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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 91 92 93 94 95 96 97 98 99 100 101 102 | import OAI.Computability.DegreeRigidity.SetModels.InternalRelationCollapse
namespace OAI
namespace TuringRigidity.RelationCollapse
open TransitiveNameModel BoundedSetTheory
universe u
theorem internal_graph (M d r : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M)
(hS : Separation M) (hR : SigmaReplacement M)
(hd : d ∈ M) (hr : r ∈ M) (wf : WellFounded (Rel d r)) :
∃ f ∈ M, Graph d r d (iterUnion 2 f) f := by
obtain ⟨q,hqM,hq⟩ := internal_power M hM hPow hd
let e := cons q (cons d (cons r (fun _ => d)))
have he : ∀ i, e i ∈ M := by
intro i; rcases i with _|i; exact hqM
rcases i with _|i; exact hd
rcases i with _|i; exact hr
exact hd
have cert (x f : ZFSet.{u}) (hx : x ∈ d) (hf : f ∈ M) :
certificate.Realize M (cons f (cons x e)) ↔
∃ b ∈ M, Graph d r (RelationHull.hull d r q x) b f :=
realize_certificate M d r q hM hS hd hr hqM hq hx hf
obtain ⟨B,hB,hBdef⟩ := hR certificate e he d hd (by
intro x hx
obtain ⟨f,hf,hfg⟩ := internal_hull_graph M d r q hM hP hU hS hR hd hr hqM hq wf x hx
refine ⟨f,hf,(cert x f hx hf).mpr ⟨_,iterUnion_mem M hM hU hf 2,hfg⟩,?_⟩
intro g hg hcg
obtain ⟨b,_,hgg⟩ := (cert x g hx hg).mp hcg
exact hgg.unique wf hfg)
have hBg (f : ZFSet.{u}) : f ∈ B ↔
∃ x ∈ d, ∃ b ∈ M, Graph d r (RelationHull.hull d r q x) b f := by
constructor
· intro hf
have hfM := hM B hB f hf
obtain ⟨x,hx,hc⟩ := (hBdef f hfM).mp hf
exact ⟨x,hx,(cert x f hx hfM).mp hc⟩
· rintro ⟨x,hx,b,hb,hg⟩
obtain ⟨g,hgM,hgg⟩ := internal_hull_graph M d r q hM hP hU hS hR hd hr hqM hq wf x hx
have hfM : f ∈ M := hg.unique wf hgg ▸ hgM
exact (hBdef f hfM).mpr ⟨x,hx,(cert x f hx hfM).mpr ⟨b,hb,hg⟩⟩
refine ⟨ZFSet.sUnion B,union_mem M hM hU hB,
graph_of_mem_iff d r d _ wf (fun _ h => h) (fun _ _ _ h _ => h) ?_⟩
intro z
rw [ZFSet.mem_sUnion]
constructor
· rintro ⟨f,hf,hzf⟩
obtain ⟨x,_,b,_,hfg⟩ := (hBg f).mp hf
obtain ⟨y,hy,hz⟩ := (hfg.mem_iff wf z).mp hzf
exact ⟨y,RelationHull.hull_subset d r q x hy,hz⟩
· rintro ⟨x,hx,hz⟩
obtain ⟨f,hf,hfg⟩ := internal_hull_graph M d r q hM hP hU hS hR hd hr hqM hq wf x hx
exact ⟨f,(hBg f).mpr ⟨x,hx,_,iterUnion_mem M hM hU hf 2,hfg⟩,
(hfg.mem_iff wf z).mpr ⟨x,RelationHull.self_mem d r q hx,hz⟩⟩
def On (d r : ZFSet.{u}) : Prop := r ⊆ ZFSet.prod d d
theorem right_mem {d r x y : ZFSet.{u}} (hr : On d r) (hyx : ZFSet.pair y x ∈ r) : x ∈ d := by
obtain ⟨a,_,b,hb,heq⟩ := ZFSet.mem_prod.mp (hr hyx)
exact (ZFSet.pair_inj.mp heq).2 ▸ hb
theorem Graph.wellFounded {d r b f : ZFSet.{u}} (hr : On d r)
(hg : Graph d r d b f) : WellFounded (Rel d r) := by
classical
let F : ZFSet.{u} → ZFSet.{u} := fun x =>
if hx : x ∈ d then (hg.2.2.2.1 x hx).choose else ∅
have hF (x : ZFSet.{u}) (hx : x ∈ d) :
F x ∈ b ∧ ZFSet.pair x (F x) ∈ f := by
simpa only [F,dite_eq_left hx] using (hg.2.2.2.1 x hx).choose_spec |>.imp_right And.left
have wf : WellFounded (fun y x => F y ∈ F x) := InvImage.wf F ZFSet.mem_wf
apply wf.mono
intro y x hyx
have hx := right_mem hr hyx.2
exact (hg.2.2.2.2 x hx (F x) (hF x hx).1 (hF x hx).2).2
y hyx.1 hyx.2 (F y) (hF y hyx.1).1 (hF y hyx.1).2
namespace Code
def full (d r f : ℕ) : SigmaFormula :=
.existsSet (.bounded (graph (d+1) (r+1) (d+1) 0 (f+1)))
theorem realize_full (M : ZFSet.{u}) (hM : Transitive M)
(e : ℕ → ZFSet.{u}) (he : ∀ i, e i ∈ M) (d r f : ℕ) :
(full d r f).Realize M e ↔ ∃ b ∈ M, Graph (e d) (e r) (e d) b (e f) := by
change (∃ b ∈ M, _) ↔ _
apply exists_congr
intro b
apply and_congr_right
intro hb
simp only [SigmaFormula.Realize]
rw [Formula.absolute _ M hM _ (by intro i; cases i with
| zero => exact hb
| succ i => exact he i)]
exact eval_graph (d+1) (r+1) (d+1) 0 (f+1) _
end Code
end TuringRigidity.RelationCollapse
end OAI
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