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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 | import OAI.Computability.DegreeRigidity.OrdinalCodes.OmegaModelCoreClosure
namespace OAI
namespace TuringRigidity.OmegaModelCore.OmegaData
open BoundedSetTheory
universe u
noncomputable section
variable {α : Type u} (S : OmegaData α)
def push {β : Type*} (x : β) (e : ℕ → β) : ℕ → β
| 0 => x
| n+1 => e n
def Eval (e : ℕ → α) : Formula → Prop
| .equal i j => e i = e j
| .member i j => S.mem (e i) (e j)
| .conj p q => Eval e p ∧ Eval e q
| .neg p => ¬ Eval e p
| .existsMem i p => ∃ x, S.mem x (e i) ∧ Eval (push x e) p
theorem collapse_eval (p : Formula) (e : ℕ → S.Core) :
p.Eval (fun i => S.collapse (e i)) ↔ S.Eval (fun i => (e i).val) p := by
induction p generalizing e with
| equal i j =>
change S.collapse (e i) = S.collapse (e j) ↔ (e i).val = (e j).val
exact ⟨fun h => congrArg Subtype.val (S.collapse_injective h),
fun h => congrArg S.collapse (Subtype.ext h)⟩
| member i j => exact S.collapse_mem_iff (e i) (e j)
| conj p q hp hq => exact and_congr (hp e) (hq e)
| neg p hp => exact not_congr (hp e)
| existsMem i p hp =>
have hstep (y : S.Core) : p.Eval (cons (S.collapse y) (fun i => S.collapse (e i))) ↔
S.Eval (push y.val (fun i => (e i).val)) p := by
have he : (fun i => S.collapse (push y e i)) =
cons (S.collapse y) (fun i => S.collapse (e i)) := by
funext i
cases i <;> rfl
have hv : (fun i => (push y e i).val) = push y.val (fun i => (e i).val) := by
funext i
cases i <;> rfl
simpa only [he,hv] using hp (push y e)
constructor
· rintro ⟨x,hx,hp'⟩
obtain ⟨y,hy,rfl⟩ := (S.mem_collapse (e i) x).mp hx
exact ⟨y.val,hy,(hstep y).mp hp'⟩
· rintro ⟨x,hx,hp'⟩
let y : S.Core := ⟨x,S.core_transitive (e i).property hx⟩
exact ⟨S.collapse y,(S.mem_collapse (e i) _).mpr ⟨y,hx,rfl⟩,
(hstep y).mpr hp'⟩
def PairingAxiom : Prop := ∀ a b : α, ∃ c, ∀ x, S.mem x c ↔ x=a ∨ x=b
def UnionAxiom : Prop := ∀ a : α, ∃ b, ∀ x, S.mem x b ↔ ∃ y, S.mem y a ∧ S.mem x y
def SeparationAxiom : Prop := ∀ (p : Formula) (e : ℕ → α) (a : α),
∃ b, ∀ x, S.mem x b ↔ S.mem x a ∧ S.Eval (push x e) p
end
end TuringRigidity.OmegaModelCore.OmegaData
end OAI
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