Add foldrImplGoLemma to Data.Vect (#2835)

Co-authored-by: CodingCellist <teh6@st-andrews.ac.uk>

CHANGELOG.md

@@ -154,6 +154,7 @@
   assumption in `setByte` that the value is between 0 and 255.
 
 * Adds RefC support for 16- and 32-bit access in `Data.Buffer`.
+
 * Add `Show` instance to `Data.Vect.Quantifiers.All` and add a few helpers for listy
   computations on the `All` type.
 * Add an alias for `HVect` to `All id` in `Data.Vect.Quantifiers.All`. This is the
@@ -248,6 +249,8 @@
 * A more generalised way of applicative mapping of `TTImp` expression was added,
   called `mapATTImp`; the original `mapMTTimp` was implemented through the new one.
 
+* Adds `Data.Vect.foldrImplGoLemma`.
+
 #### System
 
 * Changes `getNProcessors` to return the number of online processors rather than

CONTRIBUTORS

@@ -7,6 +7,7 @@ Aleksei Volkov
 Alex Gryzlov
 Alex Silva
 Alissa Tung
+Andor Penzes
 Andre Kuhlenschmidt
 André Videla
 Andy Lok

libs/base/Data/Vect.idr

@@ -6,8 +6,9 @@ import Data.Nat
 import public Data.Fin
 import public Data.Zippable
 
-import Decidable.Equality
 import Control.Function
+import Decidable.Equality
+import Syntax.PreorderReasoning
 
 %default total
 
@@ -413,6 +414,17 @@ foldrImpl : (t -> acc -> acc) -> acc -> (acc -> acc) -> Vect n t -> acc
 foldrImpl f e go [] = go e
 foldrImpl f e go (x::xs) = foldrImpl f e (go . (f x)) xs
 
+export
+foldrImplGoLemma
+  :  (x : a) -> (xs : Vect n a) -> (f : a -> b -> b) -> (e : b) -> (go : b -> b)
+  -> go (foldrImpl f e (f x) xs) === foldrImpl f e (go . (f x)) xs
+foldrImplGoLemma z []        f e go = Refl
+foldrImplGoLemma z (y :: ys) f e go = Calc $
+  |~ go (foldrImpl f e ((f z) . (f y)) ys)
+  ~~ go ((f z) (foldrImpl f e (f y) ys))   ... (cong go (sym (foldrImplGoLemma y ys f e (f z))))
+  ~~ (go . (f z)) (foldrImpl f e (f y) ys) ... (cong go Refl)
+  ~~ foldrImpl f e (go . (f z) . (f y)) ys ... (foldrImplGoLemma y ys f e (go . (f z)))
+
 public export
 implementation Foldable (Vect n) where
   foldr f e xs = foldrImpl f e id xs