* Step 1: DependencyPairs WORST_CASE(?,O(1)) + Considered Problem: - Strict TRS: f(k(a()),k(b()),X) -> f(X,X,X) g(X) -> u(h(X),h(X),X) h(d()) -> c(a()) h(d()) -> c(b()) u(d(),c(Y),X) -> k(Y) - Signature: {f/3,g/1,h/1,u/3} / {a/0,b/0,c/1,d/0,k/1} - Obligation: innermost runtime complexity wrt. defined symbols {f,g,h,u} and constructors {a,b,c,d,k} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs f#(k(a()),k(b()),X) -> c_1(f#(X,X,X)) g#(X) -> c_2(u#(h(X),h(X),X),h#(X),h#(X)) h#(d()) -> c_3() h#(d()) -> c_4() u#(d(),c(Y),X) -> c_5() Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: f#(k(a()),k(b()),X) -> c_1(f#(X,X,X)) g#(X) -> c_2(u#(h(X),h(X),X),h#(X),h#(X)) h#(d()) -> c_3() h#(d()) -> c_4() u#(d(),c(Y),X) -> c_5() - Weak TRS: f(k(a()),k(b()),X) -> f(X,X,X) g(X) -> u(h(X),h(X),X) h(d()) -> c(a()) h(d()) -> c(b()) u(d(),c(Y),X) -> k(Y) - Signature: {f/3,g/1,h/1,u/3,f#/3,g#/1,h#/1,u#/3} / {a/0,b/0,c/1,d/0,k/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#,h#,u#} and constructors {a,b,c,d,k} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: h(d()) -> c(a()) h(d()) -> c(b()) f#(k(a()),k(b()),X) -> c_1(f#(X,X,X)) g#(X) -> c_2(u#(h(X),h(X),X),h#(X),h#(X)) h#(d()) -> c_3() h#(d()) -> c_4() u#(d(),c(Y),X) -> c_5() * Step 3: Trivial WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: f#(k(a()),k(b()),X) -> c_1(f#(X,X,X)) g#(X) -> c_2(u#(h(X),h(X),X),h#(X),h#(X)) h#(d()) -> c_3() h#(d()) -> c_4() u#(d(),c(Y),X) -> c_5() - Weak TRS: h(d()) -> c(a()) h(d()) -> c(b()) - Signature: {f/3,g/1,h/1,u/3,f#/3,g#/1,h#/1,u#/3} / {a/0,b/0,c/1,d/0,k/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#,h#,u#} and constructors {a,b,c,d,k} + Applied Processor: Trivial + Details: Consider the dependency graph 1:S:f#(k(a()),k(b()),X) -> c_1(f#(X,X,X)) 2:S:g#(X) -> c_2(u#(h(X),h(X),X),h#(X),h#(X)) -->_1 u#(d(),c(Y),X) -> c_5():5 -->_3 h#(d()) -> c_4():4 -->_2 h#(d()) -> c_4():4 -->_3 h#(d()) -> c_3():3 -->_2 h#(d()) -> c_3():3 3:S:h#(d()) -> c_3() 4:S:h#(d()) -> c_4() 5:S:u#(d(),c(Y),X) -> c_5() The dependency graph contains no loops, we remove all dependency pairs. * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: h(d()) -> c(a()) h(d()) -> c(b()) - Signature: {f/3,g/1,h/1,u/3,f#/3,g#/1,h#/1,u#/3} / {a/0,b/0,c/1,d/0,k/1,c_1/1,c_2/3,c_3/0,c_4/0,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#,h#,u#} and constructors {a,b,c,d,k} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(1))