diff --git a/SciLean/Data/IndexType/Fold.lean b/SciLean/Data/IndexType/Fold.lean
index 2f6ea2f7..2f1e268d 100644
--- a/SciLean/Data/IndexType/Fold.lean
+++ b/SciLean/Data/IndexType/Fold.lean
@@ -1,5 +1,6 @@
 import SciLean.Analysis.Calculus.RevFDeriv
 import SciLean.Analysis.Calculus.FwdFDeriv
+import SciLean.Analysis.Calculus.RevFDerivProj
 import SciLean.Data.IndexType.Operations
 import SciLean.Tactic.Autodiff
 import SciLean.Data.DataArray.DataArray
@@ -184,51 +185,50 @@ theorem IndexType.Range.foldl.arg_opinit.revFDeriv_rule_closures (r : Range I)
         dw + dw') := sorry_proof
 
 
-/-- Reverse derivative of fold - version storing every point - use DataArray if possible -/
+/-- Reverse derivative of fold - version storing every point - store in Array if DataArray is not
+available for `X` -/
 @[fun_trans]
-theorem IndexType.Range.foldl.arg_opinit.revFDeriv_rule_data_array
-    {I: Type} [IndexType I]
-    {X : Type} [NormedAddCommGroup X] [AdjointSpace R X] [CompleteSpace X] [PlainDataType X]
+theorem IndexType.Range.foldl.arg_opinit.revFDeriv_rule_array (r : Range I)
     (op : W → X → I → X) (hop : ∀ i, Differentiable R (fun (w,x) => op w x i))
     (init : W → X) (hinit : Differentiable R init) :
-    revFDeriv R (fun w => (.full : Range I).foldl (op w) (init w))
+    revFDeriv R (fun w => r.foldl (op w) (init w))
     =
     fun w =>
       let idi := revFDeriv R init w
-      let xsx := (.full : Range I).foldl (fun (xs,x) i =>
-        let xs := xs.set i x
+      let xsx := r.foldl (fun (xs,x) i =>
+        let xs := xs.push (x,i)
         let x := op w x i
-        (xs,x)) ((0 : X^[I]), idi.1)
+        (xs,x)) ((#[] : Array (X×I)), idi.1)
       let xs := xsx.1
       let x := xsx.2
       (x, fun dx =>
-        let dwx := (.full : Range I).reverse.foldl (fun (dw,dx) i  =>
-          let x := xs[i]
+        let dwx := xs.foldr (fun (x,i) (dw,dx) =>
           let dwx := (revFDeriv R (fun (w,x) => op w x i) (w,x)).2 dx
           (dw + dwx.1, dwx.2)) (0, dx)
         let dw' := idi.2 dwx.2
         dwx.1 + dw') := sorry_proof
 
 
-
-/-- Reverse derivative of fold - version storing every point - store in Array if DataArray is not
-available for `X` -/
+/-- Reverse derivative of fold - version storing every point - use DataArray if possible -/
 @[fun_trans]
-theorem IndexType.Range.foldl.arg_opinit.revFDeriv_rule_array (r : Range I)
+theorem IndexType.Range.foldl.arg_opinit.revFDeriv_rule_data_array
+    {I: Type} [IndexType I]
+    {X : Type} [NormedAddCommGroup X] [AdjointSpace R X] [CompleteSpace X] [PlainDataType X]
     (op : W → X → I → X) (hop : ∀ i, Differentiable R (fun (w,x) => op w x i))
     (init : W → X) (hinit : Differentiable R init) :
-    revFDeriv R (fun w => r.foldl (op w) (init w))
+    revFDeriv R (fun w => (.full : Range I).foldl (op w) (init w))
     =
     fun w =>
       let idi := revFDeriv R init w
-      let xsx := r.foldl (fun (xs,x) i =>
-        let xs := xs.push (x,i)
+      let xsx := (.full : Range I).foldl (fun (xs,x) i =>
+        let xs := xs.set i x
         let x := op w x i
-        (xs,x)) ((#[] : Array (X×I)), idi.1)
+        (xs,x)) ((0 : X^[I]), idi.1)
       let xs := xsx.1
       let x := xsx.2
       (x, fun dx =>
-        let dwx := xs.foldr (fun (x,i) (dw,dx) =>
+        let dwx := (.full : Range I).reverse.foldl (fun (dw,dx) i  =>
+          let x := xs[i]
           let dwx := (revFDeriv R (fun (w,x) => op w x i) (w,x)).2 dx
           (dw + dwx.1, dwx.2)) (0, dx)
         let dw' := idi.2 dwx.2
@@ -255,4 +255,32 @@ theorem IndexType.Range.foldl.arg_opinit.revFDeriv_rule_linear (r : Range I)
         let dw' := idi.2 dwx.2
         dwx.1 + dw') := sorry_proof
 
+
+@[fun_trans]
+theorem IndexType.Range.foldl.arg_opinit.revFDerivProj_rule_data_array
+    {R : Type} [RCLike R]
+    {I : Type} [IndexType I]
+    {X : Type} [NormedAddCommGroup X] [AdjointSpace R X] [CompleteSpace X] [PlainDataType X]
+    {W : Type} [NormedAddCommGroup W] [AdjointSpace R W] [CompleteSpace W]
+    (op : W → X → I → X) (hop : ∀ i, Differentiable R (fun (w,x) => op w x i))
+    (init : W → X) (hinit : Differentiable R init) :
+    revFDerivProj R Unit (fun w => (.full : Range I).foldl (op w) (init w))
+    =
+    fun w =>
+      let idi := revFDeriv R init w
+      let xsx := (.full : Range I).foldl (fun (xs,x) i =>
+        let xs := xs.set i x
+        let x := op w x i
+        (xs,x)) ((0 : X^[I]), idi.1)
+      let xs := xsx.1
+      let x := xsx.2
+      (x, fun _ dx =>
+        let dwx := (.full : Range I).reverse.foldl (fun (dw,dx) i  =>
+          let x := xs[i]
+          let dwx : W×X := (revFDerivProjUpdate R Unit (fun (w,x) => op w x i) (w,x)).2 () dx (dw,0)
+          dwx) (0, dx)
+        let dw' := idi.2 dwx.2
+        dwx.1 + dw') := sorry_proof
+
+
 -- TODO: add checkpointing version
diff --git a/SciLean/Data/IndexType/Iterator.lean b/SciLean/Data/IndexType/Iterator.lean
index 30b78d51..9a6653e3 100644
--- a/SciLean/Data/IndexType/Iterator.lean
+++ b/SciLean/Data/IndexType/Iterator.lean
@@ -62,5 +62,5 @@ def sprod (i : Iterator I) (j : Iterator J) [FirstLast I I] [FirstLast J J] :
 
 
 -- todo: implement this and provide a better implementation of IndexType instance for Sum
-private def ofSum [FirstLast α α] [FirstLast β β] (s : Iterator (α ⊕ β)) :
-    ((Iterator α × Range β)) ⊕ ((Iterator β × Range α)) := sorry
+-- private def ofSum [FirstLast α α] [FirstLast β β] (s : Iterator (α ⊕ β)) :
+--     ((Iterator α × Range β)) ⊕ ((Iterator β × Range α)) := sorry
diff --git a/SciLean/Data/IndexType/Operations.lean b/SciLean/Data/IndexType/Operations.lean
index afebc936..5d5e5845 100644
--- a/SciLean/Data/IndexType/Operations.lean
+++ b/SciLean/Data/IndexType/Operations.lean
@@ -30,12 +30,13 @@ def reduceMD {m} [Monad m] (r : Range ι) (f : ι → α) (op : α → α → m
   match first? r with
   | none => return default
   | .some fst => do
-    r.foldlM (fun a i => op a (f i)) (f fst)
+    let mut a := (f fst)
+    for i in Iterator.val fst r do
+      a ← op a (f i)
+    return a
 
 def reduceD (r : Range ι) (f : ι → α) (op : α → α → α) (default : α) : α := Id.run do
-  match first? r with
-  | none => return default
-  | .some fst => do r.foldl (fun a i => op a (f i)) (f fst)
+  r.reduceMD f (fun x y => pure (op x y)) default
 
 abbrev reduce [Inhabited α] (r : Range ι) (f : ι → α) (op : α → α → α) : α :=
   r.reduceD f op default
@@ -51,7 +52,7 @@ variable {ι : Type*} [IndexType ι]
 abbrev foldlM {m} [Monad m] (op : α → ι → m α) (init : α) : m α :=
   Range.full.foldlM op init
 
-abbrev foldl (op : α → ι → α) (init : α) : α := Id.run do
+abbrev foldl (op : α → ι → α) (init : α) : α :=
   Range.full.foldl op init
 
 abbrev reduceMD {m} [Monad m] (f : ι → α) (op : α → α → m α) (default : α) : m α :=