CHICAGO (CBS) — A war of words between Mayor Lori Lightfoot and U.S. Sen. Ted Cruz (R-Texas) continued on Twitter Tuesday – with Cruz firing back and insisting that the pro-gun rights policy he favors would be a better idea for Chicago.

After the mass shooting in West Texas, Cruz tweeted: a link to a Breitbart News article, which broke down the number of shootings in Chicago this Labor Day weekend and other recent weekends, as well as some details about the holiday weekend incidents – while repeatedly making reference to “Democrat-controlled” Chicago.

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With the link, Cruz tweeted: “Gun control doesn’t work. Look at Chicago. Disarming law-abiding citizens isn’t the answer.” He went on to call for stopping violent criminals before they commit more crimes.

Mayor Lightfoot in turn tweeted a bar graph with a statistic saying 60 percent of illegal firearms recovered in Chicago come outside Illinois. Addressing Cruz, Lightfoot said the guns come “mostly from states dominated by coward Republicans like you who refuse to enact commonsense gun legislation. Keep our name out of your mouth.”

In another tweet, Lightfoot continued to defend “common sense gun policies” and take Cruz and other Republican lawmakers to task for “dismissing” them.

As CBS 2’s Jim Williams reported, Lightfoot backed up her tweet on Tuesday with a tongue-lashing from the podium.

“For him to make a political point, or try to – I should say try to – make a political point out of the horrible tragedy that’s been unfolding in Texas is really offensive – not just to Chicago, but to the people in Texas who still and will be forever changed because of the violence there,” Lightfoot said.

Cruz responded with five tweets.

First, he wrote: “Mayor, your anger is misplaced. Virtually every year, Chicago has the most murders of any city in America. In 2017, the top five were: Chicago (650 murders), Baltimore (343), Philadelphia (317), New York (290), and Los Angeles (286).”

Multiple sources have pointed out that Chicago’s murder rate per capita is not near the top for the country – St. Louis came in first with 66.1 homicides per 100,000 people in 2017; Chicago was ninth with 24.1. But in terms of the total number of murders per city for 2017, Cruz’s numbers are correct.

Cruz went on: “All five of those cities have had Democratic mayors for decades and aggressive gun control policies—none seems to be working. Your explanation (‘blame Indiana!’) doesn’t hold water. There are 27 states with GOP governors. Chicago is not alone in having neighboring red states.

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“But Chicago’s murder totals are often DOUBLE the next closest city. So how do you explain the big cities in red states, without aggressive gun control laws, where murder rates are MUCH lower than Chicago?” he continued.

Cruz said Lightfoot should put aside “partisan talking points” and “failed gun-control policies,” instead calling for preventing “felons and fugitives” from illegally obtaining firearms. He said the city should “ALLOW law-abiding citizens (who are right now obeying Chicago’s prohibition) to exercise their constitutional right to protect their own lives and their families.”

Lightfoot in turn replied calling on Cruz and the Senate to support universal background checks and an assault weapons ban.

Republicans, including President Donald Trump, often point to Chicago in response to calls for tougher gun control.

President Trump two years ago:

“The city with the strongest gun laws in our nation is Chicago and Chicago is a disaster.”

But it should be noted that Chicagoans are allowed to obtain FOID cards and concealed carry licenses.

Chicago did once have an ordinance on the books banning handguns, but that was law was overturned in the McDonald v. Chicago U.S. Supreme Court ruling in 2010.

In 2012, the U.S. Seventh Circuit Court of Appeals also struck down a ban on carrying concealed weapons in Illinois – which at the time was the only state that had such a ban, but not anymore.

Meanwhile, Mayor Lightfoot is in favor of that a uniform, federal gun control law that covers all the states.

“We ought to try to come together and solve one of the core problems with gun violence, which is this patch work of systems in different states – different neighboring states having different systems – because the federal government hasn’t stepped up to do its job,” Lightfoot said.

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When asked if she’d be willing to meet with Senator Cruz, Lightfoot said she’d be willing to sit down with anyone who comes forward in good faith and wants to solve a problem.