Artificial Intelligence A* algorithm proofs:

Fact 1: if h(n) <= h*(n) , A* terminates by finding an optimal path(if a solution exists)

Prove it.

Hint: look at start node, goal node, and heuristic merit of middle node f(n) =g(n) + h(n)

Fact 2: if A* uses h(n) <= h*(n) heuristic, then f is non-decreasing along any path.

Prove it.

Fact 3: if h satisfies the monotone restriction (MR) then whenever A* expands a node, it has already found an optimal path to this node.

Prove it.