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Coq is a formal proof management system, also called an interactive theorem prover. It is used to express mathematical assertions, mechanically check proofs of these assertions, find formal proofs, and extract certified programs.
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Function extensionality: does it make a difference? why would one keep it out of the axioms?
statement that if two functions $f$ and $g$ on the same domain satisfy $f\left(x\right) = g\left(x\right)$ for all $x$ in the domain, then $f = g$) is not an axiom in the standard constructive logic of Coq … (I must say the eta-expansion in Coq feels rather weird, too -- it is triggered by the reflexivity tactic. I expected it to be a tactic on its own...) …