openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /OrdinalCodes /OmegaModelArithmetic.lean
| import OAI.Computability.DegreeRigidity.OrdinalCodes.OmegaModelAxiomTransport | |
| import OAI.Computability.DegreeRigidity.SetModels.SetModelArithmeticClosure | |
| namespace OAI | |
| namespace TuringRigidity.OmegaModelCore.OmegaData | |
| open BoundedSetTheory SetModelReals SetModelSyntax | |
| universe u | |
| noncomputable section | |
| variable {α : Type u} (S : OmegaData α) | |
| def RealCode (a : α) (A : Oracle) : Prop := | |
| S.mem a (S.power S.omega) ∧ ∀ n, S.mem (S.num n) a ↔ A n = true | |
| theorem realCode_collapse {a : α} {A : Oracle} (ha : S.RealCode a A) : | |
| S.collapse (⟨a,1,ha.1⟩ : S.Core) = realSet A := by | |
| obtain ⟨B,hB,hBc⟩ := S.collapse_real a ha.1 | |
| have he : B=A := by | |
| funext n | |
| have hn := (hBc n).trans (ha.2 n) | |
| cases hb : B n <;> cases ha : A n <;> simp_all | |
| exact hB.trans (congrArg realSet he) | |
| theorem represented_real_iff (A : Oracle) : | |
| (∃ a : α, S.RealCode a A) ↔ A ∈ reals S.collapsedModel := by | |
| constructor | |
| · rintro ⟨a,ha⟩ | |
| exact (S.mem_collapsedModel _).mpr ⟨⟨a,1,ha.1⟩,(S.realCode_collapse ha).symm⟩ | |
| · intro hA | |
| change realSet A ∈ S.collapsedModel at hA | |
| obtain ⟨a,ha⟩ := (S.mem_collapsedModel _).mp hA | |
| have hsub : S.collapse a ⊆ S.collapse S.omegaNode := by | |
| rw [S.collapse_omega,←ha] | |
| exact realSet_subset A | |
| refine ⟨a.val,(S.mem_power _ _).mpr ((S.collapse_subset_iff a S.omegaNode).mp hsub),?_⟩ | |
| intro n | |
| change S.Rel (S.numNode n) a ↔ A n = true | |
| rw [←S.collapse_mem_iff,S.collapse_num,←ha,nat_mem_realSet] | |
| theorem omega_arithmetic_comprehension (hP : S.PairingAxiom) (hU : S.UnionAxiom) | |
| (hS : S.SeparationAxiom) {P : UniformArithmetic.Predicate} (hp : UniformArithmetic.Arith P) | |
| (O : UniformArithmetic.Oracles) (hO : ∀ i, ∃ a, S.RealCode a (O i)) | |
| (A : Oracle) (hA : ∀ n, P O n ↔ A n = true) : ∃ a, S.RealCode a A := by | |
| apply (S.represented_real_iff A).mpr | |
| exact arithmetic_comprehension (S.collapsed_context hP hU hS) hp O | |
| (fun i => (S.represented_real_iff _).mp (hO i)) A hA | |
| theorem omega_lower (hP : S.PairingAxiom) (hU : S.UnionAxiom) (hS : S.SeparationAxiom) | |
| {A B : Oracle} (hB : ∃ b, S.RealCode b B) (hAB : Reduces A B) : ∃ a, S.RealCode a A := | |
| (S.represented_real_iff _).mpr (lower_mem (S.collapsed_context hP hU hS) | |
| ((S.represented_real_iff _).mp hB) hAB) | |
| theorem omega_jump (hP : S.PairingAxiom) (hU : S.UnionAxiom) (hS : S.SeparationAxiom) | |
| {A : Oracle} (hA : ∃ a, S.RealCode a A) : ∃ b, S.RealCode b (OracleJump.jump A) := | |
| (S.represented_real_iff _).mpr (jump_mem (S.collapsed_context hP hU hS) | |
| ((S.represented_real_iff _).mp hA)) | |
| end | |
| end TuringRigidity.OmegaModelCore.OmegaData | |
| end OAI | |