diff --git a/src/Lean/Elab/App.lean b/src/Lean/Elab/App.lean
index 4ba28f696ead..350e833df365 100644
--- a/src/Lean/Elab/App.lean
+++ b/src/Lean/Elab/App.lean
@@ -1140,8 +1140,9 @@ inductive LValResolution where
   | projIdx  (structName : Name) (idx : Nat)
   /-- When applied to `f`, effectively expands to `constName ... (Struct.toBase f)`, with the argument placed in the correct
   positional argument if possible, or otherwise as a named argument. The `Struct.toBase` is not present if `baseStructName == structName`,
-  in which case these do not need to be structures. Supports generalized field notation. -/
-  | const    (baseStructName : Name) (structName : Name) (constName : Name)
+  in which case these do not need to be structures. Supports generalized field notation.
+  If `useFirstExplicit` is true, then `f` is inserted as the first explicit argument to `constName`. -/
+  | const    (baseStructName : Name) (structName : Name) (constName : Name) (useFirstExplicit : Bool := false)
   /-- Like `const`, but with `fvar` instead of `constName`.
   The `fullName` is the name of the recursive function, and `baseName` is the base name of the type to search for in the parameter list. -/
   | localRec (baseName : Name) (fullName : Name) (fvar : Expr)
@@ -1150,47 +1151,49 @@ private def throwLValError (e : Expr) (eType : Expr) (msg : MessageData) : TermE
   throwError "{msg}{indentExpr e}\nhas type{indentExpr eType}"
 
 /--
-`findMethod? S fName` tries the following for each namespace `S'` in the resolution order for `S`:
-- If `env` contains `S' ++ fName`, returns `(S', S' ++ fName)`
-- Otherwise if `env` contains private name `prv` for `S' ++ fName`, returns `(S', prv)`
+`findMethod? S fName` tries the for each namespace `S'` in the resolution order for `S` to resolve the name `S'.fname`.
+If it resolves to `name`, returns `(S', name)`.
 -/
 private partial def findMethod? (structName fieldName : Name) : MetaM (Option (Name × Name)) := do
   let env ← getEnv
   let find? structName' : MetaM (Option (Name × Name)) := do
     let fullName := structName' ++ fieldName
-    if env.contains fullName then
-      return some (structName', fullName)
-    let fullNamePrv := mkPrivateName env fullName
-    if env.contains fullNamePrv then
-      return some (structName', fullNamePrv)
-    return none
+    -- We do not want to make use of the current namespace for resolution.
+    let candidates := ResolveName.resolveGlobalName (← getEnv) Name.anonymous (← getOpenDecls) fullName
+      |>.filter (fun (_, fieldList) => fieldList.isEmpty)
+      |>.map Prod.fst
+    match candidates with
+    | []          => return none
+    | [fullName'] => return some (structName', fullName')
+    | _ => throwError "\
+      invalid field notation '{fieldName}', the name '{fullName}' is ambiguous, possible interpretations: \
+      {MessageData.joinSep (candidates.map (m!"'{.ofConstName ·}'")) ", "}"
   -- Optimization: the first element of the resolution order is `structName`,
   -- so we can skip computing the resolution order in the common case
   -- of the name resolving in the `structName` namespace.
   find? structName <||> do
     let resolutionOrder ← if isStructure env structName then getStructureResolutionOrder structName else pure #[structName]
-    for h : i in [1:resolutionOrder.size] do
-      if let some res ← find? resolutionOrder[i] then
+    for ns in resolutionOrder[1:resolutionOrder.size] do
+      if let some res ← find? ns then
         return res
     return none
 
