openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /OrdinalCodes /LevelOrdinalHeight.lean
| import OAI.Computability.DegreeRigidity.OrdinalCodes.OrdinalCut | |
| namespace OAI | |
| namespace TuringRigidity.RelativeConstructible | |
| open ElementaryModel BoundedSetTheory TransitiveNameModel | |
| universe u | |
| theorem ordinalHeight_level (R : ZFSet.{u}) (o : Ordinal.{u}) : | |
| ordinalHeight (level R o) = ordinalHeight (seed R) + o := by | |
| induction o using Ordinal.limitRecOn with | |
| | zero => simp only [level_zero,add_zero] | |
| | add_one o ih => | |
| rw [level_succ,ordinalHeight_definablePower _ (level_transitive R o),ih,add_assoc] | |
| | limit o ho ih => | |
| apply le_antisymm | |
| · by_contra h | |
| have hx := (ordinal_mem_iff_height _ (level_transitive R o) _).mpr (lt_of_not_ge h) | |
| obtain ⟨i,hi,hx⟩ := (mem_level_limit R _ o ho).mp hx | |
| have hx' := (ordinal_mem_iff_height _ (level_transitive R i) _).mp hx | |
| rw [ih i hi] at hx' | |
| exact (not_lt_of_ge (add_le_add (le_refl (ordinalHeight (seed R))) hi.le)) hx' | |
| · by_contra h | |
| obtain ⟨i,hi,hi'⟩ := Ordinal.lt_add_iff_of_isSuccLimit ho |>.mp (lt_of_not_ge h) | |
| rw [← ih i hi] at hi' | |
| have hx := (ordinal_mem_iff_height _ (level_transitive R i) _).mpr hi' | |
| have hx' := level_mono R hi.le hx | |
| exact (lt_irrefl _) ((ordinal_mem_iff_height _ (level_transitive R o) _).mp hx') | |
| theorem ordinal_mem_level_iff (R : ZFSet.{u}) (o a : Ordinal.{u}) : | |
| a.toZFSet ∈ level R o ↔ a < ordinalHeight (seed R) + o := by | |
| rw [ordinal_mem_iff_height _ (level_transitive R o),ordinalHeight_level] | |
| theorem largest_ordinal_level_succ (R : ZFSet.{u}) (o : Ordinal.{u}) : | |
| (ordinalHeight (seed R) + o).toZFSet ∈ level R (o+1) ∧ | |
| ∀ a : Ordinal.{u}, a.toZFSet ∈ level R (o+1) ↔ a ≤ ordinalHeight (seed R) + o := by | |
| have h (a : Ordinal.{u}) : a.toZFSet ∈ level R (o+1) ↔ a ≤ ordinalHeight (seed R) + o := by | |
| rw [ordinal_mem_level_iff,← add_assoc] | |
| exact Order.lt_add_one_iff | |
| exact ⟨(h _).mpr le_rfl,h⟩ | |
| theorem largest_ordinal_at_difference (R : ZFSet.{u}) (a : Ordinal.{u}) | |
| (ha : ordinalHeight (seed R) ≤ a) : | |
| a.toZFSet ∈ level R (a - ordinalHeight (seed R) + 1) ∧ | |
| ∀ b : Ordinal.{u}, b.toZFSet ∈ level R (a - ordinalHeight (seed R) + 1) ↔ b ≤ a := by | |
| simpa only [Ordinal.add_sub_cancel_of_le ha] using | |
| largest_ordinal_level_succ R (a - ordinalHeight (seed R)) | |
| end TuringRigidity.RelativeConstructible | |
| end OAI | |