Skip to content

[ add ] Nat lemmas with _∸_, _⊔_ and _⊓_ #2924

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
MatthewDaggitt merged 1 commit into from
Mar 3, 2026
Merged

[ add ] Nat lemmas with _∸_, _⊔_ and _⊓_ #2924

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
5 changes: 5 additions & 0 deletions CHANGELOG.md
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -309,6 +309,11 @@ Additions to existing modules
∸-suc : .(m ≤ n) → suc n ∸ m ≡ suc (n ∸ m)
^-distribʳ-* : ∀ m n o → (n * o) ^ m ≡ n ^ m * o ^ m
2*suc[n]≡2+n+n : ∀ n → 2 * (suc n) ≡ 2 + (n + n)
m∸n+o≡m∸[n∸o] : ∀ {m n o} → .(n ≤ m) → .(o ≤ n) → (m ∸ n) + o ≡ m ∸ (n ∸ o)
m∸n≤m⊔n : ∀ m n → m ∸ n ≤ m ⊔ n
m⊔n∸[m∸n]≡n : ∀ m n → m ⊔ n ∸ (m ∸ n) ≡ n
m⊔n≡m∸n+n : ∀ m n → m ⊔ n ≡ m ∸ n + n
∣m-n∣≡m⊔n∸m⊓n : ∀ m n → ∣ m - n ∣ ≡ m ⊔ n ∸ m ⊓ n
```

* In `Data.Product.Properties`:
Expand Down
39 changes: 39 additions & 0 deletions src/Data/Nat/Properties.agda
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -1640,6 +1640,15 @@ m≤n⇒n∸m≤n (s≤s m≤n) = m≤n⇒m≤1+n (m≤n⇒n∸m≤n m≤n)
suc ((m + n) ∸ o) ≡⟨ cong suc (+-∸-assoc m o≤n) ⟩
suc (m + (n ∸ o)) ∎

m∸n+o≡m∸[n∸o] : ∀ {m n o} → .(n ≤ m) → .(o ≤ n) → (m ∸ n) + o ≡ m ∸ (n ∸ o)
m∸n+o≡m∸[n∸o] {m} {zero} {zero} _ _ = +-identityʳ m
m∸n+o≡m∸[n∸o] {suc m} {suc n} {zero} _ _ = +-identityʳ (m ∸ n)
m∸n+o≡m∸[n∸o] {suc m} {suc n} {suc o} sn≤sm so≤sn = begin-equality
suc m ∸ suc n + suc o ≡⟨ +-suc (m ∸ n) o ⟩
suc (m ∸ n + o) ≡⟨ cong suc (m∸n+o≡m∸[n∸o] (s≤s⁻¹ sn≤sm) (s≤s⁻¹ so≤sn)) ⟩
suc (m ∸ (n ∸ o)) ≡⟨ ∸-suc (≤-trans (m∸n≤m n o) (s≤s⁻¹ sn≤sm)) ⟨
suc m ∸ (n ∸ o) ∎

m≤n+o⇒m∸n≤o : ∀ m n {o} → m ≤ n + o → m ∸ n ≤ o
m≤n+o⇒m∸n≤o m zero le = le
m≤n+o⇒m∸n≤o zero (suc n) _ = z≤n
Expand Down Expand Up @@ -1728,11 +1737,30 @@ even≢odd (suc m) (suc n) eq = even≢odd m n (suc-injective (begin-equality
------------------------------------------------------------------------
-- Properties of _∸_ and _⊓_ and _⊔_

m∸n≤m⊔n : ∀ m n → m ∸ n ≤ m ⊔ n
Copy link
Member

@Taneb Taneb Jan 27, 2026

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

This property doesn't feel natural to me, and the definition is short enough I don't really think it needs a name

Copy link
Contributor Author

@jkopanski jkopanski Jan 27, 2026

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

It looks kind of similar to the m⊔n≤m+n that is already here.

Copy link
Collaborator

@JacquesCarette JacquesCarette Feb 4, 2026

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

I'm okay - lets one do proofs at a higher level with fewer steps.

