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c270adf | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 | 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
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