WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: F(0()) -> c1() F(1()) -> c2() F(s(z0)) -> c3(F(z0)) G(z0,c(z1)) -> c6(G(z0,z1)) G(z0,c(z1)) -> c7(G(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF(f(z0),c(g(s(z0),z1)),c(z1)),F(z0)) G(z0,c(z1)) -> c8(G(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF(f(z0),c(g(s(z0),z1)),c(z1)),G(s(z0),z1)) IF(false(),s(z0),s(z1)) -> c5() IF(true(),s(z0),s(z1)) -> c4() - Weak TRS: f(0()) -> true() f(1()) -> false() f(s(z0)) -> f(z0) g(z0,c(z1)) -> c(g(z0,z1)) g(z0,c(z1)) -> g(z0,if(f(z0),c(g(s(z0),z1)),c(z1))) if(false(),s(z0),s(z1)) -> s(z1) if(true(),s(z0),s(z1)) -> s(z0) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1,c7/3,c8/3,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {F,G,IF,f,g,if} and constructors {0,1,c,c1,c2,c3,c4,c5,c6 ,c7,c8,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: F(0()) -> c1() F(1()) -> c2() F(s(z0)) -> c3(F(z0)) G(z0,c(z1)) -> c6(G(z0,z1)) G(z0,c(z1)) -> c7(G(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF(f(z0),c(g(s(z0),z1)),c(z1)),F(z0)) G(z0,c(z1)) -> c8(G(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF(f(z0),c(g(s(z0),z1)),c(z1)),G(s(z0),z1)) IF(false(),s(z0),s(z1)) -> c5() IF(true(),s(z0),s(z1)) -> c4() - Weak TRS: f(0()) -> true() f(1()) -> false() f(s(z0)) -> f(z0) g(z0,c(z1)) -> c(g(z0,z1)) g(z0,c(z1)) -> g(z0,if(f(z0),c(g(s(z0),z1)),c(z1))) if(false(),s(z0),s(z1)) -> s(z1) if(true(),s(z0),s(z1)) -> s(z0) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1,c7/3,c8/3,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {F,G,IF,f,g,if} and constructors {0,1,c,c1,c2,c3,c4,c5,c6 ,c7,c8,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: F(0()) -> c1() F(1()) -> c2() F(s(z0)) -> c3(F(z0)) G(z0,c(z1)) -> c6(G(z0,z1)) G(z0,c(z1)) -> c7(G(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF(f(z0),c(g(s(z0),z1)),c(z1)),F(z0)) G(z0,c(z1)) -> c8(G(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF(f(z0),c(g(s(z0),z1)),c(z1)),G(s(z0),z1)) IF(false(),s(z0),s(z1)) -> c5() IF(true(),s(z0),s(z1)) -> c4() - Weak TRS: f(0()) -> true() f(1()) -> false() f(s(z0)) -> f(z0) g(z0,c(z1)) -> c(g(z0,z1)) g(z0,c(z1)) -> g(z0,if(f(z0),c(g(s(z0),z1)),c(z1))) if(false(),s(z0),s(z1)) -> s(z1) if(true(),s(z0),s(z1)) -> s(z0) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1,c7/3,c8/3,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {F,G,IF,f,g,if} and constructors {0,1,c,c1,c2,c3,c4,c5,c6 ,c7,c8,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: F(x){x -> s(x)} = F(s(x)) ->^+ c3(F(x)) = C[F(x) = F(x){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: F(0()) -> c1() F(1()) -> c2() F(s(z0)) -> c3(F(z0)) G(z0,c(z1)) -> c6(G(z0,z1)) G(z0,c(z1)) -> c7(G(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF(f(z0),c(g(s(z0),z1)),c(z1)),F(z0)) G(z0,c(z1)) -> c8(G(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF(f(z0),c(g(s(z0),z1)),c(z1)),G(s(z0),z1)) IF(false(),s(z0),s(z1)) -> c5() IF(true(),s(z0),s(z1)) -> c4() - Weak TRS: f(0()) -> true() f(1()) -> false() f(s(z0)) -> f(z0) g(z0,c(z1)) -> c(g(z0,z1)) g(z0,c(z1)) -> g(z0,if(f(z0),c(g(s(z0),z1)),c(z1))) if(false(),s(z0),s(z1)) -> s(z1) if(true(),s(z0),s(z1)) -> s(z0) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1,c7/3,c8/3,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {F,G,IF,f,g,if} and constructors {0,1,c,c1,c2,c3,c4,c5,c6 ,c7,c8,false,s,true} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs F#(0()) -> c_1() F#(1()) -> c_2() F#(s(z0)) -> c_3(F#(z0)) G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_5(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),F#(z0)) G#(z0,c(z1)) -> c_6(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),G#(s(z0),z1)) IF#(false(),s(z0),s(z1)) -> c_7() IF#(true(),s(z0),s(z1)) -> c_8() Weak DPs f#(0()) -> c_9() f#(1()) -> c_10() f#(s(z0)) -> c_11(f#(z0)) g#(z0,c(z1)) -> c_12(g#(z0,z1)) g#(z0,c(z1)) -> c_13(g#(z0,if(f(z0),c(g(s(z0),z1)),c(z1)))) if#(false(),s(z0),s(z1)) -> c_14() if#(true(),s(z0),s(z1)) -> c_15() and mark the set of starting terms. ** Step 1.b:2: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(0()) -> c_1() F#(1()) -> c_2() F#(s(z0)) -> c_3(F#(z0)) G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_5(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),F#(z0)) G#(z0,c(z1)) -> c_6(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),G#(s(z0),z1)) IF#(false(),s(z0),s(z1)) -> c_7() IF#(true(),s(z0),s(z1)) -> c_8() - Strict TRS: F(0()) -> c1() F(1()) -> c2() F(s(z0)) -> c3(F(z0)) G(z0,c(z1)) -> c6(G(z0,z1)) G(z0,c(z1)) -> c7(G(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF(f(z0),c(g(s(z0),z1)),c(z1)),F(z0)) G(z0,c(z1)) -> c8(G(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF(f(z0),c(g(s(z0),z1)),c(z1)),G(s(z0),z1)) IF(false(),s(z0),s(z1)) -> c5() IF(true(),s(z0),s(z1)) -> c4() - Weak DPs: f#(0()) -> c_9() f#(1()) -> c_10() f#(s(z0)) -> c_11(f#(z0)) g#(z0,c(z1)) -> c_12(g#(z0,z1)) g#(z0,c(z1)) -> c_13(g#(z0,if(f(z0),c(g(s(z0),z1)),c(z1)))) if#(false(),s(z0),s(z1)) -> c_14() if#(true(),s(z0),s(z1)) -> c_15() - Weak TRS: f(0()) -> true() f(1()) -> false() f(s(z0)) -> f(z0) g(z0,c(z1)) -> c(g(z0,z1)) g(z0,c(z1)) -> g(z0,if(f(z0),c(g(s(z0),z1)),c(z1))) if(false(),s(z0),s(z1)) -> s(z1) if(true(),s(z0),s(z1)) -> s(z0) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/3,c_6/3,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: f(0()) -> true() f(1()) -> false() f(s(z0)) -> f(z0) g(z0,c(z1)) -> c(g(z0,z1)) g(z0,c(z1)) -> g(z0,if(f(z0),c(g(s(z0),z1)),c(z1))) F#(0()) -> c_1() F#(1()) -> c_2() F#(s(z0)) -> c_3(F#(z0)) G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_5(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),F#(z0)) G#(z0,c(z1)) -> c_6(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),G#(s(z0),z1)) IF#(false(),s(z0),s(z1)) -> c_7() IF#(true(),s(z0),s(z1)) -> c_8() f#(0()) -> c_9() f#(1()) -> c_10() f#(s(z0)) -> c_11(f#(z0)) g#(z0,c(z1)) -> c_12(g#(z0,z1)) g#(z0,c(z1)) -> c_13(g#(z0,if(f(z0),c(g(s(z0),z1)),c(z1)))) if#(false(),s(z0),s(z1)) -> c_14() if#(true(),s(z0),s(z1)) -> c_15() ** Step 1.b:3: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(0()) -> c_1() F#(1()) -> c_2() F#(s(z0)) -> c_3(F#(z0)) G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_5(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),F#(z0)) G#(z0,c(z1)) -> c_6(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),G#(s(z0),z1)) IF#(false(),s(z0),s(z1)) -> c_7() IF#(true(),s(z0),s(z1)) -> c_8() - Weak DPs: f#(0()) -> c_9() f#(1()) -> c_10() f#(s(z0)) -> c_11(f#(z0)) g#(z0,c(z1)) -> c_12(g#(z0,z1)) g#(z0,c(z1)) -> c_13(g#(z0,if(f(z0),c(g(s(z0),z1)),c(z1)))) if#(false(),s(z0),s(z1)) -> c_14() if#(true(),s(z0),s(z1)) -> c_15() - Weak