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[Merged by Bors] - feat(Data/NNReal): nnabs n = n for n : ℕ #39577

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[Merged by Bors] - feat(Data/NNReal): nnabs n = n for n : ℕ #39577

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feat(Data/NNReal): `nnabs n = n` for `n : ℕ`
YaelDillies May 19, 2026
89df590
simp high
YaelDillies May 20, 2026
851c7fb
ofNat too
YaelDillies May 20, 2026
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9 changes: 7 additions & 2 deletions Mathlib/Data/NNReal/Defs.lean
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Expand Up @@ -355,13 +355,15 @@ theorem mk_natCast (n : ℕ) : NNReal.mk (n : ℝ) (n.cast_nonneg) = n :=
NNReal.eq (NNReal.coe_natCast n).symm

@[simp]
theorem _root_.Real.toNNReal_coe_nat (n : ℕ) : Real.toNNReal n = n :=
theorem _root_.Real.toNNReal_natCast (n : ℕ) : Real.toNNReal n = n :=
NNReal.eq <| by simp [Real.coe_toNNReal]

@[deprecated (since := "2026-05-19")] alias _root_.Real.toNNReal_coe_nat := Real.toNNReal_natCast

@[simp]
theorem _root_.Real.toNNReal_ofNat (n : ℕ) [n.AtLeastTwo] :
Real.toNNReal ofNat(n) = OfNat.ofNat n :=
Real.toNNReal_coe_nat n
Real.toNNReal_natCast n

/-- `Real.toNNReal` and `NNReal.toReal : ℝ≥0 → ℝ` form a Galois insertion. -/
def gi : GaloisInsertion Real.toNNReal (↑) :=
Expand Down Expand Up @@ -910,6 +912,9 @@ theorem coe_toNNReal_le (x : ℝ) : (toNNReal x : ℝ) ≤ |x| :=

@[simp] lemma toNNReal_abs (x : ℝ) : |x|.toNNReal = nnabs x := NNReal.coe_injective <| by simp

@[simp high] lemma nnabs_natCast (n : ℕ) : nnabs n = n := by simp
@[simp high] lemma nnabs_ofNat (n : ℕ) [n.AtLeastTwo] : nnabs ofNat(n) = ofNat(n) := by simp

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theorem cast_natAbs_eq_nnabs_cast (n : ℤ) : (n.natAbs : ℝ≥0) = nnabs n := by
ext
rw [NNReal.coe_natCast, Nat.cast_natAbs, Real.coe_nnabs, Int.cast_abs]
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