MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) f(0()) -> 0() f(s(0())) -> s(0()) f(s(s(x))) -> +(p(g(x)),q(g(x))) f(s(s(x))) -> p(h(g(x))) g(0()) -> pair(s(0()),s(0())) g(s(x)) -> h(g(x)) g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x))) h(x) -> pair(+(p(x),q(x)),p(x)) p(pair(x,y)) -> x q(pair(x,y)) -> y - Signature: {+/2,f/1,g/1,h/1,p/1,q/1} / {0/0,pair/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,f,g,h,p,q} and constructors {0,pair,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs +#(x,0()) -> c_1() +#(x,s(y)) -> c_2(+#(x,y)) f#(0()) -> c_3() f#(s(0())) -> c_4() f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x)) f#(s(s(x))) -> c_6(p#(h(g(x))),h#(g(x)),g#(x)) g#(0()) -> c_7() g#(s(x)) -> c_8(h#(g(x)),g#(x)) g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)) h#(x) -> c_10(+#(p(x),q(x)),p#(x),q#(x),p#(x)) p#(pair(x,y)) -> c_11() q#(pair(x,y)) -> c_12() Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: +#(x,0()) -> c_1() +#(x,s(y)) -> c_2(+#(x,y)) f#(0()) -> c_3() f#(s(0())) -> c_4() f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x)) f#(s(s(x))) -> c_6(p#(h(g(x))),h#(g(x)),g#(x)) g#(0()) -> c_7() g#(s(x)) -> c_8(h#(g(x)),g#(x)) g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)) h#(x) -> c_10(+#(p(x),q(x)),p#(x),q#(x),p#(x)) p#(pair(x,y)) -> c_11() q#(pair(x,y)) -> c_12() - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) f(0()) -> 0() f(s(0())) -> s(0()) f(s(s(x))) -> +(p(g(x)),q(g(x))) f(s(s(x))) -> p(h(g(x))) g(0()) -> pair(s(0()),s(0())) g(s(x)) -> h(g(x)) g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x))) h(x) -> pair(+(p(x),q(x)),p(x)) p(pair(x,y)) -> x q(pair(x,y)) -> y - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/5 ,c_6/3,c_7/0,c_8/2,c_9/7,c_10/4,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),s(0())) g(s(x)) -> h(g(x)) g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x))) h(x) -> pair(+(p(x),q(x)),p(x)) p(pair(x,y)) -> x q(pair(x,y)) -> y +#(x,0()) -> c_1() +#(x,s(y)) -> c_2(+#(x,y)) f#(0()) -> c_3() f#(s(0())) -> c_4() f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x)) f#(s(s(x))) -> c_6(p#(h(g(x))),h#(g(x)),g#(x)) g#(0()) -> c_7() g#(s(x)) -> c_8(h#(g(x)),g#(x)) g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)) h#(x) -> c_10(+#(p(x),q(x)),p#(x),q#(x),p#(x)) p#(pair(x,y)) -> c_11() q#(pair(x,y)) -> c_12() * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: +#(x,0()) -> c_1() +#(x,s(y)) -> c_2(+#(x,y)) f#(0()) -> c_3() f#(s(0())) -> c_4() f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x)) f#(s(s(x))) -> c_6(p#(h(g(x))),h#(g(x)),g#(x)) g#(0()) -> c_7() g#(s(x)) -> c_8(h#(g(x)),g#(x)) g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)) h#(x) -> c_10(+#(p(x),q(x)),p#(x),q#(x),p#(x)) p#(pair(x,y)) -> c_11() q#(pair(x,y)) -> c_12() - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),s(0())) g(s(x)) -> h(g(x)) g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x))) h(x) -> pair(+(p(x),q(x)),p(x)) p(pair(x,y)) -> x q(pair(x,y)) -> y - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/5 ,c_6/3,c_7/0,c_8/2,c_9/7,c_10/4,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,4,7,11,12} by application of Pre({1,3,4,7,11,12}) = {2,5,6,8,9,10}. Here rules are labelled as follows: 1: +#(x,0()) -> c_1() 2: +#(x,s(y)) -> c_2(+#(x,y)) 3: f#(0()) -> c_3() 4: f#(s(0())) -> c_4() 5: f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x)) 6: f#(s(s(x))) -> c_6(p#(h(g(x))),h#(g(x)),g#(x)) 7: g#(0()) -> c_7() 8: g#(s(x)) -> c_8(h#(g(x)),g#(x)) 9: g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)) 10: h#(x) -> c_10(+#(p(x),q(x)),p#(x),q#(x),p#(x)) 11: p#(pair(x,y)) -> c_11() 12: q#(pair(x,y)) -> c_12() * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: +#(x,s(y)) -> c_2(+#(x,y)) f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x)) f#(s(s(x))) -> c_6(p#(h(g(x))),h#(g(x)),g#(x)) g#(s(x)) -> c_8(h#(g(x)),g#(x)) g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)) h#(x) -> c_10(+#(p(x),q(x)),p#(x),q#(x),p#(x)) - Weak DPs: +#(x,0()) -> c_1() f#(0()) -> c_3() f#(s(0())) -> c_4() g#(0()) -> c_7() p#(pair(x,y)) -> c_11() q#(pair(x,y)) -> c_12() - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),s(0())) g(s(x)) -> h(g(x)) g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x))) h(x) -> pair(+(p(x),q(x)),p(x)) p(pair(x,y)) -> x q(pair(x,y)) -> y - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/5 ,c_6/3,c_7/0,c_8/2,c_9/7,c_10/4,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:+#(x,s(y)) -> c_2(+#(x,y)) -->_1 +#(x,0()) -> c_1():7 -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1 2:S:f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x)) -->_5 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)):5 -->_3 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)):5 -->_5 g#(s(x)) -> c_8(h#(g(x)),g#(x)):4 -->_3 g#(s(x)) -> c_8(h#(g(x)),g#(x)):4 -->_4 q#(pair(x,y)) -> c_12():12 -->_2 p#(pair(x,y)) -> c_11():11 -->_5 g#(0()) -> c_7():10 -->_3 g#(0()) -> c_7():10 -->_1 +#(x,0()) -> c_1():7 -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1 3:S:f#(s(s(x))) -> c_6(p#(h(g(x))),h#(g(x)),g#(x)) -->_2 h#(x) -> c_10(+#(p(x),q(x)),p#(x),q#(x),p#(x)):6 -->_3 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)):5 -->_3 g#(s(x)) -> c_8(h#(g(x)),g#(x)):4 -->_1 p#(pair(x,y)) -> c_11():11 -->_3 g#(0()) -> c_7():10 4:S:g#(s(x)) -> c_8(h#(g(x)),g#(x)) -->_1 h#(x) -> c_10(+#(p(x),q(x)),p#(x),q#(x),p#(x)):6 -->_2 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)):5 -->_2 g#(0()) -> c_7():10 -->_2 g#(s(x)) -> c_8(h#(g(x)),g#(x)):4 5:S:g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)) -->_4 q#(pair(x,y)) -> c_12():12 -->_6 p#(pair(x,y)) -> c_11():11 -->_2 p#(pair(x,y)) -> c_11():11 -->_7 g#(0()) -> c_7():10 -->_5 g#(0()) -> c_7():10 -->_3 g#(0()) -> c_7():10 -->_1 +#(x,0()) -> c_1():7 -->_7 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)):5 -->_5 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)):5 -->_3 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)):5 -->_7 g#(s(x)) -> c_8(h#(g(x)),g#(x)):4 -->_5 g#(s(x)) -> c_8(h#(g(x)),g#(x)):4 -->_3 g#(s(x)) -> c_8(h#(g(x)),g#(x)):4 -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1 6:S:h#(x) -> c_10(+#(p(x),q(x)),p#(x),q#(x),p#(x)) -->_3 q#(pair(x,y)) -> c_12():12 -->_4 p#(pair(x,y)) -> c_11():11 -->_2 p#(pair(x,y)) -> c_11():11 -->_1 +#(x,0()) -> c_1():7 -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1 7:W:+#(x,0()) -> c_1() 8:W:f#(0()) -> c_3() 9:W:f#(s(0())) -> c_4() 10:W:g#(0()) -> c_7() 11:W:p#(pair(x,y)) -> c_11() 12:W:q#(pair(x,y)) -> c_12() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: f#(s(0())) -> c_4() 8: f#(0()) -> c_3() 10: g#(0()) -> c_7() 11: p#(pair(x,y)) -> c_11() 12: q#(pair(x,y)) -> c_12() 7: +#(x,0()) -> c_1() * Step 5: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: +#(x,s(y)) -> c_2(+#(x,y)) f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x)) f#(s(s(x))) -> c_6(p#(h(g(x))),h#(g(x)),g#(x)) g#(s(x)) -> c_8(h#(g(x)),g#(x)) g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)) h#(x) -> c_10(+#(p(x),q(x)),p#(x),q#(x),p#(x)) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),s(0())) g(s(x)) -> h(g(x)) g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x))) h(x) -> pair(+(p(x),q(x)),p(x)) p(pair(x,y)) -> x q(pair(x,y)) -> y - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/5 ,c_6/3,c_7/0,c_8/2,c_9/7,c_10/4,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:+#(x,s(y)) -> c_2(+#(x,y)) -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1 2:S:f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x)) -->_5 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)):5 -->_3 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)):5 -->_5 g#(s(x)) -> c_8(h#(g(x)),g#(x)):4 -->_3 g#(s(x)) -> c_8(h#(g(x)),g#(x)):4 -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1 3:S:f#(s(s(x))) -> c_6(p#(h(g(x))),h#(g(x)),g#(x)) -->_2 h#(x) -> c_10(+#(p(x),q(x)),p#(x),q#(x),p#(x)):6 -->_3 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)):5 -->_3 g#(s(x)) -> c_8(h#(g(x)),g#(x)):4 4:S:g#(s(x)) -> c_8(h#(g(x)),g#(x)) -->_1 h#(x) -> c_10(+#(p(x),q(x)),p#(x),q#(x),p#(x)):6 -->_2 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)):5 -->_2 g#(s(x)) -> c_8(h#(g(x)),g#(x)):4 5:S:g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)) -->_7 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)):5 -->_5 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)):5 -->_3 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),p#(g(x)),g#(x),q#(g(x)),g#(x),p#(g(x)),g#(x)):5 -->_7 g#(s(x)) -> c_8(h#(g(x)),g#(x)):4 -->_5 g#(s(x)) -> c_8(h#(g(x)),g#(x)):4 -->_3 g#(s(x)) -> c_8(h#(g(x)),g#(x)):4 -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1 6:S:h#(x) -> c_10(+#(p(x),q(x)),p#(x),q#(x),p#(x)) -->_1 +#(x,s(y)) -> c_2(+#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),g#(x),g#(x)) f#(s(s(x))) -> c_6(h#(g(x)),g#(x)) g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)) h#(x) -> c_10(+#(p(x),q(x))) * Step 6: Decompose MAYBE + Considered Problem: - Strict DPs: +#(x,s(y)) -> c_2(+#(x,y)) f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),g#(x),g#(x)) f#(s(s(x))) -> c_6(h#(g(x)),g#(x)) g#(s(x)) -> c_8(h#(g(x)),g#(x)) g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)) h#(x) -> c_10(+#(p(x),q(x))) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),s(0())) g(s(x)) -> h(g(x)) g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x))) h(x) -> pair(+(p(x),q(x)),p(x)) p(pair(x,y)) -> x q(pair(x,y)) -> y - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/3 ,c_6/2,c_7/0,c_8/2,c_9/4,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: +#(x,s(y)) -> c_2(+#(x,y)) - Weak DPs: f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),g#(x),g#(x)) f#(s(s(x))) -> c_6(h#(g(x)),g#(x)) g#(s(x)) -> c_8(h#(g(x)),g#(x)) g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)) h#(x) -> c_10(+#(p(x),q(x))) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),s(0())) g(s(x)) -> h(g(x)) g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x))) h(x) -> pair(+(p(x),q(x)),p(x)) p(pair(x,y)) -> x q(pair(x,y)) -> y - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/3 ,c_6/2,c_7/0,c_8/2,c_9/4,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} Problem (S) - Strict DPs: f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),g#(x),g#(x)) f#(s(s(x))) -> c_6(h#(g(x)),g#(x)) g#(s(x)) -> c_8(h#(g(x)),g#(x)) g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)) h#(x) -> c_10(+#(p(x),q(x))) - Weak DPs: +#(x,s(y)) -> c_2(+#(x,y)) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),s(0())) g(s(x)) -> h(g(x)) g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x))) h(x) -> pair(+(p(x),q(x)),p(x)) p(pair(x,y)) -> x q(pair(x,y)) -> y - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/3 ,c_6/2,c_7/0,c_8/2,c_9/4,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} ** Step 6.a:1: Failure MAYBE + Considered Problem: - Strict DPs: +#(x,s(y)) -> c_2(+#(x,y)) - Weak DPs: f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),g#(x),g#(x)) f#(s(s(x))) -> c_6(h#(g(x)),g#(x)) g#(s(x)) -> c_8(h#(g(x)),g#(x)) g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)) h#(x) -> c_10(+#(p(x),q(x))) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),s(0())) g(s(x)) -> h(g(x)) g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x))) h(x) -> pair(+(p(x),q(x)),p(x)) p(pair(x,y)) -> x q(pair(x,y)) -> y - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/3 ,c_6/2,c_7/0,c_8/2,c_9/4,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. ** Step 6.b:1: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),g#(x),g#(x)) f#(s(s(x))) -> c_6(h#(g(x)),g#(x)) g#(s(x)) -> c_8(h#(g(x)),g#(x)) g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)) h#(x) -> c_10(+#(p(x),q(x))) - Weak DPs: +#(x,s(y)) -> c_2(+#(x,y)) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),s(0())) g(s(x)) -> h(g(x)) g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x))) h(x) -> pair(+(p(x),q(x)),p(x)) p(pair(x,y)) -> x q(pair(x,y)) -> y - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/3 ,c_6/2,c_7/0,c_8/2,c_9/4,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {5} by application of Pre({5}) = {2,3}. Here rules are labelled as follows: 1: f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),g#(x),g#(x)) 2: f#(s(s(x))) -> c_6(h#(g(x)),g#(x)) 3: g#(s(x)) -> c_8(h#(g(x)),g#(x)) 4: g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)) 5: h#(x) -> c_10(+#(p(x),q(x))) 6: +#(x,s(y)) -> c_2(+#(x,y)) ** Step 6.b:2: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),g#(x),g#(x)) f#(s(s(x))) -> c_6(h#(g(x)),g#(x)) g#(s(x)) -> c_8(h#(g(x)),g#(x)) g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)) - Weak DPs: +#(x,s(y)) -> c_2(+#(x,y)) h#(x) -> c_10(+#(p(x),q(x))) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),s(0())) g(s(x)) -> h(g(x)) g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x))) h(x) -> pair(+(p(x),q(x)),p(x)) p(pair(x,y)) -> x q(pair(x,y)) -> y - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/3 ,c_6/2,c_7/0,c_8/2,c_9/4,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),g#(x),g#(x)) -->_1 +#(x,s(y)) -> c_2(+#(x,y)):5 -->_3 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)):4 -->_2 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)):4 -->_3 g#(s(x)) -> c_8(h#(g(x)),g#(x)):3 -->_2 g#(s(x)) -> c_8(h#(g(x)),g#(x)):3 2:S:f#(s(s(x))) -> c_6(h#(g(x)),g#(x)) -->_1 h#(x) -> c_10(+#(p(x),q(x))):6 -->_2 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)):4 -->_2 g#(s(x)) -> c_8(h#(g(x)),g#(x)):3 3:S:g#(s(x)) -> c_8(h#(g(x)),g#(x)) -->_1 h#(x) -> c_10(+#(p(x),q(x))):6 -->_2 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)):4 -->_2 g#(s(x)) -> c_8(h#(g(x)),g#(x)):3 4:S:g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)) -->_1 +#(x,s(y)) -> c_2(+#(x,y)):5 -->_4 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)):4 -->_3 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)):4 -->_2 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)):4 -->_4 g#(s(x)) -> c_8(h#(g(x)),g#(x)):3 -->_3 g#(s(x)) -> c_8(h#(g(x)),g#(x)):3 -->_2 g#(s(x)) -> c_8(h#(g(x)),g#(x)):3 5:W:+#(x,s(y)) -> c_2(+#(x,y)) -->_1 +#(x,s(y)) -> c_2(+#(x,y)):5 6:W:h#(x) -> c_10(+#(p(x),q(x))) -->_1 +#(x,s(y)) -> c_2(+#(x,y)):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: h#(x) -> c_10(+#(p(x),q(x))) 5: +#(x,s(y)) -> c_2(+#(x,y)) ** Step 6.b:3: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),g#(x),g#(x)) f#(s(s(x))) -> c_6(h#(g(x)),g#(x)) g#(s(x)) -> c_8(h#(g(x)),g#(x)) g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),s(0())) g(s(x)) -> h(g(x)) g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x))) h(x) -> pair(+(p(x),q(x)),p(x)) p(pair(x,y)) -> x q(pair(x,y)) -> y - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/3 ,c_6/2,c_7/0,c_8/2,c_9/4,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:f#(s(s(x))) -> c_5(+#(p(g(x)),q(g(x))),g#(x),g#(x)) -->_3 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)):4 -->_2 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)):4 -->_3 g#(s(x)) -> c_8(h#(g(x)),g#(x)):3 -->_2 g#(s(x)) -> c_8(h#(g(x)),g#(x)):3 2:S:f#(s(s(x))) -> c_6(h#(g(x)),g#(x)) -->_2 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)):4 -->_2 g#(s(x)) -> c_8(h#(g(x)),g#(x)):3 3:S:g#(s(x)) -> c_8(h#(g(x)),g#(x)) -->_2 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)):4 -->_2 g#(s(x)) -> c_8(h#(g(x)),g#(x)):3 4:S:g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)) -->_4 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)):4 -->_3 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)):4 -->_2 g#(s(x)) -> c_9(+#(p(g(x)),q(g(x))),g#(x),g#(x),g#(x)):4 -->_4 g#(s(x)) -> c_8(h#(g(x)),g#(x)):3 -->_3 g#(s(x)) -> c_8(h#(g(x)),g#(x)):3 -->_2 g#(s(x)) -> c_8(h#(g(x)),g#(x)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: f#(s(s(x))) -> c_5(g#(x),g#(x)) f#(s(s(x))) -> c_6(g#(x)) g#(s(x)) -> c_8(g#(x)) g#(s(x)) -> c_9(g#(x),g#(x),g#(x)) ** Step 6.b:4: UsableRules MAYBE + Considered Problem: - Strict DPs: f#(s(s(x))) -> c_5(g#(x),g#(x)) f#(s(s(x))) -> c_6(g#(x)) g#(s(x)) -> c_8(g#(x)) g#(s(x)) -> c_9(g#(x),g#(x),g#(x)) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) g(0()) -> pair(s(0()),s(0())) g(s(x)) -> h(g(x)) g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x))) h(x) -> pair(+(p(x),q(x)),p(x)) p(pair(x,y)) -> x q(pair(x,y)) -> y - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2 ,c_6/1,c_7/0,c_8/1,c_9/3,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: f#(s(s(x))) -> c_5(g#(x),g#(x)) f#(s(s(x))) -> c_6(g#(x)) g#(s(x)) -> c_8(g#(x)) g#(s(x)) -> c_9(g#(x),g#(x),g#(x)) ** Step 6.b:5: Decompose MAYBE + Considered Problem: - Strict DPs: f#(s(s(x))) -> c_5(g#(x),g#(x)) f#(s(s(x))) -> c_6(g#(x)) g#(s(x)) -> c_8(g#(x)) g#(s(x)) -> c_9(g#(x),g#(x),g#(x)) - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2 ,c_6/1,c_7/0,c_8/1,c_9/3,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: f#(s(s(x))) -> c_5(g#(x),g#(x)) g#(s(x)) -> c_8(g#(x)) g#(s(x)) -> c_9(g#(x),g#(x),g#(x)) - Weak DPs: f#(s(s(x))) -> c_6(g#(x)) - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2 ,c_6/1,c_7/0,c_8/1,c_9/3,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} Problem (S) - Strict DPs: f#(s(s(x))) -> c_6(g#(x)) - Weak DPs: f#(s(s(x))) -> c_5(g#(x),g#(x)) g#(s(x)) -> c_8(g#(x)) g#(s(x)) -> c_9(g#(x),g#(x),g#(x)) - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2 ,c_6/1,c_7/0,c_8/1,c_9/3,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} *** Step 6.b:5.a:1: NaturalPI MAYBE + Considered Problem: - Strict DPs: f#(s(s(x))) -> c_5(g#(x),g#(x)) g#(s(x)) -> c_8(g#(x)) g#(s(x)) -> c_9(g#(x),g#(x),g#(x)) - Weak DPs: f#(s(s(x))) -> c_6(g#(x)) - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2 ,c_6/1,c_7/0,c_8/1,c_9/3,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} + Applied Processor: NaturalPI {shape = Mixed 3, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(3)): The following argument positions are considered usable: uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1,2,3} Following symbols are considered usable: {+#,f#,g#,h#,p#,q#} TcT has computed the following interpretation: p(+) = x1 + 2*x1*x2 + x1^2 + x1^2*x2 + x1^3 + x2 p(0) = 1 p(f) = 1 + 4*x1 + x1^2 p(g) = 1 + 2*x1 p(h) = 0 p(p) = 1 + x1 + 4*x1^2 + x1^3 p(pair) = 4 p(q) = 4 + x1 + 4*x1^2 p(s) = 0 p(+#) = 1 + 2*x1 + 2*x1^2 + x1^3 + 4*x2^2 + x2^3 p(f#) = 5 + x1^3 p(g#) = 0 p(h#) = 1 p(p#) = 0 p(q#) = 4*x1 + x1^3 p(c_1) = 0 p(c_2) = 4 p(c_3) = 4 p(c_4) = 0 p(c_5) = 1 + x1 + x2 p(c_6) = x1 p(c_7) = 0 p(c_8) = x1 p(c_9) = x1 + x2 + x3 p(c_10) = 1 p(c_11) = 0 p(c_12) = 0 Following rules are strictly oriented: f#(s(s(x))) = 5 > 1 = c_5(g#(x),g#(x)) Following rules are (at-least) weakly oriented: f#(s(s(x))) = 5 >= 0 = c_6(g#(x)) g#(s(x)) = 0 >= 0 = c_8(g#(x)) g#(s(x)) = 0 >= 0 = c_9(g#(x),g#(x),g#(x)) *** Step 6.b:5.a:2: Failure MAYBE + Considered Problem: - Strict DPs: g#(s(x)) -> c_8(g#(x)) g#(s(x)) -> c_9(g#(x),g#(x),g#(x)) - Weak DPs: f#(s(s(x))) -> c_5(g#(x),g#(x)) f#(s(s(x))) -> c_6(g#(x)) - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2 ,c_6/1,c_7/0,c_8/1,c_9/3,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. *** Step 6.b:5.b:1: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: f#(s(s(x))) -> c_6(g#(x)) - Weak DPs: f#(s(s(x))) -> c_5(g#(x),g#(x)) g#(s(x)) -> c_8(g#(x)) g#(s(x)) -> c_9(g#(x),g#(x),g#(x)) - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2 ,c_6/1,c_7/0,c_8/1,c_9/3,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {}. Here rules are labelled as follows: 1: f#(s(s(x))) -> c_6(g#(x)) 2: f#(s(s(x))) -> c_5(g#(x),g#(x)) 3: g#(s(x)) -> c_8(g#(x)) 4: g#(s(x)) -> c_9(g#(x),g#(x),g#(x)) *** Step 6.b:5.b:2: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f#(s(s(x))) -> c_5(g#(x),g#(x)) f#(s(s(x))) -> c_6(g#(x)) g#(s(x)) -> c_8(g#(x)) g#(s(x)) -> c_9(g#(x),g#(x),g#(x)) - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2 ,c_6/1,c_7/0,c_8/1,c_9/3,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:f#(s(s(x))) -> c_5(g#(x),g#(x)) -->_2 g#(s(x)) -> c_9(g#(x),g#(x),g#(x)):4 -->_1 g#(s(x)) -> c_9(g#(x),g#(x),g#(x)):4 -->_2 g#(s(x)) -> c_8(g#(x)):3 -->_1 g#(s(x)) -> c_8(g#(x)):3 2:W:f#(s(s(x))) -> c_6(g#(x)) -->_1 g#(s(x)) -> c_9(g#(x),g#(x),g#(x)):4 -->_1 g#(s(x)) -> c_8(g#(x)):3 3:W:g#(s(x)) -> c_8(g#(x)) -->_1 g#(s(x)) -> c_9(g#(x),g#(x),g#(x)):4 -->_1 g#(s(x)) -> c_8(g#(x)):3 4:W:g#(s(x)) -> c_9(g#(x),g#(x),g#(x)) -->_3 g#(s(x)) -> c_9(g#(x),g#(x),g#(x)):4 -->_2 g#(s(x)) -> c_9(g#(x),g#(x),g#(x)):4 -->_1 g#(s(x)) -> c_9(g#(x),g#(x),g#(x)):4 -->_3 g#(s(x)) -> c_8(g#(x)):3 -->_2 g#(s(x)) -> c_8(g#(x)):3 -->_1 g#(s(x)) -> c_8(g#(x)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: f#(s(s(x))) -> c_6(g#(x)) 1: f#(s(s(x))) -> c_5(g#(x),g#(x)) 4: g#(s(x)) -> c_9(g#(x),g#(x),g#(x)) 3: g#(s(x)) -> c_8(g#(x)) *** Step 6.b:5.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {+/2,f/1,g/1,h/1,p/1,q/1,+#/2,f#/1,g#/1,h#/1,p#/1,q#/1} / {0/0,pair/2,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/2 ,c_6/1,c_7/0,c_8/1,c_9/3,c_10/1,c_11/0,c_12/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,f#,g#,h#,p#,q#} and constructors {0,pair,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). MAYBE