Needed Narrowing in Prolog

– [root property] The minimum element, referred to as the root, of T is f(X1, . . . , Xn), where X1, . . . , Xn are distinct variables. – [leaves property] The maximal elements, referred to as the leaves, of T are all and only (variants of) the left hand sides of the rules defining f . Non-maximal elements of T are referred to as branches. – [parent property] If π is a pattern of T different from the root, there exists in T a unique pattern π′ strictly preceding π such that there exists no other pattern strictly between π and π′. π′ is referred to as the parent of π and π as a child of π′. – [induction property] All the children of a same parent differ from each other only at the position of a variable, referred to as inductive, of their parent.