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[Merged by Bors] - feat(Order/Floor): x - 1 / 2 < round x #14964

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[Merged by Bors] - feat(Order/Floor): x - 1 / 2 < round x #14964

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76a32c4
Update Floor.lean
Colin166 Jul 21, 2024
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Update Mathlib/Algebra/Order/Floor.lean
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Update Mathlib/Algebra/Order/Floor.lean
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8 changes: 8 additions & 0 deletions Mathlib/Algebra/Order/Floor.lean
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Expand Up @@ -1395,6 +1395,14 @@ theorem abs_sub_round_div_natCast_eq {m n : ℕ} :
rw [abs_sub_round_eq_min, Nat.cast_min, ← min_div_div_right hn'.le,
fract_div_natCast_eq_div_natCast_mod, Nat.cast_sub (m.mod_lt hn).le, sub_div, div_self hn'.ne']

theorem sub_half_lt_round (x : α) : x - 1 / 2 < round x := by
rw [round_eq x, show x - 1 / 2 = x + 1 / 2 - 1 by nlinarith]
exact Int.sub_one_lt_floor (x + 1 / 2)

theorem round_le_add_half (x : α) : round x ≤ x + 1 / 2 := by
rw [round_eq x]
exact Int.floor_le (x + 1 / 2)

end LinearOrderedField

end round
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