| import OAI.NumberTheory.Ostmann.Arithmetic.AtomIntervalRanges | |
| namespace OAI | |
| /-! # The original interval guards have bounded complexity at every finite depth -/ | |
| namespace Ostmann | |
| open scoped Classical | |
| theorem atomIntervalRanges_singleton {I : Type*} [Fintype I] | |
| (role : I → CopyScheduleRole) (lo hi : I → ℕ) (n : ℕ) | |
| (r : ScheduleAtomRange role n) (hr : r ∈ atomIntervalRanges role lo hi n) : | |
| r.atoms.length = 1 := by | |
| obtain ⟨v, _, rfl⟩ := List.mem_map.mp hr | |
| rfl | |
| theorem atomIntervalRanges_length_le {I : Type*} [Fintype I] | |
| (role : I → CopyScheduleRole) (lo hi : I → ℕ) (k n : ℕ) (hn : n ≤ k) : | |
| (atomIntervalRanges role lo hi n).length ≤ 3 ^ k * Fintype.card I := by | |
| have he : (atomIntervalRanges role lo hi n).length = Fintype.card (CopyScheduleAtoms role n) := by | |
| simp only [atomIntervalRanges, List.length_map, Finset.length_toList, Finset.card_univ] | |
| rw [he] | |
| exact (copyScheduleAtoms_card_le role n).trans | |
| (Nat.mul_le_mul_right _ (Nat.pow_le_pow_right (by omega : 1 ≤ 3) hn)) | |
| end Ostmann | |
| end OAI | |