File size: 6,327 Bytes
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 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 | import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.TreeMachine
namespace OAI
namespace MatchingFPRAS.TreeMachine
open TreeData RegisterProc Turing MachineCost
variable {R Λ : Type} [DecidableEq R]
inductive Effect : Instr R Λ → Env R → Option Λ → Env R → Prop
| halt (E) : Effect .halt E none E
| go (E l) : Effect (.go l) E (some l) E
| nilBranch (E i l r) (h : E i=.nil) : Effect (.branch i l r) E (some l) E
| consBranch (E i l r x y) (h : E i=.cons x y) : Effect (.branch i l r) E (some r) E
| nil (E i l) : Effect (.nil i l) E (some l) (Function.update E i .nil)
| copy (E i j h l) : Effect (.copy i j h l) E (some l) (putPair E i j (E i) (E i))
| cons (E i j h l) : Effect (.cons i j h l) E (some l) (putPair E i j .nil (.cons (E i) (E j)))
| split (E i j h l x y) (he : E j=.cons x y) : Effect (.split i j h l) E (some l) (putPair E i j x y)
abbrev Label (M : Λ → Instr R Λ) := Λ ⊕ ((l : Λ) × (procedure (M l)).Label)
instance (M : Λ → Instr R Λ) [Fintype Λ] : Fintype (Label M) := by unfold Label; infer_instance
def code (M : Λ → Instr R Λ) : Label M → TM2.Stmt (fun _ : K R=>Bool) (Label M) Store
| .inl l => match M l with
| .halt => .halt
| .go k => .goto (fun _=>.inl k)
| .branch i a b => .peek (.inl i) (fun _ x=>x) (.branch (fun x=>x.getD false)
(.load (fun _=>none) (.goto (fun _=>.inl b)))
(.load (fun _=>none) (.goto (fun _=>.inl a))))
| _ => .goto (fun _=>.inr ⟨l,default⟩)
| .inr ⟨l,u⟩ => StackComposition.relabel (fun u=>.inr ⟨l,u⟩)
((successor (M l)).map Sum.inl) ((procedure (M l)).code u)
def compile [Fintype Λ] [Inhabited Λ] (M : Λ → Instr R Λ) : Proc (K R) :=
{ Label := Label M
entry := ⟨.inl default⟩
code := code M }
def cfg (M : Λ → Instr R Λ) (l : Option Λ) (E : Env R) : TM2.Cfg (fun _ : K R=>Bool) (Label M) Store :=
⟨l.map Sum.inl,none,encodeEnv E⟩
def SizeBound (E : Env R) (B : ℕ) : Prop := ∀ i,(encode (E i)).length≤B
lemma run_procedure (M : Λ → Instr R Λ) (l : Λ) (E F : Env R) (n : ℕ)
(h : Exec (procedure (M l)) (encodeEnv E) (encodeEnv F) n)
(hd : code M (.inl l)=.goto (fun _=>.inr ⟨l,default⟩)) :
∃ k≤n+1,MachineCost.run (TM2.step (code M)) k (cfg M (some l) E)=
cfg M (successor (M l)) F := by
obtain ⟨j,hj,he⟩:=h
obtain ⟨k,hk,hs⟩:=StackComposition.run_relabel_to_halt (procedure (M l)).code (code M)
(fun u=>Sum.inr ⟨l,u⟩) ((successor (M l)).map Sum.inl) (fun _=>rfl)
j ⟨some default,none,encodeEnv E⟩ none (encodeEnv F) he
refine ⟨k+1,by omega,?_⟩
rw [MachineCost.run]
have ha : advance (TM2.step (code M)) (cfg M (some l) E)=
StackComposition.mapCfg (fun u=>Sum.inr ⟨l,u⟩) ((successor (M l)).map Sum.inl)
⟨some default,none,encodeEnv E⟩ := by
simp only [advance,TM2.step,cfg,Option.map_some,hd,TM2.stepAux,Option.getD_some,
StackComposition.mapCfg_some]
rw [ha,hs]
rfl
theorem effect_compiles (M : Λ → Instr R Λ) (l : Λ) (I : Instr R Λ) (hI : M l=I)
(E F : Env R) (next : Option Λ) (h : Effect I E next F) (B : ℕ) (hB : SizeBound E B) :
∃ k≤5*(B+1),MachineCost.run (TM2.step (code M)) k (cfg M (some l) E)=cfg M next F := by
cases h with
| halt E =>
refine ⟨1,by omega,?_⟩
simp only [MachineCost.run,advance,TM2.step,cfg,Option.map_some,code,hI,TM2.stepAux,
Option.getD_some,Option.map_none]
| go E a =>
refine ⟨1,by omega,?_⟩
simp only [MachineCost.run,advance,TM2.step,cfg,Option.map_some,code,hI,TM2.stepAux,Option.getD_some]
| nilBranch E i a b hi =>
refine ⟨1,by omega,?_⟩
simp only [MachineCost.run,advance,TM2.step,cfg,Option.map_some,code,hI,TM2.stepAux,
encodeEnv,hi,encode,List.head?_cons,Option.getD_some,Bool.cond_false]
| consBranch E i a b x y hi =>
refine ⟨1,by omega,?_⟩
simp only [MachineCost.run,advance,TM2.step,cfg,Option.map_some,code,hI,TM2.stepAux,
encodeEnv,hi,encode,List.head?_cons,Option.getD_some,Bool.cond_true]
| nil E i a =>
have hp:=nil_exec E i a
rw [←hI] at hp
obtain ⟨k,hk,he⟩:=run_procedure M l E _ _ hp (by simp only [code,hI])
refine ⟨k,?_,?_⟩
· have hi:=hB i; omega
· simpa only [hI,successor] using he
| copy E i j hij a =>
have hp:=copy_exec E i j hij a
rw [←hI] at hp
obtain ⟨k,hk,he⟩:=run_procedure M l E _ _ hp (by simp only [code,hI])
refine ⟨k,?_,?_⟩
· have hi:=hB i; have hj:=hB j; omega
· simpa only [hI,successor] using he
| cons E i j hij a =>
have hp:=cons_exec E i j hij a
rw [←hI] at hp
obtain ⟨k,hk,he⟩:=run_procedure M l E _ _ hp (by simp only [code,hI])
refine ⟨k,?_,?_⟩
· have hi:=hB i; omega
· simpa only [hI,successor] using he
| split E i j hij a x y hy =>
have hp:=split_exec E i j hij a x y hy
rw [←hI] at hp
obtain ⟨k,hk,he⟩:=run_procedure M l E _ _ hp (by simp only [code,hI])
refine ⟨k,?_,?_⟩
· have hi:=hB i; have hj:=hB j
rw [hy] at hj
simp only [encode,List.length_cons,List.length_append] at hj
omega
· simpa only [hI,successor] using he
inductive Runs (M : Λ → Instr R Λ) (B : ℕ) : Option Λ → Env R → Env R → ℕ → Prop
| done (E) : Runs M B none E E 0
| step (l E next F G n) (hB : SizeBound E B) (h : Effect (M l) E next F)
(rest : Runs M B next F G n) : Runs M B (some l) E G (n+1)
theorem runs_compiles (M : Λ → Instr R Λ) (B n : ℕ) (l : Option Λ) (E F : Env R)
(h : Runs M B l E F n) :
∃ k≤n*(5*(B+1)),MachineCost.run (TM2.step (code M)) k (cfg M l E)=cfg M none F := by
induction h with
| done E => exact ⟨0,by simp,rfl⟩
| step l E next F G n hB h rest ih =>
obtain ⟨i,hi,he⟩:=effect_compiles M l (M l) rfl E F next h B hB
obtain ⟨j,hj,hf⟩:=ih
refine ⟨i+j,?_,?_⟩
· calc
i+j≤5*(B+1)+n*(5*(B+1)) := Nat.add_le_add hi hj
_=(n+1)*(5*(B+1)) := by ring
· rw [run_add,he,hf]
theorem compile_exec [Fintype Λ] [Inhabited Λ] (M : Λ → Instr R Λ) (B n : ℕ) (E F : Env R)
(h : Runs M B (some default) E F n) :
Exec (compile M) (encodeEnv E) (encodeEnv F) (n*(5*(B+1))) :=
runs_compiles M B n (some default) E F h
end MatchingFPRAS.TreeMachine
end OAI
|