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[Merged by Bors] - chore(Analysis/SpecialFunctions/Log/PosLog): golf a proof using grw #30704

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[Merged by Bors] - chore(Analysis/SpecialFunctions/Log/PosLog): golf a proof using grw #30704

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chore(Analysis/SpecialFunctions/Log/PosLog): golf a proof using `grw`
eric-wieser Oct 19, 2025
4f24a27
Update PosLog.lean
eric-wieser Oct 19, 2025
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12 changes: 5 additions & 7 deletions Mathlib/Analysis/SpecialFunctions/Log/PosLog.lean
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Expand Up @@ -91,6 +91,10 @@ theorem monotoneOn_posLog : MonotoneOn log⁺ (Set.Ici 0) := by
simp only [this, and_true, ge_iff_le]
linarith

@[gcongr]
lemma posLog_le_posLog {x y : ℝ} (hx : 0 ≤ x) (hxy : x ≤ y) : log⁺ x ≤ log⁺ y :=
monotoneOn_posLog hx (hx.trans hxy) hxy

/-!
## Estimates for Products
-/
Expand Down Expand Up @@ -169,12 +173,6 @@ monotonicity of `log⁺` and the triangle inequality.
-/
lemma posLog_norm_add_le {E : Type*} [NormedAddCommGroup E] (a b : E) :
log⁺ ‖a + b‖ ≤ log⁺ ‖a‖ + log⁺ ‖b‖ + log 2 := by
calc log⁺ ‖a + b‖
_ ≤ log⁺ (‖a‖ + ‖b‖) := by
apply monotoneOn_posLog _ _ (norm_add_le a b)
<;> simp [add_nonneg (norm_nonneg a) (norm_nonneg b)]
_ ≤ log⁺ ‖a‖ + log⁺ ‖b‖ + log 2 := by
convert posLog_add using 1
ring
grw [norm_add_le, posLog_add, add_rotate]

end Real
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