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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 | import OAI.Computability.DegreeRigidity.OrdinalCodes.NaturalDefinitions
namespace OAI
namespace TuringRigidity.BoundedSetTheory
universe u
namespace Formula
def naturalProjection (right : Bool) : Formula := .existsMem 2 (.existsMem 3
(.conj (naturalPair 4 5 1 0 3) (.equal 2 (if right then 0 else 1))))
theorem naturalProjection_spec (right : Bool) : DefinesNatural.{u} (naturalProjection right)
(fun n => if right then (Nat.unpair n).2 else (Nat.unpair n).1) := by
intro e ho hQ n hn
have select (a b : ZFSet.{u}) : cons b (cons a e) (if right then 0 else 1) =
if right then b else a := by cases right <;> rfl
have hmem (k : ℕ) : natSet.{u} k ∈ e 2 := by rw [ho]; exact (mem_omega _).mpr ⟨k,rfl⟩
simp only [naturalProjection,Formula.Eval,cons_succ,select]
constructor
· rintro ⟨a,ha,b,hb,hpair,hy⟩
rw [ho] at ha hb
obtain ⟨a,rfl⟩ := (mem_omega a).mp ha
obtain ⟨b,rfl⟩ := (mem_omega b).mp hb
have hp := (naturalPair_spec 4 5 1 0 3 (cons (natSet b) (cons (natSet a) e))
ho hQ a b rfl rfl).mp hpair
have he : n = Nat.pair a b := natSet_injective (hn.symm.trans hp)
subst n
cases right <;> simpa only [Bool.false_eq_true,ite_false,ite_true,Nat.unpair_pair] using hy
· intro hy
refine ⟨natSet (Nat.unpair n).1,hmem _,natSet (Nat.unpair n).2,hmem _,?_,?_⟩
· apply (naturalPair_spec 4 5 1 0 3
(cons (natSet (Nat.unpair n).2) (cons (natSet (Nat.unpair n).1) e))
ho hQ _ _ rfl rfl).mpr
simpa only [cons_succ,Nat.pair_unpair] using hn
· cases right <;> simpa only [Bool.false_eq_true,ite_false,ite_true] using hy
end Formula
end TuringRigidity.BoundedSetTheory
end OAI
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