openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /PersistentCountability.lean
| import OAI.Computability.DegreeRigidity.Constructibility.PersistentPresentation | |
| namespace OAI | |
| namespace TuringRigidity.PersistentCountability | |
| open EncodedForcing OracleJump ArithmeticHierarchy PersistentRestrictions PersistentPresentation | |
| noncomputable section | |
| attribute [local instance] Classical.propDecidable | |
| theorem presentation_exists (I : CountableIdeal) : ∃ H, Presented I H := by | |
| obtain ⟨f,hf⟩ := I.countable.exists_eq_range I.nonempty | |
| choose A hA using (fun n => degree_surjective (f n)) | |
| refine ⟨CountableBound.sum A,fun x => ?_⟩ | |
| have hc (k : ℕ) : columns (CountableBound.sum A) k = A k := by | |
| funext n | |
| simp only [columns,CountableBound.sum,Nat.unpair_pair] | |
| simp only [hc,hA,hf,Set.mem_range] | |
| def graphOracle {I : CountableIdeal} {H : Oracle} (hpres : Presented I H) | |
| (ρ : I ≃o I) : Oracle := fun n => decide (Graph ρ hpres n) | |
| theorem graphOracle_injective {I : CountableIdeal} {H : Oracle} | |
| (hpres : Presented I H) : Function.Injective (graphOracle hpres) := by | |
| intro ρ σ he | |
| apply OrderIso.ext | |
| funext x | |
| obtain ⟨n,hn⟩ := (hpres x.val).mp x.property | |
| obtain ⟨j,hj⟩ := (hpres (ρ x).val).mp (ρ x).property | |
| have hx : entry hpres n = x := Subtype.ext hn | |
| have hp : Graph ρ hpres (Nat.pair n j) := by | |
| simp only [PersistentPresentation.Graph,Nat.unpair_pair,hx,hj] | |
| have hq : Graph σ hpres (Nat.pair n j) := by | |
| by_contra hn | |
| have hh := congrFun he (Nat.pair n j) | |
| simp only [graphOracle,decide_eq_true hp,decide_eq_false hn] at hh | |
| cases hh | |
| apply Subtype.ext | |
| simpa only [PersistentPresentation.Graph,Nat.unpair_pair,hx,hj] using hq.symm | |
| theorem graphOracle_reduces {I : CountableIdeal} {H Y : Oracle} | |
| (hpres : Presented I H) (ρ : I ≃o I) (h : RecursivePred Y (Graph ρ hpres)) : | |
| Reduces (graphOracle hpres ρ) Y := by | |
| apply RecursiveIn.iff_nat.mpr | |
| exact h.of_eq (fun n => by simp [oracleFunction,graphOracle]) | |
| theorem arithmetic_maps_countable {I : CountableIdeal} {H : Oracle} | |
| (hpres : Presented I H) (Y : Oracle) : | |
| {ρ : I ≃o I | RecursivePred Y (Graph ρ hpres)}.Countable := by | |
| apply ((countable_reduces Y).preimage (graphOracle_injective hpres)).mono | |
| intro ρ hρ | |
| exact graphOracle_reduces hpres ρ hρ | |
| theorem source_4_1_9 (I : CountableIdeal) | |
| (hz : degree (jump FixedArithmetic.zero) ∈ I.carrier) : | |
| {ρ : I ≃o I | Persistent I ρ}.Countable := by | |
| obtain ⟨H,hpres⟩ := presentation_exists I | |
| apply (arithmetic_maps_countable hpres (iterate H 11)).mono | |
| intro ρ hp | |
| exact (source_4_1_8 ρ hp hz hpres).2 | |
| end | |
| end TuringRigidity.PersistentCountability | |
| end OAI | |