openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /SetModels /ElementaryDefinableSets.lean
| import OAI.Computability.DegreeRigidity.SetModels.ElementaryRealModel | |
| import OAI.Computability.DegreeRigidity.Representation.NativeModelCompatibility | |
| namespace OAI | |
| namespace TuringRigidity.ElementaryModel | |
| open BoundedSetTheory TransitiveNameModel RelationCollapse SentenceForm SetDegreeDecoding | |
| universe u | |
| theorem unique_definable_mem (a : ℕ → ZFSet.{u}) (p : SentenceForm) (e : ℕ → ZFSet.{u}) | |
| (he : ∀ i, e i ∈ hullSet a) (b : ZFSet.{u}) (hb : p.Sat Set.univ (cons b e)) | |
| (hu : ∀ x, p.Sat Set.univ (cons x e) → x = b) : b ∈ hullSet a := by | |
| have hen : ∀ i, e i ∈ hull a := fun i => (mem_hullSet a _).mp (he i) | |
| have hs := (hull_elementary a (.ex p) e hen).mpr ⟨b,Set.mem_univ _,hb⟩ | |
| obtain ⟨x,hx,hp⟩ := hs | |
| have ht := (hull_elementary a p (cons x e) | |
| (by intro i; cases i <;> simp [cons,hen,hx])).mp hp | |
| exact hu x ht ▸ (mem_hullSet a x).mpr hx | |
| theorem unorderedPair_mem_hull (a : ℕ → ZFSet.{u}) {x y : ZFSet.{u}} | |
| (hx : x ∈ hullSet a) (hy : y ∈ hullSet a) : ({x,y} : ZFSet.{u}) ∈ hullSet a := by | |
| apply unique_definable_mem a (fromBounded (.unorderedPair 0 1 2)) (cons x (fun _ => y)) | |
| (by intro i; cases i <;> assumption) | |
| · exact (bounded_univ _ _).mpr ((Formula.eval_unorderedPair _ _ _ _).mpr rfl) | |
| · intro z hz | |
| exact (Formula.eval_unorderedPair _ _ _ _).mp ((bounded_univ _ _).mp hz) | |
| theorem orderedPair_mem_hull (a : ℕ → ZFSet.{u}) {x y : ZFSet.{u}} | |
| (hx : x ∈ hullSet a) (hy : y ∈ hullSet a) : ZFSet.pair x y ∈ hullSet a := by | |
| have hxx := unorderedPair_mem_hull a hx hx | |
| have hxy := unorderedPair_mem_hull a hx hy | |
| simpa [ZFSet.pair] using unorderedPair_mem_hull a hxx hxy | |
| theorem collapse_unorderedPair (a : ℕ → ZFSet.{u}) {x y : ZFSet.{u}} | |
| (hx : x ∈ hullSet a) (hy : y ∈ hullSet a) : | |
| structureMap (hullSet a) ({x,y} : ZFSet.{u}) = | |
| ({structureMap (hullSet a) x,structureMap (hullSet a) y} : ZFSet.{u}) := by | |
| apply ZFSet.ext; intro z | |
| rw [structureMap,mem_value,ZFSet.mem_pair] | |
| constructor | |
| · rintro ⟨w,_,hw,hzw⟩ | |
| have hwp := ((membershipRelation_pair _ _ _).mp hw).2.2 | |
| rcases ZFSet.mem_pair.mp hwp with rfl|rfl | |
| · exact Or.inl hzw | |
| · exact Or.inr hzw | |
| · intro hz | |
| have hp := unorderedPair_mem_hull a hx hy | |
| rcases hz with hxz|hyz | |
| · exact ⟨x,hx,(membershipRelation_pair _ _ _).mpr ⟨hx,hp,ZFSet.mem_pair.mpr (Or.inl rfl)⟩,hxz⟩ | |
| · exact ⟨y,hy,(membershipRelation_pair _ _ _).mpr ⟨hy,hp,ZFSet.mem_pair.mpr (Or.inr rfl)⟩,hyz⟩ | |
