diff --git a/src/Lean/Expr.lean b/src/Lean/Expr.lean
index 44e1f7deb2cf..9b6ef5d9b66e 100644
--- a/src/Lean/Expr.lean
+++ b/src/Lean/Expr.lean
@@ -1332,6 +1332,14 @@ lambda expression. See docstring for `betaRev` for examples.
 def beta (f : Expr) (args : Array Expr) : Expr :=
   betaRev f args.reverse
 
+/--
+Count the number of lambdas at the head of the given expression.
+-/
+def getNumHeadLambdas : Expr → Nat
+  | .lam _ _ b _ => getNumHeadLambdas b + 1
+  | .mdata _ b => getNumHeadLambdas b
+  | _ => 0
+
 /--
 Return true if the given expression is the function of an expression that is target for (head) beta reduction.
 If `useZeta = true`, then `let`-expressions are visited. That is, it assumes
diff --git a/src/Lean/Meta/Tactic/Simp/Main.lean b/src/Lean/Meta/Tactic/Simp/Main.lean
index 3a5498261960..b7aa54fe4a42 100644
--- a/src/Lean/Meta/Tactic/Simp/Main.lean
+++ b/src/Lean/Meta/Tactic/Simp/Main.lean
@@ -194,6 +194,23 @@ private def unfold? (e : Expr) : SimpM (Option Expr) := do
       if eNew == e then return none
       trace[Meta.Tactic.simp.ground] "{e}\n---->\n{eNew}"
       return some eNew
+  let rec unfoldDeclToUnfold? : SimpM (Option Expr) := do
+    let options ← getOptions
+    let cfg ← getConfig
+    -- Support for issue #2042
+    if cfg.unfoldPartialApp -- If we are unfolding partial applications, ignore issue #2042
+       -- When smart unfolding is enabled, and `f` supports it, we don't need to worry about issue #2042
+       || (smartUnfolding.get options && (← getEnv).contains (mkSmartUnfoldingNameFor fName)) then
+      withDefault <| unfoldDefinition? e
+    else
+      -- `We are not unfolding partial applications, and `fName` does not have smart unfolding support.
+      -- Thus, we must check whether the arity of the function >= number of arguments.
+      let some cinfo := (← getEnv).find? fName | return none
+      let some value := cinfo.value? | return none
+      let arity := value.getNumHeadLambdas
+      -- Partially applied function, return `none`. See issue #2042
+      if arity > e.getAppNumArgs then return none
+      withDefault <| unfoldDefinition? e
   if let some eNew ← unfoldGround? then
     return some eNew
   else if (← isProjectionFn fName) then
@@ -210,7 +227,7 @@ private def unfold? (e : Expr) : SimpM (Option Expr) := do
     else
       return none
   else if ctx.isDeclToUnfold fName then
-    withDefault <| unfoldDefinition? e
+    unfoldDeclToUnfold?
   else
     return none
 
@@ -352,18 +369,18 @@ where
 
   simpStep (e : Expr) : M Result := do
     match e with
-    | Expr.mdata m e   => let r ← simp e; return { r with expr := mkMData m r.expr }
-    | Expr.proj ..     => simpProj e
-    | Expr.app ..      => simpApp e
-    | Expr.lam ..      => simpLambda e
-    | Expr.forallE ..  => simpForall e
-    | Expr.letE ..     => simpLet e
-    | Expr.const ..    => simpConst e
-    | Expr.bvar ..     => unreachable!
-    | Expr.sort ..     => return { expr := e }
-    | Expr.lit ..      => simpLit e
-    | Expr.mvar ..     => return { expr := (← instantiateMVars e) }
-    | Expr.fvar ..     => return { expr := (← reduceFVar (← getConfig) (← getSimpTheorems) e) }
+    | .mdata m e   => let r ← simp e; return { r with expr := mkMData m r.expr }
+    | .proj ..     => simpProj e
+    | .app ..      => simpApp e
+    | .lam ..      => simpLambda e
+    | .forallE ..  => simpForall e
+    | .letE ..     => simpLet e
+    | .const ..    => simpConst e
+    | .bvar ..     => unreachable!
+    | .sort ..     => return { expr := e }
+    | .lit ..      => simpLit e
+    | .mvar ..     => return { expr := (← instantiateMVars e) }
+    | .fvar ..     => return { expr := (← reduceFVar (← getConfig) (← getSimpTheorems) e) }
 
   simpLit (e : Expr) : M Result := do
     match e.natLit? with
@@ -768,7 +785,7 @@ where
       return { expr := (← dsimp e) }
 
   simpLet (e : Expr) : M Result := do
-    let Expr.letE n t v b _ := e | unreachable!
+    let .letE n t v b _ := e | unreachable!
     if (← getConfig).zeta then
       return { expr := b.instantiate1 v }
     else
diff --git a/tests/lean/run/2042.lean b/tests/lean/run/2042.lean
new file mode 100644
index 000000000000..01fa29d12e15
--- /dev/null
+++ b/tests/lean/run/2042.lean
@@ -0,0 +1,21 @@
+@[simp] def foo (a : Nat) : Nat :=
+  2 * a
+
+example : foo = fun a => a + a :=
+by
+  fail_if_success simp -- should not unfold `foo` into a lambda
+  funext x
+  simp -- unfolds `foo`
+  trace_state
+  simp_arith
+
+@[simp] def boo : Nat → Nat
+  | a => 2 * a
+
+example : boo = fun a => a + a :=
+by
+  fail_if_success simp -- should not unfold `boo` into a lambda
+  funext x
+  simp -- unfolds `boo`
+  trace_state
+  simp_arith