MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: if(false(),b,x,y) -> if2(b,x,y) if(true(),b,x,y) -> log_undefined() if2(false(),x,y) -> log2(quot(x,s(s(0()))),y) if2(true(),x,s(y)) -> y inc(0()) -> 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) -> log2(x,0()) log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y)) minus(x,0()) -> x minus(0(),y) -> 0() 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,if2/3,inc/1,le/2,log/1,log2/2,minus/2,quot/2} / {0/0,false/0,log_undefined/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if,if2,inc,le,log,log2,minus,quot} and constructors {0 ,false,log_undefined,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs if#(false(),b,x,y) -> c_1(if2#(b,x,y)) if#(true(),b,x,y) -> c_2() if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))) if2#(true(),x,s(y)) -> c_4() inc#(0()) -> c_5() inc#(s(x)) -> c_6(inc#(x)) le#(0(),y) -> c_7() le#(s(x),0()) -> c_8() le#(s(x),s(y)) -> c_9(le#(x,y)) log#(x) -> c_10(log2#(x,0())) log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)) minus#(x,0()) -> c_12() minus#(0(),y) -> c_13() minus#(s(x),s(y)) -> c_14(minus#(x,y)) quot#(0(),s(y)) -> c_15() quot#(s(x),s(y)) -> c_16(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(if2#(b,x,y)) if#(true(),b,x,y) -> c_2() if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))) if2#(true(),x,s(y)) -> c_4() inc#(0()) -> c_5() inc#(s(x)) -> c_6(inc#(x)) le#(0(),y) -> c_7() le#(s(x),0()) -> c_8() le#(s(x),s(y)) -> c_9(le#(x,y)) log#(x) -> c_10(log2#(x,0())) log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)) minus#(x,0()) -> c_12() minus#(0(),y) -> c_13() minus#(s(x),s(y)) -> c_14(minus#(x,y)) quot#(0(),s(y)) -> c_15() quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: if(false(),b,x,y) -> if2(b,x,y) if(true(),b,x,y) -> log_undefined() if2(false(),x,y) -> log2(quot(x,s(s(0()))),y) if2(true(),x,s(y)) -> y inc(0()) -> 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) -> log2(x,0()) log2(x,y) -> if(le(x,0()),le(x,s(0())),x,inc(y)) minus(x,0()) -> x minus(0(),y) -> 0() 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,if2/3,inc/1,le/2,log/1,log2/2,minus/2,quot/2,if#/4,if2#/3,inc#/1,le#/2,log#/1,log2#/2,minus#/2 ,quot#/2} / {0/0,false/0,log_undefined/0,s/1,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1 ,c_10/1,c_11/4,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {if#,if2#,inc#,le#,log#,log2#,minus# ,quot#} and constructors {0,false,log_undefined,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: inc(0()) -> 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(0(),y) -> 0() 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(if2#(b,x,y)) if#(true(),b,x,y) -> c_2() if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))) if2#(true(),x,s(y)) -> c_4() inc#(0()) -> c_5() inc#(s(x)) -> c_6(inc#(x)) le#(0(),y) -> c_7() le#(s(x),0()) -> c_8() le#(s(x),s(y)) -> c_9(le#(x,y)) log#(x) -> c_10(log2#(x,0())) log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)) minus#(x,0()) -> c_12() minus#(0(),y) -> c_13() minus#(s(x),s(y)) -> c_14(minus#(x,y)) quot#(0(),s(y)) -> c_15() quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: if#(false(),b,x,y) -> c_1(if2#(b,x,y)) if#(true(),b,x,y) -> c_2() if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))) if2#(true(),x,s(y)) -> c_4() inc#(0()) -> c_5() inc#(s(x)) -> c_6(inc#(x)) le#(0(),y) -> c_7() le#(s(x),0()) -> c_8() le#(s(x),s(y)) -> c_9(le#(x,y)) log#(x) -> c_10(log2#(x,0())) log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)) minus#(x,0()) -> c_12() minus#(0(),y) -> c_13() minus#(s(x),s(y)) -> c_14(minus#(x,y)) quot#(0(),s(y)) -> c_15() quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: inc(0()) -> 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(0(),y) -> 0() 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,if2/3,inc/1,le/2,log/1,log2/2,minus/2,quot/2,if#/4,if2#/3,inc#/1,le#/2,log#/1,log2#/2,minus#/2 ,quot#/2} / {0/0,false/0,log_undefined/0,s/1,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1 ,c_10/1,c_11/4,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {if#,if2#,inc#,le#,log#,log2#,minus# ,quot#} and constructors {0,false,log_undefined,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,4,5,7,8,12,13,15} by application of Pre({2,4,5,7,8,12,13,15}) = {1,3,6,9,11,14,16}. Here rules are labelled as follows: 1: if#(false(),b,x,y) -> c_1(if2#(b,x,y)) 