President Trump declared success after a summit with North Korean leader Kim Jong Un that resulted in an understanding that Pyongyang would work toward denuclearization and the U.S. would end joint military exercises with South Korea. Mr. Trump and Kim signed a document agreeing to a handful of key provisions.

But the future remains unclear -- the U.S. and North Korea did not reach any agreement on the details of how to achieve or verify that denuclearization, Mr. Trump said. "We'll be verifying," the president remarked in a rare, extended news conference after the summit. Mr. Trump also claimed Kim told him North Korea has destroyed a "major" missile testing site, although he didn't expand on that.

Mr. Trump said the U.S. will be continuing sanctions until denuclearization occurs -- but suggested sanctions might be lifted earlier than many experts thought possible.

Earlier in the day, the president claimed the summit went "better than anybody could have expected." The president also claimed Kim has a great personality, and said he will "absolutely" invite Kim to the White House.

For a detailed account of the summit, visit CBS News' earlier live blog.

Follow along below for updates:

Scalise on Pence meeting

CBS News' Alan He reports that Majority Whip Steve Scalise was asked about what the Vice President said about military exercises given the earlier confusion between Sen. Cory Gardner and Pence.

Scalise pivoted, telling reporters, "Well you know there were a number of things that they talked about winding down."

"President Kim Jong Un first of all has for months now withheld any kind of missile test, and that's a big win already. Of course we got the release of the 3 hostages. But Kim Jong Un also agreed that he would stop these missile engine tests, which is another big win that came out of the talks yesterday, and President Trump talked about winding down some of these war games, and all of this is subject to Kim Jong Un continuing to follow through on concrete steps to denuclearize," Scalise added.

Joe Biden says Trump-Kim agreement "light on details"

Vice President Joe Biden said in a statement on Tuesday that he welcomes efforts to "de-escalate tensions and pursue diplomacy, but proof of success will only come when we see substantive and verifiable evidence that North Korea is eliminating its nuclear arsenal."

He criticized the declaration reached between Mr. Trump and Kim as being "very light on details" and "is the beginning of a long negotiation process."

"It is troubling, however, that the Trump administration has given the North Korean regime many sought-after wins up front without getting anything in return," Biden said.

He added, "Talking to dictators is one thing; embracing them is another. So far, this is not a deal that advantages the United States or makes us safer."

Cory Gardner says training to continue despite "war games" comment

Sen. Cory Gardner, R-Colorado, told reporters on Capitol Hill that military "exercises will continue with South Korea" despite the president's assurances that such "war games" will halt in light of the agreement reached with Kim.

"I think what the vice president said today will continue to clarify what the president had talked about. Exercises will continue with South Korea. Look forward to further comment and clarification from the president when he gets here," Gardner told reporters.

While many were left confused on the change in message, Gardner later tweeted that the Vice President "was very clear" that "regular readiness training and training exchanges will continue." Pence's press secretary Alyssa Farah, however, disputed that the vice president ever made such comments at the Senate lunch.

Some 30 minutes after Gardner's original tweet, the Senator added that the vice president "went on to say while this readiness training and exchanges will occur, war games will not."

Pence on Hill to brief lawmakers

Vice President Mike Pence is on Capitol Hill briefing lawmakers on the president's work in North Korea.

Pence is briefing Rep. Kevin McCarthy, R-California and Whip Steve Scalise on the summit. House Speaker Paul Ryan is scheduled to speak with the Vice President later Tuesday afternoon. CBS News' Rebecca Kaplan reports when asked about the summit, Pence waved and gave reporters a thumbs up.

Following Senate policy lunches, Sen. John Barrasso told reporters that Mr. Trump had separately called into the Republicans meeting from Air Force One to discuss the events in North Korea.

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Graham warns Kim against "playing" Trump

"Look at this way, if Kim shakes Trump's hand and receives lavish praise from President Trump and tries to play President Trump, look what happened to Canada. Bad things are gonna happen. The worst thing you can do to President Trump if you're a foreign leader is try to play him," Sen. Lindsey Graham, R-South Carolina told reporters on Capitol Hill on Tuesday.

Earlier, Graham told "CBS this Morning" that the document signed in Singapore is simply an "agreement in principle" that must first come to Congress for approval.

"This is a first step, this is a good start but we're a long ways away from an agreement," Graham told "CBS This Morning" on Tuesday.

While Graham said he congratulated Mr. Trump on bringing about such an "historic opportunity" to help end the Korean war and force the regime to give up its nuclear capabilities, he noted, "They've promised to give up their nuclear weapons, they've done this twice."

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Schumer says Trump-Kim agreement "short on details"

Senate Minority Leader Chuck Schumer took to the Senate floor to discuss the U.S.-North Korea summit, saying the agreement reached between Mr. Trump and Kim is "short on details" and "vague" on concrete action.

"We must get action not just photo ops," urged Schumer.

Schumer says the joint agreement views denuclearization as a "far off goal" with no clear pathway on how to achieve it or verify that North Korea has in fact disarmed.

He noted however that it was a "welcomed improvement to see them have a dialogue rather than engaging in name calling,"

"We are all rooting for diplomacy to success, we must be clear-eyed about what a diplomatic success looks like with North Korea."

Meanwhile, Senate Majority Leader Mitch McConnell said in his floor speech that he supports the goals contained in the joint statement reached between Mr. Trump and Kim, but says the "next steps will test whether we can get to a verifiable deal."

"Resolving this 65 year old international challenge will take a great deal of hard work," said McConnell. "Today is the beginning of the arduous process, our eyes are wide open."

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Trump campaign hails summit as making "history"

Trump campaign adviser and Mr. Trump's daughter-in-law Lara Trump said in a statement following the summit that the "historic" meeting with Kim "was an end product of President Trump's bold and vigilant leadership on behalf of the American people."

"Already, the President has achieved more than expected, with an agreement from North Korea to return the remains of American POWs and destroy a missile testing site, while economic sanctions remain in place for the time being," the statement added.

Trump says that the developments with the North Korean regime "are yet another validation that the American people were right to entrust Donald Trump to change the course and direction of our country that had been commanded by the political class in Washington for decades."

"President Trump will continue to succeed in dramatic ways because he will always put America First and wishes only to succeed on behalf of the American people," she added.

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U.S. defense community responds to end of "war games"

Members of the Defense community appear to be caught off guard following Mr. Trump's decree that the U.S. would be halting its joint exercises in South Korea.

Christopher Sherwood, spokesperson for the Department of Defense said in a statement to CBS News on Tuesday, "The Department of Defense continues to work with the White House, the interagency, and our allies and partners on the way forward following the U.S./DPRK summit."

Meanwhile, the United States Forces Korea who conducts the exercises, said in a statement that they have received "no updated guidance on execution or cessation of training exercises."

"In coordination with our ROK partners, we will continue with our current military posture until we receive updated guidance from the Department of Defense (DoD) and/or Indo-Pacific Command (INDOPACOM)," the statement added.

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Japan's Abe says he "highly respects" efforts in Singapore

Per a Japanese government official, Trump and Prime Minister Abe will speak tonight. And Japanese Prime Minister Shinzo Abe told reporters at the conclusion of the summit that he expresses his "heartfelt respect for the leadership and efforts of President Trump leading up to his meeting with Chairman Kim Jong Un."

Abe added,"I support the summit meeting today that has reconfirmed Chairman Kim Jong Un's commitment to the complete denuclearization of the Korean Peninsula in a written document as a step toward the comprehensive resolutions of outstanding issues of concern."

The prime minister said that he "highly appreciated" Mr. Trump efforts in raising the issue of Japanese abductions "which is important for Japan," he noted.

"With regard to the abductions issue, Japan is determined to directly face North Korea to resolve this issue," Abe added.

The two leaders are expected to talk over the phone later tonight.

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Trump departs Singapore

After a lengthy news conference Mr. Trump departs Singapore for Guam. "There was nothing more we could have done. We had the agreement done. The only thing he could have done is gone to the news conference," Mr. Trump told reporters as to why he chose to leave in the night instead of the next day.

