That sounds different from what you're implying in your other emails.  Using sentences here is confusing.

The current behavior is:

Case 1

input: ~//bin

shell: ~/bin

default-directory: /bin

I'm suggesting

When you say "I think it should change default-directory to the directory to which

But your other emails suggest that you want:

 

Assuming that you want Case 3, I think that is horrible for multiple reasons.  Firstly, a user would expect their shell to behave as in Case 2; Case 3 is a surprising and semantically different rewrite of user input.  Furthermore, we would either have to rewrite all paths not just paths to `cd`, which would be infeasible and error-prone, or surprise the user again that Emacs is rewriting paths to `cd` specifically, but not, say, `rm` or `mv` or many other commands (e.g., `mv ~//bin ~//lib`).  Furthermore, the only shell input rewriting that Emacs currently does is for history substitutions, and the syntax for this is identical to common shells; therefore, users can expect that Emacs does not do any surprising semantic rewrites of their shell input.  Case 3 would be violating this expectation.

For completeness, there is also

However, I am aware of zero shells in common use that behave like Case 4.  Even if we consider it, it should not be the default.

Also for completeness, there is a nuclear solution option, which is to replace all of the cd, pushd, popd logic that shell-mode currently implements and always run M-x dirs after every cd, pushd, popd command.  This naturally results in the behavior of Case 2, but also supports Case 4 for hypothetical shells that have said behavior.