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