| import Mathlib | |
| namespace OAI | |
| namespace LargeIndependentSets.ShortestPaths | |
| open scoped ENNReal | |
| lemma integer_edge_test {D : ℕ} [NeZero D] (c : ℕ∞) : | |
| (c:ℝ≥0∞)/(D:ℝ≥0∞) ≤ ENNReal.ofReal (1/8) ↔ 8*c ≤ (D:ℕ∞) := by | |
| induction c using ENat.recTopCoe with | |
| | top => simp [ENNReal.top_div_of_ne_top (ENNReal.natCast_ne_top D)] | |
| | coe c => | |
| have hd : 0 < (D:ℝ) := by exact_mod_cast (NeZero.pos D) | |
| have hD : (D:ℝ≥0∞) ≠ 0 := by exact_mod_cast (NeZero.ne D) | |
| rw [← ENNReal.toReal_le_toReal (ENNReal.div_ne_top (by simp) hD) (by simp)] | |
| simp only [ENat.toENNReal_coe,ENNReal.toReal_div,ENNReal.toReal_natCast, | |
| ENNReal.toReal_ofReal (by norm_num : (0:ℝ) ≤ 1/8)] | |
| rw [div_le_iff₀ hd] | |
| have hr : (c:ℝ) ≤ 1/8*(D:ℝ) ↔ 8*(c:ℝ) ≤ (D:ℝ) := by constructor <;> intro h <;> linarith | |
| rw [hr] | |
| have he : (8:ℕ∞)*(c:ℕ∞) = ((8*c:ℕ):ℕ∞) := by norm_cast | |
| rw [he,ENat.natCast_le_natCast] | |
| exact_mod_cast Iff.rfl | |
| end LargeIndependentSets.ShortestPaths | |
| end OAI | |