diff --git a/mypy/build.py b/mypy/build.py
index 931cac47dccb..0eca45d6ea62 100644
--- a/mypy/build.py
+++ b/mypy/build.py
@@ -391,7 +391,8 @@ def __init__(self, data_dir: str,
         check_untyped_defs = CHECK_UNTYPED_DEFS in self.flags
         self.semantic_analyzer = SemanticAnalyzer(lib_path, self.errors,
                                                   pyversion=pyversion,
-                                                  check_untyped_defs=check_untyped_defs)
+                                                  check_untyped_defs=check_untyped_defs,
+                                                  lightweight_type_check=(target >= TYPE_CHECK))
         self.modules = self.semantic_analyzer.modules
         self.semantic_analyzer_pass3 = ThirdPass(self.modules, self.errors)
         self.type_checker = TypeChecker(self.errors,
diff --git a/mypy/semanal.py b/mypy/semanal.py
index 75e32184d068..b3ae3bb4553e 100644
--- a/mypy/semanal.py
+++ b/mypy/semanal.py
@@ -63,6 +63,7 @@
     YieldFromExpr, NamedTupleExpr, NonlocalDecl,
     SetComprehension, DictionaryComprehension, TYPE_ALIAS, TypeAliasExpr,
     YieldExpr, ExecStmt, Argument, BackquoteExpr, ImportBase, COVARIANT, CONTRAVARIANT,
+    IntExpr, FloatExpr, UnicodeExpr,
     INVARIANT, UNBOUND_IMPORTED
 )
 from mypy.visitor import NodeVisitor
@@ -169,6 +170,10 @@ class SemanticAnalyzer(NodeVisitor):
     bound_tvars = None  # type: List[SymbolTableNode]
     # Stack of type variables that were bound by outer classess
     tvar_stack = None  # type: List[List[SymbolTableNode]]
+    # Do weak type checking in this file
+    weak_opts = set()        # type: Set[str]
+    # Do lightweight type checking
+    lightweight_type_check = False  # type: bool
 
     # Stack of functions being analyzed
     function_stack = None  # type: List[FuncItem]
@@ -193,7 +198,8 @@ def __init__(self,
                  lib_path: List[str],
                  errors: Errors,
                  pyversion: Tuple[int, int],
-                 check_untyped_defs: bool) -> None:
+                 check_untyped_defs: bool,
+                 lightweight_type_check: bool = False) -> None:
         """Construct semantic analyzer.
 
         Use lib_path to search for modules, and report analysis errors
@@ -214,6 +220,7 @@ def __init__(self,
         self.modules = {}
         self.pyversion = pyversion
         self.check_untyped_defs = check_untyped_defs
+        self.lightweight_type_check = lightweight_type_check
         self.postpone_nested_functions_stack = [FUNCTION_BOTH_PHASES]
         self.postponed_functions_stack = []
         self.all_exports = set()  # type: Set[str]
@@ -224,6 +231,7 @@ def visit_file(self, file_node: MypyFile, fnam: str) -> None:
         self.cur_mod_id = file_node.fullname()
         self.is_stub_file = fnam.lower().endswith('.pyi')
         self.globals = file_node.names
+        self.weak_opts = file_node.weak_opts
 
         if 'builtins' in self.modules:
             self.globals['__builtins__'] = SymbolTableNode(
@@ -1043,8 +1051,11 @@ def visit_assignment_stmt(self, s: AssignmentStmt) -> None:
             s.type = self.anal_type(s.type, allow_tuple_literal)
         else:
             # For simple assignments, allow binding type aliases.
+            # Also set the type if the rvalue is a simple literal.
             if (s.type is None and len(s.lvalues) == 1 and
                     isinstance(s.lvalues[0], NameExpr)):
+                if s.lvalues[0].is_def:
+                    s.type = self.analyze_simple_literal_type(s.rvalue)
                 res = analyze_type_alias(s.rvalue,
                                          self.lookup_qualified,
                                          self.lookup_fully_qualified,
@@ -1070,6 +1081,29 @@ def visit_assignment_stmt(self, s: AssignmentStmt) -> None:
                 isinstance(s.rvalue, (ListExpr, TupleExpr))):
             self.add_exports(*s.rvalue.items)
 
+    def analyze_simple_literal_type(self, rvalue: Node) -> Optional[Type]:
+        """Return builtins.int if rvalue is an int literal, etc."""
+        if self.weak_opts or not self.lightweight_type_check or self.function_stack:
+            # Skip this if any weak options are set.
+            # Also skip if lightweight type check not requested.
+            # This is mostly to avoid breaking unit tests.
+            # Also skip inside a function; this is to avoid confusing
+            # the code that handles dead code due to isinstance()
+            # inside type variables with value restrictions (like
+            # AnyStr).
+            return None
+        if isinstance(rvalue, IntExpr):
+            return self.named_type_or_none('builtins.int')
+        if isinstance(rvalue, FloatExpr):
+            return self.named_type_or_none('builtins.float')
+        if isinstance(rvalue, StrExpr):
+            return self.named_type_or_none('builtins.str')
+        if isinstance(rvalue, BytesExpr):
+            return self.named_type_or_none('builtins.bytes')
+        if isinstance(rvalue, UnicodeExpr):
+            return self.named_type_or_none('builtins.unicode')
+        return None
+
     def check_and_set_up_type_alias(self, s: AssignmentStmt) -> None:
         """Check if assignment creates a type alias and set it up as needed."""
         # For now, type aliases only work at the top level of a module.
diff --git a/test-data/unit/check-classes.test b/test-data/unit/check-classes.test
index bf501819a3c6..176c808f8e90 100644
--- a/test-data/unit/check-classes.test
+++ b/test-data/unit/check-classes.test
@@ -575,8 +575,9 @@ A.B = None # E: Cannot assign to a type
 
 [case testAccessingClassAttributeWithTypeInferenceIssue]
 x = C.x # E: Cannot determine type of 'x'
+def f() -> int: return 1
 class C:
-    x = 1
+    x = f()
 [builtins fixtures/list.py]
 
 [case testAccessingClassAttributeWithTypeInferenceIssue2]
diff --git a/test-data/unit/check-expressions.test b/test-data/unit/check-expressions.test
index 1e2e78062dff..96faa414e6c0 100644
--- a/test-data/unit/check-expressions.test
+++ b/test-data/unit/check-expressions.test
@@ -1492,4 +1492,4 @@ reveal_type("foo") # E: Argument 1 to "reveal_type" has incompatible type "str";
 
 [case testRevealTypeVar]
 reveal_type = 1
-1 + "foo" # E: Unsupported left operand type for + ("int")
+1 + "foo" # E: Unsupported operand types for + ("int" and "str")
diff --git a/test-data/unit/check-functions.test b/test-data/unit/check-functions.test
index 0f3f90bd833a..9106232841b9 100644
--- a/test-data/unit/check-functions.test
+++ b/test-data/unit/check-functions.test
@@ -1043,7 +1043,7 @@ f('x') # fail
 [out]
 main: note: In function "f":
 main:5: error: Incompatible types in assignment (expression has type "str", variable has type "int")
-main:9: error: Unsupported left operand type for + ("int")
+main:9: error: Unsupported operand types for + ("int" and "str")
 main: note: At top level:
 main:12: error: Argument 1 to "f" has incompatible type "str"; expected "int"
 
@@ -1064,7 +1064,7 @@ def top() -> None:
 [out]
 main: note: In function "f":
 main:6: error: Incompatible types in assignment (expression has type "str", variable has type "int")
-main:10: error: Unsupported left operand type for + ("int")
+main:10: error: Unsupported operand types for + ("int" and "str")
 main: note: In function "top":
 main:13: error: Argument 1 to "f" has incompatible type "str"; expected "int"
 
@@ -1227,7 +1227,7 @@ A().f('x') # fail
 [out]
 main: note: In member "f" of class "A":
 main:6: error: Incompatible types in assignment (expression has type "str", variable has type "int")
-main:10: error: Unsupported left operand type for + ("int")
+main:10: error: Unsupported operand types for + ("int" and "str")
 main: note: At top level:
 main:13: error: Argument 1 to "f" of "A" has incompatible type "str"; expected "int"
 
@@ -1272,7 +1272,7 @@ def f(x: Callable[..., int]) -> None:
     x()
     x(1)
     x(z=1)
-    x() + '' # E: Unsupported left operand type for + ("int")
+    x() + '' # E: Unsupported operand types for + ("int" and "str")
 [out]
 main: note: In function "f":
 
@@ -1285,7 +1285,7 @@ def f(x: Callable[..., int]) -> None:
 [case testCastWithCallableAndArbitraryArgs]
 from typing import Callable, cast
 f = cast(Callable[..., int], None)
-f(x=4) + '' # E: Unsupported left operand type for + ("int")
+f(x=4) + '' # E: Unsupported operand types for + ("int" and "str")
 
