(CNN) Everyone's lying to Robert Mueller -- and he knows it.

The court filings released Friday by Robert Mueller and federal prosecutors detail the alleged lies Donald Trump's former lawyer Michael Cohen and former campaign chairman Paul Manafort told both publicly and to the special counsel's investigators.

And for the first time, prosecutors publicly endorsed Cohen's earlier statements in court that he acted at the direction of Trump when he committed two crimes in 2016: making payments to the two women alleging affairs with Trump in order to keep them silent.

The court documents provide new insights into the Mueller probe itself, including why the proposed Trump Tower Moscow project was relevant to Russian election meddling and potential new contacts between Cohen and Russian officials seeking to arrange a meeting between Trump and Russian President Vladimir Putin. They reveal that a Russian national who claimed connections to the Russian government spoke with Cohen in 2015 and offered "political synergy" with the Trump campaign.

In addition, the documents also detail how Manafort was directly in touch with a senior Trump administration official earlier this year as well as his contact with Konstantin Kilimnik, a Russian national who Mueller's team says has ties to Russian intelligence services, though the details of those interactions are redacted.

Here are the key takeaways:

Prosecutors say Trump directed Cohen

For the first time, federal prosecutors say that Trump directed Cohen to make payments designed to silence women who claimed affairs with Trump.

When Cohen pleaded guilty in August to campaign finance violations connected to the payments, Cohen stated in court that he had been directed by Trump.

But the sentencing memo from the Manhattan US Attorney's office, which was filed separately from Mueller's team, was the first time the government has stated Trump's role in the payments. Trump has denied the affairs, and he has not been accused of any crimes related to the payments.

"In particular, and as Cohen himself has now admitted, with respect to both payments, he acted in coordination with and at the direction of Individual-1," prosecutors wrote in a reference to Trump.

Lying liars

Mueller's team documents a number of lies told by Cohen and Manafort -- and has the proof to back it up.

Manafort's legal team has disputed that he lied to the special counsel since entering into a cooperation agreement -- which would violate that agreement -- but Mueller has texts. The court filing cites a text exchange in which Manafort authorized an associate to speak to a Trump administration official on his behalf.

In all, Mueller accuses Manafort of lying on five separate matters.

Cohen's lies, some of which he admitted to in a guilty plea last week, included lying to the special counsel investigators about the Trump Tower Moscow project after offering to cooperate. Mueller's team says he eventually took responsibility for his lies, later explaining he was trying not to contradict his congressional testimony.

There are now three officials in Trump's orbit accused of lying to the special counsel: Manafort, Cohen and former Trump campaign official Rick Gates who pleaded guilty to lying during one of his early meetings with Mueller's team.

Trump Tower Moscow is relevant to meddling

Mueller's memo lays out how the Trump Tower Moscow project is relevant to Russia's election meddling during the 2016 campaign.

The fact that Cohen continued to work on the proposal through June 2016, not ending in January as he falsely testified -- and he discussed it with Trump -- was material to both the ongoing congressional and special counsel investigations, the special counsel said.

Mueller said that the continued work on Trump Tower Moscow "occurred at a time of sustained efforts by the Russian government to interfere with the US presidential election."

The special counsel memo states that Cohen's false statements to investigators about the Moscow project "obscured the fact that the Moscow Project was a lucrative business opportunity that sought, and likely required, the assistance of the Russian government."

The argument in Cohen's filing underscores that the overlap of business and politics is relevant to the special counsel's inquiry.

False public statements also key to investigation

The special counsel's office made clear that it sees not just lies to investigators but also false public statements as significant in the Cohen sentencing memo.

Prosecutors explained that Cohen's effort to lie about the Moscow project continuing through June 2016 were an effort to alter the investigation.

"The defendant amplified his false statements by releasing and repeating his lies to the public, including to other potential witnesses. By publicly presenting this false narrative, the defendant deliberately shifted the timeline of what occurred in the hopes of limiting the investigation into possible Russian interference in the 2016 US presidential election -- an issue of heightened national interest."

Earlier in the week, the special counsel stressed that Michael Flynn's false statement also impacted what others said publicly.

"Several senior members of the transition publicly repeated false information conveyed to them by the defendant about communications between [Flynn] and the Russian ambassador regarding the sanctions," Mueller wrote.

Interestingly, Mueller redacts the rest of the paragraph. So if the impact was explained, it is still a secret.

The emphasis on public statements could offer clues at how prosecutors view the relevance of efforts to misdirect in formulating the first public statement about the 2016 Trump Tower meeting aboard Air Force One the next year.

One of the questions that Mueller wanted to ask Trump, as seen in the questions that were given to the President's lawyers earlier this year, focused just on that: "What involvement did you have in the communication strategy, including the release of Donald Trump Jr.'s emails?"

Why did Manafort try to hide contacts with administration officials?

One of the mysteries in the Manafort court filing: Who was he allegedly contacting in the Trump administration during 2018 -- and why?

Mueller accuses Manafort of lying about multiple contacts with Trump administration officials, including a senior administration official. Mueller cites a May 2018 text exchange authorizing an associate to contact one official, and a Manafort colleague describing a February 2018 contact with the senior official.

It's not clear why Manafort was conducting the outreach -- but Mueller's team signals in the filing that it likely has the answer.

"A review of documents recovered from a search of Manafort's electronic documents demonstrations additional contacts with Administration officials," the filing states.

Why Mueller still thinks Cohen's credible

While the special counsel's office does not make a recommendation to give Cohen reduced prison time, Mueller still says that Cohen was a cooperative witness.

And Mueller's team makes a reference to Cohen's credibility, which is poised to be a key question if any of the information Cohen provides is used in other cases the special counsel is pursuing -- especially given that Mueller documents how Cohen even lied to the special counsel.

In the court filing, Mueller's team notes that the information that Cohen provided "has been credible and consistent with other evidence obtained" in the special counsel's ongoing investigation, which signals the special counsel investigators independently corroborated what Cohen told them.

Mueller outlined four areas where Cohen cooperated, which include his own contacts with "Russian interests" during the 2016 campaign, "useful information" concerning Russia-related matters, contacts with people connected to the Trump White House and "the circumstances of preparing and circulating his response to the congressional inquiries."

Cohen was less useful to Manhattan US Attorney's office, which said that the information Cohen provided to the New York Attorney General's office in their ongoing case against the Trump Foundation "warrants little to no consideration as a mitigating factor."

"After cheating the IRS for years, lying to banks and to Congress, and seeking to criminally influence the Presidential election, Cohen's decision to plead guilty -- rather than seek a pardon for his manifold crimes -- does not make him a hero," prosecutors wrote.