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8df23f4 | 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 97 98 99 100 101 102 103 104 105 106 107 108 | import OAI.Computability.DegreeRigidity.Syntax.AtomicTruth
import OAI.Computability.DegreeRigidity.Syntax.BoundedRelations
namespace OAI
namespace TuringRigidity.BoundedForcing
open Set RecursiveNames CountableForcing AtomicForcing
open BoundedSetTheory (Formula)
universe u
variable {P : Type u} [Preorder P]
def push {α : Sort*} (x : α) (e : ℕ → α) : ℕ → α
| 0 => x
| n+1 => e n
@[simp] theorem push_zero {α : Sort*} (x : α) (e : ℕ → α) : push x e 0 = x := rfl
@[simp] theorem push_succ {α : Sort*} (x : α) (e : ℕ → α) (n : ℕ) : push x e (n+1) = e n := rfl
omit [Preorder P] in
theorem child_relation (a : Name P) (i : a.arity) : Name.Child (a.child i) a := by
cases a
exact ⟨i,rfl⟩
def Forces (e : ℕ → Name P) : Formula → P → Prop
| .equal i j, p => EqForces (e i) (e j) p
| .member i j, p => MemForces (e i) (e j) p
| .conj φ ψ, p => Forces e φ p ∧ Forces e ψ p
| .neg φ, p => ∀ q, q ≤ p → ¬ Forces e φ q
| .existsMem i φ, p => ∃ j, p ≤ (e i).tag j ∧ Forces (push ((e i).child j) e) φ p
theorem forces_mono (φ : Formula) (e : ℕ → Name P) {p q : P} (hqp : q ≤ p)
(h : Forces e φ p) : Forces e φ q := by
induction φ generalizing e with
| equal i j => exact eq_mono _ _ hqp h
| member i j => exact mem_mono _ _ hqp h
| conj φ ψ ihφ ihψ => exact ⟨ihφ e h.1,ihψ e h.2⟩
| neg φ ih => exact fun r hr => h r (hr.trans hqp)
| existsMem i φ ih =>
obtain ⟨j,hj,hf⟩ := h
exact ⟨j,hqp.trans hj,ih _ hf⟩
def Decision (e : ℕ → Name P) (φ : Formula) : Set P := {p | Forces e φ p ∨ Forces e (.neg φ) p}
theorem decision_dense (e : ℕ → Name P) (φ : Formula) : Dense (Decision e φ) := by
classical
intro p
by_cases h : ∃ q, q ≤ p ∧ Forces e φ q
· obtain ⟨q,hq,hf⟩ := h
exact ⟨q,hq,Or.inl hf⟩
· exact ⟨p,le_rfl,Or.inr (fun q hq hf => h ⟨q,hq,hf⟩)⟩
def Generic (N : Set (Name P)) (G : GenericFilter P) : Prop :=
∀ φ e, (∀ i, e i ∈ N) → ∃ p ∈ G.carrier, p ∈ Decision e φ
theorem truth {N : Set (Name P)} (hN : ChildClosed N) (G : GenericFilter P)
(hA : AtomicForcing.Generic N G) (hG : Generic N G) (φ : Formula)
(e : ℕ → Name P) (he : ∀ i, e i ∈ N) :
φ.Eval (fun i => (e i).val G.carrier) ↔ ∃ p ∈ G.carrier, Forces e φ p := by
induction φ generalizing e with
| equal i j => exact eq_truth hN G hA _ _ (he i) (he j)
| member i j => exact mem_truth hN G hA _ _ (he i) (he j)
| conj φ ψ ihφ ihψ =>
constructor
· rintro ⟨hφ,hψ⟩
obtain ⟨p,hp,hfp⟩ := (ihφ e he).mp hφ
obtain ⟨q,hq,hfq⟩ := (ihψ e he).mp hψ
obtain ⟨r,hr,hrp,hrq⟩ := G.directed hp hq
exact ⟨r,hr,forces_mono φ e hrp hfp,forces_mono ψ e hrq hfq⟩
· rintro ⟨p,hp,hφ,hψ⟩
exact ⟨(ihφ e he).mpr ⟨p,hp,hφ⟩,(ihψ e he).mpr ⟨p,hp,hψ⟩⟩
| neg φ ih =>
constructor
· intro hn
obtain ⟨p,hp,hf | hf⟩ := hG φ e he
· exact False.elim (hn ((ih e he).mpr ⟨p,hp,hf⟩))
· exact ⟨p,hp,hf⟩
· rintro ⟨p,hp,hn⟩ hv
obtain ⟨q,hq,hf⟩ := (ih e he).mp hv
obtain ⟨r,hr,hrp,hrq⟩ := G.directed hp hq
exact hn r hrp (forces_mono φ e hrq hf)
| existsMem i φ ih =>
have hej (j) : ∀ n, push ((e i).child j) e n ∈ N := by
intro n
cases n with
| zero =>
exact hN (e i) (he i) _ (child_relation (e i) j)
| succ n => exact he n
have env (j) : (fun n => (push ((e i).child j) e n).val G.carrier) =
BoundedSetTheory.cons ((e i).child j |>.val G.carrier) (fun n => (e n).val G.carrier) := by
funext n
cases n <;> rfl
constructor
· rintro ⟨x,hx,hφ⟩
obtain ⟨j,hj,hxj⟩ := ((e i).mem_val_children G.carrier x).mp hx
subst x
rw [←env j] at hφ
obtain ⟨p,hp,hf⟩ := (ih _ (hej j)).mp hφ
obtain ⟨q,hq,hqp,hqj⟩ := G.directed hp hj
exact ⟨q,hq,j,hqj,forces_mono φ _ hqp hf⟩
· rintro ⟨p,hp,j,hj,hf⟩
refine ⟨_,((e i).mem_val_children G.carrier _).mpr ⟨j,G.upper hj hp,rfl⟩,?_⟩
rw [←env j]
exact (ih _ (hej j)).mpr ⟨p,hp,hf⟩
end TuringRigidity.BoundedForcing
end OAI
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