MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: if(false(),b,x,y) -> logZeroError() if(true(),false(),x,s(y)) -> y if(true(),true(),x,y) -> logIter(x,y) inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) log(x) -> logIter(x,0()) logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y)) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if/4,inc/1,le/2,log/1,logIter/2,minus/2,quot/2} / {0/0,false/0,logZeroError/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if,inc,le,log,logIter,minus,quot} and constructors {0 ,false,logZeroError,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs if#(false(),b,x,y) -> c_1() if#(true(),false(),x,s(y)) -> c_2() if#(true(),true(),x,y) -> c_3(logIter#(x,y)) inc#(0()) -> c_4() inc#(s(x)) -> c_5(inc#(x)) le#(0(),y) -> c_6() le#(s(x),0()) -> c_7() le#(s(x),s(y)) -> c_8(le#(x,y)) log#(x) -> c_9(logIter#(x,0())) logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,quot#(x,s(s(0()))) ,inc#(y)) minus#(x,0()) -> c_11() minus#(s(x),s(y)) -> c_12(minus#(x,y)) quot#(0(),s(y)) -> c_13() quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: if#(false(),b,x,y) -> c_1() if#(true(),false(),x,s(y)) -> c_2() if#(true(),true(),x,y) -> c_3(logIter#(x,y)) inc#(0()) -> c_4() inc#(s(x)) -> c_5(inc#(x)) le#(0(),y) -> c_6() le#(s(x),0()) -> c_7() le#(s(x),s(y)) -> c_8(le#(x,y)) log#(x) -> c_9(logIter#(x,0())) logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,quot#(x,s(s(0()))) ,inc#(y)) minus#(x,0()) -> c_11() minus#(s(x),s(y)) -> c_12(minus#(x,y)) quot#(0(),s(y)) -> c_13() quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: if(false(),b,x,y) -> logZeroError() if(true(),false(),x,s(y)) -> y if(true(),true(),x,y) -> logIter(x,y) inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) log(x) -> logIter(x,0()) logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y)) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if/4,inc/1,le/2,log/1,logIter/2,minus/2,quot/2,if#/4,inc#/1,le#/2,log#/1,logIter#/2,minus#/2 ,quot#/2} / {0/0,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/1 ,c_10/5,c_11/0,c_12/1,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {if#,inc#,le#,log#,logIter#,minus# ,quot#} and constructors {0,false,logZeroError,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) if#(false(),b,x,y) -> c_1() if#(true(),false(),x,s(y)) -> c_2() if#(true(),true(),x,y) -> c_3(logIter#(x,y)) inc#(0()) -> c_4() inc#(s(x)) -> c_5(inc#(x)) le#(0(),y) -> c_6() le#(s(x),0()) -> c_7() le#(s(x),s(y)) -> c_8(le#(x,y)) log#(x) -> c_9(logIter#(x,0())) logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,quot#(x,s(s(0()))) ,inc#(y)) minus#(x,0()) -> c_11() minus#(s(x),s(y)) -> c_12(minus#(x,y)) quot#(0(),s(y)) -> c_13() quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: if#(false(),b,x,y) -> c_1() if#(true(),false(),x,s(y)) -> c_2() if#(true(),true(),x,y) -> c_3(logIter#(x,y)) inc#(0()) -> c_4() inc#(s(x)) -> c_5(inc#(x)) le#(0(),y) -> c_6() le#(s(x),0()) -> c_7() le#(s(x),s(y)) -> c_8(le#(x,y)) log#(x) -> c_9(logIter#(x,0())) logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,quot#(x,s(s(0()))) ,inc#(y)) minus#(x,0()) -> c_11() minus#(s(x),s(y)) -> c_12(minus#(x,y)) quot#(0(),s(y)) -> c_13() quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if/4,inc/1,le/2,log/1,logIter/2,minus/2,quot/2,if#/4,inc#/1,le#/2,log#/1,logIter#/2,minus#/2 ,quot#/2} / {0/0,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/1 ,c_10/5,c_11/0,c_12/1,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {if#,inc#,le#,log#,logIter#,minus# ,quot#} and constructors {0,false,logZeroError,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,4,6,7,11,13} by application of Pre({1,2,4,6,7,11,13}) = {5,8,10,12,14}. Here rules are labelled as follows: 1: if#(false(),b,x,y) -> c_1() 2: if#(true(),false(),x,s(y)) -> c_2() 3: if#(true(),true(),x,y) -> c_3(logIter#(x,y)) 4: inc#(0()) -> c_4() 5: inc#(s(x)) -> c_5(inc#(x)) 6: le#(0(),y) -> c_6() 7: le#(s(x),0()) -> c_7() 8: le#(s(x),s(y)) -> c_8(le#(x,y)) 9: log#(x) -> c_9(logIter#(x,0())) 10: logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,quot#(x,s(s(0()))) ,inc#(y)) 11: minus#(x,0()) -> c_11() 12: minus#(s(x),s(y)) -> c_12(minus#(x,y)) 13: quot#(0(),s(y)) -> c_13() 14: quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: if#(true(),true(),x,y) -> c_3(logIter#(x,y)) inc#(s(x)) -> c_5(inc#(x)) le#(s(x),s(y)) -> c_8(le#(x,y)) log#(x) -> c_9(logIter#(x,0())) logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,quot#(x,s(s(0()))) ,inc#(y)) minus#(s(x),s(y)) -> c_12(minus#(x,y)) quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak DPs: if#(false(),b,x,y) -> c_1() if#(true(),false(),x,s(y)) -> c_2() inc#(0()) -> c_4() le#(0(),y) -> c_6() le#(s(x),0()) -> c_7() minus#(x,0()) -> c_11() quot#(0(),s(y)) -> c_13() - Weak TRS: inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if/4,inc/1,le/2,log/1,logIter/2,minus/2,quot/2,if#/4,inc#/1,le#/2,log#/1,logIter#/2,minus#/2 ,quot#/2} / {0/0,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/1 ,c_10/5,c_11/0,c_12/1,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {if#,inc#,le#,log#,logIter#,minus# ,quot#} and constructors {0,false,logZeroError,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:if#(true(),true(),x,y) -> c_3(logIter#(x,y)) -->_1 logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,quot#(x,s(s(0()))) ,inc#(y)):5 2:S:inc#(s(x)) -> c_5(inc#(x)) -->_1 inc#(0()) -> c_4():10 -->_1 inc#(s(x)) -> c_5(inc#(x)):2 3:S:le#(s(x),s(y)) -> c_8(le#(x,y)) -->_1 le#(s(x),0()) -> c_7():12 -->_1 le#(0(),y) -> c_6():11 -->_1 le#(s(x),s(y)) -> c_8(le#(x,y)):3 4:S:log#(x) -> c_9(logIter#(x,0())) -->_1 logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,quot#(x,s(s(0()))) ,inc#(y)):5 5:S:logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,quot#(x,s(s(0()))) ,inc#(y)) -->_4 quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)):7 -->_4 quot#(0(),s(y)) -> c_13():14 -->_3 le#(s(x),0()) -> c_7():12 -->_2 le#(s(x),0()) -> c_7():12 -->_5 inc#(0()) -> c_4():10 -->_1 if#(true(),false(),x,s(y)) -> c_2():9 -->_1 if#(false(),b,x,y) -> c_1():8 -->_3 le#(s(x),s(y)) -> c_8(le#(x,y)):3 -->_2 le#(s(x),s(y)) -> c_8(le#(x,y)):3 -->_5 inc#(s(x)) -> c_5(inc#(x)):2 -->_1 if#(true(),true(),x,y) -> c_3(logIter#(x,y)):1 6:S:minus#(s(x),s(y)) -> c_12(minus#(x,y)) -->_1 minus#(x,0()) -> c_11():13 -->_1 minus#(s(x),s(y)) -> c_12(minus#(x,y)):6 7:S:quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)) -->_1 quot#(0(),s(y)) -> c_13():14 -->_2 minus#(x,0()) -> c_11():13 -->_1 quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)):7 -->_2 