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