MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: f() -> g() f() -> h() half(0()) -> 0() half(s(0())) -> 0() half(s(s(x))) -> s(half(x)) 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) logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) logarithm(x) -> logIter(x,0()) - Signature: {f/0,half/1,if/4,inc/1,le/2,logIter/2,logarithm/1} / {0/0,false/0,g/0,h/0,logZeroError/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {f,half,if,inc,le,logIter,logarithm} and constructors {0 ,false,g,h,logZeroError,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs f#() -> c_1() f#() -> c_2() half#(0()) -> c_3() half#(s(0())) -> c_4() half#(s(s(x))) -> c_5(half#(x)) if#(false(),b,x,y) -> c_6() if#(true(),false(),x,s(y)) -> c_7() if#(true(),true(),x,y) -> c_8(logIter#(x,y)) inc#(0()) -> c_9() inc#(s(x)) -> c_10(inc#(x)) le#(0(),y) -> c_11() le#(s(x),0()) -> c_12() le#(s(x),s(y)) -> c_13(le#(x,y)) logIter#(x,y) -> c_14(if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,half#(x) ,inc#(y)) logarithm#(x) -> c_15(logIter#(x,0())) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: f#() -> c_1() f#() -> c_2() half#(0()) -> c_3() half#(s(0())) -> c_4() half#(s(s(x))) -> c_5(half#(x)) if#(false(),b,x,y) -> c_6() if#(true(),false(),x,s(y)) -> c_7() if#(true(),true(),x,y) -> c_8(logIter#(x,y)) inc#(0()) -> c_9() inc#(s(x)) -> c_10(inc#(x)) le#(0(),y) -> c_11() le#(s(x),0()) -> c_12() le#(s(x),s(y)) -> c_13(le#(x,y)) logIter#(x,y) -> c_14(if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,half#(x) ,inc#(y)) logarithm#(x) -> c_15(logIter#(x,0())) - Weak TRS: f() -> g() f() -> h() half(0()) -> 0() half(s(0())) -> 0() half(s(s(x))) -> s(half(x)) 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) logIter(x,y) -> if(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) logarithm(x) -> logIter(x,0()) - Signature: {f/0,half/1,if/4,inc/1,le/2,logIter/2,logarithm/1,f#/0,half#/1,if#/4,inc#/1,le#/2,logIter#/2 ,logarithm#/1} / {0/0,false/0,g/0,h/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/5,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,half#,if#,inc#,le#,logIter# ,logarithm#} and constructors {0,false,g,h,logZeroError,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: half(0()) -> 0() half(s(0())) -> 0() half(s(s(x))) -> s(half(x)) 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) f#() -> c_1() f#() -> c_2() half#(0()) -> c_3() half#(s(0())) -> c_4() half#(s(s(x))) -> c_5(half#(x)) if#(false(),b,x,y) -> c_6() if#(true(),false(),x,s(y)) -> c_7() if#(true(),true(),x,y) -> c_8(logIter#(x,y)) inc#(0()) -> c_9() inc#(s(x)) -> c_10(inc#(x)) le#(0(),y) -> c_11() le#(s(x),0()) -> c_12() le#(s(x),s(y)) -> c_13(le#(x,y)) logIter#(x,y) -> c_14(if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,half#(x) ,inc#(y)) logarithm#(x) -> c_15(logIter#(x,0())) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: f#() -> c_1() f#() -> c_2() half#(0()) -> c_3() half#(s(0())) -> c_4() half#(s(s(x))) -> c_5(half#(x)) if#(false(),b,x,y) -> c_6() if#(true(),false(),x,s(y)) -> c_7() if#(true(),true(),x,y) -> c_8(logIter#(x,y)) inc#(0()) -> c_9() inc#(s(x)) -> c_10(inc#(x)) le#(0(),y) -> c_11() le#(s(x),0()) -> c_12() le#(s(x),s(y)) -> c_13(le#(x,y)) logIter#(x,y) -> c_14(if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,half#(x) ,inc#(y)) logarithm#(x) -> c_15(logIter#(x,0())) - Weak