feat: Vector.getElem_flatMap (#6661)

This PR adds array indexing lemmas for `Vector.flatMap`. (These were not
available for `List` and `Array` due to variable lengths.)

src/Init/Data/Nat/Lemmas.lean

@@ -622,6 +622,14 @@ protected theorem pos_of_mul_pos_right {a b : Nat} (h : 0 < a * b) : 0 < a := by
     0 < a * b ↔ 0 < a :=
   ⟨Nat.pos_of_mul_pos_right, fun w => Nat.mul_pos w h⟩
 
+protected theorem pos_of_lt_mul_left {a b c : Nat} (h : a < b * c) : 0 < c := by
+  replace h : 0 < b * c := by omega
+  exact Nat.pos_of_mul_pos_left h
+
+protected theorem pos_of_lt_mul_right {a b c : Nat} (h : a < b * c) : 0 < b := by
+  replace h : 0 < b * c := by omega
+  exact Nat.pos_of_mul_pos_right h
+
 /-! ### div/mod -/
 
 theorem mod_two_eq_zero_or_one (n : Nat) : n % 2 = 0 ∨ n % 2 = 1 :=

src/Init/Data/Vector/Lemmas.lean

@@ -1456,6 +1456,44 @@ theorem append_eq_map_iff {f : α → β} :
       mk (L.map toArray).flatten (by simp [Function.comp_def, Array.map_const', h]) := by
   simp [flatten]
 
+@[simp] theorem getElem_flatten (l : Vector (Vector β m) n) (i : Nat) (hi : i < n * m) :
+    l.flatten[i] =
+      haveI : i / m < n := by rwa [Nat.div_lt_iff_lt_mul (Nat.pos_of_lt_mul_left hi)]
+      haveI : i % m < m := Nat.mod_lt _ (Nat.pos_of_lt_mul_left hi)
+      l[i / m][i % m] := by
+  rcases l with ⟨⟨l⟩, rfl⟩
+  simp only [flatten_mk, List.map_toArray, getElem_mk, List.getElem_toArray, Array.flatten_toArray]
+  induction l generalizing i with
+  | nil => simp at hi
+  | cons a l ih =>
+    simp only [List.map_cons, List.map_map, List.flatten_cons]
+    by_cases h : i < m
+    · rw [List.getElem_append_left (by simpa)]
+      have h₁ : i / m = 0 := Nat.div_eq_of_lt h
+      have h₂ : i % m = i := Nat.mod_eq_of_lt h
+      simp [h₁, h₂]
+    · have h₁ : a.toList.length ≤ i := by simp; omega
+      rw [List.getElem_append_right h₁]
+      simp only [Array.length_toList, size_toArray]
+      specialize ih (i - m) (by simp_all [Nat.add_one_mul]; omega)
+      have h₂ : i / m = (i - m) / m + 1 := by
+        conv => lhs; rw [show i = i - m + m by omega]
+        rw [Nat.add_div_right]
+        exact Nat.pos_of_lt_mul_left hi
+      simp only [Array.length_toList, size_toArray] at h₁
+      have h₃ : (i - m) % m = i % m := (Nat.mod_eq_sub_mod h₁).symm
+      simp_all
+
+theorem getElem?_flatten (l : Vector (Vector β m) n) (i : Nat) :
+    l.flatten[i]? =
+      if hi : i < n * m then
+        haveI : i / m < n := by rwa [Nat.div_lt_iff_lt_mul (Nat.pos_of_lt_mul_left hi)]
+        haveI : i % m < m := Nat.mod_lt _ (Nat.pos_of_lt_mul_left hi)
+        some l[i / m][i % m]
+      else
+        none := by
+  simp [getElem?_def]
+
 @[simp] theorem flatten_singleton (l : Vector α n) : #v[l].flatten = l.cast (by simp) := by
   simp [flatten]
 
@@ -1542,6 +1580,23 @@ theorem flatMap_def (l : Vector α n) (f : α → Vector β m) : l.flatMap f = f
   rcases l with ⟨l, rfl⟩
   simp [Array.flatMap_def, Function.comp_def]
 
+@[simp] theorem getElem_flatMap (l : Vector α n) (f : α → Vector β m) (i : Nat) (hi : i < n * m) :
+    (l.flatMap f)[i] =
+      haveI : i / m < n := by rwa [Nat.div_lt_iff_lt_mul (Nat.pos_of_lt_mul_left hi)]
+      haveI : i % m < m := Nat.mod_lt _ (Nat.pos_of_lt_mul_left hi)
+      (f (l[i / m]))[i % m] := by
+  rw [flatMap_def, getElem_flatten, getElem_map]
+
+theorem getElem?_flatMap (l : Vector α n) (f : α → Vector β m) (i : Nat) :
+    (l.flatMap f)[i]? =
+      if hi : i < n * m then
+        haveI : i / m < n := by rwa [Nat.div_lt_iff_lt_mul (Nat.pos_of_lt_mul_left hi)]
+        haveI : i % m < m := Nat.mod_lt _ (Nat.pos_of_lt_mul_left hi)
+        some ((f (l[i / m]))[i % m])
+      else
+        none := by
+  simp [getElem?_def]
+
 @[simp] theorem flatMap_id (l : Vector (Vector α m) n) : l.flatMap id = l.flatten := by simp [flatMap_def]
 
 @[simp] theorem flatMap_id' (l : Vector (Vector α m) n) : l.flatMap (fun a => a) = l.flatten := by simp [flatMap_def]