Question

I mean, they're registered in the machine.

Answer 1

Im assuming this is R/S since you say trainers eyes because its called match call in emerald. Hide this or comment if you mean emerald.

• You never rebattle gym leaders in R/S so they cannot be rebattled through this function
• For E4 members and the just go to ever grande city and enter the Pokemon league and battle them again. They are not battled through the pokenav.
• For trainer eyes, the PokéNav will keep track of 69 unique Trainers and alert the user when they want a rematch. The only requirement is that the player must have obtained five Badges first. There exists a hidden game mechanic that means the more often the player steps on to the route or cave where any of these unique Trainers are located, the greater chance the opponent Trainer in that area will be ready for a rematch. After a certain number of steps, there is a chance that a Trainer will be ready for a rematch. So just walk around the trainer that you want to battle and this will increase the chance that they want to battle.
  Source: http://bulbapedia.bulbagarden.net/wiki/Pokenav

Comments

Huh, they're registered in the PokeNav. Is it just for proof that you beat 'em?

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yeah sorta and also gives a little info on the gym leader but is this question for R/S?

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Yup, Ruby/Sapphire