For reference, here is the equation that we must show is false.
cat(h : t, u : v) = h : (u : (cat(t, v)))
There are many counterexamples. How about h = 1, t = [2], u = 3 and v = [4]. The left-hand side of the equation is:
| cat(1 : [2], 3 : [4]) | |||
| = cat([1, 2], [3, 4]) | |||
| = [1, 2, 3, 4]. | from the definition of what cat does | ||
The right-hand side of the equation is
| 1 : (3 : (cat([2], [4]))) | |||
| = 1 : (3 : ([2, 4])) from the definition of cat | |||
| = 1 : [3, 2, 4] | since 3 : [2, 4] = [3, 2, 4] | ||
| = [1, 3, 2, 4] | since 1 : [3, 2, 4] = [1, 3, 2, 4] | ||
The two sides are not the same for this example. So they are certainly not always the same.