Another introduction to Martin-Löf ’ s Intuitionistic Type Theory (original) (raw)

ion Γ, x :j N ⊢ K :i L Γ ⊢ ((x :j N)K) :i ((x :j N)L) application Γ ⊢ N :i ((x :j L)M) Γ ⊢ K :j L Γ ⊢ N(K) :i M [x := K] These rules by themselves are almost useless since no expression can be assigned a type because to prove the conclusion of a rule one should have already proved the premise(s). So in order to start we need some axiom. The first thing one has to do is to settle the maximum level (s)he wants to use when describing a particular theory; to this aim we will use the symbol ∗ to indicate the only type of the highest level. We can then define all the other types downward from ∗. In the case of ITT, the basic idea is to define a chain a :0 A :1 type :2 ∗ to mean that a is an element of A which is a type, i.e. an element of type, which is the only element of ∗. Thus our first axiom is: ⊢ type :2 ∗ We can now begin our description of ITT; to this aim we will follow the explanation we already used during the informal introduction of the rules in the previous sections. We star...