TRS: half(0()) -> 0() half(s(0())) -> 0() half(s(s(x))) -> s(half(x)) 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) - Signature: {f/0,half/1,if/4,inc/1,le/2,logIter/2,logarithm/1,f#/0,half#/1,if#/4,inc#/1,le#/2,logIter#/2 ,logarithm#/1} / {0/0,false/0,g/0,h/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/5,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,half#,if#,inc#,le#,logIter# ,logarithm#} and constructors {0,false,g,h,logZeroError,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,6,7,9,11,12} by application of Pre({1,2,3,4,6,7,9,11,12}) = {5,10,13,14}. Here rules are labelled as follows: 1: f#() -> c_1() 2: f#() -> c_2() 3: half#(0()) -> c_3() 4: half#(s(0())) -> c_4() 5: half#(s(s(x))) -> c_5(half#(x)) 6: if#(false(),b,x,y) -> c_6() 7: if#(true(),false(),x,s(y)) -> c_7() 8: if#(true(),true(),x,y) -> c_8(logIter#(x,y)) 9: inc#(0()) -> c_9() 10: inc#(s(x)) -> c_10(inc#(x)) 11: le#(0(),y) -> c_11() 12: le#(s(x),0()) -> c_12() 13: le#(s(x),s(y)) -> c_13(le#(x,y)) 14: logIter#(x,y) -> c_14(if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,half#(x) ,inc#(y)) 15: logarithm#(x) -> c_15(logIter#(x,0())) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: half#(s(s(x))) -> c_5(half#(x)) if#(true(),true(),x,y) -> c_8(logIter#(x,y)) inc#(s(x)) -> c_10(inc#(x)) le#(s(x),s(y)) -> c_13(le#(x,y)) logIter#(x,y) -> c_14(if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,half#(x) ,inc#(y)) logarithm#(x) -> c_15(logIter#(x,0())) - Weak DPs: f#() -> c_1() f#() -> c_2() half#(0()) -> c_3() half#(s(0())) -> c_4() if#(false(),b,x,y) -> c_6() if#(true(),false(),x,s(y)) -> c_7() inc#(0()) -> c_9() le#(0(),y) -> c_11() le#(s(x),0()) -> c_12() - Weak TRS: half(0()) -> 0() half(s(0())) -> 0() half(s(s(x))) -> s(half(x)) 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) - Signature: {f/0,half/1,if/4,inc/1,le/2,logIter/2,logarithm/1,f#/0,half#/1,if#/4,inc#/1,le#/2,logIter#/2 ,logarithm#/1} / {0/0,false/0,g/0,h/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/5,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,half#,if#,inc#,le#,logIter# ,logarithm#} and constructors {0,false,g,h,logZeroError,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:half#(s(s(x))) -> c_5(half#(x)) -->_1 half#(s(0())) -> c_4():10 -->_1 half#(0()) -> c_3():9 -->_1 half#(s(s(x))) -> c_5(half#(x)):1 2:S:if#(true(),true(),x,y) -> c_8(logIter#(x,y)) -->_1 logIter#(x,y) -> c_14(if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,half#(x) ,inc#(y)):5 3:S:inc#(s(x)) -> c_10(inc#(x)) -->_1 inc#(0()) -> c_9():13 -->_1 inc#(s(x)) -> c_10(inc#(x)):3 4:S:le#(s(x),s(y)) -> c_13(le#(x,y)) -->_1 le#(s(x),0()) -> c_12():15 -->_1 le#(0(),y) -> c_11():14 -->_1 le#(s(x),s(y)) -> c_13(le#(x,y)):4 5:S:logIter#(x,y) -> c_14(if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,half#(x) ,inc#(y)) -->_3 le#(s(x),0()) -> c_12():15 -->_2 le#(s(x),0()) -> c_12():15 -->_5 inc#(0()) -> c_9():13 -->_1 if#(true(),false(),x,s(y)) -> c_7():12 -->_1 if#(false(),b,x,y) -> c_6():11 -->_4 half#(s(0())) -> c_4():10 -->_4 half#(0()) -> c_3():9 -->_3 le#(s(x),s(y)) -> c_13(le#(x,y)):4 -->_2 le#(s(x),s(y)) -> c_13(le#(x,y)):4 -->_5 inc#(s(x)) -> c_10(inc#(x)):3 -->_1 if#(true(),true(),x,y) -> c_8(logIter#(x,y)):2 -->_4 half#(s(s(x))) -> c_5(half#(x)):1 