openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /OrdinalCodes /LargestOrdinalSyntax.lean
| import OAI.Computability.DegreeRigidity.OrdinalCodes.LevelOrdinalHeight | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open ElementaryModel BoundedSetTheory TransitiveNameModel | |
| universe u | |
| noncomputable def largestOrdinalSentence : SentenceForm := | |
| .conj ordinalSentence (.all (.imp ordinalSentence (.disj (.member 0 1) (.equal 0 1)))) | |
| theorem largestOrdinalSentence_bound : largestOrdinalSentence.bound = 1 := by rfl | |
| theorem largestOrdinalSentence_spec (A : ZFSet.{u}) (hA : Transitive A) | |
| (e : ℕ → ZFSet.{u}) (he : e 0 ∈ A) : | |
| largestOrdinalSentence.Sat (A : Set ZFSet) e ↔ | |
| (e 0).IsOrdinal ∧ ∀ x ∈ A, x.IsOrdinal → x ∈ e 0 ∨ x = e 0 := by | |
| simp only [largestOrdinalSentence,SentenceForm.Sat,SentenceForm.sat_all, | |
| SentenceForm.sat_imp,SentenceForm.sat_disj,cons_zero,cons_succ] | |
| rw [ordinalSentence_spec A hA e he] | |
| apply and_congr_right; intro _ | |
| apply forall_congr'; intro x | |
| apply forall_congr'; intro hx | |
| rw [ordinalSentence_spec A hA (cons x e) hx] | |
| rfl | |
| theorem largestOrdinalSentence_unique (A : ZFSet.{u}) (hA : Transitive A) | |
| (a : Ordinal.{u}) (ha : a.toZFSet ∈ A) | |
| (hmax : ∀ b : Ordinal.{u}, b.toZFSet ∈ A → b ≤ a) | |
| (e : ℕ → ZFSet.{u}) (he : e 0 ∈ A) : | |
| largestOrdinalSentence.Sat (A : Set ZFSet) e ↔ e 0 = a.toZFSet := by | |
| rw [largestOrdinalSentence_spec A hA e he] | |
| constructor | |
| · rintro ⟨ho,hm⟩ | |
| have heq := ho.toZFSet_rank_eq | |
| have hle := hmax (e 0).rank (by rwa [heq]) | |
| have hge : a ≤ (e 0).rank := by | |
| rcases hm a.toZFSet ha (ZFSet.isOrdinal_toZFSet a) with h|h | |
| · rw [← heq,Ordinal.toZFSet_mem_toZFSet_iff] at h | |
| exact h.le | |
| · have := congrArg ZFSet.rank h | |
| simpa only [Ordinal.rank_toZFSet] using this.le | |
| exact heq.symm.trans (congrArg Ordinal.toZFSet (le_antisymm hle hge)) | |
| · intro heq | |
| refine ⟨heq ▸ ZFSet.isOrdinal_toZFSet a,?_⟩ | |
| intro x hx ho | |
| have hle : x.rank ≤ a := hmax x.rank (by rwa [ho.toZFSet_rank_eq]) | |
| rcases lt_or_eq_of_le hle with h|h | |
| · left | |
| rw [heq,← ho.toZFSet_rank_eq] | |
| exact Ordinal.toZFSet_mem_toZFSet_iff.mpr h | |
| · right | |
| rw [← ho.toZFSet_rank_eq,h,heq] | |
| theorem largestOrdinalSentence_level (R : ZFSet.{u}) (o : Ordinal.{u}) | |
| (e : ℕ → ZFSet.{u}) (he : e 0 ∈ level R (o+1)) : | |
| largestOrdinalSentence.Sat (level R (o+1) : Set ZFSet) e ↔ | |
| e 0 = (ordinalHeight (seed R) + o).toZFSet := by | |
| have hm := largest_ordinal_level_succ R o | |
| exact largestOrdinalSentence_unique _ (level_transitive R _) _ hm.1 | |
| (fun b hb => (hm.2 b).mp hb) e he | |
| noncomputable def bindLargestOrdinal (p : SentenceForm) : SentenceForm := | |
| .ex (.conj largestOrdinalSentence p) | |
| theorem bindLargestOrdinal_bound (p : SentenceForm) : | |
| (bindLargestOrdinal p).bound = p.bound - 1 := by | |
| change max largestOrdinalSentence.bound p.bound - 1 = p.bound - 1 | |
| rw [largestOrdinalSentence_bound] | |
| omega | |
| theorem bindLargestOrdinal_level (p : SentenceForm) (R : ZFSet.{u}) (o : Ordinal.{u}) | |
| (e : ℕ → ZFSet.{u}) : | |
| (bindLargestOrdinal p).Sat (level R (o+1) : Set ZFSet) e ↔ | |
| p.Sat (level R (o+1) : Set ZFSet) (cons (ordinalHeight (seed R) + o).toZFSet e) := by | |
| change (∃ a ∈ level R (o+1), largestOrdinalSentence.Sat _ (cons a e) ∧ p.Sat _ (cons a e)) ↔ _ | |
| constructor | |
| · rintro ⟨a,ha,hm,hp⟩ | |
| have heq := (largestOrdinalSentence_level R o (cons a e) ha).mp hm | |
| change a = _ at heq | |
| simpa only [heq] using hp | |
| · intro hp | |
| have hm := largest_ordinal_level_succ R o | |
| exact ⟨_,hm.1,(largestOrdinalSentence_level R o _ hm.1).mpr rfl,hp⟩ | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |