Why is it allowed to Generate Keys inside the game? I am studying reductions to prove security of crypto systems. Generally "games" are used for the proofs. For example, the next image was extracted from the page 91 of the book Post-Quantum Cryptography. Here, an adversary is constructed against Lamport-Diffie signature using a Forger with access to an oracle. 

My question is why is it permitted to the adversary to generate keys? I make this question because the step 1 of this Algorithm calls the key generation process.
The adversary is permitted to generate the keys

because we're not restricting the adversary's internals.

(For example, the adversary is also permitted to compute the parity of y,

and to generate a random element of {45,56,62,68}.)