6:S:logarithm#(x) -> c_15(logIter#(x,0())) -->_1 logIter#(x,y) -> c_14(if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,half#(x) ,inc#(y)):5 7:W:f#() -> c_1() 8:W:f#() -> c_2() 9:W:half#(0()) -> c_3() 10:W:half#(s(0())) -> c_4() 11:W:if#(false(),b,x,y) -> c_6() 12:W:if#(true(),false(),x,s(y)) -> c_7() 13:W:inc#(0()) -> c_9() 14:W:le#(0(),y) -> c_11() 15:W:le#(s(x),0()) -> c_12() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: f#() -> c_2() 7: f#() -> c_1() 14: le#(0(),y) -> c_11() 11: if#(false(),b,x,y) -> c_6() 12: if#(true(),false(),x,s(y)) -> c_7() 13: inc#(0()) -> c_9() 15: le#(s(x),0()) -> c_12() 9: half#(0()) -> c_3() 10: half#(s(0())) -> c_4() * Step 5: RemoveHeads MAYBE + Considered Problem: - Strict DPs: half#(s(s(x))) -> c_5(half#(x)) if#(true(),true(),x,y) -> c_8(logIter#(x,y)) inc#(s(x)) -> c_10(inc#(x)) le#(s(x),s(y)) -> c_13(le#(x,y)) logIter#(x,y) -> c_14(if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,half#(x) ,inc#(y)) logarithm#(x) -> c_15(logIter#(x,0())) - Weak TRS: half(0()) -> 0() half(s(0())) -> 0() half(s(s(x))) -> s(half(x)) 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) - Signature: {f/0,half/1,if/4,inc/1,le/2,logIter/2,logarithm/1,f#/0,half#/1,if#/4,inc#/1,le#/2,logIter#/2 ,logarithm#/1} / {0/0,false/0,g/0,h/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/5,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,half#,if#,inc#,le#,logIter# ,logarithm#} and constructors {0,false,g,h,logZeroError,s,true} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:half#(s(s(x))) -> c_5(half#(x)) -->_1 half#(s(s(x))) -> c_5(half#(x)):1 2:S:if#(true(),true(),x,y) -> c_8(logIter#(x,y)) -->_1 logIter#(x,y) -> c_14(if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,half#(x) ,inc#(y)):5 3:S:inc#(s(x)) -> c_10(inc#(x)) -->_1 inc#(s(x)) -> c_10(inc#(x)):3 4:S:le#(s(x),s(y)) -> c_13(le#(x,y)) -->_1 le#(s(x),s(y)) -> c_13(le#(x,y)):4 5:S:logIter#(x,y) -> c_14(if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,half#(x) ,inc#(y)) -->_3 le#(s(x),s(y)) -> c_13(le#(x,y)):4 -->_2 le#(s(x),s(y)) -> c_13(le#(x,y)):4 -->_5 inc#(s(x)) -> c_10(inc#(x)):3 -->_1 if#(true(),true(),x,y) -> c_8(logIter#(x,y)):2 -->_4 half#(s(s(x))) -> c_5(half#(x)):1 6:S:logarithm#(x) -> c_15(logIter#(x,0())) -->_1 logIter#(x,y) -> c_14(if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,half#(x) ,inc#(y)):5 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). [(6,logarithm#(x) -> c_15(logIter#(x,0())))] * Step 6: Failure MAYBE + Considered Problem: - Strict DPs: half#(s(s(x))) -> c_5(half#(x)) if#(true(),true(),x,y) -> c_8(logIter#(x,y)) inc#(s(x)) -> c_10(inc#(x)) le#(s(x),s(y)) -> c_13(le#(x,y)) logIter#(x,y) -> c_14(if#(le(s(0()),x),le(s(s(0())),x),half(x),inc(y)) ,le#(s(0()),x) ,le#(s(s(0())),x) ,half#(x) ,inc#(y)) - Weak TRS: half(0()) -> 0() half(s(0())) -> 0() half(s(s(x))) -> s(half(x)) 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) - Signature: {f/0,half/1,if/4,inc/1,le/2,logIter/2,logarithm/1,f#/0,half#/1,if#/4,inc#/1,le#/2,logIter#/2 ,logarithm#/1} / {0/0,false/0,g/0,h/0,logZeroError/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/5,c_15/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,half#,if#,inc#,le#,logIter# ,logarithm#} and constructors {0,false,g,h,logZeroError,s,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE