diff --git a/src/Init/Data.lean b/src/Init/Data.lean
index 4b571eae65..6028db89a3 100644
--- a/src/Init/Data.lean
+++ b/src/Init/Data.lean
@@ -46,3 +46,4 @@ import Init.Data.NeZero
 import Init.Data.Function
 import Init.Data.RArray
 import Init.Data.Vector
+import Init.Data.Iterators
diff --git a/src/Init/Data/Iterators.lean b/src/Init/Data/Iterators.lean
new file mode 100644
index 0000000000..115a534b45
--- /dev/null
+++ b/src/Init/Data/Iterators.lean
@@ -0,0 +1,19 @@
+/-
+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
+import Init.Data.Iterators.Basic
+import Init.Data.Iterators.PostconditionMonad
+import Init.Data.Iterators.Consumers
+import Init.Data.Iterators.Lemmas
+import Init.Data.Iterators.Internal
+
+/-!
+# Iterators
+
+See `Std.Data.Iterators` for an overview over the iterator API.
+-/
diff --git a/src/Std/Data/Iterators/Basic.lean b/src/Init/Data/Iterators/Basic.lean
similarity index 95%
rename from src/Std/Data/Iterators/Basic.lean
rename to src/Init/Data/Iterators/Basic.lean
index 7890bbb313..cafea31f0f 100644
--- a/src/Std/Data/Iterators/Basic.lean
+++ b/src/Init/Data/Iterators/Basic.lean
@@ -3,6 +3,8 @@ 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
 import Init.Core
 import Init.Classical
@@ -31,7 +33,7 @@ See `Std.Data.Iterators.Producers` for ways to iterate over common data structur
 By convention, the monadic iterator associated with an object can be obtained via dot notation.
 For example, `List.iterM IO` creates an iterator over a list in the monad `IO`.
 
-See `Std.Data.Iterators.Consumers` for ways to use an iterator. For example, `it.toList` will
+See `Init.Data.Iterators.Consumers` for ways to use an iterator. For example, `it.toList` will
 convert a provably finite iterator `it` into a list and `it.allowNontermination.toList` will
 do so even if finiteness cannot be proved. It is also always possible to manually iterate using
 `it.step`, relying on the termination measures `it.finitelyManySteps` and `it.finitelyManySkips`.
@@ -75,7 +77,7 @@ See `Std.Data.Iterators.Producers` for ways to iterate over common data structur
 By convention, the monadic iterator associated with an object can be obtained via dot notation.
 For example, `List.iterM IO` creates an iterator over a list in the monad `IO`.
 
-See `Std.Data.Iterators.Consumers` for ways to use an iterator. For example, `it.toList` will
+See `Init.Data.Iterators.Consumers` for ways to use an iterator. For example, `it.toList` will
 convert a provably finite iterator `it` into a list and `it.allowNontermination.toList` will
 do so even if finiteness cannot be proved. It is also always possible to manually iterate using
 `it.step`, relying on the termination measures `it.finitelyManySteps` and `it.finitelyManySkips`.
@@ -111,12 +113,14 @@ structure Iter {α : Type w} (β : Type w) where
 Converts a pure iterator (`Iter β`) into a monadic iterator (`IterM Id β`) in the
 identity monad `Id`.
 -/
+@[expose]
 def Iter.toIterM {α : Type w} {β : Type w} (it : Iter (α := α) β) : IterM (α := α) Id β :=
   ⟨it.internalState⟩
 
 /--
 Converts a monadic iterator (`IterM Id β`) over `Id` into a pure iterator (`Iter β`).
 -/
+@[expose]
 def IterM.toIter {α : Type w} {β : Type w} (it : IterM (α := α) Id β) : Iter (α := α) β :=
   ⟨it.internalState⟩
 
@@ -170,6 +174,7 @@ inductive IterStep (α β) where
 Returns the succeeding iterator stored in an iterator step or `none` if the step is `.done`
 and the iterator has finished.
 -/
+@[expose]
 def IterStep.successor : IterStep α β → Option α
   | .yield it _ => some it
   | .skip it => some it
@@ -179,7 +184,7 @@ def IterStep.successor : IterStep α β → Option α
 If present, applies `f` to the iterator of an `IterStep` and replaces the iterator
 with the result of the application of `f`.
 -/
-@[always_inline, inline]
+@[always_inline, inline, expose]
 def IterStep.mapIterator {α' : Type u'} (f : α → α') : IterStep α β → IterStep α' β
   | .yield it out => .yield (f it) out
   | .skip it => .skip (f it)
@@ -224,12 +229,13 @@ of another state. Having this proof bundled up with the step is important for te
 
 See `IterM.Step` and `Iter.Step` for the concrete choice of the plausibility predicate.
 -/
+@[expose]
 def PlausibleIterStep (IsPlausibleStep : IterStep α β → Prop) := Subtype IsPlausibleStep
 
 /--
 Match pattern for the `yield` case. See also `IterStep.yield`.
 -/
-@[match_pattern, simp]
+@[match_pattern, simp, expose]
 def PlausibleIterStep.yield {IsPlausibleStep : IterStep α β → Prop}
     (it' : α) (out : β) (h : IsPlausibleStep (.yield it' out)) :
     PlausibleIterStep IsPlausibleStep :=
@@ -238,7 +244,7 @@ def PlausibleIterStep.yield {IsPlausibleStep : IterStep α β → Prop}
 /--
 Match pattern for the `skip` case. See also `IterStep.skip`.
 -/
-@[match_pattern, simp]
+@[match_pattern, simp, expose]
 def PlausibleIterStep.skip {IsPlausibleStep : IterStep α β → Prop}
     (it' : α) (h : IsPlausibleStep (.skip it')) : PlausibleIterStep IsPlausibleStep :=
   ⟨.skip it', h⟩
@@ -246,7 +252,7 @@ def PlausibleIterStep.skip {IsPlausibleStep : IterStep α β → Prop}
 /--
 Match pattern for the `done` case. See also `IterStep.done`.
 -/
-@[match_pattern, simp]
+@[match_pattern, simp, expose]
 def PlausibleIterStep.done {IsPlausibleStep : IterStep α β → Prop}
     (h : IsPlausibleStep .done) : PlausibleIterStep IsPlausibleStep :=
   ⟨.done, h⟩
@@ -283,7 +289,7 @@ section Monadic
 /--
 Converts wraps the state of an iterator into an `IterM` object.
 -/
-@[always_inline, inline]
+@[always_inline, inline, expose]
 def toIterM {α : Type w} (it : α) (m : Type w → Type w') (β : Type w) :
     IterM (α := α) m β :=
   ⟨it⟩
@@ -302,6 +308,7 @@ theorem internalState_toIterM {α m β} (it : α) :
 Asserts that certain step is plausibly the successor of a given iterator. What "plausible" means
 is up to the `Iterator` instance but it should be strong enough to allow termination proofs.
 -/
+@[expose]
 abbrev IterM.IsPlausibleStep {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β] :
     IterM (α := α) m β → IterStep (IterM (α := α) m β) β → Prop :=
   Iterator.IsPlausibleStep (α := α) (m := m)
@@ -310,6 +317,7 @@ abbrev IterM.IsPlausibleStep {α : Type w} {m : Type w → Type w'} {β : Type w
 The type of the step object returned by `IterM.step`, containing an `IterStep`
 and a proof that this is a plausible step for the given iterator.
 -/
+@[expose]
 abbrev IterM.Step {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
     (it : IterM (α := α) m β) :=
   PlausibleIterStep it.IsPlausibleStep
@@ -318,6 +326,7 @@ abbrev IterM.Step {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator
 Asserts that a certain output value could plausibly be emitted by the given iterator in its next
 step.
 -/
+@[expose]
 def IterM.IsPlausibleOutput {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
     (it : IterM (α := α) m β) (out : β) : Prop :=
   ∃ it', it.IsPlausibleStep (.yield it' out)
@@ -326,6 +335,7 @@ def IterM.IsPlausibleOutput {α : Type w} {m : Type w → Type w'} {β : Type w}
 Asserts that a certain iterator `it'` could plausibly be the directly succeeding iterator of another
 given iterator `it`.
 -/
+@[expose]
 def IterM.IsPlausibleSuccessorOf {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
     (it' it : IterM (α := α) m β) : Prop :=
   ∃ step, step.successor = some it' ∧ it.IsPlausibleStep step
@@ -334,6 +344,7 @@ def IterM.IsPlausibleSuccessorOf {α : Type w} {m : Type w → Type w'} {β : Ty
 Asserts that a certain iterator `it'` could plausibly be the directly succeeding iterator of another
 given iterator `it` while no value is emitted (see `IterStep.skip`).
 -/
+@[expose]
 def IterM.IsPlausibleSkipSuccessorOf {α : Type w} {m : Type w → Type w'} {β : Type w}
     [Iterator α m β] (it' it : IterM (α := α) m β) : Prop :=
   it.IsPlausibleStep (.skip it')
@@ -356,6 +367,7 @@ section Pure
 Asserts that certain step is plausibly the successor of a given iterator. What "plausible" means
 is up to the `Iterator` instance but it should be strong enough to allow termination proofs.
 -/
+@[expose]
 def Iter.IsPlausibleStep {α : Type w} {β : Type w} [Iterator α Id β]
     (it : Iter (α := α) β) (step : IterStep (Iter (α := α) β) β) : Prop :=
   it.toIterM.IsPlausibleStep (step.mapIterator Iter.toIterM)
@@ -364,6 +376,7 @@ def Iter.IsPlausibleStep {α : Type w} {β : Type w} [Iterator α Id β]
 The type of the step object returned by `Iter.step`, containing an `IterStep`
 and a proof that this is a plausible step for the given iterator.
 -/
+@[expose]
 def Iter.Step {α : Type w} {β : Type w} [Iterator α Id β] (it : Iter (α := α) β) :=
   PlausibleIterStep (Iter.IsPlausibleStep it)
 
@@ -378,7 +391,7 @@ def Iter.Step.toMonadic {α : Type w} {β : Type w} [Iterator α Id β] {it : It
 /--
 Converts an `IterM.Step` into an `Iter.Step`.
 -/
-@[always_inline, inline]
+@[always_inline, inline, expose]
 def IterM.Step.toPure {α : Type w} {β : Type w} [Iterator α Id β] {it : IterM (α := α) Id β}
     (step : it.Step) : it.toIter.Step :=
   ⟨step.val.mapIterator IterM.toIter, (by simp [Iter.IsPlausibleStep, step.property])⟩
@@ -402,6 +415,7 @@ theorem IterM.Step.toPure_done {α β : Type w} [Iterator α Id β] {it : IterM
 Asserts that a certain output value could plausibly be emitted by the given iterator in its next
 step.
 -/
+@[expose]
 def Iter.IsPlausibleOutput {α : Type w} {β : Type w} [Iterator α Id β]
     (it : Iter (α := α) β) (out : β) : Prop :=
   it.toIterM.IsPlausibleOutput out
@@ -410,6 +424,7 @@ def Iter.IsPlausibleOutput {α : Type w} {β : Type w} [Iterator α Id β]
 Asserts that a certain iterator `it'` could plausibly be the directly succeeding iterator of another
 given iterator `it`.
 -/
+@[expose]
 def Iter.IsPlausibleSuccessorOf {α : Type w} {β : Type w} [Iterator α Id β]
     (it' it : Iter (α := α) β) : Prop :=
   it'.toIterM.IsPlausibleSuccessorOf it.toIterM
@@ -427,7 +442,7 @@ Makes a single step with the given iterator `it`, potentially emitting a value a
 succeeding iterator. If this function is used recursively, termination can sometimes be proved with
 the termination measures `it.finitelyManySteps` and `it.finitelyManySkips`.
 -/
-@[always_inline, inline]
+@[always_inline, inline, expose]
 def Iter.step {α β : Type w} [Iterator α Id β] (it : Iter (α := α) β) : it.Step :=
   it.toIterM.step.run.toPure
 
@@ -456,6 +471,7 @@ structure IterM.TerminationMeasures.Finite
 The relation of plausible successors on `IterM.TerminationMeasures.Finite`. It is well-founded
 if there is a `Finite` instance.
 -/
+@[expose]
 def IterM.TerminationMeasures.Finite.Rel
     {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β] :
     TerminationMeasures.Finite α m → TerminationMeasures.Finite α m → Prop :=
@@ -464,12 +480,13 @@ def IterM.TerminationMeasures.Finite.Rel
 instance {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
     [Finite α m] : WellFoundedRelation (IterM.TerminationMeasures.Finite α m) where
   rel := IterM.TerminationMeasures.Finite.Rel
-  wf := (InvImage.wf _ Finite.wf).transGen
+  wf := by exact (InvImage.wf _ Finite.wf).transGen
 
 /--
 Termination measure to be used in well-founded recursive functions recursing over a finite iterator
 (see also `Finite`).
 -/
+@[expose]
 def IterM.finitelyManySteps {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
     [Finite α m] (it : IterM (α := α) m β) : IterM.TerminationMeasures.Finite α m :=
   ⟨it⟩
@@ -494,9 +511,10 @@ theorem IterM.TerminationMeasures.Finite.rel_of_skip
 macro_rules | `(tactic| decreasing_trivial) => `(tactic|
   first
   | exact IterM.TerminationMeasures.Finite.rel_of_yield ‹_›
-  | exact IterM.TerminationMeasures.Finite.rel_of_skip ‹_›)
+  | exact IterM.TerminationMeasures.Finite.rel_of_skip ‹_›
+  | fail)
 
-@[inherit_doc IterM.finitelyManySteps]
+@[inherit_doc IterM.finitelyManySteps, expose]
 def Iter.finitelyManySteps {α : Type w} {β : Type w} [Iterator α Id β] [Finite α Id]
     (it : Iter (α := α) β) : IterM.TerminationMeasures.Finite α Id :=
   it.toIterM.finitelyManySteps
@@ -521,7 +539,8 @@ theorem Iter.TerminationMeasures.Finite.rel_of_skip
 macro_rules | `(tactic| decreasing_trivial) => `(tactic|
   first
   | exact Iter.TerminationMeasures.Finite.rel_of_yield ‹_›
-  | exact Iter.TerminationMeasures.Finite.rel_of_skip ‹_›)
+  | exact Iter.TerminationMeasures.Finite.rel_of_skip ‹_›
+  | fail)
 
 theorem IterM.isPlausibleSuccessorOf_of_yield
     {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
@@ -561,6 +580,7 @@ structure IterM.TerminationMeasures.Productive
 The relation of plausible successors while skipping on `IterM.TerminationMeasures.Productive`.
 It is well-founded if there is a `Productive` instance.
 -/
+@[expose]
 def IterM.TerminationMeasures.Productive.Rel
     {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β] :
     TerminationMeasures.Productive α m → TerminationMeasures.Productive α m → Prop :=
@@ -569,12 +589,13 @@ def IterM.TerminationMeasures.Productive.Rel
 instance {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
     [Productive α m] : WellFoundedRelation (IterM.TerminationMeasures.Productive α m) where
   rel := IterM.TerminationMeasures.Productive.Rel
-  wf := (InvImage.wf _ Productive.wf).transGen
+  wf := by exact (InvImage.wf _ Productive.wf).transGen
 
 /--
 Termination measure to be used in well-founded recursive functions recursing over a productive
 iterator (see also `Productive`).
 -/
+@[expose]
 def IterM.finitelyManySkips {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
     [Productive α m] (it : IterM (α := α) m β) : IterM.TerminationMeasures.Productive α m :=
   ⟨it⟩
@@ -590,9 +611,11 @@ theorem IterM.TerminationMeasures.Productive.rel_of_skip
   .single h
 
 macro_rules | `(tactic| decreasing_trivial) => `(tactic|
-  exact IterM.TerminationMeasures.Productive.rel_of_skip ‹_›)
+  first
+    | exact IterM.TerminationMeasures.Productive.rel_of_skip ‹_›
+    | fail)
 
