Prove m ≤ n & # 8594; k ≤ l & # 8594; m + k ≤ n + l in Agda
I want to prove
{m n k l : ℕ} -> m ≤ n -> k ≤ l -> m + k ≤ n + l
in Agda. I can prove with the m + k ≤ m + l
following code
add≤ : {m n : ℕ} -> (k : ℕ) -> m ≤ n -> k + m ≤ k + n
add≤ zero exp = exp
add≤ (suc k) exp = s≤s (add≤ k exp)
Since I can prove it m + k ≤ m + l
, I want to prove it m + l ≤ n + l
. If I can do this, I will use the ≤-trans : Transitive _≤_
one I have already defined.
Can I prove m + l ≤ n + l
with help m ≤ n, k ≤ l
? Or, should I change my usage plan ≤-trans
?
+3
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