openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /CohenForcing /CheckedForcingRelation.lean
| import OAI.Computability.DegreeRigidity.Syntax.CheckedMembershipSeparation | |
| import OAI.Computability.DegreeRigidity.Syntax.SigmaDisjunction | |
| namespace OAI | |
| namespace TuringRigidity.CheckedForcingRelation | |
| open TransitiveNameModel BoundedSetTheory CountableForcing RecursiveNames AtomicForcing | |
| variable {c : ZFSet.{0}} [Preorder (Conditions c)] [Top (Conditions c)] | |
| noncomputable def relation (a : ZFSet.{0}) (τ : Name (Conditions c)) : ZFSet.{0} := | |
| (ZFSet.prod a c).sep (fun z => ∃ x ∈ a, ∃ p : Conditions c, | |
| z = ZFSet.pair x (label c p) ∧ MemForces (Name.check x) τ p) | |
| theorem pair_relation (a x : ZFSet.{0}) (τ : Name (Conditions c)) (p : Conditions c) : | |
| ZFSet.pair x (label c p) ∈ relation a τ ↔ x ∈ a ∧ MemForces (Name.check x) τ p := by | |
| rw [relation,ZFSet.mem_sep] | |
| constructor | |
| · rintro ⟨_,y,hy,q,he,hf⟩ | |
| have hpq := label_injective c (ZFSet.pair_inj.mp he).2 | |
| have hxy := (ZFSet.pair_inj.mp he).1 | |
| subst y; subst q; exact ⟨hy,hf⟩ | |
| · rintro ⟨hx,hf⟩ | |
| exact ⟨ZFSet.mem_prod.mpr ⟨x,hx,_,label_mem c p,rfl⟩,x,hx,p,rfl,hf⟩ | |
| theorem relation_mem (M a : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M) | |
| (hc : c ∈ M) (ha : a ∈ M) {o : ZFSet.{0}} (hoM : o ∈ M) | |
| (ho : ∀ p q : Conditions c, ZFSet.pair (label c p) (label c q) ∈ o ↔ p ≤ q) | |
| (τ : Name (Conditions c)) (hτ : τ.encode (label c) ∈ M) : relation a τ ∈ M := by | |
| let e := cons a (cons c (cons o (cons (τ.encode (label c)) (fun _ => label c ⊤)))) | |
| have he : ∀ i, e i ∈ M := by | |
| intro i; rcases i with _|_|_|_|i | |
| exact ha; exact hc; exact hoM; exact hτ; exact hM c hc _ (label_mem c ⊤) | |
| let ren : ℕ → ℕ := fun | 0 => 1 | 1 => 4 | 2 => 5 | 3 => 0 | 4 => 6 | _ => 7 | |
| let φ : SigmaFormula := .existsMem 1 (.existsMem 3 (.andBounded (.orderedPair 2 1 0) | |
| (FullSetForcing.checkedMembershipSigma.rename ren))) | |
| have hpM := product_mem M hM hT.pairing hT.union hT.powerSet hT.separation.finitePrefix.bounded ha hc | |
| obtain ⟨R,hRM,hR⟩ := hT.separation.finitePrefix φ e he (ZFSet.prod a c) hpM | |
| have spec (x y z : ZFSet.{0}) (hx : x ∈ M) (hy : y ∈ M) | |
| (p : Conditions c) (hpy : label c p = y) : | |
| (FullSetForcing.checkedMembershipSigma.rename ren).Realize M (cons y (cons x (cons z e))) ↔ | |
| MemForces (Name.check x) τ p := by | |
| rw [SigmaFormula.realize_rename,FullSetForcing.checkedMembershipSigma_spec] | |
| have hv : ∀ i, (cons y (cons x (cons z e)) ∘ ren) i ∈ M := by | |
| intro i; rcases i with _|_|_|_|_|i | |
| exact hx; exact hc; exact hoM; exact hy; exact hτ; exact he 4 | |
| exact FullSetForcing.checkedMembership_spec M hM hT ho τ p x _ hv | |
| rfl rfl rfl hpy.symm rfl rfl | |
| have pairSpec (x y z : ZFSet.{0}) (hx : x ∈ M) (hy : y ∈ M) (hz : z ∈ M) : | |
| (BoundedSetTheory.Formula.orderedPair 2 1 0).Realize M (cons y (cons x (cons z e))) ↔ | |
| z = ZFSet.pair x y := by | |
| have hv : ∀ i, cons y (cons x (cons z e)) i ∈ M := by | |
| intro i; rcases i with _|_|_|i; exact hy; exact hx; exact hz; exact he i | |
| exact (BoundedSetTheory.Formula.absolute _ M hM _ hv).trans | |
| (by simp only [BoundedSetTheory.Formula.eval_orderedPair,cons_zero,cons_succ]) | |
| have heq : R = relation a τ := by | |
| apply ZFSet.ext; intro z | |
| constructor | |
| · intro hzR | |
| have hz := hM R hRM z hzR | |
| obtain ⟨hzprod,hf⟩ := (hR z hz).mp hzR | |
| simp only [φ,SigmaFormula.realize_existsMem,SigmaFormula.realize_andBounded, | |
| cons_zero,cons_succ,e] at hf | |
| obtain ⟨x,hx,hxa,y,hy,hyc,hzxy,hf⟩ := hf | |
| obtain ⟨p,hpy⟩ := label_surjective c hyc | |
| exact ZFSet.mem_sep.mpr ⟨hzprod,x,hxa,p,((pairSpec x y z hx hy hz).mp hzxy).trans (congrArg (ZFSet.pair x) hpy.symm), | |
| (spec x y z hx hy p hpy).mp hf⟩ | |
| · intro hzR | |
| obtain ⟨hzprod,x,hxa,p,hzxy,hf⟩ := ZFSet.mem_sep.mp hzR | |
| have hz := hM _ hpM z hzprod | |
| have hx := hM a ha x hxa | |
| have hy := hM c hc _ (label_mem c p) | |
| apply (hR z hz).mpr | |
| refine ⟨hzprod,?_⟩ | |
| simp only [φ,SigmaFormula.realize_existsMem,SigmaFormula.realize_andBounded, | |
| cons_zero,cons_succ,e] | |
| exact ⟨x,hx,hxa,label c p,hy,label_mem c p,(pairSpec x (label c p) z hx hy hz).mpr hzxy, | |
| (spec x (label c p) z hx hy p rfl).mpr hf⟩ | |
| exact heq ▸ hRM | |
| end TuringRigidity.CheckedForcingRelation | |
| end OAI | |