-@[inherit_doc IterM.finitelyManySkips]
+@[inherit_doc IterM.finitelyManySkips, expose]
 def Iter.finitelyManySkips {α : Type w} {β : Type w} [Iterator α Id β] [Productive α Id]
     (it : Iter (α := α) β) : IterM.TerminationMeasures.Productive α Id :=
   it.toIterM.finitelyManySkips
@@ -608,7 +631,9 @@ theorem Iter.TerminationMeasures.Productive.rel_of_skip
   IterM.TerminationMeasures.Productive.rel_of_skip h
 
 macro_rules | `(tactic| decreasing_trivial) => `(tactic|
-  exact Iter.TerminationMeasures.Productive.rel_of_skip ‹_›)
+  first
+    | exact Iter.TerminationMeasures.Productive.rel_of_skip ‹_›
+    | fail)
 
 instance [Iterator α m β] [Finite α m] : Productive α m where
   wf := by
diff --git a/src/Init/Data/Iterators/Consumers.lean b/src/Init/Data/Iterators/Consumers.lean
new file mode 100644
index 0000000000..e61fab7a94
--- /dev/null
+++ b/src/Init/Data/Iterators/Consumers.lean
@@ -0,0 +1,13 @@
+/-
+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
+import Init.Data.Iterators.Consumers.Monadic
+import Init.Data.Iterators.Consumers.Access
+import Init.Data.Iterators.Consumers.Collect
+import Init.Data.Iterators.Consumers.Loop
+import Init.Data.Iterators.Consumers.Partial
diff --git a/src/Std/Data/Iterators/Consumers/Access.lean b/src/Init/Data/Iterators/Consumers/Access.lean
similarity index 97%
rename from src/Std/Data/Iterators/Consumers/Access.lean
rename to src/Init/Data/Iterators/Consumers/Access.lean
index a4a675e5a0..65173c3402 100644
--- a/src/Std/Data/Iterators/Consumers/Access.lean
+++ b/src/Init/Data/Iterators/Consumers/Access.lean
@@ -3,8 +3,10 @@ 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
-import Std.Data.Iterators.Consumers.Partial
+import Init.Data.Iterators.Consumers.Partial
 
 namespace Std.Iterators
 
diff --git a/src/Std/Data/Iterators/Consumers/Collect.lean b/src/Init/Data/Iterators/Consumers/Collect.lean
similarity index 93%
rename from src/Std/Data/Iterators/Consumers/Collect.lean
rename to src/Init/Data/Iterators/Consumers/Collect.lean
index 4e21f8f21b..fd848dfd55 100644
--- a/src/Std/Data/Iterators/Consumers/Collect.lean
+++ b/src/Init/Data/Iterators/Consumers/Collect.lean
@@ -3,10 +3,12 @@ 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
-import Std.Data.Iterators.Basic
-import Std.Data.Iterators.Consumers.Partial
-import Std.Data.Iterators.Consumers.Monadic.Collect
+import Init.Data.Iterators.Basic
+import Init.Data.Iterators.Consumers.Partial
+import Init.Data.Iterators.Consumers.Monadic.Collect
 
 /-!
 # Collectors
diff --git a/src/Std/Data/Iterators/Consumers/Loop.lean b/src/Init/Data/Iterators/Consumers/Loop.lean
similarity index 97%
rename from src/Std/Data/Iterators/Consumers/Loop.lean
rename to src/Init/Data/Iterators/Consumers/Loop.lean
index 35bc0e095a..893ffe6116 100644
--- a/src/Std/Data/Iterators/Consumers/Loop.lean
+++ b/src/Init/Data/Iterators/Consumers/Loop.lean
@@ -3,9 +3,11 @@ 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
-import Std.Data.Iterators.Consumers.Monadic.Loop
-import Std.Data.Iterators.Consumers.Partial
+import Init.Data.Iterators.Consumers.Monadic.Loop
+import Init.Data.Iterators.Consumers.Partial
 
 /-!
 # Loop consumers
diff --git a/src/Init/Data/Iterators/Consumers/Monadic.lean b/src/Init/Data/Iterators/Consumers/Monadic.lean
new file mode 100644
index 0000000000..c47ed1f435
--- /dev/null
+++ b/src/Init/Data/Iterators/Consumers/Monadic.lean
@@ -0,0 +1,11 @@
+/-
+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
+import Init.Data.Iterators.Consumers.Monadic.Collect
+import Init.Data.Iterators.Consumers.Monadic.Loop
+import Init.Data.Iterators.Consumers.Monadic.Partial
diff --git a/src/Std/Data/Iterators/Consumers/Monadic/Collect.lean b/src/Init/Data/Iterators/Consumers/Monadic/Collect.lean
similarity index 98%
rename from src/Std/Data/Iterators/Consumers/Monadic/Collect.lean
rename to src/Init/Data/Iterators/Consumers/Monadic/Collect.lean
index 226513ab88..e0e714943c 100644
--- a/src/Std/Data/Iterators/Consumers/Monadic/Collect.lean
+++ b/src/Init/Data/Iterators/Consumers/Monadic/Collect.lean
@@ -3,9 +3,11 @@ 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
-import Std.Data.Iterators.Consumers.Monadic.Partial
-import Std.Data.Internal.LawfulMonadLiftFunction
+import Init.Data.Iterators.Consumers.Monadic.Partial
+import Init.Data.Iterators.Internal.LawfulMonadLiftFunction
 
 /-!
 # Collectors
diff --git a/src/Std/Data/Iterators/Consumers/Monadic/Loop.lean b/src/Init/Data/Iterators/Consumers/Monadic/Loop.lean
similarity index 93%
rename from src/Std/Data/Iterators/Consumers/Monadic/Loop.lean
rename to src/Init/Data/Iterators/Consumers/Monadic/Loop.lean
index 7aefb897a3..650629ed14 100644
--- a/src/Std/Data/Iterators/Consumers/Monadic/Loop.lean
+++ b/src/Init/Data/Iterators/Consumers/Monadic/Loop.lean
@@ -3,10 +3,12 @@ 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
 import Init.RCases
-import Std.Data.Iterators.Basic
-import Std.Data.Iterators.Consumers.Monadic.Partial
+import Init.Data.Iterators.Basic
+import Init.Data.Iterators.Consumers.Monadic.Partial
 
 /-!
 # Loop-based consumers
@@ -62,8 +64,9 @@ class IteratorLoop (α : Type w) (m : Type w → Type w') {β : Type w} [Iterato
   forIn : ∀ (_lift : (γ : Type w) → m γ → n γ) (γ : Type w),
       (plausible_forInStep : β → γ → ForInStep γ → Prop) →
       IteratorLoop.WellFounded α m plausible_forInStep →
-      IterM (α := α) m β → γ →
-      ((b : β) → (c : γ) → n (Subtype (plausible_forInStep b c))) → n γ
+      (it : IterM (α := α) m β) → γ →
+      ((b : β) → (c : γ) → n (Subtype (plausible_forInStep b c))) →
+      n γ
 
 /--
 `IteratorLoopPartial α m` provides efficient implementations of loop-based consumers for `α`-based
@@ -76,7 +79,8 @@ provided by the standard library.
 class IteratorLoopPartial (α : Type w) (m : Type w → Type w') {β : Type w} [Iterator α m β]
     (n : Type w → Type w'') where
   forInPartial : ∀ (_lift : (γ : Type w) → m γ → n γ) {γ : Type w},
-      IterM (α := α) m β → γ → ((b : β) → (c : γ) → n (ForInStep γ)) → n γ
+      (it : IterM (α := α) m β) → γ →
+      ((b : β) → (c : γ) → n (ForInStep γ)) → n γ
 
 end Typeclasses
 
@@ -91,7 +95,7 @@ private def IteratorLoop.WFRel.mk {α : Type w} {m : Type w → Type w'} {β : T
     IteratorLoop.WFRel wf :=
   (it, c)
 
-instance {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
+private instance {α : Type w} {m : Type w → Type w'} {β : Type w} [Iterator α m β]
     {γ : Type x} {plausible_forInStep : β → γ → ForInStep γ → Prop}
     (wf : IteratorLoop.WellFounded α m plausible_forInStep) :
     WellFoundedRelation (IteratorLoop.WFRel wf) where
@@ -116,9 +120,13 @@ def IterM.DefaultConsumers.forIn {m : Type w → Type w'} {α : Type w} {β : Ty
     match ← it.step with
     | .yield it' out _ =>
       match ← f out init with
-      | ⟨.yield c, _⟩ => IterM.DefaultConsumers.forIn lift _ plausible_forInStep wf it' c f
+      | ⟨.yield c, _⟩ =>
+        IterM.DefaultConsumers.forIn lift _ plausible_forInStep wf it' c
+          (fun out acc => f out acc)
       | ⟨.done c, _⟩ => return c
-    | .skip it' _ => IterM.DefaultConsumers.forIn lift _ plausible_forInStep wf it' init f
+    | .skip it' _ =>
+      IterM.DefaultConsumers.forIn lift _ plausible_forInStep wf it' init
+        (fun out acc => f out acc)
     | .done _ => return init
 termination_by IteratorLoop.WFRel.mk wf it init
 decreasing_by
@@ -159,9 +167,13 @@ partial def IterM.DefaultConsumers.forInPartial {m : Type w → Type w'} {α : T
     match ← it.step with
     | .yield it' out _ =>
       match ← f out init with
-      | .yield c => IterM.DefaultConsumers.forInPartial lift _ it' c f
+      | .yield c =>
+        IterM.DefaultConsumers.forInPartial lift _ it' c
+          fun out acc => f out acc
       | .done c => return c
-    | .skip it' _ => IterM.DefaultConsumers.forInPartial lift _ it' init f
+    | .skip it' _ =>
+      IterM.DefaultConsumers.forInPartial lift _ it' init
+        fun out acc => f out acc
     | .done _ => return init
 
 /--
@@ -206,12 +218,13 @@ def IteratorLoop.finiteForIn {m : Type w → Type w'} {n : Type w → Type w''}
   forIn {γ} [Monad n] it init f :=
     IteratorLoop.forIn (α := α) (m := m) lift γ (fun _ _ _ => True)
       wellFounded_of_finite
-      it init ((⟨·, .intro⟩) <$> f · ·)
+      it init (fun out acc => (⟨·, .intro⟩) <$> f out acc)
 
 instance {m : Type w → Type w'} {n : Type w → Type w''}
     {α : Type w} {β : Type w} [Iterator α m β] [Finite α m] [IteratorLoop α m n]
     [MonadLiftT m n] :
-    ForIn n (IterM (α := α) m β) β := IteratorLoop.finiteForIn (fun _ => monadLift)
+    ForIn n (IterM (α := α) m β) β :=
+  IteratorLoop.finiteForIn (fun _ => monadLift)
 
 instance {m : Type w → Type w'} {n : Type w → Type w''}
     {α : Type w} {β : Type w} [Iterator α m β] [IteratorLoopPartial α m n] [MonadLiftT m n] :
diff --git a/src/Std/Data/Iterators/Consumers/Monadic/Partial.lean b/src/Init/Data/Iterators/Consumers/Monadic/Partial.lean
similarity index 95%
rename from src/Std/Data/Iterators/Consumers/Monadic/Partial.lean
rename to src/Init/Data/Iterators/Consumers/Monadic/Partial.lean
index 0729615bdc..5f87ab2cb5 100644
--- a/src/Std/Data/Iterators/Consumers/Monadic/Partial.lean
+++ b/src/Init/Data/Iterators/Consumers/Monadic/Partial.lean
@@ -3,8 +3,10 @@ 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
-import Std.Data.Iterators.Basic
+import Init.Data.Iterators.Basic
 
 namespace Std.Iterators
 
diff --git a/src/Std/Data/Iterators/Consumers/Partial.lean b/src/Init/Data/Iterators/Consumers/Partial.lean
similarity index 95%
rename from src/Std/Data/Iterators/Consumers/Partial.lean
rename to src/Init/Data/Iterators/Consumers/Partial.lean
index 2d16d5b93a..2823d318b2 100644
--- a/src/Std/Data/Iterators/Consumers/Partial.lean
+++ b/src/Init/Data/Iterators/Consumers/Partial.lean
@@ -3,8 +3,10 @@ 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
-import Std.Data.Iterators.Basic
+import Init.Data.Iterators.Basic
 
 namespace Std.Iterators
 
diff --git a/src/Init/Data/Iterators/Internal.lean b/src/Init/Data/Iterators/Internal.lean
new file mode 100644
index 0000000000..2728123f13
--- /dev/null
+++ b/src/Init/Data/Iterators/Internal.lean
@@ -0,0 +1,10 @@
+/-
+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
+import Init.Data.Iterators.Internal.LawfulMonadLiftFunction
+import Init.Data.Iterators.Internal.Termination
diff --git a/src/Std/Data/Internal/LawfulMonadLiftFunction.lean b/src/Init/Data/Iterators/Internal/LawfulMonadLiftFunction.lean
similarity index 99%
rename from src/Std/Data/Internal/LawfulMonadLiftFunction.lean
rename to src/Init/Data/Iterators/Internal/LawfulMonadLiftFunction.lean
index 28508c7d6a..b011065a5c 100644
--- a/src/Std/Data/Internal/LawfulMonadLiftFunction.lean
+++ b/src/Init/Data/Iterators/Internal/LawfulMonadLiftFunction.lean
@@ -3,6 +3,8 @@ 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
 import Init.Control.Basic
 import Init.Control.Lawful.Basic
diff --git a/src/Std/Data/Iterators/Internal/Termination.lean b/src/Init/Data/Iterators/Internal/Termination.lean
similarity index 97%
rename from src/Std/Data/Iterators/Internal/Termination.lean
rename to src/Init/Data/Iterators/Internal/Termination.lean
index 7da422b1d0..0229464884 100644
--- a/src/Std/Data/Iterators/Internal/Termination.lean
+++ b/src/Init/Data/Iterators/Internal/Termination.lean
@@ -3,8 +3,10 @@ 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
-import Std.Data.Iterators.Basic
+import Init.Data.Iterators.Basic
 
 /-!
 This is an internal module used by iterator implementations.
diff --git a/src/Std/Data/Iterators/Internal.lean b/src/Init/Data/Iterators/Lemmas.lean
similarity index 75%
rename from src/Std/Data/Iterators/Internal.lean
rename to src/Init/Data/Iterators/Lemmas.lean
index aa5ab7f1cd..65a041ee83 100644
--- a/src/Std/Data/Iterators/Internal.lean
+++ b/src/Init/Data/Iterators/Lemmas.lean
@@ -3,5 +3,7 @@ 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
-import Std.Data.Iterators.Internal.Termination
+import Init.Data.Iterators.Lemmas.Consumers
diff --git a/src/Std/Data/Iterators/Lemmas/Basic.lean b/src/Init/Data/Iterators/Lemmas/Basic.lean
similarity index 97%
rename from src/Std/Data/Iterators/Lemmas/Basic.lean
rename to src/Init/Data/Iterators/Lemmas/Basic.lean
index f9f208f025..dc8733e83d 100644
--- a/src/Std/Data/Iterators/Lemmas/Basic.lean
+++ b/src/Init/Data/Iterators/Lemmas/Basic.lean
@@ -3,8 +3,10 @@ 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
-import Std.Data.Iterators.Basic
+import Init.Data.Iterators.Basic
 
 namespace Std.Iterators
 
diff --git a/src/Init/Data/Iterators/Lemmas/Consumers.lean b/src/Init/Data/Iterators/Lemmas/Consumers.lean
new file mode 100644
index 0000000000..a363ee747f
--- /dev/null
+++ b/src/Init/Data/Iterators/Lemmas/Consumers.lean
@@ -0,0 +1,11 @@
+/-
+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
+import Init.Data.Iterators.Lemmas.Consumers.Monadic
+import Init.Data.Iterators.Lemmas.Consumers.Collect
+import Init.Data.Iterators.Lemmas.Consumers.Loop
diff --git a/src/Init/Data/Iterators/Lemmas/Consumers/Collect.lean b/src/Init/Data/Iterators/Lemmas/Consumers/Collect.lean
new file mode 100644
index 0000000000..cb257f54ba
--- /dev/null
+++ b/src/Init/Data/Iterators/Lemmas/Consumers/Collect.lean
@@ -0,0 +1,114 @@
+/-
+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
+import Init.Data.Iterators.Lemmas.Basic
+import Init.Data.Iterators.Lemmas.Consumers.Monadic.Collect
+import all Init.Data.Iterators.Consumers.Access
+import all Init.Data.Iterators.Consumers.Collect
+
+namespace Std.Iterators
+
+theorem Iter.toArray_eq_toArray_toIterM {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
+    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
+    it.toArray = it.toIterM.toArray.run :=
+  (rfl)
+
+theorem Iter.toList_eq_toList_toIterM {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
+    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
+    it.toList = it.toIterM.toList.run :=
+  (rfl)
+
+theorem Iter.toListRev_eq_toListRev_toIterM {α β} [Iterator α Id β] [Finite α Id]
+    {it : Iter (α := α) β} :
+    it.toListRev = it.toIterM.toListRev.run :=
+  (rfl)
+
+@[simp]
+theorem IterM.toList_toIter {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
+    {it : IterM (α := α) Id β} :
+    it.toIter.toList = it.toList.run :=
+  (rfl)
+
+@[simp]
+theorem IterM.toListRev_toIter {α β} [Iterator α Id β] [Finite α Id]
+    {it : IterM (α := α) Id β} :
+    it.toIter.toListRev = it.toListRev.run :=
+  (rfl)
+
+theorem Iter.toList_toArray {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
+    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
+    it.toArray.toList = it.toList := by
+  simp [toArray_eq_toArray_toIterM, toList_eq_toList_toIterM, ← IterM.toList_toArray]
+
+theorem Iter.toArray_toList {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
+    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
+    it.toList.toArray = it.toArray := by
+  simp [toArray_eq_toArray_toIterM, toList_eq_toList_toIterM, ← IterM.toArray_toList]
+
+@[simp]
+theorem Iter.reverse_toListRev [Iterator α Id β] [Finite α Id]
+    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    {it : Iter (α := α) β} :
+    it.toListRev.reverse = it.toList := by
+  simp [toListRev_eq_toListRev_toIterM, toList_eq_toList_toIterM, ← IterM.reverse_toListRev]
+
+theorem Iter.toListRev_eq {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
+    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
+    it.toListRev = it.toList.reverse := by
+  simp [Iter.toListRev_eq_toListRev_toIterM, Iter.toList_eq_toList_toIterM, IterM.toListRev_eq]
+
+theorem Iter.toArray_eq_match_step {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
+    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
+    it.toArray = match it.step with
+      | .yield it' out _ => #[out] ++ it'.toArray
+      | .skip it' _ => it'.toArray
+      | .done _ => #[] := by
+  simp only [Iter.toArray_eq_toArray_toIterM, Iter.step]
+  rw [IterM.toArray_eq_match_step, Id.run_bind]
+  generalize it.toIterM.step.run = step
+  cases step using PlausibleIterStep.casesOn <;> simp
+
+theorem Iter.toList_eq_match_step {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
+    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
+    it.toList = match it.step with
+      | .yield it' out _ => out :: it'.toList
+      | .skip it' _ => it'.toList
+      | .done _ => [] := by
+  rw [← Iter.toList_toArray, Iter.toArray_eq_match_step]
+  split <;> simp [Iter.toList_toArray]
+
+theorem Iter.toListRev_eq_match_step {α β} [Iterator α Id β] [Finite α Id] {it : Iter (α := α) β} :
+    it.toListRev = match it.step with
+      | .yield it' out _ => it'.toListRev ++ [out]
+      | .skip it' _ => it'.toListRev
+      | .done _ => [] := by
+  rw [Iter.toListRev_eq_toListRev_toIterM, IterM.toListRev_eq_match_step, Iter.step, Id.run_bind]
+  generalize it.toIterM.step.run = step
+  cases step using PlausibleIterStep.casesOn <;> simp
+
+theorem Iter.getElem?_toList_eq_atIdxSlow? {α β}
+    [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    {it : Iter (α := α) β} {k : Nat} :
+    it.toList[k]? = it.atIdxSlow? k := by
+  induction it using Iter.inductSteps generalizing k with | step it ihy ihs =>
+  rw [toList_eq_match_step, atIdxSlow?]
+  obtain ⟨step, h⟩ := it.step
+  cases step
+  · cases k <;> simp [ihy h]
+  · simp [ihs h]
+  · simp
+
+theorem Iter.toList_eq_of_atIdxSlow?_eq {α₁ α₂ β}
+    [Iterator α₁ Id β] [Finite α₁ Id] [IteratorCollect α₁ Id Id] [LawfulIteratorCollect α₁ Id Id]
+    [Iterator α₂ Id β] [Finite α₂ Id] [IteratorCollect α₂ Id Id] [LawfulIteratorCollect α₂ Id Id]
+    {it₁ : Iter (α := α₁) β} {it₂ : Iter (α := α₂) β}
+    (h : ∀ k, it₁.atIdxSlow? k = it₂.atIdxSlow? k) :
+    it₁.toList = it₂.toList := by
+  ext; simp [getElem?_toList_eq_atIdxSlow?, h]
+
+end Std.Iterators
diff --git a/src/Init/Data/Iterators/Lemmas/Consumers/Loop.lean b/src/Init/Data/Iterators/Lemmas/Consumers/Loop.lean
new file mode 100644
index 0000000000..27faae68e8
--- /dev/null
+++ b/src/Init/Data/Iterators/Lemmas/Consumers/Loop.lean
@@ -0,0 +1,180 @@
+/-
+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
+import Init.Data.Iterators.Lemmas.Consumers.Collect
+import all Init.Data.Iterators.Lemmas.Consumers.Monadic.Loop
+import all Init.Data.Iterators.Consumers.Loop
+
+namespace Std.Iterators
+
+theorem Iter.forIn_eq {α β : Type w} [Iterator α Id β] [Finite α Id]
+    {m : Type w → Type w''} [Monad m] [IteratorLoop α Id m] [hl : LawfulIteratorLoop α Id m]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ}
+    {f : (b : β) → γ → m (ForInStep γ)} :
+    ForIn.forIn it init f =
+      IterM.DefaultConsumers.forIn (fun _ c => pure c.run) γ (fun _ _ _ => True)
+        IteratorLoop.wellFounded_of_finite it.toIterM init
+          (fun out acc => (⟨·, .intro⟩) <$>
+            f out acc) := by
+  cases hl.lawful; rfl
+
+theorem Iter.forIn_eq_forIn_toIterM {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ}
+    {f : β → γ → m (ForInStep γ)} :
+    ForIn.forIn it init f =
+      letI : MonadLift Id m := ⟨Std.Internal.idToMonad (α := _)⟩
+      ForIn.forIn it.toIterM init f := by
+  rfl
+
+theorem Iter.forIn_eq_match_step {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ}
+    {f : β → γ → m (ForInStep γ)} :
+    ForIn.forIn it init f = (do
+      match it.step with
+      | .yield it' out _ =>
+        match ← f out init with
+        | .yield c => ForIn.forIn it' c f
+        | .done c => return c
+      | .skip it' _ => ForIn.forIn it' init f
+      | .done _ => return init) := by
+  rw [Iter.forIn_eq_forIn_toIterM, @IterM.forIn_eq_match_step, Iter.step]
+  simp only [liftM, monadLift, pure_bind]
+  generalize it.toIterM.step = step
+  cases step using PlausibleIterStep.casesOn
+  · apply bind_congr
+    intro forInStep
+    rfl
+  · rfl
+  · rfl
+
+theorem Iter.forIn_toList {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ}
+    {f : β → γ → m (ForInStep γ)} :
+    ForIn.forIn it.toList init f = ForIn.forIn it init f := by
+  rw [List.forIn_eq_foldlM]
+  induction it using Iter.inductSteps generalizing init with case step it ihy ihs =>
+  rw [forIn_eq_match_step, Iter.toList_eq_match_step]
+  simp only [map_eq_pure_bind]
+  generalize it.step = step
+  cases step using PlausibleIterStep.casesOn
+  · rename_i it' out h
+    simp only [List.foldlM_cons, bind_pure_comp, map_bind]
+    apply bind_congr
+    intro forInStep
+    cases forInStep
+    · induction it'.toList <;> simp [*]
+    · simp only [ForIn.forIn, forIn', List.forIn'] at ihy
+      simp [ihy h, forIn_eq_forIn_toIterM]
+  · rename_i it' h
+    simp only [bind_pure_comp]
+    rw [ihs h]
+  · simp
+
+theorem Iter.foldM_eq_forIn {α β γ : Type w} [Iterator α Id β] [Finite α Id] {m : Type w → Type w'}
+    [Monad m] [IteratorLoop α Id m] {f : γ → β → m γ}
+    {init : γ} {it : Iter (α := α) β} :
+    it.foldM (init := init) f = ForIn.forIn it init (fun x acc => ForInStep.yield <$> f acc x) :=
+  (rfl)
+
+theorem Iter.foldM_eq_foldM_toIterM {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ} {f : γ → β → m γ} :
+    it.foldM (init := init) f = letI : MonadLift Id m := ⟨pure⟩; it.toIterM.foldM (init := init) f :=
+  (rfl)
+
+theorem Iter.forIn_yield_eq_foldM {α β γ δ : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m] [IteratorLoop α Id m]
+    [LawfulIteratorLoop α Id m] {f : β → γ → m δ} {g : β → γ → δ → γ} {init : γ}
+    {it : Iter (α := α) β} :
+    ForIn.forIn it init (fun c b => (fun d => .yield (g c b d)) <$> f c b) =
+      it.foldM (fun b c => g c b <$> f c b) init := by
+  simp [Iter.foldM_eq_forIn]
+
+theorem Iter.foldM_eq_match_step {α β γ : Type w} [Iterator α Id β] [Finite α Id]
+    {m : Type w → Type w'} [Monad m] [LawfulMonad m] [IteratorLoop α Id m]
+    [LawfulIteratorLoop α Id m] {f : γ → β → m γ} {init : γ} {it : Iter (α := α) β} :
+    it.foldM (init := init) f = (do
+      match it.step with
+      | .yield it' out _ => it'.foldM (init := ← f init out) f
+      | .skip it' _ => it'.foldM (init := init) f
+      | .done _ => return init) := by
+  rw [Iter.foldM_eq_forIn, Iter.forIn_eq_match_step]
+  generalize it.step = step
+  cases step using PlausibleIterStep.casesOn <;> simp [foldM_eq_forIn]
+
+theorem Iter.foldlM_toList {α β γ : Type w} [Iterator α Id β] [Finite α Id] {m : Type w → Type w'}
+    [Monad m] [LawfulMonad m] [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    {f : γ → β → m γ}
+    {init : γ} {it : Iter (α := α) β} :
+    it.toList.foldlM (init := init) f = it.foldM (init := init) f := by
+  rw [Iter.foldM_eq_forIn, ← Iter.forIn_toList]
+  simp only [List.forIn_yield_eq_foldlM, id_map']
+
+theorem IterM.forIn_eq_foldM {α β : Type w} [Iterator α Id β]
+    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
+    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
+    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    {γ : Type w} {it : Iter (α := α) β} {init : γ}
+    {f : β → γ → m (ForInStep γ)} :
+    forIn it init f = ForInStep.value <$>
+      it.foldM (fun c b => match c with
+        | .yield c => f b c
+        | .done c => pure (.done c)) (ForInStep.yield init) := by
+  simp only [← Iter.forIn_toList, List.forIn_eq_foldlM, ← Iter.foldlM_toList]; rfl
+
+theorem Iter.fold_eq_forIn {α β γ : Type w} [Iterator α Id β]
+    [Finite α Id] [IteratorLoop α Id Id] {f : γ → β → γ} {init : γ} {it : Iter (α := α) β} :
+    it.fold (init := init) f =
+      (ForIn.forIn (m := Id) it init (fun x acc => pure (ForInStep.yield (f acc x)))).run := by
+  rfl
+
+theorem Iter.fold_eq_foldM {α β γ : Type w} [Iterator α Id β]
+    [Finite α Id] [IteratorLoop α Id Id] {f : γ → β → γ} {init : γ}
+    {it : Iter (α := α) β} :
+    it.fold (init := init) f = (it.foldM (m := Id) (init := init) (pure <| f · ·)).run := by
+  simp [foldM_eq_forIn, fold_eq_forIn]
+
+@[simp]
+theorem Iter.forIn_pure_yield_eq_fold {α β γ : Type w} [Iterator α Id β]
+    [Finite α Id] [IteratorLoop α Id Id]
+    [LawfulIteratorLoop α Id Id] {f : β → γ → γ} {init : γ}
+    {it : Iter (α := α) β} :
+    ForIn.forIn (m := Id) it init (fun c b => pure (.yield (f c b))) =
+      pure (it.fold (fun b c => f c b) init) := by
+  simp only [fold_eq_forIn]
+  rfl
+
+theorem Iter.fold_eq_match_step {α β γ : Type w} [Iterator α Id β] [Finite α Id]
+    [IteratorLoop α Id Id] [LawfulIteratorLoop α Id Id]
+    {f : γ → β → γ} {init : γ} {it : Iter (α := α) β} :
+    it.fold (init := init) f = (match it.step with
+      | .yield it' out _ => it'.fold (init := f init out) f
+      | .skip it' _ => it'.fold (init := init) f
+      | .done _ => init) := by
+  rw [fold_eq_foldM, foldM_eq_match_step]
+  simp only [fold_eq_foldM]
+  generalize it.step = step
+  cases step using PlausibleIterStep.casesOn <;> simp
+
+theorem Iter.foldl_toList {α β γ : Type w} [Iterator α Id β] [Finite α Id]
+    [IteratorLoop α Id Id] [LawfulIteratorLoop α Id Id]
+    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
+    {f : γ → β → γ} {init : γ} {it : Iter (α := α) β} :
+    it.toList.foldl (init := init) f = it.fold (init := init) f := by
+  rw [fold_eq_foldM, List.foldl_eq_foldlM, ← Iter.foldlM_toList]
+
+end Std.Iterators
diff --git a/src/Std/Data/Iterators/Consumers/Monadic.lean b/src/Init/Data/Iterators/Lemmas/Consumers/Monadic.lean
similarity index 51%
rename from src/Std/Data/Iterators/Consumers/Monadic.lean
rename to src/Init/Data/Iterators/Lemmas/Consumers/Monadic.lean
index 6e861afb81..bcbc5fd29f 100644
--- a/src/Std/Data/Iterators/Consumers/Monadic.lean
+++ b/src/Init/Data/Iterators/Lemmas/Consumers/Monadic.lean
@@ -3,7 +3,8 @@ 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
-import Std.Data.Iterators.Consumers.Monadic.Collect
-import Std.Data.Iterators.Consumers.Monadic.Loop
-import Std.Data.Iterators.Consumers.Monadic.Partial
+import Init.Data.Iterators.Lemmas.Consumers.Monadic.Collect
+import Init.Data.Iterators.Lemmas.Consumers.Monadic.Loop
diff --git a/src/Init/Data/Iterators/Lemmas/Consumers/Monadic/Collect.lean b/src/Init/Data/Iterators/Lemmas/Consumers/Monadic/Collect.lean
new file mode 100644
index 0000000000..7f45f163c0
--- /dev/null
+++ b/src/Init/Data/Iterators/Lemmas/Consumers/Monadic/Collect.lean
@@ -0,0 +1,157 @@
+/-
+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
+import Init.Data.Array.Lemmas
+import Init.Data.Iterators.Lemmas.Monadic.Basic
+import all Init.Data.Iterators.Consumers.Monadic.Collect
+
+namespace Std.Iterators
+
+variable {α β γ : Type w} {m : Type w → Type w'} {n : Type w → Type w''}
+  {lift : ⦃δ : Type w⦄ → m δ → n δ} {f : β → n γ} {it : IterM (α := α) m β}
+
+theorem IterM.DefaultConsumers.toArrayMapped.go.aux₁ [Monad n] [LawfulMonad n] [Iterator α m β]
+    [Finite α m] {b : γ} {bs : Array γ} :
+    IterM.DefaultConsumers.toArrayMapped.go lift f it (#[b] ++ bs) (m := m) =
+      (#[b] ++ ·) <$> IterM.DefaultConsumers.toArrayMapped.go lift f it bs (m := m) := by
+  induction it, bs using IterM.DefaultConsumers.toArrayMapped.go.induct
+  next it bs ih₁ ih₂ =>
+  rw [go, map_eq_pure_bind, go, bind_assoc]
+  apply bind_congr
+  intro step
+  split
+  · simp [ih₁ _ _ ‹_›]
+  · simp [ih₂ _ ‹_›]
+  · simp
+
+theorem IterM.DefaultConsumers.toArrayMapped.go.aux₂ [Monad n] [LawfulMonad n] [Iterator α m β]
+    [Finite α m] {acc : Array γ} :
+    IterM.DefaultConsumers.toArrayMapped.go lift f it acc (m := m) =
+      (acc ++ ·) <$> IterM.DefaultConsumers.toArrayMapped lift f it (m := m) := by
+  rw [← Array.toArray_toList (xs := acc)]
+  generalize acc.toList = acc
+  induction acc with
+  | nil => simp [toArrayMapped]
+  | cons x xs ih =>
+    rw [List.toArray_cons, IterM.DefaultConsumers.toArrayMapped.go.aux₁, ih]
+    simp only [Functor.map_map, Array.append_assoc]
+
+theorem IterM.DefaultConsumers.toArrayMapped_eq_match_step [Monad n] [LawfulMonad n]
+    [Iterator α m β] [Finite α m] :
+    IterM.DefaultConsumers.toArrayMapped lift f it (m := m) = letI : MonadLift m n := ⟨lift (δ := _)⟩; (do
+      match ← it.step with
+      | .yield it' out _ =>
+        return #[← f out] ++ (← IterM.DefaultConsumers.toArrayMapped lift f it' (m := m))
+      | .skip it' _ => IterM.DefaultConsumers.toArrayMapped lift f it' (m := m)
+      | .done _ => return #[]) := by
+  rw [IterM.DefaultConsumers.toArrayMapped, IterM.DefaultConsumers.toArrayMapped.go]
+  apply bind_congr
+  intro step
+  split <;> simp [IterM.DefaultConsumers.toArrayMapped.go.aux₂]
+
+theorem IterM.toArray_eq_match_step [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
+    [IteratorCollect α m m] [LawfulIteratorCollect α m m] :
+    it.toArray = (do
+      match ← it.step with
+      | .yield it' out _ => return #[out] ++ (← it'.toArray)
+      | .skip it' _ => it'.toArray
+      | .done _ => return #[]) := by
+  simp only [IterM.toArray, LawfulIteratorCollect.toArrayMapped_eq]
+  rw [IterM.DefaultConsumers.toArrayMapped_eq_match_step]
+  simp [bind_pure_comp, pure_bind, toArray]
+
+theorem IterM.toList_toArray [Monad m] [Iterator α m β] [Finite α m] [IteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    Array.toList <$> it.toArray = it.toList := by
+  simp [IterM.toList]
+
+theorem IterM.toArray_toList [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
+    [IteratorCollect α m m] {it : IterM (α := α) m β} :
+    List.toArray <$> it.toList = it.toArray := by
+  simp [IterM.toList]
+
+theorem IterM.toList_eq_match_step [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
+    [IteratorCollect α m m] [LawfulIteratorCollect α m m] {it : IterM (α := α) m β} :
+    it.toList = (do
+      match ← it.step with
+      | .yield it' out _ => return out :: (← it'.toList)
+      | .skip it' _ => it'.toList
+      | .done _ => return []) := by
+  simp [← IterM.toList_toArray]
+  rw [IterM.toArray_eq_match_step, map_eq_pure_bind, bind_assoc]
+  apply bind_congr
+  intro step
+  split <;> simp
+
+theorem IterM.toListRev.go.aux₁ [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
+    {it : IterM (α := α) m β} {b : β} {bs : List β} :
+    IterM.toListRev.go it (bs ++ [b]) = (· ++ [b]) <$> IterM.toListRev.go it bs:= by
+  induction it, bs using IterM.toListRev.go.induct
+  next it bs ih₁ ih₂ =>
+  rw [go, go, map_eq_pure_bind, bind_assoc]
+  apply bind_congr
+  intro step
+  simp only [List.cons_append] at ih₁
+  split <;> simp [*]
+
+theorem IterM.toListRev.go.aux₂ [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
+    {it : IterM (α := α) m β} {acc : List β} :
+    IterM.toListRev.go it acc = (· ++ acc) <$> it.toListRev := by
+  rw [← List.reverse_reverse (as := acc)]
+  generalize acc.reverse = acc
+  induction acc with
+  | nil => simp [toListRev]
+  | cons x xs ih => simp [IterM.toListRev.go.aux₁, ih]
+
+theorem IterM.toListRev_eq_match_step [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
+    {it : IterM (α := α) m β} :
+    it.toListRev = (do
+      match ← it.step with
+      | .yield it' out _ => return (← it'.toListRev) ++ [out]
+      | .skip it' _ => it'.toListRev
+      | .done _ => return []) := by
+  simp [IterM.toListRev]
+  rw [toListRev.go]
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn <;> simp [IterM.toListRev.go.aux₂]
+
+theorem IterM.reverse_toListRev [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
+    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    List.reverse <$> it.toListRev = it.toList := by
+  apply Eq.symm
+  induction it using IterM.inductSteps
+  rename_i it ihy ihs
+  rw [toListRev_eq_match_step, toList_eq_match_step, map_eq_pure_bind, bind_assoc]
+  apply bind_congr
+  intro step
+  split <;> simp (discharger := assumption) [ihy, ihs]
+
+theorem IterM.toListRev_eq [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
+    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    it.toListRev = List.reverse <$> it.toList := by
+  rw [← IterM.reverse_toListRev]
+  simp
+
+theorem LawfulIteratorCollect.toArray_eq {α β : Type w} {m : Type w → Type w'}
+    [Monad m] [Iterator α m β] [Finite α m] [IteratorCollect α m m]
+    [hl : LawfulIteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    it.toArray = (letI : IteratorCollect α m m := .defaultImplementation; it.toArray) := by
+  simp only [IterM.toArray, toArrayMapped_eq]
+
+theorem LawfulIteratorCollect.toList_eq {α β : Type w} {m : Type w → Type w'}
+    [Monad m] [Iterator α m β] [Finite α m] [IteratorCollect α m m]
+    [hl : LawfulIteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    it.toList = (letI : IteratorCollect α m m := .defaultImplementation; it.toList) := by
+  simp [IterM.toList, toArray_eq]
+
+end Std.Iterators
diff --git a/src/Init/Data/Iterators/Lemmas/Consumers/Monadic/Loop.lean b/src/Init/Data/Iterators/Lemmas/Consumers/Monadic/Loop.lean
new file mode 100644
index 0000000000..e594b61fab
--- /dev/null
+++ b/src/Init/Data/Iterators/Lemmas/Consumers/Monadic/Loop.lean
@@ -0,0 +1,239 @@
+/-
+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
+import Init.Data.Iterators.Lemmas.Consumers.Monadic.Collect
+import all Init.Data.Iterators.Consumers.Monadic.Loop
+
+namespace Std.Iterators
+
+theorem IterM.DefaultConsumers.forIn_eq_match_step {α β : Type w} {m : Type w → Type w'}
+    [Iterator α m β]
+    {n : Type w → Type w''} [Monad n]
+    {lift : ∀ γ, m γ → n γ} {γ : Type w}
+    {plausible_forInStep : β → γ → ForInStep γ → Prop}
+    {wf : IteratorLoop.WellFounded α m plausible_forInStep}
+    {it : IterM (α := α) m β} {init : γ}
+    {f : (b : β) → (c : γ) → n (Subtype (plausible_forInStep b c))} :
+    IterM.DefaultConsumers.forIn lift γ plausible_forInStep wf it init f = (do
+      match ← lift _ it.step with
+      | .yield it' out _ =>
+        match ← f out init with
+        | ⟨.yield c, _⟩ =>
+          IterM.DefaultConsumers.forIn lift _ plausible_forInStep wf it' c
+            fun out acc => f out acc
+        | ⟨.done c, _⟩ => return c
+      | .skip it' _ =>
+        IterM.DefaultConsumers.forIn lift _ plausible_forInStep wf it' init
+          fun out acc => f out acc
+      | .done _ => return init) := by
+  rw [forIn]
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn <;> rfl
+
+theorem IterM.forIn_eq {α β : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
+    {n : Type w → Type w''} [Monad n] [IteratorLoop α m n] [hl : LawfulIteratorLoop α m n]
+    [MonadLiftT m n] {γ : Type w} {it : IterM (α := α) m β} {init : γ}
+    {f : β → γ → n (ForInStep γ)} :
+    ForIn.forIn it init f = IterM.DefaultConsumers.forIn (fun _ => monadLift) γ (fun _ _ _ => True)
+        IteratorLoop.wellFounded_of_finite it init (fun out acc => (⟨·, .intro⟩) <$> f out acc) := by
+  cases hl.lawful; rfl
+
+theorem IterM.forIn_eq_match_step {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] {n : Type w → Type w''} [Monad n] [LawfulMonad n]
+    [IteratorLoop α m n] [LawfulIteratorLoop α m n]
+    [MonadLiftT m n] {γ : Type w} {it : IterM (α := α) m β} {init : γ}
+    {f : β → γ → n (ForInStep γ)} :
+    ForIn.forIn it init f = (do
+      match ← it.step with
+      | .yield it' out _ =>
+        match ← f out init with
+        | .yield c => ForIn.forIn it' c f
+        | .done c => return c
+      | .skip it' _ => ForIn.forIn it' init f
+      | .done _ => return init) := by
+  rw [IterM.forIn_eq, DefaultConsumers.forIn_eq_match_step]
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn
+  · simp only [map_eq_pure_bind, bind_assoc]
+    apply bind_congr
+    intro forInStep
+    cases forInStep <;> simp [IterM.forIn_eq]
+  · simp [IterM.forIn_eq]
+  · simp
+
+theorem IterM.forM_eq_forIn {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] {n : Type w → Type w''} [Monad n] [LawfulMonad n]
+    [IteratorLoop α m n] [LawfulIteratorLoop α m n]
+    [MonadLiftT m n] {it : IterM (α := α) m β}
+    {f : β → n PUnit} :
+    ForM.forM it f = ForIn.forIn it PUnit.unit (fun out _ => do f out; return .yield .unit) :=
+  rfl
+
+theorem IterM.forM_eq_match_step {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] {n : Type w → Type w''} [Monad n] [LawfulMonad n]
+    [IteratorLoop α m n] [LawfulIteratorLoop α m n]
+    [MonadLiftT m n] {it : IterM (α := α) m β}
+    {f : β → n PUnit} :
+    ForM.forM it f = (do
+      match ← it.step with
+      | .yield it' out _ =>
+        f out
+        ForM.forM it' f
+      | .skip it' _ => ForM.forM it' f
+      | .done _ => return) := by
+  rw [forM_eq_forIn, forIn_eq_match_step]
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn <;> simp [forM_eq_forIn]
+
+theorem IterM.foldM_eq_forIn {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
+    {n : Type w → Type w''} [Monad n] [IteratorLoop α m n] [MonadLiftT m n] {f : γ → β → n γ}
+    {init : γ} {it : IterM (α := α) m β} :
+    it.foldM (init := init) f = ForIn.forIn it init (fun x acc => ForInStep.yield <$> f acc x) :=
+  (rfl)
+
+theorem IterM.forIn_yield_eq_foldM {α β γ δ : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] {n : Type w → Type w''} [Monad n] [LawfulMonad n] [IteratorLoop α m n]
+    [LawfulIteratorLoop α m n] [MonadLiftT m n] {f : β → γ → n δ} {g : β → γ → δ → γ} {init : γ}
+    {it : IterM (α := α) m β} :
+    ForIn.forIn it init (fun c b => (fun d => .yield (g c b d)) <$> f c b) =
+      it.foldM (fun b c => g c b <$> f c b) init := by
+  simp [IterM.foldM_eq_forIn]
+
+theorem IterM.foldM_eq_match_step {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
+    {n : Type w → Type w''} [Monad n] [LawfulMonad n] [IteratorLoop α m n] [LawfulIteratorLoop α m n]
+    [MonadLiftT m n] {f : γ → β → n γ} {init : γ} {it : IterM (α := α) m β} :
+    it.foldM (init := init) f = (do
+      match ← it.step with
+      | .yield it' out _ => it'.foldM (init := ← f init out) f
+      | .skip it' _ => it'.foldM (init := init) f
+      | .done _ => return init) := by
+  rw [IterM.foldM_eq_forIn, IterM.forIn_eq_match_step]
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn <;> simp [foldM_eq_forIn]
+
+theorem IterM.fold_eq_forIn {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] [Monad m]
+    [IteratorLoop α m m] {f : γ → β → γ} {init : γ} {it : IterM (α := α) m β} :
+    it.fold (init := init) f =
+      ForIn.forIn (m := m) it init (fun x acc => pure (ForInStep.yield (f acc x))) := by
+  rfl
+
+theorem IterM.fold_eq_foldM {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m] {f : γ → β → γ} {init : γ}
+    {it : IterM (α := α) m β} :
+    it.fold (init := init) f = it.foldM (init := init) (pure <| f · ·) := by
+  simp [foldM_eq_forIn, fold_eq_forIn]
+
+@[simp]
+theorem IterM.forIn_pure_yield_eq_fold {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m]
+    [LawfulIteratorLoop α m m] {f : β → γ → γ} {init : γ}
+    {it : IterM (α := α) m β} :
+    ForIn.forIn it init (fun c b => pure (.yield (f c b))) =
+      it.fold (fun b c => f c b) init := by
+  simp [IterM.fold_eq_forIn]
+
+theorem IterM.fold_eq_match_step {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
+    [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
+    {f : γ → β → γ} {init : γ} {it : IterM (α := α) m β} :
+    it.fold (init := init) f = (do
+      match ← it.step with
+      | .yield it' out _ => it'.fold (init := f init out) f
+      | .skip it' _ => it'.fold (init := init) f
+      | .done _ => return init) := by
+  rw [fold_eq_foldM, foldM_eq_match_step]
+  simp only [fold_eq_foldM]
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn <;> simp
+
+theorem IterM.toList_eq_fold {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
+    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    it.toList = it.fold (init := []) (fun l out => l ++ [out]) := by
+  suffices h : ∀ l' : List β, (l' ++ ·) <$> it.toList =
+      it.fold (init := l') (fun l out => l ++ [out]) by
+    specialize h []
+    simpa using h
+  induction it using IterM.inductSteps with | step it ihy ihs =>
+  intro l'
+  rw [IterM.toList_eq_match_step, IterM.fold_eq_match_step]
+  simp only [map_eq_pure_bind, bind_assoc]
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn
+  · rename_i it' out h
+    specialize ihy h (l' ++ [out])
+    simpa using ihy
+  · rename_i it' h
+    simp [ihs h]
+  · simp
+
+theorem IterM.drain_eq_fold {α β : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
+    [Monad m] [IteratorLoop α m m] {it : IterM (α := α) m β} :
+    it.drain = it.fold (init := PUnit.unit) (fun _ _ => .unit) :=
+  (rfl)
+
+theorem IterM.drain_eq_foldM {α β : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
+    [Monad m] [LawfulMonad m] [IteratorLoop α m m] {it : IterM (α := α) m β} :
+    it.drain = it.foldM (init := PUnit.unit) (fun _ _ => pure .unit) := by
+  simp [IterM.drain_eq_fold, IterM.fold_eq_foldM]
+
+theorem IterM.drain_eq_forIn {α β : Type w} {m : Type w → Type w'}  [Iterator α m β] [Finite α m]
+    [Monad m] [IteratorLoop α m m] {it : IterM (α := α) m β} :
+    it.drain = ForIn.forIn (m := m) it PUnit.unit (fun _ _ => pure (ForInStep.yield .unit)) := by
+  simp [IterM.drain_eq_fold, IterM.fold_eq_forIn]
+
+theorem IterM.drain_eq_match_step {α β : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
+    [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
+    {it : IterM (α := α) m β} :
+    it.drain = (do
+      match ← it.step with
+      | .yield it' _ _ => it'.drain
+      | .skip it' _ => it'.drain
+      | .done _ => return .unit) := by
+  rw [IterM.drain_eq_fold, IterM.fold_eq_match_step]
+  simp [IterM.drain_eq_fold]
+
+theorem IterM.drain_eq_map_toList {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
+    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    it.drain = (fun _ => .unit) <$> it.toList := by
+  induction it using IterM.inductSteps with | step it ihy ihs =>
+  rw [IterM.drain_eq_match_step, IterM.toList_eq_match_step]
+  simp only [map_eq_pure_bind, bind_assoc]
+  apply bind_congr
+  intro step
+  cases step using PlausibleIterStep.casesOn
+  · rename_i it' out h
+    simp [ihy h]
+  · rename_i it' h
+    simp [ihs h]
+  · simp
+
+theorem IterM.drain_eq_map_toListRev {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
+    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    it.drain = (fun _ => .unit) <$> it.toListRev := by
+  simp [IterM.drain_eq_map_toList, IterM.toListRev_eq]
+
+theorem IterM.drain_eq_map_toArray {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
+    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
+    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
+    {it : IterM (α := α) m β} :
+    it.drain = (fun _ => .unit) <$> it.toList := by
+  simp [IterM.drain_eq_map_toList]
+
+end Std.Iterators
diff --git a/src/Std/Data/Iterators/Lemmas/Monadic/Basic.lean b/src/Init/Data/Iterators/Lemmas/Monadic/Basic.lean
similarity index 97%
rename from src/Std/Data/Iterators/Lemmas/Monadic/Basic.lean
rename to src/Init/Data/Iterators/Lemmas/Monadic/Basic.lean
index cabd7e96ef..3753c26f9d 100644
--- a/src/Std/Data/Iterators/Lemmas/Monadic/Basic.lean
+++ b/src/Init/Data/Iterators/Lemmas/Monadic/Basic.lean
@@ -3,8 +3,10 @@ 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
-import Std.Data.Iterators.Basic
+import Init.Data.Iterators.Basic
 
