Progress towards UFDs (#88)

Lists/Concat.agda

@@ -0,0 +1,25 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Functions
+open import Lists.Definition
+open import Lists.Length
+open import Numbers.Naturals.Semiring
+
+module Lists.Concat where
+
+appendEmptyList : {a : _} → {A : Set a} → (l : List A) → (l ++ [] ≡ l)
+appendEmptyList [] = refl
+appendEmptyList (x :: l) = applyEquality (_::_ x) (appendEmptyList l)
+
+concatAssoc : {a : _} → {A : Set a} → (x : List A) → (y : List A) → (z : List A) → ((x ++ y) ++ z) ≡ x ++ (y ++ z)
+concatAssoc [] m n = refl
+concatAssoc (x :: l) m n = applyEquality (_::_ x) (concatAssoc l m n)
+
+canMovePrepend : {a : _} → {A : Set a} → (l : A) → (x : List A) (y : List A) → ((l :: x) ++ y ≡ l :: (x ++ y))
+canMovePrepend l [] n = refl
+canMovePrepend l (x :: m) n = refl
+
+lengthConcat : {a : _} {A : Set a} (l1 l2 : List A) → length (l1 ++ l2) ≡ length l1 +N length l2
+lengthConcat [] l2 = refl
+lengthConcat (x :: l1) l2 = applyEquality succ (lengthConcat l1 l2)