| 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 |
|
|
