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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 | import OAI.Computability.DegreeRigidity.Syntax.AtomicFormula
namespace OAI
namespace TuringRigidity.AtomicForcing
open TransitiveNameModel BoundedSetTheory
universe u
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
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⟩
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))
end TuringRigidity.AtomicForcing
end OAI
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