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The LSP:

Let φ(x) be a property provable about objects x of type T. Then φ(y) should be true for objects y of type S where S is a subtype of T.

What is provable about Animal? you don't provide code or return type, Presumably something like:

(Animal.Hungry = false AND Food.State = Consumed) OR InvalidArgumentException thrown

Is it still true for Cat? Yes. (assuming you add the state changes)

Now what about Food and Meat? You don't prove any code, so we can assume the only difference is the Type. No violation is possible.

What about that type checking statement in Cat though? this looks bad, its not a LSP violation, but it does raise questions about whether Food has a violation in it. If it doesn't then I shouldn't need to check the Type right?

Really the code smell is pointing out that Food should have some sort of FoodType property or CanBeEatenBy(Animal animal) method, rather than using SubTypes to identify the type.

Edit: it seems the correct design is in scope...

I'm going to use c#, which has a "flaw" in that you cant specify what exceptions might be thrown in the class defintion. That being the case, and respecting the normal rule about not using exceptions for logic, I will try to avoid all exceptions.

Obviously this isnt the only way to do this, its just a way to avoid your code smell and problems with exceptions.

public enum AnimalType
{
    Omnivore,
    Herbivore,
    Carnivore
}

public class Food
{
    public bool IsEdibleBy(Animal a)
    {
        return true;
    }
}

public class EatenFood
{
    public override bool IsEdibleBy(Animal a)
    {
        return false;
    }
}

public class Animal
{
    public readonly AnimalType Type = AnimalType.Omnivore;
    public Food Eat(Food f)
    {
         if(f.IsEdibleBy(this))
         {
              return new EatenFood(f);
         }
         else
         {
              return f;
         }
    }
}

What is proveable?

Eat always returns a type of Food 
IsEdibleBy always returns true or false
AnimalType is one of the enum values

Now we add Cat:

public class Cat : Animal
{
    public Cat() { this.Type = AnimalType.Carnivore; }
}

Public class Meat : Food
{
   public override bool IsEdibleBy(Animal a)
   {
       return a.Type == AnimalType.Omnivore || 
              a.Type == AnimalType.Carnivore;
   }
}

Now you have avoided all exceptions and LSP questions, you can extend you food types indefinitely, although you may have issues if you add more AnimalType enums. It would be important to make sure that that list is complete at the start