[TYPES] strict unit type

[Vladimir Voevodsky]

Hello,

let Pt be the unit type ("one point"). In Coq-like systems it is introduced through an eliminator and the associated iota-reduction rule. This is in practice sufficient for proving all its expected properties modulo propositional equality (or identity types) but does not create a terminal object in the category of contexts. 

I wonder if any one has considered type theories where Pt is introduced as having a distinguished object tt together with reduction rules of the following form (in the absence of dependent sums, otherwise one needs more):

1. any object of Pt other than tt reduces to tt,

2. Prod x : T , Pt reduces to Pt,

3. Prod x : Pt , T reduces to T.

Thanks,
Vladimir.