| import OAI.Computability.DegreeRigidity.Effective.FiniteSumDomains |
| import OAI.Computability.DegreeRigidity.Effective.FiniteSumGraph |
| import OAI.Computability.DegreeRigidity.SetModels.FinitePresentation |
|
|
| namespace OAI |
|
|
| namespace TuringRigidity.OrdinalArithmetic |
| open TransitiveNameModel BoundedSetTheory RelationCollapse ElementaryModel RelativeConstructible OrdinalCoding |
| universe u |
|
|
| theorem sum_certificates_at_add_five (R : ZFSet.{u}) (γ a b : Ordinal.{u}) |
| (hω : ZFSet.omega.{u} ∈ level R γ) |
| (ha : a.toZFSet ∈ level R γ) (hb : b.toZFSet ∈ level R γ) |
| (hlocal : ∀ t < b, (a+t).toZFSet ∈ level R γ ∧ SumCertificates (level R γ) a t) : |
| SumCertificates (level R (γ+5)) a b := by |
| have h0 := level_transitive R γ _ hω _ ZFSet.omega_zero |
| have h1 : ({∅} : ZFSet.{u}) ∈ level R γ := by |
| have h := level_transitive R γ _ hω _ ((mem_omega _).mpr ⟨1,rfl⟩) |
| simpa [natSet] using h |
| have h02 : γ ≤ γ+2 := le_self_add |
| have h24 : γ+2 ≤ γ+4 := add_le_add le_rfl (by exact_mod_cast (show (2 : ℕ) ≤ 4 by decide) : (2 : Ordinal.{u}) ≤ 4) |
| have h04 := h02.trans h24 |
| have hd2 : sumDomain a.toZFSet b.toZFSet ⊆ level R (γ+2) := by |
| intro x hx |
| rcases (mem_sumDomain _ _ _).mp hx with ⟨t,ht,rfl⟩|⟨t,ht,rfl⟩ |
| · exact orderedPair_mem_level_add_two R γ h0 (level_transitive R γ _ ha _ ht) |
| · exact orderedPair_mem_level_add_two R γ h1 (level_transitive R γ _ hb _ ht) |
| have hd4 : sumDomain a.toZFSet b.toZFSet ⊆ level R (γ+4) := |
| fun _ h => level_mono R h24 (hd2 h) |
| have hr4 : sumRelation a.toZFSet b.toZFSet ⊆ level R (γ+4) := by |
| intro z hz |
| obtain ⟨x,hx,y,hy,rfl⟩ := ZFSet.mem_prod.mp (sumRelation_on _ _ hz) |
| simpa only [add_assoc,show (2 : Ordinal.{u})+2=4 from by norm_num] using |
| orderedPair_mem_level_add_two R (γ+2) (hd2 hx) (hd2 hy) |
| have hv (x : ZFSet.{u}) (hx : x ∈ sumDomain a.toZFSet b.toZFSet) : |
| sumValue a x ∈ level R γ := by |
| rcases (mem_sumDomain _ _ _).mp hx with ⟨t,ht,rfl⟩|⟨t,ht,rfl⟩ |
| · simpa only [sumValue_left] using level_transitive R γ _ ha _ ht |
| · obtain ⟨t,ht,rfl⟩ := Ordinal.mem_toZFSet_iff.mp ht |
| simpa only [sumValue_right,Ordinal.rank_toZFSet] using (hlocal t (Ordinal.toZFSet_mem_toZFSet_iff.mp ht)).1 |
| have hpairs (x : ZFSet.{u}) (hx : x ∈ sumDomain a.toZFSet b.toZFSet) : |
| ZFSet.pair x (sumValue a x) ∈ level R (γ+4) := by |
| simpa only [add_assoc,show (2 : Ordinal.{u})+2=4 from by norm_num] using |
| orderedPair_mem_level_add_two R (γ+2) (hd2 hx) (level_mono R h02 (hv x hx)) |
| have ha4 := level_mono R h04 ha |
| have hb4 := level_mono R h04 hb |
| have hz4 := level_mono R h04 h0 |
| have ho4 := level_mono R h04 h1 |
| have hω4 := level_mono R h04 hω |
| have hc4 (t : Ordinal.{u}) (ht : t < b) : |
| (a+t).toZFSet ∈ level R (γ+4) ∧ SumCertificates (level R (γ+4)) a t := |
| ⟨level_mono R h04 (hlocal t ht).1,(hlocal t ht).2.mono (level_mono R h04)⟩ |
| have hd := sum_domain_at_successor_of_subset R (γ+4) ha4 hb4 hz4 ho4 hd4 |
| have hr := sum_relation_at_successor_of_subset R (γ+4) ha4 hb4 hz4 ho4 hd4 hr4 |
| have hf := presentation_graph_at_successor_of_pairs R (γ+4) _ _ (a+b) (sumValue a) |
| hpairs (sumValue_mem a b) (sumValue_surjective a b) (sumValue_relation a b) |
| trimmedSumGraphSentence (cons a.toZFSet (cons b.toZFSet (cons ∅ (fun _ => {∅})))) |
| (fun i => by rcases i with _|_|_|i; exact ha4; exact hb4; exact hz4; exact ho4) |
| (sum_graph_spec_with_internal_nodes _ (level_transitive R (γ+4)) a b |
| hω4 ha4 hb4 hz4 ho4 hc4 hd4) |
| simpa only [SumCertificates,add_assoc,show (4 : Ordinal.{u})+1=5 from by norm_num] using And.intro hd (And.intro hr hf) |
|
|
| end TuringRigidity.OrdinalArithmetic |
|
|
| end OAI |
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|
