diff --git a/SciLean/Core/FunctionTransformations/RevCDeriv.lean b/SciLean/Core/FunctionTransformations/RevCDeriv.lean
index 808e2ad0..30710488 100644
--- a/SciLean/Core/FunctionTransformations/RevCDeriv.lean
+++ b/SciLean/Core/FunctionTransformations/RevCDeriv.lean
@@ -1146,6 +1146,20 @@ by
   sorry_proof
 
 
+-- this should not apply for `a0 = (fun x => x)`
+-- @[ftrans] 
+theorem SciLean.cderiv.arg_a3.semiAdjoint_rule
+  (f : X → Y) (x : X) (a0 : W → X) (ha0 : HasSemiAdjoint K a0)
+  : semiAdjoint K (fun w => cderiv K f x (a0 w)) 
+    =
+    fun dy => 
+      let dx := semiAdjoint K (cderiv K f x) dy
+      semiAdjoint K a0 dx :=
+by
+  sorry_proof
+
+
+set_option trace.Meta.Tactic.simp.rewrite true in
 @[ftrans]
 theorem SciLean.semiAdjoint.arg_a3.revCDeriv_rule
   (f : X → Y) (a0 : W → Y) (hf : HasSemiAdjoint K f) (ha0 : HasAdjDiff K a0)
diff --git a/SciLean/Core/FunctionTransformations/RevDerivUpdate.lean b/SciLean/Core/FunctionTransformations/RevDerivUpdate.lean
index 1797ceb9..b0044e2f 100644
--- a/SciLean/Core/FunctionTransformations/RevDerivUpdate.lean
+++ b/SciLean/Core/FunctionTransformations/RevDerivUpdate.lean
@@ -1,10 +1,4 @@
-import SciLean.Core.FunctionPropositions.HasAdjDiffAt
-import SciLean.Core.FunctionPropositions.HasAdjDiff
-
-import SciLean.Core.FunctionTransformations.SemiAdjoint
-
-import SciLean.Tactic.LetNormalize
-
+import SciLean.Core.FunctionTransformations.RevCDeriv
 
 set_option linter.unusedVariables false
 
@@ -32,7 +26,7 @@ namespace revDerivUpdate
 --------------------------------------------------------------------------------
 
 variable (X)
-theorem id_rule 
+theorem id_rule
   : revDerivUpdate K (fun x : X => x) = fun x => (x, fun dx' k dx => dx + k • dx') :=
 by
   unfold revDerivUpdate
@@ -44,9 +38,9 @@ theorem const_rule (y : Y)
 by
   unfold revDerivUpdate
   funext _; ftrans; ftrans
-variable{X}
+variable {X}
 
-variable(E)
+variable (E)
 theorem proj_rule (i : ι)
   : revDerivUpdate K (fun (x : (i:ι) → E i) => x i)
     = 
@@ -75,6 +69,21 @@ by
   unfold revDerivUpdate
   funext _; ftrans; ftrans; simp
 
+theorem comp_rule'
+  (f : Y → Z) (g : X → Y) 
+  (hf : HasAdjDiff K f) (hg : HasAdjDiff K g)
+  : revDerivUpdate K (fun x : X => f (g x))
+    = 
+    fun x =>
+      let ydg := revDerivUpdate K g x
+      let zdf := revDerivUpdate K (fun x' => f (ydg.1 + semiAdjoint K (ydg.2 · 1 0) (x' -x))) x
+      zdf := 
+by
+  have ⟨_,_⟩ := hf
+  have ⟨_,_⟩ := hg
+  unfold revDerivUpdate
+  funext _; simp; ftrans
+
 
 theorem let_rule 
   (f : X → Y → Z) (g : X → Y) 
@@ -99,6 +108,21 @@ by
   have h : IsLinearMap K (semiAdjoint K (cderiv K g x)) := sorry_proof
   rw[h.map_smul]
 
+theorem let_rule'
+  (f : X → Y → Z) (g : X → Y) 
+  (hf : HasAdjDiff K (fun (xy : X×Y) => f xy.1 xy.2)) (hg : HasAdjDiff K g)
+  : revDerivUpdate K (fun x : X => let y := g x; f x y) 
+    = 
+    fun x => 
+      let ydg := revDerivUpdate K g x
+      let zdf := revDerivUpdate K (fun x' => f x' (ydg.1 + semiAdjoint K (ydg.2 · 1 0) (x' - x))) x
+      zdf :=
+by
+  have ⟨_,_⟩ := hf
+  have ⟨_,_⟩ := hg
+  unfold revDerivUpdate
+  funext x; simp; ftrans
+
 
 @[inline] 
 def fun_fold {ι : Type _} [EnumType ι] (f : ι → X → X) (x₀ : X) : X := Id.run do
@@ -216,7 +240,7 @@ end SciLean
 
 --------------------------------------------------------------------------------
 -- Function Rules --------------------------------------------------------------
---------------------------------------------------------------------------------b
+--------------------------------------------------------------------------------
 
 open SciLean 
 
@@ -230,7 +254,7 @@ variable
   {E : ι → Type _} [∀ i, SemiInnerProductSpace K (E i)]
 
 
--- Prod.mk -----------------------------------v---------------------------------
+-- Prod.mk ---------------------------------------------------------------------
 --------------------------------------------------------------------------------
 
 @[ftrans]
@@ -247,7 +271,18 @@ by
   have ⟨_,_⟩ := hf
   have ⟨_,_⟩ := hg
   unfold revDerivUpdate; simp; ftrans; ftrans; simp[add_assoc]
+
  
+theorem Prod.mk.arg_fstsnd.revDerivUpdate_rule_simple
+  : revDerivUpdate K (fun xy : X × Y => (xy.1, xy.2))
+    =
+    fun xy => 
+      (xy, fun (dx',dy') k (dx,dy) => (dx+k•dx', dy+k•dy')) := 
+by 
+  unfold revDerivUpdate; 
+  funext (x,y); simp
+  funext (dx',dy') k (dx,dy); ftrans; ftrans; simp
+
 
 -- Prod.fst --------------------------------------------------------------------
 --------------------------------------------------------------------------------
@@ -265,6 +300,17 @@ by
   unfold revDerivUpdate; ftrans; ftrans; simp
 
 
+theorem Prod.fst.arg_self.revDerivUpdate_rule_simple
+  : revDerivUpdate K (fun xy : X×Y => xy.1)
+    =
+    fun xy => 
+      (xy.1, fun dx' k (dx,dy) => (dx+k•dx', dy)) := 
+by 
+  unfold revDerivUpdate;
+  funext (x,y); simp; ftrans;
+  funext dx' k (dx,dy); ftrans; ftrans; simp
+
+
 -- Prod.snd --------------------------------------------------------------------
 --------------------------------------------------------------------------------
 
@@ -280,6 +326,16 @@ by
   have ⟨_,_⟩ := hf
   unfold revDerivUpdate; ftrans; ftrans; simp
 
+theorem Prod.snd.arg_self.revDerivUpdate_simple_rule
+  : revDerivUpdate K (fun xy : X×Y => xy.2)
+    =
+    fun xy => 
+      (xy.2, fun dy' k (dx,dy) => (dx, dy + k•dy')) := 
+by 
+  unfold revDerivUpdate;
+  funext (x,y); simp; ftrans;
+  funext dy' k (dx,dy); ftrans; ftrans; simp
+
 
 -- Function.comp ---------------------------------------------------------------
 --------------------------------------------------------------------------------