Here is a new US export: President Donald Trump's exaggerations about his record.

In his speech in Saudi Arabia on Sunday during his first foreign trip as president, Mr Trump claimed to have accomplished record spending on the armed forces, even though Congress has yet to pass a budget that reflects his plans and promises.

Mr Trump will release a detailed budget proposal on Tuesday after having come up only with an outline before, and nothing is achieved until and unless Congress passes something.

Mr Trump often takes credit for accomplishments that have yet to be realised or which were the work of his predecessor, as he did last week when boasting about a Coast Guard icebreaker that the Obama administration started. But it was his first opportunity to do so abroad.

Mr Trump's foreign trip came as something of a break from the storm over the investigation into his 2016 campaign's relationship with Russia. That episode prompted a number of questionable statements by the president and his aides. Here is a review of claims on various matters over the past week:

:: Mr Trump in a speech in Riyadh on Sunday: "In just a few months, we have created almost a million new jobs ... and made record investments in our military."

The facts: He is getting ahead of developments on military spending, with no budget passed. He also not proposing a record increase in military spending as his remarks might imply.

The 10% increase he called for in his March budget outline has been exceeded three times in recent history - the base military budget went up by 14.3% in 2002, 11.3% in 2003 and 10.9% in 2008, according to the Pentagon. Looked at another way and deeper into history, military spending consumed 43% of the economy in 1944, during the Second World War, and 15% in 1952, during the Korean War. It was 3.3% in 2015, says the World Bank.

Mr Trump's claim that almost one million jobs have been added is in the ballpark, though it has taken more than a "few months" and Barack Obama was president for most of one of them - January.

Job creation has averaged 185,000 a month from January to April. But that is the same pace of hiring as occurred in 2016, when Mr Obama was president, and slower than in 2014 and 2015, when more than 225,000 jobs a month were added, on average.

:: Mr Trump in a Coast Guard Academy speech last Wednesday: "I'm proud to say that under my administration, as you just heard, we will be building the first new heavy icebreakers the United States has seen in over 40 years."

The facts: Mr Trump is claiming credit for something that started under his predecessor. Mr Obama's homeland security secretary, Jeh Johnson, spoke about modernisation of the Coast Guard fleet and design work on a new heavy polar icebreaker a year ago in a speech to graduating Coast Guard cadets.

:: Mr Trump, on his decision to fire FBI director James Comey in a news conference on Thursday with his Colombian counterpart, Juan Manuel Santos: "I actually thought when I made that decision - and I also got a very, very strong recommendation, as you know, from the deputy attorney general, Rod Rosenstein."

The facts: The recommendation he cites came after Mr Trump decided to fire Mr Comey, according to Mr Rosenstein and to Mr Trump's own previous statement taking sole ownership of the decision.

In an interview with NBC two days after Mr Comey's May 9 dismissal, Mr Trump said he had been planning to fire the FBI chief for months, and linked it with the FBI's Russia investigation. "In fact when I decided to just do it, I said to myself, I said 'You know, this Russia thing with Trump and Russia is a made-up story'."

On Thursday, Mr Rosenstein told senators in a closed-door briefing that he had been informed of Mr Trump's decision to fire Mr Comey before he wrote his memo providing a rationale for that act, said Illinois Democratic Senator Dick Durbin.

:: Mr Trump in Thursday's news conference: "Even my enemies have said there is no collusion."

The facts: Democrats have not absolved Mr Trump on whether his campaign and Russian officials co-ordinated efforts last year to disadvantage his Democratic rival, Hillary Clinton. Several have said they have not seen evidence of collusion, but that is not to say they are satisfied it did not happen.

Mr Trump has cited James Clapper, the director of national intelligence until Mr Trump took office on January 20, among others, as being "convinced" there was no collusion.

Mr Clapper said this past week that, while a report he issued in January did not uncover collusion, he did not know at the time that the FBI was digging deeply into "potential political collusion between the Trump campaign and the Russians" and he was unaware of what the bureau might have found. The FBI inquiry continues, as do congressional investigations and, now, one by the special counsel.

