/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: cond1(true(),x) -> cond2(even(x),x) cond2(false(),x) -> cond1(neq(x,0()),p(x)) cond2(true(),x) -> cond1(neq(x,0()),div2(x)) div2(0()) -> 0() div2(s(0())) -> 0() div2(s(s(x))) -> s(div2(x)) even(0()) -> true() even(s(0())) -> false() even(s(s(x))) -> even(x) neq(0(),0()) -> false() neq(0(),s(x)) -> true() neq(s(x),0()) -> true() neq(s(x),s(y())) -> neq(x,y()) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/2,div2/1,even/1,neq/2,p/1} / {0/0,false/0,s/1,true/0,y/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1,cond2,div2,even,neq,p} and constructors {0,false,s ,true,y} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: cond1(true(),x) -> cond2(even(x),x) cond2(false(),x) -> cond1(neq(x,0()),p(x)) cond2(true(),x) -> cond1(neq(x,0()),div2(x)) div2(0()) -> 0() div2(s(0())) -> 0() div2(s(s(x))) -> s(div2(x)) even(0()) -> true() even(s(0())) -> false() even(s(s(x))) -> even(x) neq(0(),0()) -> false() neq(0(),s(x)) -> true() neq(s(x),0()) -> true() neq(s(x),s(y())) -> neq(x,y()) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/2,div2/1,even/1,neq/2,p/1} / {0/0,false/0,s/1,true/0,y/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1,cond2,div2,even,neq,p} and constructors {0,false,s ,true,y} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: cond1(true(),x) -> cond2(even(x),x) cond2(false(),x) -> cond1(neq(x,0()),p(x)) cond2(true(),x) -> cond1(neq(x,0()),div2(x)) div2(0()) -> 0() div2(s(0())) -> 0() div2(s(s(x))) -> s(div2(x)) even(0()) -> true() even(s(0())) -> false() even(s(s(x))) -> even(x) neq(0(),0()) -> false() neq(0(),s(x)) -> true() neq(s(x),0()) -> true() neq(s(x),s(y())) -> neq(x,y()) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/2,div2/1,even/1,neq/2,p/1} / {0/0,false/0,s/1,true/0,y/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1,cond2,div2,even,neq,p} and constructors {0,false,s ,true,y} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: div2(x){x -> s(s(x))} = div2(s(s(x))) ->^+ s(div2(x)) = C[div2(x) = div2(x){}] ** Step 1.b:1: DependencyPairs. MAYBE + Considered Problem: - Strict TRS: cond1(true(),x) -> cond2(even(x),x) cond2(false(),x) -> cond1(neq(x,0()),p(x)) cond2(true(),x) -> cond1(neq(x,0()),div2(x)) div2(0()) -> 0() div2(s(0())) -> 0() div2(s(s(x))) -> s(div2(x)) even(0()) -> true() even(s(0())) -> false() even(s(s(x))) -> even(x) neq(0(),0()) -> false() neq(0(),s(x)) -> true() neq(s(x),0()) -> true() neq(s(x),s(y())) -> neq(x,y()) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/2,div2/1,even/1,neq/2,p/1} / {0/0,false/0,s/1,true/0,y/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1,cond2,div2,even,neq,p} and constructors {0,false,s ,true,y} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)) cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x)),neq#(x,0()),p#(x)) cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),neq#(x,0()),div2#(x)) div2#(0()) -> c_4() div2#(s(0())) -> c_5() div2#(s(s(x))) -> c_6(div2#(x)) even#(0()) -> c_7() even#(s(0())) -> c_8() even#(s(s(x))) -> c_9(even#(x)) neq#(0(),0()) -> c_10() neq#(0(),s(x)) -> c_11() neq#(s(x),0()) -> c_12() neq#(s(x),s(y())) -> c_13(neq#(x,y())) p#(0()) -> c_14() p#(s(x)) -> c_15() Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. MAYBE + Considered Problem: - Strict DPs: cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)) cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x)),neq#(x,0()),p#(x)) cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),neq#(x,0()),div2#(x)) div2#(0()) -> c_4() div2#(s(0())) -> c_5() div2#(s(s(x))) -> c_6(div2#(x)) even#(0()) -> c_7() even#(s(0())) -> c_8() even#(s(s(x))) -> c_9(even#(x)) neq#(0(),0()) -> c_10() neq#(0(),s(x)) -> c_11() neq#(s(x),0()) -> c_12() neq#(s(x),s(y())) -> c_13(neq#(x,y())) p#(0()) -> c_14() p#(s(x)) -> c_15() - Weak TRS: cond1(true(),x) -> cond2(even(x),x) cond2(false(),x) -> cond1(neq(x,0()),p(x)) cond2(true(),x) -> cond1(neq(x,0()),div2(x)) div2(0()) -> 0() div2(s(0())) -> 0() div2(s(s(x))) -> s(div2(x)) even(0()) -> true() even(s(0())) -> false() even(s(s(x))) -> even(x) neq(0(),0()) -> false() neq(0(),s(x)) -> true() neq(s(x),0()) -> true() neq(s(x),s(y())) -> neq(x,y()) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/2,div2/1,even/1,neq/2,p/1,cond1#/2,cond2#/2,div2#/1,even#/1,neq#/2,p#/1} / {0/0,false/0,s/1 ,true/0,y/0,c_1/2,c_2/3,c_3/3,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,div2#,even#,neq#,p#} and constructors {0 ,false,s,true,y} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {4,5,7,8,10,11,12,13,14,15} by application of Pre({4,5,7,8,10,11,12,13,14,15}) = {1,2,3,6,9}. Here rules are labelled as follows: 1: cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)) 2: cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x)),neq#(x,0()),p#(x)) 3: cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),neq#(x,0()),div2#(x)) 4: div2#(0()) -> c_4() 5: div2#(s(0())) -> c_5() 6: div2#(s(s(x))) -> c_6(div2#(x)) 7: even#(0()) -> c_7() 8: even#(s(0())) -> c_8() 9: even#(s(s(x))) -> c_9(even#(x)) 10: neq#(0(),0()) -> c_10() 11: neq#(0(),s(x)) -> c_11() 12: neq#(s(x),0()) -> c_12() 13: neq#(s(x),s(y())) -> c_13(neq#(x,y())) 14: p#(0()) -> c_14() 15: p#(s(x)) -> c_15() ** Step 1.b:3: RemoveWeakSuffixes. MAYBE + Considered Problem: - Strict DPs: cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)) cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x)),neq#(x,0()),p#(x)) cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),neq#(x,0()),div2#(x)) div2#(s(s(x))) -> c_6(div2#(x)) even#(s(s(x))) -> c_9(even#(x)) - Weak DPs: div2#(0()) -> c_4() div2#(s(0())) -> c_5() even#(0()) -> c_7() even#(s(0())) -> c_8() neq#(0(),0()) -> c_10() neq#(0(),s(x)) -> c_11() neq#(s(x),0()) -> c_12() neq#(s(x),s(y())) -> c_13(neq#(x,y())) p#(0()) -> c_14() p#(s(x)) -> c_15() - Weak TRS: cond1(true(),x) -> cond2(even(x),x) cond2(false(),x) -> cond1(neq(x,0()),p(x)) cond2(true(),x) -> cond1(neq(x,0()),div2(x)) div2(0()) -> 0() div2(s(0())) -> 0() div2(s(s(x))) -> s(div2(x)) even(0()) -> true() even(s(0())) -> false() even(s(s(x))) -> even(x) neq(0(),0()) -> false() neq(0(),s(x)) -> true() neq(s(x),0()) -> true() neq(s(x),s(y())) -> neq(x,y()) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/2,div2/1,even/1,neq/2,p/1,cond1#/2,cond2#/2,div2#/1,even#/1,neq#/2,p#/1} / {0/0,false/0,s/1 ,true/0,y/0,c_1/2,c_2/3,c_3/3,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,div2#,even#,neq#,p#} and constructors {0 ,false,s,true,y} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)) -->_2 even#(s(s(x))) -> c_9(even#(x)):5 -->_1 cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),neq#(x,0()),div2#(x)):3 -->_1 cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x)),neq#(x,0()),p#(x)):2 -->_2 even#(s(0())) -> c_8():9 -->_2 even#(0()) -> c_7():8 2:S:cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x)),neq#(x,0()),p#(x)) -->_3 p#(s(x)) -> c_15():15 -->_3 p#(0()) -> c_14():14 -->_2 neq#(s(x),0()) -> c_12():12 -->_2 neq#(0(),0()) -> c_10():10 -->_1 cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)):1 3:S:cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),neq#(x,0()),div2#(x)) -->_3 div2#(s(s(x))) -> c_6(div2#(x)):4 -->_2 neq#(s(x),0()) -> c_12():12 -->_2 neq#(0(),0()) -> c_10():10 -->_3 div2#(s(0())) -> c_5():7 -->_3 div2#(0()) -> c_4():6 -->_1 cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)):1 4:S:div2#(s(s(x))) -> c_6(div2#(x)) -->_1 div2#(s(0())) -> c_5():7 -->_1 div2#(0()) -> c_4():6 -->_1 div2#(s(s(x))) -> c_6(div2#(x)):4 5:S:even#(s(s(x))) -> c_9(even#(x)) -->_1 even#(s(0())) -> c_8():9 -->_1 even#(0()) -> c_7():8 -->_1 even#(s(s(x))) -> c_9(even#(x)):5 6:W:div2#(0()) -> c_4() 7:W:div2#(s(0())) -> c_5() 8:W:even#(0()) -> c_7() 9:W:even#(s(0())) -> c_8() 10:W:neq#(0(),0()) -> c_10() 11:W:neq#(0(),s(x)) -> c_11() 12:W:neq#(s(x),0()) -> c_12() 13:W:neq#(s(x),s(y())) -> c_13(neq#(x,y())) 14:W:p#(0()) -> c_14() 15:W:p#(s(x)) -> c_15() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 13: neq#(s(x),s(y())) -> c_13(neq#(x,y())) 11: neq#(0(),s(x)) -> c_11() 14: p#(0()) -> c_14() 15: p#(s(x)) -> c_15() 10: neq#(0(),0()) -> c_10() 12: neq#(s(x),0()) -> c_12() 6: div2#(0()) -> c_4() 7: div2#(s(0())) -> c_5() 8: even#(0()) -> c_7() 9: even#(s(0())) -> c_8() ** Step 1.b:4: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)) cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x)),neq#(x,0()),p#(x)) cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),neq#(x,0()),div2#(x)) div2#(s(s(x))) -> c_6(div2#(x)) even#(s(s(x))) -> c_9(even#(x)) - Weak TRS: cond1(true(),x) -> cond2(even(x),x) cond2(false(),x) -> cond1(neq(x,0()),p(x)) cond2(true(),x) -> cond1(neq(x,0()),div2(x)) div2(0()) -> 0() div2(s(0())) -> 0() div2(s(s(x))) -> s(div2(x)) even(0()) -> true() even(s(0())) -> false() even(s(s(x))) -> even(x) neq(0(),0()) -> false() neq(0(),s(x)) -> true() neq(s(x),0()) -> true() neq(s(x),s(y())) -> neq(x,y()) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/2,div2/1,even/1,neq/2,p/1,cond1#/2,cond2#/2,div2#/1,even#/1,neq#/2,p#/1} / {0/0,false/0,s/1 ,true/0,y/0,c_1/2,c_2/3,c_3/3,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,div2#,even#,neq#,p#} and constructors {0 ,false,s,true,y} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)) -->_2 even#(s(s(x))) -> c_9(even#(x)):5 -->_1 cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),neq#(x,0()),div2#(x)):3 -->_1 cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x)),neq#(x,0()),p#(x)):2 2:S:cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x)),neq#(x,0()),p#(x)) -->_1 cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)):1 3:S:cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),neq#(x,0()),div2#(x)) -->_3 div2#(s(s(x))) -> c_6(div2#(x)):4 -->_1 cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)):1 4:S:div2#(s(s(x))) -> c_6(div2#(x)) -->_1 div2#(s(s(x))) -> c_6(div2#(x)):4 5:S:even#(s(s(x))) -> c_9(even#(x)) -->_1 even#(s(s(x))) -> c_9(even#(x)):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x))) cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),div2#(x)) ** Step 1.b:5: UsableRules. MAYBE + Considered Problem: - Strict DPs: cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)) cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x))) cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),div2#(x)) div2#(s(s(x))) -> c_6(div2#(x)) even#(s(s(x))) -> c_9(even#(x)) - Weak TRS: cond1(true(),x) -> cond2(even(x),x) cond2(false(),x) -> cond1(neq(x,0()),p(x)) cond2(true(),x) -> cond1(neq(x,0()),div2(x)) div2(0()) -> 0() div2(s(0())) -> 0() div2(s(s(x))) -> s(div2(x)) even(0()) -> true() even(s(0())) -> false() even(s(s(x))) -> even(x) neq(0(),0()) -> false() neq(0(),s(x)) -> true() neq(s(x),0()) -> true() neq(s(x),s(y())) -> neq(x,y()) p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/2,div2/1,even/1,neq/2,p/1,cond1#/2,cond2#/2,div2#/1,even#/1,neq#/2,p#/1} / {0/0,false/0,s/1 ,true/0,y/0,c_1/2,c_2/1,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,div2#,even#,neq#,p#} and constructors {0 ,false,s,true,y} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: div2(0()) -> 0() div2(s(0())) -> 0() div2(s(s(x))) -> s(div2(x)) even(0()) -> true() even(s(0())) -> false() even(s(s(x))) -> even(x) neq(0(),0()) -> false() neq(s(x),0()) -> true() p(0()) -> 0() p(s(x)) -> x cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)) cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x))) cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),div2#(x)) div2#(s(s(x))) -> c_6(div2#(x)) even#(s(s(x))) -> c_9(even#(x)) ** Step 1.b:6: DecomposeDG. MAYBE + Considered Problem: - Strict DPs: cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)) cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x))) cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),div2#(x)) div2#(s(s(x))) -> c_6(div2#(x)) even#(s(s(x))) -> c_9(even#(x)) - Weak TRS: div2(0()) -> 0() div2(s(0())) -> 0() div2(s(s(x))) -> s(div2(x)) even(0()) -> true() even(s(0())) -> false() even(s(s(x))) -> even(x) neq(0(),0()) -> false() neq(s(x),0()) -> true() p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/2,div2/1,even/1,neq/2,p/1,cond1#/2,cond2#/2,div2#/1,even#/1,neq#/2,p#/1} / {0/0,false/0,s/1 ,true/0,y/0,c_1/2,c_2/1,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,div2#,even#,neq#,p#} and constructors {0 ,false,s,true,y} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)) cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x))) cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),div2#(x)) and a lower component div2#(s(s(x))) -> c_6(div2#(x)) even#(s(s(x))) -> c_9(even#(x)) Further, following extension rules are added to the lower component. cond1#(true(),x) -> cond2#(even(x),x) cond1#(true(),x) -> even#(x) cond2#(false(),x) -> cond1#(neq(x,0()),p(x)) cond2#(true(),x) -> cond1#(neq(x,0()),div2(x)) cond2#(true(),x) -> div2#(x) *** Step 1.b:6.a:1: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)) cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x))) cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),div2#(x)) - Weak TRS: div2(0()) -> 0() div2(s(0())) -> 0() div2(s(s(x))) -> s(div2(x)) even(0()) -> true() even(s(0())) -> false() even(s(s(x))) -> even(x) neq(0(),0()) -> false() neq(s(x),0()) -> true() p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/2,div2/1,even/1,neq/2,p/1,cond1#/2,cond2#/2,div2#/1,even#/1,neq#/2,p#/1} / {0/0,false/0,s/1 ,true/0,y/0,c_1/2,c_2/1,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,div2#,even#,neq#,p#} and constructors {0 ,false,s,true,y} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)) -->_1 cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),div2#(x)):3 -->_1 cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x))):2 2:S:cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x))) -->_1 cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)):1 3:S:cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x)),div2#(x)) -->_1 cond1#(true(),x) -> c_1(cond2#(even(x),x),even#(x)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: cond1#(true(),x) -> c_1(cond2#(even(x),x)) cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x))) *** Step 1.b:6.a:2: WeightGap. MAYBE + Considered Problem: - Strict DPs: cond1#(true(),x) -> c_1(cond2#(even(x),x)) cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x))) cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x))) - Weak TRS: div2(0()) -> 0() div2(s(0())) -> 0() div2(s(s(x))) -> s(div2(x)) even(0()) -> true() even(s(0())) -> false() even(s(s(x))) -> even(x) neq(0(),0()) -> false() neq(s(x),0()) -> true() p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/2,div2/1,even/1,neq/2,p/1,cond1#/2,cond2#/2,div2#/1,even#/1,neq#/2,p#/1} / {0/0,false/0,s/1 ,true/0,y/0,c_1/1,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,div2#,even#,neq#,p#} and constructors {0 ,false,s,true,y} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(s) = {1}, uargs(cond1#) = {1,2}, uargs(cond2#) = {1}, uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [14] p(cond1) = [0] p(cond2) = [0] p(div2) = [1] x1 + [0] p(even) = [0] p(false) = [0] p(neq) = [13] p(p) = [1] x1 + [2] p(s) = [1] x1 + [2] p(true) = [0] p(y) = [1] p(cond1#) = [1] x1 + [1] x2 + [9] p(cond2#) = [1] x1 + [1] x2 + [0] p(div2#) = [0] p(even#) = [0] p(neq#) = [0] p(p#) = [0] p(c_1) = [1] x1 + [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [2] p(c_15) = [1] Following rules are strictly oriented: cond1#(true(),x) = [1] x + [9] > [1] x + [0] = c_1(cond2#(even(x),x)) Following rules are (at-least) weakly oriented: cond2#(false(),x) = [1] x + [0] >= [1] x + [24] = c_2(cond1#(neq(x,0()),p(x))) cond2#(true(),x) = [1] x + [0] >= [1] x + [22] = c_3(cond1#(neq(x,0()),div2(x))) div2(0()) = [14] >= [14] = 0() div2(s(0())) = [16] >= [14] = 0() div2(s(s(x))) = [1] x + [4] >= [1] x + [2] = s(div2(x)) even(0()) = [0] >= [0] = true() even(s(0())) = [0] >= [0] = false() even(s(s(x))) = [0] >= [0] = even(x) neq(0(),0()) = [13] >= [0] = false() neq(s(x),0()) = [13] >= [0] = true() p(0()) = [16] >= [14] = 0() p(s(x)) = [1] x + [4] >= [1] x + [0] = x Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:6.a:3: Failure MAYBE + Considered Problem: - Strict DPs: cond2#(false(),x) -> c_2(cond1#(neq(x,0()),p(x))) cond2#(true(),x) -> c_3(cond1#(neq(x,0()),div2(x))) - Weak DPs: cond1#(true(),x) -> c_1(cond2#(even(x),x)) - Weak TRS: div2(0()) -> 0() div2(s(0())) -> 0() div2(s(s(x))) -> s(div2(x)) even(0()) -> true() even(s(0())) -> false() even(s(s(x))) -> even(x) neq(0(),0()) -> false() neq(s(x),0()) -> true() p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/2,div2/1,even/1,neq/2,p/1,cond1#/2,cond2#/2,div2#/1,even#/1,neq#/2,p#/1} / {0/0,false/0,s/1 ,true/0,y/0,c_1/1,c_2/1,c_3/1,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,div2#,even#,neq#,p#} and constructors {0 ,false,s,true,y} + Applied Processor: EmptyProcessor + Details: The problem is still open. *** Step 1.b:6.b:1: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div2#(s(s(x))) -> c_6(div2#(x)) even#(s(s(x))) -> c_9(even#(x)) - Weak DPs: cond1#(true(),x) -> cond2#(even(x),x) cond1#(true(),x) -> even#(x) cond2#(false(),x) -> cond1#(neq(x,0()),p(x)) cond2#(true(),x) -> cond1#(neq(x,0()),div2(x)) cond2#(true(),x) -> div2#(x) - Weak TRS: div2(0()) -> 0() div2(s(0())) -> 0() div2(s(s(x))) -> s(div2(x)) even(0()) -> true() even(s(0())) -> false() even(s(s(x))) -> even(x) neq(0(),0()) -> false() neq(s(x),0()) -> true() p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/2,div2/1,even/1,neq/2,p/1,cond1#/2,cond2#/2,div2#/1,even#/1,neq#/2,p#/1} / {0/0,false/0,s/1 ,true/0,y/0,c_1/2,c_2/1,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,div2#,even#,neq#,p#} and constructors {0 ,false,s,true,y} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(s) = {1}, uargs(cond1#) = {1,2}, uargs(cond2#) = {1}, uargs(c_6) = {1}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(cond1) = [1] x1 + [2] p(cond2) = [4] x1 + [0] p(div2) = [1] x1 + [0] p(even) = [3] p(false) = [3] p(neq) = [1] x2 + [2] p(p) = [1] x1 + [0] p(s) = [1] x1 + [4] p(true) = [3] p(y) = [0] p(cond1#) = [1] x1 + [1] x2 + [0] p(cond2#) = [1] x1 + [1] x2 + [0] p(div2#) = [0] p(even#) = [1] x1 + [2] p(neq#) = [4] p(p#) = [2] p(c_1) = [1] x1 + [1] p(c_2) = [1] p(c_3) = [2] x1 + [0] p(c_4) = [1] p(c_5) = [0] p(c_6) = [1] x1 + [1] p(c_7) = [4] p(c_8) = [0] p(c_9) = [1] x1 + [6] p(c_10) = [0] p(c_11) = [1] p(c_12) = [1] p(c_13) = [4] p(c_14) = [1] p(c_15) = [2] Following rules are strictly oriented: even#(s(s(x))) = [1] x + [10] > [1] x + [8] = c_9(even#(x)) Following rules are (at-least) weakly oriented: cond1#(true(),x) = [1] x + [3] >= [1] x + [3] = cond2#(even(x),x) cond1#(true(),x) = [1] x + [3] >= [1] x + [2] = even#(x) cond2#(false(),x) = [1] x + [3] >= [1] x + [3] = cond1#(neq(x,0()),p(x)) cond2#(true(),x) = [1] x + [3] >= [1] x + [3] = cond1#(neq(x,0()),div2(x)) cond2#(true(),x) = [1] x + [3] >= [0] = div2#(x) div2#(s(s(x))) = [0] >= [1] = c_6(div2#(x)) div2(0()) = [1] >= [1] = 0() div2(s(0())) = [5] >= [1] = 0() div2(s(s(x))) = [1] x + [8] >= [1] x + [4] = s(div2(x)) even(0()) = [3] >= [3] = true() even(s(0())) = [3] >= [3] = false() even(s(s(x))) = [3] >= [3] = even(x) neq(0(),0()) = [3] >= [3] = false() neq(s(x),0()) = [3] >= [3] = true() p(0()) = [1] >= [1] = 0() p(s(x)) = [1] x + [4] >= [1] x + [0] = x Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:6.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div2#(s(s(x))) -> c_6(div2#(x)) - Weak DPs: cond1#(true(),x) -> cond2#(even(x),x) cond1#(true(),x) -> even#(x) cond2#(false(),x) -> cond1#(neq(x,0()),p(x)) cond2#(true(),x) -> cond1#(neq(x,0()),div2(x)) cond2#(true(),x) -> div2#(x) even#(s(s(x))) -> c_9(even#(x)) - Weak TRS: div2(0()) -> 0() div2(s(0())) -> 0() div2(s(s(x))) -> s(div2(x)) even(0()) -> true() even(s(0())) -> false() even(s(s(x))) -> even(x) neq(0(),0()) -> false() neq(s(x),0()) -> true() p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/2,div2/1,even/1,neq/2,p/1,cond1#/2,cond2#/2,div2#/1,even#/1,neq#/2,p#/1} / {0/0,false/0,s/1 ,true/0,y/0,c_1/2,c_2/1,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,div2#,even#,neq#,p#} and constructors {0 ,false,s,true,y} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_6) = {1}, uargs(c_9) = {1} Following symbols are considered usable: {div2,p,cond1#,cond2#,div2#,even#,neq#,p#} TcT has computed the following interpretation: p(0) = [5] p(cond1) = [2] x2 + [1] p(cond2) = [2] x1 + [1] x2 + [0] p(div2) = [1] x1 + [0] p(even) = [1] x1 + [0] p(false) = [4] p(neq) = [1] x1 + [13] p(p) = [1] x1 + [0] p(s) = [1] x1 + [4] p(true) = [1] p(y) = [8] p(cond1#) = [2] x2 + [0] p(cond2#) = [2] x2 + [0] p(div2#) = [2] x1 + [0] p(even#) = [0] p(neq#) = [1] x1 + [4] x2 + [8] p(p#) = [2] x1 + [4] p(c_1) = [1] x1 + [2] p(c_2) = [2] x1 + [1] p(c_3) = [2] x1 + [2] x2 + [2] p(c_4) = [0] p(c_5) = [1] p(c_6) = [1] x1 + [15] p(c_7) = [1] p(c_8) = [4] p(c_9) = [4] x1 + [0] p(c_10) = [0] p(c_11) = [1] p(c_12) = [4] p(c_13) = [0] p(c_14) = [1] p(c_15) = [1] Following rules are strictly oriented: div2#(s(s(x))) = [2] x + [16] > [2] x + [15] = c_6(div2#(x)) Following rules are (at-least) weakly oriented: cond1#(true(),x) = [2] x + [0] >= [2] x + [0] = cond2#(even(x),x) cond1#(true(),x) = [2] x + [0] >= [0] = even#(x) cond2#(false(),x) = [2] x + [0] >= [2] x + [0] = cond1#(neq(x,0()),p(x)) cond2#(true(),x) = [2] x + [0] >= [2] x + [0] = cond1#(neq(x,0()),div2(x)) cond2#(true(),x) = [2] x + [0] >= [2] x + [0] = div2#(x) even#(s(s(x))) = [0] >= [0] = c_9(even#(x)) div2(0()) = [5] >= [5] = 0() div2(s(0())) = [9] >= [5] = 0() div2(s(s(x))) = [1] x + [8] >= [1] x + [4] = s(div2(x)) p(0()) = [5] >= [5] = 0() p(s(x)) = [1] x + [4] >= [1] x + [0] = x *** Step 1.b:6.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: cond1#(true(),x) -> cond2#(even(x),x) cond1#(true(),x) -> even#(x) cond2#(false(),x) -> cond1#(neq(x,0()),p(x)) cond2#(true(),x) -> cond1#(neq(x,0()),div2(x)) cond2#(true(),x) -> div2#(x) div2#(s(s(x))) -> c_6(div2#(x)) even#(s(s(x))) -> c_9(even#(x)) - Weak TRS: div2(0()) -> 0() div2(s(0())) -> 0() div2(s(s(x))) -> s(div2(x)) even(0()) -> true() even(s(0())) -> false() even(s(s(x))) -> even(x) neq(0(),0()) -> false() neq(s(x),0()) -> true() p(0()) -> 0() p(s(x)) -> x - Signature: {cond1/2,cond2/2,div2/1,even/1,neq/2,p/1,cond1#/2,cond2#/2,div2#/1,even#/1,neq#/2,p#/1} / {0/0,false/0,s/1 ,true/0,y/0,c_1/2,c_2/1,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/0,c_12/0,c_13/1,c_14/0,c_15/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond1#,cond2#,div2#,even#,neq#,p#} and constructors {0 ,false,s,true,y} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),?)