TRS: f(0()) -> true() f(1()) -> false() f(s(z0)) -> f(z0) g(z0,c(z1)) -> c(g(z0,z1)) g(z0,c(z1)) -> g(z0,if(f(z0),c(g(s(z0),z1)),c(z1))) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/3,c_6/3,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,7,8} by application of Pre({1,2,7,8}) = {3,5}. Here rules are labelled as follows: 1: F#(0()) -> c_1() 2: F#(1()) -> c_2() 3: F#(s(z0)) -> c_3(F#(z0)) 4: G#(z0,c(z1)) -> c_4(G#(z0,z1)) 5: G#(z0,c(z1)) -> c_5(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),F#(z0)) 6: G#(z0,c(z1)) -> c_6(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),G#(s(z0),z1)) 7: IF#(false(),s(z0),s(z1)) -> c_7() 8: IF#(true(),s(z0),s(z1)) -> c_8() 9: f#(0()) -> c_9() 10: f#(1()) -> c_10() 11: f#(s(z0)) -> c_11(f#(z0)) 12: g#(z0,c(z1)) -> c_12(g#(z0,z1)) 13: g#(z0,c(z1)) -> c_13(g#(z0,if(f(z0),c(g(s(z0),z1)),c(z1)))) 14: if#(false(),s(z0),s(z1)) -> c_14() 15: if#(true(),s(z0),s(z1)) -> c_15() ** Step 1.b:4: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(s(z0)) -> c_3(F#(z0)) G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_5(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),F#(z0)) G#(z0,c(z1)) -> c_6(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),G#(s(z0),z1)) - Weak DPs: F#(0()) -> c_1() F#(1()) -> c_2() IF#(false(),s(z0),s(z1)) -> c_7() IF#(true(),s(z0),s(z1)) -> c_8() f#(0()) -> c_9() f#(1()) -> c_10() f#(s(z0)) -> c_11(f#(z0)) g#(z0,c(z1)) -> c_12(g#(z0,z1)) g#(z0,c(z1)) -> c_13(g#(z0,if(f(z0),c(g(s(z0),z1)),c(z1)))) if#(false(),s(z0),s(z1)) -> c_14() if#(true(),s(z0),s(z1)) -> c_15() - Weak TRS: f(0()) -> true() f(1()) -> false() f(s(z0)) -> f(z0) g(z0,c(z1)) -> c(g(z0,z1)) g(z0,c(z1)) -> g(z0,if(f(z0),c(g(s(z0),z1)),c(z1))) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/3,c_6/3,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:F#(s(z0)) -> c_3(F#(z0)) -->_1 F#(1()) -> c_2():6 -->_1 F#(0()) -> c_1():5 -->_1 F#(s(z0)) -> c_3(F#(z0)):1 2:S:G#(z0,c(z1)) -> c_4(G#(z0,z1)) -->_1 G#(z0,c(z1)) -> c_6(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))) ,IF#(f(z0),c(g(s(z0),z1)),c(z1)) ,G#(s(z0),z1)):4 -->_1 G#(z0,c(z1)) -> c_5(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),F#(z0)):3 -->_1 G#(z0,c(z1)) -> c_4(G#(z0,z1)):2 3:S:G#(z0,c(z1)) -> c_5(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),F#(z0)) -->_3 F#(1()) -> c_2():6 -->_3 F#(0()) -> c_1():5 -->_3 F#(s(z0)) -> c_3(F#(z0)):1 4:S:G#(z0,c(z1)) -> c_6(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),G#(s(z0),z1)) -->_3 G#(z0,c(z1)) -> c_6(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))) ,IF#(f(z0),c(g(s(z0),z1)),c(z1)) ,G#(s(z0),z1)):4 -->_3 G#(z0,c(z1)) -> c_5(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),F#(z0)):3 -->_3 G#(z0,c(z1)) -> c_4(G#(z0,z1)):2 5:W:F#(0()) -> c_1() 6:W:F#(1()) -> c_2() 7:W:IF#(false(),s(z0),s(z1)) -> c_7() 8:W:IF#(true(),s(z0),s(z1)) -> c_8() 9:W:f#(0()) -> c_9() 10:W:f#(1()) -> c_10() 11:W:f#(s(z0)) -> c_11(f#(z0)) -->_1 f#(s(z0)) -> c_11(f#(z0)):11 -->_1 f#(1()) -> c_10():10 -->_1 f#(0()) -> c_9():9 12:W:g#(z0,c(z1)) -> c_12(g#(z0,z1)) -->_1 g#(z0,c(z1)) -> c_13(g#(z0,if(f(z0),c(g(s(z0),z1)),c(z1)))):13 -->_1 g#(z0,c(z1)) -> c_12(g#(z0,z1)):12 13:W:g#(z0,c(z1)) -> c_13(g#(z0,if(f(z0),c(g(s(z0),z1)),c(z1)))) 14:W:if#(false(),s(z0),s(z1)) -> c_14() 15:W:if#(true(),s(z0),s(z1)) -> c_15() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 15: if#(true(),s(z0),s(z1)) -> c_15() 14: if#(false(),s(z0),s(z1)) -> c_14() 12: g#(z0,c(z1)) -> c_12(g#(z0,z1)) 13: g#(z0,c(z1)) -> c_13(g#(z0,if(f(z0),c(g(s(z0),z1)),c(z1)))) 11: f#(s(z0)) -> c_11(f#(z0)) 10: f#(1()) -> c_10() 9: f#(0()) -> c_9() 8: IF#(true(),s(z0),s(z1)) -> c_8() 7: IF#(false(),s(z0),s(z1)) -> c_7() 5: F#(0()) -> c_1() 6: F#(1()) -> c_2() ** Step 1.b:5: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(s(z0)) -> c_3(F#(z0)) G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_5(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),F#(z0)) G#(z0,c(z1)) -> c_6(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),G#(s(z0),z1)) - Weak TRS: f(0()) -> true() f(1()) -> false() f(s(z0)) -> f(z0) g(z0,c(z1)) -> c(g(z0,z1)) g(z0,c(z1)) -> g(z0,if(f(z0),c(g(s(z0),z1)),c(z1))) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/3,c_6/3,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:F#(s(z0)) -> c_3(F#(z0)) -->_1 F#(s(z0)) -> c_3(F#(z0)):1 2:S:G#(z0,c(z1)) -> c_4(G#(z0,z1)) -->_1 G#(z0,c(z1)) -> c_6(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))) ,IF#(f(z0),c(g(s(z0),z1)),c(z1)) ,G#(s(z0),z1)):4 -->_1 G#(z0,c(z1)) -> c_5(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),F#(z0)):3 -->_1 G#(z0,c(z1)) -> c_4(G#(z0,z1)):2 3:S:G#(z0,c(z1)) -> c_5(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),F#(z0)) -->_3 F#(s(z0)) -> c_3(F#(z0)):1 4:S:G#(z0,c(z1)) -> c_6(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),G#(s(z0),z1)) -->_3 G#(z0,c(z1)) -> c_6(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))) ,IF#(f(z0),c(g(s(z0),z1)),c(z1)) ,G#(s(z0),z1)):4 -->_3 G#(z0,c(z1)) -> c_5(G#(z0,if(f(z0),c(g(s(z0),z1)),c(z1))),IF#(f(z0),c(g(s(z0),z1)),c(z1)),F#(z0)):3 -->_3 G#(z0,c(z1)) -> c_4(G#(z0,z1)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: G#(z0,c(z1)) -> c_5(F#(z0)) G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) ** Step 1.b:6: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(s(z0)) -> c_3(F#(z0)) G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_5(F#(z0)) G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) - Weak TRS: f(0()) -> true() f(1()) -> false() f(s(z0)) -> f(z0) g(z0,c(z1)) -> c(g(z0,z1)) g(z0,c(z1)) -> g(z0,if(f(z0),c(g(s(z0),z1)),c(z1))) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: F#(s(z0)) -> c_3(F#(z0)) G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_5(F#(z0)) G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) ** Step 1.b:7: Decompose. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(s(z0)) -> c_3(F#(z0)) G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_5(F#(z0)) G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,s,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: F#(s(z0)) -> c_3(F#(z0)) - Weak DPs: G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_5(F#(z0)) G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3 ,c4,c5,c6,c7,c8,false,s,true} Problem (S) - Strict DPs: G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_5(F#(z0)) G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) - Weak DPs: F#(s(z0)) -> c_3(F#(z0)) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3 ,c4,c5,c6,c7,c8,false,s,true} *** Step 1.b:7.a:1: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(s(z0)) -> c_3(F#(z0)) - Weak DPs: G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_5(F#(z0)) G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,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)) -> c_3(F#(z0)) The strictly oriented rules are moved into the weak component. **** Step 1.b:7.a:1.