,Tape.mk'_head,addBottom_head_fst, |
| Bool.cond_true,Option.getD_some] |
| rw [hf] at hlast |
| refine ⟨(S k).length+1+(S' k).length+1+t,b,?_,?_,?_⟩ |
| · rw [accesses_run] |
| have h₁:=hS k |
| have h₂:=hS' k |
| nlinarith |
| · simpa only [step_run] using hb |
| · simpa only [trNormal_run,TM1.stepAux] using (((hgo.trans hstep).trans hret).trans hlast).trans he |
| | load a q ih => exact ih (a v) S L H hS hL |
| | branch f q r ihq ihr => |
| cases hp:f v |
| · obtain ⟨t,b,ht,hb,he⟩:=ihr v S L H hS hL |
| refine ⟨t,b,ht.trans ?_,?_,?_⟩ |
| · simp only [accesses]; gcongr <;> exact le_max_right _ _ |
| · simpa only [TM2.stepAux,hp,Bool.cond_false] using hb |
| · simpa only [trNormal,TM1.stepAux,hp,Bool.cond_false] using he |
| · obtain ⟨t,b,ht,hb,he⟩:=ihq v S L H hS hL |
| refine ⟨t,b,ht.trans ?_,?_,?_⟩ |
| · simp only [accesses]; gcongr <;> exact le_max_left _ _ |
| · simpa only [TM2.stepAux,hp,Bool.cond_true] using hb |
| · simpa only [trNormal,TM1.stepAux,hp,Bool.cond_true] using he |
| | goto f => exact ⟨0,_,Nat.zero_le _,⟨L,hL⟩,.refl _⟩ |
| | halt => exact ⟨0,_,Nat.zero_le _,⟨L,hL⟩,.refl _⟩ |
|
|
| lemma one_step_path (A : ℕ) (hA : ∀ l,accesses (M l)≤ A) |
| {c : TM2.Cfg Γ Λ σ} {a : TM1.Cfg (Γ' K Γ) (Λ' K Γ Λ σ) σ} |
| (hca : TrCfg c a) (hc : ∀ l,c.l=some l→p l) (H : ℕ) (hS : StackBound c.stk H) : |
| ∃ (t : ℕ) (b : TM1.Cfg (Γ' K Γ) (Λ' K Γ Λ σ) σ), |
| t≤ 1+2*A*(H+A+1) ∧ TrCfg (advance (TM2.step M) c) b ∧ Path (tr M) (safe p) t a b := by |
| rcases hca with ⟨L,hL⟩ |
| rename_i l v S |
| cases l with |
| | none => exact ⟨0,_,Nat.zero_le _,⟨L,hL⟩,.refl _⟩ |
| | some l => |
| obtain ⟨t,b,ht,hb,he⟩:=statement_path M p (M l) v S L H hS hL |
| refine ⟨t+1,b,?_,hb,?_⟩ |
| · have hh : 2*accesses (M l)*(H+accesses (M l)+1)≤ 2*A*(H+A+1) := by |
| gcongr <;> exact hA l |
| omega |
| · apply Path.cons (by intro l' hl; cases hl; exact hc l rfl) |
| simpa only [advance,TM1.step,Option.map_some,tr,Option.getD_some] using he |
|
|
| omit [DecidableEq K] in |
| lemma coins_none {c : CoinTM1.Coins Λ} {l : Λ' K Γ Λ σ} |
| (h : safe (fun l=>c l=none) l) : coins c l=none := by |
| cases l with |
| | normal l => simp only [coins,safe] at *; rw [h]; rfl |
| | go k o q => rfl |
| | ret q => rfl |
| end MatchingFPRAS.CoinStack |
|
|
| end OAI |
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