diff --git a/CHANGELOG.md b/CHANGELOG.md
index e56fc6dcd8..01966abff4 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1790,6 +1790,11 @@ Other minor changes
 
   ℕ-ℕ≡toℕ‿ℕ-         : n ℕ-ℕ i ≡ toℕ (n ℕ- i)
 
+  punchIn-mono-≤     : ∀ i (j k : Fin n) → j ≤ k → punchIn i j ≤ punchIn i k
+  punchIn-cancel-≤   : ∀ i (j k : Fin n) → punchIn i j ≤ punchIn i k → j ≤ k
+  punchOut-mono-≤    : (i≢j : i ≢ j) (i≢k : i ≢ k) → j ≤ k → punchOut i≢j ≤ punchOut i≢k
+  punchOut-cancel-≤  : (i≢j : i ≢ j) (i≢k : i ≢ k) → punchOut i≢j ≤ punchOut i≢k → j ≤ k
+
   lower₁-injective   : lower₁ i n≢i ≡ lower₁ j n≢j → i ≡ j
   pinch-injective    : suc i ≢ j → suc i ≢ k → pinch i j ≡ pinch i k → j ≡ k
 
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index cb19ae7ab4..f730458038 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -802,6 +802,16 @@ punchIn-injective (suc i) (suc j) (suc k) ↑j+1≡↑k+1 =
 punchInᵢ≢i : ∀ i (j : Fin n) → punchIn i j ≢ i
 punchInᵢ≢i (suc i) (suc j) = punchInᵢ≢i i j ∘ suc-injective
 
+punchIn-mono-≤ : ∀ i (j k : Fin n) → j ≤ k → punchIn i j ≤ punchIn i k
+punchIn-mono-≤ zero    _        _      j≤k       = s≤s j≤k
+punchIn-mono-≤ (suc _) zero    _       z≤n       = z≤n
+punchIn-mono-≤ (suc i) (suc j) (suc k) (s≤s j≤k) = s≤s (punchIn-mono-≤ i j k j≤k)
+
+punchIn-cancel-≤ : ∀ i (j k : Fin n) → punchIn i j ≤ punchIn i k → j ≤ k
+punchIn-cancel-≤ zero    _       _       (s≤s j≤k)   = j≤k
+punchIn-cancel-≤ (suc _) zero    _       z≤n         = z≤n
+punchIn-cancel-≤ (suc i) (suc j) (suc k) (s≤s ↑j≤↑k) = s≤s (punchIn-cancel-≤ i j k ↑j≤↑k)
+
 ------------------------------------------------------------------------
 -- punchOut
 ------------------------------------------------------------------------
@@ -835,6 +845,22 @@ punchOut-injective {suc n} {suc i}  {zero}  {zero}  _   _    _    = refl
 punchOut-injective {suc n} {suc i}  {suc j} {suc k} i≢j i≢k pⱼ≡pₖ =
   cong suc (punchOut-injective (i≢j ∘ cong suc) (i≢k ∘ cong suc) (suc-injective pⱼ≡pₖ))
 
+punchOut-mono-≤ : ∀ {i j k : Fin (suc n)} (i≢j : i ≢ j) (i≢k : i ≢ k) →
+                  j ≤ k → punchOut i≢j ≤ punchOut i≢k
+punchOut-mono-≤ {_    } {zero } {zero } {_    } 0≢0 _   z≤n       = contradiction refl 0≢0
+punchOut-mono-≤ {_    } {zero } {suc _} {suc _} _   _   (s≤s j≤k) = j≤k
+punchOut-mono-≤ {suc _} {suc _} {zero } {_    } _   _   z≤n       = z≤n
+punchOut-mono-≤ {suc _} {suc _} {suc _} {suc _} i≢j i≢k (s≤s j≤k) = s≤s (punchOut-mono-≤ (i≢j ∘ cong suc) (i≢k ∘ cong suc) j≤k)
+
+punchOut-cancel-≤ : ∀ {i j k : Fin (suc n)} (i≢j : i ≢ j) (i≢k : i ≢ k) →
+                    punchOut i≢j ≤ punchOut i≢k → j ≤ k
+punchOut-cancel-≤ {_    } {zero } {zero } {_    } 0≢0 _   _           = contradiction refl 0≢0
+punchOut-cancel-≤ {_    } {zero } {suc _} {zero } _   0≢0 _           = contradiction refl 0≢0
+punchOut-cancel-≤ {suc _} {zero } {suc _} {suc _} _   _   pⱼ≤pₖ       = s≤s pⱼ≤pₖ
+punchOut-cancel-≤ {_    } {suc _} {zero } {_    } _   _   _           = z≤n
+punchOut-cancel-≤ {suc _} {suc _} {suc _} {zero } _   _   ()
+punchOut-cancel-≤ {suc _} {suc _} {suc _} {suc _} i≢j i≢k (s≤s pⱼ≤pₖ) = s≤s (punchOut-cancel-≤ (i≢j ∘ cong suc) (i≢k ∘ cong suc) pⱼ≤pₖ)
+
 punchIn-punchOut : ∀ {i j : Fin (suc n)} (i≢j : i ≢ j) →
                    punchIn i (punchOut i≢j) ≡ j
 punchIn-punchOut {_}     {zero}   {zero}  0≢0 = contradiction refl 0≢0