transitivity (tri n₂) using LT.lt, LE.le
case h.right
m₁ n₁ m₂ n₂ : Nat
h₁ : m₁ ≤ n₁
h₂ : m₂ ≤ n₂
h : n₁.tri + m₁ = n₂.tri + m₂
x✝ : Decidable (n₁ ≤ n₂)
hlt : ¬n₁ ≥ n₂
⊢ n₁.tri + n₁ < n₂.tri + m₂
case h.right.left
m₁ n₁ m₂ n₂ : Nat
h₁ : m₁ ≤ n₁
h₂ : m₂ ≤ n₂
h : n₁.tri + m₁ = n₂.tri + m₂
x✝ : Decidable (n₁ ≤ n₂)
hlt : ¬n₁ ≥ n₂
⊢ n₁.tri + n₁ < n₂.tri

case h.right.right
m₁ n₁ m₂ n₂ : Nat
h₁ : m₁ ≤ n₁
h₂ : m₂ ≤ n₂
h : n₁.tri + m₁ = n₂.tri + m₂
x✝ : Decidable (n₁ ≤ n₂)
hlt : ¬n₁ ≥ n₂
⊢ n₂.tri ≤ n₂.tri + m₂