Player A picks up the dribble with both feet on the ground. He then steps (jabs) with his right several time to the right. (This establishes the left foot as the pivot foot.) He then steps to the left with his right foot with the left (pivot) foot planted on the ground. He then lifts the left (pivot) foot while the right foot is on the ground, jumps off the right foot and shoots the ball before the pivot foot is returned to the floor.

Several of my friends say this is a travel, even though the rule states that once the pivot foot is established, the player may lift it as long as the player shoots or passes the ball before it returns to the floor.

The difference of opinion here between my friends and I is that, they say he can only lift the pivot foot if the other foot is lifted at the same time. (i.e. he can jump of both feet but not lift the pivot and then jump off the other foot).

Can you please clarify?

Its different than a layup. I pick up my dribble, pivot on left. Lift left foot (balancing one footed on right) and shoot of the right.