 [case testCallableWithArbitraryArgsInErrorMessage]
 from typing import Callable
diff --git a/test-data/unit/check-inference.test b/test-data/unit/check-inference.test
index 5dad20e5fb98..63342c9f27a7 100644
--- a/test-data/unit/check-inference.test
+++ b/test-data/unit/check-inference.test
@@ -1523,7 +1523,7 @@ main: note: In member "f" of class "A":
 [case testMultipassAndTopLevelVariable]
 y = x # E: Cannot determine type of 'x'
 y()
-x = 1
+x = 1+0
 [out]
 
 [case testMultipassAndDecoratedMethod]
diff --git a/test-data/unit/check-isinstance.test b/test-data/unit/check-isinstance.test
index c5cf467397d4..26736b2f92a1 100644
--- a/test-data/unit/check-isinstance.test
+++ b/test-data/unit/check-isinstance.test
@@ -1118,11 +1118,10 @@ def f(x: Union[str, int]):
 from typing import Any
 def f(x: Any):
     assert isinstance(x, int)  # this should narrow x to type int
-    x + "foo"
+    x + "foo"  # E: Unsupported operand types for + ("int" and "str")
 [builtins fixtures/isinstance.py]
 [out]
 main: note: In function "f":
-main:4: error: Unsupported left operand type for + ("int")
 
 [case testIsinstanceOfGenericClassRetainsParameters]
 from typing import List, Union
diff --git a/test-data/unit/check-modules.test b/test-data/unit/check-modules.test
index 6674f91ee0e0..f8f35d255d83 100644
--- a/test-data/unit/check-modules.test
+++ b/test-data/unit/check-modules.test
@@ -486,13 +486,12 @@ x = 1
 x = 1
 
 [case testAssignToFuncDefViaImport]
-from m import *
-# TODO: This is bad, we don't see what's coming from m.
+from m import *  # E: Incompatible import of "x" (imported name has type "int", local name has type "str")
 f = None # E: Need type annotation for variable
 x = ''
 [file m.py]
 def f(): pass
-x = 1
+x = 1+0
 [out]
 
 
@@ -816,9 +815,11 @@ x = 0
 -- Test stability under import cycles
 -- ----------------------------------
 
--- The two tests are identical except one main has 'import x' and the other 'import y'.
--- Previously (before build.order_ascc() was added) one of these would fail because the
--- imports were processed in the (reverse) order in which the files were encountered.
+-- The first two tests are identical except one main has 'import x'
+-- and the other 'import y'.  Previously (before build.order_ascc()
+-- was added) one of these would fail because the imports were
+-- processed in the (reverse) order in which the files were
+-- encountered.
 
 [case testImportCycleStability1]
 import x
@@ -847,3 +848,114 @@ import x
 class Sub(x.Base):
     attr = x.Base.attr
 [out]
+
+-- This case isn't fixed by order_ascc(), but is fixed by the
+-- lightweight type inference added to semanal.py
+-- (analyze_simple_literal_type()).
+
+[case testImportCycleStability3]
+import y
+[file x.py]
+class Base:
+    pass
+def foo() -> int:
+    import y
+    reveal_type(y.Sub.attr)
+    return y.Sub.attr
+[file y.py]
+import x
+class Sub(x.Base):
+    attr = 0
+[out]
+tmp/y.py:1: note: In module imported here,
+main:1: note: ... from here:
+tmp/x.py: note: In function "foo":
+tmp/x.py:5: error: Revealed type is 'builtins.int'
+
+-- This case has a symmetrical cycle, so it doesn't matter in what
+-- order the files are processed.  It depends on the lightweight type
+-- interference.
+
+[case testImportCycleStability4]
+import x
+[file x.py]
+import y
+class C:
+    attr = ''
+def foo() -> int:
+    return y.D.attr
+[file y.py]
+import x
+class D:
+    attr = 0
+def bar() -> str:
+    return x.C.attr
+
+-- These cases test all supported literal types.
+
+[case testImportCycleStability5]
+import y
+[file x.py]
+class Base:
+    pass
+def foo() -> None:
+    import y
+    i = y.Sub.iattr  # type: int
+    f = y.Sub.fattr  # type: float
+    s = y.Sub.sattr  # type: str
+    b = y.Sub.battr  # type: bytes
+[file y.py]
+import x
+class Sub(x.Base):
+    iattr = 0
+    fattr = 0.0
+    sattr = ''
+    battr = b''
+[out]
+
+[case testImportCycleStability6_python2]
+import y
+[file x.py]
+class Base:
+    pass
+def foo() -> None:
+    import y
+    i = y.Sub.iattr  # type: int
+    f = y.Sub.fattr  # type: float
+    s = y.Sub.sattr  # type: str
+    u = y.Sub.uattr  # type: unicode
+[file y.py]
+import x
+class Sub(x.Base):
+    iattr = 0
+    fattr = 0.0
+    sattr = ''
+    uattr = u''
+[out]
+
+-- This case tests module-level variables.
+
+[case testImportCycleStability7]
+import x
+[file x.py]
+def foo() -> int:
+    import y
+    reveal_type(y.value)
+    return y.value
+[file y.py]
+import x
+value = 12
+[out]
+main:1: note: In module imported here:
+tmp/x.py: note: In function "foo":
+tmp/x.py:3: error: Revealed type is 'builtins.int'
+
+-- This is not really cycle-related but still about the lightweight
+-- type checker.
+
+[case testImportCycleStability8]
+x = 1  # type: str
+reveal_type(x)
+[out]
+main:1: error: Incompatible types in assignment (expression has type "int", variable has type "str")
+main:2: error: Revealed type is 'builtins.str'
diff --git a/test-data/unit/check-weak-typing.test b/test-data/unit/check-weak-typing.test
index 35b4b73dd382..8aa14db53b99 100644
--- a/test-data/unit/check-weak-typing.test
+++ b/test-data/unit/check-weak-typing.test
@@ -120,7 +120,7 @@ main: note: In function "f":
 [case testWeakFunction3]
 # mypy: weak
 def f():
-    1 + 'a' # E: Unsupported left operand type for + ("int")
+    1 + 'a' # E: Unsupported operand types for + ("int" and "str")
 [out]
 main: note: In function "f":
 [case testWeakFunctionCall]
diff --git a/test-data/unit/fixtures/isinstance.py b/test-data/unit/fixtures/isinstance.py
index 18542713da83..c155a9724ad4 100644
--- a/test-data/unit/fixtures/isinstance.py
+++ b/test-data/unit/fixtures/isinstance.py
@@ -14,7 +14,9 @@ class function: pass
 
