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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 | import OAI.Computability.DegreeRigidity.CohenForcing.BoundedForcingCode
import OAI.Computability.DegreeRigidity.SetModels.InternalContainer
namespace OAI
namespace TuringRigidity.BoundedForcing
open Set RecursiveNames TransitiveNameModel BoundedSetTheory
universe u
def bound : BoundedSetTheory.Formula → ℕ
| .equal i j => max (i+1) (j+1)
| .member i j => max (i+1) (j+1)
| .conj φ ψ => max (bound φ) (bound ψ)
| .neg φ => bound φ
| .existsMem i φ => max (i+1) (bound φ - 1)
theorem forces_congr {P : Type u} [Preorder P] (φ : BoundedSetTheory.Formula)
(e e' : ℕ → Name P) (h : ∀ i, i < bound φ → e i = e' i) (p : P) :
Forces e φ p ↔ Forces e' φ p := by
induction φ generalizing e e' p with
| equal i j => simp only [Forces,h i (by simp [bound]),h j (by simp [bound])]
| member i j => simp only [Forces,h i (by simp [bound]),h j (by simp [bound])]
| conj φ ψ ihφ ihψ =>
exact and_congr (ihφ e e' (fun i hi => h i (by simp only [bound]; omega)) p)
(ihψ e e' (fun i hi => h i (by simp only [bound]; omega)) p)
| neg φ ih =>
exact forall_congr' (fun q => imp_congr_right (fun _ => not_congr (ih e e' h q)))
| existsMem i φ ih =>
have hei := h i (by simp only [bound]; omega)
change (fun a : Name P => ∃ j : a.arity, p ≤ a.tag j ∧ Forces (push (a.child j) e) φ p) (e i) ↔ _
rw [hei]
apply exists_congr
intro j
apply and_congr_right
intro _
apply ih
intro n hn
cases n with
| zero => rfl
| succ n =>
simp only [push_succ]
exact h n (by simp only [bound]; omega)
theorem eval_congr (φ : BoundedSetTheory.Formula) (e e' : ℕ → ZFSet.{u})
(h : ∀ i, i < bound φ → e i = e' i) : φ.Eval e ↔ φ.Eval e' := by
induction φ generalizing e e' with
| equal i j => simp only [Formula.Eval,h i (by simp [bound]),h j (by simp [bound])]
| member i j => simp only [Formula.Eval,h i (by simp [bound]),h j (by simp [bound])]
| conj φ ψ ihφ ihψ =>
exact and_congr (ihφ e e' (fun i hi => h i (by simp only [bound]; omega)))
(ihψ e e' (fun i hi => h i (by simp only [bound]; omega)))
| neg φ ih => exact not_congr (ih e e' h)
| existsMem i φ ih =>
have hei := h i (by simp only [bound]; omega)
simp only [Formula.Eval,hei]
apply exists_congr
intro x
apply and_congr_right
intro _
apply ih
intro n hn
cases n with
| zero => rfl
| succ n => exact h n (by simp only [bound]; omega)
def truncate {α : Sort*} (n : ℕ) (e : ℕ → α) (i : ℕ) : α := if i < n then e i else e 0
theorem truncate_forces {P : Type u} [Preorder P] (φ : BoundedSetTheory.Formula)
(e : ℕ → Name P) (p : P) : Forces (truncate (bound φ) e) φ p ↔ Forces e φ p :=
forces_congr φ _ e (fun _ hi => ite_eq_left hi) p
theorem finite_container (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hS : Separation M)
(hR : SigmaReplacement M) (hI : Infinity M)
(e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) (n : ℕ) :
∃ d ∈ M, Transitive d ∧ e 0 ∈ d ∧ ∀ i, i < n → e i ∈ d := by
induction n with
| zero =>
obtain ⟨d,hd,hdT,he0⟩ := internal_transitive_container M hM hP hU hS hR hI (he 0)
exact ⟨d,hd,hdT,he0,fun i hi => False.elim (Nat.not_lt_zero i hi)⟩
| succ n ih =>
obtain ⟨d,hd,hdT,he0,hen⟩ := ih
obtain ⟨v,hv,hvT,hvpair⟩ := internal_transitive_container M hM hP hU hS hR hI
(pair_mem M hM hP hd (he n))
have hdv : d ∈ v := hvT _ hvpair _ (ZFSet.mem_pair.mpr (Or.inl rfl))
have hnv : e n ∈ v := hvT _ hvpair _ (ZFSet.mem_pair.mpr (Or.inr rfl))
refine ⟨v,hv,hvT,hvT d hdv _ he0,fun i hi => ?_⟩
rcases Nat.lt_succ_iff_lt_or_eq.mp hi with hi | rfl
· exact hvT d hdv _ (hen i hi)
· exact hnv
end TuringRigidity.BoundedForcing
end OAI
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