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[Merged by Bors] - refactor(NumberTheory): golf Mathlib/NumberTheory/PythagoreanTriples #38454

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yuanyi-350 wants to merge 2 commits into from
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[Merged by Bors] - refactor(NumberTheory): golf Mathlib/NumberTheory/PythagoreanTriples #38454

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refactor(NumberTheory): golf `Mathlib/NumberTheory/PythagoreanTriples`
yuanyi-350 Apr 24, 2026
93a8c76
Refactor handling of zero in Pythagorean triples
yuanyi-350 Apr 25, 2026
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26 changes: 5 additions & 21 deletions Mathlib/NumberTheory/PythagoreanTriples.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -143,16 +143,11 @@ theorem even_odd_of_coprime (hc : Int.gcd x y = 1) :

theorem gcd_dvd : (Int.gcd x y : ℤ) ∣ z := by
by_cases h0 : Int.gcd x y = 0
· have hx : x = 0 := by
apply Int.natAbs_eq_zero.mp
apply Nat.eq_zero_of_gcd_eq_zero_left h0
have hy : y = 0 := by
apply Int.natAbs_eq_zero.mp
apply Nat.eq_zero_of_gcd_eq_zero_right h0
· obtain ⟨hx, hy⟩ := Int.gcd_eq_zero_iff.mp h0
have hz : z = 0 := by
simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
or_self_iff] using h
simp only [hz, dvd_zero]
simp [h0, hz]
Comment on lines -155 to +150
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@Vierkantor Vierkantor Apr 27, 2026

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This change by itself would not be worth including. I have a weak preference towards simp at the end of tactic blocks so it is not a regression, but it is not much clearer either.

Since it is included in a set of good changes, it can lift along.

obtain ⟨k, x0, y0, _, h2, rfl, rfl⟩ :
∃ (k : ℕ) (x0 y0 : _), 0 < k ∧ Int.gcd x0 y0 = 1 ∧ x = x0 * k ∧ y = y0 * k :=
Int.exists_gcd_one' (Nat.pos_of_ne_zero h0)
Expand All @@ -163,17 +158,11 @@ theorem gcd_dvd : (Int.gcd x y : ℤ) ∣ z := by

theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by
by_cases h0 : Int.gcd x y = 0
· have hx : x = 0 := by
apply Int.natAbs_eq_zero.mp
apply Nat.eq_zero_of_gcd_eq_zero_left h0
have hy : y = 0 := by
apply Int.natAbs_eq_zero.mp
apply Nat.eq_zero_of_gcd_eq_zero_right h0
· obtain ⟨hx, hy⟩ := Int.gcd_eq_zero_iff.mp h0
have hz : z = 0 := by
simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
or_self_iff] using h
simp only [hx, hy, hz]
exact zero
simpa [h0, hx, hy, hz] using zero
rcases h.gcd_dvd with ⟨z0, rfl⟩
obtain ⟨k, x0, y0, k0, h2, rfl, rfl⟩ :
∃ (k : ℕ) (x0 y0 : _), 0 < k ∧ Int.gcd x0 y0 = 1 ∧ x = x0 * k ∧ y = y0 * k :=
Expand Down Expand Up @@ -528,12 +517,7 @@ theorem isPrimitiveClassified_of_coprime (hc : Int.gcd x y = 1) : h.IsPrimitiveC

theorem classified : h.IsClassified := by
by_cases h0 : Int.gcd x y = 0
· have hx : x = 0 := by
apply Int.natAbs_eq_zero.mp
apply Nat.eq_zero_of_gcd_eq_zero_left h0
have hy : y = 0 := by
apply Int.natAbs_eq_zero.mp
apply Nat.eq_zero_of_gcd_eq_zero_right h0
· obtain ⟨hx, hy⟩ := Int.gcd_eq_zero_iff.mp h0
use 0, 1, 0
simp [hx, hy]
apply h.isClassified_of_normalize_isPrimitiveClassified
Expand Down
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