help orphan modules

src/Init/Data/Iterators/Lemmas/Consumers/Collect.lean

@@ -55,6 +55,12 @@ theorem Iter.toListRev_ensureTermination_eq_toListRev {α β} [Iterator α Id β
     {it : Iter (α := α) β} : it.ensureTermination.toListRev = it.toListRev :=
   (rfl)
 
+@[simp]
+theorem IterM.toArray_toIter {α β} [Iterator α Id β] [Finite α Id]
+    {it : IterM (α := α) Id β} :
+    it.toIter.toArray = it.toArray.run :=
+  (rfl)
+
 @[simp]
 theorem IterM.toList_toIter {α β} [Iterator α Id β] [Finite α Id]
     {it : IterM (α := α) Id β} :

src/Std/Data/Iterators/Lemmas/Producers.lean

@@ -8,6 +8,7 @@ module
 prelude
 public import Std.Data.Iterators.Lemmas.Producers.Monadic
 public import Std.Data.Iterators.Lemmas.Producers.Array
+public import Std.Data.Iterators.Lemmas.Producers.Vector
 public import Std.Data.Iterators.Lemmas.Producers.Empty
 public import Std.Data.Iterators.Lemmas.Producers.Repeat
 public import Std.Data.Iterators.Lemmas.Producers.Range

src/Std/Data/Iterators/Lemmas/Producers/Array.lean

@@ -7,6 +7,8 @@ module
 
 prelude
 public import Std.Data.Iterators.Lemmas.Consumers.Collect
+import Std.Data.Iterators.Lemmas.Consumers.Loop
+import Init.Data.List.TakeDrop
 public import Std.Data.Iterators.Producers.Array
 public import Init.Data.Iterators.Producers.List
 public import Std.Data.Iterators.Lemmas.Producers.Monadic.Array
@@ -76,7 +78,7 @@ theorem Array.toArray_iterFromIdx {array : Array β} {pos : Nat} :
   simp [iterFromIdx_eq_toIter_iterFromIdxM, Iter.toArray]
 
 @[simp, grind =]
-theorem Array.toArray_toIter {array : Array β} :
+theorem Array.toArray_iter {array : Array β} :
     array.iter.toArray = array := by
   simp [Array.iter_eq_iterFromIdx]
 
@@ -86,10 +88,20 @@ theorem Array.toListRev_iterFromIdx {array : Array β} {pos : Nat} :
   simp [Iter.toListRev_eq, Array.toList_iterFromIdx]
 
 @[simp, grind =]
-theorem Array.toListRev_toIter {array : Array β} :
+theorem Array.toListRev_iter {array : Array β} :
     array.iter.toListRev = array.toListRev := by
   simp [Array.iter_eq_iterFromIdx]
 
+@[simp, grind =]
+theorem Array.length_iterFromIdx {array : Array β} {pos : Nat} :
+    (array.iterFromIdx pos).length = array.size - pos := by
+  simp [← Iter.length_toList_eq_length]
+
+@[simp, grind =]
+theorem Array.length_iter {array : Array β} :
+    array.iter.length = array.size := by
+  simp [← Iter.length_toList_eq_length]
+
 section Equivalence
 
 theorem Array.iterFromIdx_equiv_iter_drop_toList {α : Type w} {array : Array α}

src/Std/Data/Iterators/Lemmas/Producers/Monadic.lean

@@ -7,5 +7,6 @@ module
 
 prelude
 public import Std.Data.Iterators.Lemmas.Producers.Monadic.Array
+public import Std.Data.Iterators.Lemmas.Producers.Monadic.Vector
 public import Std.Data.Iterators.Lemmas.Producers.Monadic.Empty
 public import Std.Data.Iterators.Lemmas.Producers.Monadic.List

src/Std/Data/Iterators/Lemmas/Producers/Monadic/Array.lean

@@ -141,7 +141,7 @@ theorem Array.toArray_iterFromIdxM [LawfulMonad m] {array : Array β} {pos : Nat
   rw [← List.toArray_drop]
 
 @[simp, grind =]
-theorem Array.toArray_toIterM [LawfulMonad m] {array : Array β} :
+theorem Array.toArray_iterM [LawfulMonad m] {array : Array β} :
     (array.iterM m).toArray = pure array := by
   simp [Array.iterM_eq_iterFromIdxM, Array.toArray_iterFromIdxM]
 
