| import OAI.Computability.DegreeRigidity.Syntax.AtomicRequirements |
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| namespace OAI |
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| namespace TuringRigidity.AtomicForcing |
| open Set RecursiveNames CountableForcing |
| universe u |
| variable {P : Type u} [Preorder P] |
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| private theorem mem_truth_step {N : Set (Name P)} {G : GenericFilter P} |
| (hG : Generic N G) (a : Name P) (κ : Type u) (b : κ → Name P) (t : κ → P) |
| (ha : a ∈ N) (hb : Name.mk κ b t ∈ N) |
| (he : ∀ j, Name.val G.carrier a = Name.val G.carrier (b j) ↔ |
| ∃ p ∈ G.carrier, EqForces a (b j) p) : |
| Name.val G.carrier a ∈ Name.val G.carrier (.mk κ b t) ↔ |
| ∃ p ∈ G.carrier, MemForces a (.mk κ b t) p := by |
| constructor |
| · intro hm |
| obtain ⟨j,hj,hv⟩ := (Name.mem_val _ _ _ _ _).mp hm |
| obtain ⟨p,hp,hf⟩ := (he j).mp hv.symm |
| obtain ⟨r,hr,hrp,hrj⟩ := G.directed hp hj |
| exact ⟨r,hr,witness_mem _ _ _ ⟨j,hrj,eq_mono _ _ hrp hf⟩⟩ |
| · rintro ⟨p,hp,hm⟩ |
| obtain ⟨q,hq,j,hj,hf⟩ := generic_witness hG ha hb hp hm |
| exact (Name.mem_val _ _ _ _ _).mpr ⟨j,G.upper hj hq,((he j).mpr ⟨q,hq,hf⟩).symm⟩ |
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| theorem eq_truth {N : Set (Name P)} (hN : ChildClosed N) (G : GenericFilter P) |
| (hG : Generic N G) (a b : Name P) (ha : a ∈ N) (hb : b ∈ N) : |
| Name.val G.carrier a = Name.val G.carrier b ↔ |
| ∃ p ∈ G.carrier, EqForces a b p := by |
| induction a generalizing b with |
| | mk ι a s ih => |
| cases b with |
| | mk κ b t => |
| have ha' (i) : a i ∈ N := hN _ ha _ ⟨i,rfl⟩ |
| have hb' (j) : b j ∈ N := hN _ hb _ ⟨j,rfl⟩ |
| have left (i) : Name.val G.carrier (a i) ∈ Name.val G.carrier (.mk κ b t) ↔ |
| ∃ p ∈ G.carrier, MemForces (a i) (.mk κ b t) p := |
| mem_truth_step hG (a i) κ b t (ha' i) hb (fun j => ih i (b j) (ha' i) (hb' j)) |
| have right (j) : Name.val G.carrier (b j) ∈ Name.val G.carrier (.mk ι a s) ↔ |
| ∃ p ∈ G.carrier, MemForces (b j) (.mk ι a s) p := by |
| apply mem_truth_step hG (b j) ι a s (hb' j) ha |
| intro i |
| constructor |
| · intro he |
| obtain ⟨p,hp,hf⟩ := (ih i (b j) (ha' i) (hb' j)).mp he.symm |
| exact ⟨p,hp,eq_symm _ _ _ hf⟩ |
| · rintro ⟨p,hp,hf⟩ |
| exact ((ih i (b j) (ha' i) (hb' j)).mpr ⟨p,hp,eq_symm _ _ _ hf⟩).symm |
| constructor |
| · intro hv |
| obtain ⟨p,hp,he | hn⟩ := hG.eqTest _ ha _ hb |
| · exact ⟨p,hp,he⟩ |
| · rcases hn with ⟨i,hi,hn⟩ | ⟨j,hj,hn⟩ |
| · have hm : Name.val G.carrier (a i) ∈ Name.val G.carrier (.mk κ b t) := by |
| rw [←hv] |
| exact (Name.mem_val _ _ _ _ _).mpr ⟨i,G.upper hi hp,rfl⟩ |
| obtain ⟨q,hq,hf⟩ := (left i).mp hm |
| obtain ⟨r,hr,hrp,hrq⟩ := G.directed hp hq |
| exact False.elim (hn r hrp (mem_mono _ _ hrq hf)) |
| · have hm : Name.val G.carrier (b j) ∈ Name.val G.carrier (.mk ι a s) := by |
| rw [hv] |
| exact (Name.mem_val _ _ _ _ _).mpr ⟨j,G.upper hj hp,rfl⟩ |
| obtain ⟨q,hq,hf⟩ := (right j).mp hm |
| obtain ⟨r,hr,hrp,hrq⟩ := G.directed hp hq |
| exact False.elim (hn r hrp (mem_mono _ _ hrq hf)) |
| · rintro ⟨p,hp,he⟩ |
| apply ZFSet.ext |
| intro x |
| constructor |
| · intro hx |
| obtain ⟨i,hi,rfl⟩ := (Name.mem_val _ _ _ _ _).mp hx |
| obtain ⟨r,hr,hrp,hri⟩ := G.directed hp hi |
| exact (left i).mpr ⟨r,hr,fun q hq => he.1 i q (hq.trans hrp) (hq.trans hri)⟩ |
| · intro hx |
| obtain ⟨j,hj,rfl⟩ := (Name.mem_val _ _ _ _ _).mp hx |
| obtain ⟨r,hr,hrp,hrj⟩ := G.directed hp hj |
| apply (right j).mpr |
| refine ⟨r,hr,?_⟩ |
| intro q hq |
| obtain ⟨w,hw,i,hi,hf⟩ := he.2 j q (hq.trans hrp) (hq.trans hrj) |
| exact ⟨w,hw,i,hi,eq_symm _ _ _ hf⟩ |
|
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| theorem mem_truth {N : Set (Name P)} (hN : ChildClosed N) (G : GenericFilter P) |
| (hG : Generic N G) (a b : Name P) (ha : a ∈ N) (hb : b ∈ N) : |
| Name.val G.carrier a ∈ Name.val G.carrier b ↔ |
| ∃ p ∈ G.carrier, MemForces a b p := by |
| cases b with |
| | mk κ b t => |
| exact mem_truth_step hG a κ b t ha hb |
| (fun j => eq_truth hN G hG a (b j) ha (hN _ hb _ ⟨j,rfl⟩)) |
|
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| end TuringRigidity.AtomicForcing |
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| end OAI |
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