| import OAI.Computability.DegreeRigidity.SetModels.CollapseCertificate |
| import OAI.Computability.DegreeRigidity.Syntax.SigmaSeparationName |
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| namespace OAI |
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| namespace TuringRigidity.RelationCollapse |
| open TransitiveNameModel BoundedSetTheory |
| universe u |
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| 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⟩⟩ |
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| 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⟩) |
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| end TuringRigidity.RelationCollapse |
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| end OAI |
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