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Gravity compresses the atmosphere.

Because of the influence of gravity, the atmosphere is compressed to a certain extent. The atmosphere is not continually being compressed further.

Gasses under pressure generate heat.

Heat is generated during the compression process. Simply being under pressure (such as sitting in a pressure vessel or at the bottom of the atmosphere) does not generate heat.

The atmosphere does make the surface temperature higher than it would be otherwise (see Greenhouse effect), and when air is moved up and down in the atmosphere, heat is exchanged. But to say that the gravitational compression of the atmosphere increases the surface temperature is incorrect.