[isabelle] Can I define direct sum of sets in R^n and R^m



Dear Isabelle Users

If A and B are sets, direct sum C of A and B is a set of pairs (a, b)  where a \in A and b \in B. 
My first question – is this direct sum defined somewhere in Isabelle?

Now, in my case A is a subset of R^n and B is a subset of R^m. In this case, clearly, C is a subset of R^{n+m}. Currently, finite Cartesian product  R^n formalized as type real^'n where 'n is some finite type. Now, is it possible to say that C is a subset of real^('n+'m) ? It seems that I can add types (theory Sum_Type),  so I would imagine definition of direct sum as a function from  (real^'n,  real^'m)  to real^('n+'m) such that first n coordinates of  sum(x, y) coincides with x, and the next m coordinates - with y. Any ideas how to write down such a definition, which theories to look at, or may be suggestions how to formalize this in a different, more convenient way, would be appreciated.    

Sincerely,
Bogdan Grechuk





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