@@ -151,6 +151,16 @@ theorem Array.toListRev_iterFromIdxM [LawfulMonad m] {array : Array β} {pos : N
   simp [IterM.toListRev_eq, Array.toList_iterFromIdxM]
 
 @[simp, grind =]
-theorem Array.toListRev_toIterM [LawfulMonad m] {array : Array β} :
+theorem Array.toListRev_iterM [LawfulMonad m] {array : Array β} :
     (array.iterM m).toListRev = pure array.toListRev := by
   simp [Array.iterM_eq_iterFromIdxM, Array.toListRev_iterFromIdxM]
+
+@[simp, grind =]
+theorem Array.length_iterFromIdxM [LawfulMonad m] {array : Array β} {pos : Nat} :
+    (array.iterFromIdxM m pos).length = pure (.up (array.size - pos)) := by
+  simp [← IterM.up_length_toList_eq_length]
+
+@[simp, grind =]
+theorem Array.length_iterM [LawfulMonad m] {array : Array β} :
+    (array.iterM m).length = pure (.up array.size) := by
+  simp [← IterM.up_length_toList_eq_length]

src/Std/Data/Iterators/Lemmas/Producers/Monadic/Vector.lean

@@ -0,0 +1,148 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Paul Reichert
+-/
+module
+
+prelude
+public import Std.Data.Iterators.Lemmas.Producers.Monadic.Array
+public import Std.Data.Iterators.Producers.Monadic.Vector
+public import Std.Data.Iterators.Lemmas.Consumers.Monadic
+public import Std.Data.Iterators.Lemmas.Producers.Monadic.List
+public import Init.Data.Array.Lemmas
+import Init.Data.List.Nat.TakeDrop
+import Init.Data.List.TakeDrop
+import Init.Omega
+import Init.Data.Vector.Lemmas
+
+set_option doc.verso true
+
+open Std Std.Iterators
+
+@[expose] public section
+
+/-!
+# Lemmas about vector iterators
+
+This module provides lemmas about the interactions of {name}`Vector.iterM` with {name}`IterM.step`
+and various
+collectors.
+-/
+
+variable {m : Type w → Type w'} [Monad m] {β : Type w} {n : Nat}
+
+theorem Vector.iterM_eq_iterFromIdxM {xs : Vector β n} :
+    xs.iterM m = xs.iterFromIdxM m 0 :=
+  rfl
+
+theorem Std.Iterators.Vector.isPlausibleStep_iterFromIdxM_of_lt {xs : Vector β n} {pos : Nat}
+    (h : pos < n) :
+    (xs.iterFromIdxM m pos).IsPlausibleStep (.yield (xs.iterFromIdxM m (pos + 1)) xs[pos]) := by
+  simp only [IterM.IsPlausibleStep, Iterator.IsPlausibleStep]
+  refine ⟨rfl, rfl, ?_, rfl⟩
+  simpa [Vector.iterFromIdxM, Array.iterFromIdxM] using h
+
+theorem Std.Iterators.Vector.isPlausibleStep_iterFromIdxM_of_not_lt {xs : Vector β n} {pos : Nat}
+    (h : ¬ pos < n) :
+    (xs.iterFromIdxM m pos).IsPlausibleStep .done := by
+  simp only [IterM.IsPlausibleStep, Iterator.IsPlausibleStep]
+  simpa [Vector.iterFromIdxM, Array.iterFromIdxM, Nat.not_lt] using h
+
+theorem Std.Iterators.Vector.isPlausibleStep_iterM_of_pos {xs : Vector β n}
+    (h : 0 < n) :
+    (xs.iterM m).IsPlausibleStep (.yield (xs.iterFromIdxM m 1) xs[0]) := by
+  simp only [Vector.iterM_eq_iterFromIdxM]
+  exact isPlausibleStep_iterFromIdxM_of_lt h
+
+theorem Std.Iterators.Vector.isPlausibleStep_iterM_of_not_pos {xs : Vector β n}
+    (h : ¬ 0 < n) :
+    (xs.iterM m).IsPlausibleStep .done := by
+  simp only [Vector.iterM_eq_iterFromIdxM]
+  exact isPlausibleStep_iterFromIdxM_of_not_lt h
+
+@[simp, grind =]
+theorem Vector.iterFromIdxM_toArray {xs : Vector β n} {pos : Nat} :
+    xs.toArray.iterFromIdxM m pos = xs.iterFromIdxM m pos :=
+  rfl
+
+@[simp, grind =]
+theorem Vector.iterM_toArray {xs : Vector β n} :
+    xs.toArray.iterM m = xs.iterM m :=
+  rfl
+
+theorem Vector.step_iterFromIdxM {xs : Vector β n} {pos : Nat} :
+    (xs.iterFromIdxM m pos).step = (pure <| .deflate <| if h : pos < n then
+        .yield
+          (xs.iterFromIdxM m (pos + 1))
+          xs[pos]
+          (Vector.isPlausibleStep_iterFromIdxM_of_lt h)
+      else
+        .done (Vector.isPlausibleStep_iterFromIdxM_of_not_lt h)) := by
+  simp [← Vector.iterFromIdxM_toArray, Array.step_iterFromIdxM]
+
+theorem Vector.step_iterM {xs : Vector β n} :
+    (xs.iterM m).step = (pure <| .deflate <| if h : 0 < n then
+        .yield
+          (xs.iterFromIdxM m 1)
+          xs[0]
+          (Vector.isPlausibleStep_iterM_of_pos h)
+      else
+        .done (Vector.isPlausibleStep_iterM_of_not_pos h)) := by
+  simp [← Vector.iterFromIdxM_toArray, ← Vector.iterM_toArray, Array.step_iterM]
+
+section Equivalence
+
+theorem Vector.iterFromIdxM_equiv_iterM_drop_toList {α : Type w} {xs : Vector α n}
+    {m : Type w → Type w'} [Monad m] [LawfulMonad m] {pos : Nat} :
+    (xs.iterFromIdxM m pos).Equiv ((xs.toList.drop pos).iterM m) := by
+  simp only [← Vector.iterFromIdxM_toArray]
+  apply Array.iterFromIdxM_equiv_iterM_drop_toList
+
+theorem Vector.iterM_equiv_iterM_toList {α : Type w} {xs : Vector α n} {m : Type w → Type w'}
+    [Monad m] [LawfulMonad m] :
+    (xs.iterM m).Equiv (xs.toList.iterM m) := by
+  simp only [← Vector.iterM_toArray]
+  apply Array.iterM_equiv_iterM_toList
+
+end Equivalence
+
+@[simp, grind =]
+theorem Vector.toList_iterFromIdxM [LawfulMonad m] {xs : Vector β n} {pos : Nat} :
+    (xs.iterFromIdxM m pos).toList = pure (xs.toList.drop pos) := by
+  simp [← Vector.iterFromIdxM_toArray, Vector.toList_toArray]
+
+@[simp, grind =]
+theorem Vector.toList_iterM [LawfulMonad m] {xs : Vector β n} :
+    (xs.iterM m).toList = pure xs.toList := by
+  simp [← Vector.iterM_toArray, Vector.toList_toArray]
+
+@[simp, grind =]
+theorem Vector.toArray_iterFromIdxM [LawfulMonad m] {xs : Vector β n} {pos : Nat} :
+    (xs.iterFromIdxM m pos).toArray = pure (xs.toArray.extract pos n) := by
+  simp [← Vector.iterFromIdxM_toArray]
+
+@[simp, grind =]
+theorem Vector.toArray_iterM [LawfulMonad m] {xs : Vector β n} :
+    (xs.iterM m).toArray = pure xs.toArray := by
+  simp [← Vector.iterM_toArray]
+
+@[simp, grind =]
+theorem Vector.toListRev_iterFromIdxM [LawfulMonad m] {xs : Vector β n} {pos : Nat} :
+    (xs.iterFromIdxM m pos).toListRev = pure (xs.toList.drop pos).reverse := by
+  simp [← Vector.iterFromIdxM_toArray, Vector.toList_toArray]
+
+@[simp, grind =]
+theorem Vector.toListRev_iterM [LawfulMonad m] {xs : Vector β n} :
+    (xs.iterM m).toListRev = pure xs.toList.reverse := by
+  simp [← Vector.iterM_toArray, Vector.toList_toArray]
+
+@[simp, grind =]
+theorem Vector.length_iterFromIdxM [LawfulMonad m] {xs : Vector β n} {pos : Nat} :
+    (xs.iterFromIdxM m pos).length = pure (.up (n - pos)) := by
+  simp [← IterM.up_length_toList_eq_length]
+
+@[simp, grind =]
+theorem Vector.length_iterM [LawfulMonad m] {xs : Vector β n} :
+    (xs.iterM m).length = pure (.up n) := by
+  simp [← IterM.up_length_toList_eq_length]

