Lexicographic ordering of pairs/lists in Agda using the standard library - standard-library

Question

The Agda standard library contains some modules Relation.Binary.*.(Non)StrictLex (currently only for Product and List). We can use these modules to easily construct an instance of, for example, IsStrictTotalOrder for pairs of natural numbers (i.e. ℕ × ℕ).
open import Data.Nat as ℕ using (ℕ; _<_)
open import Data.Nat.Properties as ℕ
open import Relation.Binary using (module StrictTotalOrder; IsStrictTotalOrder)
open import Relation.Binary.PropositionalEquality using (_≡_)
open import Relation.Binary.Product.StrictLex using (×-Lex; _×-isStrictTotalOrder_)
open import Relation.Binary.Product.Pointwise using (_×-Rel_)
ℕ-isSTO : IsStrictTotalOrder _≡_ _<_
ℕ-isSTO = StrictTotalOrder.isStrictTotalOrder ℕ.strictTotalOrder
ℕ×ℕ-isSTO : IsStrictTotalOrder (_≡_ ×-Rel _≡_) (×-Lex _≡_ _<_ _<_)
ℕ×ℕ-isSTO = ℕ-isSTO ×-isStrictTotalOrder ℕ-isSTO
This creates an instance using the pointwise equality _≡_ ×-Rel _≡_. In the case of propositional equality, this should be equivalent to using just propositional equality.
Is there an easy way of converting the instance above to an instance of type IsStrictTotalOrder _≡_ (×-Lex _≡_ _<_ _<_), using normal propositional equality?

Answer 1

The kit required isn't too hard to assemble:
open import Data.Product
open import Function using (_∘_; case_of_)
open import Relation.Binary
_⇔₂_ : ∀ {a ℓ₁ ℓ₂} {A : Set a} → Rel A ℓ₁ → Rel A ℓ₂ → Set _
_≈_ ⇔₂ _≈′_ = (∀ {x y} → x ≈ y → x ≈′ y) × (∀ {x y} → x ≈′ y → x ≈ y)
-- I was unable to write this nicely using Data.Product.map...
-- hence it is moved here to a toplevel where it can pattern-match
-- on the product of proofs
transform-resp : ∀ {a ℓ₁ ℓ₂ ℓ} {A : Set a} {≈ : Rel A ℓ₁} {≈′ : Rel A ℓ₂} {< : Rel A ℓ} →
≈ ⇔₂ ≈′ →
< Respects₂ ≈ → < Respects₂ ≈′
transform-resp (to , from) = λ { (resp₁ , resp₂) → (resp₁ ∘ from , resp₂ ∘ from) }
transform-isSTO : ∀ {a ℓ₁ ℓ₂ ℓ} {A : Set a} {≈ : Rel A ℓ₁} {≈′ : Rel A ℓ₂} {< : Rel A ℓ} →
≈ ⇔₂ ≈′ →
IsStrictTotalOrder ≈ < → IsStrictTotalOrder ≈′ <
transform-isSTO {≈′ = ≈′} {< = <} (to , from) isSTO = record
{ isEquivalence = let open IsEquivalence (IsStrictTotalOrder.isEquivalence isSTO)
in record { refl = to refl
; sym = to ∘ sym ∘ from
; trans = λ x y → to (trans (from x) (from y))
}
; trans = IsStrictTotalOrder.trans isSTO
; compare = compare
; <-resp-≈ = transform-resp (to , from) (IsStrictTotalOrder.<-resp-≈ isSTO)
}
where
compare : Trichotomous ≈′ <
compare x y with IsStrictTotalOrder.compare isSTO x y
compare x y | tri< a ¬b ¬c = tri< a (¬b ∘ from) ¬c
compare x y | tri≈ ¬a b ¬c = tri≈ ¬a (to b) ¬c
compare x y | tri> ¬a ¬b c = tri> ¬a (¬b ∘ from) c
Then we can use this to solve your original problem:
ℕ×ℕ-isSTO′ : IsStrictTotalOrder _≡_ (×-Lex _≡_ _<_ _<_)
ℕ×ℕ-isSTO′ = transform-isSTO (to , from) ℕ×ℕ-isSTO
where
open import Function using (_⟨_⟩_)
open import Relation.Binary.PropositionalEquality
to : ∀ {a b} {A : Set a} {B : Set b}
{x x′ : A} {y y′ : B} → (x , y) ⟨ _≡_ ×-Rel _≡_ ⟩ (x′ , y′) → (x , y) ≡ (x′ , y′)
to (refl , refl) = refl
from : ∀ {a b} {A : Set a} {B : Set b}
{x x′ : A} {y y′ : B} → (x , y) ≡ (x′ , y′) → (x , y) ⟨ _≡_ ×-Rel _≡_ ⟩ (x′ , y′)
from refl = refl , refl