:: Mr Trump, also at Thursday's news conference: "Obamacare is collapsing. It's dead. It's gone. There's nothing to compare anything to because we don't have healthcare in this country. You just look at what's happening. Aetna just pulled out. Other insurance companies are pulling out. We don't have healthcare. Obamacare is a fallacy. It's gone."

The facts: He is venting and not to be taken literally. Mr Obama's healthcare law remains in effect and people are using it. As of last count 12.2 million signed up for private health plans through HealthCare.gov and state markets that offer federally subsidised coverage. Separately another estimated 12 million were made eligible for Medicaid through the law's expansion of that programme. It is true that many people who buy their own health insurance are facing another year of big premium increases and shrinking choices.

Mr Trump worked with House Republicans to pass a Bill which would roll back much of the health law and the Senate is considering the legislation.

:: Mr Trump, speaking of the MS-13 gang presence in the US: "A horrible, horrible large group of gangs that have been let into our country over a fairly short period of time ... They've literally taken over towns and cities of the United States."

The facts: His depiction of the gang as a foreign one "let into" the US is not accurate.

The gang actually began in Los Angeles, according to a factsheet from Mr Trump's own Justice Department, and "spread quickly across the country". And it started not recently, but in the 1980s, according to that same factsheet.

The department indirectly credits the Obama administration, in its early years, with helping to rein in the group, largely made up of first-generation Salvadoran-Americans and Salvadoran nationals.

It said: "Through the combined efforts of federal, state and local law enforcement, great progress was made diminishing or severely (disrupting) the gang within certain targeted areas of the US by 2009 and 2010."

The US carried out record deportations during the Obama administration and, on MS-13 specifically, took the unprecedented action of labelling the street gang a transnational criminal organisation and announcing a freeze on its US assets. Such actions were not enough to bring down the group and the Trump administration says it will do more.

:: Treasury Secretary Steven Mnuchin at a Senate Banking, Housing and Urban Affairs Committee hearing on Thursday: "I believe that a goal of 3% GDP or higher economic growth is achievable if we make historic reforms to both taxes and regulation."

The facts: Several quarters or a year of 3% growth may be possible, but few economists expect the changes Mr Mnuchin has proposed would result in sustained growth at that pace. That is because the US economy is facing long-term constraints. As baby boomers retire, fewer people are working. In addition, workers' productivity is growing at historically weak levels. An economy can only grow as fast as the size and productivity of its workforce. If Mr Trump's policies reduce immigration, the US workforce would grow even more slowly.

Mr Trump's goal of cutting corporate taxes could encourage companies to spend more on computers and machinery, making employees more productive, accelerating growth and lifting wages. Liberal economists argue that corporate profits are already high and any tax cut probably would go to shareholders instead of equipment.

:: Mr Trump in a speech to the Coast Guard Academy: "I won't talk about how much I saved you on the F-35 fighter jet. I won't even talk about it."

The facts: He should not. He has repeatedly taken credit for cost savings which began on the jet before his presidency. Pentagon officials took steps before the election to reduce costs on the Lockheed contract and announced savings on December 19, a month before Mr Trump was sworn in.

:: Ambassador to the UN Nikki Haley in an interview on the Christian Broadcasting Network last Wednesday: "I believe the Western Wall is part of Israel and I think that that is how, you know, we've always seen it and that's how we should pursue it ... we've always thought the Western Wall was part of Israel."

The facts: That is a mis-statement of US policy and diplomatic history. The wall is in the Old City, a part of east Jerusalem, which the US and most of the world consider to be occupied territory. So the US position is that the wall is part of Jerusalem, not specifically Israel. Since Israel's founding, the US has maintained that no state has sovereignty over Jerusalem and its ultimate status must be resolved through Israeli-Palestinian negotiations. That stance has not changed.

In addition to mis-stating US policy, Ms Haley stepped outside diplomatic norms in asserting a personal view at variance with that policy - that the Western Wall is or should be considered part of Israel.