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: F#(s(z0)) -> c_3(F#(z0)) - Weak DPs: G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_5(F#(z0)) G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,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_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {F#,G#,IF#,f#,g#,if#} TcT has computed the following interpretation: p(0) = [1] p(1) = [1] p(F) = [0] p(G) = [2] x1 + [1] x2 + [1] p(IF) = [1] x3 + [0] p(c) = [1] x1 + [4] p(c1) = [1] p(c2) = [2] p(c3) = [2] p(c4) = [0] p(c5) = [0] p(c6) = [1] x1 + [0] p(c7) = [1] x1 + [1] x2 + [1] x3 + [0] p(c8) = [1] x1 + [1] x3 + [0] p(f) = [0] p(false) = [0] p(g) = [0] p(if) = [1] x3 + [0] p(s) = [1] x1 + [1] p(true) = [1] p(F#) = [1] x1 + [0] p(G#) = [4] x1 + [1] x2 + [14] p(IF#) = [1] x2 + [2] x3 + [0] p(f#) = [0] p(g#) = [1] p(if#) = [2] x1 + [1] x2 + [0] p(c_1) = [0] p(c_2) = [2] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [4] p(c_5) = [4] x1 + [2] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [1] p(c_9) = [0] p(c_10) = [1] p(c_11) = [1] x1 + [2] p(c_12) = [8] x1 + [2] p(c_13) = [1] x1 + [0] p(c_14) = [1] p(c_15) = [0] Following rules are strictly oriented: F#(s(z0)) = [1] z0 + [1] > [1] z0 + [0] = c_3(F#(z0)) Following rules are (at-least) weakly oriented: G#(z0,c(z1)) = [4] z0 + [1] z1 + [18] >= [4] z0 + [1] z1 + [18] = c_4(G#(z0,z1)) G#(z0,c(z1)) = [4] z0 + [1] z1 + [18] >= [4] z0 + [2] = c_5(F#(z0)) G#(z0,c(z1)) = [4] z0 + [1] z1 + [18] >= [4] z0 + [1] z1 + [18] = c_6(G#(s(z0),z1)) **** Step 1.b:7.a:1.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: F#(s(z0)) -> c_3(F#(z0)) G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_5(F#(z0)) G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () **** Step 1.b:7.a:1.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: F#(s(z0)) -> c_3(F#(z0)) G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_5(F#(z0)) G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:F#(s(z0)) -> c_3(F#(z0)) -->_1 F#(s(z0)) -> c_3(F#(z0)):1 2:W:G#(z0,c(z1)) -> c_4(G#(z0,z1)) -->_1 G#(z0,c(z1)) -> c_6(G#(s(z0),z1)):4 -->_1 G#(z0,c(z1)) -> c_5(F#(z0)):3 -->_1 G#(z0,c(z1)) -> c_4(G#(z0,z1)):2 3:W:G#(z0,c(z1)) -> c_5(F#(z0)) -->_1 F#(s(z0)) -> c_3(F#(z0)):1 4:W:G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) -->_1 G#(z0,c(z1)) -> c_6(G#(s(z0),z1)):4 -->_1 G#(z0,c(z1)) -> c_5(F#(z0)):3 -->_1 G#(z0,c(z1)) -> c_4(G#(z0,z1)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: G#(z0,c(z1)) -> c_4(G#(z0,z1)) 4: G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) 3: G#(z0,c(z1)) -> c_5(F#(z0)) 1: F#(s(z0)) -> c_3(F#(z0)) **** Step 1.b:7.a:1.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:7.b:1: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_5(F#(z0)) G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) - Weak DPs: F#(s(z0)) -> c_3(F#(z0)) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2} by application of Pre({2}) = {1,3}. Here rules are labelled as follows: 1: G#(z0,c(z1)) -> c_4(G#(z0,z1)) 2: G#(z0,c(z1)) -> c_5(F#(z0)) 3: G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) 4: F#(s(z0)) -> c_3(F#(z0)) *** Step 1.b:7.b:2: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) - Weak DPs: F#(s(z0)) -> c_3(F#(z0)) G#(z0,c(z1)) -> c_5(F#(z0)) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:G#(z0,c(z1)) -> c_4(G#(z0,z1)) -->_1 G#(z0,c(z1)) -> c_5(F#(z0)):4 -->_1 G#(z0,c(z1)) -> c_6(G#(s(z0),z1)):2 -->_1 G#(z0,c(z1)) -> c_4(G#(z0,z1)):1 2:S:G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) -->_1 G#(z0,c(z1)) -> c_5(F#(z0)):4 -->_1 G#(z0,c(z1)) -> c_6(G#(s(z0),z1)):2 -->_1 G#(z0,c(z1)) -> c_4(G#(z0,z1)):1 3:W:F#(s(z0)) -> c_3(F#(z0)) -->_1 F#(s(z0)) -> c_3(F#(z0)):3 4:W:G#(z0,c(z1)) -> c_5(F#(z0)) -->_1 F#(s(z0)) -> c_3(F#(z0)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: G#(z0,c(z1)) -> c_5(F#(z0)) 3: F#(s(z0)) -> c_3(F#(z0)) *** Step 1.b:7.b:3: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,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: G#(z0,c(z1)) -> c_4(G#(z0,z1)) 2: G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) The strictly oriented rules are moved into the weak component. **** Step 1.b:7.b:3.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,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_4) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {F#,G#,IF#,f#,g#,if#} TcT has computed the following interpretation: p(0) = [0] p(1) = [0] p(F) = [0] p(G) = [0] p(IF) = [0] p(c) = [1] x1 + [2] p(c1) = [0] p(c2) = [0] p(c3) = [0] p(c4) = [0] p(c5) = [0] p(c6) = [1] x1 + [0] p(c7) = [1] x2 + [0] p(c8) = [1] x1 + [1] x2 + [0] p(f) = [0] p(false) = [0] p(g) = [2] x1 + [4] x2 + [0] p(if) = [8] x1 + [1] x2 + [0] p(s) = [1] x1 + [2] p(true) = [0] p(F#) = [0] p(G#) = [1] x1 + [2] x2 + [2] p(IF#) = [2] x1 + [2] x3 + [0] p(f#) = [1] x1 + [0] p(g#) = [2] x1 + [8] x2 + [0] p(if#) = [1] x1 + [1] x2 + [2] x3 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [2] x1 + [0] p(c_4) = [1] x1 + [2] p(c_5) = [1] x1 + [2] p(c_6) = [1] x1 + [1] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] p(c_12) = [0] p(c_13) = [1] x1 + [0] p(c_14) = [0] p(c_15) = [0] Following rules are strictly oriented: G#(z0,c(z1)) = [1] z0 + [2] z1 + [6] > [1] z0 + [2] z1 + [4] = c_4(G#(z0,z1)) G#(z0,c(z1)) = [1] z0 + [2] z1 + [6] > [1] z0 + [2] z1 + [5] = c_6(G#(s(z0),z1)) Following rules are (at-least) weakly oriented: **** Step 1.b:7.b:3.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () **** Step 1.b:7.b:3.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: G#(z0,c(z1)) -> c_4(G#(z0,z1)) G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:G#(z0,c(z1)) -> c_4(G#(z0,z1)) -->_1 G#(z0,c(z1)) -> c_6(G#(s(z0),z1)):2 -->_1 G#(z0,c(z1)) -> c_4(G#(z0,z1)):1 2:W:G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) -->_1 G#(z0,c(z1)) -> c_6(G#(s(z0),z1)):2 -->_1 G#(z0,c(z1)) -> c_4(G#(z0,z1)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: G#(z0,c(z1)) -> c_4(G#(z0,z1)) 2: G#(z0,c(z1)) -> c_6(G#(s(z0),z1)) **** Step 1.b:7.b:3.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Signature: {F/1,G/2,IF/3,f/1,g/2,if/3,F#/1,G#/2,IF#/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,c1/0,c2/0,c3/1,c4/0,c5/0,c6/1 ,c7/3,c8/3,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1,c_12/1 ,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {F#,G#,IF#,f#,g#,if#} and constructors {0,1,c,c1,c2,c3,c4 ,c5,c6,c7,c8,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))