diff --git a/RELEASES.md b/RELEASES.md
index 41afd578e86b..bc7d826cac12 100644
--- a/RELEASES.md
+++ b/RELEASES.md
@@ -66,6 +66,21 @@ v4.7.0 (development in progress)
     rw [h]
   ```
 
+* When adding new local theorems to `simp`, the system assumes that the function application arguments
+  have been annotated with `no_index`. This modification, which addresses issue [#2670](https://github.com/leanprover/lean4/issues/2670),
+  restores the Lean 3 behavior that users expect. With this modification, the following examples are now operational:
+  ```
+  example {α β : Type} {f : α × β → β → β} (h : ∀ p : α × β, f p p.2 = p.2)
+    (a : α) (b : β) : f (a, b) b = b := by
+    simp [h]
+
+  example {α β : Type} {f : α × β → β → β}
+    (a : α) (b : β) (h : f (a,b) (a,b).2 = (a,b).2) : f (a, b) b = b := by
+    simp [h]
+  ```
+  In both cases, `h` is applicable because `simp` does not index f-arguments anymore when adding `h` to the `simp`-set.
+  It's important to note, however, that global theorems continue to be indexed in the usual manner.
+
 Breaking changes:
 * `Lean.withTraceNode` and variants got a stronger `MonadAlwaysExcept` assumption to
   fix trace trees not being built on elaboration runtime exceptions. Instances for most elaboration
diff --git a/src/Lean/Meta/DiscrTree.lean b/src/Lean/Meta/DiscrTree.lean
index 9b569b7f0cbe..c4dad896c53e 100644
--- a/src/Lean/Meta/DiscrTree.lean
+++ b/src/Lean/Meta/DiscrTree.lean
@@ -326,7 +326,26 @@ def reduceDT (e : Expr) (root : Bool) (config : WhnfCoreConfig) : MetaM Expr :=
 
 /- Remark: we use `shouldAddAsStar` only for nested terms, and `root == false` for nested terms -/
 
-private def pushArgs (root : Bool) (todo : Array Expr) (e : Expr) (config : WhnfCoreConfig) : MetaM (Key × Array Expr) := do
+
+/--
+Append `n` wildcards to `todo`
+-/
+private def pushWildcards (n : Nat) (todo : Array Expr) : Array Expr :=
+  match n with
+  | 0   => todo
+  | n+1 => pushWildcards n (todo.push tmpStar)
+
+/--
+When `noIndexAtArgs := true`, `pushArgs` assumes function application arguments have a `no_index` annotation.
+That is, `f a b` is indexed as it was `f (no_index a) (no_index b)`.
+This feature is used when indexing local proofs in the simplifier. This is useful in examples like the one described on issue #2670.
+In this issue, we have a local hypotheses `(h : ∀ p : α × β, f p p.2 = p.2)`, and users expect it to be applicable to
+`f (a, b) b = b`. This worked in Lean 3 since no indexing was used. We can retrieve Lean 3 behavior by writing
+`(h : ∀ p : α × β, f p (no_index p.2) = p.2)`, but this is very inconvenient when the hypotheses was not written by the user in first place.
