| import OAI.Computability.DegreeRigidity.Syntax.AtomicEncoding | |
| import OAI.Computability.DegreeRigidity.SetModels.NameExtension | |
| namespace OAI | |
| namespace TuringRigidity.AtomicForcing | |
| open Set RecursiveNames CountableForcing | |
| universe u | |
| variable {P : Type u} [Preorder P] | |
| def Witness (a : Name P) : Name P → P → Prop | |
| | .mk _ b t, p => ∃ j, p ≤ t j ∧ EqForces a (b j) p | |
| def NegMem (a b : Name P) (p : P) : Prop := ∀ q, q ≤ p → ¬ MemForces a b q | |
| def MemTest (a b : Name P) : Set P := {p | Witness a b p ∨ NegMem a b p} | |
| def Refute : Name P → Name P → P → Prop | |
| | .mk _ a s, .mk _ b t, p => | |
| (∃ i, p ≤ s i ∧ NegMem (a i) (.mk _ b t) p) ∨ | |
| (∃ j, p ≤ t j ∧ NegMem (b j) (.mk _ a s) p) | |
| def EqTest (a b : Name P) : Set P := {p | EqForces a b p ∨ Refute a b p} | |
| theorem witness_mono (a b : Name P) {p q : P} (hq : q ≤ p) | |
| (h : Witness a b p) : Witness a b q := by | |
| cases b with | |
| | mk κ b t => | |
| obtain ⟨j,hj,he⟩ := h | |
| exact ⟨j,hq.trans hj,eq_mono _ _ hq he⟩ | |
| theorem witness_mem (a b : Name P) (p : P) (h : Witness a b p) : MemForces a b p := by | |
| cases b with | |
| | mk κ b t => | |
| intro q hq | |
| exact ⟨q,le_rfl,witness_mono _ _ hq h⟩ | |
| theorem memTest_dense (a b : Name P) : Dense (MemTest a b) := by | |
| classical | |
| intro p | |
| by_cases h : ∃ q, q ≤ p ∧ Witness a b q | |
| · obtain ⟨q,hq,hw⟩ := h | |
| exact ⟨q,hq,Or.inl hw⟩ | |
| · refine ⟨p,le_rfl,Or.inr ?_⟩ | |
| intro q hq hm | |
| cases b with | |
| | mk κ b t => | |
| obtain ⟨r,hr,hw⟩ := hm q le_rfl | |
| exact h ⟨r,hr.trans hq,hw⟩ | |
| theorem eqTest_dense (a b : Name P) : Dense (EqTest a b) := by | |
| classical | |
| cases a with | |
| | mk ι a s => | |
| cases b with | |
| | mk κ b t => | |
| intro p | |
| by_cases he : EqForces (.mk ι a s) (.mk κ b t) p | |
| · exact ⟨p,le_rfl,Or.inl he⟩ | |
| · change ¬ ((∀ i, Below p (s i) _) ∧ (∀ j, Below p (t j) _)) at he | |
| simp only [not_and_or, not_forall, Below] at he | |
| rcases he with ⟨i,q,hq⟩ | ⟨j,q,hq⟩ | |
| · obtain ⟨hqp,hqs,hn⟩ := hq | |
| refine ⟨q,hqp,Or.inr (Or.inl ⟨i,hqs,?_⟩)⟩ | |
| intro r hr hm | |
| obtain ⟨w,hw,k,hk,hf⟩ := hm r le_rfl | |
| exact hn ⟨w,hw.trans hr,k,hk,hf⟩ | |
| · obtain ⟨hqp,hqt,hn⟩ := hq | |
| refine ⟨q,hqp,Or.inr (Or.inr ⟨j,hqt,?_⟩)⟩ | |
| intro r hr hm | |
| obtain ⟨w,hw,i,hi,hf⟩ := hm r le_rfl | |
| exact hn ⟨w,hw.trans hr,i,hi,eq_symm _ _ _ hf⟩ | |
| structure Generic (N : Set (Name P)) (G : GenericFilter P) : Prop where | |
| memTest : ∀ a ∈ N, ∀ b ∈ N, ∃ p ∈ G.carrier, p ∈ MemTest a b | |
| eqTest : ∀ a ∈ N, ∀ b ∈ N, ∃ p ∈ G.carrier, p ∈ EqTest a b | |
| theorem generic_witness {N : Set (Name P)} {G : GenericFilter P} | |
| (hG : Generic N G) {a b : Name P} (ha : a ∈ N) (hb : b ∈ N) | |
| {p : P} (hp : p ∈ G.carrier) (hm : MemForces a b p) : | |
| ∃ q ∈ G.carrier, Witness a b q := by | |
| obtain ⟨q,hq,hw | hn⟩ := hG.memTest a ha b hb | |
| · exact ⟨q,hq,hw⟩ | |
| · obtain ⟨r,hr,hrp,hrq⟩ := G.directed hp hq | |
| exact False.elim (hn r hrq (mem_mono _ _ hrp hm)) | |
| end TuringRigidity.AtomicForcing | |
| end OAI | |