openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /Constructibility /ContextDefinablePower.lean
| import OAI.Computability.DegreeRigidity.Constructibility.DefSystemCertificate | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open ElementaryModel BoundedSetTheory TransitiveNameModel SentenceCoding | |
| open BoundedDefinability SetModelFunctions | |
| universe u | |
| theorem tupleSpace_mem_of_context (M A : ZFSet.{u}) (C : Context M) (hA : A ∈ M) : tupleSpace A ∈ M := by | |
| have hS := C.separation | |
| have hω := C.omega_mem | |
| obtain ⟨Q,hQM,hQdef⟩ := internal_power M C.transitive C.power | |
| (product_mem M C.transitive C.pairing C.union C.power hS hω hA) | |
| have hQ (n : ℕ) (v : Fin n → ZFSet.{u}) (hv : ∀ i, v i ∈ A) : tupleGraph v ∈ Q := by | |
| apply (hQdef _).mpr | |
| refine ⟨tupleGraph_mem M C.transitive C.pairing C.union hω v (fun i => C.transitive A hA _ (hv i)),?_⟩ | |
| intro z hz | |
| obtain ⟨i,rfl⟩ := ZFSet.mem_range.mp hz | |
| exact ZFSet.mem_prod.mpr ⟨_,(mem_omega _).mpr ⟨i.val,rfl⟩,_,hv i,rfl⟩ | |
| let e := cons ZFSet.omega (cons A (fun _ => Q)) | |
| have he : ∀ i, e i ∈ M := by intro i; rcases i with _|_|i; exact hω; exact hA; exact hQM | |
| have hs := sep_mem M C.transitive hS (tupleFormula 1 2 3 0) e he | |
| (product_mem M C.transitive C.pairing C.union C.power hS hω hQM) | |
| have heq : (ZFSet.prod ZFSet.omega Q).sep | |
| (fun t => (tupleFormula 1 2 3 0).Eval (cons t e)) = tupleSpace A := by | |
| apply ZFSet.ext; intro t | |
| rw [ZFSet.mem_sep,tupleFormula_spec 1 2 3 0 (cons t e) rfl hQ] | |
| change (t ∈ ZFSet.prod ZFSet.omega Q ∧ t ∈ tupleSpace A) ↔ t ∈ tupleSpace A | |
| refine ⟨And.right,fun ht => ⟨?_,ht⟩⟩ | |
| obtain ⟨n,v,hv,rfl⟩ := (mem_tupleSpace A t).mp ht | |
| exact ZFSet.mem_prod.mpr ⟨natSet n,(mem_omega _).mpr ⟨n,rfl⟩,tupleGraph v,hQ n v hv,rfl⟩ | |
| exact heq ▸ hs | |
| theorem truthSlice_mem_of_context (M A : ZFSet.{u}) (C : Context M) (hA : A ∈ M) (p : SentenceForm) : truthSlice A p ∈ M := by | |
| have hS := C.separation | |
| have hω := C.omega_mem | |
| have hn (n : ℕ) : natSet.{u} n ∈ M := C.transitive _ hω _ ((mem_omega _).mpr ⟨n,rfl⟩) | |
| obtain ⟨Q,hQM,hQdef⟩ := internal_power M C.transitive C.power | |
| (product_mem M C.transitive C.pairing C.union C.power hS hω hA) | |
| have hQ {n : ℕ} (v : Fin n → ZFSet.{u}) (hv : ∀ i, v i ∈ A) : tupleGraph v ∈ Q := by | |
| apply (hQdef _).mpr | |
| refine ⟨tupleGraph_mem M C.transitive C.pairing C.union hω v (fun i => C.transitive A hA _ (hv i)),?_⟩ | |
| intro z hz | |
| obtain ⟨i,rfl⟩ := ZFSet.mem_range.mp hz | |
| exact ZFSet.mem_prod.mpr ⟨_,(mem_omega _).mpr ⟨i.val,rfl⟩,_,hv i,rfl⟩ | |
| let e := cons ZFSet.omega (cons A (cons Q (cons (natSet p.bound) natSet))) | |
| have he : ∀ i, e i ∈ M := by | |
| intro i; rcases i with _|_|_|_|i | |
| exact hω; exact hA; exact hQM; exact hn _; exact hn _ | |
| let φ := tupleTruth p 1 2 3 4 0 (fun i => i+5) | |
| have hs := sep_mem M C.transitive hS φ e he (tupleSpace_mem_of_context M A C hA) | |
| have heq : (tupleSpace A).sep (fun t => φ.Eval (cons t e)) = truthSlice A p := by | |
| apply ZFSet.ext; intro t | |
| rw [truthSlice,ZFSet.mem_sep,ZFSet.mem_sep] | |
| apply and_congr_right; intro ht | |
| obtain ⟨n,v,hv,rfl⟩ := (mem_tupleSpace A t).mp ht | |
| exact (tupleTruth_spec p 1 2 3 4 0 (fun i => i+5) (cons (tupleCode v) e) | |
| rfl rfl (fun _ _ => rfl) v hv rfl (hQ v hv)).trans (satisfactionSet_record A p v hv).symm | |
| exact heq ▸ hs | |
| theorem finiteSatisfaction_mem_of_context (M A : ZFSet.{u}) (C : Context M) (hA : A ∈ M) (ps : List SentenceForm) : finiteSatisfaction A ps ∈ M := by | |
| have hω := C.omega_mem | |
| have hn (n : ℕ) : natSet.{u} n ∈ M := C.transitive _ hω _ ((mem_omega _).mpr ⟨n,rfl⟩) | |
| apply union_mem M C.transitive C.union | |