+For example, it was introduced by another tactic. Thus, when populating the discrimination tree explicit arguments provided to `simp` (e.g., `simp [h]`),
+we use `noIndexAtArgs := true`. See comment: https://github.com/leanprover/lean4/issues/2670#issuecomment-1758889365
+-/
+private def pushArgs (root : Bool) (todo : Array Expr) (e : Expr) (config : WhnfCoreConfig) (noIndexAtArgs : Bool) : MetaM (Key × Array Expr) := do
   if hasNoindexAnnotation e then
     return (.star, todo)
   else
@@ -334,7 +353,10 @@ private def pushArgs (root : Bool) (todo : Array Expr) (e : Expr) (config : Whnf
     let fn := e.getAppFn
     let push (k : Key) (nargs : Nat) (todo : Array Expr): MetaM (Key × Array Expr) := do
       let info ← getFunInfoNArgs fn nargs
-      let todo ← pushArgsAux info.paramInfo (nargs-1) e todo
+      let todo ← if noIndexAtArgs then
+        pure <| pushWildcards nargs todo
+      else
+        pushArgsAux info.paramInfo (nargs-1) e todo
       return (k, todo)
     match fn with
     | .lit v     =>
@@ -378,22 +400,24 @@ private def pushArgs (root : Bool) (todo : Array Expr) (e : Expr) (config : Whnf
     | _ =>
       return (.other, todo)
 
-partial def mkPathAux (root : Bool) (todo : Array Expr) (keys : Array Key) (config : WhnfCoreConfig) : MetaM (Array Key) := do
+@[inherit_doc pushArgs]
+partial def mkPathAux (root : Bool) (todo : Array Expr) (keys : Array Key) (config : WhnfCoreConfig) (noIndexAtArgs : Bool) : MetaM (Array Key) := do
   if todo.isEmpty then
     return keys
   else
     let e    := todo.back
     let todo := todo.pop
-    let (k, todo) ← pushArgs root todo e config
-    mkPathAux false todo (keys.push k) config
+    let (k, todo) ← pushArgs root todo e config noIndexAtArgs
+    mkPathAux false todo (keys.push k) config noIndexAtArgs
 
 private def initCapacity := 8
 
-def mkPath (e : Expr) (config : WhnfCoreConfig) : MetaM (Array Key) := do
+@[inherit_doc pushArgs]
+def mkPath (e : Expr) (config : WhnfCoreConfig) (noIndexAtArgs := false) : MetaM (Array Key) := do
   withReducible do
     let todo : Array Expr := .mkEmpty initCapacity
     let keys : Array Key := .mkEmpty initCapacity
-    mkPathAux (root := true) (todo.push e) keys config
+    mkPathAux (root := true) (todo.push e) keys config noIndexAtArgs
 
 private partial def createNodes (keys : Array Key) (v : α) (i : Nat) : Trie α :=
   if h : i < keys.size then
@@ -447,22 +471,21 @@ def insertCore [BEq α] (d : DiscrTree α) (keys : Array Key) (v : α) : DiscrTr
       let c := insertAux keys v 1 c
       { root := d.root.insert k c }
 
-def insert [BEq α] (d : DiscrTree α) (e : Expr) (v : α) (config : WhnfCoreConfig) : MetaM (DiscrTree α) := do
-  let keys ← mkPath e config
+def insert [BEq α] (d : DiscrTree α) (e : Expr) (v : α) (config : WhnfCoreConfig) (noIndexAtArgs := false) : MetaM (DiscrTree α) := do
+  let keys ← mkPath e config noIndexAtArgs
   return d.insertCore keys v
 
 /--
 Inserts a value into a discrimination tree,
 but only if its key is not of the form `#[*]` or `#[=, *, *, *]`.
 -/
-def insertIfSpecific [BEq α] (d : DiscrTree α) (e : Expr) (v : α) (config : WhnfCoreConfig) : MetaM (DiscrTree α) := do
-  let keys ← mkPath e config
+def insertIfSpecific [BEq α] (d : DiscrTree α) (e : Expr) (v : α) (config : WhnfCoreConfig) (noIndexAtArgs := false) : MetaM (DiscrTree α) := do
+  let keys ← mkPath e config noIndexAtArgs
