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_ = WIDP} + Details: We add the following weak innermost dependency pairs: 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: UsableRules 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)) - 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,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: UsableRules + Details: We replace rewrite rules by usable rules: 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)) * Step 3: 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)) - 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 4: 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() - 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 5: 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)) - 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 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: Decompose 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: 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_1#(x) -> c_2(g_1#(x,x)) g_1#(s(x),y) -> c_7(g_1#(x,y)) - Weak DPs: 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_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} Problem (S) - Strict DPs: 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_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_1#(x) -> c_2(g_1#(x,x)) g_1#(s(x),y) -> c_7(g_1#(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} ** Step 7.a:1: DecomposeDG WORST_CASE(?,O(n^5)) + 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) -> 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_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 = Just someStrategy, 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.a:1: PredecessorEstimationCP 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) The strictly oriented rules are moved into the weak component. **** Step 7.a:1.a:1.a:1: NaturalMI 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: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_11) = {2} 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) = [1] p(b) = [0] p(f_0) = [1] 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 + [2] p(f_0#) = [2] x1 + [2] p(f_1#) = [1] p(f_2#) = [8] x1 + [2] p(f_3#) = [1] p(f_4#) = [1] p(f_5#) = [1] p(g_1#) = [1] x2 + [2] p(g_2#) = [1] x1 + [4] x2 + [0] p(g_3#) = [2] x1 + [0] p(g_4#) = [1] x2 + [0] p(g_5#) = [8] x1 + [2] p(c_1) = [0] p(c_2) = [1] p(c_3) = [2] p(c_4) = [1] x1 + [1] p(c_5) = [1] p(c_6) = [4] x1 + [1] p(c_7) = [1] p(c_8) = [1] p(c_9) = [1] x2 + [8] p(c_10) = [2] x2 + [2] p(c_11) = [8] x1 + [1] x2 + [5] Following rules are strictly oriented: g_5#(s(x),y) = [8] x + [18] > [8] x + [15] = c_11(f_4#(y),g_5#(x,y)) Following rules are (at-least) weakly oriented: **** Step 7.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 7.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W: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 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) **** Step 7.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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). *** Step 7.a:1.b:1: DecomposeDG WORST_CASE(?,O(n^4)) + 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) -> 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_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) -> 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 = Just someStrategy, 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.a:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) - Weak DPs: f_4#(x) -> c_5(g_4#(x,x)) 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) The strictly oriented rules are moved into the weak component. ***** Step 7.a:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) - Weak DPs: f_4#(x) -> c_5(g_4#(x,x)) 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: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: 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) = {2} 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) = [1] x1 + [4] p(f_0#) = [0] p(f_1#) = [0] p(f_2#) = [0] p(f_3#) = [6] p(f_4#) = [4] x1 + [8] p(f_5#) = [0] p(g_1#) = [0] p(g_2#) = [0] p(g_3#) = [0] p(g_4#) = [4] x1 + [8] p(g_5#) = [4] x1 + [6] x2 + [11] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [2] x1 + [1] x2 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [1] x2 + [5] p(c_11) = [1] x1 + [0] Following rules are strictly oriented: g_4#(s(x),y) = [4] x + [24] > [4] x + [19] = c_10(f_3#(y),g_4#(x,y)) Following rules are (at-least) weakly oriented: f_4#(x) = [4] x + [8] >= [4] x + [8] = c_5(g_4#(x,x)) g_5#(s(x),y) = [4] x + [6] y + [27] >= [4] y + [8] = f_4#(y) g_5#(s(x),y) = [4] x + [6] y + [27] >= [4] x + [6] y + [11] = g_5#(x,y) ***** Step 7.a:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: 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) - 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 7.a:1.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: 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) - 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W: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:W: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 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: g_5#(s(x),y) -> g_5#(x,y) 3: g_5#(s(x),y) -> f_4#(y) 1: f_4#(x) -> c_5(g_4#(x,x)) 2: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) ***** Step 7.a:1.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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). **** Step 7.a:1.b:1.b:1: DecomposeDG WORST_CASE(?,O(n^3)) + 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) -> c_3(g_2#(x,x)) f_3#(x) -> c_4(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) -> 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) - 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 = Just someStrategy, 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.a:1.b:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) - Weak DPs: f_3#(x) -> c_4(g_3#(x,x)) 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) The strictly oriented rules are moved into the weak component. ****** Step 7.a:1.b:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) - Weak DPs: f_3#(x) -> c_4(g_3#(x,x)) 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: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: 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) = {2} 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) = [2] x2 + [0] p(g_4) = [1] x2 + [1] p(g_5) = [2] x1 + [8] x2 + [0] p(s) = [1] x1 + [8] p(f_0#) = [2] p(f_1#) = [8] p(f_2#) = [0] p(f_3#) = [4] x1 + [2] p(f_4#) = [11] x1 + [4] p(f_5#) = [1] x1 + [1] p(g_1#) = [0] p(g_2#) = [2] x2 + [0] p(g_3#) = [2] x1 + [2] x2 + [0] p(g_4#) = [3] x1 + [8] x2 + [4] p(g_5#) = [1] x1 + [11] x2 + [5] p(c_1) = [1] p(c_2) = [2] p(c_3) = [0] p(c_4) = [1] x1 + [2] p(c_5) = [2] p(c_6) = [2] p(c_7) = [8] x1 + [1] p(c_8) = [1] x1 + [1] p(c_9) = [8] x1 + [1] x2 + [15] p(c_10) = [1] x1 + [1] x2 + [2] p(c_11) = [8] Following rules are strictly oriented: g_3#(s(x),y) = [2] x + [2] y + [16] > [2] x + [2] y + [15] = c_9(f_2#(y),g_3#(x,y)) Following rules are (at-least) weakly oriented: f_3#(x) = [4] x + [2] >= [4] x + [2] = c_4(g_3#(x,x)) f_4#(x) = [11] x + [4] >= [11] x + [4] = g_4#(x,x) g_4#(s(x),y) = [3] x + [8] y + [28] >= [4] y + [2] = f_3#(y) g_4#(s(x),y) = [3] x + [8] y + [28] >= [3] x + [8] y + [4] = g_4#(x,y) g_5#(s(x),y) = [1] x + [11] y + [13] >= [11] y + [4] = f_4#(y) g_5#(s(x),y) = [1] x + [11] y + [13] >= [1] x + [11] y + [5] = g_5#(x,y) ****** Step 7.a:1.b:1.b:1.a:1.a:2: Assumption 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(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) - 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 7.a:1.b:1.b:1.a:1.b:1: RemoveWeakSuffixes 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(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) - 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:f_3#(x) -> c_4(g_3#(x,x)) -->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):3 2: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 3:W: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)):3 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):2 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 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: g_5#(s(x),y) -> g_5#(x,y) 6: g_5#(s(x),y) -> f_4#(y) 2: f_4#(x) -> g_4#(x,x) 5: g_4#(s(x),y) -> g_4#(x,y) 4: g_4#(s(x),y) -> f_3#(y) 1: f_3#(x) -> c_4(g_3#(x,x)) 3: g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) ****** Step 7.a:1.b:1.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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). ***** Step 7.a:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + 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) -> 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) - 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: g_1#(s(x),y) -> c_7(g_1#(x,y)) The strictly oriented rules are moved into the weak component. ****** Step 7.a:1.b:1.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + 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) -> 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) - 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: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1,2} 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) = 1 p(b) = 2 + x1 p(f_0) = 1 + x1 p(f_1) = 1 + x1^2 p(f_2) = 2 p(f_3) = 0 p(f_4) = 1 + x1 p(f_5) = 1 p(g_1) = 1 + x1*x2 p(g_2) = 4 + 2*x1 + x2 p(g_3) = 4 + x1 + 4*x2 + x2^2 p(g_4) = 1 + x1 + x2 p(g_5) = 1 + x1 + 2*x1^2 p(s) = 2 + x1 p(f_0#) = 0 p(f_1#) = 5 + 5*x1 p(f_2#) = 6 + 2*x1 + 6*x1^2 p(f_3#) = 7 + 2*x1 + 6*x1^2 p(f_4#) = 7 + 7*x1^2 p(f_5#) = x1 p(g_1#) = 4*x1 + x2 p(g_2#) = 2*x1 + 4*x1*x2 + x1^2 + x2^2 p(g_3#) = 6 + 2*x2 + 6*x2^2 p(g_4#) = 7 + x1*x2 + 6*x2^2 p(g_5#) = 1 + 5*x1 + 4*x2 + 7*x2^2 p(c_1) = 1 p(c_2) = 5 + x1 p(c_3) = 6 + x1 p(c_4) = 0 p(c_5) = 4 + x1 p(c_6) = 1 + x1 p(c_7) = 2 + x1 p(c_8) = 1 + x1 + x2 p(c_9) = 1 + x2 p(c_10) = x1 p(c_11) = 2 + x2 Following rules are strictly oriented: g_1#(s(x),y) = 8 + 4*x + y > 2 + 4*x + y = c_7(g_1#(x,y)) Following rules are (at-least) weakly oriented: f_1#(x) = 5 + 5*x >= 5 + 5*x = c_2(g_1#(x,x)) f_2#(x) = 6 + 2*x + 6*x^2 >= 6 + 2*x + 6*x^2 = c_3(g_2#(x,x)) f_3#(x) = 7 + 2*x + 6*x^2 >= 6 + 2*x + 6*x^2 = g_3#(x,x) f_4#(x) = 7 + 7*x^2 >= 7 + 7*x^2 = g_4#(x,x) g_2#(s(x),y) = 8 + 6*x + 4*x*y + x^2 + 8*y + y^2 >= 6 + 2*x + 4*x*y + x^2 + 5*y + y^2 = c_8(f_1#(y),g_2#(x,y)) g_3#(s(x),y) = 6 + 2*y + 6*y^2 >= 6 + 2*y + 6*y^2 = f_2#(y) g_3#(s(x),y) = 6 + 2*y + 6*y^2 >= 6 + 2*y + 6*y^2 = g_3#(x,y) g_4#(s(x),y) = 7 + x*y + 2*y + 6*y^2 >= 7 + 2*y + 6*y^2 = f_3#(y) g_4#(s(x),y) = 7 + x*y + 2*y + 6*y^2 >= 7 + x*y + 6*y^2 = g_4#(x,y) g_5#(s(x),y) = 11 + 5*x + 4*y + 7*y^2 >= 7 + 7*y^2 = f_4#(y) g_5#(s(x),y) = 11 + 5*x + 4*y + 7*y^2 >= 1 + 5*x + 4*y + 7*y^2 = g_5#(x,y) ****** Step 7.a:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: f_1#(x) -> c_2(g_1#(x,x)) - 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_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) -> 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 7.a:1.b:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f_1#(x) -> c_2(g_1#(x,x)) - 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_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) -> 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: 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(g_1#(x,y)):5 2:W:f_2#(x) -> c_3(g_2#(x,x)) -->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):6 3:W:f_3#(x) -> g_3#(x,x) -->_1 g_3#(s(x),y) -> g_3#(x,y):8 -->_1 g_3#(s(x),y) -> f_2#(y):7 4:W:f_4#(x) -> g_4#(x,x) -->_1 g_4#(s(x),y) -> g_4#(x,y):10 -->_1 g_4#(s(x),y) -> f_3#(y):9 5:W:g_1#(s(x),y) -> c_7(g_1#(x,y)) -->_1 g_1#(s(x),y) -> c_7(g_1#(x,y)):5 6:W: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)):6 -->_1 f_1#(x) -> c_2(g_1#(x,x)):1 7:W:g_3#(s(x),y) -> f_2#(y) -->_1 f_2#(x) -> c_3(g_2#(x,x)):2 8:W:g_3#(s(x),y) -> g_3#(x,y) -->_1 g_3#(s(x),y) -> g_3#(x,y):8 -->_1 g_3#(s(x),y) -> f_2#(y):7 9:W:g_4#(s(x),y) -> f_3#(y) -->_1 f_3#(x) -> g_3#(x,x):3 10:W:g_4#(s(x),y) -> g_4#(x,y) -->_1 g_4#(s(x),y) -> g_4#(x,y):10 -->_1 g_4#(s(x),y) -> f_3#(y):9 11:W:g_5#(s(x),y) -> f_4#(y) -->_1 f_4#(x) -> g_4#(x,x):4 12:W:g_5#(s(x),y) -> g_5#(x,y) -->_1 g_5#(s(x),y) -> g_5#(x,y):12 -->_1 g_5#(s(x),y) -> f_4#(y):11 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: g_1#(s(x),y) -> c_7(g_1#(x,y)) ****** Step 7.a:1.b:1.b:1.b:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f_1#(x) -> c_2(g_1#(x,x)) - 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(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) - 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_1#(x) -> c_2(g_1#(x,x)) 2:W:f_2#(x) -> c_3(g_2#(x,x)) -->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):6 3:W:f_3#(x) -> g_3#(x,x) -->_1 g_3#(s(x),y) -> g_3#(x,y):8 -->_1 g_3#(s(x),y) -> f_2#(y):7 4:W:f_4#(x) -> g_4#(x,x) -->_1 g_4#(s(x),y) -> g_4#(x,y):10 -->_1 g_4#(s(x),y) -> f_3#(y):9 6:W: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)):6 -->_1 f_1#(x) -> c_2(g_1#(x,x)):1 7:W:g_3#(s(x),y) -> f_2#(y) -->_1 f_2#(x) -> c_3(g_2#(x,x)):2 8:W:g_3#(s(x),y) -> g_3#(x,y) -->_1 g_3#(s(x),y) -> g_3#(x,y):8 -->_1 g_3#(s(x),y) -> f_2#(y):7 9:W:g_4#(s(x),y) -> f_3#(y) -->_1 f_3#(x) -> g_3#(x,x):3 10:W:g_4#(s(x),y) -> g_4#(x,y) -->_1 g_4#(s(x),y) -> g_4#(x,y):10 -->_1 g_4#(s(x),y) -> f_3#(y):9 11:W:g_5#(s(x),y) -> f_4#(y) -->_1 f_4#(x) -> g_4#(x,x):4 12:W:g_5#(s(x),y) -> g_5#(x,y) -->_1 g_5#(s(x),y) -> g_5#(x,y):12 -->_1 g_5#(s(x),y) -> f_4#(y):11 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: f_1#(x) -> c_2() ****** Step 7.a:1.b:1.b:1.b:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f_1#(x) -> c_2() - 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(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) - 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_2/0,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: f_1#(x) -> c_2() The strictly oriented rules are moved into the weak component. ******* Step 7.a:1.b:1.b:1.b:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f_1#(x) -> c_2() - 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(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) - 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_2/0,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: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and 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) = {1,2} 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) = [1] x1 + [8] p(f_0#) = [0] p(f_1#) = [4] p(f_2#) = [8] x1 + [9] p(f_3#) = [12] x1 + [0] p(f_4#) = [14] x1 + [0] p(f_5#) = [2] x1 + [0] p(g_1#) = [8] x2 + [1] p(g_2#) = [1] x1 + [4] x2 + [8] p(g_3#) = [2] x1 + [9] x2 + [0] p(g_4#) = [2] x1 + [12] x2 + [0] p(g_5#) = [14] x2 + [2] p(c_1) = [0] p(c_2) = [3] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [2] p(c_5) = [2] x1 + [2] p(c_6) = [1] x1 + [1] p(c_7) = [1] p(c_8) = [1] x1 + [1] x2 + [4] p(c_9) = [1] x1 + [4] p(c_10) = [2] x1 + [1] x2 + [2] p(c_11) = [2] x1 + [1] x2 + [1] Following rules are strictly oriented: f_1#(x) = [4] > [3] = c_2() Following rules are (at-least) weakly oriented: f_2#(x) = [8] x + [9] >= [5] x + [8] = c_3(g_2#(x,x)) f_3#(x) = [12] x + [0] >= [11] x + [0] = g_3#(x,x) f_4#(x) = [14] x + [0] >= [14] x + [0] = g_4#(x,x) g_2#(s(x),y) = [1] x + [4] y + [16] >= [1] x + [4] y + [16] = c_8(f_1#(y),g_2#(x,y)) g_3#(s(x),y) = [2] x + [9] y + [16] >= [8] y + [9] = f_2#(y) g_3#(s(x),y) = [2] x + [9] y + [16] >= [2] x + [9] y + [0] = g_3#(x,y) g_4#(s(x),y) = [2] x + [12] y + [16] >= [12] y + [0] = f_3#(y) g_4#(s(x),y) = [2] x + [12] y + [16] >= [2] x + [12] y + [0] = g_4#(x,y) g_5#(s(x),y) = [14] y + [2] >= [14] y + [0] = f_4#(y) g_5#(s(x),y) = [14] y + [2] >= [14] y + [2] = g_5#(x,y) ******* Step 7.a:1.b:1.b:1.b:1.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f_1#(x) -> c_2() 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) - 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_2/0,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 7.a:1.b:1.b:1.b:1.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f_1#(x) -> c_2() 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) - 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_2/0,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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:f_1#(x) -> c_2() 2:W:f_2#(x) -> c_3(g_2#(x,x)) -->_1 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):5 3:W:f_3#(x) -> g_3#(x,x) -->_1 g_3#(s(x),y) -> g_3#(x,y):7 -->_1 g_3#(s(x),y) -> f_2#(y):6 4:W:f_4#(x) -> g_4#(x,x) -->_1 g_4#(s(x),y) -> g_4#(x,y):9 -->_1 g_4#(s(x),y) -> f_3#(y):8 5:W: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)):5 -->_1 f_1#(x) -> c_2():1 6:W:g_3#(s(x),y) -> f_2#(y) -->_1 f_2#(x) -> c_3(g_2#(x,x)):2 7:W:g_3#(s(x),y) -> g_3#(x,y) -->_1 g_3#(s(x),y) -> g_3#(x,y):7 -->_1 g_3#(s(x),y) -> f_2#(y):6 8:W:g_4#(s(x),y) -> f_3#(y) -->_1 f_3#(x) -> g_3#(x,x):3 9:W:g_4#(s(x),y) -> g_4#(x,y) -->_1 g_4#(s(x),y) -> g_4#(x,y):9 -->_1 g_4#(s(x),y) -> f_3#(y):8 10:W:g_5#(s(x),y) -> f_4#(y) -->_1 f_4#(x) -> g_4#(x,x):4 11:W:g_5#(s(x),y) -> g_5#(x,y) -->_1 g_5#(s(x),y) -> g_5#(x,y):11 -->_1 g_5#(s(x),y) -> f_4#(y):10 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: g_5#(s(x),y) -> g_5#(x,y) 10: g_5#(s(x),y) -> f_4#(y) 4: f_4#(x) -> g_4#(x,x) 9: g_4#(s(x),y) -> g_4#(x,y) 8: g_4#(s(x),y) -> f_3#(y) 3: f_3#(x) -> g_3#(x,x) 7: g_3#(s(x),y) -> g_3#(x,y) 6: g_3#(s(x),y) -> f_2#(y) 2: f_2#(x) -> c_3(g_2#(x,x)) 5: g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) 1: f_1#(x) -> c_2() ******* Step 7.a:1.b:1.b:1.b:1.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_2/0,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). ** Step 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: 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_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_1#(x) -> c_2(g_1#(x,x)) g_1#(s(x),y) -> c_7(g_1#(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: RemoveWeakSuffixes + 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)):4 2: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)):5 3: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)):6 4:S:g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)) -->_1 f_1#(x) -> c_2(g_1#(x,x)):8 -->_2 g_2#(s(x),y) -> c_8(f_1#(y),g_2#(x,y)):4 5: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)):5 -->_1 f_2#(x) -> c_3(g_2#(x,x)):1 6: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)):6 -->_1 f_3#(x) -> c_4(g_3#(x,x)):2 7: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)):7 -->_1 f_4#(x) -> c_5(g_4#(x,x)):3 8:W:f_1#(x) -> c_2(g_1#(x,x)) -->_1 g_1#(s(x),y) -> c_7(g_1#(x,y)):9 9:W:g_1#(s(x),y) -> c_7(g_1#(x,y)) -->_1 g_1#(s(x),y) -> c_7(g_1#(x,y)):9 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: f_1#(x) -> c_2(g_1#(x,x)) 9: g_1#(s(x),y) -> c_7(g_1#(x,y)) ** Step 7.b:2: SimplifyRHS WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: 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_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: 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)):4 2: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)):5 3: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)):6 4: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)):4 5: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)):5 -->_1 f_2#(x) -> c_3(g_2#(x,x)):1 6: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)):6 -->_1 f_3#(x) -> c_4(g_3#(x,x)):2 7: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)):7 -->_1 f_4#(x) -> c_5(g_4#(x,x)):3 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:3: Decompose WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: 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_2#(s(x),y) -> c_8(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_8/1,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: 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_2#(x) -> c_3(g_2#(x,x)) g_2#(s(x),y) -> c_8(g_2#(x,y)) - Weak DPs: f_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) 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_8/1 ,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} Problem (S) - Strict DPs: f_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) 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_2#(x) -> c_3(g_2#(x,x)) g_2#(s(x),y) -> c_8(g_2#(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} *** Step 7.b:3.a:1: DecomposeDG WORST_CASE(?,O(n^4)) + 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) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) 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_8/1,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: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, 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_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_2#(s(x),y) -> c_8(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.b:3.a:1.a:1: PredecessorEstimationCP 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_8/1,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) The strictly oriented rules are moved into the weak component. ***** Step 7.b:3.a:1.a:1.a:1: NaturalMI 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_8/1,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 first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_11) = {2} 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) = [1] p(b) = [1] x2 + [1] p(f_0) = [1] x1 + [1] p(f_1) = [8] 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) = [2] x2 + [1] p(g_5) = [2] p(s) = [1] x1 + [2] p(f_0#) = [1] x1 + [1] p(f_1#) = [0] p(f_2#) = [0] p(f_3#) = [0] p(f_4#) = [0] p(f_5#) = [1] p(g_1#) = [2] x2 + [0] p(g_2#) = [1] x1 + [1] x2 + [1] p(g_3#) = [1] p(g_4#) = [1] x1 + [2] x2 + [0] p(g_5#) = [1] x1 + [0] p(c_1) = [4] p(c_2) = [1] x1 + [0] p(c_3) = [1] p(c_4) = [1] x1 + [0] p(c_5) = [2] x1 + [0] p(c_6) = [2] x1 + [0] p(c_7) = [2] x1 + [2] p(c_8) = [1] x1 + [4] p(c_9) = [1] x2 + [0] p(c_10) = [1] x1 + [1] x2 + [1] p(c_11) = [1] x2 + [0] Following rules are strictly oriented: g_5#(s(x),y) = [1] x + [2] > [1] x + [0] = c_11(f_4#(y),g_5#(x,y)) Following rules are (at-least) weakly oriented: ***** Step 7.b:3.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak 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_8/1,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 7.b:3.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak 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_8/1,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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W: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 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) ***** Step 7.b:3.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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:3.a:1.b:1: DecomposeDG WORST_CASE(?,O(n^3)) + 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) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) 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) -> 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: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, 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_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> c_4(g_3#(x,x)) g_2#(s(x),y) -> c_8(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:3.a:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) - Weak DPs: f_4#(x) -> c_5(g_4#(x,x)) 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) The strictly oriented rules are moved into the weak component. ****** Step 7.b:3.a:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) - Weak DPs: f_4#(x) -> c_5(g_4#(x,x)) 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 first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: 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) = {2} 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) = [1] x1 + [2] p(f_0#) = [0] p(f_1#) = [0] p(f_2#) = [0] p(f_3#) = [0] p(f_4#) = [9] x1 + [3] p(f_5#) = [1] x1 + [0] p(g_1#) = [1] x1 + [1] x2 + [0] p(g_2#) = [1] x1 + [1] x2 + [0] p(g_3#) = [1] x2 + [0] p(g_4#) = [9] x1 + [1] p(g_5#) = [9] x2 + [3] p(c_1) = [0] p(c_2) = [2] x1 + [0] p(c_3) = [2] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [1] p(c_6) = [2] x1 + [0] p(c_7) = [2] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [2] p(c_10) = [1] x2 + [10] p(c_11) = [1] x2 + [0] Following rules are strictly oriented: g_4#(s(x),y) = [9] x + [19] > [9] x + [11] = c_10(f_3#(y),g_4#(x,y)) Following rules are (at-least) weakly oriented: f_4#(x) = [9] x + [3] >= [9] x + [2] = c_5(g_4#(x,x)) g_5#(s(x),y) = [9] y + [3] >= [9] y + [3] = f_4#(y) g_5#(s(x),y) = [9] y + [3] >= [9] y + [3] = g_5#(x,y) ****** Step 7.b:3.a:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: 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) - 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 7.b:3.a:1.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: 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) - 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W: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:W: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 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: g_5#(s(x),y) -> g_5#(x,y) 3: g_5#(s(x),y) -> f_4#(y) 1: f_4#(x) -> c_5(g_4#(x,x)) 2: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) ****** Step 7.b:3.a:1.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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:3.a:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + 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) -> 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) - 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: f_2#(x) -> c_3(g_2#(x,x)) 2: g_2#(s(x),y) -> c_8(g_2#(x,y)) The strictly oriented rules are moved into the weak component. ****** Step 7.b:3.a:1.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + 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) -> 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) - 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: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_8) = {1}, uargs(c_9) = {1,2} 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) = 4 + x2 p(f_0) = 0 p(f_1) = 1 + 2*x1^2 p(f_2) = 2 + 4*x1 + x1^2 p(f_3) = 2*x1^2 p(f_4) = 4 + x1 + 2*x1^2 p(f_5) = 1 + x1 + x1^2 p(g_1) = 1 + 2*x1^2 + x2^2 p(g_2) = 1 + x1 + x1*x2 + 2*x2 + x2^2 p(g_3) = 2 + x1 + x1*x2 + 4*x1^2 + x2 + x2^2 p(g_4) = x1^2 + 2*x2 + 4*x2^2 p(g_5) = 2*x1 + 4*x1*x2 + x2^2 p(s) = 3 + x1 p(f_0#) = 1 + 4*x1^2 p(f_1#) = 1 + x1 p(f_2#) = 7 + 4*x1 p(f_3#) = 5*x1 + 6*x1^2 p(f_4#) = 4 + 3*x1 + 7*x1^2 p(f_5#) = x1 + 4*x1^2 p(g_1#) = 4*x1 + x1*x2 + 4*x1^2 + 4*x2 p(g_2#) = 5 + x1 p(g_3#) = 3*x1 + 4*x1*x2 + 2*x2 + 2*x2^2 p(g_4#) = 2 + x1 + x1*x2 + 2*x2 + 6*x2^2 p(g_5#) = 2 + x1 + x1*x2 + x1^2 + x2 + 7*x2^2 p(c_1) = 1 p(c_2) = 1 p(c_3) = x1 p(c_4) = x1 p(c_5) = 0 p(c_6) = 0 p(c_7) = 1 p(c_8) = 2 + x1 p(c_9) = 1 + x1 + x2 p(c_10) = 2 + x2 p(c_11) = 0 Following rules are strictly oriented: f_2#(x) = 7 + 4*x > 5 + x = c_3(g_2#(x,x)) g_2#(s(x),y) = 8 + x > 7 + x = c_8(g_2#(x,y)) Following rules are (at-least) weakly oriented: f_3#(x) = 5*x + 6*x^2 >= 5*x + 6*x^2 = c_4(g_3#(x,x)) f_4#(x) = 4 + 3*x + 7*x^2 >= 2 + 3*x + 7*x^2 = g_4#(x,x) g_3#(s(x),y) = 9 + 3*x + 4*x*y + 14*y + 2*y^2 >= 8 + 3*x + 4*x*y + 6*y + 2*y^2 = c_9(f_2#(y),g_3#(x,y)) g_4#(s(x),y) = 5 + x + x*y + 5*y + 6*y^2 >= 5*y + 6*y^2 = f_3#(y) g_4#(s(x),y) = 5 + x + x*y + 5*y + 6*y^2 >= 2 + x + x*y + 2*y + 6*y^2 = g_4#(x,y) g_5#(s(x),y) = 14 + 7*x + x*y + x^2 + 4*y + 7*y^2 >= 4 + 3*y + 7*y^2 = f_4#(y) g_5#(s(x),y) = 14 + 7*x + x*y + x^2 + 4*y + 7*y^2 >= 2 + x + x*y + x^2 + y + 7*y^2 = g_5#(x,y) ****** Step 7.b:3.a:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> c_4(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) -> 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) - 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 7.b:3.a:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f_2#(x) -> c_3(g_2#(x,x)) f_3#(x) -> c_4(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) -> 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) - 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:f_2#(x) -> c_3(g_2#(x,x)) -->_1 g_2#(s(x),y) -> c_8(g_2#(x,y)):4 2:W:f_3#(x) -> c_4(g_3#(x,x)) -->_1 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):5 3:W:f_4#(x) -> g_4#(x,x) -->_1 g_4#(s(x),y) -> g_4#(x,y):7 -->_1 g_4#(s(x),y) -> f_3#(y):6 4:W:g_2#(s(x),y) -> c_8(g_2#(x,y)) -->_1 g_2#(s(x),y) -> c_8(g_2#(x,y)):4 5:W: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)):5 -->_1 f_2#(x) -> c_3(g_2#(x,x)):1 6:W:g_4#(s(x),y) -> f_3#(y) -->_1 f_3#(x) -> c_4(g_3#(x,x)):2 7:W:g_4#(s(x),y) -> g_4#(x,y) -->_1 g_4#(s(x),y) -> g_4#(x,y):7 -->_1 g_4#(s(x),y) -> f_3#(y):6 8:W:g_5#(s(x),y) -> f_4#(y) -->_1 f_4#(x) -> g_4#(x,x):3 9:W:g_5#(s(x),y) -> g_5#(x,y) -->_1 g_5#(s(x),y) -> g_5#(x,y):9 -->_1 g_5#(s(x),y) -> f_4#(y):8 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: g_5#(s(x),y) -> g_5#(x,y) 8: g_5#(s(x),y) -> f_4#(y) 3: f_4#(x) -> g_4#(x,x) 7: g_4#(s(x),y) -> g_4#(x,y) 6: g_4#(s(x),y) -> f_3#(y) 2: f_3#(x) -> c_4(g_3#(x,x)) 5: g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) 1: f_2#(x) -> c_3(g_2#(x,x)) 4: g_2#(s(x),y) -> c_8(g_2#(x,y)) ****** Step 7.b:3.a:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: f_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) 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_2#(x) -> c_3(g_2#(x,x)) g_2#(s(x),y) -> c_8(g_2#(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: RemoveWeakSuffixes + 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)):3 2: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)):4 3:S:g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)) -->_1 f_2#(x) -> c_3(g_2#(x,x)):6 -->_2 g_3#(s(x),y) -> c_9(f_2#(y),g_3#(x,y)):3 4: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)):4 -->_1 f_3#(x) -> c_4(g_3#(x,x)):1 5: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)):5 -->_1 f_4#(x) -> c_5(g_4#(x,x)):2 6:W:f_2#(x) -> c_3(g_2#(x,x)) -->_1 g_2#(s(x),y) -> c_8(g_2#(x,y)):7 7:W:g_2#(s(x),y) -> c_8(g_2#(x,y)) -->_1 g_2#(s(x),y) -> c_8(g_2#(x,y)):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: f_2#(x) -> c_3(g_2#(x,x)) 7: g_2#(s(x),y) -> c_8(g_2#(x,y)) *** Step 7.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: f_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) 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_8/1,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: 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)):3 2: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)):4 3: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)):3 4: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)):4 -->_1 f_3#(x) -> c_4(g_3#(x,x)):1 5: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)):5 -->_1 f_4#(x) -> c_5(g_4#(x,x)):2 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:3.b:3: Decompose WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: f_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) g_3#(s(x),y) -> c_9(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_8/1,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: 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_3#(x) -> c_4(g_3#(x,x)) g_3#(s(x),y) -> c_9(g_3#(x,y)) - Weak DPs: 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) -> 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_8/1 ,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} Problem (S) - 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)) g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) - Weak DPs: f_3#(x) -> c_4(g_3#(x,x)) g_3#(s(x),y) -> c_9(g_3#(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_9/1,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} **** Step 7.b:3.b:3.a:1: DecomposeDG WORST_CASE(?,O(n^3)) + 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) -> 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) -> 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_8/1,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: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, 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_3#(x) -> c_4(g_3#(x,x)) f_4#(x) -> c_5(g_4#(x,x)) g_3#(s(x),y) -> c_9(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.b:3.b:3.a:1.a:1: PredecessorEstimationCP 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_8/1,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) The strictly oriented rules are moved into the weak component. ****** Step 7.b:3.b:3.a:1.a:1.a:1: NaturalMI 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_8/1,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: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_11) = {2} 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) = [1] x2 + [0] p(g_4) = [1] x1 + [0] p(g_5) = [1] x1 + [1] x2 + [1] p(s) = [1] x1 + [2] p(f_0#) = [0] p(f_1#) = [1] x1 + [0] p(f_2#) = [1] p(f_3#) = [8] x1 + [1] p(f_4#) = [1] p(f_5#) = [1] p(g_1#) = [2] x1 + [1] x2 + [1] p(g_2#) = [1] x1 + [1] p(g_3#) = [8] x1 + [0] p(g_4#) = [1] x1 + [1] p(g_5#) = [8] x1 + [2] x2 + [0] p(c_1) = [1] p(c_2) = [0] p(c_3) = [2] p(c_4) = [1] x1 + [2] p(c_5) = [1] x1 + [2] p(c_6) = [0] p(c_7) = [1] p(c_8) = [1] x1 + [1] p(c_9) = [1] x1 + [4] p(c_10) = [4] x1 + [1] x2 + [1] p(c_11) = [1] x1 + [1] x2 + [13] Following rules are strictly oriented: g_5#(s(x),y) = [8] x + [2] y + [16] > [8] x + [2] y + [14] = c_11(f_4#(y),g_5#(x,y)) Following rules are (at-least) weakly oriented: ****** Step 7.b:3.b:3.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak 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_8/1,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 7.b:3.b:3.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak 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_8/1,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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W: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 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) ****** Step 7.b:3.b:3.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_9/1 ,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:3.b:3.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + 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) -> 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) - 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_9/1 ,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: g_3#(s(x),y) -> c_9(g_3#(x,y)) The strictly oriented rules are moved into the weak component. ****** Step 7.b:3.b:3.a:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + 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) -> 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) - 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_9/1 ,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: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1,2} 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) = 0 p(f_0) = 4*x1 p(f_1) = 1 + x1^2 p(f_2) = 1 + 2*x1 + 2*x1^2 p(f_3) = 2 + x1 + x1^2 p(f_4) = 1 p(f_5) = 4 p(g_1) = x1^2 + x2 + 2*x2^2 p(g_2) = x1*x2 + 4*x2 p(g_3) = x2^2 p(g_4) = x1*x2 + 4*x2 p(g_5) = 1 + 4*x1^2 p(s) = 2 + x1 p(f_0#) = 0 p(f_1#) = x1 + x1^2 p(f_2#) = 1 p(f_3#) = 2*x1 p(f_4#) = 7 + 7*x1 + 5*x1^2 p(f_5#) = 2 + 2*x1 + x1^2 p(g_1#) = 4 p(g_2#) = x1 + x1*x2 p(g_3#) = x1 p(g_4#) = 2*x1*x2 + 4*x2 p(g_5#) = 1 + 3*x1 + 4*x1*x2 + 2*x2 + 5*x2^2 p(c_1) = 2 p(c_2) = 1 p(c_3) = 0 p(c_4) = x1 p(c_5) = 7 + x1 p(c_6) = 0 p(c_7) = x1 p(c_8) = 0 p(c_9) = x1 p(c_10) = x1 + x2 p(c_11) = 1 + x2 Following rules are strictly oriented: g_3#(s(x),y) = 2 + x > x = c_9(g_3#(x,y)) Following rules are (at-least) weakly oriented: f_3#(x) = 2*x >= x = c_4(g_3#(x,x)) f_4#(x) = 7 + 7*x + 5*x^2 >= 7 + 4*x + 2*x^2 = c_5(g_4#(x,x)) g_4#(s(x),y) = 2*x*y + 8*y >= 2*x*y + 6*y = c_10(f_3#(y),g_4#(x,y)) g_5#(s(x),y) = 7 + 3*x + 4*x*y + 10*y + 5*y^2 >= 7 + 7*y + 5*y^2 = f_4#(y) g_5#(s(x),y) = 7 + 3*x + 4*x*y + 10*y + 5*y^2 >= 1 + 3*x + 4*x*y + 2*y + 5*y^2 = g_5#(x,y) ****** Step 7.b:3.b:3.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: f_3#(x) -> c_4(g_3#(x,x)) - Weak DPs: f_4#(x) -> c_5(g_4#(x,x)) g_3#(s(x),y) -> c_9(g_3#(x,y)) 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) - 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_9/1 ,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 7.b:3.b:3.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f_3#(x) -> c_4(g_3#(x,x)) - Weak DPs: f_4#(x) -> c_5(g_4#(x,x)) g_3#(s(x),y) -> c_9(g_3#(x,y)) 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) - 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_9/1 ,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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:f_3#(x) -> c_4(g_3#(x,x)) -->_1 g_3#(s(x),y) -> c_9(g_3#(x,y)):3 2:W:f_4#(x) -> c_5(g_4#(x,x)) -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):4 3:W:g_3#(s(x),y) -> c_9(g_3#(x,y)) -->_1 g_3#(s(x),y) -> c_9(g_3#(x,y)):3 4:W: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)):4 -->_1 f_3#(x) -> c_4(g_3#(x,x)):1 5:W:g_5#(s(x),y) -> f_4#(y) -->_1 f_4#(x) -> c_5(g_4#(x,x)):2 6:W:g_5#(s(x),y) -> g_5#(x,y) -->_1 g_5#(s(x),y) -> g_5#(x,y):6 -->_1 g_5#(s(x),y) -> f_4#(y):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: g_3#(s(x),y) -> c_9(g_3#(x,y)) ****** Step 7.b:3.b:3.a:1.b:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f_3#(x) -> c_4(g_3#(x,x)) - Weak DPs: 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) - 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_9/1 ,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: SimplifyRHS + Details: Consider the dependency graph 1:S:f_3#(x) -> c_4(g_3#(x,x)) 2:W:f_4#(x) -> c_5(g_4#(x,x)) -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):4 4:W: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)):4 -->_1 f_3#(x) -> c_4(g_3#(x,x)):1 5:W:g_5#(s(x),y) -> f_4#(y) -->_1 f_4#(x) -> c_5(g_4#(x,x)):2 6:W:g_5#(s(x),y) -> g_5#(x,y) -->_1 g_5#(s(x),y) -> g_5#(x,y):6 -->_1 g_5#(s(x),y) -> f_4#(y):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: f_3#(x) -> c_4() ****** Step 7.b:3.b:3.a:1.b:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f_3#(x) -> c_4() - Weak DPs: 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) - 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_4/0,c_5/1,c_6/1,c_7/1,c_8/1,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: f_3#(x) -> c_4() The strictly oriented rules are moved into the weak component. ******* Step 7.b:3.b:3.a:1.b:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f_3#(x) -> c_4() - Weak DPs: 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) - 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_4/0,c_5/1,c_6/1,c_7/1,c_8/1,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: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: 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) = {1,2} 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) = [1] x1 + [2] p(f_0#) = [0] p(f_1#) = [0] p(f_2#) = [0] p(f_3#) = [10] p(f_4#) = [8] x1 + [4] p(f_5#) = [0] p(g_1#) = [0] p(g_2#) = [0] p(g_3#) = [1] x1 + [4] x2 + [2] p(g_4#) = [6] x1 + [1] p(g_5#) = [2] x1 + [8] x2 + [0] p(c_1) = [1] p(c_2) = [0] p(c_3) = [2] p(c_4) = [8] p(c_5) = [1] x1 + [3] p(c_6) = [1] p(c_7) = [2] p(c_8) = [1] p(c_9) = [8] p(c_10) = [1] x1 + [1] x2 + [2] p(c_11) = [2] Following rules are strictly oriented: f_3#(x) = [10] > [8] = c_4() Following rules are (at-least) weakly oriented: f_4#(x) = [8] x + [4] >= [6] x + [4] = c_5(g_4#(x,x)) g_4#(s(x),y) = [6] x + [13] >= [6] x + [13] = c_10(f_3#(y),g_4#(x,y)) g_5#(s(x),y) = [2] x + [8] y + [4] >= [8] y + [4] = f_4#(y) g_5#(s(x),y) = [2] x + [8] y + [4] >= [2] x + [8] y + [0] = g_5#(x,y) ******* Step 7.b:3.b:3.a:1.b:1.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f_3#(x) -> c_4() 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) - 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_4/0,c_5/1,c_6/1,c_7/1,c_8/1,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 7.b:3.b:3.a:1.b:1.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: f_3#(x) -> c_4() 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) - 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_4/0,c_5/1,c_6/1,c_7/1,c_8/1,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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:f_3#(x) -> c_4() 2:W:f_4#(x) -> c_5(g_4#(x,x)) -->_1 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):3 3:W: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)):3 -->_1 f_3#(x) -> c_4():1 4:W:g_5#(s(x),y) -> f_4#(y) -->_1 f_4#(x) -> c_5(g_4#(x,x)):2 5:W:g_5#(s(x),y) -> g_5#(x,y) -->_1 g_5#(s(x),y) -> g_5#(x,y):5 -->_1 g_5#(s(x),y) -> f_4#(y):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: g_5#(s(x),y) -> g_5#(x,y) 4: g_5#(s(x),y) -> f_4#(y) 2: f_4#(x) -> c_5(g_4#(x,x)) 3: g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)) 1: f_3#(x) -> c_4() ******* Step 7.b:3.b:3.a:1.b:1.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_4/0,c_5/1,c_6/1,c_7/1,c_8/1,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:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + 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)) g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) - Weak DPs: f_3#(x) -> c_4(g_3#(x,x)) g_3#(s(x),y) -> c_9(g_3#(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_9/1 ,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: RemoveWeakSuffixes + 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)) -->_1 f_3#(x) -> c_4(g_3#(x,x)):4 -->_2 g_4#(s(x),y) -> c_10(f_3#(y),g_4#(x,y)):2 3: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)):3 -->_1 f_4#(x) -> c_5(g_4#(x,x)):1 4:W:f_3#(x) -> c_4(g_3#(x,x)) -->_1 g_3#(s(x),y) -> c_9(g_3#(x,y)):5 5:W:g_3#(s(x),y) -> c_9(g_3#(x,y)) -->_1 g_3#(s(x),y) -> c_9(g_3#(x,y)):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: f_3#(x) -> c_4(g_3#(x,x)) 5: g_3#(s(x),y) -> c_9(g_3#(x,y)) **** Step 7.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + 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)) 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_8/1,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: 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: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)):3 -->_1 f_4#(x) -> c_5(g_4#(x,x)):1 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:3.b:3.b:3: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict 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) -> 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_8/1,c_9/1 ,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: 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_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) -> 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_8/1 ,c_9/1,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} Problem (S) - Strict DPs: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) - Weak DPs: f_4#(x) -> c_5(g_4#(x,x)) g_4#(s(x),y) -> c_10(g_4#(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_9/1,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} ***** Step 7.b:3.b:3.b:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + 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) -> 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_8/1,c_9/1 ,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: f_4#(x) -> c_5(g_4#(x,x)) 2: g_4#(s(x),y) -> c_10(g_4#(x,y)) The strictly oriented rules are moved into the weak component. ****** Step 7.b:3.b:3.b:3.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + 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) -> 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_8/1,c_9/1 ,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: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_5) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1,2} 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) = x1 p(f_0) = x1 + 2*x1^2 p(f_1) = 1 + 2*x1 + 4*x1^2 p(f_2) = x1 + x1^2 p(f_3) = 4*x1 + 2*x1^2 p(f_4) = x1 + 2*x1^2 p(f_5) = 1 + x1 + x1^2 p(g_1) = 1 + 4*x1 + 2*x1*x2 + x1^2 p(g_2) = 1 + 8*x1 + x1*x2 + 2*x2 + 8*x2^2 p(g_3) = x1 + 2*x1^2 + x2^2 p(g_4) = x1 + x1*x2 + x1^2 + x2^2 p(g_5) = 2*x1*x2 + x1^2 + 4*x2 + x2^2 p(s) = 2 + x1 p(f_0#) = x1 p(f_1#) = x1 + 2*x1^2 p(f_2#) = 2 p(f_3#) = 1 + x1 p(f_4#) = 8 + 7*x1 p(f_5#) = 1 + x1 + 4*x1^2 p(g_1#) = x1*x2 + x2 p(g_2#) = x1 + 4*x1^2 + 2*x2^2 p(g_3#) = x1^2 + x2 p(g_4#) = 4*x1 + 3*x2 p(g_5#) = 2*x1 + 11*x1*x2 + 3*x1^2 + 2*x2 + 2*x2^2 p(c_1) = 1 p(c_2) = 2 + x1 p(c_3) = 1 p(c_4) = 0 p(c_5) = x1 p(c_6) = 1 + x1 p(c_7) = 0 p(c_8) = 0 p(c_9) = 2 + x1 p(c_10) = 2 + x1 p(c_11) = 2 + x1 + x2 Following rules are strictly oriented: f_4#(x) = 8 + 7*x > 7*x = c_5(g_4#(x,x)) g_4#(s(x),y) = 8 + 4*x + 3*y > 2 + 4*x + 3*y = c_10(g_4#(x,y)) Following rules are (at-least) weakly oriented: g_5#(s(x),y) = 16 + 14*x + 11*x*y + 3*x^2 + 24*y + 2*y^2 >= 10 + 2*x + 11*x*y + 3*x^2 + 9*y + 2*y^2 = c_11(f_4#(y),g_5#(x,y)) ****** Step 7.b:3.b:3.b:3.a:1.a:2: Assumption 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) -> 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_8/1,c_9/1 ,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 7.b:3.b:3.b:3.a:1.b:1: RemoveWeakSuffixes 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) -> 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_8/1,c_9/1 ,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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:f_4#(x) -> c_5(g_4#(x,x)) -->_1 g_4#(s(x),y) -> c_10(g_4#(x,y)):2 2:W:g_4#(s(x),y) -> c_10(g_4#(x,y)) -->_1 g_4#(s(x),y) -> c_10(g_4#(x,y)):2 3:W: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)):3 -->_1 f_4#(x) -> c_5(g_4#(x,x)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) 1: f_4#(x) -> c_5(g_4#(x,x)) 2: g_4#(s(x),y) -> c_10(g_4#(x,y)) ****** Step 7.b:3.b:3.b:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_9/1 ,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:3.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) - Weak DPs: f_4#(x) -> c_5(g_4#(x,x)) g_4#(s(x),y) -> c_10(g_4#(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_9/1 ,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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)) -->_1 f_4#(x) -> c_5(g_4#(x,x)):2 -->_2 g_5#(s(x),y) -> c_11(f_4#(y),g_5#(x,y)):1 2:W:f_4#(x) -> c_5(g_4#(x,x)) -->_1 g_4#(s(x),y) -> c_10(g_4#(x,y)):3 3:W:g_4#(s(x),y) -> c_10(g_4#(x,y)) -->_1 g_4#(s(x),y) -> c_10(g_4#(x,y)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: f_4#(x) -> c_5(g_4#(x,x)) 3: g_4#(s(x),y) -> c_10(g_4#(x,y)) ***** Step 7.b:3.b:3.b:3.b:2: 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_8/1,c_9/1 ,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: 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.b:3.b:3.b:3.b:3: PredecessorEstimationCP 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_8/1,c_9/1 ,c_10/1,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: g_5#(s(x),y) -> c_11(g_5#(x,y)) The strictly oriented rules are moved into the weak component. ****** Step 7.b:3.b:3.b:3.b:3.a:1: NaturalMI 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_8/1,c_9/1 ,c_10/1,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: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: 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: {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) = [1] p(b) = [1] x1 + [1] p(f_0) = [1] p(f_1) = [1] p(f_2) = [2] p(f_3) = [1] p(f_4) = [2] x1 + [1] p(f_5) = [1] p(g_1) = [0] p(g_2) = [0] p(g_3) = [0] p(g_4) = [0] p(g_5) = [0] p(s) = [1] x1 + [6] 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#) = [1] x1 + [1] p(g_3#) = [2] x2 + [8] p(g_4#) = [0] p(g_5#) = [2] x1 + [0] p(c_1) = [1] p(c_2) = [2] p(c_3) = [1] x1 + [1] p(c_4) = [2] x1 + [0] p(c_5) = [1] p(c_6) = [8] x1 + [0] p(c_7) = [2] p(c_8) = [4] x1 + [0] p(c_9) = [1] x1 + [0] p(c_10) = [8] p(c_11) = [1] x1 + [10] Following rules are strictly oriented: g_5#(s(x),y) = [2] x + [12] > [2] x + [10] = c_11(g_5#(x,y)) Following rules are (at-least) weakly oriented: ****** Step 7.b:3.b:3.b:3.b:3.a:2: Assumption 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_8/1,c_9/1 ,c_10/1,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 7.b:3.b:3.b:3.b:3.b:1: RemoveWeakSuffixes 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_8/1,c_9/1 ,c_10/1,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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:g_5#(s(x),y) -> c_11(g_5#(x,y)) -->_1 g_5#(s(x),y) -> c_11(g_5#(x,y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: g_5#(s(x),y) -> c_11(g_5#(x,y)) ****** Step 7.b:3.b:3.b:3.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_9/1 ,c_10/1,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). WORST_CASE(?,O(n^5))