WORST_CASE(?,O(n^1)) * Step 1: Sum. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: F(0(),z0,0(),z1) -> c2() F(0(),z0,s(z1),z2) -> c3() F(s(z0),0(),z1,z2) -> c4(F(z0,z2,minus(z1,s(z0)),z2)) F(s(z0),s(z1),z2,z3) -> c5(F(s(z0),minus(z1,z0),z2,z3)) F(s(z0),s(z1),z2,z3) -> c6(F(z0,z3,z2,z3)) PERFECTP(0()) -> c() PERFECTP(s(z0)) -> c1(F(z0,s(0()),s(z0),s(z0))) - Weak TRS: f(0(),z0,0(),z1) -> true() f(0(),z0,s(z1),z2) -> false() f(s(z0),0(),z1,z2) -> f(z0,z2,minus(z1,s(z0)),z2) f(s(z0),s(z1),z2,z3) -> if(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) perfectp(0()) -> false() perfectp(s(z0)) -> f(z0,s(0()),s(z0),s(z0)) - Signature: {F/4,PERFECTP/1,f/4,perfectp/1} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/1,c6/1,false/0,if/3,le/2,minus/2,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {F,PERFECTP,f,perfectp} and constructors {0,c,c1,c2,c3,c4 ,c5,c6,false,if,le,minus,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: F(0(),z0,0(),z1) -> c2() F(0(),z0,s(z1),z2) -> c3() F(s(z0),0(),z1,z2) -> c4(F(z0,z2,minus(z1,s(z0)),z2)) F(s(z0),s(z1),z2,z3) -> c5(F(s(z0),minus(z1,z0),z2,z3)) F(s(z0),s(z1),z2,z3) -> c6(F(z0,z3,z2,z3)) PERFECTP(0()) -> c() PERFECTP(s(z0)) -> c1(F(z0,s(0()),s(z0),s(z0))) - Weak TRS: f(0(),z0,0(),z1) -> true() f(0(),z0,s(z1),z2) -> false() f(s(z0),0(),z1,z2) -> f(z0,z2,minus(z1,s(z0)),z2) f(s(z0),s(z1),z2,z3) -> if(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) perfectp(0()) -> false() perfectp(s(z0)) -> f(z0,s(0()),s(z0),s(z0)) - Signature: {F/4,PERFECTP/1,f/4,perfectp/1} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/1,c6/1,false/0,if/3,le/2,minus/2,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {F,PERFECTP,f,perfectp} and constructors {0,c,c1,c2,c3,c4 ,c5,c6,false,if,le,minus,s,true} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs F#(0(),z0,0(),z1) -> c_1() F#(0(),z0,s(z1),z2) -> c_2() F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)) F#(s(z0),s(z1),z2,z3) -> c_4(F#(s(z0),minus(z1,z0),z2,z3)) F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)) PERFECTP#(0()) -> c_6() PERFECTP#(s(z0)) -> c_7(F#(z0,s(0()),s(z0),s(z0))) Weak DPs f#(0(),z0,0(),z1) -> c_8() f#(0(),z0,s(z1),z2) -> c_9() f#(s(z0),0(),z1,z2) -> c_10(f#(z0,z2,minus(z1,s(z0)),z2)) f#(s(z0),s(z1),z2,z3) -> c_11(f#(s(z0),minus(z1,z0),z2,z3),f#(z0,z3,z2,z3)) perfectp#(0()) -> c_12() perfectp#(s(z0)) -> c_13(f#(z0,s(0()),s(z0),s(z0))) and mark the set of starting terms. * Step 3: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(0(),z0,0(),z1) -> c_1() F#(0(),z0,s(z1),z2) -> c_2() F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)) F#(s(z0),s(z1),z2,z3) -> c_4(F#(s(z0),minus(z1,z0),z2,z3)) F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)) PERFECTP#(0()) -> c_6() PERFECTP#(s(z0)) -> c_7(F#(z0,s(0()),s(z0),s(z0))) - Strict TRS: F(0(),z0,0(),z1) -> c2() F(0(),z0,s(z1),z2) -> c3() F(s(z0),0(),z1,z2) -> c4(F(z0,z2,minus(z1,s(z0)),z2)) F(s(z0),s(z1),z2,z3) -> c5(F(s(z0),minus(z1,z0),z2,z3)) F(s(z0),s(z1),z2,z3) -> c6(F(z0,z3,z2,z3)) PERFECTP(0()) -> c() PERFECTP(s(z0)) -> c1(F(z0,s(0()),s(z0),s(z0))) - Weak DPs: f#(0(),z0,0(),z1) -> c_8() f#(0(),z0,s(z1),z2) -> c_9() f#(s(z0),0(),z1,z2) -> c_10(f#(z0,z2,minus(z1,s(z0)),z2)) f#(s(z0),s(z1),z2,z3) -> c_11(f#(s(z0),minus(z1,z0),z2,z3),f#(z0,z3,z2,z3)) perfectp#(0()) -> c_12() perfectp#(s(z0)) -> c_13(f#(z0,s(0()),s(z0),s(z0))) - Weak TRS: f(0(),z0,0(),z1) -> true() f(0(),z0,s(z1),z2) -> false() f(s(z0),0(),z1,z2) -> f(z0,z2,minus(z1,s(z0)),z2) f(s(z0),s(z1),z2,z3) -> if(le(z0,z1),f(s(z0),minus(z1,z0),z2,z3),f(z0,z3,z2,z3)) perfectp(0()) -> false() perfectp(s(z0)) -> f(z0,s(0()),s(z0),s(z0)) - Signature: {F/4,PERFECTP/1,f/4,perfectp/1,F#/4,PERFECTP#/1,f#/4,perfectp#/1} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/1,c6/1 ,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/2 ,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,PERFECTP#,f#,perfectp#} and constructors {0,c,c1,c2,c3 ,c4,c5,c6,false,if,le,minus,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: F#(0(),z0,0(),z1) -> c_1() F#(0(),z0,s(z1),z2) -> c_2() F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)) F#(s(z0),s(z1),z2,z3) -> c_4(F#(s(z0),minus(z1,z0),z2,z3)) F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)) PERFECTP#(0()) -> c_6() PERFECTP#(s(z0)) -> c_7(F#(z0,s(0()),s(z0),s(z0))) f#(0(),z0,0(),z1) -> c_8() f#(0(),z0,s(z1),z2) -> c_9() f#(s(z0),0(),z1,z2) -> c_10(f#(z0,z2,minus(z1,s(z0)),z2)) f#(s(z0),s(z1),z2,z3) -> c_11(f#(s(z0),minus(z1,z0),z2,z3),f#(z0,z3,z2,z3)) perfectp#(0()) -> c_12() perfectp#(s(z0)) -> c_13(f#(z0,s(0()),s(z0),s(z0))) * Step 4: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(0(),z0,0(),z1) -> c_1() F#(0(),z0,s(z1),z2) -> c_2() F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)) F#(s(z0),s(z1),z2,z3) -> c_4(F#(s(z0),minus(z1,z0),z2,z3)) F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)) PERFECTP#(0()) -> c_6() PERFECTP#(s(z0)) -> c_7(F#(z0,s(0()),s(z0),s(z0))) - Weak DPs: f#(0(),z0,0(),z1) -> c_8() f#(0(),z0,s(z1),z2) -> c_9() f#(s(z0),0(),z1,z2) -> c_10(f#(z0,z2,minus(z1,s(z0)),z2)) f#(s(z0),s(z1),z2,z3) -> c_11(f#(s(z0),minus(z1,z0),z2,z3),f#(z0,z3,z2,z3)) perfectp#(0()) -> c_12() perfectp#(s(z0)) -> c_13(f#(z0,s(0()),s(z0),s(z0))) - Signature: {F/4,PERFECTP/1,f/4,perfectp/1,F#/4,PERFECTP#/1,f#/4,perfectp#/1} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/1,c6/1 ,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/2 ,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,PERFECTP#,f#,perfectp#} and constructors {0,c,c1,c2,c3 ,c4,c5,c6,false,if,le,minus,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,4,6} by application of Pre({1,2,4,6}) = {3,5,7}. Here rules are labelled as follows: 1: F#(0(),z0,0(),z1) -> c_1() 2: F#(0(),z0,s(z1),z2) -> c_2() 3: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)) 4: F#(s(z0),s(z1),z2,z3) -> c_4(F#(s(z0),minus(z1,z0),z2,z3)) 5: F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)) 6: PERFECTP#(0()) -> c_6() 7: PERFECTP#(s(z0)) -> c_7(F#(z0,s(0()),s(z0),s(z0))) 8: f#(0(),z0,0(),z1) -> c_8() 9: f#(0(),z0,s(z1),z2) -> c_9() 10: f#(s(z0),0(),z1,z2) -> c_10(f#(z0,z2,minus(z1,s(z0)),z2)) 11: f#(s(z0),s(z1),z2,z3) -> c_11(f#(s(z0),minus(z1,z0),z2,z3),f#(z0,z3,z2,z3)) 12: perfectp#(0()) -> c_12() 13: perfectp#(s(z0)) -> c_13(f#(z0,s(0()),s(z0),s(z0))) * Step 5: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)) F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)) PERFECTP#(s(z0)) -> c_7(F#(z0,s(0()),s(z0),s(z0))) - Weak DPs: F#(0(),z0,0(),z1) -> c_1() F#(0(),z0,s(z1),z2) -> c_2() F#(s(z0),s(z1),z2,z3) -> c_4(F#(s(z0),minus(z1,z0),z2,z3)) PERFECTP#(0()) -> c_6() f#(0(),z0,0(),z1) -> c_8() f#(0(),z0,s(z1),z2) -> c_9() f#(s(z0),0(),z1,z2) -> c_10(f#(z0,z2,minus(z1,s(z0)),z2)) f#(s(z0),s(z1),z2,z3) -> c_11(f#(s(z0),minus(z1,z0),z2,z3),f#(z0,z3,z2,z3)) perfectp#(0()) -> c_12() perfectp#(s(z0)) -> c_13(f#(z0,s(0()),s(z0),s(z0))) - Signature: {F/4,PERFECTP/1,f/4,perfectp/1,F#/4,PERFECTP#/1,f#/4,perfectp#/1} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/1,c6/1 ,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/2 ,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,PERFECTP#,f#,perfectp#} and constructors {0,c,c1,c2,c3 ,c4,c5,c6,false,if,le,minus,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)) -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)):2 -->_1 F#(s(z0),s(z1),z2,z3) -> c_4(F#(s(z0),minus(z1,z0),z2,z3)):6 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)):1 2:S:F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)) -->_1 F#(s(z0),s(z1),z2,z3) -> c_4(F#(s(z0),minus(z1,z0),z2,z3)):6 -->_1 F#(0(),z0,s(z1),z2) -> c_2():5 -->_1 F#(0(),z0,0(),z1) -> c_1():4 -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)):2 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)):1 3:S:PERFECTP#(s(z0)) -> c_7(F#(z0,s(0()),s(z0),s(z0))) -->_1 F#(s(z0),s(z1),z2,z3) -> c_4(F#(s(z0),minus(z1,z0),z2,z3)):6 -->_1 F#(0(),z0,s(z1),z2) -> c_2():5 -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)):2 4:W:F#(0(),z0,0(),z1) -> c_1() 5:W:F#(0(),z0,s(z1),z2) -> c_2() 6:W:F#(s(z0),s(z1),z2,z3) -> c_4(F#(s(z0),minus(z1,z0),z2,z3)) 7:W:PERFECTP#(0()) -> c_6() 8:W:f#(0(),z0,0(),z1) -> c_8() 9:W:f#(0(),z0,s(z1),z2) -> c_9() 10:W:f#(s(z0),0(),z1,z2) -> c_10(f#(z0,z2,minus(z1,s(z0)),z2)) -->_1 f#(s(z0),s(z1),z2,z3) -> c_11(f#(s(z0),minus(z1,z0),z2,z3),f#(z0,z3,z2,z3)):11 -->_1 f#(s(z0),0(),z1,z2) -> c_10(f#(z0,z2,minus(z1,s(z0)),z2)):10 11:W:f#(s(z0),s(z1),z2,z3) -> c_11(f#(s(z0),minus(z1,z0),z2,z3),f#(z0,z3,z2,z3)) -->_2 f#(s(z0),s(z1),z2,z3) -> c_11(f#(s(z0),minus(z1,z0),z2,z3),f#(z0,z3,z2,z3)):11 -->_2 f#(s(z0),0(),z1,z2) -> c_10(f#(z0,z2,minus(z1,s(z0)),z2)):10 -->_2 f#(0(),z0,s(z1),z2) -> c_9():9 -->_2 f#(0(),z0,0(),z1) -> c_8():8 12:W:perfectp#(0()) -> c_12() 13:W:perfectp#(s(z0)) -> c_13(f#(z0,s(0()),s(z0),s(z0))) -->_1 f#(s(z0),s(z1),z2,z3) -> c_11(f#(s(z0),minus(z1,z0),z2,z3),f#(z0,z3,z2,z3)):11 -->_1 f#(0(),z0,s(z1),z2) -> c_9():9 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 13: perfectp#(s(z0)) -> c_13(f#(z0,s(0()),s(z0),s(z0))) 12: perfectp#(0()) -> c_12() 10: f#(s(z0),0(),z1,z2) -> c_10(f#(z0,z2,minus(z1,s(z0)),z2)) 11: f#(s(z0),s(z1),z2,z3) -> c_11(f#(s(z0),minus(z1,z0),z2,z3),f#(z0,z3,z2,z3)) 9: f#(0(),z0,s(z1),z2) -> c_9() 8: f#(0(),z0,0(),z1) -> c_8() 7: PERFECTP#(0()) -> c_6() 4: F#(0(),z0,0(),z1) -> c_1() 5: F#(0(),z0,s(z1),z2) -> c_2() 6: F#(s(z0),s(z1),z2,z3) -> c_4(F#(s(z0),minus(z1,z0),z2,z3)) * Step 6: RemoveHeads. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)) F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)) PERFECTP#(s(z0)) -> c_7(F#(z0,s(0()),s(z0),s(z0))) - Signature: {F/4,PERFECTP/1,f/4,perfectp/1,F#/4,PERFECTP#/1,f#/4,perfectp#/1} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/1,c6/1 ,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/2 ,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,PERFECTP#,f#,perfectp#} and constructors {0,c,c1,c2,c3 ,c4,c5,c6,false,if,le,minus,s,true} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)) -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)):2 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)):1 2:S:F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)) -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)):2 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)):1 3:S:PERFECTP#(s(z0)) -> c_7(F#(z0,s(0()),s(z0),s(z0))) -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)):2 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). [(3,PERFECTP#(s(z0)) -> c_7(F#(z0,s(0()),s(z0),s(z0))))] * Step 7: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)) F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)) - Signature: {F/4,PERFECTP/1,f/4,perfectp/1,F#/4,PERFECTP#/1,f#/4,perfectp#/1} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/1,c6/1 ,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/2 ,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,PERFECTP#,f#,perfectp#} and constructors {0,c,c1,c2,c3 ,c4,c5,c6,false,if,le,minus,s,true} + 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#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)) 2: F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)) The strictly oriented rules are moved into the weak component. ** Step 7.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)) F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)) - Signature: {F/4,PERFECTP/1,f/4,perfectp/1,F#/4,PERFECTP#/1,f#/4,perfectp#/1} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/1,c6/1 ,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/2 ,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,PERFECTP#,f#,perfectp#} and constructors {0,c,c1,c2,c3 ,c4,c5,c6,false,if,le,minus,s,true} + 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_5) = {1} Following symbols are considered usable: {F#,PERFECTP#,f#,perfectp#} TcT has computed the following interpretation: p(0) = [0] p(F) = [1] x3 + [1] x4 + [2] p(PERFECTP) = [1] x1 + [2] p(c) = [1] p(c1) = [0] p(c2) = [1] p(c3) = [1] p(c4) = [0] p(c5) = [1] x1 + [1] p(c6) = [1] x1 + [1] p(f) = [1] x1 + [2] x2 + [1] x3 + [1] x4 + [1] p(false) = [0] p(if) = [0] p(le) = [1] p(minus) = [0] p(perfectp) = [1] p(s) = [1] x1 + [2] p(true) = [0] p(F#) = [1] x1 + [1] x4 + [0] p(PERFECTP#) = [4] x1 + [1] p(f#) = [1] x1 + [1] x2 + [4] x3 + [1] x4 + [1] p(perfectp#) = [1] x1 + [2] p(c_1) = [2] p(c_2) = [1] p(c_3) = [1] x1 + [1] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [1] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [1] x1 + [1] p(c_12) = [0] p(c_13) = [1] x1 + [0] Following rules are strictly oriented: F#(s(z0),0(),z1,z2) = [1] z0 + [1] z2 + [2] > [1] z0 + [1] z2 + [1] = c_3(F#(z0,z2,minus(z1,s(z0)),z2)) F#(s(z0),s(z1),z2,z3) = [1] z0 + [1] z3 + [2] > [1] z0 + [1] z3 + [0] = c_5(F#(z0,z3,z2,z3)) Following rules are (at-least) weakly oriented: ** Step 7.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)) F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)) - Signature: {F/4,PERFECTP/1,f/4,perfectp/1,F#/4,PERFECTP#/1,f#/4,perfectp#/1} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/1,c6/1 ,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/2 ,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,PERFECTP#,f#,perfectp#} and constructors {0,c,c1,c2,c3 ,c4,c5,c6,false,if,le,minus,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () ** Step 7.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)) F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)) - Signature: {F/4,PERFECTP/1,f/4,perfectp/1,F#/4,PERFECTP#/1,f#/4,perfectp#/1} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/1,c6/1 ,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/2 ,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,PERFECTP#,f#,perfectp#} and constructors {0,c,c1,c2,c3 ,c4,c5,c6,false,if,le,minus,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)) -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)):2 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)):1 2:W:F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)) -->_1 F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)):2 -->_1 F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: F#(s(z0),0(),z1,z2) -> c_3(F#(z0,z2,minus(z1,s(z0)),z2)) 2: F#(s(z0),s(z1),z2,z3) -> c_5(F#(z0,z3,z2,z3)) ** Step 7.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Signature: {F/4,PERFECTP/1,f/4,perfectp/1,F#/4,PERFECTP#/1,f#/4,perfectp#/1} / {0/0,c/0,c1/1,c2/0,c3/0,c4/1,c5/1,c6/1 ,false/0,if/3,le/2,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/2 ,c_12/0,c_13/1} - Obligation: innermost runtime complexity wrt. defined symbols {F#,PERFECTP#,f#,perfectp#} and constructors {0,c,c1,c2,c3 ,c4,c5,c6,false,if,le,minus,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))