feat: add Nat.gcd_left_comm and Int.gcd_left_comm (#11901)

This PR adds `gcd_left_comm` lemmas for both `Nat` and `Int`:

- `Nat.gcd_left_comm`: `gcd m (gcd n k) = gcd n (gcd m k)`
- `Int.gcd_left_comm`: `gcd a (gcd b c) = gcd b (gcd a c)`

These lemmas establish the left-commutativity property for gcd,
complementing the existing `gcd_comm` and `gcd_assoc` lemmas.

Upstreamed from
https://github.com/leanprover-community/mathlib4/pull/33235

🤖 Prepared with Claude Code

Co-authored-by: Claude Opus 4.5 <noreply@anthropic.com>

src/Init/Data/Int/Gcd.lean

@@ -113,6 +113,8 @@ theorem gcd_eq_right_iff_dvd (hb : 0 ≤ b) : gcd a b = b ↔ b ∣ a := by
 
 theorem gcd_assoc (a b c : Int) : gcd (gcd a b) c = gcd a (gcd b c) := Nat.gcd_assoc ..
 
+theorem gcd_left_comm (a b c : Int) : gcd a (gcd b c) = gcd b (gcd a c) := Nat.gcd_left_comm ..
+
 theorem gcd_mul_left (m n k : Int) : gcd (m * n) (m * k) = m.natAbs * gcd n k := by
   simp [gcd_eq_natAbs_gcd_natAbs, Nat.gcd_mul_left, natAbs_mul]
 

src/Init/Data/Nat/Gcd.lean

@@ -129,6 +129,9 @@ theorem gcd_assoc (m n k : Nat) : gcd (gcd m n) k = gcd m (gcd n k) :=
       (Nat.dvd_trans (gcd_dvd_right m (gcd n k)) (gcd_dvd_right n k)))
 instance : Std.Associative gcd := ⟨gcd_assoc⟩
 
+theorem gcd_left_comm (m n k : Nat) : gcd m (gcd n k) = gcd n (gcd m k) := by
+  rw [← gcd_assoc, ← gcd_assoc, gcd_comm m n]
+
 @[simp] theorem gcd_one_right (n : Nat) : gcd n 1 = 1 := (gcd_comm n 1).trans (gcd_one_left n)
 
 theorem gcd_mul_left (m n k : Nat) : gcd (m * n) (m * k) = m * gcd n k := by