propositions as types
This week, we will continue our type series by revisiting dependent types, this time in the context of type theory and theorem proving. In previous post type systems and dependent types, we mentioned agda as an example of a dependently typed language. One of the uses of agda — if not the primary use — is theorem proving. How can a language — which after a quick syntax examination somewhat resembles haskell — be used to write proofs? In this post, we will build a correspondence between mathematical proofs and computer programs using the Curry-Howard isomorphism. The isomorphism contains two parts, which will be the start of our mapping and that is the interpretation of propositions as types and proofs as programs (written in the lambda calculus syntax). Under this interpretation, to write a proof of a proposition (mathematical statement) is to write a program of a corresponding type, leading to proofs that can be run.
To start mapping logic into types, we will first consider logical connectives. We shall start with implication. Here, a good first step is to look at implication, , as an object that can be used in the rule known as modus ponens, i.e. an object which produces once is provided. This optic naturally leads to a type based on the suggestive symbol, the function type . The mentioned modus ponens rule becomes a simple function application:
To prove we need to supply and to prove from , we simply write .
Other common connectives are logical conjunction and disjunction . By conjunction we mean that both and hold. To prove said statement, one simply proves , then proves . If we transform this process into types, one needs to provide a term , a term , then package these terms into a box capable of holding all the necessary data. This naturally leads to a type of a container capable of holding two terms of two types e.g. a pair type .
To create a proof of one simply constructs a pair from and . The standard implications and correspond to projections and .
By disjunction we mean that at least one of and holds. Here, it is enough, to prove either or . This can be mapped to the type of a coproduct (e.g. Either in haskell), sometimes called tagged or discriminated union. In haskell, the tag is implemented through value constructors and , a common names for the injective constructors used in type theory are and . These correspond to the tautologies and .
Another aspect of logic we need to encode into types are constants, namely true and false . Let us start by finding the true type. The true type should be trivially provable, meaning producing a term of said type ‘should not require any work’ and the type should not have any further structure — the type should have a single accessible element:
We map true to the (unit) type (eg. in haskell) with the single nullary constructor .
To map false, we need a type that has no elements as it should not be provable. This can be achieved by an empty type, which we will denote (sometimes, the name
voidis used, but it does not correspond to the notion of
voidused in C and C++ — to avoid confusion, we will use ). Here, an interesting observation is that ex falso quodlibet (from falsehood, anything follows) corresponds to absurd functions , which exist for all . Once you have a term , you can — through the corresponding absurd function — get a term of any type.
We would like to be able to state that something is not true. In logic, this of course corresponds to negation, as stating is the same as stating that is not true.
So far, we have been avoiding a particular logic, but this is a good point to make a choice: from now on, we will use intuitionistic (constructive) logic, which uses as its definition of negation (i.e. it is a rewrite/derivation rule – you might have noticed that we have scrupulously avoided truth tables in our operator definitions). In other words:
To prove a negation , one needs to produce a proof which, if we also assume , leads to a contradiction (this kind of proof is in constructive mathematics unsurprisingly called proof of negation).
By mapping to our function type and (false) to , we get the type . How do we write down a function that returns an element of , which is a type with no elements? One way to prove negation is to produce an element of a type we know leads to a contradiction. Suppose, you want to prove, assuming . The proof can be easily written in terms of the following lemma :
which is inhabited by the following term:
Which is nothing else but (specialised) function composition; then clearly
as desired. Our current state of affairs is hence this:
Let us quickly check that the above mapping is consistent with standard behaviour of logic. We know, that is . In the type realm, this corresponds to a claim that ‘is’ . But what does ‘is’ mean?
For our purposes, it can be defined as an isomorphism1 . To state then requires a pair of functions and which are inverses of each other. To show , for instance, we construct the following pair of inverse functions:
- is the function which defines ,
- is the absurd function for .
- is the first projection from the product,
- is (again) the absurd function, this time for .
Another example is . Here, the argument is somewhat less straightforward. Consider the absurd function again. To be used, such a function would need to be given an element of type . Since there is no such element, it will never be applied, meaning the right-hand side does not matter – extensionally (point-wise), all possible definitions are the same function.
Thus types of these (absurd) functions are contractible (roughly speaking, a type is contractible if there is an element and all elements are ‘equal’ to ). Perhaps more importantly, they are isomorphic to the unit type (since all their inhabiting functions are the same). In a stroke of luck, this is exactly the type which corresponds to (true) in our established mapping.
Let's look at another property of implication: . Here, we are producing a function but all functions of this type are just constant functions which return , meaning once again, the type is contractible and thus isomorphic to (unit) – the type-theoretic version of true.
As a final example, we shall consider the proof of a more complicated statement:
which, rewritten as a type, becomes:
- assume ,
- we need some ,
- which we can define using a case split2 as:
- if for some then ,
- if for some then .
Equality of types is a topic rather complicated and heavily discussed in type theories such as HoTT. We think, that for the illustrative purposes isomorphism will do.
A case split is a term of type constructed from three sub-terms: , and . The meaning is almost obvious given the types: ‘take apart and apply either or to get the result’ or in other words ‘give me a recipe to build a from , a recipe to build a from and I will give you a recipe for building a from ’.
So far so good, but more importantly so far so non-dependent. Where do dependent types fit into this? One feature of first-order logic that we did not deal with yet are quantifiers: the universal and the existential .
Let us start by considering the former. We would like the quantifier to express the notion ‘a proposition about a value holds no matter the value we choose’. Since propositions correspond to types, to make a statement about a value, one needs to have a type which can incorporate values: a dependent type.
Let us recall, that a type family is a family of types where . We say that is indexed by . An example of a type family is the family of integer vectors types indexed by , where is a type of vectors of length e.g. .
To find the mapping of the universal quantifier, we use the idea that to prove a statement where is some predicate on , we need to be able to prove the statement about no matter the you ‘get’. This seems an awful lot like a function, and more precisely a dependent function: we need to produce for every argument .
There are two commonly used notations for dependent functions:
- – called a pi type in a type-theoretic setting,
- – as used in Agda.
- and are type families indexed by natural numbers,
is inhabited iff
- we can take if for some ,
- is inhabited iff is odd (same idea as above).
Another example can be