module Cat.Solver where
Solver for Categories🔗
This module is split pretty cleanly into two halves: the first half implements an algorithm for reducing, in a systematic way, problems involving associativity and identity of composition in a precategory. The latter half, significantly more cursed, uses this infrastructure to automatically solve equality goals of this form.
With a precategory in hand, we start by defining a language of composition.
module NbE (Cat : Precategory o h) where open Precategory Cat
data Expr : Ob → Ob → Type (o ⊔ h) where `id : Expr A A _`∘_ : Expr B C → Expr A B → Expr A C _↑ : Hom A B → Expr A B infixr 40 _`∘_ infix 50 _↑
A term of type Expr represents, in a symbolic way, a composite of morphisms in our category . What this means is that, while is some unknowable inhabitant of Hom, represents an inhabitant of Hom which is known to be a composition of (the trees that represent) and . We can now define “two” ways of computing the morphism that an Expr represents. The first is a straightforward embedding:
embed : Expr A B → Hom A B embed `id = id embed (f ↑) = f embed (f `∘ g) = embed f ∘ embed g
The second computation is a bit less obvious. If you’re a programmer, it should be familiar under the name “continuation passing style”. Categorically, it can be seen as embedding into the presheaf category of . In either case, the difference is that instead of computing a single morphism, we compute a transformation of hom-spaces:
eval : Expr B C → Hom A B → Hom A C eval `id f = f eval (f ↑) g = f ∘ g eval (f `∘ g) h = eval f (eval g h) nf : Expr A B → Hom A B nf e = eval e id
Working this out in a back-of-the-envelope calculation, one sees that
eval f id should compute the same morphism as
embed f. Indeed, that’s the case! Since embed is the “intended semantics”, and eval is an “optimised evaluator”, we call
this result soundness. We can prove it by induction on the
expression, by first generalising over id:
eval-sound-k : (e : Expr B C) (f : Hom A B) → eval e f ≡ embed e ∘ f eval-sound-k `id f = sym (idl _) -- f ≡ id ∘ f eval-sound-k (f `∘ g) h = eval f (eval g h) ≡⟨ eval-sound-k f _ ⟩≡ embed f ∘ eval g h ≡⟨ ap (embed f ∘_) (eval-sound-k g _) ⟩≡ embed f ∘ embed g ∘ h ≡⟨ assoc _ _ _ ⟩≡ (embed f ∘ embed g) ∘ h ∎ eval-sound-k (x ↑) f = refl -- x ∘ f ≡ x ∘ f eval-sound : (e : Expr A B) → nf e ≡ embed e eval-sound e = eval-sound-k e id ∙ idr _
We now have a general theorem for solving associativity and identity problems! If two expressions compute the same transformation of hom-sets, then they represent the same morphism.
abstract solve : (f g : Expr A B) → nf f ≡ nf g → embed f ≡ embed g solve f g p = sym (eval-sound f) ·· p ·· (eval-sound g) solve-filler : (f g : Expr A B) → (p : nf f ≡ nf g) → Square (eval-sound f) p (solve f g p) (eval-sound g) solve-filler f g p j i = ··-filler (sym (eval-sound f)) p (eval-sound g) j i
The cursed part🔗
module Reflection where pattern category-args xs = _ hm∷ _ hm∷ _ v∷ xs pattern “id” = def (quote Precategory.id) (category-args (_ h∷ [])) pattern “∘” f g = def (quote Precategory._∘_) (category-args (_ h∷ _ h∷ _ h∷ f v∷ g v∷ [])) mk-category-args : Term → List (Arg Term) → List (Arg Term) mk-category-args cat xs = unknown h∷ unknown h∷ cat v∷ xs “solve” : Term → Term → Term → Term “solve” cat lhs rhs = def (quote NbE.solve) (mk-category-args cat $ infer-hidden 2 $ lhs v∷ rhs v∷ def (quote refl) [] v∷ []) “nf” : Term → Term → Term “nf” cat e = def (quote NbE.nf) (mk-category-args cat $ infer-hidden 2 $ e v∷ []) build-expr : Term → Term build-expr “id” = con (quote NbE.`id) [] build-expr (“∘” f g) = con (quote NbE._`∘_) (build-expr f v∷ build-expr g v∷ [] ) build-expr f = con (quote NbE._↑) (f v∷ []) dont-reduce : List Name dont-reduce = quote Precategory.id ∷ quote Precategory._∘_ ∷ [] cat-solver : Term → SimpleSolver cat-solver cat .SimpleSolver.dont-reduce = dont-reduce cat-solver cat .SimpleSolver.build-expr tm = returnTC $ build-expr tm cat-solver cat .SimpleSolver.invoke-solver = “solve” cat cat-solver cat .SimpleSolver.invoke-normaliser = “nf” cat repr-macro : Term → Term → Term → TC ⊤ repr-macro cat f _ = mk-simple-repr (cat-solver cat) f simplify-macro : Term → Term → Term → TC ⊤ simplify-macro cat f hole = mk-simple-normalise (cat-solver cat) f hole solve-macro : Term → Term → TC ⊤ solve-macro cat hole = mk-simple-solver (cat-solver cat) hole macro repr-cat! : Term → Term → Term → TC ⊤ repr-cat! cat f = Reflection.repr-macro cat f simpl-cat! : Term → Term → Term → TC ⊤ simpl-cat! cat f = Reflection.simplify-macro cat f cat! : Term → Term → TC ⊤ cat! = Reflection.solve-macro
Demo🔗
As a quick demonstration (and sanity check/future proofing/integration testing/what have you):
module _ (C : Precategory o h) where private module C = Precategory C variable A B : C.Ob a b c d : C.Hom A B test : a C.∘ (b C.∘ (c C.∘ C.id) C.∘ C.id C.∘ (d C.∘ C.id)) ≡ a C.∘ b C.∘ c C.∘ d test = cat! C