For every , and , we define pointwise equality as
The axiom of weak extensionality is stated as the existence of a function
Recall that the axiom of (strong) function extensionality is stated as the fact that the canonical map
In this exercise, we show that weak function extensionality implies strong funextionality.
Recall that a retraction between two types and is given by
As usual, denotes the dependent sum, with first and second projections noted and .
Show that singletons are contractible, that is for every type and , we have
Show assuming that for every type , and , we have
Show that if a type is contractible and is a retract of , then is contractible.
Show for every type , and assuming , that if
Show that from any , we can define such that
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