 def isinstance(x: object, t: Union[type, Tuple[type, ...]]) -> bool: pass
 
-class int: pass
+class int:
+    def __add__(self, other: 'int') -> 'int': pass
 class float: pass
 class bool(int): pass
-class str: pass
+class str:
+    def __add__(self, other: 'str') -> 'str': pass
diff --git a/test-data/unit/lib-stub/__builtin__.py b/test-data/unit/lib-stub/__builtin__.py
index 11540f50d70e..47e518d14dd0 100644
--- a/test-data/unit/lib-stub/__builtin__.py
+++ b/test-data/unit/lib-stub/__builtin__.py
@@ -12,7 +12,10 @@ def __init__(self, x):
 
 # These are provided here for convenience.
 class int: pass
+class float: pass
+
 class str: pass
+class unicode: pass
 
 class tuple: pass
 class function: pass
diff --git a/test-data/unit/lib-stub/builtins.py b/test-data/unit/lib-stub/builtins.py
index 29dc26d81091..9a636bf6ff92 100644
--- a/test-data/unit/lib-stub/builtins.py
+++ b/test-data/unit/lib-stub/builtins.py
@@ -7,8 +7,13 @@ class type:
     def __init__(self, x: Any) -> None: pass
 
 # These are provided here for convenience.
-class int: pass
-class str: pass
+class int:
+    def __add__(self, other: 'int') -> 'int': pass
+class float: pass
+
+class str:
+    def __add__(self, other: 'str') -> 'str': pass
+class bytes: pass
 
 class tuple: pass
 class function: pass
diff --git a/test-data/unit/typexport-basic.test b/test-data/unit/typexport-basic.test
index 8a947b569c01..15c772c23663 100644
--- a/test-data/unit/typexport-basic.test
+++ b/test-data/unit/typexport-basic.test
@@ -268,11 +268,11 @@ NameExpr(4) : builtins.bool
 
 [case testInferSingleType]
 import typing
-x = 1
+x = ()
 [builtins fixtures/primitives.py]
 [out]
-IntExpr(2) : builtins.int
-NameExpr(2) : builtins.int
+NameExpr(2) : Tuple[]
+TupleExpr(2) : Tuple[]
 
 [case testInferTwoTypes]
 ## NameExpr
@@ -287,11 +287,11 @@ NameExpr(4) : builtins.int
 [case testInferSingleLocalVarType]
 import typing
 def f() -> None:
-    x = 1
+    x = ()
 [builtins fixtures/primitives.py]
 [out]
-IntExpr(3) : builtins.int
-NameExpr(3) : builtins.int
+NameExpr(3) : Tuple[]
+TupleExpr(3) : Tuple[]
 
 
 -- Basic generics