File size: 3,227 Bytes
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 | import OAI.Computability.DegreeRigidity.SetModels.CollapseCertificate
import OAI.Computability.DegreeRigidity.Syntax.SigmaSeparationName
namespace OAI
namespace TuringRigidity.RelationCollapse
open TransitiveNameModel BoundedSetTheory
universe u
theorem collect_graphs (M d : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M)
(hS : SigmaSeparation M) (hR : SigmaReplacement M) (hd : d ∈ M) :
∃ B ∈ M, ∀ f, f ∈ B ↔ f ∈ M ∧ ∃ r ∈ M, On d r ∧ ∃ b ∈ M, Graph d r d b f := by
obtain ⟨q,hq,hqdef⟩ := internal_power M hM hPow
(product_mem M hM hP hU hPow hS.bounded hd hd)
let e := fun _ : ℕ => d
let success := SigmaFormula.existsSet (Code.full 2 1 0)
let a := ZFSet.sep (fun r => success.Realize M (cons r e)) q
have ha : a ∈ M := sigma_sep_mem M hM hS success e (fun _ => hd) hq
have cert (r f : ZFSet.{u}) (hr : r ∈ M) (hf : f ∈ M) :
(Code.full 2 1 0).Realize M (cons f (cons r e)) ↔ ∃ b ∈ M, Graph d r d b f :=
Code.realize_full M hM _ (by
intro i; rcases i with _|i; exact hf
rcases i with _|i; exact hr
exact hd) 2 1 0
have ha_def (r : ZFSet.{u}) : r ∈ a ↔
r ∈ M ∧ On d r ∧ ∃ f ∈ M, ∃ b ∈ M, Graph d r d b f := by
rw [ZFSet.mem_sep,hqdef]
change (r ∈ M ∧ On d r) ∧ (∃ f ∈ M, _) ↔ _
constructor
· rintro ⟨⟨hr,hrd⟩,f,hf,hc⟩
exact ⟨hr,hrd,f,hf,(cert r f hr hf).mp hc⟩
· rintro ⟨hr,hrd,f,hf,b,hb,hg⟩
exact ⟨⟨hr,hrd⟩,f,hf,(cert r f hr hf).mpr ⟨b,hb,hg⟩⟩
obtain ⟨B,hB,hBdef⟩ := hR (Code.full 2 1 0) e (fun _ => hd) a ha (by
intro r hr
obtain ⟨hrM,hrd,f,hf,b,hb,hfg⟩ := (ha_def r).mp hr
refine ⟨f,hf,(cert r f hrM hf).mpr ⟨b,hb,hfg⟩,?_⟩
intro g hg hcg
obtain ⟨b',_,hgg⟩ := (cert r g hrM hg).mp hcg
exact hgg.unique (hfg.wellFounded hrd) hfg)
refine ⟨B,hB,fun f => ?_⟩
constructor
· intro hf
have hfM := hM B hB f hf
obtain ⟨r,hr,hc⟩ := (hBdef f hfM).mp hf
obtain ⟨hrM,hrd,_⟩ := (ha_def r).mp hr
exact ⟨hfM,r,hrM,hrd,(cert r f hrM hfM).mp hc⟩
· rintro ⟨hf,r,hr,hrd,b,hb,hg⟩
exact (hBdef f hf).mpr ⟨r,(ha_def r).mpr ⟨hr,hrd,f,hf,b,hb,hg⟩,
(cert r f hr hf).mpr ⟨b,hb,hg⟩⟩
theorem internal_value_bound (M d : ZFSet.{u}) (hM : Transitive M)
(hP : Pairing M) (hU : BoundedSetTheory.Union M) (hPow : PowerSet M)
(hS : SigmaSeparation M) (hR : SigmaReplacement M) (hd : d ∈ M) :
∃ W ∈ M, ∀ r ∈ M, On d r → ∀ wf : WellFounded (Rel d r),
∀ x ∈ d, value d r wf x ∈ W := by
obtain ⟨B,hB,hBdef⟩ := collect_graphs M d hM hP hU hPow hS hR hd
let W := iterUnion 2 (ZFSet.sUnion B)
have hW : W ∈ M := iterUnion_mem M hM hU (union_mem M hM hU hB) 2
refine ⟨W,hW,?_⟩
intro r hr hrd wf x hx
obtain ⟨f,hf,hfg⟩ := internal_graph M d r hM hP hU hPow hS.bounded hR hd hr wf
have hfB := (hBdef f).mpr ⟨hf,r,hr,hrd,_,iterUnion_mem M hM hU hf 2,hfg⟩
have hpair := (hfg.mem_iff wf _).mpr ⟨x,hx,rfl⟩
exact second_mem_doubleUnion (ZFSet.mem_sUnion.mpr ⟨f,hfB,hpair⟩)
end TuringRigidity.RelationCollapse
end OAI
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