src/Std/Data/Iterators/Lemmas/Producers/Vector.lean

@@ -0,0 +1,153 @@
+/-
+Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Paul Reichert
+-/
+module
+
+prelude
+public import Std.Data.Iterators.Lemmas.Producers.Array
+public import Std.Data.Iterators.Producers.Vector
+public import Std.Data.Iterators.Producers.Monadic.Vector
+import Init.Data.Vector.Lemmas
+import Std.Data.Iterators.Lemmas.Consumers.Collect
+import Std.Data.Iterators.Lemmas.Consumers.Loop
+import Init.Data.List.TakeDrop
+import Std.Data.Iterators.Lemmas.Producers.Monadic.Vector
+
+set_option doc.verso true
+
+open Std Std.Iterators
+
+@[expose] public section
+
+/-!
+# Lemmas about vector iterators
+
+This module provides lemmas about the interactions of {name}`Vector.iter` with {name}`Iter.step` and
+various collectors.
+-/
+
+variable {β : Type w} {n : Nat}
+
+theorem Vector.iter_eq_toIter_iterM {xs : Vector β n} :
+    xs.iter = (xs.iterM Id).toIter :=
+  rfl
+
+theorem Vector.iter_eq_iterFromIdx {xs : Vector β n} :
+    xs.iter = xs.iterFromIdx 0 :=
+  rfl
+
+theorem Vector.iterFromIdx_eq_toIter_iterFromIdxM {xs : Vector β n} {pos : Nat} :
+    xs.iterFromIdx pos = (xs.iterFromIdxM Id pos).toIter :=
+  rfl
+
+theorem Std.Iterators.Vector.isPlausibleStep_iterFromIdx_of_lt {xs : Vector β n} {pos : Nat}
+    (h : pos < n) :
+    (xs.iterFromIdx pos).IsPlausibleStep (.yield (xs.iterFromIdx (pos + 1)) xs[pos]) := by
+  simp only [Iter.IsPlausibleStep, IterM.IsPlausibleStep, Iterator.IsPlausibleStep]
+  refine ⟨rfl, rfl, ?_, rfl⟩
+  simpa [Vector.iterFromIdx, Array.iterFromIdxM, Array.iterFromIdx] using h
+
+theorem Std.Iterators.Vector.isPlausibleStep_iterFromIdx_of_not_lt {xs : Vector β n} {pos : Nat}
+    (h : ¬ pos < n) :
+    (xs.iterFromIdx pos).IsPlausibleStep .done := by
+  simp only [Iter.IsPlausibleStep, IterM.IsPlausibleStep, Iterator.IsPlausibleStep]
+  simpa [Vector.iterFromIdx, Array.iterFromIdxM, Array.iterFromIdx, Nat.not_lt] using h
+
+theorem Std.Iterators.Vector.isPlausibleStep_iter_of_pos {xs : Vector β n}
+    (h : 0 < n) :
+    (xs.iter).IsPlausibleStep (.yield (xs.iterFromIdx 1) xs[0]) := by
+  simp only [Vector.iter_eq_iterFromIdx]
+  exact isPlausibleStep_iterFromIdx_of_lt h
+
+theorem Std.Iterators.Vector.isPlausibleStep_iter_of_not_pos {xs : Vector β n}
+    (h : ¬ 0 < n) :
+    (xs.iter).IsPlausibleStep .done := by
+  simp only [Vector.iter_eq_iterFromIdx]
+  exact isPlausibleStep_iterFromIdx_of_not_lt h
+
+@[simp, grind =]
+theorem Vector.iterFromIdx_toArray {xs : Vector β n} {pos : Nat} :
+    xs.toArray.iterFromIdx pos = xs.iterFromIdx pos :=
+  rfl
+
+@[simp, grind =]
+theorem Vector.iter_toArray {xs : Vector β n} :
+    xs.toArray.iter = xs.iter :=
+  rfl
+
+theorem Vector.step_iterFromIdx {xs : Vector β n} {pos : Nat} :
+    (xs.iterFromIdx pos).step = if h : pos < n then
+        .yield
+          (xs.iterFromIdx (pos + 1))
+          xs[pos]
+          (Vector.isPlausibleStep_iterFromIdx_of_lt h)
+      else
+        .done (Vector.isPlausibleStep_iterFromIdx_of_not_lt h) := by
+  split <;> simp [Vector.iterFromIdx_eq_toIter_iterFromIdxM, Iter.step, Vector.step_iterFromIdxM, *]
+
+theorem Vector.step_iter {xs : Vector β n} :
+    xs.iter.step = if h : 0 < n then
+        .yield
+          (xs.iterFromIdx 1)
+          xs[0]
+          (Vector.isPlausibleStep_iter_of_pos h)
+      else
+        .done (Vector.isPlausibleStep_iter_of_not_pos h) := by
+  simp [iter_eq_iterFromIdx, step_iterFromIdx]
+
+@[simp, grind =]
+theorem Vector.toList_iterFromIdx {xs : Vector β n} {pos : Nat} :
+    (xs.iterFromIdx pos).toList = xs.toList.drop pos := by
+  simp [Iter.toList, Vector.iterFromIdx_eq_toIter_iterFromIdxM, Iter.toIterM_toIter,
+    Vector.toList_iterFromIdxM]
+
+@[simp, grind =]
+theorem Vector.toList_iter {xs : Vector β n} :
+    xs.iter.toList = xs.toList := by
+  simp [← iter_toArray, Vector.toList_toArray]
+
+@[simp, grind =]
+theorem Vector.toArray_iterFromIdx {xs : Vector β n} {pos : Nat} :
+    (xs.iterFromIdx pos).toArray = xs.toArray.extract pos n := by
+  simp [← iterFromIdx_toArray]
+
+@[simp, grind =]
+theorem Vector.toArray_iter {xs : Vector β n} :
+    xs.iter.toArray = xs.toArray := by
+  simp [← iter_toArray]
+
+@[simp, grind =]
+theorem Vector.toListRev_iterFromIdx {xs : Vector β n} {pos : Nat} :
+    (xs.iterFromIdx pos).toListRev = (xs.toList.drop pos).reverse := by
+  simp [Iter.toListRev_eq, Vector.toList_iterFromIdx]
+
+@[simp, grind =]
+theorem Vector.toListRev_toIter {xs : Vector β n} :
+    xs.iter.toListRev = xs.toList.reverse := by
+  simp [Vector.iter_eq_iterFromIdx]
+
+@[simp, grind =]
+theorem Vector.length_iterFromIdx {xs : Vector β n} {pos : Nat} :
+    (xs.iterFromIdx pos).length = n - pos := by
+  simp [← Iter.length_toList_eq_length]
+
+@[simp, grind =]
+theorem Vector.length_iter {xs : Vector β n} :
+    xs.iter.length = n := by
+  simp [← Iter.length_toList_eq_length]
+
+section Equivalence
+
+theorem Vector.iterFromIdx_equiv_iter_drop_toList {α : Type w} {xs : Vector α n}
+    {pos : Nat} : (xs.iterFromIdx pos).Equiv (xs.toList.drop pos).iter := by
+  apply IterM.Equiv.toIter
+  exact Vector.iterFromIdxM_equiv_iterM_drop_toList
+
+theorem Vector.iter_equiv_iter_toList {α : Type w} {xs : Vector α n} :
+    xs.iter.Equiv xs.toList.iter := by
+  rw [Vector.iter_eq_iterFromIdx]
+  simpa using iterFromIdx_equiv_iter_drop_toList
+
+end Equivalence