| theorem collapse_orderedPair (a : ℕ → ZFSet.{u}) {x y : ZFSet.{u}} | |
| (hx : x ∈ hullSet a) (hy : y ∈ hullSet a) : | |
| structureMap (hullSet a) (ZFSet.pair x y) = | |
| ZFSet.pair (structureMap (hullSet a) x) (structureMap (hullSet a) y) := by | |
| have hxx := unorderedPair_mem_hull a hx hx | |
| have hxy := unorderedPair_mem_hull a hx hy | |
| have hh := collapse_unorderedPair a hxx hxy | |
| rw [collapse_unorderedPair a hx hx,collapse_unorderedPair a hx hy] at hh | |
| simpa [ZFSet.pair] using hh | |
| noncomputable def ambientRealSet : ZFSet.{u} := ZFSet.powerset ZFSet.omega | |
| noncomputable def ambientGraphBound : ZFSet.{u} := ZFSet.powerset (ZFSet.prod ZFSet.omega ZFSet.omega) | |
| theorem ambient_real_decode (x : ZFSet.{u}) (hx : x ∈ ambientRealSet) : | |
| ∃ B : Oracle, realCode B = x := | |
| ⟨decodeReal x,realCode_decodeReal (ZFSet.mem_powerset.mp hx)⟩ | |
| theorem ambient_graph_bound (f : ℕ → ℕ) (k : ℕ) : finiteNaturalGraph.{u} f k ∈ ambientGraphBound := by | |
| apply ZFSet.mem_powerset.mpr | |
| intro x hx | |
| obtain ⟨n,_,rfl⟩ := (mem_finiteNaturalGraph f k x).mp hx | |
| exact ZFSet.mem_prod.mpr ⟨natSet n,(mem_omega _).mpr ⟨n,rfl⟩, | |
| natSet (f n),(mem_omega _).mpr ⟨f n,rfl⟩,rfl⟩ | |
| theorem ambient_degreeCode (A : Oracle) : degreeCode ambientRealSet.{u} A = degreeSet (degree A) := by | |
| apply ZFSet.ext; intro x | |
| rw [degreeCode,ZFSet.mem_sep,mem_degreeSet] | |
| exact ⟨And.right,fun ⟨B,hB,hBA⟩ => | |
| ⟨hB.symm ▸ ZFSet.mem_powerset.mpr (realCode_subset B),B,hB,hBA⟩⟩ | |
| theorem degreeSet_mem_hull (a : ℕ → ZFSet.{u}) | |
| (hω : ZFSet.omega ∈ hullSet a) (h0 : natSet 0 ∈ hullSet a) | |
| (hR : ambientRealSet ∈ hullSet a) (hQ : ambientGraphBound ∈ hullSet a) | |
| {A : Oracle} (hA : realCode A ∈ hullSet a) : degreeSet (degree A) ∈ hullSet a := by | |
| obtain ⟨φ,hφ⟩ := degree_equality_bounded.{u} | |
| let e := cons ZFSet.omega (cons ambientGraphBound (cons (natSet 0) (cons ambientRealSet (fun _ => realCode A)))) | |
| have he : ∀ i, e i ∈ hullSet a := by | |
| intro i | |
| rcases i with _|i; exact hω | |
| rcases i with _|i; exact hQ | |
| rcases i with _|i; exact h0 | |
| rcases i with _|i; exact hR | |
| exact hA | |
| let p := fromBounded (Formula.degreeSet φ 1 2 3 4 5 0) | |
| have hp (x : ZFSet.{u}) : p.Sat Set.univ (cons x e) ↔ x = degreeSet (degree A) := by | |
| rw [bounded_univ,Formula.degreeSet_spec hφ 1 2 3 4 5 0 _ rfl ambient_graph_bound rfl ambient_real_decode A rfl] | |
| simp only [cons_zero,cons_succ,e,ambient_degreeCode] | |
| exact unique_definable_mem a p e he _ ((hp _).mpr rfl) (fun x hx => (hp x).mp hx) | |
| end TuringRigidity.ElementaryModel | |
| end OAI | |