2: if#(true(),b,x,y) -> c_2() 3: if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))) 4: if2#(true(),x,s(y)) -> c_4() 5: inc#(0()) -> c_5() 6: inc#(s(x)) -> c_6(inc#(x)) 7: le#(0(),y) -> c_7() 8: le#(s(x),0()) -> c_8() 9: le#(s(x),s(y)) -> c_9(le#(x,y)) 10: log#(x) -> c_10(log2#(x,0())) 11: log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)) 12: minus#(x,0()) -> c_12() 13: minus#(0(),y) -> c_13() 14: minus#(s(x),s(y)) -> c_14(minus#(x,y)) 15: quot#(0(),s(y)) -> c_15() 16: quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: if#(false(),b,x,y) -> c_1(if2#(b,x,y)) if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))) inc#(s(x)) -> c_6(inc#(x)) le#(s(x),s(y)) -> c_9(le#(x,y)) log#(x) -> c_10(log2#(x,0())) log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)) minus#(s(x),s(y)) -> c_14(minus#(x,y)) quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak DPs: if#(true(),b,x,y) -> c_2() if2#(true(),x,s(y)) -> c_4() inc#(0()) -> c_5() le#(0(),y) -> c_7() le#(s(x),0()) -> c_8() minus#(x,0()) -> c_12() minus#(0(),y) -> c_13() quot#(0(),s(y)) -> c_15() - Weak TRS: inc(0()) -> 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(0(),y) -> 0() 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,if2/3,inc/1,le/2,log/1,log2/2,minus/2,quot/2,if#/4,if2#/3,inc#/1,le#/2,log#/1,log2#/2,minus#/2 ,quot#/2} / {0/0,false/0,log_undefined/0,s/1,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1 ,c_10/1,c_11/4,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {if#,if2#,inc#,le#,log#,log2#,minus# ,quot#} and constructors {0,false,log_undefined,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:if#(false(),b,x,y) -> c_1(if2#(b,x,y)) -->_1 if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))):2 -->_1 if2#(true(),x,s(y)) -> c_4():10 2:S:if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))) -->_2 quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)):8 -->_1 log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)):6 -->_2 quot#(0(),s(y)) -> c_15():16 3:S:inc#(s(x)) -> c_6(inc#(x)) -->_1 inc#(0()) -> c_5():11 -->_1 inc#(s(x)) -> c_6(inc#(x)):3 4:S:le#(s(x),s(y)) -> c_9(le#(x,y)) -->_1 le#(s(x),0()) -> c_8():13 -->_1 le#(0(),y) -> c_7():12 -->_1 le#(s(x),s(y)) -> c_9(le#(x,y)):4 5:S:log#(x) -> c_10(log2#(x,0())) -->_1 log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)):6 6:S:log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)) -->_2 le#(s(x),0()) -> c_8():13 -->_3 le#(0(),y) -> c_7():12 -->_2 le#(0(),y) -> c_7():12 -->_4 inc#(0()) -> c_5():11 -->_1 if#(true(),b,x,y) -> c_2():9 -->_3 le#(s(x),s(y)) -> c_9(le#(x,y)):4 -->_4 inc#(s(x)) -> c_6(inc#(x)):3 -->_1 if#(false(),b,x,y) -> c_1(if2#(b,x,y)):1 7:S:minus#(s(x),s(y)) -> c_14(minus#(x,y)) -->_1 minus#(0(),y) -> c_13():15 -->_1 minus#(x,0()) -> c_12():14 -->_1 minus#(s(x),s(y)) -> c_14(minus#(x,y)):7 8:S:quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)) -->_1 quot#(0(),s(y)) -> c_15():16 -->_2 minus#(0(),y) -> c_13():15 -->_2 minus#(x,0()) -> c_12():14 -->_1 quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)):8 -->_2 minus#(s(x),s(y)) -> c_14(minus#(x,y)):7 9:W:if#(true(),b,x,y) -> c_2() 10:W:if2#(true(),x,s(y)) -> c_4() 11:W:inc#(0()) -> c_5() 12:W:le#(0(),y) -> c_7() 13:W:le#(s(x),0()) -> c_8() 14:W:minus#(x,0()) -> c_12() 15:W:minus#(0(),y) -> c_13() 16:W:quot#(0(),s(y)) -> c_15() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: if2#(true(),x,s(y)) -> c_4() 9: if#(true(),b,x,y) -> c_2() 11: inc#(0()) -> c_5() 12: le#(0(),y) -> c_7() 13: le#(s(x),0()) -> c_8() 14: minus#(x,0()) -> c_12() 15: minus#(0(),y) -> c_13() 16: quot#(0(),s(y)) -> c_15() * Step 5: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: if#(false(),b,x,y) -> c_1(if2#(b,x,y)) if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))) inc#(s(x)) -> c_6(inc#(x)) le#(s(x),s(y)) -> c_9(le#(x,y)) log#(x) -> c_10(log2#(x,0())) log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)) minus#(s(x),s(y)) -> c_14(minus#(x,y)) quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: inc(0()) -> 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(0(),y) -> 0() 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,if2/3,inc/1,le/2,log/1,log2/2,minus/2,quot/2,if#/4,if2#/3,inc#/1,le#/2,log#/1,log2#/2,minus#/2 ,quot#/2} / {0/0,false/0,log_undefined/0,s/1,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1 ,c_10/1,c_11/4,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {if#,if2#,inc#,le#,log#,log2#,minus# ,quot#} and constructors {0,false,log_undefined,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:if#(false(),b,x,y) -> c_1(if2#(b,x,y)) -->_1 if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))):2 2:S:if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))) -->_2 quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)):8 -->_1 log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)):6 3:S:inc#(s(x)) -> c_6(inc#(x)) -->_1 inc#(s(x)) -> c_6(inc#(x)):3 4:S:le#(s(x),s(y)) -> c_9(le#(x,y)) -->_1 le#(s(x),s(y)) -> c_9(le#(x,y)):4 5:S:log#(x) -> c_10(log2#(x,0())) -->_1 log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)):6 6:S:log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,0()),le#(x,s(0())),inc#(y)) -->_3 le#(s(x),s(y)) -> c_9(le#(x,y)):4 -->_4 inc#(s(x)) -> c_6(inc#(x)):3 -->_1 if#(false(),b,x,y) -> c_1(if2#(b,x,y)):1 7:S:minus#(s(x),s(y)) -> c_14(minus#(x,y)) -->_1 minus#(s(x),s(y)) -> c_14(minus#(x,y)):7 8:S:quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)) -->_1 quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)):8 -->_2 minus#(s(x),s(y)) -> c_14(minus#(x,y)):7 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,s(0())),inc#(y)) * Step 6: RemoveHeads MAYBE + Considered Problem: - Strict DPs: if#(false(),b,x,y) -> c_1(if2#(b,x,y)) if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))) inc#(s(x)) -> c_6(inc#(x)) le#(s(x),s(y)) -> c_9(le#(x,y)) log#(x) -> c_10(log2#(x,0())) log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,s(0())),inc#(y)) minus#(s(x),s(y)) -> c_14(minus#(x,y)) quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: inc(0()) -> 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(0(),y) -> 0() 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,if2/3,inc/1,le/2,log/1,log2/2,minus/2,quot/2,if#/4,if2#/3,inc#/1,le#/2,log#/1,log2#/2,minus#/2 ,quot#/2} / {0/0,false/0,log_undefined/0,s/1,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1 ,c_10/1,c_11/3,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {if#,if2#,inc#,le#,log#,log2#,minus# ,quot#} and constructors {0,false,log_undefined,s,true} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:if#(false(),b,x,y) -> c_1(if2#(b,x,y)) -->_1 if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))):2 2:S:if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))) -->_2 quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)):8 -->_1 log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,s(0())),inc#(y)):6 3:S:inc#(s(x)) -> c_6(inc#(x)) -->_1 inc#(s(x)) -> c_6(inc#(x)):3 4:S:le#(s(x),s(y)) -> c_9(le#(x,y)) -->_1 le#(s(x),s(y)) -> c_9(le#(x,y)):4 5:S:log#(x) -> c_10(log2#(x,0())) -->_1 log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,s(0())),inc#(y)):6 6:S:log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,s(0())),inc#(y)) -->_2 le#(s(x),s(y)) -> c_9(le#(x,y)):4 -->_3 inc#(s(x)) -> c_6(inc#(x)):3 -->_1 if#(false(),b,x,y) -> c_1(if2#(b,x,y)):1 7:S:minus#(s(x),s(y)) -> c_14(minus#(x,y)) -->_1 minus#(s(x),s(y)) -> c_14(minus#(x,y)):7 8:S:quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)) -->_1 quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)):8 -->_2 minus#(s(x),s(y)) -> c_14(minus#(x,y)):7 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). [(5,log#(x) -> c_10(log2#(x,0())))] * Step 7: Failure MAYBE + Considered Problem: - Strict DPs: if#(false(),b,x,y) -> c_1(if2#(b,x,y)) if2#(false(),x,y) -> c_3(log2#(quot(x,s(s(0()))),y),quot#(x,s(s(0())))) inc#(s(x)) -> c_6(inc#(x)) le#(s(x),s(y)) -> c_9(le#(x,y)) log2#(x,y) -> c_11(if#(le(x,0()),le(x,s(0())),x,inc(y)),le#(x,s(0())),inc#(y)) minus#(s(x),s(y)) -> c_14(minus#(x,y)) quot#(s(x),s(y)) -> c_16(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: inc(0()) -> 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(0(),y) -> 0() 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,if2/3,inc/1,le/2,log/1,log2/2,minus/2,quot/2,if#/4,if2#/3,inc#/1,le#/2,log#/1,log2#/2,minus#/2 ,quot#/2} / {0/0,false/0,log_undefined/0,s/1,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1 ,c_10/1,c_11/3,c_12/0,c_13/0,c_14/1,c_15/0,c_16/2} - Obligation: innermost runtime complexity wrt. defined symbols {if#,if2#,inc#,le#,log#,log2#,minus# ,quot#} and constructors {0,false,log_undefined,s,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE