/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + 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: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + 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: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + 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: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: le(x,y){x -> s(x),y -> s(y)} = le(s(x),s(y)) ->^+ le(x,y) = C[le(x,y) = le(x,y){}] ** Step 1.b: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 1.b:2: 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: 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: 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 1.b:3: 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: 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: 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 1.b:4: 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: 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: 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 1.b:5: 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#(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: 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_1/3,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: 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#(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)) ** Step 1.b:6: DecomposeDG. 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: 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 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)))) and a lower component le#(s(x),s(y)) -> c_12(le#(x,y)) Further, following extension rules are added to the lower component. average#(x,y) -> if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) average#(x,y) -> le#(y,s(0())) average#(x,y) -> le#(y,s(s(0()))) 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) if3#(false(),b3,x,y) -> if4#(b3,x,y) if4#(false(),x,y) -> average#(s(x),p(p(y))) *** Step 1.b:6.a:1: 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#(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)))) - 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: 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#(y,s(0())) ,le#(y,s(s(0())))) -->_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))):2 2:S:if#(false(),b1,b2,b3,x,y) -> c_2(average#(p(x),s(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#(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)))):6 6:S:if4#(false(),x,y) -> c_8(average#(s(x),p(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#(y,s(0())) ,le#(y,s(s(0())))):1 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)) *** Step 1.b:6.a:2: WeightGap. 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)) 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)))) - 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_1/1,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: 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(p) = {1}, uargs(average#) = {1,2}, uargs(if#) = {1,2,3,4}, uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1}, uargs(c_8) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(average) = [4] x1 + [0] p(false) = [0] p(if) = [1] x1 + [4] x2 + [4] x3 + [1] x5 + [2] p(if2) = [2] x1 + [1] x3 + [2] p(if3) = [1] x2 + [1] p(if4) = [4] x2 + [1] p(le) = [1] p(p) = [1] x1 + [0] p(s) = [1] x1 + [0] p(true) = [0] p(average#) = [1] x1 + [1] x2 + [0] p(if#) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] x6 + [6] p(if2#) = [1] x4 + [1] x5 + [1] p(if3#) = [1] x3 + [1] x4 + [4] p(if4#) = [1] x2 + [1] x3 + [4] p(le#) = [4] p(p#) = [1] x1 + [0] p(c_1) = [1] x1 + [1] p(c_2) = [1] x1 + [1] p(c_3) = [1] x1 + [3] p(c_4) = [1] x1 + [0] p(c_5) = [2] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [1] x1 + [1] p(c_9) = [2] p(c_10) = [0] p(c_11) = [0] p(c_12) = [2] x1 + [2] p(c_13) = [1] p(c_14) = [0] Following rules are strictly oriented: if#(false(),b1,b2,b3,x,y) = [1] b1 + [1] b2 + [1] b3 + [1] x + [1] y + [6] > [1] x + [1] y + [1] = c_2(average#(p(x),s(y))) if#(true(),b1,b2,b3,x,y) = [1] b1 + [1] b2 + [1] b3 + [1] x + [1] y + [6] > [1] x + [1] y + [4] = c_3(if2#(b1,b2,b3,x,y)) if4#(false(),x,y) = [1] x + [1] y + [4] > [1] x + [1] y + [1] = c_8(average#(s(x),p(p(y)))) Following rules are (at-least) weakly oriented: average#(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [11] = c_1(if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y)) if2#(false(),b2,b3,x,y) = [1] x + [1] y + [1] >= [1] x + [1] y + [4] = c_4(if3#(b2,b3,x,y)) if3#(false(),b3,x,y) = [1] x + [1] y + [4] >= [1] x + [1] y + [4] = c_6(if4#(b3,x,y)) le(0(),y) = [1] >= [0] = true() le(s(x),0()) = [1] >= [0] = false() le(s(x),s(y)) = [1] >= [1] = le(x,y) p(0()) = [1] >= [1] = 0() p(s(x)) = [1] x + [0] >= [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: WeightGap. 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)) if2#(false(),b2,b3,x,y) -> c_4(if3#(b2,b3,x,y)) if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) - Weak DPs: 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)) if4#(false(),x,y) -> c_8(average#(s(x),p(p(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_1/1,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: 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(p) = {1}, uargs(average#) = {1,2}, uargs(if#) = {1,2,3,4}, uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1}, uargs(c_8) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(average) = [0] p(false) = [2] p(if) = [0] p(if2) = [2] x3 + [0] p(if3) = [0] p(if4) = [0] p(le) = [2] p(p) = [1] x1 + [0] p(s) = [1] x1 + [4] p(true) = [2] p(average#) = [1] x1 + [1] x2 + [1] p(if#) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] x6 + [3] p(if2#) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [2] p(if3#) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [2] p(if4#) = [1] x1 + [1] x2 + [1] x3 + [6] p(le#) = [1] p(p#) = [1] x1 + [0] p(c_1) = [1] x1 + [1] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [3] p(c_4) = [1] x1 + [1] p(c_5) = [1] p(c_6) = [1] x1 + [3] p(c_7) = [1] p(c_8) = [1] x1 + [2] p(c_9) = [1] p(c_10) = [1] p(c_11) = [1] p(c_12) = [1] x1 + [4] p(c_13) = [1] p(c_14) = [0] Following rules are strictly oriented: if2#(false(),b2,b3,x,y) = [1] b2 + [1] b3 + [1] x + [1] y + [4] > [1] b2 + [1] b3 + [1] x + [1] y + [3] = c_4(if3#(b2,b3,x,y)) Following rules are (at-least) weakly oriented: average#(x,y) = [1] x + [1] y + [1] >= [1] x + [1] y + [12] = c_1(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) = [1] b1 + [1] b2 + [1] b3 + [1] x + [1] y + [5] >= [1] x + [1] y + [5] = c_2(average#(p(x),s(y))) if#(true(),b1,b2,b3,x,y) = [1] b1 + [1] b2 + [1] b3 + [1] x + [1] y + [5] >= [1] b1 + [1] b2 + [1] b3 + [1] x + [1] y + [5] = c_3(if2#(b1,b2,b3,x,y)) if3#(false(),b3,x,y) = [1] b3 + [1] x + [1] y + [4] >= [1] b3 + [1] x + [1] y + [9] = c_6(if4#(b3,x,y)) if4#(false(),x,y) = [1] x + [1] y + [8] >= [1] x + [1] y + [7] = c_8(average#(s(x),p(p(y)))) le(0(),y) = [2] >= [2] = true() le(s(x),0()) = [2] >= [2] = false() le(s(x),s(y)) = [2] >= [2] = le(x,y) p(0()) = [0] >= [0] = 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:4: WeightGap. 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)) if3#(false(),b3,x,y) -> c_6(if4#(b3,x,y)) - Weak DPs: 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)) if4#(false(),x,y) -> c_8(average#(s(x),p(p(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_1/1,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: 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(p) = {1}, uargs(average#) = {1,2}, uargs(if#) = {1,2,3,4}, uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1}, uargs(c_8) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [5] p(average) = [1] p(false) = [0] p(if) = [1] x2 + [1] x4 + [2] x6 + [2] p(if2) = [1] x1 + [2] x2 + [1] p(if3) = [1] x2 + [2] x3 + [4] p(if4) = [4] x2 + [1] x3 + [2] p(le) = [2] p(p) = [1] x1 + [0] p(s) = [1] x1 + [0] p(true) = [1] p(average#) = [1] x1 + [1] x2 + [0] p(if#) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [1] x6 + [2] p(if2#) = [1] x2 + [1] x4 + [1] x5 + [2] p(if3#) = [1] x3 + [1] x4 + [1] p(if4#) = [1] x2 + [1] x3 + [0] p(le#) = [1] x2 + [0] p(p#) = [1] p(c_1) = [1] x1 + [1] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] p(c_6) = [1] x1 + [0] p(c_7) = [1] p(c_8) = [1] x1 + [0] p(c_9) = [1] p(c_10) = [0] p(c_11) = [1] p(c_12) = [1] p(c_13) = [0] p(c_14) = [0] Following rules are strictly oriented: if3#(false(),b3,x,y) = [1] x + [1] y + [1] > [1] x + [1] y + [0] = c_6(if4#(b3,x,y)) Following rules are (at-least) weakly oriented: average#(x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [11] = c_1(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) = [1] b1 + [1] b2 + [1] b3 + [1] x + [1] y + [2] >= [1] x + [1] y + [0] = c_2(average#(p(x),s(y))) if#(true(),b1,b2,b3,x,y) = [1] b1 + [1] b2 + [1] b3 + [1] x + [1] y + [3] >= [1] b2 + [1] x + [1] y + [2] = c_3(if2#(b1,b2,b3,x,y)) if2#(false(),b2,b3,x,y) = [1] b2 + [1] x + [1] y + [2] >= [1] x + [1] y + [1] = c_4(if3#(b2,b3,x,y)) if4#(false(),x,y) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = c_8(average#(s(x),p(p(y)))) le(0(),y) = [2] >= [1] = true() le(s(x),0()) = [2] >= [0] = false() le(s(x),s(y)) = [2] >= [2] = le(x,y) p(0()) = [5] >= [5] = 0() p(s(x)) = [1] x + [0] >= [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:5: 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)) - Weak DPs: 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)))) - 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_1/1,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. *** Step 1.b:6.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: le#(s(x),s(y)) -> c_12(le#(x,y)) - Weak DPs: average#(x,y) -> if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) average#(x,y) -> le#(y,s(0())) average#(x,y) -> le#(y,s(s(0()))) 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) if3#(false(),b3,x,y) -> if4#(b3,x,y) if4#(false(),x,y) -> average#(s(x),p(p(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: 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_12) = {1} Following symbols are considered usable: {average#,if#,if2#,if3#,if4#,le#,p#} TcT has computed the following interpretation: p(0) = [2] p(average) = [1] x1 + [1] x2 + [1] p(false) = [0] p(if) = [1] x4 + [4] x5 + [1] x6 + [1] p(if2) = [1] x2 + [2] x4 + [1] p(if3) = [1] x1 + [1] x4 + [2] p(if4) = [0] p(le) = [2] x1 + [5] x2 + [6] p(p) = [4] x1 + [0] p(s) = [1] x1 + [1] p(true) = [1] p(average#) = [10] p(if#) = [10] p(if2#) = [10] p(if3#) = [10] p(if4#) = [10] p(le#) = [1] x2 + [0] p(p#) = [1] x1 + [0] p(c_1) = [4] p(c_2) = [1] p(c_3) = [1] x1 + [1] p(c_4) = [1] x1 + [4] p(c_5) = [1] p(c_6) = [1] x1 + [1] p(c_7) = [0] p(c_8) = [2] p(c_9) = [1] p(c_10) = [1] p(c_11) = [0] p(c_12) = [1] x1 + [0] p(c_13) = [1] p(c_14) = [0] Following rules are strictly oriented: le#(s(x),s(y)) = [1] y + [1] > [1] y + [0] = c_12(le#(x,y)) Following rules are (at-least) weakly oriented: average#(x,y) = [10] >= [10] = if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) average#(x,y) = [10] >= [3] = le#(y,s(0())) average#(x,y) = [10] >= [4] = le#(y,s(s(0()))) if#(false(),b1,b2,b3,x,y) = [10] >= [10] = average#(p(x),s(y)) if#(true(),b1,b2,b3,x,y) = [10] >= [10] = if2#(b1,b2,b3,x,y) if2#(false(),b2,b3,x,y) = [10] >= [10] = if3#(b2,b3,x,y) if3#(false(),b3,x,y) = [10] >= [10] = if4#(b3,x,y) if4#(false(),x,y) = [10] >= [10] = average#(s(x),p(p(y))) *** Step 1.b:6.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: average#(x,y) -> if#(le(x,0()),le(y,0()),le(y,s(0())),le(y,s(s(0()))),x,y) average#(x,y) -> le#(y,s(0())) average#(x,y) -> le#(y,s(s(0()))) 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) if3#(false(),b3,x,y) -> if4#(b3,x,y) if4#(false(),x,y) -> 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 already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),?)