   return if keys == #[Key.star] || keys == #[Key.const `Eq 3, Key.star, Key.star, Key.star] then
     d
   else
     d.insertCore keys v
 
-
 private def getKeyArgs (e : Expr) (isMatch root : Bool) (config : WhnfCoreConfig) : MetaM (Key × Array Expr) := do
   let e ← reduceDT e root config
   unless root do
diff --git a/src/Lean/Meta/Tactic/Simp/SimpTheorems.lean b/src/Lean/Meta/Tactic/Simp/SimpTheorems.lean
index 63e66ae03110..44816e98a6e1 100644
--- a/src/Lean/Meta/Tactic/Simp/SimpTheorems.lean
+++ b/src/Lean/Meta/Tactic/Simp/SimpTheorems.lean
@@ -319,7 +319,7 @@ private def checkTypeIsProp (type : Expr) : MetaM Unit :=
   unless (← isProp type) do
     throwError "invalid 'simp', proposition expected{indentExpr type}"
 
-private def mkSimpTheoremCore (origin : Origin) (e : Expr) (levelParams : Array Name) (proof : Expr) (post : Bool) (prio : Nat) : MetaM SimpTheorem := do
+private def mkSimpTheoremCore (origin : Origin) (e : Expr) (levelParams : Array Name) (proof : Expr) (post : Bool) (prio : Nat) (noIndexAtArgs : Bool) : MetaM SimpTheorem := do
   assert! origin != .fvar ⟨.anonymous⟩
   let type ← instantiateMVars (← inferType e)
   withNewMCtxDepth do
@@ -327,7 +327,7 @@ private def mkSimpTheoremCore (origin : Origin) (e : Expr) (levelParams : Array
     let type ← whnfR type
     let (keys, perm) ←
       match type.eq? with
-      | some (_, lhs, rhs) => pure (← DiscrTree.mkPath lhs simpDtConfig, ← isPerm lhs rhs)
+      | some (_, lhs, rhs) => pure (← DiscrTree.mkPath lhs simpDtConfig noIndexAtArgs, ← isPerm lhs rhs)
       | none => throwError "unexpected kind of 'simp' theorem{indentExpr type}"
     return { origin, keys, perm, post, levelParams, proof, priority := prio, rfl := (← isRflProof proof) }
 
@@ -341,10 +341,10 @@ private def mkSimpTheoremsFromConst (declName : Name) (post : Bool) (inv : Bool)
       let mut r := #[]
       for (val, type) in (← preprocess val type inv (isGlobal := true)) do
         let auxName ← mkAuxLemma cinfo.levelParams type val
-        r := r.push <| (← mkSimpTheoremCore (.decl declName post inv) (mkConst auxName (cinfo.levelParams.map mkLevelParam)) #[] (mkConst auxName) post prio)
+        r := r.push <| (← mkSimpTheoremCore (.decl declName post inv) (mkConst auxName (cinfo.levelParams.map mkLevelParam)) #[] (mkConst auxName) post prio (noIndexAtArgs := false))
       return r
     else
-      return #[← mkSimpTheoremCore (.decl declName post inv) (mkConst declName (cinfo.levelParams.map mkLevelParam)) #[] (mkConst declName) post prio]
+      return #[← mkSimpTheoremCore (.decl declName post inv) (mkConst declName (cinfo.levelParams.map mkLevelParam)) #[] (mkConst declName) post prio (noIndexAtArgs := false)]
 
 inductive SimpEntry where
   | thm      : SimpTheorem → SimpEntry
@@ -455,7 +455,7 @@ private def preprocessProof (val : Expr) (inv : Bool) : MetaM (Array Expr) := do
 /-- Auxiliary method for creating simp theorems from a proof term `val`. -/
 def mkSimpTheorems (id : Origin) (levelParams : Array Name) (proof : Expr) (post := true) (inv := false) (prio : Nat := eval_prio default) : MetaM (Array SimpTheorem) :=
   withReducible do
-    (← preprocessProof proof inv).mapM fun val => mkSimpTheoremCore id val levelParams val post prio
+    (← preprocessProof proof inv).mapM fun val => mkSimpTheoremCore id val levelParams val post prio (noIndexAtArgs := true)
 
 /-- Auxiliary method for adding a local simp theorem to a `SimpTheorems` datastructure. -/
 def SimpTheorems.add (s : SimpTheorems) (id : Origin) (levelParams : Array Name) (proof : Expr) (inv := false) (post := true) (prio : Nat := eval_prio default) : MetaM SimpTheorems := do
diff --git a/tests/lean/run/2670.lean b/tests/lean/run/2670.lean
new file mode 100644
index 000000000000..64443b27b569
--- /dev/null
+++ b/tests/lean/run/2670.lean
@@ -0,0 +1,23 @@
+example {α β : Type} {f : α × β → β → β} (h : ∀ p : α × β, f p p.2 = p.2)
+  (a : α) (b : β) : f (a, b) b = b := by
+  simp [h] -- should not fail
+
+example {α β : Type} {f : α × β → β → β}
+  (a : α) (b : β) (h : f (a,b) (a,b).2 = (a,b).2) : f (a, b) b = b := by
+  simp [h] -- should not fail
+
+def enumFromTR' (n : Nat) (l : List α) : List (Nat × α) :=
+  let arr := l.toArray
+  (arr.foldr (fun a (n, acc) => (n-1, (n-1, a) :: acc)) (n + arr.size, [])).2
+
+open List in
+theorem enumFrom_eq_enumFromTR' : @enumFrom = @enumFromTR' := by
+  funext α n l; simp [enumFromTR', -Array.size_toArray]
+  let f := fun (a : α) (n, acc) => (n-1, (n-1, a) :: acc)
+  let rec go : ∀ l n, l.foldr f (n + l.length, []) = (n, enumFrom n l)
+    | [], n => rfl
+    | a::as, n => by
+      rw [← show _ + as.length = n + (a::as).length from Nat.succ_add .., List.foldr, go as]
+      simp [enumFrom, f]
+  rw [Array.foldr_eq_foldr_data]
+  simp [go] -- Should close the goal
diff --git a/tests/lean/run/ac_rfl.lean b/tests/lean/run/ac_rfl.lean
index 5d0cc9f73060..eb1bc983ac65 100644
--- a/tests/lean/run/ac_rfl.lean
+++ b/tests/lean/run/ac_rfl.lean
@@ -17,9 +17,9 @@ instance : LawfulCommIdentity HMul.hMul 1 where right_id := Nat.mul_one
 set_option linter.unusedVariables false in
 theorem le_ext : ∀ {x y : Nat} (h : ∀ z, z ≤ x ↔ z ≤ y), x = y
   | 0, 0, _ => rfl
-  | x+1, y+1, h => congrArg (· + 1) <| le_ext fun z => have := h (z + 1); by simp_all
+  | x+1, y+1, h => congrArg (· + 1) <| le_ext fun z => have := h (z + 1); by simp at this; assumption
   | 0, y+1, h => have := h 1; by clear h; simp_all
-  | x+1, 0, h => have := h 1; by simp_all
+  | x+1, 0, h => have := h 1; by simp at this
 
 theorem le_or_le : ∀ (a b : Nat), a ≤ b ∨ b ≤ a
   | x+1, y+1 => by simp [le_or_le x y]
diff --git a/tests/lean/run/meta3.lean b/tests/lean/run/meta3.lean
index ea2e4b456fa0..645f941b8eaa 100644
--- a/tests/lean/run/meta3.lean
+++ b/tests/lean/run/meta3.lean
@@ -34,11 +34,11 @@ def add  := mkAppN (mkConst `Add.add [levelZero]) #[nat, mkConst `Nat.add]
 def tst1 : MetaM Unit :=
 do let d : DiscrTree Nat := {};
    let mvar ← mkFreshExprMVar nat;
-   let d ← d.insert (mkAppN add #[mvar, mkNatLit 10]) 1 {};
-   let d ← d.insert (mkAppN add #[mkNatLit 0, mkNatLit 10]) 2 {};
-   let d ← d.insert (mkAppN (mkConst `Nat.add) #[mkNatLit 0, mkNatLit 20]) 3 {};
-   let d ← d.insert (mkAppN add #[mvar, mkNatLit 20]) 4 {};
-   let d ← d.insert mvar 5 {};
+   let d ← d.insert (mkAppN add #[mvar, mkNatLit 10]) 1 {}
+   let d ← d.insert (mkAppN add #[mkNatLit 0, mkNatLit 10]) 2 {}
+   let d ← d.insert (mkAppN (mkConst `Nat.add) #[mkNatLit 0, mkNatLit 20]) 3 {}
+   let d ← d.insert (mkAppN add #[mvar, mkNatLit 20]) 4 {}
+   let d ← d.insert mvar 5 {}
    print (format d);
    let vs ← d.getMatch (mkAppN add #[mkNatLit 1, mkNatLit 10]) {};
    print (format vs);
@@ -56,4 +56,5 @@ do let d : DiscrTree Nat := {};
    print (format vs);
    pure ()
 
-#eval run #[`Init.Data.Nat] tst1
+set_option trace.Meta.debug true in
+run_meta tst1