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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 | import OAI.Computability.DegreeRigidity.SetModels.ModelReals
namespace OAI
namespace TuringRigidity.BoundedSetTheory
open TransitiveNameModel
universe u
noncomputable def finiteModelGraph (f : ℕ → ZFSet.{u}) : ℕ → ZFSet.{u}
| 0 => {ZFSet.pair (natSet 0) (f 0)}
| n+1 => finiteModelGraph f n ∪ ({ZFSet.pair (natSet (n+1)) (f (n+1))} : ZFSet.{u})
theorem mem_finiteModelGraph (f : ℕ → ZFSet.{u}) (n : ℕ) (z : ZFSet.{u}) :
z ∈ finiteModelGraph f n ↔ ∃ i, i ≤ n ∧ z = ZFSet.pair (natSet i) (f i) := by
induction n with
| zero =>
rw [finiteModelGraph,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 [finiteModelGraph,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)
theorem finiteModelGraph_pair (f : ℕ → ZFSet.{u}) (n i : ℕ) (y : ZFSet.{u}) :
ZFSet.pair (natSet i) y ∈ finiteModelGraph f n ↔ i ≤ n ∧ y = f i := by
rw [mem_finiteModelGraph]
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⟩
theorem finiteModelGraph_function (R : ZFSet.{u}) (f : ℕ → ZFSet.{u}) (hf : ∀ i, f i ∈ R) (n : ℕ) :
TransitiveNameModel.FunctionGraph (natSet (n+1)) R (finiteModelGraph.{u} f n) := by
constructor
· intro z hz
obtain ⟨i,hi,rfl⟩ := (mem_finiteModelGraph f n z).mp hz
exact ⟨natSet i,(natSet_mem_natSet i (n+1)).mpr (by omega),f i,
hf i,rfl⟩
· intro x hx
obtain ⟨i,hi,rfl⟩ := (mem_natSet (n+1) x).mp hx
have hin : i ≤ n := by omega
refine ⟨f i,hf i,
(finiteModelGraph_pair f n i _).mpr ⟨hin,rfl⟩,?_⟩
intro y _ hy
exact ((finiteModelGraph_pair f n i y).mp hy).2
theorem finiteModelGraph_mem (M : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hω : ZFSet.omega.{u} ∈ M)
(f : ℕ → ZFSet.{u}) (hf : ∀ i, f i ∈ M) (n : ℕ) : finiteModelGraph 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) (hf 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)) (hf (n+1))))
theorem internal_finite_model_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) (R : ZFSet.{u}) (hR : R ∈ M) :
∃ Q ∈ M, (∀ F, F ∈ Q ↔ F ∈ M ∧ F ⊆ ZFSet.prod ZFSet.omega R) ∧
∀ f : ℕ → ZFSet.{u}, (∀ i, f i ∈ R) → ∀ n, finiteModelGraph f n ∈ Q := by
obtain ⟨Q,hQ,hQdef⟩ := internal_power M hM hPow (product_mem M hM hP hU hPow hS hω hR)
refine ⟨Q,hQ,hQdef,?_⟩
intro f hf n
apply (hQdef _).mpr
refine ⟨finiteModelGraph_mem M hM hP hU hω f (fun i => hM _ hR _ (hf i)) n,?_⟩
intro z hz
obtain ⟨i,_,rfl⟩ := (mem_finiteModelGraph f n z).mp hz
exact ZFSet.mem_prod.mpr ⟨_,(mem_omega _).mpr ⟨i,rfl⟩,_,hf i,rfl⟩
end TuringRigidity.BoundedSetTheory
end OAI
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