No compiler in existence makes these kinds of checks on anything except maybe literal values in special cases. What you propose would entail evaluating the program's runtime behavior at compile time. Ensuring anything that gets put in an integer field is an integer is easy assuming you have type annotations on everything.
For the cases you mention the checks would simply be: integer * integer = integer, integer + integer = integer. Problems only arise when real/floating point numbers are introduced and how to handle integer division.
>No compiler in existence makes these kinds of checks
If your claim is that no compiler in existence can make these kinds of checks at compile time, that's not true. Any dependently-typed language can do this (Idris, Coq, Agda, ATS). In fact, even a language with refinement types can do this, like Liquid Haskell.
And a number of languages do these checks at runtime: Ada and Nimrod are two that come to mind.
For the cases you mention the checks would simply be: integer * integer = integer, integer + integer = integer. Problems only arise when real/floating point numbers are introduced and how to handle integer division.