The president told reporters on board Air Force One looking to speak further with him "lets see how we feel, it's 26 hours now, you become vulnerable at 26."

Asked why the first lady did not join him for the trip, Mr. Trump said that Melania Trump was given orders to not fly within 30 days of her surgery for her kidney condition. "30 days she's not supposed to fly," he said, remarking that she did however attend a function at Fords Theatre back in D.C.

He told reporters that the first lady is "doing great."

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Trump credits rhetoric for Kim summit

In an interview set to air Tuesday night, Mr. Trump tells Fox News' Sean Hannity that without his strong rhetoric toward North Korea and its leader "we wouldn't have been here."

"You know, we did sanctions and all the things you would do but I think without the rhetoric, you know, other administrations, I don't want to get specific on that, but you know they had a policy of silence. If they said something very bad and very threatening and horrible just don't answer. That's not the answer. That's not what you have to do. So I think the rhetoric, I hated to do it, sometimes I felt foolish doing it, but we had no choice," he said in excerpts of the interview.

Mr. Trump told Hannity that the two leaders got along "very well" in their private meeting with translators.

"We got along better than I would have assumed right from the beginning. We got a lot more done today than I ever thought possible."

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A closer look at Trump's "war games" comments

Mr. Trump claimed that he made no concessions to Kim, but is ending the "war games" consisting of joint military exercises with South Korea. On top of that, CBS News correspondent Ed O'Keefe points out, Mr. Trump has invited Kim to the White House.

"I don't quite understand why Trump felt the need to stop the war games," a senior fellow at the Asia Society's Center on U.S.-China Relations. "They are something that shows U.S. resolve in the region, not only to North Korea but also more importantly to China. And it seems like there's this whole idea of Trump as this great negotiator and yet he gave up one of the most important cards we have in the region, seemingly for free. So I think I am baffled by that and I am baffled by a lot of things that came out of Trump's mouth."

Mr. Trump also suggested his desire to remove troops from South Korea eventually, although that wasn't a part of the agreement Mr. Trump and Kim signed.

O'Keefe said there may be "measured support" from some Republicans in Congress for what Mr. Trump did in Singapore, but skepticism as well.

"Some of his biggest defenders on Capitol Hill are also some of this regime's biggest critics," O'Keefe said.

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Trump to depart soon en route to U.S.

After a press conference that lasted more than an hour, Mr. Trump is about to leave Singapore for the U.S.

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Here is what Trump and Kim agreed to

Mr. Trump and Kim agreed to a basic framework, in an unexpected meeting between the two leaders.

Here are the four components of the agreement outlined in the document:

• "The United States and the DPRK commit to establish new U.S.-DPRK relations in accordance with the desire of the peoples of the two countries for peace and prosperity."
• "The United States and the DPRK will join their efforts to build a lasting and stable peace regime on the Korean peninsula."
• "Reaffirming the April 27, 2018 Panmunjom Declaration, the DPRK commits to work toward complete denuclearization of the Korean peninsula."
• "The United States and the DPRK commit to recovering POW/MIA remains, including the immediate repatriation of those already identified."

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"I do trust him," Trump says of Kim

Mr. Trump, in an interview he gave to ABC's George Stephanopoulos shortly after meeting with Kim, said he trusts the North Korean dictator.

"I do trust him," Mr. Trump said.

"He's de-nuking the whole place and I think he's going to start very quickly," the president added.

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Highlights from the Trump-Kim meeting

After two lengthy meetings (one in private and one with their respective delegations) and a hearty lunch discussion over short ribs, pork and Häagen-Dazs ice cream, Mr. Trump and Kim ended their summit with a signed document that the president described as "important" and "comprehensive."

The document, which offers few details about how the aims will be accomplished, says North Korea commits to working towards denuclearization

Mr. Trump said he "absolutely" will invite Kim to the White House. Mr. Trump said he thinks the summit went "better than anybody could have expected."

Upon closing the summit, the president called the meeting a "tremendous" 24 hours.

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