# Is $\min(h_1(s),\ h_2(s))$ consistent?

If $$h_1(s)$$ is a consistent heuristic and $$h_2(s)$$ is a admissible heuristic, is $$\min(h_1(s),\ h_2(s))$$ consistent?

You can easily find a counterexample. Suppose that there are three nodes $$s$$, $$p$$, and $$goal$$ such that $$s \rightarrow p \rightarrow goal$$. The real cost of going from $$s$$ to $$p$$ is $$c(s,p) = 10$$ and $$c(p, goal) = 10$$. Also, $$h_1(s) = 18$$, $$h_1(p) = 9$$, $$h_1(goal) = 0$$, $$h_2(s) = 17$$, $$h_2(p) = 1$$.On the other hand, $$h^*(s) = 19$$ and $$h^*(p) = 10$$.
Now, $$h_1(s) \leqslant c(s,p) + h_1(p)$$ and $$h_1(s) \leqslant c(s, goal) + h_1(goal)$$ that satisfies the consistency constraint of $$h_1$$. Also, $$h_2(s) \leqslant h^*(s)$$ and $$h_2(s) \leqslant h^*(p)$$ that approve $$h_2$$ is admissible.
However, for any node $$n$$, if we define $$h_3(n) = \min(h_1(n), h_2(n))$$:
$$h_3(s) = \min(h_1(s), h_2(s)) = 17 \not\leqslant c(s,p) + \min(h_1(p), h_2(p)) =$$ $$c(s,p) + h_3(p) = 10 + 1 = 11.$$ It means that $$h_3$$ as a heuristic function is not consistent.