First suggestion is that you not let it end up in court. If it does, you'll both lose out.
If I were faced with the situation, I'd probably ask for an agreement which would work out as:
([Net equity - (initial deposit + legal fees + refurbishment costs)] / 2) + (initial deposit + legal fees + refurbishment costs)
with net equity defined as the capital sum realised from the sale after deduction of legal and estate agent's fees.
But I'd make it a negotiation with the other person and not via the courts: if you go down the court route, you'll stand to lose a lot more in legal fees than you would lose by splitting down the middle, which should be [in my strictly amateur none-of-this-is-legal-advice opinion] your worst case fall back position. The other person may be able to argue intangible contributions (whether it's valid or not, it can be used in court to assert a greater claim) to do with maintenance / cleaning / gardening / cooking / decoration... and believe me, if you want to take the thing to court, it'll cost you.
I definitely wouldn't try to enforce the initial additional contribution as determining proportion of equity - you'll just end up with a fight on your hands. You could possibly approach it by saying that look, you feel you could press for the initial contribution to determine proportion of current equity but that you won't because you're keen to come to an amicable agreement. If negotiating doesn't work, at least if you then go the legal route you may be perceived to have attempted a reasonable solution in what are presumably, though you don't relate the background, difficult circumstances.