WORST_CASE(?,O(n^5)) * Step 1: DependencyPairs WORST_CASE(?,O(n^5)) + Considered Problem: - Strict TRS: f_0(x) -> a() f_1(x) -> g_1(x,x) f_2(x) -> g_2(x,x) f_3(x) -> g_3(x,x) f_4(x) -> g_4(x,x) f_5(x) -> g_5(x,x) g_1(s(x),y) -> b(f_0(y),g_1(x,y)) g_2(s(x),y) -> b(f_1(y),g_2(x,y)) g_3(s(x),y) -> b(f_2(y),g_3(x,y)) g_4(s(x),y) -> b(f_3(y),g_4(x,y)) g_5(s(x),y) -> b(f_4(y),g_5(x,y)) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2} / {a/0,b/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f_0,f_1,f_2,f_3,f_4,f_5,g_1,g_2,g_3,g_4 ,g_5} and constructors {a,b,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs f_0#(x) -> c_1() f_1#(x) -> c_2(g_1#(x,x)) f_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) f_5#(x) -> c_6(g_5#(x,x)) g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)) g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^5)) + Considered Problem: - Strict DPs: f_0#(x) -> c_1() f_1#(x) -> c_2(g_1#(x,x)) f_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) f_5#(x) -> c_6(g_5#(x,x)) g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)) g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) - Weak TRS: f_0(x) -> a() f_1(x) -> g_1(x,x) f_2(x) -> g_2(x,x) f_3(x) -> g_3(x,x) f_4(x) -> g_4(x,x) f_5(x) -> g_5(x,x) g_1(s(x),y) -> b(f_0(y),g_1(x,y)) g_2(s(x),y) -> b(f_1(y),g_2(x,y)) g_3(s(x),y) -> b(f_2(y),g_3(x,y)) g_4(s(x),y) -> b(f_3(y),g_4(x,y)) g_5(s(x),y) -> b(f_4(y),g_5(x,y)) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/2,c_8/2,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {7}. Here rules are labelled as follows: 1: f_0#(x) -> c_1() 2: f_1#(x) -> c_2(g_1#(x,x)) 3: f_2#(x) -> c_3(g_2#(x,x)) 4: f_3#(x) -> c_4(g_3#(x,x)) 5: f_4#(x) -> c_5(g_4#(x,x)) 6: f_5#(x) -> c_6(g_5#(x,x)) 7: g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)) 8: g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) 9: g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) 10: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) 11: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^5)) + Considered Problem: - Strict DPs: f_1#(x) -> c_2(g_1#(x,x)) f_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) f_5#(x) -> c_6(g_5#(x,x)) g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)) g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) - Weak DPs: f_0#(x) -> c_1() - Weak TRS: f_0(x) -> a() f_1(x) -> g_1(x,x) f_2(x) -> g_2(x,x) f_3(x) -> g_3(x,x) f_4(x) -> g_4(x,x) f_5(x) -> g_5(x,x) g_1(s(x),y) -> b(f_0(y),g_1(x,y)) g_2(s(x),y) -> b(f_1(y),g_2(x,y)) g_3(s(x),y) -> b(f_2(y),g_3(x,y)) g_4(s(x),y) -> b(f_3(y),g_4(x,y)) g_5(s(x),y) -> b(f_4(y),g_5(x,y)) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/2,c_8/2,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:f_1#(x) -> c_2(g_1#(x,x)) -->_1 g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)):6 2:S:f_2#(x) -> c_3(g_2#(x,x)) -->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7 3:S:f_3#(x) -> c_4(g_3#(x,x)) -->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8 4:S:f_4#(x) -> c_5(g_4#(x,x)) -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9 5:S:f_5#(x) -> c_6(g_5#(x,x)) -->_1 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10 6:S:g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)) -->_1 f_0#(x) -> c_1():11 -->_2 g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)):6 7:S:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) -->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7 -->_1 f_1#(x) -> c_2(g_1#(x,x)):1 8:S:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) -->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8 -->_1 f_2#(x) -> c_3(g_2#(x,x)):2 9:S:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9 -->_1 f_3#(x) -> c_4(g_3#(x,x)):3 10:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10 -->_1 f_4#(x) -> c_5(g_4#(x,x)):4 11:W:f_0#(x) -> c_1() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: f_0#(x) -> c_1() * Step 4: SimplifyRHS WORST_CASE(?,O(n^5)) + Considered Problem: - Strict DPs: f_1#(x) -> c_2(g_1#(x,x)) f_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) f_5#(x) -> c_6(g_5#(x,x)) g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)) g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) - Weak TRS: f_0(x) -> a() f_1(x) -> g_1(x,x) f_2(x) -> g_2(x,x) f_3(x) -> g_3(x,x) f_4(x) -> g_4(x,x) f_5(x) -> g_5(x,x) g_1(s(x),y) -> b(f_0(y),g_1(x,y)) g_2(s(x),y) -> b(f_1(y),g_2(x,y)) g_3(s(x),y) -> b(f_2(y),g_3(x,y)) g_4(s(x),y) -> b(f_3(y),g_4(x,y)) g_5(s(x),y) -> b(f_4(y),g_5(x,y)) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/2,c_8/2,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:f_1#(x) -> c_2(g_1#(x,x)) -->_1 g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)):6 2:S:f_2#(x) -> c_3(g_2#(x,x)) -->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7 3:S:f_3#(x) -> c_4(g_3#(x,x)) -->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8 4:S:f_4#(x) -> c_5(g_4#(x,x)) -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9 5:S:f_5#(x) -> c_6(g_5#(x,x)) -->_1 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10 6:S:g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)) -->_2 g_1#(s(x),y) -> c_7(f_0#(y),g_1#(x,y)):6 7:S:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) -->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7 -->_1 f_1#(x) -> c_2(g_1#(x,x)):1 8:S:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) -->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8 -->_1 f_2#(x) -> c_3(g_2#(x,x)):2 9:S:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9 -->_1 f_3#(x) -> c_4(g_3#(x,x)):3 10:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10 -->_1 f_4#(x) -> c_5(g_4#(x,x)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: g_1#(s(x),y) -> c_7(g_1#(x,y)) * Step 5: UsableRules WORST_CASE(?,O(n^5)) + Considered Problem: - Strict DPs: f_1#(x) -> c_2(g_1#(x,x)) f_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) f_5#(x) -> c_6(g_5#(x,x)) g_1#(s(x),y) -> c_7(g_1#(x,y)) g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) - Weak TRS: f_0(x) -> a() f_1(x) -> g_1(x,x) f_2(x) -> g_2(x,x) f_3(x) -> g_3(x,x) f_4(x) -> g_4(x,x) f_5(x) -> g_5(x,x) g_1(s(x),y) -> b(f_0(y),g_1(x,y)) g_2(s(x),y) -> b(f_1(y),g_2(x,y)) g_3(s(x),y) -> b(f_2(y),g_3(x,y)) g_4(s(x),y) -> b(f_3(y),g_4(x,y)) g_5(s(x),y) -> b(f_4(y),g_5(x,y)) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: f_1#(x) -> c_2(g_1#(x,x)) f_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) f_5#(x) -> c_6(g_5#(x,x)) g_1#(s(x),y) -> c_7(g_1#(x,y)) g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) * Step 6: RemoveHeads WORST_CASE(?,O(n^5)) + Considered Problem: - Strict DPs: f_1#(x) -> c_2(g_1#(x,x)) f_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) f_5#(x) -> c_6(g_5#(x,x)) g_1#(s(x),y) -> c_7(g_1#(x,y)) g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,s} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:f_1#(x) -> c_2(g_1#(x,x)) -->_1 g_1#(s(x),y) -> c_7(g_1#(x,y)):6 2:S:f_2#(x) -> c_3(g_2#(x,x)) -->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7 3:S:f_3#(x) -> c_4(g_3#(x,x)) -->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8 4:S:f_4#(x) -> c_5(g_4#(x,x)) -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9 5:S:f_5#(x) -> c_6(g_5#(x,x)) -->_1 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10 6:S:g_1#(s(x),y) -> c_7(g_1#(x,y)) -->_1 g_1#(s(x),y) -> c_7(g_1#(x,y)):6 7:S:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) -->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):7 -->_1 f_1#(x) -> c_2(g_1#(x,x)):1 8:S:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) -->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):8 -->_1 f_2#(x) -> c_3(g_2#(x,x)):2 9:S:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):9 -->_1 f_3#(x) -> c_4(g_3#(x,x)):3 10:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):10 -->_1 f_4#(x) -> c_5(g_4#(x,x)):4 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(5,f_5#(x) -> c_6(g_5#(x,x)))] * Step 7: DecomposeDG WORST_CASE(?,O(n^5)) + Considered Problem: - Strict DPs: f_1#(x) -> c_2(g_1#(x,x)) f_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) g_1#(s(x),y) -> c_7(g_1#(x,y)) g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,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 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) and a lower component f_1#(x) -> c_2(g_1#(x,x)) f_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) g_1#(s(x),y) -> c_7(g_1#(x,y)) g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) Further, following extension rules are added to the lower component. g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) ** Step 7.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: g_5#(s(x),y) -> c_11(g_5#(x,y)) ** Step 7.a:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g_5#(s(x),y) -> c_11(g_5#(x,y)) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/2,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,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(c_11) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] p(b) = [1] x1 + [1] x2 + [0] p(f_0) = [0] p(f_1) = [2] x1 + [0] p(f_2) = [2] p(f_3) = [0] p(f_4) = [0] p(f_5) = [0] p(g_1) = [0] p(g_2) = [0] p(g_3) = [0] p(g_4) = [4] x1 + [2] p(g_5) = [0] p(s) = [1] x1 + [10] p(f_0#) = [0] p(f_1#) = [0] p(f_2#) = [0] p(f_3#) = [0] p(f_4#) = [0] p(f_5#) = [0] p(g_1#) = [0] p(g_2#) = [0] p(g_3#) = [0] p(g_4#) = [0] p(g_5#) = [1] x1 + [5] p(c_1) = [0] p(c_2) = [0] p(c_3) = [2] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] x1 + [3] Following rules are strictly oriented: g_5#(s(x),y) = [1] x + [15] > [1] x + [8] = c_11(g_5#(x,y)) Following rules are (at-least) weakly oriented: Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 7.a:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: g_5#(s(x),y) -> c_11(g_5#(x,y)) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/2,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 7.b:1: DecomposeDG WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: f_1#(x) -> c_2(g_1#(x,x)) f_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) g_1#(s(x),y) -> c_7(g_1#(x,y)) g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) - Weak DPs: g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,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_4#(x) -> c_5(g_4#(x,x)) g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) and a lower component f_1#(x) -> c_2(g_1#(x,x)) f_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> c_4(g_3#(x,x)) g_1#(s(x),y) -> c_7(g_1#(x,y)) g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) Further, following extension rules are added to the lower component. f_4#(x) -> g_4#(x,x) g_4#(s(x),y) -> f_3#(y) g_4#(s(x),y) -> g_4#(x,y) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) *** Step 7.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f_4#(x) -> c_5(g_4#(x,x)) g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) - Weak DPs: g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:f_4#(x) -> c_5(g_4#(x,x)) -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):2 2:S:g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):2 3:W:g_5#(s(x),y) -> f_4#(y) -->_1 f_4#(x) -> c_5(g_4#(x,x)):1 4:W:g_5#(s(x),y) -> g_5#(x,y) -->_1 g_5#(s(x),y) -> g_5#(x,y):4 -->_1 g_5#(s(x),y) -> f_4#(y):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: g_4#(s(x),y) -> c_10(g_4#(x,y)) *** Step 7.b:1.a:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f_4#(x) -> c_5(g_4#(x,x)) g_4#(s(x),y) -> c_10(g_4#(x,y)) - Weak DPs: g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/1,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,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(c_5) = {1}, uargs(c_10) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] p(b) = [1] x1 + [1] x2 + [0] p(f_0) = [0] p(f_1) = [0] p(f_2) = [0] p(f_3) = [0] p(f_4) = [0] p(f_5) = [0] p(g_1) = [0] p(g_2) = [0] p(g_3) = [0] p(g_4) = [0] p(g_5) = [0] p(s) = [1] x1 + [1] p(f_0#) = [0] p(f_1#) = [0] p(f_2#) = [0] p(f_3#) = [1] p(f_4#) = [9] x1 + [0] p(f_5#) = [0] p(g_1#) = [0] p(g_2#) = [0] p(g_3#) = [0] p(g_4#) = [5] x1 + [4] x2 + [0] p(g_5#) = [9] x2 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [1] x2 + [0] p(c_9) = [1] x2 + [8] p(c_10) = [1] x1 + [0] p(c_11) = [0] Following rules are strictly oriented: g_4#(s(x),y) = [5] x + [4] y + [5] > [5] x + [4] y + [0] = c_10(g_4#(x,y)) Following rules are (at-least) weakly oriented: f_4#(x) = [9] x + [0] >= [9] x + [0] = c_5(g_4#(x,x)) g_5#(s(x),y) = [9] y + [0] >= [9] y + [0] = f_4#(y) g_5#(s(x),y) = [9] y + [0] >= [9] y + [0] = g_5#(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 7.b:1.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f_4#(x) -> c_5(g_4#(x,x)) - Weak DPs: g_4#(s(x),y) -> c_10(g_4#(x,y)) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/1,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,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(c_5) = {1}, uargs(c_10) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] p(b) = [1] x1 + [1] x2 + [0] p(f_0) = [0] p(f_1) = [0] p(f_2) = [0] p(f_3) = [0] p(f_4) = [0] p(f_5) = [0] p(g_1) = [1] p(g_2) = [0] p(g_3) = [0] p(g_4) = [0] p(g_5) = [0] p(s) = [1] x1 + [0] p(f_0#) = [1] x1 + [0] p(f_1#) = [0] p(f_2#) = [1] p(f_3#) = [0] p(f_4#) = [1] p(f_5#) = [0] p(g_1#) = [0] p(g_2#) = [0] p(g_3#) = [0] p(g_4#) = [0] p(g_5#) = [1] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] Following rules are strictly oriented: f_4#(x) = [1] > [0] = c_5(g_4#(x,x)) Following rules are (at-least) weakly oriented: g_4#(s(x),y) = [0] >= [0] = c_10(g_4#(x,y)) g_5#(s(x),y) = [1] >= [1] = f_4#(y) g_5#(s(x),y) = [1] >= [1] = g_5#(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 7.b:1.a:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f_4#(x) -> c_5(g_4#(x,x)) g_4#(s(x),y) -> c_10(g_4#(x,y)) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/1,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 7.b:1.b:1: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: f_1#(x) -> c_2(g_1#(x,x)) f_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> c_4(g_3#(x,x)) g_1#(s(x),y) -> c_7(g_1#(x,y)) g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) - Weak DPs: f_4#(x) -> g_4#(x,x) g_4#(s(x),y) -> f_3#(y) g_4#(s(x),y) -> g_4#(x,y) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,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_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> g_4#(x,x) g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) g_4#(s(x),y) -> f_3#(y) g_4#(s(x),y) -> g_4#(x,y) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) and a lower component f_1#(x) -> c_2(g_1#(x,x)) f_2#(x) -> c_3(g_2#(x,x)) g_1#(s(x),y) -> c_7(g_1#(x,y)) g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) Further, following extension rules are added to the lower component. f_3#(x) -> g_3#(x,x) f_4#(x) -> g_4#(x,x) g_3#(s(x),y) -> f_2#(y) g_3#(s(x),y) -> g_3#(x,y) g_4#(s(x),y) -> f_3#(y) g_4#(s(x),y) -> g_4#(x,y) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) **** Step 7.b:1.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f_3#(x) -> c_4(g_3#(x,x)) g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) - Weak DPs: f_4#(x) -> g_4#(x,x) g_4#(s(x),y) -> f_3#(y) g_4#(s(x),y) -> g_4#(x,y) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:f_3#(x) -> c_4(g_3#(x,x)) -->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):2 2:S:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) -->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):2 3:W:f_4#(x) -> g_4#(x,x) -->_1 g_4#(s(x),y) -> g_4#(x,y):5 -->_1 g_4#(s(x),y) -> f_3#(y):4 4:W:g_4#(s(x),y) -> f_3#(y) -->_1 f_3#(x) -> c_4(g_3#(x,x)):1 5:W:g_4#(s(x),y) -> g_4#(x,y) -->_1 g_4#(s(x),y) -> g_4#(x,y):5 -->_1 g_4#(s(x),y) -> f_3#(y):4 6:W:g_5#(s(x),y) -> f_4#(y) -->_1 f_4#(x) -> g_4#(x,x):3 7:W:g_5#(s(x),y) -> g_5#(x,y) -->_1 g_5#(s(x),y) -> g_5#(x,y):7 -->_1 g_5#(s(x),y) -> f_4#(y):6 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: g_3#(s(x),y) -> c_9(g_3#(x,y)) **** Step 7.b:1.b:1.a:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f_3#(x) -> c_4(g_3#(x,x)) g_3#(s(x),y) -> c_9(g_3#(x,y)) - Weak DPs: f_4#(x) -> g_4#(x,x) g_4#(s(x),y) -> f_3#(y) g_4#(s(x),y) -> g_4#(x,y) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,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(c_4) = {1}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] p(b) = [1] x1 + [1] x2 + [0] p(f_0) = [0] p(f_1) = [1] p(f_2) = [0] p(f_3) = [0] p(f_4) = [0] p(f_5) = [0] p(g_1) = [0] p(g_2) = [0] p(g_3) = [0] p(g_4) = [0] p(g_5) = [0] p(s) = [1] x1 + [1] p(f_0#) = [0] p(f_1#) = [0] p(f_2#) = [0] p(f_3#) = [1] x1 + [11] p(f_4#) = [1] x1 + [11] p(f_5#) = [0] p(g_1#) = [0] p(g_2#) = [0] p(g_3#) = [1] x1 + [0] p(g_4#) = [1] x2 + [11] p(g_5#) = [4] x1 + [1] x2 + [7] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [2] p(c_7) = [1] x1 + [1] p(c_8) = [2] x2 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [2] x1 + [1] x2 + [0] p(c_11) = [1] x1 + [8] x2 + [1] Following rules are strictly oriented: f_3#(x) = [1] x + [11] > [1] x + [0] = c_4(g_3#(x,x)) g_3#(s(x),y) = [1] x + [1] > [1] x + [0] = c_9(g_3#(x,y)) Following rules are (at-least) weakly oriented: f_4#(x) = [1] x + [11] >= [1] x + [11] = g_4#(x,x) g_4#(s(x),y) = [1] y + [11] >= [1] y + [11] = f_3#(y) g_4#(s(x),y) = [1] y + [11] >= [1] y + [11] = g_4#(x,y) g_5#(s(x),y) = [4] x + [1] y + [11] >= [1] y + [11] = f_4#(y) g_5#(s(x),y) = [4] x + [1] y + [11] >= [4] x + [1] y + [7] = g_5#(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 7.b:1.b:1.a:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> g_4#(x,x) g_3#(s(x),y) -> c_9(g_3#(x,y)) g_4#(s(x),y) -> f_3#(y) g_4#(s(x),y) -> g_4#(x,y) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/1 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 7.b:1.b:1.b:1: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: f_1#(x) -> c_2(g_1#(x,x)) f_2#(x) -> c_3(g_2#(x,x)) g_1#(s(x),y) -> c_7(g_1#(x,y)) g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) - Weak DPs: f_3#(x) -> g_3#(x,x) f_4#(x) -> g_4#(x,x) g_3#(s(x),y) -> f_2#(y) g_3#(s(x),y) -> g_3#(x,y) g_4#(s(x),y) -> f_3#(y) g_4#(s(x),y) -> g_4#(x,y) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,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_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> g_3#(x,x) f_4#(x) -> g_4#(x,x) g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) g_3#(s(x),y) -> f_2#(y) g_3#(s(x),y) -> g_3#(x,y) g_4#(s(x),y) -> f_3#(y) g_4#(s(x),y) -> g_4#(x,y) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) and a lower component f_1#(x) -> c_2(g_1#(x,x)) g_1#(s(x),y) -> c_7(g_1#(x,y)) Further, following extension rules are added to the lower component. f_2#(x) -> g_2#(x,x) f_3#(x) -> g_3#(x,x) f_4#(x) -> g_4#(x,x) g_2#(s(x),y) -> f_1#(y) g_2#(s(x),y) -> g_2#(x,y) g_3#(s(x),y) -> f_2#(y) g_3#(s(x),y) -> g_3#(x,y) g_4#(s(x),y) -> f_3#(y) g_4#(s(x),y) -> g_4#(x,y) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) ***** Step 7.b:1.b:1.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f_2#(x) -> c_3(g_2#(x,x)) g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) - Weak DPs: f_3#(x) -> g_3#(x,x) f_4#(x) -> g_4#(x,x) g_3#(s(x),y) -> f_2#(y) g_3#(s(x),y) -> g_3#(x,y) g_4#(s(x),y) -> f_3#(y) g_4#(s(x),y) -> g_4#(x,y) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:f_2#(x) -> c_3(g_2#(x,x)) -->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):2 2:S:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) -->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):2 3:W:f_3#(x) -> g_3#(x,x) -->_1 g_3#(s(x),y) -> g_3#(x,y):6 -->_1 g_3#(s(x),y) -> f_2#(y):5 4:W:f_4#(x) -> g_4#(x,x) -->_1 g_4#(s(x),y) -> g_4#(x,y):8 -->_1 g_4#(s(x),y) -> f_3#(y):7 5:W:g_3#(s(x),y) -> f_2#(y) -->_1 f_2#(x) -> c_3(g_2#(x,x)):1 6:W:g_3#(s(x),y) -> g_3#(x,y) -->_1 g_3#(s(x),y) -> g_3#(x,y):6 -->_1 g_3#(s(x),y) -> f_2#(y):5 7:W:g_4#(s(x),y) -> f_3#(y) -->_1 f_3#(x) -> g_3#(x,x):3 8:W:g_4#(s(x),y) -> g_4#(x,y) -->_1 g_4#(s(x),y) -> g_4#(x,y):8 -->_1 g_4#(s(x),y) -> f_3#(y):7 9:W:g_5#(s(x),y) -> f_4#(y) -->_1 