∀ {n : Type u_1} {R : Type u_2} [inst : Ring R] [inst_1 : LinearOrder R] {A : Matrix n n R} [inst_2 : Fintype n]
  [IsOrderedRing R] [PosMulStrictMono R] [Nontrivial R] [inst_6 : DecidableEq n],
  (∀ (i j : n), 0 ≤ A i j) → (A.IsIrreducible ↔ ∀ (i j : n), ∃ k > 0, 0 < (A ^ k) i j)