Add String.length_join and List.length_join (#770)

* Add `String.length_join` and `List.length_join`

* Update Std/Data/List/Lemmas.lean

Update Std/Data/String/Lemmas.lean

Co-Authored-By: Kim Morrison <kim@tqft.net>

---------

Co-authored-by: Kim Morrison <kim@tqft.net>

Std/Data/List/Lemmas.lean

@@ -365,6 +365,9 @@ theorem exists_of_mem_join : a ∈ join L → ∃ l, l ∈ L ∧ a ∈ l := mem_
 
 theorem mem_join_of_mem (lL : l ∈ L) (al : a ∈ l) : a ∈ join L := mem_join.2 ⟨l, lL, al⟩
 
+theorem length_join (l : List (List α)) : (join l).length = Nat.sum (l.map length) := by
+  induction l <;> simp [*]
+
 /-! ### bind -/
 
 theorem mem_bind {f : α → List β} {b} {l : List α} : b ∈ l.bind f ↔ ∃ a, a ∈ l ∧ b ∈ f a := by

Std/Data/String/Lemmas.lean

@@ -516,6 +516,11 @@ theorem join_eq (ss : List String) : join ss = ⟨(ss.map data).join⟩ := go ss
 @[simp] theorem data_join (ss : List String) : (join ss).data = (ss.map data).join := by
   rw [join_eq]
 
+@[simp] theorem length_data (s : String) : s.data.length = s.length := rfl
+
+theorem length_join (l : List String) : (join l).length = Nat.sum (l.map length) := by
+  simp [length, List.length_join, List.map_map, (funext length_data : List.length ∘ data = length)]
+
 theorem singleton_eq (c : Char) : singleton c = ⟨[c]⟩ := rfl
 
 @[simp] theorem data_singleton (c : Char) : (singleton c).data = [c] := rfl