Another introduction to Martin-Löf’s Intuitionistic Type Theory

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 start by stating an axiom which introduces a constant for each type constructor in correspondence with each formation rule. For instance suppose we want to describe the type A → B; to this aim we will add the axiom ⊢→:1 (X :1 type)(Y :1 type) type. which means that→ is a type-constructor constant which gives a new type when applied to the types X and Y . The next step corresponds to the introduction rules: we will add a new axiom for each kind of canonical element. Let us consider again the case of the type A → B; then we put ⊢ λ :0 (X : type)(Y : type)(y : (x : X) Y ) X → Y. which states that, supposing X and Y be two types and y be a function from X to Y , λ(X,Y, y) is an element of the type X → Y . Also the elimination rule becomes a new axiom; it introduces the constant we used in the elimination rule. For instance for the type A → B we put ⊢ F :0 (X : type)(Y : type)(Z : (z : X → Y ) type) (c : X → Y )(d : (y : (x : X)Y )Z(λ(X,Y, y))) Z(c) which states that, supposing X and Y are two types, Z is a propositional function on elements of X → Y , c is an element of X → Y and d is a method which maps any function y from X to Y into an element of Z(λ(X,Y, y)), F (X,Y, c, d) is an element of Z(c). In this way any rule of ITT becomes an axiom of the multi-level typed λ-calculus. By means of the rules we have considered so far we can not deal with the equality rules of ITT. To deal also with them let us add some equality rules to our multi-level typed λ-calculus. We can start from the equality rules for the simple typed λ-calculus and enrich them by specifying the levels. var-equality Γ ⊢ x :i N Γ ⊢ x = x :i N const-equality Γ ⊢ c :i N Γ ⊢ c = c :i N app-equality Γ ⊢ K1 = K2 :i ((x :j N)M) Γ ⊢ L1 = L2 :j N Γ ⊢ K1(L1) = K2(L2) :i M [x := K1] ξ-equality Γ, x :j N ⊢ K = L :i M Γ ⊢ ((x :j N)K) = ((x :j N)L) :i ((x :j N)M) α-equality Γ, y :j N ⊢ K :i M Γ ⊢ ((y :j N)K) = ((x :j N)K[y := x]) :i ((y :j N)M) β-equality Γ, x :j N ⊢ K :i M Γ ⊢ L :j N Γ ⊢ ((x :j N)K)(L) = K[x := L] :i M [x := L] η-equality Γ ⊢ K :i ((x :j N)M) Γ ⊢ ((x :j N)K(x)) = K :i ((x :j N)M) where in the α-equality rule and in the η-equality rule we assume that x does not appear in K. Using these rules we will obtain all the rules of ITT which state that the type-constructor constants and the element-constructor constants are functions. On the other hand to deal with the equality rules characteristic of ITT we have to add specific axioms. For instance we add X : type, Y : type, b : (x : X) Y, d : (y : (x : X) Y )Z(λ(X,Y, y)) ⊢ F (λ(X,Y, b), d) = d(b) : Z(λ(X,Y, b)) if we want to deal with the equality rule for the type A → B. We can show that the multi-level typed λ-calculus satisfies all our requirements on a well-writing theory [V95]. Let us recall that the β-reduction relation between expressions is obtained extending in the obvious way the relation defined by ((x : N) K)(M) ⇒ K[x := M ] Then, provided we say that an expression is in normal form if it contains no sub-expression which can be β-reduced, we can prove the following theorem. Theorem A.1 (Normalization) Any expression can be reduced into an equivalent one in normal form. One of the consequences of this result is a theorem on the decidability of the equality between expressions. In fact, supposing to consider only the general rules on equality, we can prove the following theorem. Theorem A.2 (Decidability of equality) Given two expressions M and N of the same type, one can effectively decide if M and N are equal.