Shortest-proof-game challenge

That was too easy. Let's try it another way.
What's the shortest sequence of legal moves ending with mate by the white c-pawn before it has promoted or made any captures?

And that's the shortest. Now try this version.
What's the shortest sequence of legal moves ending with mate by the white c-pawn on its first move?

The 5 moves can be shortened to 4 if it's Black's c-pawn that mates (not necessarily on its first move)!

Here is a legal position. How short a proof game can you construct for it?
how is it legal??
oh the black queen captured something on g3,which was blocking the check.
that can only be a knight,since all the squares around the square are blocked so only a knight can block.
okay so the previous ply are possible.
no actually the knight was there before the queen.
looks legal but ridiculously precise
No, it must still be a pawn after the mate. I should have specified an unpromoted white c-pawn.