 namespace Std.Iterators
 
diff --git a/src/Std/Data/Iterators/PostConditionMonad.lean b/src/Init/Data/Iterators/PostconditionMonad.lean
similarity index 90%
rename from src/Std/Data/Iterators/PostConditionMonad.lean
rename to src/Init/Data/Iterators/PostconditionMonad.lean
index 245b10b5ca..eca755026a 100644
--- a/src/Std/Data/Iterators/PostConditionMonad.lean
+++ b/src/Init/Data/Iterators/PostconditionMonad.lean
@@ -3,6 +3,8 @@ 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
 import Init.Control.Lawful.Basic
 import Init.Data.Subtype
@@ -48,6 +50,7 @@ def PostconditionT.lift {α : Type w} {m : Type w → Type w'} [Functor m] (x :
     PostconditionT m α :=
   ⟨fun _ => True, (⟨·, .intro⟩) <$> x⟩
 
+@[always_inline, inline]
 protected def PostconditionT.pure {m : Type w → Type w'} [Pure m] {α : Type w}
     (a : α) : PostconditionT m α :=
   ⟨fun y => a = y, pure <| ⟨a, rfl⟩⟩
@@ -117,16 +120,20 @@ instance {m : Type w → Type w'} [Monad m] : Monad (PostconditionT m) where
   pure := PostconditionT.pure
   bind := PostconditionT.bind
 
-@[simp]
-theorem PostconditionT.computation_pure {m : Type w → Type w'} [Monad m] {α : Type w}
-    {x : α} :
-    (pure x : PostconditionT m α).operation = pure ⟨x, rfl⟩ :=
+theorem PostconditionT.pure_eq_pure {m : Type w → Type w'} [Monad m] {α} {a : α} :
+    pure a = PostconditionT.pure (m := m) a :=
   rfl
 
 @[simp]
 theorem PostconditionT.property_pure {m : Type w → Type w'} [Monad m] {α : Type w}
     {x : α} :
-    (pure x : PostconditionT m α).Property = (x = ·) :=
+    (pure x : PostconditionT m α).Property = (x = ·) := by
+  rfl
+
+@[simp]
+theorem PostconditionT.operation_pure {m : Type w → Type w'} [Monad m] {α : Type w}
+    {x : α} :
+    (pure x : PostconditionT m α).operation = pure ⟨x, property_pure (m := m) ▸ rfl⟩ := by
   rfl
 
 theorem PostconditionT.ext {m : Type w → Type w'} [Monad m] [LawfulMonad m]
@@ -209,12 +216,19 @@ theorem PostconditionT.property_map {m : Type w → Type w'} [Functor m] {α : T
 @[simp]
 theorem PostconditionT.operation_map {m : Type w → Type w'} [Functor m] {α : Type w} {β : Type w}
     {x : PostconditionT m α} {f : α → β} :
-    (x.map f).operation = (fun a => ⟨_, a, rfl⟩) <$> x.operation :=
+    (x.map f).operation =
+      (fun a => ⟨_, (property_map (m := m)).mpr ⟨a.1, rfl, a.2⟩⟩) <$> x.operation := by
   rfl
 
 @[simp]
-theorem PostconditionT.operation_lift {m : Type w →Type w'} [Functor m] {α : Type w}
-    {x : m α} : (lift x : PostconditionT m α).operation = (⟨·, True.intro⟩) <$> x :=
+theorem PostconditionT.property_lift {m : Type w → Type w'} [Functor m] {α : Type w}
+    {x : m α} : (lift x : PostconditionT m α).Property = (fun _ => True) := by
+  rfl
+
+@[simp]
+theorem PostconditionT.operation_lift {m : Type w → Type w'} [Functor m] {α : Type w}
+    {x : m α} : (lift x : PostconditionT m α).operation =
+      (⟨·, property_lift (m := m) ▸ True.intro⟩) <$> x := by
   rfl
 
 end Std.Iterators
diff --git a/src/Std/Data/Iterators.lean b/src/Std/Data/Iterators.lean
index 2e4c17e3bb..2aa9f61c6a 100644
--- a/src/Std/Data/Iterators.lean
+++ b/src/Std/Data/Iterators.lean
@@ -4,13 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
 prelude
-import Std.Data.Iterators.Basic
+import Init.Data.Iterators.Basic
+import Init.Data.Iterators.Consumers
+import Init.Data.Iterators.PostconditionMonad
+import Init.Data.Iterators.Internal
 import Std.Data.Iterators.Producers
-import Std.Data.Iterators.Consumers
 import Std.Data.Iterators.Combinators
 import Std.Data.Iterators.Lemmas
-import Std.Data.Iterators.PostConditionMonad
-import Std.Data.Iterators.Internal
 
 /-!
 # Iterators
@@ -79,7 +79,9 @@ that needs to wait for a language feature.
 A distinction that cuts through the whole module is that between pure and monadic
 iterators. Each submodule contains a dedicated `Monadic` sub-submodule.
 
