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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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Machines.PhysicalPost
namespace OAI
namespace MatchingFPRAS.PhysicalPost
open Turing PhysicalStream
variable {Γ Q : Type} [Inhabited Γ]
def Supported (M : Table Γ Q) (S : Finset Q) : Prop :=
∀ q ∈ S, ∀ a b q' act,M q a b=some (q',act) → q'∈S
noncomputable def restrict (M : Table Γ Q) (S : Finset Q) (_h : Supported M S) :
Table Γ {q // q∈S} := by
classical
exact fun q a b=>(M q.val a b).map (fun qa=>
((if hm:qa.1∈S then ⟨qa.1,hm⟩ else q),qa.2))
def liftCfg (S : Finset Q) (c : Cfg Γ {q // q∈S}) : Cfg Γ Q :=
⟨c.state.map Subtype.val,c.tape⟩
lemma tick_lift (M : Table Γ Q) (S : Finset Q) (h : Supported M S)
(c : Cfg Γ {q // q∈S}) (b : Bool) :
tick M (liftCfg S c) b=liftCfg S (tick (restrict M S h) c b) := by
cases c with
| mk state T => cases state with
| none => rfl
| some q =>
cases e:M q.val T.head b with
| none => simp only [tick,liftCfg,Option.map_some,restrict,e,Option.map_none]
| some qa =>
rcases qa with ⟨q',act⟩
have hq' : q'∈S:=h q.val q.property T.head b q' act e
simp only [tick,liftCfg,Option.map_some,restrict,e,dite_eq_left hq']
lemma execute_lift (M : Table Γ Q) (S : Finset Q) (h : Supported M S)
(n : ℕ) (c : Cfg Γ {q // q∈S}) (b : Stream) :
execute M n (liftCfg S c) b=liftCfg S (execute (restrict M S h) n c b) := by
induction n generalizing c b with
| zero => rfl
| succ n ih => rw [execute,tick_lift,ih,execute]
noncomputable def restrictedMachine (M : Table Symbol Q) (S : Finset Q)
(h : Supported M S) (q₀ : Q) (hq₀ : q₀∈S) : RandomMachine :=
letI : Inhabited {q //q∈S}:=⟨⟨q₀,hq₀⟩⟩
machine (restrict M S h)
lemma restrictedMachine_run (M : Table Symbol Q) (S : Finset Q)
(h : Supported M S) (q₀ : Q) (hq₀ : q₀∈S)
(input : List Symbol) (n : ℕ) (b : Bits n) :
letI : Inhabited {q //q∈S}:=⟨⟨q₀,hq₀⟩⟩
MatchingFPRAS.run (restrictedMachine M S h q₀ hq₀) input b=
sealCfg (restrict M S h)
(execute (restrict M S h) n ⟨some default,Tape.mk₁ input⟩ (extend b)) :=
by
let : Inhabited {q //q∈S}:=⟨⟨q₀,hq₀⟩⟩
exact run_seal (restrict M S h) input n b
end MatchingFPRAS.PhysicalPost
end OAI
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