Hypothetical, formerly-C++0x concepts questions

(Preamble: I am a late follower to the C++0x game and the recent controversy regarding the removal of concepts from the C++0x standard has motivated me to learn more about them. While I understand that all of my questions are completely hypothetical -- insofar as concepts won't be valid C++ code for some time to come, if at all -- I am still interested in learning more about concepts, especially given how it would help me understand more fully the merits behind the recent decision and the controversy that has followed)

After having read some introductory material on concepts as C++0x (until recently) proposed them, I am having trouble wrapping my mind around some syntactical issues. Without further ado, here are my questions:

1) Would a type that supports a particular derived concept (either implicitly, via the auto keyword, or explicitly via concept_maps) also need to support the base concept indepdendently? In other words, does the act of deriving a concept from another (e.g. concept B<typename T> : A<T>) implicitly include an 'invisible' requires statement (within B, requires A<T>;)? The confusion arises from the Wikipedia page on concepts which states:

Like in class inheritance, types that meet the requirements of the derived concept also meet the requirements of the base concept.

That seems to say that a type only needs to satisfy the derived concept's requirements and not necessarily the base concept's requirements, which makes no sense to me. I understand that Wikipedia is far from a definitive source; is the above description just a poor choice of words?

2) Can a concept which lists typenames be 'auto'? If so, how would the compiler map these typenames automatically? If not, are there any other occasions where it would be invalid to use 'auto' on a concept?

To clarify, consider the following hypothetical code:

template<typename Type>
class Dummy {};

class Dummy2 { public: typedef int Type; };

auto concept SomeType<typename T>
{
     typename Type;
}

template<typename T> requires SomeType<T>
void function(T t)
{}

int main()
{
    function(Dummy<int>()); //would this match SomeType?
    function(Dummy2()); //how about this?
    return 0;
}

Would either of those classes match SomeType? Or is a concept_map necessary for concepts involving typenames?

3) Finally, I'm having a hard time understanding what axioms would be allowed to define. For example, could I have a concept define an axiom which is logically inconsistent, such as

concept SomeConcept<typename T>
{
    T operator*(T&, int);

    axiom Inconsistency(T a)
    {
         a * 1 == a * 2;
    }
}

What would that do? Is that even valid?

I appreciate that this is a very long set of questions and so I thank you in advance.

I've used the most recent C++0x draft, N2914 (which still has concepts wording in it) as a reference for the following answer.

1) Concepts are like interfaces in that. If your type supports a concept, it should also support all "base" concepts. Wikipedia statement you quote makes sense from the point of view of a type's client - if he knows that T satisfies concept Derived<T>, then he also knows that it satisfies concept Base<T>. From type author perspective, this naturally means that both have to be implemented. See 14.10.3/2.

2) Yes, a concept with typename members can be auto. Such members can be automatically deduced if they are used in definitions of function members in the same concept. For example, value_type for iterator can be deduced as a return type of its operator*. However, if a type member is not used anywhere, it will not be deduced, and thus will not be implicitly defined. In your example, there's no way to deduce SomeType<T>::Type for either Dummy or Dummy1, as Type isn't used by other members of the concept, so neither class will map to the concept (and, in fact, no class could possibly auto-map to it). See 14.10.1.2/11 and 14.10.2.2/4.

3) Axioms were a weak point of the spec, and they were being constantly updated to make some (more) sense. Just before concepts were pulled from the draft, there was a paper that changed quite a bit - read it and see if it makes more sense to you, or you still have questions regarding it.

For your specific example (accounting for syntactic difference), it would mean that compiler would be permitted to consider expression (a*1) to be the same as (a*2), for the purpose of the "as-if" rule of the language (i.e. the compiler permitted to do any optimizations it wants, so long as the result behaves as if there were none). However, the compiler is not in any way required to validate the correctness of axioms (hence why they're called axioms!) - it just takes them for what they are.

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