MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: average(x,y) -> if(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) if(false(),b1,b2,b3,x,y) -> average(p(x),s(y)) if(true(),b1,b2,b3,x,y) -> if2(b1,b2,b3,x,y) if2(false(),b2,b3,x,y) -> if3(b2,b3,x,y) if2(true(),b2,b3,x,y) -> 0() if3(false(),b3,x,y) -> if4(b3,x,y) if3(true(),b3,x,y) -> 0() if4(false(),x,y) -> average(s(x),p(p(y))) if4(true(),x,y) -> s(0()) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {average,if,if2,if3,if4,le,p} and constructors {0,false,s ,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) ,le#(x,0()) ,le#(y,0()) ,le#(y,s(0())) ,le#(y,s(s(0())))) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y)),p#(x)) if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if2#(true(),b2,b3,x,y) -> c_5() if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if3#(true(),b3,x,y) -> c_7() if4#(false(),x,y) -> c_8(average#(s(x),p(p(y))),p#(p(y)),p#(y)) if4#(true(),x,y) -> c_9() le#(0(),y) -> c_10() le#(s(x),0()) -> c_11() le#(s(x),s(y)) -> c_12(le#(x,y)) p#(0()) -> c_13() p#(s(x)) -> c_14() Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) ,le#(x,0()) ,le#(y,0()) ,le#(y,s(0())) ,le#(y,s(s(0())))) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y)),p#(x)) if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if2#(true(),b2,b3,x,y) -> c_5() if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if3#(true(),b3,x,y) -> c_7() if4#(false(),x,y) -> c_8(average#(s(x),p(p(y))),p#(p(y)),p#(y)) if4#(true(),x,y) -> c_9() le#(0(),y) -> c_10() le#(s(x),0()) -> c_11() le#(s(x),s(y)) -> c_12(le#(x,y)) p#(0()) -> c_13() p#(s(x)) -> c_14() - Weak TRS: average(x,y) -> if(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) if(false(),b1,b2,b3,x,y) -> average(p(x),s(y)) if(true(),b1,b2,b3,x,y) -> if2(b1,b2,b3,x,y) if2(false(),b2,b3,x,y) -> if3(b2,b3,x,y) if2(true(),b2,b3,x,y) -> 0() if3(false(),b3,x,y) -> if4(b3,x,y) if3(true(),b3,x,y) -> 0() if4(false(),x,y) -> average(s(x),p(p(y))) if4(true(),x,y) -> s(0()) le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/5,c_2/2,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/3,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) ,le#(x,0()) ,le#(y,0()) ,le#(y,s(0())) ,le#(y,s(s(0())))) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y)),p#(x)) if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if2#(true(),b2,b3,x,y) -> c_5() if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if3#(true(),b3,x,y) -> c_7() if4#(false(),x,y) -> c_8(average#(s(x),p(p(y))),p#(p(y)),p#(y)) if4#(true(),x,y) -> c_9() le#(0(),y) -> c_10() le#(s(x),0()) -> c_11() le#(s(x),s(y)) -> c_12(le#(x,y)) p#(0()) -> c_13() p#(s(x)) -> c_14() * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) ,le#(x,0()) ,le#(y,0()) ,le#(y,s(0())) ,le#(y,s(s(0())))) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y)),p#(x)) if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if2#(true(),b2,b3,x,y) -> c_5() if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if3#(true(),b3,x,y) -> c_7() if4#(false(),x,y) -> c_8(average#(s(x),p(p(y))),p#(p(y)),p#(y)) if4#(true(),x,y) -> c_9() le#(0(),y) -> c_10() le#(s(x),0()) -> c_11() le#(s(x),s(y)) -> c_12(le#(x,y)) p#(0()) -> c_13() p#(s(x)) -> c_14() - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/5,c_2/2,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/3,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {5,7,9,10,11,13,14} by application of Pre({5,7,9,10,11,13,14}) = {1,2,3,4,6,8,12}. Here rules are labelled as follows: 1: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) ,le#(x,0()) ,le#(y,0()) ,le#(y,s(0())) ,le#(y,s(s(0())))) 2: if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y)),p#(x)) 3: if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) 4: if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) 5: if2#(true(),b2,b3,x,y) -> c_5() 6: if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) 7: if3#(true(),b3,x,y) -> c_7() 8: if4#(false(),x,y) -> c_8(average#(s(x),p(p(y))),p#(p(y)),p#(y)) 9: if4#(true(),x,y) -> c_9() 10: le#(0(),y) -> c_10() 11: le#(s(x),0()) -> c_11() 12: le#(s(x),s(y)) -> c_12(le#(x,y)) 13: p#(0()) -> c_13() 14: p#(s(x)) -> c_14() * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) ,le#(x,0()) ,le#(y,0()) ,le#(y,s(0())) ,le#(y,s(s(0())))) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y)),p#(x)) if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if4#(false(),x,y) -> c_8(average#(s(x),p(p(y))),p#(p(y)),p#(y)) le#(s(x),s(y)) -> c_12(le#(x,y)) - Weak DPs: if2#(true(),b2,b3,x,y) -> c_5() if3#(true(),b3,x,y) -> c_7() if4#(true(),x,y) -> c_9() le#(0(),y) -> c_10() le#(s(x),0()) -> c_11() p#(0()) -> c_13() p#(s(x)) -> c_14() - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/5,c_2/2,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/3,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) ,le#(x,0()) ,le#(y,0()) ,le#(y,s(0())) ,le#(y,s(s(0())))) -->_5 le#(s(x),s(y)) -> c_12(le#(x,y)):7 -->_4 le#(s(x),s(y)) -> c_12(le#(x,y)):7 -->_1 if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)):3 -->_1 if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y)),p#(x)):2 -->_3 le#(s(x),0()) -> c_11():12 -->_2 le#(s(x),0()) -> c_11():12 -->_5 le#(0(),y) -> c_10():11 -->_4 le#(0(),y) -> c_10():11 -->_3 le#(0(),y) -> c_10():11 -->_2 le#(0(),y) -> c_10():11 2:S:if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y)),p#(x)) -->_2 p#(s(x)) -> c_14():14 -->_2 p#(0()) -> c_13():13 -->_1 average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) ,le#(x,0()) ,le#(y,0()) ,le#(y,s(0())) ,le#(y,s(s(0())))):1 3:S:if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) -->_1 if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)):4 -->_1 if2#(true(),b2,b3,x,y) -> c_5():8 4:S:if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) -->_1 if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)):5 -->_1 if3#(true(),b3,x,y) -> c_7():9 5:S:if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) -->_1 if4#(false(),x,y) -> c_8(average#(s(x),p(p(y))),p#(p(y)),p#(y)):6 -->_1 if4#(true(),x,y) -> c_9():10 6:S:if4#(false(),x,y) -> c_8(average#(s(x),p(p(y))),p#(p(y)),p#(y)) -->_3 p#(s(x)) -> c_14():14 -->_2 p#(s(x)) -> c_14():14 -->_3 p#(0()) -> c_13():13 -->_2 p#(0()) -> c_13():13 -->_1 average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) ,le#(x,0()) ,le#(y,0()) ,le#(y,s(0())) ,le#(y,s(s(0())))):1 7:S:le#(s(x),s(y)) -> c_12(le#(x,y)) -->_1 le#(s(x),0()) -> c_11():12 -->_1 le#(0(),y) -> c_10():11 -->_1 le#(s(x),s(y)) -> c_12(le#(x,y)):7 8:W:if2#(true(),b2,b3,x,y) -> c_5() 9:W:if3#(true(),b3,x,y) -> c_7() 10:W:if4#(true(),x,y) -> c_9() 11:W:le#(0(),y) -> c_10() 12:W:le#(s(x),0()) -> c_11() 13:W:p#(0()) -> c_13() 14:W:p#(s(x)) -> c_14() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: if2#(true(),b2,b3,x,y) -> c_5() 9: if3#(true(),b3,x,y) -> c_7() 10: if4#(true(),x,y) -> c_9() 13: p#(0()) -> c_13() 14: p#(s(x)) -> c_14() 11: le#(0(),y) -> c_10() 12: le#(s(x),0()) -> c_11() * Step 5: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) ,le#(x,0()) ,le#(y,0()) ,le#(y,s(0())) ,le#(y,s(s(0())))) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y)),p#(x)) if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if4#(false(),x,y) -> c_8(average#(s(x),p(p(y))),p#(p(y)),p#(y)) le#(s(x),s(y)) -> c_12(le#(x,y)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/5,c_2/2,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/3,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) ,le#(x,0()) ,le#(y,0()) ,le#(y,s(0())) ,le#(y,s(s(0())))) -->_5 le#(s(x),s(y)) -> c_12(le#(x,y)):7 -->_4 le#(s(x),s(y)) -> c_12(le#(x,y)):7 -->_1 if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)):3 -->_1 if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y)),p#(x)):2 2:S:if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y)),p#(x)) -->_1 average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) ,le#(x,0()) ,le#(y,0()) ,le#(y,s(0())) ,le#(y,s(s(0())))):1 3:S:if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) -->_1 if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)):4 4:S:if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) -->_1 if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)):5 5:S:if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) -->_1 if4#(false(),x,y) -> c_8(average#(s(x),p(p(y))),p#(p(y)),p#(y)):6 6:S:if4#(false(),x,y) -> c_8(average#(s(x),p(p(y))),p#(p(y)),p#(y)) -->_1 average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) ,le#(x,0()) ,le#(y,0()) ,le#(y,s(0())) ,le#(y,s(s(0())))):1 7:S:le#(s(x),s(y)) -> c_12(le#(x,y)) -->_1 le#(s(x),s(y)) -> c_12(le#(x,y)):7 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) ,le#(y,s(0())) ,le#(y,s(s(0())))) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) * Step 6: Failure MAYBE + Considered Problem: - Strict DPs: average#(x,y) -> c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) ,le#(y,s(0())) ,le#(y,s(s(0())))) if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(y))) if#(true(),b1,b2,b3,x,y) -> c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) if4#(false(),x,y) -> c_8(average#(s(x),p(p(y)))) le#(s(x),s(y)) -> c_12(le#(x,y)) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) p(0()) -> 0() p(s(x)) -> x - Signature: {average/2,if/6,if2/5,if3/4,if4/3,le/2,p/1,average#/2,if#/6,if2#/5,if3#/4,if4#/3,le#/2,p#/1} / {0/0,false/0 ,s/1,true/0,c_1/3,c_2/1,c_3/1,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/0,c_11/0,c_12/1,c_13/0,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {average#,if#,if2#,if3#,if4#,le#,p#} and constructors {0 ,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE