| import OAI.Computability.DegreeRigidity.Syntax.AtomicFormula |
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| namespace OAI |
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| namespace TuringRigidity.AtomicForcing |
| open TransitiveNameModel BoundedSetTheory |
| universe u |
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| def Graph (d c o f : ZFSet.{u}) : Prop := |
| f ⊆ ZFSet.prod (ZFSet.prod d d) c ∧ |
| ∀ x ∈ d, ∀ y ∈ d, ∀ q ∈ c, |
| ZFSet.pair (ZFSet.pair x y) q ∈ f ↔ Step d c (ZFSet.prod d d) o f x y q |
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| theorem vertex_pair (d c o f x y q : ZFSet.{u}) (hx : x ∈ d) (hy : y ∈ d) (hq : q ∈ c) : |
| Vertex d c (ZFSet.prod d d) o f (ZFSet.pair (ZFSet.pair x y) q) ↔ |
| Step d c (ZFSet.prod d d) o f x y q := by |
| constructor |
| · rintro ⟨a,ha,b,hb,r,hr,v,hv,hve,hze,hs⟩ |
| obtain ⟨hp,hq'⟩ := ZFSet.pair_inj.mp hze |
| have hab := ZFSet.pair_inj.mp (hp.trans hve) |
| rcases hab with ⟨rfl,rfl⟩ |
| simpa only [hq'] using hs |
| · intro hs |
| exact ⟨x,hx,y,hy,q,hq,_,ZFSet.mem_prod.mpr ⟨x,hx,y,hy,rfl⟩,rfl,rfl,hs⟩ |
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| theorem internal_atomic_graph (M : ZFSet.{u}) (hM : Transitive M) |
| (hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M) (hS : Separation M) |
| {d c o : ZFSet.{u}} (hd : d ∈ M) (hc : c ∈ M) (ho : o ∈ M) : |
| ∃ f ∈ M, Graph d c o f := by |
| let k := ZFSet.prod d d |
| let a := ZFSet.prod k c |
| have hk : k ∈ M := product_mem M hM hP hU hPow hS hd hd |
| have ha : a ∈ M := product_mem M hM hP hU hPow hS hk hc |
| let e := cons d (cons c (cons k (cons o (fun _ => d)))) |
| have he : ∀ i, e i ∈ M := by |
| intro i |
| rcases i with _|i; exact hd |
| rcases i with _|i; exact hc |
| rcases i with _|i; exact hk |
| rcases i with _|i; exact ho |
| exact hd |
| let φ := AtomicFormula.vertex 2 3 4 5 1 0 |
| have hφ (f z : ZFSet.{u}) : φ.Eval (cons z (cons f e)) ↔ Vertex d c k o f z := |
| AtomicFormula.eval_vertex 2 3 4 5 1 0 _ |
| obtain ⟨f,hf,hfa,hfdef⟩ := internal_fixed_point M hM hU hPow hS φ e he a ha (by |
| intro f g _ _ hfg z _ h |
| exact (hφ g z).mpr (vertex_mono _ _ _ _ _ _ _ hfg ((hφ f z).mp h))) |
| refine ⟨f,hf,hfa,fun x hx y hy q hq => ?_⟩ |
| have hz : ZFSet.pair (ZFSet.pair x y) q ∈ a := |
| ZFSet.mem_prod.mpr ⟨_,ZFSet.mem_prod.mpr ⟨x,hx,y,hy,rfl⟩,q,hq,rfl⟩ |
| exact (hfdef _ hz).trans ((hφ f _).trans (vertex_pair d c o f x y q hx hy hq)) |
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| end TuringRigidity.AtomicForcing |
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| end OAI |
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