[ base ] implement decEq for SnocList (#2630)

Co-authored-by: Guillaume Allais <guillaume.allais@ens-lyon.org>

libs/base/Control/Function.idr

@@ -36,7 +36,7 @@ public export
 ||| This is more strict than injectivity on each of arguments.
 ||| For instance, list appending is injective on both arguments but is not biinjective.
 public export
-interface Biinjective (f : a -> b -> c) | f where
+interface Biinjective (0 f : a -> b -> c) | f where
   constructor MkBiinjective
   biinjective : {x, y : a} -> {v, w : b} -> f x v = f y w -> (x = y, v = w)
 

libs/base/Data/SnocList.idr

@@ -21,6 +21,14 @@ public export
 asList : SnocList type -> List type
 asList = (reverse . cast)
 
+export
+Uninhabited (Lin = x :< xs) where
+  uninhabited Refl impossible
+
+export
+Uninhabited (x :< xs = Lin) where
+  uninhabited Refl impossible
+
 ||| True iff input is Lin
 public export
 isLin : SnocList a -> Bool
@@ -87,6 +95,11 @@ Alternative SnocList where
   empty = Lin
   xs <|> ys = xs ++ ys
 
+-- Why can't we just use an implementation here?!
+export %hint
+SnocBiinjective : Biinjective (:<)
+SnocBiinjective = MkBiinjective $ \case Refl => (Refl, Refl)
+
 ||| Find the rightmost element of the snoc-list that satisfies the predicate.
 public export
 find : (a -> Bool) -> SnocList a -> Maybe a

libs/base/Decidable/Equality.idr

@@ -7,6 +7,7 @@ import Data.Nat
 import Data.List
 import Data.List1
 import Data.List1.Properties
+import Data.SnocList
 import Data.These
 
 import public Decidable.Equality.Core as Decidable.Equality
@@ -111,6 +112,17 @@ public export
 DecEq a => DecEq (List1 a) where
   decEq (x ::: xs) (y ::: ys) = decEqCong2 (decEq x y) (decEq xs ys)
 
+--------------------------------------------------------------------------------
+-- SnocList
+--------------------------------------------------------------------------------
+
+public export
+DecEq a => DecEq (SnocList a) where
+  decEq Lin Lin = Yes Refl
+  decEq (xs :< x) Lin = No absurd
+  decEq Lin (xs :< x) = No absurd
+  decEq (xs :< x) (ys :< y) = decEqCong2 (decEq xs ys) (decEq x y)
+
 -- TODO: Other prelude data types
 
 -- For the primitives, we have to cheat because we don't have access to their