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