About the "lies." Punishing a child for a lie, especially one you don't know about first-hand, is tricky. Ignoring that it happened when you have it on good authority isn't a good idea, either.
Most of all, keep in mind that it may not be a lie. He may have convinced himself this is what happened. Don't try to trap him in his lie(t) and don't entice him to lie on top of a lie. Instead, tell him, "I talked to the teacher. She told me that this is what happened...." Pause. "Is that the way you remember it?" Maybe he'll remember it differently now, maybe he won't.
If he doesn't, tell him, "I'm not saying you made up a story. I'm not saying you lied. But in our family, we really value honesty and truthfulness." Period. Move on.
Ultimately, the message you want him to learn is that lying erodes trust. At another time removed from the first conversation, let him know that even if he feels badly about something he did, it's OK to tell you the truth.
There's potentially another piece of the story: If he doesn't change his story, it may be because he didn't lie, at least not in his head. That could just be a developmental response -- kids this age engage in magical thinking all the time -- but it's worth wondering what kind of coping mechanisms he has when life doesn't go his way. So Here's more food for thought:
Does he always get way at home? Do you "snow plow" problems away for him so he doesn't learn coping skills for dealing with disappointment? You know the Rolling Stones' line, "You can't always get what you want....." I used to sing that to my son when he was this age and would get unhappy. It diffused the tension not just because I have a bad voice but because it was the Rolling Stones saying what I wanted to say.
How do you model for him when you are dealing with a minor disappointment? Here's one strategy: Mutter to yourself, but loud enough for him to hear: "Oh boy. This isn't the way I wanted this to go. OK, gotta regroup. Let me see what my choices are....." The first time he overhears, he'll likely ignore you, but at some point, he's gonna say, "Mom, why are you talking to yourself." And you can say, in an almost distracted way, "You know, sometimes, when I'm annoyed at things, I talk to myself in my head about what I can do and sometimes I talk out loud."
Also, ask him directly, once the topic comes up: what do you do when you're disappointed, or you don't like the way things are going, like at school? See where the conversation goes.