Answer to Question equation-1
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.