minus#(s(x),s(y)) -> c_12(minus#(x,y)):6 8:W:if#(false(),b,x,y) -> c_1() 9:W:if#(true(),false(),x,s(y)) -> c_2() 10:W:inc#(0()) -> c_4() 11:W:le#(0(),y) -> c_6() 12:W:le#(s(x),0()) -> c_7() 13:W:minus#(x,0()) -> c_11() 14:W:quot#(0(),s(y)) -> c_13() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: le#(0(),y) -> c_6() 8: if#(false(),b,x,y) -> c_1() 9: if#(true(),false(),x,s(y)) -> c_2() 10: inc#(0()) -> c_4() 12: le#(s(x),0()) -> c_7() 13: minus#(x,0()) -> c_11() 14: quot#(0(),s(y)) -> c_13() * Step 5: RemoveHeads MAYBE + Considered Problem: - Strict DPs: if#(true(),true(),x,y) -> c_3(logIter#(x,y)) inc#(s(x)) -> c_5(inc#(x)) le#(s(x),s(y)) -> c_8(le#(x,y)) log#(x) -> c_9(logIter#(x,0())) logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,quot#(x,s(s(0()))) ,inc#(y)) minus#(s(x),s(y)) -> c_12(minus#(x,y)) quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if/4,inc/1,le/2,log/1,logIter/2,minus/2,quot/2,if#/4,inc#/1,le#/2,log#/1,logIter#/2,minus#/2 ,quot#/2} / {0/0,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/1 ,c_10/5,c_11/0,c_12/1,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {if#,inc#,le#,log#,logIter#,minus# ,quot#} and constructors {0,false,logZeroError,s,true} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:if#(true(),true(),x,y) -> c_3(logIter#(x,y)) -->_1 logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,quot#(x,s(s(0()))) ,inc#(y)):5 2:S:inc#(s(x)) -> c_5(inc#(x)) -->_1 inc#(s(x)) -> c_5(inc#(x)):2 3:S:le#(s(x),s(y)) -> c_8(le#(x,y)) -->_1 le#(s(x),s(y)) -> c_8(le#(x,y)):3 4:S:log#(x) -> c_9(logIter#(x,0())) -->_1 logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,quot#(x,s(s(0()))) ,inc#(y)):5 5:S:logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,quot#(x,s(s(0()))) ,inc#(y)) -->_4 quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)):7 -->_3 le#(s(x),s(y)) -> c_8(le#(x,y)):3 -->_2 le#(s(x),s(y)) -> c_8(le#(x,y)):3 -->_5 inc#(s(x)) -> c_5(inc#(x)):2 -->_1 if#(true(),true(),x,y) -> c_3(logIter#(x,y)):1 6:S:minus#(s(x),s(y)) -> c_12(minus#(x,y)) -->_1 minus#(s(x),s(y)) -> c_12(minus#(x,y)):6 7:S:quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)) -->_1 quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)):7 -->_2 minus#(s(x),s(y)) -> c_12(minus#(x,y)):6 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). [(4,log#(x) -> c_9(logIter#(x,0())))] * Step 6: Failure MAYBE + Considered Problem: - Strict DPs: if#(true(),true(),x,y) -> c_3(logIter#(x,y)) inc#(s(x)) -> c_5(inc#(x)) le#(s(x),s(y)) -> c_8(le#(x,y)) logIter#(x,y) -> c_10(if#(le(s(0()),x),le(s(s(0())),x),quot(x,s(s(0()))),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,quot#(x,s(s(0()))) ,inc#(y)) minus#(s(x),s(y)) -> c_12(minus#(x,y)) quot#(s(x),s(y)) -> c_14(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: inc(0()) -> s(0()) inc(s(x)) -> s(inc(x)) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if/4,inc/1,le/2,log/1,logIter/2,minus/2,quot/2,if#/4,inc#/1,le#/2,log#/1,logIter#/2,minus#/2 ,quot#/2} / {0/0,false/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0,c_8/1,c_9/1 ,c_10/5,c_11/0,c_12/1,c_13/0,c_14/2} - Obligation: innermost runtime complexity wrt. defined symbols {if#,inc#,le#,log#,logIter#,minus# ,quot#} and constructors {0,false,logZeroError,s,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE