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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 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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Complexity.GuardMachineBase
import OAI.Combinatorics.MatchingCount.Machines.PhysicalPost
namespace OAI
namespace MatchingFPRAS.CoinTM1
open Turing MachineCost PhysicalStream
variable {Γ Λ σ : Type} [Inhabited Γ]
abbrev Node (Γ Λ σ : Type) := Λ ⊕ TM1.Stmt Γ Λ σ
abbrev State (Γ Λ σ : Type) := Node Γ Λ σ × σ
abbrev Coins (Λ : Type) := Λ → Option (Bool → Λ)
def aux (a : Γ) : TM1.Stmt Γ Λ σ → σ → Option (State Γ Λ σ × TM0.Stmt Γ)
| .move d q,v => some ((.inr q,v),.move d)
| .write f q,v => some ((.inr q,v),.write (f a v))
| .load f q,v => aux a q (f a v)
| .branch f q r,v => if f a v then aux a q v else aux a r v
| .goto f,v => some ((.inl (f a v),v),.write a)
| .halt,_ => none
def table (M : Λ → TM1.Stmt Γ Λ σ) (coin : Coins Λ) : PhysicalPost.Table Γ (State Γ Λ σ)
| (.inl l,v),a,b => match coin l with
| none => aux a (M l) v
| some f => some ((.inl (f b),v),.write a)
| (.inr q,v),a,_ => aux a q v
def pending (q : TM1.Stmt Γ Λ σ) (v : σ) (T : Tape Γ) : PhysicalPost.Cfg Γ (State Γ Λ σ) :=
⟨some (.inr q,v),T⟩
def trCfg (c : TM1.Cfg Γ Λ σ) : PhysicalPost.Cfg Γ (State Γ Λ σ) :=
⟨c.l.map (fun l=>(Sum.inl l,c.var)),c.Tape⟩
lemma execute_first_eq (M : PhysicalPost.Table Γ (State Γ Λ σ))
(x y : PhysicalPost.Cfg Γ (State Γ Λ σ)) (h : ∀ b,PhysicalPost.tick M x b=PhysicalPost.tick M y b)
{n : ℕ} (hn : 0 < n) (b : Stream) : PhysicalPost.execute M n x b=PhysicalPost.execute M n y b := by
cases n with
| zero => omega
| succ n => simp only [PhysicalPost.execute,h]
theorem statement_ticks (M : Λ → TM1.Stmt Γ Λ σ) (coin : Coins Λ)
(q : TM1.Stmt Γ Λ σ) (v : σ) (T : Tape Γ) :
∃ k,0 < k ∧ k ≤ statementCost q ∧ ∀ b : Stream,
PhysicalPost.execute (table M coin) k (pending q v T) b=trCfg (TM1.stepAux q v T) := by
induction q generalizing v T with
| move d q ih =>
obtain ⟨k,hk,hc,he⟩:=ih v (T.move d)
refine ⟨k+1,by omega,by simp only [statementCost]; omega,fun b=>?_⟩
simpa only [PhysicalPost.execute,PhysicalPost.tick,pending,table,aux,TM1.stepAux] using he (shift 1 b)
| write f q ih =>
obtain ⟨k,hk,hc,he⟩:=ih v (T.write (f T.head v))
refine ⟨k+1,by omega,by simp only [statementCost]; omega,fun b=>?_⟩
simpa only [PhysicalPost.execute,PhysicalPost.tick,pending,table,aux,TM1.stepAux] using he (shift 1 b)
| load f q ih =>
obtain ⟨k,hk,hc,he⟩:=ih (f T.head v) T
refine ⟨k,hk,hc,fun b=>?_⟩
rw [execute_first_eq (table M coin) (pending (.load f q) v T) (pending q (f T.head v) T) (fun _=>rfl) hk]
exact he b
| branch f q r ihq ihr =>
cases h:f T.head v with
| false =>
obtain ⟨k,hk,hc,he⟩:=ihr v T
refine ⟨k,hk,hc.trans (le_max_right _ _),fun b=>?_⟩
rw [execute_first_eq (table M coin) (pending (.branch f q r) v T) (pending r v T) (fun _=>by
simp only [PhysicalPost.tick,pending,table,aux,h,Bool.false_eq_true,ite_false]) hk]
simpa only [TM1.stepAux,h,Bool.cond_false] using he b
| true =>
obtain ⟨k,hk,hc,he⟩:=ihq v T
refine ⟨k,hk,hc.trans (le_max_left _ _),fun b=>?_⟩
rw [execute_first_eq (table M coin) (pending (.branch f q r) v T) (pending q v T) (fun _=>by
simp only [PhysicalPost.tick,pending,table,aux,h,ite_true]) hk]
simpa only [TM1.stepAux,h,Bool.cond_true] using he b
| goto f =>
refine ⟨1,by omega,le_rfl,fun b=>?_⟩
simp only [PhysicalPost.execute,PhysicalPost.tick,pending,table,aux,TM1.stepAux,trCfg,Option.map_some,Tape.write_self]
| halt =>
refine ⟨1,by omega,le_rfl,fun b=>?_⟩
rfl
lemma entry_ticks (M : Λ → TM1.Stmt Γ Λ σ) (coin : Coins Λ)
(l : Λ) (v : σ) (T : Tape Γ) (hl : coin l=none) :
∃ k,0 < k ∧ k ≤ statementCost (M l) ∧ ∀ b : Stream,
PhysicalPost.execute (table M coin) k (trCfg ⟨some l,v,T⟩) b=
trCfg (TM1.stepAux (M l) v T) := by
obtain ⟨k,hk,hc,he⟩:=statement_ticks M coin (M l) v T
refine ⟨k,hk,hc,fun b=>?_⟩
rw [execute_first_eq (table M coin) (trCfg ⟨some l,v,T⟩) (pending (M l) v T) (fun _=>by
simp only [PhysicalPost.tick,trCfg,Option.map_some,table,hl,pending]) hk]
exact he b
lemma coin_tick (M : Λ → TM1.Stmt Γ Λ σ) (coin : Coins Λ)
(l : Λ) (v : σ) (T : Tape Γ) (f : Bool → Λ) (hl : coin l=some f) (b : Bool) :
PhysicalPost.tick (table M coin) (trCfg ⟨some l,v,T⟩) b=trCfg ⟨some (f b),v,T⟩ := by
simp only [PhysicalPost.tick,trCfg,Option.map_some,table,hl,Tape.write_self]
end MatchingFPRAS.CoinTM1
end OAI
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