src/Std/Data/Iterators/Producers.lean

@@ -8,6 +8,7 @@ module
 prelude
 public import Std.Data.Iterators.Producers.Monadic
 public import Std.Data.Iterators.Producers.Array
+public import Std.Data.Iterators.Producers.Vector
 public import Std.Data.Iterators.Producers.Empty
 public import Std.Data.Iterators.Producers.Range
 public import Std.Data.Iterators.Producers.Repeat

src/Std/Data/Iterators/Producers/Monadic.lean

@@ -7,4 +7,5 @@ module
 
 prelude
 public import Std.Data.Iterators.Producers.Monadic.Array
+public import Std.Data.Iterators.Producers.Monadic.Vector
 public import Std.Data.Iterators.Producers.Monadic.Empty

src/Std/Data/Iterators/Producers/Monadic/Vector.lean

@@ -0,0 +1,57 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Paul Reichert
+-/
+module
+
+prelude
+public import Init.Data.Vector.Basic
+public import Std.Data.Iterators.Producers.Monadic.Array
+
+set_option doc.verso true
+
+/-!
+# Vector iterator
+
+This module provides an iterator for vectors that is accessible via
+{name (scope := "Std.Data.Iterators.Producers.Monadic.Vector")}`Vector.iterM`.
+-/
+
+@[expose] public section
+
+open Std Std.Iterators
+
+/--
+Returns a finite monadic iterator for the given vector starting at the given index.
+The iterator yields the elements of the vector in order and then terminates.
+
+The pure version of this iterator is
+{name (scope := "Std.Data.Iterators.Producers.Vector")}`Vector.iterFromIdx`.
+
+**Termination properties:**
+
+* {name}`Finite` instance: always
+* {name}`Productive` instance: always
+-/
+@[always_inline, inline, match_pattern]
+def _root_.Vector.iterFromIdxM (xs : Vector α n) (m : Type w → Type w')
+    (pos : Nat) [Pure m] :=
+  (xs.toArray.iterFromIdxM m pos : IterM m α)
+
+/--
+Returns a finite monadic iterator for the given vector.
+The iterator yields the elements of the vector in order and then terminates. There are no side
+effects.
+
+The pure version of this iterator is
+{name (scope := "Std.Data.Iterators.Producers.Vector")}`Vector.iter`.
+
+**Termination properties:**
+
+* {name}`Finite` instance: always
+* {name}`Productive` instance: always
+-/
+@[inline]
+def Vector.iterM (xs : Vector α n) (m : _) [Pure m] :=
+  (xs.toArray.iterM m : IterM m α)

src/Std/Data/Iterators/Producers/Vector.lean

@@ -0,0 +1,55 @@
+/-
+Copyright (c) 2026 Lean FRO, LLC. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Paul Reichert
+-/
+module
+
+prelude
+public import Init.Data.Vector.Basic
+public import Std.Data.Iterators.Producers.Array
+
+set_option doc.verso true
+
+/-!
+# Vector iterator
+
+This module provides an iterator for vectors that is accessible via
+{name (scope := "Std.Data.Iterators.Producers.Vector")}`Vector.iter`.
+-/
+
+@[expose] public section
+
+open Std Std.Iterators
+
+/--
+Returns a finite iterator for the given vector starting at the given index.
+The iterator yields the elements of the vector in order and then terminates.
+
+The monadic version of this iterator is
+{name (scope := "Std.Data.Iterators.Producers.Monadic.Vector")}`Vector.iterFromIdxM`.
+
+**Termination properties:**
+
+* {name}`Finite` instance: always
+* {name}`Productive` instance: always
+-/
+@[always_inline, inline]
+def Vector.iterFromIdx {α : Type w} (xs : Vector α n) (pos : Nat) :=
+  (xs.toArray.iterFromIdx pos : Iter α)
+
+/--
+Returns a finite iterator for the given vector.
+The iterator yields the elements of the vector in order and then terminates.
+
+The monadic version of this iterator is
+{name (scope := "Std.Data.Iterators.Producers.Monadic.Vector")}`Vector.iterM`.
+
+**Termination properties:**
+
+* {name}`Finite` instance: always
+* {name}`Productive` instance: always
+-/
+@[always_inline, inline]
+def Vector.iter {α : Type w} (xs : Vector α n) :=
+  (xs.toArray.iter : Iter α)