Here's a small theorem in mathematics:
Suppose u is not an element of A, and v is not an element of B, and f is an injective function from A to B. Let A' = A union {u} and B' = B union {v}, and define g: A' -> B' by g(x) = f(x) if x is in A, and g(u) = v. Then g is injective as well.
If I were writing OCaml-like code, I'd represent A and B as types, and f as an A->B function, something like
module type Q =
sig
type 'a
type 'b
val f: 'a -> 'b
end
and then define a functor
module Extend (M : Q) : Q =
struct
type a = OrdinaryA of M.a | ExoticA
type b = OrdinaryB of M.b | ExoticB
let f x = match x with
OrdinaryA t -> OrdinaryB ( M.f t)
| Exotic A -> ExoticB
end;;
and my theorem would be that if Q.f
is injective, then so is (Extend Q).f
, where I'm hoping I've gotten the syntax more or less correct.
I'd like to do the same thing in Isabelle/Isar. Normally, that'd mean writing something like
definition injective :: "('a ⇒ 'b) ⇒ bool"
where "injective f ⟷ ( ∀ P Q. (f(P) = f(Q)) ⟷ (P = Q))"
proposition: "injective f ⟹ injective (Q(f))"
and Q
is ... something. I don't know how to make, in Isabelle a single operation analogous to the functor Q
in OCaml that creates two new datatypes and a function between them. The proof of injectivity seems as if it'd be fairly straightforward --- merely a four-case split. But I'd like help defining the new function that I've called Q f
, given the function f
.