Peter Bergen is CNN's national security analyst, a vice president at New America and a professor of practice at Arizona State University. He is the author of " United States of Jihad: Investigating America's Homegrown Terrorists ."

(CNN) At a rally in Florida on Wednesday, Donald Trump described President Obama as the "founder" of ISIS, going on to say the terrorist organization "is honoring President Obama."

Really?

On Thursday on CNBC he amplified these comments, saying Hillary Clinton was the "co-founder" of ISIS.

Like so much else that Trump has said, these claims are false. The founder of ISIS is Abu Bakr al-Baghdadi, a shadowy Iraqi cleric who President Obama is doing everything in his power to kill.

Trump has claimed he knows more about ISIS than America's leading generals. Clearly this is also total nonsense; he doesn't seem to have done the slightest thing to educate himself about ISIS.

In June, Gen. David Petraeus, formerly the commander of all US forces in Iraq, predicted the largest city that ISIS now holds, the Iraqi city of Mosul, could fall to US-supported Iraqi forces before President Obama leaves office in January.

Petraeus characterized this as "a very big deal" and went on to say, "no question ISIS is a loser in Iraq and, increasingly, Syria," in an interview with me at New America's offices in Manhattan.

The facts on the ground bear this out. ISIS has lost just under half the territory it once controlled in Iraq and around a fifth of what it had controlled in Syria.

45,000 ISIS fighters killed

Just in the past few days ISIS lost the city of Manbij , in northern Syria, a significant victory because it controls key routes to ISIS' de facto Syrian capital, Raqqa.

On Wednesday Lt. Gen. Sean MacFarland, who leads the anti-ISIS campaign, said 45,000 ISIS fighters have been killed so far by the US-led coalition. "We estimate that over the past 11 months, we've killed about 25,000 enemy fighters. When you add that to the 20,000 estimated killed (previously), that's 45,000 enemy (fighters) taken off the battlefield."

That's an astonishing amount of attrition for a force MacFarland estimates has a remaining strength of 15,000 to 30,000 fighters now.

U.S. intelligence estimates the US-led coalition has also killed at least 135 of ISIS leaders.

Meanwhile, the flow of Americans going to join ISIS or attempting to do so has slowed to a trickle from an average of six to one a month, according to US intelligence estimates.

The U.S. military has also stepped up the air campaign against ISIS' wealth, for instance, bombing a bank in Iraq in January in which ISIS had stored millions in cash.

U.S. bombers have also repeatedly struck trucks carrying oil that ISIS has extracted from oil fields in the shrinking area it now controls, striking 83 tankers along the Syria-Iraq border as recently as Sunday.

These attacks on ISIS' cash supply and revenue streams have had real effects on ISIS' bottom line. ISIS has had to halve the salaries of its foot soldiers, according to documents that leaked from the terrorist army earlier this year.

Who is ultimately responsible for all the damage that has been done to ISIS? The supposed founder of ISIS, President Obama.

As to the charge that Clinton co-founded ISIS, she left the State Department in January 2013 and ISIS wasn't founded until three months later.

US exit from Iraq

Trump's charge does raise an interesting question, which is how best to assign responsibility for the rise of ISIS, including the issue of how might the Obama administration's exit from Iraq at the end of 2011 have helped smooth the path for ISIS?

The rise of ISIS starts with a Jordanian thug named Abu Musab al-Zarqawi who founded ISIS' parent organization, al Qaeda in Iraq. What gave Zarqawi the opportunity to create al Qaeda in Iraq? It was, of course, George W. Bush's decision to invade Iraq in 2003. Iraqi dictator Saddam Hussein brutally repressed all forms of opposition to his regime and before the Iraq War, al Qaeda had no presence in Iraq.

None of these factors can be easily ascribed to Obama or to Clinton, although certainly they did preside over the withdrawal of all US troops from Iraq at the end of 2011, a plan bequeathed to them by Bush.

Could the Obama administration have fought harder to retain a US troop presence in Iraq? Possibly.

Would the Iraqi government have allowed such a presence? Not likely as the Iraq government was by the time of the U.S. withdrawal more aligned with Iran than with the United States.

Obama was also slow to recognize the threat posed by ISIS, famously referring to the Islamist terrorist group gaining ground in Syria and Iraq as a "JV" team in a January 2014 interview.

Could the Obama administration do more to counter ISIS? Yes. In northern Syria, a no-fly zone targeted at Assad's air force and safe zones for refugees fleeing the fighting would help halt the massive flow of refugees out of Syria, many of whom are heading to Europe, and also help reduce whatever remains of ISIS' appeal to the Sunni population of Syria, to whom it can pose as their defenders against the Syrian regime.

It is Hillary Clinton, not Donald Trump, who is advocating for this common sense measure, which is a policy that could actually help tamp down the violence and killing in which ISIS thrives. Of course, Trump has no real policy ideas about what to do about ISIS because he seems incapable of producing them, other than empty slogans like he would "bomb the s...t out of" ISIS.

As Petraeus acidly observed in June, "I think carpet bombing is an absolutely tremendous idea if the enemy accommodates you by laying himself out like a carpet in the middle of the desert, without any civilians or infrastructure around him. Sadly, the Islamic State has learned that that is a losing proposition and does not accommodate us in that way. In fact, the Islamic State is very much underground now in places like Mosul and Raqqa."