Jeremy Corbyn took on Owen Smith in a head-to-head debate for the first time this week.

The two men met in Cardiff for the first of nine (!) official debates before the party chooses its leader on 24 September.

Supporters of both campaigns have accused the other side of peddling untruths.

Corbyn: ‘I didn’t say that’

Owen Smith said he wants a second EU referendum, and said Corbyn failed to fight hard enough to keep Britain in.

He accused Mr Corbyn of making a “real mistake” by apparently calling for the government to trigger Article 50 of the Treaty on European Union, beginning the legal process of leaving the bloc.

“No, I didn’t say that”, Mr Corbyn interrupted.

It didn’t take long for Smith supporters to unearth a video of Mr Corbyn saying something very close to that in an interview with the BBC on 24 June, the morning of the referendum result.

Mr Corbyn in fact said: “The British people have made their decision. We must respect that result and Article 50 has to be invoked now so that we negotiate an exit from European Union.”

Later in the debate he said his remarks had been wrongly interpreted.

“I said it was inevitable Article 50 was going to be triggered. This was immediately interpreted as being that it should be triggered immediately.

“Maybe I can choose my words more carefully at the point. It is going to be triggered at some point.”

As a section of the crowd began to jeer, Mr Corbyn added: “All right, okay, well… mea culpa.”

Corbyn: ‘We were ahead in May’

Mr Corbyn was forced to defend Labour’s record in the various elections that have been held during his leadership after Mr Smith said he was making it harder for the party to win.

The Labour leader suggested that it was the unrest caused by Labour MPs’ efforts to oust him that had led to a slump in support in the polls.

Mr Corbyn said: “We won all four by-elections, three with a big swing to Labour. We won four mayoral contests.

“We’ve picked up a lot of support over the general election of 2015 and so I think as an opposition we’ve done very well.

“We were ahead in May. Then came the wave of resignations. Then came the threat to unity in the party. And that is what has put us behind in the polls.” [*See update below]

But no major polling company put Labour ahead in May.

Since Mr Corbyn became leader last September, only three voter intention polls out of 86 have put Labour ahead of the Tories, one in March and two in April.

It’s true that the Tories’ lead has grown since May, but it’s impossible to say whether this is the result of disunity in the Labour party as opposed to the Brexit result or Theresa May becoming Prime Minister.

Mr Corbyn’s broader point that Labour have “done very well” as an opposition party is highly debatable.

It’s true that Labour has experienced a surge in membership since Corbyn became leader. Labour now has more than half a million members, dwarfing every other UK party, according to the House of Commons Library.

There’s no post-war precedent for such a sharp rise, so it’s hard to say whether this will affect the party’s electoral fortunes.

It’s also true that Labour have won mayoral contests and by-elections, though few psephologists would draw any conclusions from this small number of victories about the party’s chances at the next general election.

Experts do take an interest in the results of local elections though.

Labour did better than the polls predicted in the 2016 English council elections, losing a tiny number of council seats rather than the hundreds expected.

But it was hard to spin this as a success, since opposition parties almost always gain seats in local elections.

Professor Tony Travers from the LSE notes that since 1979, only opposition parties that gain a massive lead over the incumbent party in local elections go on to form the next government.

Models that use local election results to predict the outcome of the next general election, like this one from Chris Prosser of the British Election Study – also suggest an emphatic Conservative victory.

Smith ‘victory’ poll was a hoax

Some Smith supporters got very excited about a snap poll circulating on Twitter that appeared to show their man as the winner of the first debate.

The research was attributed to YouGov, but the polling company quickly squashed it as a fabrication.

No reputable pollster has published a snap poll, so it’s hard to call the first debate for either candidate.

For what it’s worth, bookmaker William Hill said that they had shortened the odds slightly on a Corbyn victory after the debate, suggesting punters preferred him to Owen Smith.

[Update: a spokesman for Jeremy Corbyn says the words “we were ahead in May” referred to the results of the English council elections rather than opinion polling.

The spokesman said: “According to the BBC, Labour won 31 per cent of the national share of the vote at the May local elections compared with 30 per cent for the Tories.

“Therefore, when it really counted – at an actual election – we were ahead in May, as Jeremy was correct to state.”]