f_4#(x) -> g_4#(x,x):4 10:W:g_5#(s(x),y) -> g_5#(x,y) -->_1 g_5#(s(x),y) -> g_5#(x,y):10 -->_1 g_5#(s(x),y) -> f_4#(y):9 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: g_2#(s(x),y) -> c_8(g_2#(x,y)) ***** Step 7.b:1.b:1.b:1.a:2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f_2#(x) -> c_3(g_2#(x,x)) g_2#(s(x),y) -> c_8(g_2#(x,y)) - Weak DPs: f_3#(x) -> g_3#(x,x) f_4#(x) -> g_4#(x,x) g_3#(s(x),y) -> f_2#(y) g_3#(s(x),y) -> g_3#(x,y) g_4#(s(x),y) -> f_3#(y) g_4#(s(x),y) -> g_4#(x,y) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_8) = {1} Following symbols are considered usable: {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#,g_5#} TcT has computed the following interpretation: p(a) = [0] p(b) = [1] x1 + [1] x2 + [0] p(f_0) = [0] p(f_1) = [0] p(f_2) = [0] p(f_3) = [0] p(f_4) = [0] p(f_5) = [0] p(g_1) = [0] p(g_2) = [0] p(g_3) = [0] p(g_4) = [0] p(g_5) = [0] p(s) = [0] p(f_0#) = [0] p(f_1#) = [0] p(f_2#) = [9] p(f_3#) = [9] p(f_4#) = [9] p(f_5#) = [0] p(g_1#) = [0] p(g_2#) = [1] p(g_3#) = [9] p(g_4#) = [9] p(g_5#) = [9] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [1] x1 + [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] Following rules are strictly oriented: f_2#(x) = [9] > [1] = c_3(g_2#(x,x)) Following rules are (at-least) weakly oriented: f_3#(x) = [9] >= [9] = g_3#(x,x) f_4#(x) = [9] >= [9] = g_4#(x,x) g_2#(s(x),y) = [1] >= [1] = c_8(g_2#(x,y)) g_3#(s(x),y) = [9] >= [9] = f_2#(y) g_3#(s(x),y) = [9] >= [9] = g_3#(x,y) g_4#(s(x),y) = [9] >= [9] = f_3#(y) g_4#(s(x),y) = [9] >= [9] = g_4#(x,y) g_5#(s(x),y) = [9] >= [9] = f_4#(y) g_5#(s(x),y) = [9] >= [9] = g_5#(x,y) ***** Step 7.b:1.b:1.b:1.a:3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g_2#(s(x),y) -> c_8(g_2#(x,y)) - Weak DPs: f_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> g_3#(x,x) f_4#(x) -> g_4#(x,x) g_3#(s(x),y) -> f_2#(y) g_3#(s(x),y) -> g_3#(x,y) g_4#(s(x),y) -> f_3#(y) g_4#(s(x),y) -> g_4#(x,y) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_8) = {1} Following symbols are considered usable: {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4#,g_5#} TcT has computed the following interpretation: p(a) = [0] p(b) = [1] x1 + [1] x2 + [0] p(f_0) = [0] p(f_1) = [0] p(f_2) = [0] p(f_3) = [0] p(f_4) = [0] p(f_5) = [0] p(g_1) = [0] p(g_2) = [0] p(g_3) = [0] p(g_4) = [0] p(g_5) = [1] x1 + [1] x2 + [4] p(s) = [1] x1 + [2] p(f_0#) = [1] p(f_1#) = [1] x1 + [1] p(f_2#) = [4] x1 + [8] p(f_3#) = [12] x1 + [1] p(f_4#) = [13] x1 + [1] p(f_5#) = [8] p(g_1#) = [8] p(g_2#) = [1] x1 + [2] p(g_3#) = [8] x1 + [4] x2 + [1] p(g_4#) = [13] x2 + [1] p(g_5#) = [12] x1 + [13] x2 + [3] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [6] p(c_4) = [1] x1 + [8] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [4] p(c_10) = [1] x1 + [1] p(c_11) = [1] x1 + [1] Following rules are strictly oriented: g_2#(s(x),y) = [1] x + [4] > [1] x + [2] = c_8(g_2#(x,y)) Following rules are (at-least) weakly oriented: f_2#(x) = [4] x + [8] >= [1] x + [8] = c_3(g_2#(x,x)) f_3#(x) = [12] x + [1] >= [12] x + [1] = g_3#(x,x) f_4#(x) = [13] x + [1] >= [13] x + [1] = g_4#(x,x) g_3#(s(x),y) = [8] x + [4] y + [17] >= [4] y + [8] = f_2#(y) g_3#(s(x),y) = [8] x + [4] y + [17] >= [8] x + [4] y + [1] = g_3#(x,y) g_4#(s(x),y) = [13] y + [1] >= [12] y + [1] = f_3#(y) g_4#(s(x),y) = [13] y + [1] >= [13] y + [1] = g_4#(x,y) g_5#(s(x),y) = [12] x + [13] y + [27] >= [13] y + [1] = f_4#(y) g_5#(s(x),y) = [12] x + [13] y + [27] >= [12] x + [13] y + [3] = g_5#(x,y) ***** Step 7.b:1.b:1.b:1.a:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> g_3#(x,x) f_4#(x) -> g_4#(x,x) g_2#(s(x),y) -> c_8(g_2#(x,y)) g_3#(s(x),y) -> f_2#(y) g_3#(s(x),y) -> g_3#(x,y) g_4#(s(x),y) -> f_3#(y) g_4#(s(x),y) -> g_4#(x,y) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/1,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ***** Step 7.b:1.b:1.b:1.b:1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f_1#(x) -> c_2(g_1#(x,x)) g_1#(s(x),y) -> c_7(g_1#(x,y)) - Weak DPs: f_2#(x) -> g_2#(x,x) f_3#(x) -> g_3#(x,x) f_4#(x) -> g_4#(x,x) g_2#(s(x),y) -> f_1#(y) g_2#(s(x),y) -> g_2#(x,y) g_3#(s(x),y) -> f_2#(y) g_3#(s(x),y) -> g_3#(x,y) g_4#(s(x),y) -> f_3#(y) g_4#(s(x),y) -> g_4#(x,y) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,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(c_2) = {1}, uargs(c_7) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [0] p(b) = [1] x1 + [1] x2 + [4] p(f_0) = [0] p(f_1) = [2] x1 + [1] p(f_2) = [4] p(f_3) = [4] x1 + [2] p(f_4) = [2] p(f_5) = [2] x1 + [0] p(g_1) = [1] x1 + [2] x2 + [0] p(g_2) = [0] p(g_3) = [0] p(g_4) = [0] p(g_5) = [0] p(s) = [1] x1 + [1] p(f_0#) = [0] p(f_1#) = [1] x1 + [7] p(f_2#) = [1] x1 + [7] p(f_3#) = [1] x1 + [7] p(f_4#) = [1] x1 + [7] p(f_5#) = [0] p(g_1#) = [1] x1 + [4] p(g_2#) = [1] x2 + [7] p(g_3#) = [1] x2 + [7] p(g_4#) = [1] x2 + [7] p(g_5#) = [5] x1 + [1] x2 + [2] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] x2 + [0] p(c_9) = [4] x1 + [1] x2 + [1] p(c_10) = [2] x1 + [0] p(c_11) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: f_1#(x) = [1] x + [7] > [1] x + [4] = c_2(g_1#(x,x)) g_1#(s(x),y) = [1] x + [5] > [1] x + [4] = c_7(g_1#(x,y)) Following rules are (at-least) weakly oriented: f_2#(x) = [1] x + [7] >= [1] x + [7] = g_2#(x,x) f_3#(x) = [1] x + [7] >= [1] x + [7] = g_3#(x,x) f_4#(x) = [1] x + [7] >= [1] x + [7] = g_4#(x,x) g_2#(s(x),y) = [1] y + [7] >= [1] y + [7] = f_1#(y) g_2#(s(x),y) = [1] y + [7] >= [1] y + [7] = g_2#(x,y) g_3#(s(x),y) = [1] y + [7] >= [1] y + [7] = f_2#(y) g_3#(s(x),y) = [1] y + [7] >= [1] y + [7] = g_3#(x,y) g_4#(s(x),y) = [1] y + [7] >= [1] y + [7] = f_3#(y) g_4#(s(x),y) = [1] y + [7] >= [1] y + [7] = g_4#(x,y) g_5#(s(x),y) = [5] x + [1] y + [7] >= [1] y + [7] = f_4#(y) g_5#(s(x),y) = [5] x + [1] y + [7] >= [5] x + [1] y + [2] = g_5#(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ***** Step 7.b:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f_1#(x) -> c_2(g_1#(x,x)) f_2#(x) -> g_2#(x,x) f_3#(x) -> g_3#(x,x) f_4#(x) -> g_4#(x,x) g_1#(s(x),y) -> c_7(g_1#(x,y)) g_2#(s(x),y) -> f_1#(y) g_2#(s(x),y) -> g_2#(x,y) g_3#(s(x),y) -> f_2#(y) g_3#(s(x),y) -> g_3#(x,y) g_4#(s(x),y) -> f_3#(y) g_4#(s(x),y) -> g_4#(x,y) g_5#(s(x),y) -> f_4#(y) g_5#(s(x),y) -> g_5#(x,y) - Signature: {f_0/1,f_1/1,f_2/1,f_3/1,f_4/1,f_5/1,g_1/2,g_2/2,g_3/2,g_4/2,g_5/2,f_0#/1,f_1#/1,f_2#/1,f_3#/1,f_4#/1,f_5#/1 ,g_1#/2,g_2#/2,g_3#/2,g_4#/2,g_5#/2} / {a/0,b/2,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1,c_7/1,c_8/2,c_9/2 ,c_10/2,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {f_0#,f_1#,f_2#,f_3#,f_4#,f_5#,g_1#,g_2#,g_3#,g_4# ,g_5#} and constructors {a,b,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^5))