-/--
-  Return `some (structName', fullName)` if `structName ++ fieldName` is an alias for `fullName`, and
-  `fullName` is of the form `structName' ++ fieldName`.
-
-  TODO: if there is more than one applicable alias, it returns `none`. We should consider throwing an error or
-  warning.
--/
-private def findMethodAlias? (env : Environment) (structName fieldName : Name) : Option (Name × Name) :=
-  let fullName := structName ++ fieldName
-  -- We never skip `protected` aliases when resolving dot-notation.
-  let aliasesCandidates := getAliases env fullName (skipProtected := false) |>.filterMap fun alias =>
-    match alias.eraseSuffix? fieldName with
-    | none => none
-    | some structName' => some (structName', alias)
-  match aliasesCandidates with
-  | [r] => some r
-  | _   => none
+private def findTopLevelMethod? (fieldName : Name) : MetaM (Option Name) := do
+  let env ← getEnv
+  if env.contains fieldName then
+    return some fieldName
+  let prvName := mkPrivateName env fieldName
+  if env.contains prvName then
+    return some prvName
+  -- There should be no `protected` top-level aliases, but no need to filter.
+  let aliasCandidates := getAliases env fieldName (skipProtected := false)
+  match aliasCandidates with
+  | [] => pure ()
+  | [alias] => return some alias
+  | _ => throwError "\
+    invalid field notation '{fieldName}', the name '{fieldName}' is ambigous, possible interpretations: \
+    {MessageData.joinSep (aliasCandidates.map (m!"'{.ofConstName ·}'")) ", "}"
+  return none
 
 private def throwInvalidFieldNotation (e eType : Expr) : TermElabM α :=
   throwLValError e eType "invalid field notation, type is not of the form (C ...) where C is a constant"
@@ -1201,7 +1204,7 @@ private def resolveLValAux (e : Expr) (eType : Expr) (lval : LVal) : TermElabM L
     | LVal.fieldName _ fieldName _ _ =>
       let fullName := Name.str `Function fieldName
       if (← getEnv).contains fullName then
-        return LValResolution.const `Function `Function fullName
+        return LValResolution.const `Function `Function fullName (useFirstExplicit := true)
     | _ => pure ()
   match eType.getAppFn.constName?, lval with
   | some structName, LVal.fieldIdx _ idx =>
@@ -1223,30 +1226,27 @@ private def resolveLValAux (e : Expr) (eType : Expr) (lval : LVal) : TermElabM L
         throwLValError e eType m!"invalid projection, structure has only {numFields} field(s)"
   | some structName, LVal.fieldName _ fieldName _ _ =>
     let env ← getEnv
-    let searchEnv : Unit → TermElabM LValResolution := fun _ => do
-      if let some (baseStructName, fullName) ← findMethod? structName (.mkSimple fieldName) then
-        return LValResolution.const baseStructName structName fullName
-      else if let some (structName', fullName) := findMethodAlias? env structName (.mkSimple fieldName) then
-        return LValResolution.const structName' structName' fullName
-      else
-        throwLValError e eType
-          m!"invalid field '{fieldName}', the environment does not contain '{Name.mkStr structName fieldName}'"
-    -- search local context first, then environment
-    let searchCtx : Unit → TermElabM LValResolution := fun _ => do
-      let fullName := Name.mkStr structName fieldName
-      for localDecl in (← getLCtx) do
-        if localDecl.isAuxDecl then
-          if let some localDeclFullName := (← read).auxDeclToFullName.find? localDecl.fvarId then
-            if fullName == (privateToUserName? localDeclFullName).getD localDeclFullName then
-              /- LVal notation is being used to make a "local" recursive call. -/
-              return LValResolution.localRec structName fullName localDecl.toExpr
-      searchEnv ()
     if isStructure env structName then
-      match findField? env structName (Name.mkSimple fieldName) with
-      | some baseStructName => return LValResolution.projFn baseStructName structName (Name.mkSimple fieldName)
-      | none                => searchCtx ()
-    else
-      searchCtx ()
+      if let some baseStructName := findField? env structName (Name.mkSimple fieldName) then
+        return LValResolution.projFn baseStructName structName (Name.mkSimple fieldName)
+    -- Search the local context first
+    let fullName := Name.mkStr structName fieldName
+    for localDecl in (← getLCtx) do
+      if localDecl.isAuxDecl then
+        if let some localDeclFullName := (← read).auxDeclToFullName.find? localDecl.fvarId then
+          if fullName == (privateToUserName? localDeclFullName).getD localDeclFullName then
+            /- LVal notation is being used to make a "local" recursive call. -/
+            return LValResolution.localRec structName fullName localDecl.toExpr
+    -- Then search the environment
+    if let some (baseStructName, fullName) ← findMethod? structName (.mkSimple fieldName) then
+      return LValResolution.const baseStructName structName fullName
+    -- Otherwise search for a top-level declaration named `fieldName`.
+    -- We only do this if `eType` does not unfold so that this is done as a last resort (`resolveLValLoop` loops so long as the type unfolds).
+    if (← unfoldDefinition? eType).isNone then
+      if let some fullName ← findTopLevelMethod? (.mkSimple fieldName) then
+        return LValResolution.const structName structName fullName (useFirstExplicit := true)
+    throwLValError e eType
+      m!"invalid field '{fieldName}', the environment does not contain '{Name.mkStr structName fieldName}'"
   | none, LVal.fieldName _ _ (some suffix) _ =>
     if e.isConst then
       throwUnknownConstant (e.constName! ++ suffix)
@@ -1326,7 +1326,8 @@ Otherwise, if there isn't another parameter with the same name, we add `e` to `n
 
