General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

the axiom says that any object that is not represented by a constructor expression is not a member
of N1 and L[N].
Operations on Lists: Append
The append function @: L[N] × L[N] → L[N] concatenates lists
Defining equations: nil@l = l and cons(n, l)@r = cons(n, l@r)
Example 119 [3, 2, 1]@[1, 2] = [3, 2, 1, 1, 2] and []@[1, 2, 3] = [1, 2, 3] = [1, 2, 3]@[]
Lemma 120 For all l, r ∈ L[N], there is exactly one s ∈ L[N] with s = l@r.
Proof: by induction on l. (what does this mean?)
P.1 we have two cases
P.1.1 base case: l = nil:
must have s = r.
P.1.2 step case: l = cons(n, k) for some list k:
P.1.2.1 Assume that here is a unique s ′ with s ′ = k@r,
P.1.2.2 then s = cons(n, k)@r = cons(n, k@r) = cons(n, s ′ ).
Corollary 121 Append is a function (see, this just worked fine!)
c○: Michael Kohlhase 79
You should have noticed that this proof looks exactly like the one for addition. In fact, wherever
we have used an axiom P i there, we have used an axiom LPi here. It seems that we can do
anything we could for unary natural numbers for lists now, in particular, programming by recursive
equations.
Operations on Lists: more examples
Definition 122 λ(nil) = o and λ(cons(n, l)) = s(λ(l))
Definition 123 ρ(nil) = nil and ρ(cons(n, l)) = ρ(l)@cons(n, nil).
c○: Michael Kohlhase 80
Now, we have seen that “inductively defined sets” are a basis for computation, we will turn to the
programming language see them at work in concrete setting.
2.3.3 Inductively Defined Sets in SML
We are about to introduce one of the most powerful aspects of SML, its ability to define data
types. After all, we have claimed that types in SML are first-class objects, so we have to have a
means of constructing them.
We have seen above, that the main feature of an inductively defined set is that it has Peano
Axioms that enable us to use it for computation. Note that specifying them, we only need to
know the constructors (and their types). Therefore the datatype constructor in SML only needs
to specify this information as well. Moreover, note that if we have a set of constructors of an
inductively defined set — e.g. zero : mynat and suc : mynat -> mynat for the set mynat, then
their codomain type is always the same: mynat. Therefore, we can condense the syntax even
further by leaving that implicit.
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