| import OAI.Computability.DegreeRigidity.SetModels.ModelNaturalNumbers |
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| namespace OAI |
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| namespace TuringRigidity.BoundedSetTheory |
| open TransitiveNameModel |
| universe u |
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| noncomputable def finiteNaturalGraph (f : ℕ → ℕ) : ℕ → ZFSet.{u} |
| | 0 => {ZFSet.pair (natSet 0) (natSet (f 0))} |
| | n+1 => finiteNaturalGraph f n ∪ ({ZFSet.pair (natSet (n+1)) (natSet (f (n+1)))} : ZFSet.{u}) |
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| theorem mem_finiteNaturalGraph (f : ℕ → ℕ) (n : ℕ) (z : ZFSet.{u}) : |
| z ∈ finiteNaturalGraph f n ↔ ∃ i, i ≤ n ∧ z = ZFSet.pair (natSet i) (natSet (f i)) := by |
| induction n with |
| | zero => |
| rw [finiteNaturalGraph,ZFSet.mem_singleton] |
| constructor |
| · intro h; exact ⟨0,le_rfl,h⟩ |
| · rintro ⟨i,hi,h⟩ |
| have he : i = 0 := Nat.eq_zero_of_le_zero hi |
| simpa only [he] using h |
| | succ n ih => |
| rw [finiteNaturalGraph,ZFSet.mem_union,ZFSet.mem_singleton,ih] |
| constructor |
| · rintro (⟨i,hi,hz⟩|hz) |
| · exact ⟨i,Nat.le_succ_of_le hi,hz⟩ |
| · exact ⟨n+1,le_rfl,hz⟩ |
| · rintro ⟨i,hi,hz⟩ |
| by_cases hin : i ≤ n |
| · exact Or.inl ⟨i,hin,hz⟩ |
| · have he : i = n+1 := by omega |
| exact Or.inr (he ▸ hz) |
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| theorem finiteNaturalGraph_pair (f : ℕ → ℕ) (n i : ℕ) (y : ZFSet.{u}) : |
| ZFSet.pair (natSet i) y ∈ finiteNaturalGraph f n ↔ i ≤ n ∧ y = natSet (f i) := by |
| rw [mem_finiteNaturalGraph] |
| constructor |
| · rintro ⟨j,hj,he⟩ |
| obtain ⟨hij,hy⟩ := ZFSet.pair_inj.mp he |
| have heq := natSet_injective hij |
| subst j |
| exact ⟨hj,hy⟩ |
| · rintro ⟨hi,rfl⟩ |
| exact ⟨i,hi,rfl⟩ |
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| theorem finiteNaturalGraph_function (f : ℕ → ℕ) (n : ℕ) : |
| TransitiveNameModel.FunctionGraph (natSet (n+1)) ZFSet.omega (finiteNaturalGraph.{u} f n) := by |
| constructor |
| · intro z hz |
| obtain ⟨i,hi,rfl⟩ := (mem_finiteNaturalGraph f n z).mp hz |
| exact ⟨natSet i,(natSet_mem_natSet i (n+1)).mpr (by omega),natSet (f i), |
| (mem_omega _).mpr ⟨f i,rfl⟩,rfl⟩ |
| · intro x hx |
| obtain ⟨i,hi,rfl⟩ := (mem_natSet (n+1) x).mp hx |
| have hin : i ≤ n := by omega |
| refine ⟨natSet (f i),(mem_omega _).mpr ⟨f i,rfl⟩, |
| (finiteNaturalGraph_pair f n i _).mpr ⟨hin,rfl⟩,?_⟩ |
| intro y _ hy |
| exact ((finiteNaturalGraph_pair f n i y).mp hy).2 |
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| theorem finiteNaturalGraph_mem (M : ZFSet.{u}) (hM : Transitive M) |
| (hP : Pairing M) (hU : BoundedSetTheory.Union M) (hω : ZFSet.omega.{u} ∈ M) |
| (f : ℕ → ℕ) (n : ℕ) : finiteNaturalGraph f n ∈ M := by |
| have hn (i : ℕ) : natSet.{u} i ∈ M := hM _ hω _ ((mem_omega _).mpr ⟨i,rfl⟩) |
| induction n with |
| | zero => exact singleton_mem M hM hP (orderedPair_mem M hM hP (hn 0) (hn (f 0))) |
| | succ n ih => |
| exact binary_union_mem M hM hP hU ih |
| (singleton_mem M hM hP (orderedPair_mem M hM hP (hn (n+1)) (hn (f (n+1))))) |
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| theorem internal_finite_natural_graph_bound (M : ZFSet.{u}) (hM : Transitive M) |
| (hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M) |
| (hS : Separation M) (hω : ZFSet.omega.{u} ∈ M) : |
| ∃ Q ∈ M, (∀ F, F ∈ Q ↔ F ∈ M ∧ F ⊆ ZFSet.prod ZFSet.omega ZFSet.omega) ∧ |
| ∀ f : ℕ → ℕ, ∀ n, finiteNaturalGraph f n ∈ Q := by |
| obtain ⟨Q,hQ,hQdef⟩ := internal_power M hM hPow (product_mem M hM hP hU hPow hS hω hω) |
| refine ⟨Q,hQ,hQdef,?_⟩ |
| intro f n |
| apply (hQdef _).mpr |
| refine ⟨finiteNaturalGraph_mem M hM hP hU hω f n,?_⟩ |
| intro z hz |
| obtain ⟨i,_,rfl⟩ := (mem_finiteNaturalGraph f n z).mp hz |
| exact ZFSet.mem_prod.mpr ⟨_,(mem_omega _).mpr ⟨i,rfl⟩,_,(mem_omega _).mpr ⟨f i,rfl⟩,rfl⟩ |
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| end TuringRigidity.BoundedSetTheory |
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| end OAI |
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