-All of the following module names are prefixed with `Std.Data.Iterators`.
+Depending on whether functionality is needed in `Init`, modules might exist in `Init` or in `Std`.
+The following module names can then be found under `Init.Data.Iterators`, `Std.Data.Iterators` or
+both.
 
 ### Basic iterator API
 
diff --git a/src/Std/Data/Iterators/Combinators/Monadic/Drop.lean b/src/Std/Data/Iterators/Combinators/Monadic/Drop.lean
index 3297883175..856ec7724c 100644
--- a/src/Std/Data/Iterators/Combinators/Monadic/Drop.lean
+++ b/src/Std/Data/Iterators/Combinators/Monadic/Drop.lean
@@ -4,10 +4,10 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
 prelude
-import Std.Data.Iterators.Basic
-import Std.Data.Iterators.Consumers.Collect
-import Std.Data.Iterators.Consumers.Loop
-import Std.Data.Iterators.Internal.Termination
+import Init.Data.Iterators.Basic
+import Init.Data.Iterators.Consumers.Collect
+import Init.Data.Iterators.Consumers.Loop
+import Init.Data.Iterators.Internal.Termination
 
 /-!
 This file provides the iterator combinator `IterM.drop`.
diff --git a/src/Std/Data/Iterators/Combinators/Monadic/DropWhile.lean b/src/Std/Data/Iterators/Combinators/Monadic/DropWhile.lean
index 1ce9df4516..32c5d15cd8 100644
--- a/src/Std/Data/Iterators/Combinators/Monadic/DropWhile.lean
+++ b/src/Std/Data/Iterators/Combinators/Monadic/DropWhile.lean
@@ -6,11 +6,11 @@ Authors: Paul Reichert
 prelude
 import Init.Data.Nat.Lemmas
 import Init.RCases
-import Std.Data.Iterators.Basic
-import Std.Data.Iterators.Consumers.Monadic.Collect
-import Std.Data.Iterators.Consumers.Monadic.Loop
-import Std.Data.Iterators.Internal.Termination
-import Std.Data.Iterators.PostConditionMonad
+import Init.Data.Iterators.Basic
+import Init.Data.Iterators.Consumers.Monadic.Collect
+import Init.Data.Iterators.Consumers.Monadic.Loop
+import Init.Data.Iterators.Internal.Termination
+import Init.Data.Iterators.PostconditionMonad
 
 /-!
 # Monadic `dropWhile` iterator combinator
diff --git a/src/Std/Data/Iterators/Combinators/Monadic/FilterMap.lean b/src/Std/Data/Iterators/Combinators/Monadic/FilterMap.lean
index c309fa3148..8e3ccacc86 100644
--- a/src/Std/Data/Iterators/Combinators/Monadic/FilterMap.lean
+++ b/src/Std/Data/Iterators/Combinators/Monadic/FilterMap.lean
@@ -4,11 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
 prelude
-import Std.Data.Iterators.Basic
-import Std.Data.Iterators.Consumers.Collect
-import Std.Data.Iterators.Consumers.Loop
-import Std.Data.Iterators.PostConditionMonad
-import Std.Data.Iterators.Internal.Termination
+import Init.Data.Iterators.Basic
+import Init.Data.Iterators.Consumers.Collect
+import Init.Data.Iterators.Consumers.Loop
+import Init.Data.Iterators.PostconditionMonad
+import Init.Data.Iterators.Internal.Termination
 
 /-!
 
diff --git a/src/Std/Data/Iterators/Combinators/Monadic/Take.lean b/src/Std/Data/Iterators/Combinators/Monadic/Take.lean
index 19d5675faf..e0b1232d79 100644
--- a/src/Std/Data/Iterators/Combinators/Monadic/Take.lean
+++ b/src/Std/Data/Iterators/Combinators/Monadic/Take.lean
@@ -6,10 +6,10 @@ Authors: Paul Reichert
 prelude
 import Init.Data.Nat.Lemmas
 import Init.RCases
-import Std.Data.Iterators.Basic
-import Std.Data.Iterators.Consumers.Monadic.Collect
-import Std.Data.Iterators.Consumers.Monadic.Loop
-import Std.Data.Iterators.Internal.Termination
+import Init.Data.Iterators.Basic
+import Init.Data.Iterators.Consumers.Monadic.Collect
+import Init.Data.Iterators.Consumers.Monadic.Loop
+import Init.Data.Iterators.Internal.Termination
 
 /-!
 This module provides the iterator combinator `IterM.take`.
@@ -143,15 +143,9 @@ instance Take.instIteratorLoop [Monad m] [Monad n] [Iterator α m β]
     IteratorLoop (Take α m β) m n :=
   .defaultImplementation
 
-instance Take.instIteratorForPartial [Monad m] [Monad n] [Iterator α m β]
+instance Take.instIteratorLoopPartial [Monad m] [Monad n] [Iterator α m β]
     [IteratorLoopPartial α m n] [MonadLiftT m n] :
-    IteratorLoopPartial (Take α m β) m n where
-  forInPartial lift {γ} it init f := do
-    Prod.fst <$> IteratorLoopPartial.forInPartial lift it.internalState.inner (γ := γ × Nat)
-        (init, it.internalState.remaining)
-        fun out acc =>
-          match acc.snd with
-          | 0 => pure <| .done acc
-          | n + 1 => (fun | .yield x => .yield ⟨x, n⟩ | .done x => .done ⟨x, n⟩) <$> f out acc.fst
+    IteratorLoopPartial (Take α m β) m n :=
+  .defaultImplementation
 
 end Std.Iterators
diff --git a/src/Std/Data/Iterators/Combinators/Monadic/TakeWhile.lean b/src/Std/Data/Iterators/Combinators/Monadic/TakeWhile.lean
index 4b19fe458d..9d2dab13ad 100644
--- a/src/Std/Data/Iterators/Combinators/Monadic/TakeWhile.lean
+++ b/src/Std/Data/Iterators/Combinators/Monadic/TakeWhile.lean
@@ -6,11 +6,11 @@ Authors: Paul Reichert
 prelude
 import Init.Data.Nat.Lemmas
 import Init.RCases
-import Std.Data.Iterators.Basic
-import Std.Data.Iterators.Consumers.Monadic.Collect
-import Std.Data.Iterators.Consumers.Monadic.Loop
-import Std.Data.Iterators.Internal.Termination
-import Std.Data.Iterators.PostConditionMonad
+import Init.Data.Iterators.Basic
+import Init.Data.Iterators.Consumers.Monadic.Collect
+import Init.Data.Iterators.Consumers.Monadic.Loop
+import Init.Data.Iterators.Internal.Termination
+import Init.Data.Iterators.PostconditionMonad
 
 /-!
 # Monadic `takeWhile` iterator combinator
diff --git a/src/Std/Data/Iterators/Combinators/Monadic/Zip.lean b/src/Std/Data/Iterators/Combinators/Monadic/Zip.lean
index 1105ab1a12..9fa557280e 100644
--- a/src/Std/Data/Iterators/Combinators/Monadic/Zip.lean
+++ b/src/Std/Data/Iterators/Combinators/Monadic/Zip.lean
@@ -5,10 +5,10 @@ Authors: Paul Reichert
 -/
 prelude
 import Init.Data.Option.Lemmas
-import Std.Data.Iterators.Basic
-import Std.Data.Iterators.Consumers.Collect
-import Std.Data.Iterators.Consumers.Loop
-import Std.Data.Iterators.Internal.Termination
+import Init.Data.Iterators.Basic
+import Init.Data.Iterators.Consumers.Collect
+import Init.Data.Iterators.Consumers.Loop
+import Init.Data.Iterators.Internal.Termination
 
 /-!
 
diff --git a/src/Std/Data/Iterators/Consumers.lean b/src/Std/Data/Iterators/Consumers.lean
deleted file mode 100644
index f835cc73e8..0000000000
--- a/src/Std/Data/Iterators/Consumers.lean
+++ /dev/null
@@ -1,11 +0,0 @@
-/-
-Copyright (c) 2025 Lean FRO, LLC. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Paul Reichert
--/
-prelude
-import Std.Data.Iterators.Consumers.Monadic
-import Std.Data.Iterators.Consumers.Access
-import Std.Data.Iterators.Consumers.Collect
-import Std.Data.Iterators.Consumers.Loop
-import Std.Data.Iterators.Consumers.Partial
diff --git a/src/Std/Data/Iterators/Lemmas.lean b/src/Std/Data/Iterators/Lemmas.lean
index 834c639833..927b232152 100644
--- a/src/Std/Data/Iterators/Lemmas.lean
+++ b/src/Std/Data/Iterators/Lemmas.lean
@@ -4,9 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
 prelude
-import Std.Data.Iterators.Lemmas.Basic
+import Init.Data.Iterators.Lemmas.Basic
 import Std.Data.Iterators.Lemmas.Monadic
-import Std.Data.Iterators.Lemmas.Consumers
 import Std.Data.Iterators.Lemmas.Combinators
 import Std.Data.Iterators.Lemmas.Producers
 import Std.Data.Iterators.Lemmas.Equivalence
diff --git a/src/Std/Data/Iterators/Lemmas/Combinators/Drop.lean b/src/Std/Data/Iterators/Lemmas/Combinators/Drop.lean
index d17200b2c9..235089465b 100644
--- a/src/Std/Data/Iterators/Lemmas/Combinators/Drop.lean
+++ b/src/Std/Data/Iterators/Lemmas/Combinators/Drop.lean
@@ -7,7 +7,7 @@ prelude
 import Std.Data.Iterators.Combinators.Drop
 import Std.Data.Iterators.Lemmas.Combinators.Monadic.Drop
 import Std.Data.Iterators.Lemmas.Combinators.Take
-import Std.Data.Iterators.Lemmas.Consumers
+import Init.Data.Iterators.Lemmas.Consumers
 
 namespace Std.Iterators
 
diff --git a/src/Std/Data/Iterators/Lemmas/Combinators/DropWhile.lean b/src/Std/Data/Iterators/Lemmas/Combinators/DropWhile.lean
index eddbf2203c..0eb19541e8 100644
--- a/src/Std/Data/Iterators/Lemmas/Combinators/DropWhile.lean
+++ b/src/Std/Data/Iterators/Lemmas/Combinators/DropWhile.lean
@@ -6,7 +6,7 @@ Authors: Paul Reichert
 prelude
 import Std.Data.Iterators.Combinators.DropWhile
 import Std.Data.Iterators.Lemmas.Combinators.Monadic.DropWhile
-import Std.Data.Iterators.Lemmas.Consumers
+import Init.Data.Iterators.Lemmas.Consumers
 
 namespace Std.Iterators
 
diff --git a/src/Std/Data/Iterators/Lemmas/Combinators/FilterMap.lean b/src/Std/Data/Iterators/Lemmas/Combinators/FilterMap.lean
index 0042dc9127..d987a71d82 100644
--- a/src/Std/Data/Iterators/Lemmas/Combinators/FilterMap.lean
+++ b/src/Std/Data/Iterators/Lemmas/Combinators/FilterMap.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
 prelude
-import Std.Data.Iterators.Lemmas.Consumers
+import Init.Data.Iterators.Lemmas.Consumers
 import Std.Data.Iterators.Lemmas.Combinators.Monadic.FilterMap
 import Std.Data.Iterators.Combinators.FilterMap
 
diff --git a/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/Drop.lean b/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/Drop.lean
index f52ab64a48..afd4a6b8a6 100644
--- a/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/Drop.lean
+++ b/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/Drop.lean
@@ -5,7 +5,7 @@ Authors: Paul Reichert
 -/
 prelude
 import Std.Data.Iterators.Combinators.Monadic.Drop
-import Std.Data.Iterators.Lemmas.Consumers.Monadic
+import Init.Data.Iterators.Lemmas.Consumers.Monadic
 
 namespace Std.Iterators
 
diff --git a/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/DropWhile.lean b/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/DropWhile.lean
index 0585fbdcf4..f342319918 100644
--- a/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/DropWhile.lean
+++ b/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/DropWhile.lean
@@ -5,7 +5,7 @@ Authors: Paul Reichert
 -/
 prelude
 import Std.Data.Iterators.Combinators.Monadic.DropWhile
-import Std.Data.Iterators.Lemmas.Consumers.Monadic
+import Init.Data.Iterators.Lemmas.Consumers.Monadic
 
 namespace Std.Iterators
 
diff --git a/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/FilterMap.lean b/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/FilterMap.lean
index c709df1303..c6d68f8c8c 100644
--- a/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/FilterMap.lean
+++ b/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/FilterMap.lean
@@ -4,9 +4,9 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
 prelude
-import Std.Data.Internal.LawfulMonadLiftFunction
+import Init.Data.Iterators.Internal.LawfulMonadLiftFunction
 import Std.Data.Iterators.Combinators.Monadic.FilterMap
-import Std.Data.Iterators.Lemmas.Consumers.Monadic
+import Init.Data.Iterators.Lemmas.Consumers.Monadic
 import Std.Data.Iterators.Lemmas.Equivalence.StepCongr
 
 namespace Std.Iterators
diff --git a/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/Take.lean b/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/Take.lean
index 8d272a4470..1de5394ddc 100644
--- a/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/Take.lean
+++ b/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/Take.lean
@@ -5,7 +5,7 @@ Authors: Paul Reichert
 -/
 prelude
 import Std.Data.Iterators.Combinators.Monadic.Take
-import Std.Data.Iterators.Lemmas.Consumers.Monadic
+import Init.Data.Iterators.Lemmas.Consumers.Monadic
 
 namespace Std.Iterators
 
diff --git a/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/Zip.lean b/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/Zip.lean
index b438ee5835..e4aa6d1c50 100644
--- a/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/Zip.lean
+++ b/src/Std/Data/Iterators/Lemmas/Combinators/Monadic/Zip.lean
@@ -5,7 +5,7 @@ Authors: Paul Reichert
 -/
 prelude
 import Std.Data.Iterators.Combinators.Monadic.Zip
-import Std.Data.Iterators.Lemmas.Consumers.Monadic
+import Init.Data.Iterators.Lemmas.Consumers.Monadic
 
 namespace Std.Iterators
 
diff --git a/src/Std/Data/Iterators/Lemmas/Combinators/Take.lean b/src/Std/Data/Iterators/Lemmas/Combinators/Take.lean
index aacd6f29a1..1e039d97ea 100644
--- a/src/Std/Data/Iterators/Lemmas/Combinators/Take.lean
+++ b/src/Std/Data/Iterators/Lemmas/Combinators/Take.lean
@@ -5,9 +5,9 @@ Authors: Paul Reichert
 -/
 prelude
 import Std.Data.Iterators.Combinators.Take
-import Std.Data.Iterators.Consumers.Access
+import Init.Data.Iterators.Consumers.Access
 import Std.Data.Iterators.Lemmas.Combinators.Monadic.Take
-import Std.Data.Iterators.Lemmas.Consumers
+import Init.Data.Iterators.Lemmas.Consumers
 
 namespace Std.Iterators
 
diff --git a/src/Std/Data/Iterators/Lemmas/Combinators/Zip.lean b/src/Std/Data/Iterators/Lemmas/Combinators/Zip.lean
index 083fc5e93a..b4365f0944 100644
--- a/src/Std/Data/Iterators/Lemmas/Combinators/Zip.lean
+++ b/src/Std/Data/Iterators/Lemmas/Combinators/Zip.lean
@@ -6,10 +6,10 @@ Authors: Paul Reichert
 prelude
 import Std.Data.Iterators.Combinators.Take
 import Std.Data.Iterators.Combinators.Zip
-import Std.Data.Iterators.Consumers.Access
+import Init.Data.Iterators.Consumers.Access
 import Std.Data.Iterators.Lemmas.Combinators.Monadic.Zip
 import Std.Data.Iterators.Lemmas.Combinators.Take
-import Std.Data.Iterators.Lemmas.Consumers
+import Init.Data.Iterators.Lemmas.Consumers
 
 namespace Std.Iterators
 
diff --git a/src/Std/Data/Iterators/Lemmas/Consumers/Collect.lean b/src/Std/Data/Iterators/Lemmas/Consumers/Collect.lean
index bd8be165ff..4092746fac 100644
--- a/src/Std/Data/Iterators/Lemmas/Consumers/Collect.lean
+++ b/src/Std/Data/Iterators/Lemmas/Consumers/Collect.lean
@@ -4,113 +4,14 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
 prelude
-import Std.Data.Iterators.Consumers.Access
-import Std.Data.Iterators.Consumers.Collect
-import Std.Data.Iterators.Lemmas.Basic
+import Init.Data.Iterators.Consumers.Access
+import Init.Data.Iterators.Consumers.Collect
+import Init.Data.Iterators.Lemmas.Consumers.Collect
+import Init.Data.Iterators.Lemmas.Basic
 import Std.Data.Iterators.Lemmas.Consumers.Monadic.Collect
 
 namespace Std.Iterators
 
-theorem Iter.toArray_eq_toArray_toIterM {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
-    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
-    it.toArray = it.toIterM.toArray.run :=
-  rfl
-
-theorem Iter.toList_eq_toList_toIterM {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
-    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
-    it.toList = it.toIterM.toList.run :=
-  rfl
-
-theorem Iter.toListRev_eq_toListRev_toIterM {α β} [Iterator α Id β] [Finite α Id]
-    {it : Iter (α := α) β} :
-    it.toListRev = it.toIterM.toListRev.run :=
-  rfl
-
-@[simp]
-theorem IterM.toList_toIter {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
-    {it : IterM (α := α) Id β} :
-    it.toIter.toList = it.toList.run :=
-  rfl
-
-@[simp]
-theorem IterM.toListRev_toIter {α β} [Iterator α Id β] [Finite α Id]
-    {it : IterM (α := α) Id β} :
-    it.toIter.toListRev = it.toListRev.run :=
-  rfl
-
-theorem Iter.toList_toArray {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
-    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
-    it.toArray.toList = it.toList := by
-  simp [toArray_eq_toArray_toIterM, toList_eq_toList_toIterM, ← IterM.toList_toArray]
-
-theorem Iter.toArray_toList {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
-    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
-    it.toList.toArray = it.toArray := by
-  simp [toArray_eq_toArray_toIterM, toList_eq_toList_toIterM, ← IterM.toArray_toList]
-
-@[simp]
-theorem Iter.reverse_toListRev [Iterator α Id β] [Finite α Id]
-    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
-    {it : Iter (α := α) β} :
-    it.toListRev.reverse = it.toList := by
-  simp [toListRev_eq_toListRev_toIterM, toList_eq_toList_toIterM, ← IterM.reverse_toListRev]
-
-theorem Iter.toListRev_eq {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
-    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
-    it.toListRev = it.toList.reverse := by
-  simp [Iter.toListRev_eq_toListRev_toIterM, Iter.toList_eq_toList_toIterM, IterM.toListRev_eq]
-
-theorem Iter.toArray_eq_match_step {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
-    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
-    it.toArray = match it.step with
-      | .yield it' out _ => #[out] ++ it'.toArray
-      | .skip it' _ => it'.toArray
-      | .done _ => #[] := by
-  simp only [Iter.toArray_eq_toArray_toIterM, Iter.step]
-  rw [IterM.toArray_eq_match_step, Id.run_bind]
-  generalize it.toIterM.step.run = step
-  cases step using PlausibleIterStep.casesOn <;> simp
-
-theorem Iter.toList_eq_match_step {α β} [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id]
-    [LawfulIteratorCollect α Id Id] {it : Iter (α := α) β} :
-    it.toList = match it.step with
-      | .yield it' out _ => out :: it'.toList
-      | .skip it' _ => it'.toList
-      | .done _ => [] := by
-  rw [← Iter.toList_toArray, Iter.toArray_eq_match_step]
-  split <;> simp [Iter.toList_toArray]
-
-theorem Iter.toListRev_eq_match_step {α β} [Iterator α Id β] [Finite α Id] {it : Iter (α := α) β} :
-    it.toListRev = match it.step with
-      | .yield it' out _ => it'.toListRev ++ [out]
-      | .skip it' _ => it'.toListRev
-      | .done _ => [] := by
-  rw [Iter.toListRev_eq_toListRev_toIterM, IterM.toListRev_eq_match_step, Iter.step, Id.run_bind]
-  generalize it.toIterM.step.run = step
-  cases step using PlausibleIterStep.casesOn <;> simp
-
-theorem Iter.getElem?_toList_eq_atIdxSlow? {α β}
-    [Iterator α Id β] [Finite α Id] [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
-    {it : Iter (α := α) β} {k : Nat} :
-    it.toList[k]? = it.atIdxSlow? k := by
-  induction it using Iter.inductSteps generalizing k with | step it ihy ihs =>
-  rw [toList_eq_match_step, atIdxSlow?]
-  obtain ⟨step, h⟩ := it.step
-  cases step
-  · cases k <;> simp [ihy h]
-  · simp [ihs h]
-  · simp
-
-theorem Iter.toList_eq_of_atIdxSlow?_eq {α₁ α₂ β}
-    [Iterator α₁ Id β] [Finite α₁ Id] [IteratorCollect α₁ Id Id] [LawfulIteratorCollect α₁ Id Id]
-    [Iterator α₂ Id β] [Finite α₂ Id] [IteratorCollect α₂ Id Id] [LawfulIteratorCollect α₂ Id Id]
-    {it₁ : Iter (α := α₁) β} {it₂ : Iter (α := α₂) β}
-    (h : ∀ k, it₁.atIdxSlow? k = it₂.atIdxSlow? k) :
-    it₁.toList = it₂.toList := by
-  ext; simp [getElem?_toList_eq_atIdxSlow?, h]
-
-section Equivalence
-
 theorem Iter.Equiv.toListRev_eq
     [Iterator α₁ Id β] [Iterator α₂ Id β] [Finite α₁ Id] [Finite α₂ Id]
     {ita : Iter (α := α₁) β} {itb : Iter (α := α₂) β} (h : Iter.Equiv ita itb) :
@@ -133,6 +34,4 @@ theorem Iter.Equiv.toArray_eq
     ita.toArray = itb.toArray := by
   simp only [← Iter.toArray_toList, toList_eq h]
 
-end Equivalence
-
 end Std.Iterators
diff --git a/src/Std/Data/Iterators/Lemmas/Consumers/Loop.lean b/src/Std/Data/Iterators/Lemmas/Consumers/Loop.lean
index 8ed9231bd8..ee8f60d087 100644
--- a/src/Std/Data/Iterators/Lemmas/Consumers/Loop.lean
+++ b/src/Std/Data/Iterators/Lemmas/Consumers/Loop.lean
@@ -5,179 +5,15 @@ Authors: Paul Reichert
 -/
 prelude
 import Init.Data.List.Control
-import Std.Data.Iterators.Lemmas.Basic
+import Init.Data.Iterators.Lemmas.Basic
+import Init.Data.Iterators.Lemmas.Consumers.Loop
 import Std.Data.Iterators.Lemmas.Consumers.Collect
 import Std.Data.Iterators.Lemmas.Consumers.Monadic.Loop
-import Std.Data.Iterators.Consumers.Collect
-import Std.Data.Iterators.Consumers.Loop
+import Init.Data.Iterators.Consumers.Collect
+import Init.Data.Iterators.Consumers.Loop
 
 namespace Std.Iterators
 
-theorem Iter.forIn_eq {α β : Type w} [Iterator α Id β] [Finite α Id]
-    {m : Type w → Type w''} [Monad m] [IteratorLoop α Id m] [hl : LawfulIteratorLoop α Id m]
-    {γ : Type w} {it : Iter (α := α) β} {init : γ}
-    {f : β → γ → m (ForInStep γ)} :
-    ForIn.forIn it init f =
-      IterM.DefaultConsumers.forIn (fun _ c => pure c.run) γ (fun _ _ _ => True)
-        IteratorLoop.wellFounded_of_finite it.toIterM init ((⟨·, .intro⟩) <$> f · ·) := by
-  cases hl.lawful; rfl
-
-theorem Iter.forIn_eq_forIn_toIterM {α β : Type w} [Iterator α Id β]
-    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
-    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
-    {γ : Type w} {it : Iter (α := α) β} {init : γ}
-    {f : β → γ → m (ForInStep γ)} :
-    ForIn.forIn it init f =
-      letI : MonadLift Id m := ⟨Std.Internal.idToMonad (α := _)⟩
-      ForIn.forIn it.toIterM init f := by
-  rfl
-
-theorem Iter.forIn_eq_match_step {α β : Type w} [Iterator α Id β]
-    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
-    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
-    {γ : Type w} {it : Iter (α := α) β} {init : γ}
-    {f : β → γ → m (ForInStep γ)} :
-    ForIn.forIn it init f = (do
-      match it.step with
-      | .yield it' out _ =>
-        match ← f out init with
-        | .yield c => ForIn.forIn it' c f
-        | .done c => return c
-      | .skip it' _ => ForIn.forIn it' init f
-      | .done _ => return init) := by
-  rw [Iter.forIn_eq_forIn_toIterM, @IterM.forIn_eq_match_step, Iter.step]
-  simp only [liftM, monadLift, pure_bind]
-  generalize it.toIterM.step = step
-  cases step using PlausibleIterStep.casesOn
-  · apply bind_congr
-    intro forInStep
-    rfl
-  · rfl
-  · rfl
-
-theorem Iter.forIn_toList {α β : Type w} [Iterator α Id β]
-    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
-    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
-    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
-    {γ : Type w} {it : Iter (α := α) β} {init : γ}
-    {f : β → γ → m (ForInStep γ)} :
-    ForIn.forIn it.toList init f = ForIn.forIn it init f := by
-  rw [List.forIn_eq_foldlM]
-  induction it using Iter.inductSteps generalizing init with case step it ihy ihs =>
-  rw [forIn_eq_match_step, Iter.toList_eq_match_step]
-  simp only [map_eq_pure_bind]
-  generalize it.step = step
-  cases step using PlausibleIterStep.casesOn
-  · rename_i it' out h
-    simp only [List.foldlM_cons, bind_pure_comp, map_bind]
-    apply bind_congr
-    intro forInStep
-    cases forInStep
-    · induction it'.toList <;> simp [*]
-    · simp only [ForIn.forIn, forIn', List.forIn'] at ihy
-      simp [ihy h, forIn_eq_forIn_toIterM]
-  · rename_i it' h
-    simp only [bind_pure_comp]
-    rw [ihs h]
-  · simp
-
-theorem Iter.foldM_eq_forIn {α β γ : Type w} [Iterator α Id β] [Finite α Id] {m : Type w → Type w'}
-    [Monad m] [IteratorLoop α Id m] {f : γ → β → m γ}
-    {init : γ} {it : Iter (α := α) β} :
-    it.foldM (init := init) f = ForIn.forIn it init (fun x acc => ForInStep.yield <$> f acc x) :=
-  rfl
-
-theorem Iter.foldM_eq_foldM_toIterM {α β : Type w} [Iterator α Id β]
-    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
-    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
-    {γ : Type w} {it : Iter (α := α) β} {init : γ} {f : γ → β → m γ} :
-    it.foldM (init := init) f = letI : MonadLift Id m := ⟨pure⟩; it.toIterM.foldM (init := init) f :=
-  rfl
-
-theorem Iter.forIn_yield_eq_foldM {α β γ δ : Type w} [Iterator α Id β]
-    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m] [IteratorLoop α Id m]
-    [LawfulIteratorLoop α Id m] {f : β → γ → m δ} {g : β → γ → δ → γ} {init : γ}
-    {it : Iter (α := α) β} :
-    ForIn.forIn it init (fun c b => (fun d => .yield (g c b d)) <$> f c b) =
-      it.foldM (fun b c => g c b <$> f c b) init := by
-  simp [Iter.foldM_eq_forIn]
-
-theorem Iter.foldM_eq_match_step {α β γ : Type w} [Iterator α Id β] [Finite α Id]
-    {m : Type w → Type w'} [Monad m] [LawfulMonad m] [IteratorLoop α Id m]
-    [LawfulIteratorLoop α Id m] {f : γ → β → m γ} {init : γ} {it : Iter (α := α) β} :
-    it.foldM (init := init) f = (do
-      match it.step with
-      | .yield it' out _ => it'.foldM (init := ← f init out) f
-      | .skip it' _ => it'.foldM (init := init) f
-      | .done _ => return init) := by
-  rw [Iter.foldM_eq_forIn, Iter.forIn_eq_match_step]
-  generalize it.step = step
-  cases step using PlausibleIterStep.casesOn <;> simp [foldM_eq_forIn]
-
-theorem Iter.foldlM_toList {α β γ : Type w} [Iterator α Id β] [Finite α Id] {m : Type w → Type w'}
-    [Monad m] [LawfulMonad m] [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
-    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
-    {f : γ → β → m γ}
-    {init : γ} {it : Iter (α := α) β} :
-    it.toList.foldlM (init := init) f = it.foldM (init := init) f := by
-  rw [Iter.foldM_eq_forIn, ← Iter.forIn_toList]
-  simp only [List.forIn_yield_eq_foldlM, id_map']
-
-theorem IterM.forIn_eq_foldM {α β : Type w} [Iterator α Id β]
-    [Finite α Id] {m : Type w → Type w''} [Monad m] [LawfulMonad m]
-    [IteratorLoop α Id m] [LawfulIteratorLoop α Id m]
-    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
-    {γ : Type w} {it : Iter (α := α) β} {init : γ}
-    {f : β → γ → m (ForInStep γ)} :
-    forIn it init f = ForInStep.value <$>
-      it.foldM (fun c b => match c with
-        | .yield c => f b c
-        | .done c => pure (.done c)) (ForInStep.yield init) := by
-  simp only [← Iter.forIn_toList, List.forIn_eq_foldlM, ← Iter.foldlM_toList]; rfl
-
-theorem Iter.fold_eq_forIn {α β γ : Type w} [Iterator α Id β]
-    [Finite α Id] [IteratorLoop α Id Id] {f : γ → β → γ} {init : γ} {it : Iter (α := α) β} :
-    it.fold (init := init) f =
-      (ForIn.forIn (m := Id) it init (fun x acc => pure (ForInStep.yield (f acc x)))).run := by
-  rfl
-
-theorem Iter.fold_eq_foldM {α β γ : Type w} [Iterator α Id β]
-    [Finite α Id] [IteratorLoop α Id Id] {f : γ → β → γ} {init : γ}
-    {it : Iter (α := α) β} :
-    it.fold (init := init) f = (it.foldM (m := Id) (init := init) (pure <| f · ·)).run := by
-  simp [foldM_eq_forIn, fold_eq_forIn]
-
-@[simp]
-theorem Iter.forIn_pure_yield_eq_fold {α β γ : Type w} [Iterator α Id β]
-    [Finite α Id] [IteratorLoop α Id Id]
-    [LawfulIteratorLoop α Id Id] {f : β → γ → γ} {init : γ}
-    {it : Iter (α := α) β} :
-    ForIn.forIn (m := Id) it init (fun c b => pure (.yield (f c b))) =
-      pure (it.fold (fun b c => f c b) init) := by
-  simp only [fold_eq_forIn]
-  rfl
-
-theorem Iter.fold_eq_match_step {α β γ : Type w} [Iterator α Id β] [Finite α Id]
-    [IteratorLoop α Id Id] [LawfulIteratorLoop α Id Id]
-    {f : γ → β → γ} {init : γ} {it : Iter (α := α) β} :
-    it.fold (init := init) f = (match it.step with
-      | .yield it' out _ => it'.fold (init := f init out) f
-      | .skip it' _ => it'.fold (init := init) f
-      | .done _ => init) := by
-  rw [fold_eq_foldM, foldM_eq_match_step]
-  simp only [fold_eq_foldM]
-  generalize it.step = step
-  cases step using PlausibleIterStep.casesOn <;> simp
-
-theorem Iter.foldl_toList {α β γ : Type w} [Iterator α Id β] [Finite α Id]
-    [IteratorLoop α Id Id] [LawfulIteratorLoop α Id Id]
-    [IteratorCollect α Id Id] [LawfulIteratorCollect α Id Id]
-    {f : γ → β → γ} {init : γ} {it : Iter (α := α) β} :
-    it.toList.foldl (init := init) f = it.fold (init := init) f := by
-  rw [fold_eq_foldM, List.foldl_eq_foldlM, ← Iter.foldlM_toList]
-
-section Equivalence
-
 theorem Iter.Equiv.forIn_eq {α₁ α₂ β γ : Type w} {m : Type w → Type w'}
     [Iterator α₁ Id β] [Iterator α₂ Id β] [Finite α₁ Id] [Finite α₂ Id]
     [Monad m] [LawfulMonad m] [IteratorLoop α₁ Id m] [LawfulIteratorLoop α₁ Id m]
@@ -208,6 +44,4 @@ theorem Iter.Equiv.fold_eq {α₁ α₂ β γ : Type w} {m : Type w → Type w'}
     ita.fold (init := init) f = itb.fold (init := init) f := by
   simp [Iter.fold_eq_foldM, h.foldM_eq]
 
-end Equivalence
-
 end Std.Iterators
diff --git a/src/Std/Data/Iterators/Lemmas/Consumers/Monadic/Collect.lean b/src/Std/Data/Iterators/Lemmas/Consumers/Monadic/Collect.lean
index a65a445be8..d67cd59dbf 100644
--- a/src/Std/Data/Iterators/Lemmas/Consumers/Monadic/Collect.lean
+++ b/src/Std/Data/Iterators/Lemmas/Consumers/Monadic/Collect.lean
@@ -5,161 +5,13 @@ Authors: Paul Reichert
 -/
 prelude
 import Init.Data.Array.Lemmas
-import Std.Data.Iterators.Consumers.Monadic.Collect
-import Std.Data.Iterators.Lemmas.Monadic.Basic
-import Std.Data.Iterators.Producers
+import Init.Data.Iterators.Consumers.Monadic.Collect
+import Init.Data.Iterators.Lemmas.Consumers.Monadic.Collect
+import Init.Data.Iterators.Lemmas.Monadic.Basic
 import Std.Data.Iterators.Lemmas.Equivalence.StepCongr
 
 namespace Std.Iterators
 
-section Consumers
-
-variable {α β γ : Type w} {m : Type w → Type w'} {n : Type w → Type w''}
-  {lift : ⦃δ : Type w⦄ → m δ → n δ} {f : β → n γ} {it : IterM (α := α) m β}
-
-theorem IterM.DefaultConsumers.toArrayMapped.go.aux₁ [Monad n] [LawfulMonad n] [Iterator α m β]
-    [Finite α m] {b : γ} {bs : Array γ} :
-    IterM.DefaultConsumers.toArrayMapped.go lift f it (#[b] ++ bs) (m := m) =
-      (#[b] ++ ·) <$> IterM.DefaultConsumers.toArrayMapped.go lift f it bs (m := m) := by
-  induction it, bs using IterM.DefaultConsumers.toArrayMapped.go.induct
-  next it bs ih₁ ih₂ =>
-  rw [go, map_eq_pure_bind, go, bind_assoc]
-  apply bind_congr
-  intro step
-  split
-  · simp [ih₁ _ _ ‹_›]
-  · simp [ih₂ _ ‹_›]
-  · simp
-
-theorem IterM.DefaultConsumers.toArrayMapped.go.aux₂ [Monad n] [LawfulMonad n] [Iterator α m β]
-    [Finite α m] {acc : Array γ} :
-    IterM.DefaultConsumers.toArrayMapped.go lift f it acc (m := m) =
-      (acc ++ ·) <$> IterM.DefaultConsumers.toArrayMapped lift f it (m := m) := by
-  rw [← Array.toArray_toList (xs := acc)]
-  generalize acc.toList = acc
-  induction acc with
-  | nil => simp [toArrayMapped]
-  | cons x xs ih =>
-    rw [List.toArray_cons, IterM.DefaultConsumers.toArrayMapped.go.aux₁, ih]
-    simp only [Functor.map_map, Array.append_assoc]
-
-theorem IterM.DefaultConsumers.toArrayMapped_eq_match_step [Monad n] [LawfulMonad n]
-    [Iterator α m β] [Finite α m] :
-    IterM.DefaultConsumers.toArrayMapped lift f it (m := m) = letI : MonadLift m n := ⟨lift (δ := _)⟩; (do
-      match ← it.step with
-      | .yield it' out _ =>
-        return #[← f out] ++ (← IterM.DefaultConsumers.toArrayMapped lift f it' (m := m))
-      | .skip it' _ => IterM.DefaultConsumers.toArrayMapped lift f it' (m := m)
-      | .done _ => return #[]) := by
-  rw [IterM.DefaultConsumers.toArrayMapped, IterM.DefaultConsumers.toArrayMapped.go]
-  apply bind_congr
-  intro step
-  split <;> simp [IterM.DefaultConsumers.toArrayMapped.go.aux₂]
-
-theorem IterM.toArray_eq_match_step [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
-    [IteratorCollect α m m] [LawfulIteratorCollect α m m] :
-    it.toArray = (do
-      match ← it.step with
-      | .yield it' out _ => return #[out] ++ (← it'.toArray)
-      | .skip it' _ => it'.toArray
-      | .done _ => return #[]) := by
-  simp only [IterM.toArray, LawfulIteratorCollect.toArrayMapped_eq]
-  rw [IterM.DefaultConsumers.toArrayMapped_eq_match_step]
-  simp [bind_pure_comp, pure_bind, toArray]
-
-theorem IterM.toList_toArray [Monad m] [Iterator α m β] [Finite α m] [IteratorCollect α m m]
-    {it : IterM (α := α) m β} :
-    Array.toList <$> it.toArray = it.toList := by
-  simp [IterM.toList]
-
-theorem IterM.toArray_toList [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
-    [IteratorCollect α m m] {it : IterM (α := α) m β} :
-    List.toArray <$> it.toList = it.toArray := by
-  simp [IterM.toList]
-
-theorem IterM.toList_eq_match_step [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
-    [IteratorCollect α m m] [LawfulIteratorCollect α m m] {it : IterM (α := α) m β} :
-    it.toList = (do
-      match ← it.step with
-      | .yield it' out _ => return out :: (← it'.toList)
-      | .skip it' _ => it'.toList
-      | .done _ => return []) := by
-  simp [← IterM.toList_toArray]
-  rw [IterM.toArray_eq_match_step, map_eq_pure_bind, bind_assoc]
-  apply bind_congr
-  intro step
-  split <;> simp
-
-theorem IterM.toListRev.go.aux₁ [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
-    {it : IterM (α := α) m β} {b : β} {bs : List β} :
-    IterM.toListRev.go it (bs ++ [b]) = (· ++ [b]) <$> IterM.toListRev.go it bs:= by
-  induction it, bs using IterM.toListRev.go.induct
-  next it bs ih₁ ih₂ =>
-  rw [go, go, map_eq_pure_bind, bind_assoc]
-  apply bind_congr
-  intro step
-  simp only [List.cons_append] at ih₁
-  split <;> simp [*]
-
-theorem IterM.toListRev.go.aux₂ [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
-    {it : IterM (α := α) m β} {acc : List β} :
-    IterM.toListRev.go it acc = (· ++ acc) <$> it.toListRev := by
-  rw [← List.reverse_reverse (as := acc)]
-  generalize acc.reverse = acc
-  induction acc with
-  | nil => simp [toListRev]
-  | cons x xs ih => simp [IterM.toListRev.go.aux₁, ih]
-
-theorem IterM.toListRev_eq_match_step [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
-    {it : IterM (α := α) m β} :
-    it.toListRev = (do
-      match ← it.step with
-      | .yield it' out _ => return (← it'.toListRev) ++ [out]
-      | .skip it' _ => it'.toListRev
-      | .done _ => return []) := by
-  simp [IterM.toListRev]
-  rw [toListRev.go]
-  apply bind_congr
-  intro step
-  cases step using PlausibleIterStep.casesOn <;> simp [IterM.toListRev.go.aux₂]
-
-theorem IterM.reverse_toListRev [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
-    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
-    {it : IterM (α := α) m β} :
-    List.reverse <$> it.toListRev = it.toList := by
-  apply Eq.symm
-  induction it using IterM.inductSteps
-  rename_i it ihy ihs
-  rw [toListRev_eq_match_step, toList_eq_match_step, map_eq_pure_bind, bind_assoc]
-  apply bind_congr
-  intro step
-  split <;> simp (discharger := assumption) [ihy, ihs]
-
-theorem IterM.toListRev_eq [Monad m] [LawfulMonad m] [Iterator α m β] [Finite α m]
-    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
-    {it : IterM (α := α) m β} :
-    it.toListRev = List.reverse <$> it.toList := by
-  rw [← IterM.reverse_toListRev]
-  simp
-
-theorem LawfulIteratorCollect.toArray_eq {α β : Type w} {m : Type w → Type w'}
-    [Monad m] [Iterator α m β] [Finite α m] [IteratorCollect α m m]
-    [hl : LawfulIteratorCollect α m m]
-    {it : IterM (α := α) m β} :
-    it.toArray = (letI : IteratorCollect α m m := .defaultImplementation; it.toArray) := by
-  simp only [IterM.toArray, toArrayMapped_eq]
-
-theorem LawfulIteratorCollect.toList_eq {α β : Type w} {m : Type w → Type w'}
-    [Monad m] [Iterator α m β] [Finite α m] [IteratorCollect α m m]
-    [hl : LawfulIteratorCollect α m m]
-    {it : IterM (α := α) m β} :
-    it.toList = (letI : IteratorCollect α m m := .defaultImplementation; it.toList) := by
-  simp [IterM.toList, toArray_eq]
-
-end Consumers
-
-section Equivalence
-
 theorem IterM.Equiv.toListRev_eq [Monad m] [LawfulMonad m]
     [Iterator α₁ m β] [Iterator α₂ m β] [Finite α₁ m] [Finite α₂ m]
     {ita : IterM (α := α₁) m β} {itb : IterM (α := α₂) m β} (h : IterM.Equiv ita itb) :
@@ -197,6 +49,4 @@ theorem IterM.Equiv.toArray_eq [Monad m] [LawfulMonad m]
     ita.toArray = itb.toArray := by
   simp only [← IterM.toArray_toList, toList_eq h]
 
-end Equivalence
-
 end Std.Iterators
diff --git a/src/Std/Data/Iterators/Lemmas/Consumers/Monadic/Loop.lean b/src/Std/Data/Iterators/Lemmas/Consumers/Monadic/Loop.lean
index 2c11ec412a..0aee0d3df0 100644
--- a/src/Std/Data/Iterators/Lemmas/Consumers/Monadic/Loop.lean
+++ b/src/Std/Data/Iterators/Lemmas/Consumers/Monadic/Loop.lean
@@ -5,237 +5,15 @@ Authors: Paul Reichert
 -/
 prelude
 import Init.Control.Lawful.Basic
-import Std.Data.Iterators.Consumers.Monadic.Collect
-import Std.Data.Iterators.Consumers.Monadic.Loop
-import Std.Data.Iterators.Lemmas.Monadic.Basic
+import Init.Data.Iterators.Consumers.Monadic.Collect
+import Init.Data.Iterators.Consumers.Monadic.Loop
+import Init.Data.Iterators.Lemmas.Monadic.Basic
+import Init.Data.Iterators.Lemmas.Consumers.Monadic.Loop
 import Std.Data.Iterators.Lemmas.Consumers.Monadic.Collect
 import Std.Data.Iterators.Lemmas.Equivalence.StepCongr
 
 namespace Std.Iterators
 
-theorem IterM.DefaultConsumers.forIn_eq_match_step {α β : Type w} {m : Type w → Type w'}
-    [Iterator α m β]
-    {n : Type w → Type w''} [Monad n]
-    {lift : ∀ γ, m γ → n γ} {γ : Type w}
-    {plausible_forInStep : β → γ → ForInStep γ → Prop}
-    {wf : IteratorLoop.WellFounded α m plausible_forInStep}
-    {it : IterM (α := α) m β} {init : γ}
-    {f : (b : β) → (c : γ) → n (Subtype (plausible_forInStep b c))} :
-    IterM.DefaultConsumers.forIn lift γ plausible_forInStep wf it init f = (do
-      match ← lift _ it.step with
-      | .yield it' out _ =>
-        match ← f out init with
-        | ⟨.yield c, _⟩ => IterM.DefaultConsumers.forIn lift _ plausible_forInStep wf it' c f
-        | ⟨.done c, _⟩ => return c
-      | .skip it' _ => IterM.DefaultConsumers.forIn lift _ plausible_forInStep wf it' init f
-      | .done _ => return init) := by
-  rw [forIn]
-  apply bind_congr
-  intro step
-  cases step using PlausibleIterStep.casesOn <;> rfl
-
-theorem IterM.forIn_eq {α β : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
-    {n : Type w → Type w''} [Monad n] [IteratorLoop α m n] [hl : LawfulIteratorLoop α m n]
-    [MonadLiftT m n] {γ : Type w} {it : IterM (α := α) m β} {init : γ}
-    {f : β → γ → n (ForInStep γ)} :
-    ForIn.forIn it init f = IterM.DefaultConsumers.forIn (fun _ => monadLift) γ (fun _ _ _ => True)
-        IteratorLoop.wellFounded_of_finite it init ((⟨·, .intro⟩) <$> f · ·) := by
-  cases hl.lawful; rfl
-
-theorem IterM.forIn_eq_match_step {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
-    [Finite α m] {n : Type w → Type w''} [Monad n] [LawfulMonad n]
-    [IteratorLoop α m n] [LawfulIteratorLoop α m n]
-    [MonadLiftT m n] {γ : Type w} {it : IterM (α := α) m β} {init : γ}
-    {f : β → γ → n (ForInStep γ)} :
-    ForIn.forIn it init f = (do
-      match ← it.step with
-      | .yield it' out _ =>
-        match ← f out init with
-        | .yield c => ForIn.forIn it' c f
-        | .done c => return c
-      | .skip it' _ => ForIn.forIn it' init f
-      | .done _ => return init) := by
-  rw [IterM.forIn_eq, DefaultConsumers.forIn_eq_match_step]
-  apply bind_congr
-  intro step
-  cases step using PlausibleIterStep.casesOn
-  · simp only [map_eq_pure_bind, bind_assoc]
-    apply bind_congr
-    intro forInStep
-    cases forInStep <;> simp [IterM.forIn_eq]
-  · simp [IterM.forIn_eq]
-  · simp
-
-theorem IterM.forM_eq_forIn {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
-    [Finite α m] {n : Type w → Type w''} [Monad n] [LawfulMonad n]
-    [IteratorLoop α m n] [LawfulIteratorLoop α m n]
-    [MonadLiftT m n] {it : IterM (α := α) m β}
-    {f : β → n PUnit} :
-    ForM.forM it f = ForIn.forIn it PUnit.unit (fun out _ => do f out; return .yield .unit) :=
-  rfl
-
-theorem IterM.forM_eq_match_step {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
-    [Finite α m] {n : Type w → Type w''} [Monad n] [LawfulMonad n]
-    [IteratorLoop α m n] [LawfulIteratorLoop α m n]
-    [MonadLiftT m n] {it : IterM (α := α) m β}
-    {f : β → n PUnit} :
-    ForM.forM it f = (do
-      match ← it.step with
-      | .yield it' out _ =>
-        f out
-        ForM.forM it' f
-      | .skip it' _ => ForM.forM it' f
-      | .done _ => return) := by
-  rw [forM_eq_forIn, forIn_eq_match_step]
-  apply bind_congr
-  intro step
-  cases step using PlausibleIterStep.casesOn <;> simp [forM_eq_forIn]
-
-theorem IterM.foldM_eq_forIn {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
-    {n : Type w → Type w''} [Monad n] [IteratorLoop α m n] [MonadLiftT m n] {f : γ → β → n γ}
-    {init : γ} {it : IterM (α := α) m β} :
-    it.foldM (init := init) f = ForIn.forIn it init (fun x acc => ForInStep.yield <$> f acc x) :=
-  rfl
-
-theorem IterM.forIn_yield_eq_foldM {α β γ δ : Type w} {m : Type w → Type w'} [Iterator α m β]
-    [Finite α m] {n : Type w → Type w''} [Monad n] [LawfulMonad n] [IteratorLoop α m n]
-    [LawfulIteratorLoop α m n] [MonadLiftT m n] {f : β → γ → n δ} {g : β → γ → δ → γ} {init : γ}
-    {it : IterM (α := α) m β} :
-    ForIn.forIn it init (fun c b => (fun d => .yield (g c b d)) <$> f c b) =
-      it.foldM (fun b c => g c b <$> f c b) init := by
-  simp [IterM.foldM_eq_forIn]
-
-theorem IterM.foldM_eq_match_step {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
-    {n : Type w → Type w''} [Monad n] [LawfulMonad n] [IteratorLoop α m n] [LawfulIteratorLoop α m n]
-    [MonadLiftT m n] {f : γ → β → n γ} {init : γ} {it : IterM (α := α) m β} :
-    it.foldM (init := init) f = (do
-      match ← it.step with
-      | .yield it' out _ => it'.foldM (init := ← f init out) f
-      | .skip it' _ => it'.foldM (init := init) f
-      | .done _ => return init) := by
-  rw [IterM.foldM_eq_forIn, IterM.forIn_eq_match_step]
-  apply bind_congr
-  intro step
-  cases step using PlausibleIterStep.casesOn <;> simp [foldM_eq_forIn]
-
-theorem IterM.fold_eq_forIn {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β]
-    [Finite α m] [Monad m]
-    [IteratorLoop α m m] {f : γ → β → γ} {init : γ} {it : IterM (α := α) m β} :
-    it.fold (init := init) f =
-      ForIn.forIn (m := m) it init (fun x acc => pure (ForInStep.yield (f acc x))) := by
-  rfl
-
-theorem IterM.fold_eq_foldM {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β]
-    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m] {f : γ → β → γ} {init : γ}
-    {it : IterM (α := α) m β} :
-    it.fold (init := init) f = it.foldM (init := init) (pure <| f · ·) := by
-  simp [foldM_eq_forIn, fold_eq_forIn]
-
-@[simp]
-theorem IterM.forIn_pure_yield_eq_fold {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β]
-    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m]
-    [LawfulIteratorLoop α m m] {f : β → γ → γ} {init : γ}
-    {it : IterM (α := α) m β} :
-    ForIn.forIn it init (fun c b => pure (.yield (f c b))) =
-      it.fold (fun b c => f c b) init := by
-  simp [IterM.fold_eq_forIn]
-
-theorem IterM.fold_eq_match_step {α β γ : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
-    [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
-    {f : γ → β → γ} {init : γ} {it : IterM (α := α) m β} :
-    it.fold (init := init) f = (do
-      match ← it.step with
-      | .yield it' out _ => it'.fold (init := f init out) f
-      | .skip it' _ => it'.fold (init := init) f
-      | .done _ => return init) := by
-  rw [fold_eq_foldM, foldM_eq_match_step]
-  simp only [fold_eq_foldM]
-  apply bind_congr
-  intro step
-  cases step using PlausibleIterStep.casesOn <;> simp
-
-theorem IterM.toList_eq_fold {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
-    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
-    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
-    {it : IterM (α := α) m β} :
-    it.toList = it.fold (init := []) (fun l out => l ++ [out]) := by
-  suffices h : ∀ l' : List β, (l' ++ ·) <$> it.toList =
-      it.fold (init := l') (fun l out => l ++ [out]) by
-    specialize h []
-    simpa using h
-  induction it using IterM.inductSteps with | step it ihy ihs =>
-  intro l'
-  rw [IterM.toList_eq_match_step, IterM.fold_eq_match_step]
-  simp only [map_eq_pure_bind, bind_assoc]
-  apply bind_congr
-  intro step
-  cases step using PlausibleIterStep.casesOn
-  · rename_i it' out h
-    specialize ihy h (l' ++ [out])
-    simpa using ihy
-  · rename_i it' h
-    simp [ihs h]
-  · simp
-
-theorem IterM.drain_eq_fold {α β : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
-    [Monad m] [IteratorLoop α m m] {it : IterM (α := α) m β} :
-    it.drain = it.fold (init := PUnit.unit) (fun _ _ => .unit) :=
-  rfl
-
-theorem IterM.drain_eq_foldM {α β : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
-    [Monad m] [LawfulMonad m] [IteratorLoop α m m] {it : IterM (α := α) m β} :
-    it.drain = it.foldM (init := PUnit.unit) (fun _ _ => pure .unit) := by
-  simp [IterM.drain_eq_fold, IterM.fold_eq_foldM]
-
-theorem IterM.drain_eq_forIn {α β : Type w} {m : Type w → Type w'}  [Iterator α m β] [Finite α m]
-    [Monad m] [IteratorLoop α m m] {it : IterM (α := α) m β} :
-    it.drain = ForIn.forIn (m := m) it PUnit.unit (fun _ _ => pure (ForInStep.yield .unit)) := by
-  simp [IterM.drain_eq_fold, IterM.fold_eq_forIn]
-
-theorem IterM.drain_eq_match_step {α β : Type w} {m : Type w → Type w'} [Iterator α m β] [Finite α m]
-    [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
-    {it : IterM (α := α) m β} :
-    it.drain = (do
-      match ← it.step with
-      | .yield it' _ _ => it'.drain
-      | .skip it' _ => it'.drain
-      | .done _ => return .unit) := by
-  rw [IterM.drain_eq_fold, IterM.fold_eq_match_step]
-  simp [IterM.drain_eq_fold]
-
-theorem IterM.drain_eq_map_toList {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
-    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
-    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
-    {it : IterM (α := α) m β} :
-    it.drain = (fun _ => .unit) <$> it.toList := by
-  induction it using IterM.inductSteps with | step it ihy ihs =>
-  rw [IterM.drain_eq_match_step, IterM.toList_eq_match_step]
-  simp only [map_eq_pure_bind, bind_assoc]
-  apply bind_congr
-  intro step
-  cases step using PlausibleIterStep.casesOn
-  · rename_i it' out h
-    simp [ihy h]
-  · rename_i it' h
-    simp [ihs h]
-  · simp
-
-theorem IterM.drain_eq_map_toListRev {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
-    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
-    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
-    {it : IterM (α := α) m β} :
-    it.drain = (fun _ => .unit) <$> it.toListRev := by
-  simp [IterM.drain_eq_map_toList, IterM.toListRev_eq]
-
-theorem IterM.drain_eq_map_toArray {α β : Type w} {m : Type w → Type w'} [Iterator α m β]
-    [Finite α m] [Monad m] [LawfulMonad m] [IteratorLoop α m m] [LawfulIteratorLoop α m m]
-    [IteratorCollect α m m] [LawfulIteratorCollect α m m]
-    {it : IterM (α := α) m β} :
-    it.drain = (fun _ => .unit) <$> it.toList := by
-  simp [IterM.drain_eq_map_toList]
-
-section Equivalence
-
 theorem IterM.Equiv.forIn_eq {α₁ α₂ β γ : Type w} {m : Type w → Type w'}
     {n : Type w → Type w''} [Iterator α₁ m β] [Iterator α₂ m β]
     [Finite α₁ m] [Finite α₂ m]
@@ -296,6 +74,4 @@ theorem IterM.Equiv.drain_eq {α₁ α₂ β : Type w} {m : Type w → Type w'}
     ita.drain = itb.drain := by
   simp [IterM.drain_eq_fold, h.fold_eq]
 
-end Equivalence
-
 end Std.Iterators
diff --git a/src/Std/Data/Iterators/Lemmas/Equivalence/Basic.lean b/src/Std/Data/Iterators/Lemmas/Equivalence/Basic.lean
index 5700aa193d..b3574daff4 100644
--- a/src/Std/Data/Iterators/Lemmas/Equivalence/Basic.lean
+++ b/src/Std/Data/Iterators/Lemmas/Equivalence/Basic.lean
@@ -8,8 +8,8 @@ import Init.Control.Lawful.Basic
 import Init.Ext
 import Init.Internal.Order
 import Init.Core
-import Std.Data.Iterators.Basic
-import Std.Data.Iterators.PostConditionMonad
+import Init.Data.Iterators.Basic
+import Init.Data.Iterators.PostconditionMonad
 import Std.Data.Iterators.Lemmas.Equivalence.HetT
 
 namespace Std.Iterators
diff --git a/src/Std/Data/Iterators/Lemmas/Equivalence/HetT.lean b/src/Std/Data/Iterators/Lemmas/Equivalence/HetT.lean
index 0ecfd32421..ce00751a82 100644
--- a/src/Std/Data/Iterators/Lemmas/Equivalence/HetT.lean
+++ b/src/Std/Data/Iterators/Lemmas/Equivalence/HetT.lean
@@ -8,8 +8,8 @@ import Init.Control.Lawful.Basic
 import Init.Data.Subtype
 import Init.PropLemmas
 import Init.Classical
-import Std.Data.Internal.LawfulMonadLiftFunction
-import Std.Data.Iterators.PostConditionMonad
+import Init.Data.Iterators.Internal.LawfulMonadLiftFunction
+import Init.Data.Iterators.PostconditionMonad
 
 namespace Std.Internal
 
diff --git a/src/Std/Data/Iterators/Lemmas/Monadic.lean b/src/Std/Data/Iterators/Lemmas/Monadic.lean
index daa55f5fa5..cbaa515388 100644
--- a/src/Std/Data/Iterators/Lemmas/Monadic.lean
+++ b/src/Std/Data/Iterators/Lemmas/Monadic.lean
@@ -4,4 +4,4 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
 prelude
-import Std.Data.Iterators.Lemmas.Monadic.Basic
+import Init.Data.Iterators.Lemmas.Monadic.Basic
diff --git a/src/Std/Data/Iterators/Lemmas/Producers/Array.lean b/src/Std/Data/Iterators/Lemmas/Producers/Array.lean
index f74f9d7a33..9e3f0c5692 100644
--- a/src/Std/Data/Iterators/Lemmas/Producers/Array.lean
+++ b/src/Std/Data/Iterators/Lemmas/Producers/Array.lean
@@ -5,6 +5,8 @@ Authors: Paul Reichert
 -/
 prelude
 import Std.Data.Iterators.Lemmas.Consumers.Collect
+import Std.Data.Iterators.Producers.Array
+import Std.Data.Iterators.Producers.List
 import Std.Data.Iterators.Lemmas.Producers.Monadic.Array
 
 /-!
diff --git a/src/Std/Data/Iterators/Lemmas/Producers/Empty.lean b/src/Std/Data/Iterators/Lemmas/Producers/Empty.lean
index e96a100d40..6649d9cb63 100644
--- a/src/Std/Data/Iterators/Lemmas/Producers/Empty.lean
+++ b/src/Std/Data/Iterators/Lemmas/Producers/Empty.lean
@@ -5,7 +5,7 @@ Authors: Paul Reichert
 -/
 prelude
 import Std.Data.Iterators.Lemmas.Producers.Monadic.Empty
-import Std.Data.Iterators.Lemmas.Consumers
+import Init.Data.Iterators.Lemmas.Consumers
 import Std.Data.Iterators.Producers.Empty
 
 namespace Std.Iterators
diff --git a/src/Std/Data/Iterators/Lemmas/Producers/List.lean b/src/Std/Data/Iterators/Lemmas/Producers/List.lean
index e5d7b8a73f..af4adb5c8f 100644
--- a/src/Std/Data/Iterators/Lemmas/Producers/List.lean
+++ b/src/Std/Data/Iterators/Lemmas/Producers/List.lean
@@ -4,8 +4,9 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
 prelude
-import Std.Data.Iterators.Consumers
-import Std.Data.Iterators.Lemmas.Consumers.Collect
+import Init.Data.Iterators.Consumers
+import Init.Data.Iterators.Lemmas.Consumers.Collect
+import Std.Data.Iterators.Producers.List
 import Std.Data.Iterators.Lemmas.Producers.Monadic.List
 
 /-!
diff --git a/src/Std/Data/Iterators/Lemmas/Producers/Monadic/Array.lean b/src/Std/Data/Iterators/Lemmas/Producers/Monadic/Array.lean
index ba2b2d6d20..d84d4a70dd 100644
--- a/src/Std/Data/Iterators/Lemmas/Producers/Monadic/Array.lean
+++ b/src/Std/Data/Iterators/Lemmas/Producers/Monadic/Array.lean
@@ -4,9 +4,9 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
 prelude
-import Std.Data.Internal.LawfulMonadLiftFunction
+import Init.Data.Iterators.Internal.LawfulMonadLiftFunction
+import Init.Data.Iterators.Consumers
 import Std.Data.Iterators.Producers.Monadic.Array
-import Std.Data.Iterators.Consumers
 import Std.Data.Iterators.Lemmas.Consumers.Monadic
 import Std.Data.Iterators.Lemmas.Producers.Monadic.List
 import Std.Data.Iterators.Lemmas.Equivalence.Basic
diff --git a/src/Std/Data/Iterators/Lemmas/Producers/Monadic/Empty.lean b/src/Std/Data/Iterators/Lemmas/Producers/Monadic/Empty.lean
index 9bded4fd6f..3e073171e1 100644
--- a/src/Std/Data/Iterators/Lemmas/Producers/Monadic/Empty.lean
+++ b/src/Std/Data/Iterators/Lemmas/Producers/Monadic/Empty.lean
@@ -5,8 +5,8 @@ Authors: Paul Reichert
 -/
 prelude
 import Std.Data.Iterators.Producers.Monadic.Empty
-import Std.Data.Iterators.Lemmas.Consumers.Monadic
-import Std.Data.Iterators.Lemmas.Consumers.Monadic.Loop
+import Init.Data.Iterators.Lemmas.Consumers.Monadic
+import Init.Data.Iterators.Lemmas.Consumers.Monadic.Loop
 
 namespace Std.Iterators
 
diff --git a/src/Std/Data/Iterators/Lemmas/Producers/Monadic/List.lean b/src/Std/Data/Iterators/Lemmas/Producers/Monadic/List.lean
index 8052bb745a..2b7873daae 100644
--- a/src/Std/Data/Iterators/Lemmas/Producers/Monadic/List.lean
+++ b/src/Std/Data/Iterators/Lemmas/Producers/Monadic/List.lean
@@ -4,10 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
 prelude
+import Init.Data.Iterators.Consumers
+import Init.Data.Iterators.Lemmas.Consumers.Monadic
+import Init.Data.Iterators.Internal.LawfulMonadLiftFunction
 import Std.Data.Iterators.Producers.Monadic.List
-import Std.Data.Iterators.Consumers
-import Std.Data.Iterators.Lemmas.Consumers.Monadic
-import Std.Data.Internal.LawfulMonadLiftFunction
+import Std.Data.Iterators.Lemmas.Equivalence.Basic
 
 /-!
 # Lemmas about list iterators
diff --git a/src/Std/Data/Iterators/Lemmas/Producers/Repeat.lean b/src/Std/Data/Iterators/Lemmas/Producers/Repeat.lean
index 9cbbdd16a3..92b0815934 100644
--- a/src/Std/Data/Iterators/Lemmas/Producers/Repeat.lean
+++ b/src/Std/Data/Iterators/Lemmas/Producers/Repeat.lean
@@ -6,8 +6,8 @@ Authors: Paul Reichert
 prelude
 import Init.Data.Option.Lemmas
 import Std.Data.Iterators.Producers.Repeat
-import Std.Data.Iterators.Consumers.Access
-import Std.Data.Iterators.Consumers.Collect
+import Init.Data.Iterators.Consumers.Access
+import Init.Data.Iterators.Consumers.Collect
 import Std.Data.Iterators.Combinators.Take
 import Std.Data.Iterators.Lemmas.Combinators.Take
 
diff --git a/src/Std/Data/Iterators/Producers/Monadic/Array.lean b/src/Std/Data/Iterators/Producers/Monadic/Array.lean
index fb5a0d1d07..92621e86e0 100644
--- a/src/Std/Data/Iterators/Producers/Monadic/Array.lean
+++ b/src/Std/Data/Iterators/Producers/Monadic/Array.lean
@@ -6,8 +6,8 @@ Authors: Paul Reichert
 prelude
 import Init.Data.Nat.Lemmas
 import Init.RCases
-import Std.Data.Iterators.Consumers
-import Std.Data.Iterators.Internal.Termination
+import Init.Data.Iterators.Consumers
+import Init.Data.Iterators.Internal.Termination
 
 /-!
 # Array iterator
diff --git a/src/Std/Data/Iterators/Producers/Monadic/Empty.lean b/src/Std/Data/Iterators/Producers/Monadic/Empty.lean
index e952845e6c..83040e752e 100644
--- a/src/Std/Data/Iterators/Producers/Monadic/Empty.lean
+++ b/src/Std/Data/Iterators/Producers/Monadic/Empty.lean
@@ -4,9 +4,9 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
 prelude
-import Std.Data.Iterators.Consumers.Collect
-import Std.Data.Iterators.Consumers.Loop
-import Std.Data.Iterators.Internal.Termination
+import Init.Data.Iterators.Consumers.Collect
+import Init.Data.Iterators.Consumers.Loop
+import Init.Data.Iterators.Internal.Termination
 
 /-!
 This file provides an empty iterator.
diff --git a/src/Std/Data/Iterators/Producers/Monadic/List.lean b/src/Std/Data/Iterators/Producers/Monadic/List.lean
index 8af61472ef..ec95f7b0a3 100644
--- a/src/Std/Data/Iterators/Producers/Monadic/List.lean
+++ b/src/Std/Data/Iterators/Producers/Monadic/List.lean
@@ -6,8 +6,8 @@ Authors: Paul Reichert
 prelude
 import Init.Data.Nat.Lemmas
 import Init.RCases
-import Std.Data.Iterators.Consumers
-import Std.Data.Iterators.Internal.Termination
+import Init.Data.Iterators.Consumers
+import Init.Data.Iterators.Internal.Termination
 
 /-!
 # List iterator
diff --git a/src/Std/Data/Iterators/Producers/Repeat.lean b/src/Std/Data/Iterators/Producers/Repeat.lean
index 66231cebba..30b6884bef 100644
--- a/src/Std/Data/Iterators/Producers/Repeat.lean
+++ b/src/Std/Data/Iterators/Producers/Repeat.lean
@@ -4,8 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Paul Reichert
 -/
 prelude
-import Std.Data.Iterators.Consumers.Monadic
-import Std.Data.Iterators.Internal.Termination
+import Init.Data.Iterators.Consumers.Monadic
+import Init.Data.Iterators.Internal.Termination
 
 /-!
 # Function-unfolding iterator