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[Merged by Bors] - chore: use lt_add_one to simplify proof of exists_abs_lt #11046

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[Merged by Bors] - chore: use lt_add_one to simplify proof of exists_abs_lt #11046

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7 changes: 2 additions & 5 deletions Mathlib/Algebra/Order/Ring/Abs.lean
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Expand Up @@ -158,11 +158,8 @@ lemma sq_eq_sq_iff_abs_eq_abs (a b : α) : a ^ 2 = b ^ 2 ↔ |a| = |b| := by
simpa only [one_pow, abs_one] using @sq_lt_sq _ _ 1 a
#align one_lt_sq_iff_one_lt_abs one_lt_sq_iff_one_lt_abs

lemma exists_abs_lt {α : Type*} [LinearOrderedRing α] (a : α) : ∃ b > 0, |a| < b := by
refine ⟨|a| + 1, lt_of_lt_of_le zero_lt_one <| by simp, ?_⟩
cases' le_or_lt 0 a with ht ht
· simp only [abs_of_nonneg ht, lt_add_iff_pos_right, zero_lt_one]
· simp only [abs_of_neg ht, lt_add_iff_pos_right, zero_lt_one]
lemma exists_abs_lt {α : Type*} [LinearOrderedRing α] (a : α) : ∃ b > 0, |a| < b :=
⟨|a| + 1, lt_of_lt_of_le zero_lt_one <| by simp, lt_add_one |a|⟩

end LinearOrderedRing

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