Holes in Lau's story (Spoiler)

In the flashback scene where we see a version of events that transpired through the lie Lau tells the police, we see Yan get shot by Inspector Billy who uses a 9mm pistol. Then we see Billy get shot by Lau who used a .38 special revolver. The true events that transpired involved Yan being shot by Billy who used a .38 special revolver(Yan used a 9mm pistol to take Lau hostage). Afterwards Billy gives Lau Yan's 9mm pistol and it Lau uses it to kill Billy. A ballistics report would have shot down Lau's story yet it appears that there was no ballistics report done by the police which would normally be done in any investigation like this.Was there a mistake in the writing of this movie or am I missing something here?