The legal and political assurances by Theresa May on the backstop before Tuesday's vote are detailed in an exchange of letters between the Prime Minister and Jean-Claude Juncker, the European commission president, and Donald Tusk, the European council president.

What the prime minister said:

The deal is at risk … because of concerns in the UK about how we are delivering on our commitments regarding Northern Ireland's border with Ireland.

The Irish backstop solution for avoiding a hard border covers the whole of the United Kingdom. Brexiters, in particular, fear that it will turn out to be a permanent arrangement, standing in the way of pursuing an independent trade policy.

As you know, since the suspension of the debate in my parliament, I have proposed a legal commitment to have our future partnership in place by the end of 2021 at the very latest.

The prime minister has been fruitlessly seeking a deadline for the completion of trade talks.

What Tusk and Juncker say:

As you know, we are not in a position to agree to any changes or is inconsistent with the withdrawal agreement.

The EU has insisted that the deal is in the final deal, and there will not be renegotiation, or any legal interpretation which cuts across it. This includes attempts to secure a unilateral exit mechanism from the backstop, or a time-limit.

On the 13 December, the European Council (Article 50) decided on a number of additional assurances.

The 27 heads of state and government offered warm words on the temporary nature of the backstop at the last summit, and went further than expected. The EU has the pledge, which would have been legal in Parliament.

Should national ratifications be pending … the commission is ready to propose a provisional application of relevant parts of the future relationship.

The EU is in the process of becoming a member of the European Union.

The Commission confirms that the United Kingdom does not wish to return. Were it to do so, it would represent a suboptimal trading arrangement for both sides.

Rather than hoping to trap the UK into the backstop, the EU fears that the lack of parity in terms of level-playing commitments could lead to the UK having an economic advantage. A temporary shared customs territory would also be complained of.

Facilitative arrangements and technologies will be considered.

The political declaration on the future trade deal suggests that the permanent solution for avoiding a hard border would build on the customs union in the backstop. This clause in the letter emphasizes that this is not the case with the Brexiters.

The Commission is committed to providing the necessary financial resources.

The heads of state and government removed the suggestion in its December communique that the backstop would have been in place for a "short" period, for fear that it could not be due to the vagaries of the negotiations. The commission has done its best to shoehorn the phrase back in, and is offering biannual EU-UK summit to check on progress.