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9dbc64e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 | import OAI.Computability.DegreeRigidity.Syntax.AtomicRequirements
namespace OAI
namespace TuringRigidity.AtomicForcing
open Set RecursiveNames CountableForcing
universe u
variable {P : Type u} [Preorder P]
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⟩
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⟩
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⟩))
end TuringRigidity.AtomicForcing
end OAI
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