Independent experts say the North could have enough fissile material for anywhere between about a dozen and 60 nuclear bombs, and a year ago it tested long-range missiles that could reach the US.
Readiness training refers to standard military drills and training carried out by the 28,000 United States troops in South Korea, alongside their South Korean counterparts.
Trump has been on a victory tour of sorts since he returned from the historic summit, where he was the first USA president to meet with a sitting leader from North Korea. "Nobody thought we were going to have an agreement like that".
The second reference stated: "Kim Jong-un said in order to achieve peace and stability of the Korean Peninsula and realise its denuclearisation, the two countries should commit themselves to refraining from antagonising with each other out of mutual understanding, and take legal and institutional steps to guarantee it".
"Before taking office people were assuming that we were going to War with North Korea", he continued in a second post.
Max Thunder's air combat exercises so unnerved North Korea that it issued threatening statements that almost scuttled the summit. But I think so far [the negotiations with North Korea are] meeting the minimal standards for success.
Last year's talking point was that Trump, with his "fire and fury" talk, was leading the USA into a nuclear war with North Korea.
Despite those efforts, Trump announced after the summit that he had agreed to suspend USA military drills with South Korea, something North Korea has long demanded.
On the way forward, Pompeo said, "I don't know exactly what the timing will be for our next conversation with the North Koreans".
The US stations 28,500 troops in security ally South Korea to protect it from its neighbour, which invaded in 1950 to try to reunify the peninsula by force. -South Korean drills could also erode the effectiveness of the military alliance between Washington and Seoul.
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Ms Hanham also said the United States and North Korea "still don't agree on what "de-nuclearisation" means".
A White House National Security Council official said some types of training would continue even after Trump's announcement.
"I am willing to do a lot of things to get [Kim] to give up their nuclear weapons and end their missile program". He said South Korea is still trying to figure out the exact meaning and intent of Trump's comments.
"So that's good for a number of reasons, in addition to which we save a tremendous amount of money", Mr Trump said. "By the way, Korean stuffed cucumber is stuffed with Koreans - Kim is a monster".
When Baier pointed out some of the unsavory things Jong Un is accused of doing in North Korea, Trump demurred. The joint statement implicitly links denuclearisation to "new US-DPRK relations" and to mutual confidence building (that) can promote the denuclearisation of the Korean Peninsula.
"Meeting with Kim Jong-un was an interesting and very positive experience".
Experts now expect a temporary peace to continue since North Korea has probably won what it wanted from Tuesday's summit and Trump is unlikely to back down from summit deals that he wants to portray as a diplomatic triumph.
Kimmel show personality Guillermo Rodriguez and the host proceeded to sit down for a mock summit to sign their own version of the "friendship agreement" that Trump and Kim signed.
However, some are still wary of the nature of the summit, with some saying the document signed between the two leaders looks "scarily like a draft agreement".
Bridling at a question from a reporter who asked why the summit document didn't mention complete and verifiable denuclearization, Pompeo said, "I find that question insulting and ridiculous and frankly ludicrous".
In his post-summit press conference, Mr Trump made the surprise announcement that the U.S. would halt joint military exercises with its security ally Seoul - something long sought by Pyongyang, which claims the drills are a rehearsal for invasion.
Bong Youngshik, a professor at Seoul's Yonsei University, said it's too early to predict how Trump's cancellation of the military drills will play out in future nuclear negotiations.