According to new legal docs, obtained by TMZ, prosecutors in New Mexico say they intend to introduce evidence Alec was negligent with a firearm on set even before cinematographer Halyna Hutchins was shot and killed with a gun Alec was holding.

The prosecution claims at one point during filming, Alec recklessly pointed and fired a blank round at a crew member while using the crew member as a line of sight.

What's more, prosecutors claim they have evidence Alec would use the gun as a pointer when directing crew members. They also say he fired the revolver after filming was over and "cut" was called ... and they go on to allege he filmed several scenes where he placed his finger on the trigger, even when the scene didn't call for gunfire.

The prosecution also says they have evidence that shows Alec would rush the armorer to reload ... and would rush crew members to work at a faster place as well. They claim he was inattentive when armorer Hannah Gutierrez-Reed was giving firearm training sessions, and would text or FaceTime his family instead of paying attention.

Alec is also accused of engaging in horseplay with the gun and displaying erratic and aggressive behavior on set.

Following Halyna's tragic shooting, when "Rust" resumed filming and set up shop in Montana, prosecutors claim Alec was insistent he not be required to follow safety recommendations.

Remember ... Alec is supposed to show up in a New Mexico courthouse for the start of his trial in July, when he will face one count of involuntary manslaughter. He's been trying to have the case against him tossed -- but hasn't had any luck.