 Remark: `fullName` is the name of the resolved "field" access function. It is used for reporting errors
 -/
-private partial def addLValArg (baseName : Name) (fullName : Name) (e : Expr) (args : Array Arg) (namedArgs : Array NamedArg) (f : Expr) :
+private partial def addLValArg (baseName : Name) (fullName : Name) (e : Expr) (args : Array Arg) (namedArgs : Array NamedArg) (f : Expr)
+    (explicit : Bool) (useFirstExplicit : Bool := false) :
     MetaM (Array Arg × Array NamedArg) := do
   withoutModifyingState <| go f (← inferType f) 0 namedArgs (namedArgs.map (·.name)) true
 where
@@ -1351,11 +1352,11 @@ where
         /- If there is named argument with name `xDecl.userName`, then it is accounted for and we can't make use of it. -/
         remainingNamedArgs := remainingNamedArgs.eraseIdx idx
       else
-        if (← typeMatchesBaseName xDecl.type baseName) then
-          /- We found a type of the form (baseName ...).
-             First, we check if the current argument is an explicit one,
+        if ← (pure <| useFirstExplicit && bInfo.isExplicit) <||> typeMatchesBaseName xDecl.type baseName then
+          /- We found a type of the form (baseName ...), or we found the first explicit argument in useFirstExplicit mode.
+             First, we check if the current argument is one that can be used positionally,
              and if the current explicit position "fits" at `args` (i.e., it must be ≤ arg.size) -/
-          if h : argIdx ≤ args.size ∧ bInfo.isExplicit then
+          if h : argIdx ≤ args.size ∧ (explicit || bInfo.isExplicit) then
             /- We can insert `e` as an explicit argument -/
             return (args.insertIdx argIdx (Arg.expr e), namedArgs)
           else
@@ -1363,13 +1364,13 @@ where
                if there isn't an argument with the same name occurring before it. -/
             if !allowNamed || unusableNamedArgs.contains xDecl.userName then
               throwError "\
-                invalid field notation, function '{fullName}' has argument with the expected type\
+                invalid field notation, function '{.ofConstName fullName}' has argument with the expected type\
                 {indentExpr xDecl.type}\n\
                 but it cannot be used"
             else
               return (args, namedArgs.push { name := xDecl.userName, val := Arg.expr e })
         /- Advance `argIdx` and update seen named arguments. -/
-        if bInfo.isExplicit then
+        if explicit || bInfo.isExplicit then
           argIdx := argIdx + 1
         unusableNamedArgs := unusableNamedArgs.push xDecl.userName
     /- If named arguments aren't allowed, then it must still be possible to pass the value as an explicit argument.
@@ -1380,7 +1381,7 @@ where
       if let some f' ← coerceToFunction? (mkAppN f xs) then
         return ← go f' (← inferType f') argIdx remainingNamedArgs unusableNamedArgs false
     throwError "\
-      invalid field notation, function '{fullName}' does not have argument with type ({baseName} ...) that can be used, \
+      invalid field notation, function '{.ofConstName fullName}' does not have argument with type ({.ofConstName baseName} ...) that can be used, \
       it must be explicit or implicit with a unique name"
 
