1.12.2 Equivalence of path induction and based path induction
The two induction principles for the identity type introduced above are equivalent. It is easy to see that path induction follows from the based path induction principle. Indeed, let us assume the premises of path induction:
Now, given an element , we can instantiate both of the above, obtaining
Clearly, and match the premises of based path induction and hence we can construct
with the defining equality
with the required judgmental equality .
Another proof of this fact is to observe that any such can be obtained as an instance of so it suffices to define in terms of as
The other direction is a bit trickier; it is not clear how we can use a particular instance of path induction to derive a particular instance of based path induction. What we can do instead is to construct one instance of path induction which shows all possible instantiations of based path induction at once. Define
Then we can construct the function
and hence using path induction obtain
with . Unfolding the definition of , we can expand the type of :
Now given and , we can derive the conclusion of based path induction:
Notice that we also obtain the correct definitional equality.
Another proof is to observe that any use of based path induction is an instance of and to define
Note that the construction given above uses universes. That is, if we want to model with , we need to use with
since quantifies over all of the given type. While this is compatible with our definition of universes, it is also possible to derive without using universes: we can show that entails Lemma 1 (http://planetmath.org/23typefamiliesarefibrations#Thmlem1),Lemma 3.11.8 (http://planetmath.org/311contractibility#Thmprelem5), and that these two principles imply directly. We leave the details to the reader as http://planetmath.org/node/87560Exercise 1.7.
We can use either of the foregoing formulations of identity types to establish that equality is an equivalence relation, that every function preserves equality and that every family respects equality. We leave the details to the next chapter, where this will be derived and explained in the context of homotopy type theory.
|Title||1.12.2 Equivalence of path induction and based path induction|