Copy link
Member

@Taneb Taneb Feb 9, 2026

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

Can you provide an example of an "interesting" proof where this lemma is a useful step?

Copy link
Contributor Author

@jkopanski jkopanski Feb 9, 2026

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

I don't know if that counts as interesting, but the only place where I used it is down below in the m⊔n∸[m∸n]≡n.

Just to be clear I'm fine with removing that. In this PR I've just included stuff that I've tried to lookup in the stdlib first and written down once I couldn't find it.

Copy link
Collaborator

@jamesmckinna jamesmckinna Feb 18, 2026

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

Having pondered this for a while, I'm leaning towards @Taneb 's position more than @JacquesCarette 's.

Copy link
Collaborator

@jamesmckinna jamesmckinna Feb 24, 2026 •
edited
Loading

Choose a reason for hiding this comment

The reason will be displayed to describe this comment to others. Learn more.

Namely, ... TL;DR: Fairbairn threshhold, I think. It's used only once.

m∸n≤m⊔n m n = ≤-trans (m∸n≤m m n) (m≤m⊔n m n)

m⊓n+n∸m≡n : ∀ m n → (m ⊓ n) + (n ∸ m) ≡ n
m⊓n+n∸m≡n zero n = refl
m⊓n+n∸m≡n (suc m) zero = refl
m⊓n+n∸m≡n (suc m) (suc n) = cong suc $ m⊓n+n∸m≡n m n

m⊔n∸[m∸n]≡n : ∀ m n → m ⊔ n ∸ (m ∸ n) ≡ n
m⊔n∸[m∸n]≡n zero n = cong (n ∸_) (0∸n≡0 n)
m⊔n∸[m∸n]≡n (suc m) zero = n∸n≡0 m
m⊔n∸[m∸n]≡n (suc m) (suc n) = begin-equality
suc (m ⊔ n) ∸ (m ∸ n) ≡⟨ ∸-suc (m∸n≤m⊔n m n) ⟩
suc ((m ⊔ n) ∸ (m ∸ n)) ≡⟨ cong suc (m⊔n∸[m∸n]≡n m n) ⟩
suc n ∎

m⊔n≡m∸n+n : ∀ m n → m ⊔ n ≡ m ∸ n + n
m⊔n≡m∸n+n zero n = sym (cong (_+ n) (0∸n≡0 n))
m⊔n≡m∸n+n (suc m) zero = sym (cong suc (+-identityʳ m))
m⊔n≡m∸n+n (suc m) (suc n) = begin-equality
suc (m ⊔ n) ≡⟨ cong suc (m⊔n≡m∸n+n m n) ⟩
suc (m ∸ n + n) ≡⟨ +-suc (m ∸ n) n ⟨
m ∸ n + suc n ∎

[m∸n]⊓[n∸m]≡0 : ∀ m n → (m ∸ n) ⊓ (n ∸ m) ≡ 0
[m∸n]⊓[n∸m]≡0 zero zero = refl
[m∸n]⊓[n∸m]≡0 zero (suc n) = refl
Expand Down Expand Up @@ -1850,6 +1878,17 @@ m∸n≤∣m-n∣ m n with ≤-total m n
∣m-n∣≤m⊔n (suc m) zero = ≤-refl
∣m-n∣≤m⊔n (suc m) (suc n) = m≤n⇒m≤1+n (∣m-n∣≤m⊔n m n)

∣m-n∣≡m⊔n∸m⊓n : ∀ m n → ∣ m - n ∣ ≡ m ⊔ n ∸ m ⊓ n
∣m-n∣≡m⊔n∸m⊓n m n with ≤-total m n
... | inj₁ m≤n = begin-equality
∣ m - n ∣ ≡⟨ m≤n⇒∣m-n∣≡n∸m m≤n ⟩
n ∸ m ≡⟨ cong₂ _∸_ (m≤n⇒m⊔n≡n m≤n) (m≤n⇒m⊓n≡m m≤n) ⟨
m ⊔ n ∸ m ⊓ n ∎
... | inj₂ n≤m = begin-equality
∣ m - n ∣ ≡⟨ m≤n⇒∣n-m∣≡n∸m n≤m ⟩
m ∸ n ≡⟨ cong₂ _∸_ (m≥n⇒m⊔n≡m n≤m) (m≥n⇒m⊓n≡n n≤m) ⟨
m ⊔ n ∸ m ⊓ n ∎

∣-∣-identityˡ : LeftIdentity 0 ∣_-_∣
∣-∣-identityˡ x = refl

Expand Down