Progress towards UFDs (#88)

Lists/Monad.agda

@@ -0,0 +1,28 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Functions
+open import Lists.Definition
+open import Lists.Fold.Fold
+open import Lists.Concat
+open import Lists.Length
+open import Numbers.Naturals.Semiring
+
+module Lists.Monad where
+
+open import Lists.Map.Map public
+
+flatten : {a : _} {A : Set a} → (l : List (List A)) → List A
+flatten [] = []
+flatten (l :: ls) = l ++ flatten ls
+
+flatten' : {a : _} {A : Set a} → (l : List (List A)) → List A
+flatten' = fold _++_ []
+
+flatten=flatten' : {a : _} {A : Set a} (l : List (List A)) → flatten l ≡ flatten' l
+flatten=flatten' [] = refl
+flatten=flatten' (l :: ls) = applyEquality (l ++_) (flatten=flatten' ls)
+
+lengthFlatten : {a : _} {A : Set a} (l : List (List A)) → length (flatten l) ≡ (fold _+N_ zero (map length l))
+lengthFlatten [] = refl
+lengthFlatten (l :: ls) rewrite lengthConcat l (flatten ls) | lengthFlatten ls = refl