| apply finite_range_mem M C.transitive C.pairing C.union (hn 0) | |
| intro i | |
| exact product_mem M C.transitive C.pairing C.union C.power C.separation | |
| (singleton_mem M C.transitive C.pairing (hn _)) (truthSlice_mem_of_context M A C hA ps[i]) | |
| theorem internal_certificate_bound_of_context (M A : ZFSet.{u}) (C : Context M) (hA : A ∈ M) : | |
| ∃ Q ∈ M, | |
| (∀ S, S ∈ Q ↔ S ∈ M ∧ S ⊆ ZFSet.prod ZFSet.omega (tupleSpace A)) ∧ | |
| ∀ ps : List SentenceForm, finiteSatisfaction A ps ∈ Q := by | |
| obtain ⟨Q,hQ,hdef⟩ := internal_power M C.transitive C.power | |
| (product_mem M C.transitive C.pairing C.union C.power C.separation | |
| (C.omega_mem) (tupleSpace_mem_of_context M A C hA)) | |
| refine ⟨Q,hQ,hdef,fun ps => (hdef _).mpr ⟨finiteSatisfaction_mem_of_context M A C hA ps,?_⟩⟩ | |
| exact fun z hz => satisfactionSet_subset_product A (finiteSatisfaction_subset A ps hz) | |
| theorem satisfactionSet_mem_of_context (M A : ZFSet.{u}) (C : Context M) (hA : A ∈ M) : satisfactionSet A ∈ M := by | |
| obtain ⟨Q,hQ,_,hf⟩ := internal_family_bound M C.transitive C.pairing C.union C.power C.omega_mem | |
| obtain ⟨B,hB,hb⟩ := internal_support_bound C | |
| obtain ⟨G,hG,hg⟩ := internal_tuple_graph_bound C hA | |
| obtain ⟨H,hH,_,hh⟩ := internal_certificate_bound_of_context M A C hA | |
| have ht := tupleSpace_mem_of_context M A C hA | |
| let e := cons A (cons (tupleSpace A) (fun _ => ZFSet.omega)) | |
| have he : ∀ i, e i ∈ M := by | |
| intro i; rcases i with _|_|i | |
| · exact hA | |
| · exact ht | |
| · exact C.omega_mem | |
| have hd := satisfactionWitness_definable C hQ hB hG hH 1 2 0 | |
| have hs := hd.sep_mem C e he (C.prod_mem C.omega_mem ht) | |
| have heq : (ZFSet.prod ZFSet.omega (tupleSpace A)).sep | |
| (fun z => SatisfactionWitness Q B G H A (tupleSpace A) z) = satisfactionSet A := by | |
| apply ZFSet.ext; intro z | |
| rw [ZFSet.mem_sep,satisfactionWitness_spec Q B G H A z hf hb hg (fun p => hh (subformulas p))] | |
| exact ⟨And.right,fun hz => ⟨satisfactionSet_subset_product A hz,hz⟩⟩ | |
| exact heq ▸ hs | |
| theorem definablePower_mem_of_context (M A : ZFSet.{u}) (C : Context M) (hA : A ∈ M) : definablePower A ∈ M := by | |
| obtain ⟨Q,hQ,_,hf⟩ := internal_family_bound M C.transitive C.pairing C.union C.power C.omega_mem | |
| obtain ⟨B,hB,hb⟩ := internal_support_bound C | |
| obtain ⟨G,hG,hg⟩ := internal_tuple_graph_bound C hA | |
| have ht := tupleSpace_mem_of_context M A C hA | |
| have hz := satisfactionSet_mem_of_context M A C hA | |
| obtain ⟨P,hP,_,hsub⟩ := definablePower_internal_bound M A C.transitive C.separation C.power hA | |
| let e := cons A (cons (tupleSpace A) (cons (satisfactionSet A) (fun _ => ZFSet.omega))) | |
| have he : ∀ i, e i ∈ M := by | |
| intro i; rcases i with _|_|_|i | |
| · exact hA | |
| · exact ht | |
| · exact hz | |
| · exact C.omega_mem | |
| have hd := definitionWitness_definable C hQ hB hG 1 2 3 0 | |
| have hs := hd.sep_mem C e he hP | |
| have heq : P.sep (fun S => DefinitionWitness Q B G A (tupleSpace A) (satisfactionSet A) S) = | |
| definablePower A := by | |
| apply ZFSet.ext; intro S | |
| rw [ZFSet.mem_sep,definitionWitness_spec Q B G A S hf hb hg,← mem_definablePower] | |
| exact ⟨And.right,fun h => ⟨hsub h,h⟩⟩ | |
| exact heq ▸ hs | |
| theorem internal_defSystem_of_context (M : ZFSet.{u}) (C : Context 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 | |
| obtain ⟨G,hG,hg⟩ := internal_tuple_graph_bound C hA | |
| obtain ⟨H,hH,_,hh⟩ := internal_certificate_bound_of_context M A C hA | |
| exact ⟨G,hG,tupleSpace A,tupleSpace_mem_of_context M A C hA,H,hH, | |
| satisfactionSet A,satisfactionSet_mem_of_context M A C hA, | |
| defSystem_canonical Q B A G H hQ hB hg (fun p => hh (subformulas p))⟩ | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |