[14] Let k be any partial computable function in the effective enumeration 1, 2,... . Let x, y, z be arbitrary elements of N .

[14] Let φk be any partial computable function in the effective enumeration φ1, φ2,... . Let x, y, z be arbitrary elements of N . Prove the following:

(a) C(φk(x)|y) ≤ C(x|y)+2l(k) + O(1).

(b) C(y|φk(x)) ≥ C(y|x) − 2l(k) + O(1).

Assume that φk is also one-to-one. Show that

(c) |C(x) − C(φk(x))| ≤ 2l(k) + O(1).

(d) C(x|y, z) ≤ C(x|φk(y), z)+2l(k) + O(1).