/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: f(0(),x2) -> 0() f(S(x),0()) -> 0() f(S(x'),S(x)) -> h(g(x',S(x)),f(S(S(x')),x)) g(0(),x2) -> 0() g(S(x),0()) -> 0() g(S(x),S(x')) -> h(f(S(x),S(x')),g(x,S(S(x')))) h(0(),0()) -> 0() h(0(),S(x)) -> h(0(),x) h(S(x),x2) -> h(x,x2) - Signature: {f/2,g/2,h/2} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,S} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: f(0(),x2) -> 0() f(S(x),0()) -> 0() f(S(x'),S(x)) -> h(g(x',S(x)),f(S(S(x')),x)) g(0(),x2) -> 0() g(S(x),0()) -> 0() g(S(x),S(x')) -> h(f(S(x),S(x')),g(x,S(S(x')))) h(0(),0()) -> 0() h(0(),S(x)) -> h(0(),x) h(S(x),x2) -> h(x,x2) - Signature: {f/2,g/2,h/2} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,S} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: f(0(),x2) -> 0() f(S(x),0()) -> 0() f(S(x'),S(x)) -> h(g(x',S(x)),f(S(S(x')),x)) g(0(),x2) -> 0() g(S(x),0()) -> 0() g(S(x),S(x')) -> h(f(S(x),S(x')),g(x,S(S(x')))) h(0(),0()) -> 0() h(0(),S(x)) -> h(0(),x) h(S(x),x2) -> h(x,x2) - Signature: {f/2,g/2,h/2} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,S} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: f(S(x),y){y -> S(y)} = f(S(x),S(y)) ->^+ h(g(x,S(y)),f(S(S(x)),y)) = C[f(S(S(x)),y) = f(S(x),y){x -> S(x)}] ** Step 1.b:1: DependencyPairs. MAYBE + Considered Problem: - Strict TRS: f(0(),x2) -> 0() f(S(x),0()) -> 0() f(S(x'),S(x)) -> h(g(x',S(x)),f(S(S(x')),x)) g(0(),x2) -> 0() g(S(x),0()) -> 0() g(S(x),S(x')) -> h(f(S(x),S(x')),g(x,S(S(x')))) h(0(),0()) -> 0() h(0(),S(x)) -> h(0(),x) h(S(x),x2) -> h(x,x2) - Signature: {f/2,g/2,h/2} / {0/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,S} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs f#(0(),x2) -> c_1() f#(S(x),0()) -> c_2() f#(S(x'),S(x)) -> c_3(h#(g(x',S(x)),f(S(S(x')),x)),g#(x',S(x)),f#(S(S(x')),x)) g#(0(),x2) -> c_4() g#(S(x),0()) -> c_5() g#(S(x),S(x')) -> c_6(h#(f(S(x),S(x')),g(x,S(S(x')))),f#(S(x),S(x')),g#(x,S(S(x')))) h#(0(),0()) -> c_7() h#(0(),S(x)) -> c_8(h#(0(),x)) h#(S(x),x2) -> c_9(h#(x,x2)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. MAYBE + Considered Problem: - Strict DPs: f#(0(),x2) -> c_1() f#(S(x),0()) -> c_2() f#(S(x'),S(x)) -> c_3(h#(g(x',S(x)),f(S(S(x')),x)),g#(x',S(x)),f#(S(S(x')),x)) g#(0(),x2) -> c_4() g#(S(x),0()) -> c_5() g#(S(x),S(x')) -> c_6(h#(f(S(x),S(x')),g(x,S(S(x')))),f#(S(x),S(x')),g#(x,S(S(x')))) h#(0(),0()) -> c_7() h#(0(),S(x)) -> c_8(h#(0(),x)) h#(S(x),x2) -> c_9(h#(x,x2)) - Weak TRS: f(0(),x2) -> 0() f(S(x),0()) -> 0() f(S(x'),S(x)) -> h(g(x',S(x)),f(S(S(x')),x)) g(0(),x2) -> 0() g(S(x),0()) -> 0() g(S(x),S(x')) -> h(f(S(x),S(x')),g(x,S(S(x')))) h(0(),0()) -> 0() h(0(),S(x)) -> h(0(),x) h(S(x),x2) -> h(x,x2) - Signature: {f/2,g/2,h/2,f#/2,g#/2,h#/2} / {0/0,S/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/3,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,S} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,4,5,7} by application of Pre({1,2,4,5,7}) = {3,6,8,9}. Here rules are labelled as follows: 1: f#(0(),x2) -> c_1() 2: f#(S(x),0()) -> c_2() 3: f#(S(x'),S(x)) -> c_3(h#(g(x',S(x)),f(S(S(x')),x)),g#(x',S(x)),f#(S(S(x')),x)) 4: g#(0(),x2) -> c_4() 5: g#(S(x),0()) -> c_5() 6: g#(S(x),S(x')) -> c_6(h#(f(S(x),S(x')),g(x,S(S(x')))),f#(S(x),S(x')),g#(x,S(S(x')))) 7: h#(0(),0()) -> c_7() 8: h#(0(),S(x)) -> c_8(h#(0(),x)) 9: h#(S(x),x2) -> c_9(h#(x,x2)) ** Step 1.b:3: RemoveWeakSuffixes. MAYBE + Considered Problem: - Strict DPs: f#(S(x'),S(x)) -> c_3(h#(g(x',S(x)),f(S(S(x')),x)),g#(x',S(x)),f#(S(S(x')),x)) g#(S(x),S(x')) -> c_6(h#(f(S(x),S(x')),g(x,S(S(x')))),f#(S(x),S(x')),g#(x,S(S(x')))) h#(0(),S(x)) -> c_8(h#(0(),x)) h#(S(x),x2) -> c_9(h#(x,x2)) - Weak DPs: f#(0(),x2) -> c_1() f#(S(x),0()) -> c_2() g#(0(),x2) -> c_4() g#(S(x),0()) -> c_5() h#(0(),0()) -> c_7() - Weak TRS: f(0(),x2) -> 0() f(S(x),0()) -> 0() f(S(x'),S(x)) -> h(g(x',S(x)),f(S(S(x')),x)) g(0(),x2) -> 0() g(S(x),0()) -> 0() g(S(x),S(x')) -> h(f(S(x),S(x')),g(x,S(S(x')))) h(0(),0()) -> 0() h(0(),S(x)) -> h(0(),x) h(S(x),x2) -> h(x,x2) - Signature: {f/2,g/2,h/2,f#/2,g#/2,h#/2} / {0/0,S/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/3,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,S} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:f#(S(x'),S(x)) -> c_3(h#(g(x',S(x)),f(S(S(x')),x)),g#(x',S(x)),f#(S(S(x')),x)) -->_1 h#(S(x),x2) -> c_9(h#(x,x2)):4 -->_1 h#(0(),S(x)) -> c_8(h#(0(),x)):3 -->_2 g#(S(x),S(x')) -> c_6(h#(f(S(x),S(x')),g(x,S(S(x')))),f#(S(x),S(x')),g#(x,S(S(x')))):2 -->_1 h#(0(),0()) -> c_7():9 -->_2 g#(0(),x2) -> c_4():7 -->_3 f#(S(x),0()) -> c_2():6 -->_3 f#(S(x'),S(x)) -> c_3(h#(g(x',S(x)),f(S(S(x')),x)),g#(x',S(x)),f#(S(S(x')),x)):1 2:S:g#(S(x),S(x')) -> c_6(h#(f(S(x),S(x')),g(x,S(S(x')))),f#(S(x),S(x')),g#(x,S(S(x')))) -->_1 h#(S(x),x2) -> c_9(h#(x,x2)):4 -->_1 h#(0(),S(x)) -> c_8(h#(0(),x)):3 -->_1 h#(0(),0()) -> c_7():9 -->_3 g#(0(),x2) -> c_4():7 -->_3 g#(S(x),S(x')) -> c_6(h#(f(S(x),S(x')),g(x,S(S(x')))),f#(S(x),S(x')),g#(x,S(S(x')))):2 -->_2 f#(S(x'),S(x)) -> c_3(h#(g(x',S(x)),f(S(S(x')),x)),g#(x',S(x)),f#(S(S(x')),x)):1 3:S:h#(0(),S(x)) -> c_8(h#(0(),x)) -->_1 h#(0(),0()) -> c_7():9 -->_1 h#(0(),S(x)) -> c_8(h#(0(),x)):3 4:S:h#(S(x),x2) -> c_9(h#(x,x2)) -->_1 h#(0(),0()) -> c_7():9 -->_1 h#(S(x),x2) -> c_9(h#(x,x2)):4 -->_1 h#(0(),S(x)) -> c_8(h#(0(),x)):3 5:W:f#(0(),x2) -> c_1() 6:W:f#(S(x),0()) -> c_2() 7:W:g#(0(),x2) -> c_4() 8:W:g#(S(x),0()) -> c_5() 9:W:h#(0(),0()) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: g#(S(x),0()) -> c_5() 5: f#(0(),x2) -> c_1() 6: f#(S(x),0()) -> c_2() 7: g#(0(),x2) -> c_4() 9: h#(0(),0()) -> c_7() ** Step 1.b:4: UsableRules. MAYBE + Considered Problem: - Strict DPs: f#(S(x'),S(x)) -> c_3(h#(g(x',S(x)),f(S(S(x')),x)),g#(x',S(x)),f#(S(S(x')),x)) g#(S(x),S(x')) -> c_6(h#(f(S(x),S(x')),g(x,S(S(x')))),f#(S(x),S(x')),g#(x,S(S(x')))) h#(0(),S(x)) -> c_8(h#(0(),x)) h#(S(x),x2) -> c_9(h#(x,x2)) - Weak TRS: f(0(),x2) -> 0() f(S(x),0()) -> 0() f(S(x'),S(x)) -> h(g(x',S(x)),f(S(S(x')),x)) g(0(),x2) -> 0() g(S(x),0()) -> 0() g(S(x),S(x')) -> h(f(S(x),S(x')),g(x,S(S(x')))) h(0(),0()) -> 0() h(0(),S(x)) -> h(0(),x) h(S(x),x2) -> h(x,x2) - Signature: {f/2,g/2,h/2,f#/2,g#/2,h#/2} / {0/0,S/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/3,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,S} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: f(S(x),0()) -> 0() f(S(x'),S(x)) -> h(g(x',S(x)),f(S(S(x')),x)) g(0(),x2) -> 0() g(S(x),S(x')) -> h(f(S(x),S(x')),g(x,S(S(x')))) h(0(),0()) -> 0() h(0(),S(x)) -> h(0(),x) h(S(x),x2) -> h(x,x2) f#(S(x'),S(x)) -> c_3(h#(g(x',S(x)),f(S(S(x')),x)),g#(x',S(x)),f#(S(S(x')),x)) g#(S(x),S(x')) -> c_6(h#(f(S(x),S(x')),g(x,S(S(x')))),f#(S(x),S(x')),g#(x,S(S(x')))) h#(0(),S(x)) -> c_8(h#(0(),x)) h#(S(x),x2) -> c_9(h#(x,x2)) ** Step 1.b:5: DecomposeDG. MAYBE + Considered Problem: - Strict DPs: f#(S(x'),S(x)) -> c_3(h#(g(x',S(x)),f(S(S(x')),x)),g#(x',S(x)),f#(S(S(x')),x)) g#(S(x),S(x')) -> c_6(h#(f(S(x),S(x')),g(x,S(S(x')))),f#(S(x),S(x')),g#(x,S(S(x')))) h#(0(),S(x)) -> c_8(h#(0(),x)) h#(S(x),x2) -> c_9(h#(x,x2)) - Weak TRS: f(S(x),0()) -> 0() f(S(x'),S(x)) -> h(g(x',S(x)),f(S(S(x')),x)) g(0(),x2) -> 0() g(S(x),S(x')) -> h(f(S(x),S(x')),g(x,S(S(x')))) h(0(),0()) -> 0() h(0(),S(x)) -> h(0(),x) h(S(x),x2) -> h(x,x2) - Signature: {f/2,g/2,h/2,f#/2,g#/2,h#/2} / {0/0,S/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/3,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,S} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component f#(S(x'),S(x)) -> c_3(h#(g(x',S(x)),f(S(S(x')),x)),g#(x',S(x)),f#(S(S(x')),x)) g#(S(x),S(x')) -> c_6(h#(f(S(x),S(x')),g(x,S(S(x')))),f#(S(x),S(x')),g#(x,S(S(x')))) and a lower component h#(0(),S(x)) -> c_8(h#(0(),x)) h#(S(x),x2) -> c_9(h#(x,x2)) Further, following extension rules are added to the lower component. f#(S(x'),S(x)) -> f#(S(S(x')),x) f#(S(x'),S(x)) -> g#(x',S(x)) f#(S(x'),S(x)) -> h#(g(x',S(x)),f(S(S(x')),x)) g#(S(x),S(x')) -> f#(S(x),S(x')) g#(S(x),S(x')) -> g#(x,S(S(x'))) g#(S(x),S(x')) -> h#(f(S(x),S(x')),g(x,S(S(x')))) *** Step 1.b:5.a:1: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: f#(S(x'),S(x)) -> c_3(h#(g(x',S(x)),f(S(S(x')),x)),g#(x',S(x)),f#(S(S(x')),x)) g#(S(x),S(x')) -> c_6(h#(f(S(x),S(x')),g(x,S(S(x')))),f#(S(x),S(x')),g#(x,S(S(x')))) - Weak TRS: f(S(x),0()) -> 0() f(S(x'),S(x)) -> h(g(x',S(x)),f(S(S(x')),x)) g(0(),x2) -> 0() g(S(x),S(x')) -> h(f(S(x),S(x')),g(x,S(S(x')))) h(0(),0()) -> 0() h(0(),S(x)) -> h(0(),x) h(S(x),x2) -> h(x,x2) - Signature: {f/2,g/2,h/2,f#/2,g#/2,h#/2} / {0/0,S/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/3,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,S} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:f#(S(x'),S(x)) -> c_3(h#(g(x',S(x)),f(S(S(x')),x)),g#(x',S(x)),f#(S(S(x')),x)) -->_2 g#(S(x),S(x')) -> c_6(h#(f(S(x),S(x')),g(x,S(S(x')))),f#(S(x),S(x')),g#(x,S(S(x')))):2 -->_3 f#(S(x'),S(x)) -> c_3(h#(g(x',S(x)),f(S(S(x')),x)),g#(x',S(x)),f#(S(S(x')),x)):1 2:S:g#(S(x),S(x')) -> c_6(h#(f(S(x),S(x')),g(x,S(S(x')))),f#(S(x),S(x')),g#(x,S(S(x')))) -->_3 g#(S(x),S(x')) -> c_6(h#(f(S(x),S(x')),g(x,S(S(x')))),f#(S(x),S(x')),g#(x,S(S(x')))):2 -->_2 f#(S(x'),S(x)) -> c_3(h#(g(x',S(x)),f(S(S(x')),x)),g#(x',S(x)),f#(S(S(x')),x)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: f#(S(x'),S(x)) -> c_3(g#(x',S(x)),f#(S(S(x')),x)) g#(S(x),S(x')) -> c_6(f#(S(x),S(x')),g#(x,S(S(x')))) *** Step 1.b:5.a:2: UsableRules. MAYBE + Considered Problem: - Strict DPs: f#(S(x'),S(x)) -> c_3(g#(x',S(x)),f#(S(S(x')),x)) g#(S(x),S(x')) -> c_6(f#(S(x),S(x')),g#(x,S(S(x')))) - Weak TRS: f(S(x),0()) -> 0() f(S(x'),S(x)) -> h(g(x',S(x)),f(S(S(x')),x)) g(0(),x2) -> 0() g(S(x),S(x')) -> h(f(S(x),S(x')),g(x,S(S(x')))) h(0(),0()) -> 0() h(0(),S(x)) -> h(0(),x) h(S(x),x2) -> h(x,x2) - Signature: {f/2,g/2,h/2,f#/2,g#/2,h#/2} / {0/0,S/1,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,S} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: f#(S(x'),S(x)) -> c_3(g#(x',S(x)),f#(S(S(x')),x)) g#(S(x),S(x')) -> c_6(f#(S(x),S(x')),g#(x,S(S(x')))) *** Step 1.b:5.a:3: Failure MAYBE + Considered Problem: - Strict DPs: f#(S(x'),S(x)) -> c_3(g#(x',S(x)),f#(S(S(x')),x)) g#(S(x),S(x')) -> c_6(f#(S(x),S(x')),g#(x,S(S(x')))) - Signature: {f/2,g/2,h/2,f#/2,g#/2,h#/2} / {0/0,S/1,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,S} + Applied Processor: EmptyProcessor + Details: The problem is still open. *** Step 1.b:5.b:1: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: h#(0(),S(x)) -> c_8(h#(0(),x)) h#(S(x),x2) -> c_9(h#(x,x2)) - Weak DPs: f#(S(x'),S(x)) -> f#(S(S(x')),x) f#(S(x'),S(x)) -> g#(x',S(x)) f#(S(x'),S(x)) -> h#(g(x',S(x)),f(S(S(x')),x)) g#(S(x),S(x')) -> f#(S(x),S(x')) g#(S(x),S(x')) -> g#(x,S(S(x'))) g#(S(x),S(x')) -> h#(f(S(x),S(x')),g(x,S(S(x')))) - Weak TRS: f(S(x),0()) -> 0() f(S(x'),S(x)) -> h(g(x',S(x)),f(S(S(x')),x)) g(0(),x2) -> 0() g(S(x),S(x')) -> h(f(S(x),S(x')),g(x,S(S(x')))) h(0(),0()) -> 0() h(0(),S(x)) -> h(0(),x) h(S(x),x2) -> h(x,x2) - Signature: {f/2,g/2,h/2,f#/2,g#/2,h#/2} / {0/0,S/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/3,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,S} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(h) = {1,2}, uargs(h#) = {1,2}, uargs(c_8) = {1}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(S) = [1] x1 + [4] p(f) = [0] p(g) = [0] p(h) = [1] x1 + [1] x2 + [0] p(f#) = [1] x1 + [1] x2 + [0] p(g#) = [1] x1 + [1] x2 + [0] p(h#) = [1] x1 + [1] x2 + [4] p(c_1) = [0] p(c_2) = [1] p(c_3) = [2] x1 + [2] x2 + [2] p(c_4) = [4] p(c_5) = [2] p(c_6) = [1] x2 + [1] p(c_7) = [1] p(c_8) = [1] x1 + [1] p(c_9) = [1] x1 + [0] Following rules are strictly oriented: h#(0(),S(x)) = [1] x + [8] > [1] x + [5] = c_8(h#(0(),x)) h#(S(x),x2) = [1] x + [1] x2 + [8] > [1] x + [1] x2 + [4] = c_9(h#(x,x2)) Following rules are (at-least) weakly oriented: f#(S(x'),S(x)) = [1] x + [1] x' + [8] >= [1] x + [1] x' + [8] = f#(S(S(x')),x) f#(S(x'),S(x)) = [1] x + [1] x' + [8] >= [1] x + [1] x' + [4] = g#(x',S(x)) f#(S(x'),S(x)) = [1] x + [1] x' + [8] >= [4] = h#(g(x',S(x)),f(S(S(x')),x)) g#(S(x),S(x')) = [1] x + [1] x' + [8] >= [1] x + [1] x' + [8] = f#(S(x),S(x')) g#(S(x),S(x')) = [1] x + [1] x' + [8] >= [1] x + [1] x' + [8] = g#(x,S(S(x'))) g#(S(x),S(x')) = [1] x + [1] x' + [8] >= [4] = h#(f(S(x),S(x')),g(x,S(S(x')))) f(S(x),0()) = [0] >= [0] = 0() f(S(x'),S(x)) = [0] >= [0] = h(g(x',S(x)),f(S(S(x')),x)) g(0(),x2) = [0] >= [0] = 0() g(S(x),S(x')) = [0] >= [0] = h(f(S(x),S(x')),g(x,S(S(x')))) h(0(),0()) = [0] >= [0] = 0() h(0(),S(x)) = [1] x + [4] >= [1] x + [0] = h(0(),x) h(S(x),x2) = [1] x + [1] x2 + [4] >= [1] x + [1] x2 + [0] = h(x,x2) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:5.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f#(S(x'),S(x)) -> f#(S(S(x')),x) f#(S(x'),S(x)) -> g#(x',S(x)) f#(S(x'),S(x)) -> h#(g(x',S(x)),f(S(S(x')),x)) g#(S(x),S(x')) -> f#(S(x),S(x')) g#(S(x),S(x')) -> g#(x,S(S(x'))) g#(S(x),S(x')) -> h#(f(S(x),S(x')),g(x,S(S(x')))) h#(0(),S(x)) -> c_8(h#(0(),x)) h#(S(x),x2) -> c_9(h#(x,x2)) - Weak TRS: f(S(x),0()) -> 0() f(S(x'),S(x)) -> h(g(x',S(x)),f(S(S(x')),x)) g(0(),x2) -> 0() g(S(x),S(x')) -> h(f(S(x),S(x')),g(x,S(S(x')))) h(0(),0()) -> 0() h(0(),S(x)) -> h(0(),x) h(S(x),x2) -> h(x,x2) - Signature: {f/2,g/2,h/2,f#/2,g#/2,h#/2} / {0/0,S/1,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/3,c_7/0,c_8/1,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,S} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),?)