openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /DefSystemCertificate.lean
| import OAI.Computability.DegreeRigidity.Syntax.SatisfactionSystemCertificate | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding | |
| open BoundedDefinability SetModelFunctions | |
| universe u | |
| def DefSystem (Q B A G T H Z D : ZFSet.{u}) : Prop := | |
| SatisfactionSystem Q B A G T H Z ∧ | |
| (∀ S ∈ D, DefinitionWitness Q B G A T Z S) ∧ | |
| ∀ c ∈ ZFSet.omega, ∀ t ∈ T, ParameterAdequate Q B G c t → | |
| ∃ S ∈ D, DefinesSubset A G T Z c t S | |
| theorem defSystem_definable {M : ZFSet.{u}} (C : Context M) | |
| {Q B : ZFSet.{u}} (hQ : Q ∈ M) (hB : B ∈ M) (a G T H Z D : ℕ) : | |
| Definable M (fun e => DefSystem Q B (e a) (e G) (e T) (e H) (e Z) (e D)) := by | |
| have hn := defAllMem (definitionWitness_variable C hQ hB (a+1) (G+1) (T+1) (Z+1) 0) D | |
| have hc := defAllParam (defAllMem (defImp (parameterAdequate_variable C hQ hB (G+2) 1 0) | |
| ((definesSubset_variable C (a+3) (G+3) (T+3) (Z+3) 2 1 0).existsMem (D+2))) (T+1)) C.omega_mem | |
| exact (satisfactionSystem_definable C hQ hB a G T H Z).and (hn.and hc) | |
| theorem DefSystem.exact {Q B A G T H Z D : ZFSet.{u}} | |
| (hQ : ∀ p : SentenceForm, family p ∈ Q) | |
| (hB : ∀ p : SentenceForm, supportGraph p ∈ B) | |
| (h : DefSystem Q B A G T H Z D) : D = definablePower A := by | |
| obtain ⟨ht,hz,hg,_⟩ := h.1.exact hQ hB | |
| subst T; subst Z | |
| apply ZFSet.ext; intro S | |
| rw [mem_definablePower] | |
| constructor | |
| · intro hS | |
| exact (definitionWitness_spec Q B G A S hQ hB hg).mp (h.2.1 S hS) | |
| · rintro ⟨n,p,v,hp,hv,rfl⟩ | |
| obtain ⟨S,hS,hs⟩ := h.2.2 _ ((mem_omega _).mpr ⟨Encodable.encode p,rfl⟩) | |
| _ ((mem_tupleSpace A _).mpr ⟨n,v,hv,rfl⟩) | |
| ((parameterAdequate_spec Q B G _ hQ hB v (hg n v hv)).mpr ⟨p,rfl,hp⟩) | |
| exact (definesSubset_spec A G S hg p v hv hp).mp hs ▸ hS | |
| theorem defSystem_canonical (Q B A G H : ZFSet.{u}) | |
| (hQ : ∀ p : SentenceForm, family p ∈ Q) | |
| (hB : ∀ p : SentenceForm, supportGraph p ∈ B) | |
| (hG : ∀ (n : ℕ) (v : Fin n → ZFSet.{u}), (∀ i, v i ∈ A) → tupleGraph v ∈ G) | |
| (hH : ∀ p : SentenceForm, finiteSatisfaction A (subformulas p) ∈ H) : | |
| DefSystem Q B A G (tupleSpace A) H (satisfactionSet A) (definablePower A) := by | |
| refine ⟨satisfactionSystem_canonical Q B A G H hQ hB hG hH,?_,?_⟩ | |
| · intro S hS | |
| exact (definitionWitness_spec Q B G A S hQ hB hG).mpr ((mem_definablePower A S).mp hS) | |
| · intro c _ t ht ha | |
| obtain ⟨n,v,hv,rfl⟩ := (mem_tupleSpace A t).mp ht | |
| obtain ⟨p,rfl,hp⟩ := (parameterAdequate_spec Q B G c hQ hB v (hG n v hv)).mp ha | |
| exact ⟨definedSubset A p v,(mem_definablePower A _).mpr ⟨n,p,v,hp,hv,rfl⟩, | |
| (definesSubset_spec A G _ hG p v hv hp).mpr rfl⟩ | |
| theorem internal_defSystem (M : ZFSet.{u}) (hM : Transitive M) (hT : SourceT M) | |
| (Q B A : ZFSet.{u}) (hA : A ∈ M) | |
| (hQ : ∀ p : SentenceForm, family p ∈ Q) | |
| (hB : ∀ p : SentenceForm, supportGraph p ∈ B) : | |
| ∃ G ∈ M, ∃ T ∈ M, ∃ H ∈ M, ∃ Z ∈ M, | |
| DefSystem Q B A G T H Z (definablePower A) := by | |
| let C : Context M := ⟨hM,hT.pairing,hT.union,hT.powerSet,hT.separation.finitePrefix.bounded,hT.infinity⟩ | |
| obtain ⟨G,hG,hg⟩ := internal_tuple_graph_bound C hA | |
| obtain ⟨H,hH,_,hh⟩ := internal_certificate_bound M A hM hT hA | |
| exact ⟨G,hG,tupleSpace A,tupleSpace_mem M A hM hT hA,H,hH, | |
| satisfactionSet A,satisfactionSet_mem M A hM hT hA, | |
| defSystem_canonical Q B A G H hQ hB hg (fun p => hh (subformulas p))⟩ | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |