diff --git a/SciLean/Data/ArrayType/Basic.lean b/SciLean/Data/ArrayType/Basic.lean
index e9cd9196..7d1bc2f4 100644
--- a/SciLean/Data/ArrayType/Basic.lean
+++ b/SciLean/Data/ArrayType/Basic.lean
@@ -2,6 +2,7 @@ import SciLean.Util.SorryProof
 import SciLean.Data.Index
 import SciLean.Data.ListN 
 import SciLean.Data.StructType.Basic
+import SciLean.Data.Function
 
 namespace SciLean
 
@@ -278,6 +279,46 @@ theorem sum_introElem [EnumType Idx] [ArrayType Cont Idx Elem] [AddCommMonoid El
     introElem fun i => ∑ j, f j i
   := sorry_proof
 
+
+section UsefulFunctions
+
+
+variable 
+  [ArrayType Cont Idx Elem] [Index Idx]
+  [LT Elem] [∀ x y : Elem, Decidable (x < y)] [Inhabited Idx] 
+
+def argMaxCore (cont : Cont ) : Idx × Elem :=
+  Function.reduceD 
+    (fun i => (i,cont[i]))
+    (fun (i,e) (i',e') => if e < e' then (i',e') else (i,e))
+    (default, cont[default])
+
+def max (cont : Cont) : Elem :=
+  Function.reduceD 
+    (fun i => cont[i])
+    (fun e e' => if e < e' then e' else e)
+    (cont[default])
+
+def idxMax (cont : Cont) : Idx := (argMaxCore cont).1
+
+
+def argMinCore (cont : Cont ) : Idx × Elem :=
+  Function.reduceD 
+    (fun i => (i,cont[i]))
+    (fun (i,e) (i',e') => if e' < e then (i',e') else (i,e))
+    (default, cont[default])
+
+def min (cont : Cont) : Elem :=
+  Function.reduceD 
+    (fun i => cont[i])
+    (fun e e' => if e < e' then e' else e)
+    (cont[default])
+
+def idxMin (cont : Cont) : Idx := (argMinCore cont).1
+
+end UsefulFunctions
+
+
 end ArrayType
 
 
@@ -302,3 +343,4 @@ namespace ArrayType
       else y[⟨i.1-n, sorry_proof⟩]
 
 end ArrayType
+
diff --git a/SciLean/Data/ArrayType/Properties.lean b/SciLean/Data/ArrayType/Properties.lean
index bf7d2f09..58709c3e 100644
--- a/SciLean/Data/ArrayType/Properties.lean
+++ b/SciLean/Data/ArrayType/Properties.lean
@@ -925,6 +925,24 @@ theorem ArrayType.map.arg_farr.HasAdjDiff_rule
   (hf : HasAdjDiff K (fun (xe : X×Elem) => f xe.1 xe.2)) (harr : HasAdjDiff K arr)
   : HasAdjDiff K (fun x => map (f x) (arr x)) := sorry_proof
 
+@[ftrans]
+theorem ArrayType.map.arg_farr.revDeriv_rule
+  (f : X → Elem → Elem) (arr : X → Cont)
+  (hf : HasAdjDiff K (fun (x,e) => f x e)) (harr : HasAdjDiff K arr)
+  : revDeriv K (fun x => map (f x) (arr x))
+    =
+    fun x => 
+      let fdf := revDerivUpdate K (fun ((x,e) : X×Elem) => f x e)
+      let ada := revDerivUpdate K arr x
+      let a := ada.1
+      (map (f x) a, 
+       fun da => 
+         let (dx,da) := Function.repeatIdx (init:=((0 : X),da)) 
+           (fun (i : Idx) dxa => 
+             let dxai := (fdf (x,a[i])).2 dxa.2[i] (dxa.1,0)
+             (dxai.1, setElem dxa.2 i dxai.2)) 
+         ada.2 da dx) := sorry_proof
+
 
 @[ftrans]
 theorem ArrayType.map.arg_arr.revDeriv_rule
@@ -957,6 +975,41 @@ theorem ArrayType.map.arg_arr.revDerivUpdate_rule
          let da := mapIdx (fun i dai => let df := (fdf a[i]).2; df dai) da
          ada.2 da dx) := sorry_proof
 
+--------------------------------------------------------------------------------
+
+@[fprop]
+theorem ArrayType.max.arg_cont.HasAdjDiff_rule
+  [LT Elem] [∀ x y : Elem, Decidable (x < y)] [Inhabited Idx] 
+  (arr : X → Cont)
+  (hf : HasAdjDiff K arr) (hfalse : fpropParam False)
+  : HasAdjDiff K (fun x => max (arr x)) := sorry_proof
+
+
+@[ftrans]
+theorem ArrayType.max.arg_arr.revDeriv_rule
+  [LT Elem] [∀ x y : Elem, Decidable (x < y)] [Inhabited Idx]
+  (arr : X → Cont)
+  (hf : HasAdjDiff K arr) (hfalse : fpropParam False)
+  : revDeriv K (fun x => max (arr x))
+    =
+    fun x => 
+      let i := idxMax (arr x)
+      let fdf := revDerivProj K Idx arr x
+      (fdf.1[i], fun dei => fdf.2 i dei) := sorry_proof
+
+
+@[ftrans]
+theorem ArrayType.max.arg_arr.revDerivUpdate_rule
+  [LT Elem] [∀ x y : Elem, Decidable (x < y)] [Inhabited Idx] 
+  (arr : X → Cont)
+  (hf : HasAdjDiff K arr) (hfalse : fpropParam False)
+  : revDerivUpdate K (fun x => max (arr x))
+    =
+    fun x => 
+      let i := idxMax (arr x)
+      let fdf := revDerivProjUpdate K Idx arr x
+      (fdf.1[i], fun dei dx => fdf.2 i dei dx) := sorry_proof
+
 
 -- @[ftrans]
 -- theorem ArrayType.map.arg_farr.revDeriv_rule