3.4 "Propositions are types, so true is a type, but one that inhabits Prop; and it has one constant constructor and that’s the one and only proof, intro.That’s it!" --> Are you saying that the proposition can be assigned to its one and only proof? But aren't they technically different types? Or are both Props (or maybe both are in the overarching class of Type?) But both being in the same class would not make sense as it's a hierarchy of universes with different types as I understand it! #30
A proof of a proposition has the type of that specific proposition. So true.intro is of type true using and introduction rules with X and Y would give a term of type X ∧ Y. Neither term, however is of type Prop but yes the type of true or X ∧ Y is Prop
It's also worth mentioning that propositions may have multiple proofs. Or they might not have any proofs at all in the case of false or 0=1 so there are no terms of type false or 0=1
Here's how I would clarify it:
true.intro
is of typetrue
buttrue
is of typeProp
A proof of a proposition has the type of that specific proposition. So true.intro is of type
true
using and introduction rules with X and Y would give a term of typeX ∧ Y
. Neither term, however is of typeProp
but yes the type oftrue
orX ∧ Y
isProp
It's also worth mentioning that propositions may have multiple proofs. Or they might not have any proofs at all in the case of
false
or0=1
so there are no terms of typefalse
or0=1