Oh mars certainly had a much larger and thicker atmosphere.
It is believed that mars looks the way it does today due to the fact that its core slowed down, thus removing its natural electromagnetic field. This allowed the sun's radiation to destroy its (supposed) water and atmosphere.
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Didn't think of how the atmosphere was stripped, but you did a good job of explaining the most likely scenario. I would've just said something stupid, but say it eloquently to confuse people. By the way, Mars does have frozen water in its canyons. The underground water is an educated guess.