Re: Kate Stewart in Day of the Doctor (SPOILERS!)
I'm personally not all that convinced Kate was lying in TPOT; I reckon she genuinely hadn't met Eleven before then. But then we also know that history can be changed. Perhaps in a subsequent adventure, the Doctor went back in time and met Kate earlier in her personal timeline, explaining how she might recognize him in DOTD?
That's your explanation right there. In Doctor Who, history is ALWAYS changing. But, the BBC can't afford to reshoot the episodes to reflect the altered timeline...
Batman does not eat nachos.