File size: 3,550 Bytes
9dbc64e | 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 | import OAI.Computability.DegreeRigidity.Effective.UniformOracleSimulation
import OAI.Computability.DegreeRigidity.Effective.FiniteShuffle
namespace OAI
namespace TuringRigidity.FiniteOverwrite
open UniformPrograms
def overwrite (s : List Bool) (A : Oracle) (n : ℕ) : Bool :=
if n < s.length then s.getD n false else A n
theorem overwrite_below (s : List Bool) (A : Oracle) (n : ℕ) (hn : n < s.length) :
overwrite s A n = s.getD n false := by simp [overwrite,hn]
theorem overwrite_above (s : List Bool) (A : Oracle) (n : ℕ) (hn : s.length ≤ n) :
overwrite s A n = A n := by simp [overwrite,Nat.not_lt.mpr hn]
theorem overwrite_program (s : List Bool) :
Runs oracleFunction (fun A => oracleFunction (overwrite s A)) := by
have hp : Primrec (fun z : ℕ =>
if (Nat.unpair z).1 < s.length then
(if s.getD (Nat.unpair z).1 false then 1 else 0) else (Nat.unpair z).2) :=
Primrec.ite (Primrec.nat_lt.comp (Primrec.fst.comp Primrec.unpair) (Primrec.const s.length))
(Primrec.ite (Primrec.eq.comp ((Primrec.list_getD false).comp (Primrec.const s)
(Primrec.fst.comp Primrec.unpair)) (Primrec.const true)) (Primrec.const 1) (Primrec.const 0))
(Primrec.snd.comp Primrec.unpair)
have hq : Runs oracleFunction (fun A n => Part.some (if A n then 1 else 0)) := query
apply (total_comp (primrec hp) (total_pair (primrec Primrec.id) hq)).of_eq
intro A n
simp only [Nat.unpair_pair,oracleFunction,overwrite,id_eq]
split <;> rfl
theorem overwrite_reduces (s : List Bool) (A : Oracle) : Reduces (overwrite s A) A := by
obtain ⟨p,hp⟩ := overwrite_program s
exact (OracleCode.turingReducible_iff_exists_code _ _).mpr ⟨p,hp A⟩
theorem restore_initial (s : List Bool) (A : Oracle) :
overwrite (FiniteShuffle.initial A s.length) (overwrite s A) = A := by
funext n
by_cases hn : n < s.length
· rw [overwrite_below _ _ _ (by simpa using hn),FiniteShuffle.prefix_getD _ _ _ hn]
· simp [overwrite,hn]
theorem degree_overwrite (s : List Bool) (A : Oracle) : degree (overwrite s A) = degree A := by
apply (degree_eq_iff _ _).mpr
exact ⟨overwrite_reduces s A,by
simpa only [restore_initial] using
overwrite_reduces (FiniteShuffle.initial A s.length) (overwrite s A)⟩
theorem overwrite_join_program (s : List Bool) (R : Oracle) :
Runs (fun A => oracleFunction (join A R))
(fun A => oracleFunction (join (overwrite s A) R)) := by
have hp : Primrec (fun z : ℕ =>
if (Nat.unpair z).1.bodd = false ∧ (Nat.unpair z).1 / 2 < s.length then
(if s.getD ((Nat.unpair z).1 / 2) false then 1 else 0) else (Nat.unpair z).2) :=
Primrec.ite
((Primrec.eq.comp (Primrec.nat_bodd.comp (Primrec.fst.comp Primrec.unpair))
(Primrec.const false)).and
(Primrec.nat_lt.comp (Primrec.nat_div.comp (Primrec.fst.comp Primrec.unpair)
(Primrec.const 2)) (Primrec.const s.length)))
(Primrec.ite (Primrec.eq.comp ((Primrec.list_getD false).comp (Primrec.const s)
(Primrec.nat_div.comp (Primrec.fst.comp Primrec.unpair) (Primrec.const 2))) (Primrec.const true))
(Primrec.const 1) (Primrec.const 0))
(Primrec.snd.comp Primrec.unpair)
have hq : Runs (fun A => oracleFunction (join A R))
(fun A n => Part.some (if join A R n then 1 else 0)) := query
apply (total_comp (primrec hp) (total_pair (primrec Primrec.id) hq)).of_eq
intro A n
cases hb : n.bodd <;> by_cases hn : n / 2 < s.length <;>
simp [oracleFunction,join,overwrite,hb,hn]
end TuringRigidity.FiniteOverwrite
end OAI
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