 /-- Adds the `TermInfo` for the field of a projection. See `Lean.Parser.Term.identProjKind`. -/
@@ -1421,12 +1422,12 @@ private def elabAppLValsAux (namedArgs : Array NamedArg) (args : Array Arg) (exp
       else
         let f ← elabAppArgs projFn #[{ name := `self, val := Arg.expr f, suppressDeps := true }] #[] (expectedType? := none) (explicit := false) (ellipsis := false)
         loop f lvals
-    | LValResolution.const baseStructName structName constName =>
+    | LValResolution.const baseStructName structName constName useFirstExplicit =>
       let f ← if baseStructName != structName then mkBaseProjections baseStructName structName f else pure f
       let projFn ← mkConst constName
       let projFn ← addProjTermInfo lval.getRef projFn
       if lvals.isEmpty then
-        let (args, namedArgs) ← addLValArg baseStructName constName f args namedArgs projFn
+        let (args, namedArgs) ← addLValArg baseStructName constName f args namedArgs projFn explicit useFirstExplicit
         elabAppArgs projFn namedArgs args expectedType? explicit ellipsis
       else
         let f ← elabAppArgs projFn #[] #[Arg.expr f] (expectedType? := none) (explicit := false) (ellipsis := false)
@@ -1434,7 +1435,7 @@ private def elabAppLValsAux (namedArgs : Array NamedArg) (args : Array Arg) (exp
     | LValResolution.localRec baseName fullName fvar =>
       let fvar ← addProjTermInfo lval.getRef fvar
       if lvals.isEmpty then
-        let (args, namedArgs) ← addLValArg baseName fullName f args namedArgs fvar
+        let (args, namedArgs) ← addLValArg baseName fullName f args namedArgs fvar explicit
         elabAppArgs fvar namedArgs args expectedType? explicit ellipsis
       else
         let f ← elabAppArgs fvar #[] #[Arg.expr f] (expectedType? := none) (explicit := false) (ellipsis := false)
diff --git a/tests/lean/run/3031.lean b/tests/lean/run/3031.lean
new file mode 100644
index 000000000000..5fbbaddd51ed
--- /dev/null
+++ b/tests/lean/run/3031.lean
@@ -0,0 +1,119 @@
+/-!
+# Tests for generalized field notation through aliases and "top-level" dot notation
+https://github.com/leanprover/lean4/issues/3031
+-/
+
+/-!
+Alias dot notation. There used to be a different kind of alias dot notation;
+in the following example, it would have looked for an argument of type `Common.String`.
+Now it looks for one of type `String`, allowing libraries to add "extension methods" from within their own namespaces.
+-/
+def Common.String.a (s : String) : Nat := s.length
+
+export Common (String.a)
+
+/-- info: String.a "x" : Nat -/
+#guard_msgs in #check String.a "x"
+/-- info: String.a "x" : Nat -/
+#guard_msgs in #check "x".a
+
+/-!
+Declarations take precedence over aliases
+-/
+def String.a (s : String) : Nat := s.length + 100
+/-- info: "x".a : Nat -/
+#guard_msgs in #check "x".a
+/-- info: 100 -/
+#guard_msgs in #eval "".a
+
+/-!
+Private declarations take precedence over aliases
+-/
+private def String.b (s : String) : Nat := 0
+def Common.String.b (s : String) : Nat := 1
+export Common (String.b)
+/-- info: 0 -/
+#guard_msgs in #eval "".b
+
+/-!
+Multiple aliases is an error
+-/
+def Common.String.c (s : String) : Nat := 0
+def Common'.String.c (s : String) : Nat := 0
+export Common (String.c)
+export Common' (String.c)
+/--
+error: invalid field notation 'c', the name 'String.c' is ambiguous, possible interpretations: 'Common'.String.c', 'Common.String.c'
+-/
+#guard_msgs in #eval "".c
+
+/-!
+Aliases work with inheritance
+-/
+namespace Ex1
+structure A
+structure B extends A
+def Common.A.x (_ : A) : Nat := 0
+export Common (A.x)
+/-- info: fun b => A.x b.toA : B → Nat -/
+#guard_msgs in #check fun (b : B) => b.x
+end Ex1
+
+/-!
+`open` also works
+-/
+def Common.String.parse (_ : String) : List Nat := []
+
+namespace ExOpen1
+/--
+error: invalid field 'parse', the environment does not contain 'String.parse'
+  ""
+has type
+  String
+-/
+#guard_msgs in #check "".parse
+section
+open Common
+/-- info: String.parse "" : List Nat -/
+#guard_msgs in #check "".parse
+end
+section
+open Common (String.parse)
+/-- info: String.parse "" : List Nat -/
+#guard_msgs in #check "".parse
+end
+end ExOpen1
+
+
+namespace Ex2
+class A (n : Nat) where
+  x : Nat
+
+/-!
+"Top-level" dot notation. As a last resort, field notation looks for a top-level declaration or alias
+and supplies the value as the first explicit argument.
+-/
+instance : ToString (A n) where
+  toString a := s!"A.x is {a.x}"
+
+/-- info: fun a => toString a : A 2 → String -/
+#guard_msgs in #check fun (a : A 2) => a.toString
+
+/-!
+Incidental fix: `@` for generalized field notation was failing if there were implicit arguments.
+True projections were ok.
+-/
+def A.x' {n : Nat} (a : A n) := a.x
+
+/-- info: fun a => a.x' : A 2 → Nat -/
+#guard_msgs in #check fun (a : A 2) => @a.x'
+end Ex2
+
+namespace Ex3
+variable (f : α → β) (g : β → γ)
+/-!
+Functions use the "top-level" dot notation rule: they use the first explicit argument, rather than the first function argument.
+-/
+/-- info: g ∘ f : α → γ -/
+#guard_msgs in #check g.comp f
+end Ex3
diff --git a/tests/lean/run/DVec.lean b/tests/lean/run/DVec.lean
index e9f98f8b655c..0006fe7aa637 100644
--- a/tests/lean/run/DVec.lean
+++ b/tests/lean/run/DVec.lean
@@ -38,6 +38,17 @@ example (v : Vec Nat 1) : Nat :=
 
 #check @Vec.hd
 
--- works
+-- Does not work: Aliases find that `v` could be the `TypeVec` argument since `TypeVec` is an abbrev for `Vec`.
+/--
+error: application type mismatch
+  @Vec.hd ?_ v
+argument
+  v
+has type
+  Vec Nat 1 : Type
+but is expected to have type
+  TypeVec (?_ + 1) : Type (_ + 1)
+-/
+#guard_msgs in set_option pp.mvars false in
 example (v : Vec Nat 1) : Nat :=
   v.hd
diff --git a/tests/lean/run/alias.lean b/tests/lean/run/alias.lean
index 1174ec65aa63..126c8ab1e49e 100644
--- a/tests/lean/run/alias.lean
+++ b/tests/lean/run/alias.lean
@@ -3,14 +3,31 @@ def Set (α : Type) := α → Prop
 def Set.union (s₁ s₂ : Set α) : Set α :=
   fun a => s₁ a ∨ s₂ a
 
-def FinSet (n : Nat) := Fin n → Prop
+def FinSet (n : Nat) := Set (Fin n)
+
+/-!
+The type of `x` is unfolded to find `Set.union`
+-/
+example (x y : FinSet 10) : FinSet 10 :=
+  x.union y
 
 namespace FinSet
-  export Set (union)
+export Set (union)
 end FinSet
 
+/-!
+Since the types are defeq, this alias works:
+-/
 example (x y : FinSet 10) : FinSet 10 :=
   FinSet.union x y
 
+/-!
+However, this dot notation fails since there is no `FinSet` argument.
+However, unfolding is the preferred method.
+-/
+/--
+error: invalid field notation, function 'FinSet.union' does not have argument with type (FinSet ...) that can be used, it must be explicit or implicit with a unique name
+-/
+#guard_msgs in
 example (x y : FinSet 10) : FinSet 10 :=
   x.union y