Overload is used to create an "instance mock". This will "intercept" when a new instance of a class is created and the mock will be used instead. For example if this code is to be tested:

class ClassToTest {
    public function methodToTest()
    {
        $myClass = new MyClass();
        $result = $myClass->someMethod();
        return $result;
    }
}

You would create an instance mock using overload and define the expectations like this:

public function testMethodToTest()
{
     $mock = Mockery::mock('overload:MyClass');
     $mock->shouldreceive('someMethod')->andReturn('someResult');

     $classToTest = new ClassToTest();
     $result = $classToTest->methodToTest();

     $this->assertEquals('someResult', $result);
}

Alias is used to mock public static methods. For example if this code is to be tested:

class ClassToTest {

    public function methodToTest()
    {
        return MyClass::someStaticMethod();
    }
}

You would create an alias mock using alias and define the expectations like this:

public function testNewMethodToTest()
{
    $mock = Mockery::mock('alias:MyClass');
    $mock->shouldreceive('someStaticMethod')->andReturn('someResult');

    $classToTest = new ClassToTest();
    $result = $classToTest->methodToTest();

    $this->assertEquals('someResult', $result);
}

Alias mocking is persistent per test cases, so you need to deactivate it each time you use it. Just add this phpDoc to each test class there you use alias mocking:

/**
 * @runInSeparateProcess
 * @preserveGlobalState disabled
 */

You can find more information in the Mockery docs under Mocking Hard Dependencies (new Keyword).

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