openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /OrdinalCodes /NaturalProjectionFormula.lean
| 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 | |