Obfuscation of Abstract Data-Types - Rowan

Appendix A
List assertion
We want to prove that, for all finite lists xs and ys,
|xs + ys| = |xs| + |ys|
(A.1)
which is one of our assertions for the list data-type (Figure 4.1). We prove this
assertion for the different concatenation operations defined in Chapter 4.
A.1 Unobfuscated version
For standard lists, we define + as follows:
[ ] + ys = ys
(x : xs) + ys = x : (xs + ys)
We prove Equation (A.1) by structural induction on xs.
Base Case Suppose that xs = [ ]. Then
|xs + ys|
= {definition of xs}
|[ ] + ys|
= {definition of +}
|ys|
= {arithmetic}
0 + |ys|
= {definition of | |}
|[ ]| + |ys|
= {definition of xs}
|xs| + |ys|
Step Case Suppose that xs = t : ts and that ts satisfies Equation (A.1).
Then
|xs + ys|
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