MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> ys -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) avg(xs) -> quot(hd(sum(xs)),length(xs)) hd(:(x,xs)) -> x length(:(x,xs)) -> s(length(xs)) length(nil()) -> 0() quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(-(x,y),s(y))) sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys))))) sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs)) sum(:(x,nil())) -> :(x,nil()) - Signature: {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1} / {0/0,:/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,++,-,avg,hd,length,quot,sum} and constructors {0,:,nil ,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs +#(0(),y) -> c_1() +#(s(x),y) -> c_2(+#(x,y)) ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) ++#(nil(),ys) -> c_4() -#(x,0()) -> c_5() -#(0(),s(y)) -> c_6() -#(s(x),s(y)) -> c_7(-#(x,y)) avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs)) hd#(:(x,xs)) -> c_9() length#(:(x,xs)) -> c_10(length#(xs)) length#(nil()) -> c_11() quot#(0(),s(y)) -> c_12() quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) sum#(:(x,nil())) -> c_16() Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: +#(0(),y) -> c_1() +#(s(x),y) -> c_2(+#(x,y)) ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) ++#(nil(),ys) -> c_4() -#(x,0()) -> c_5() -#(0(),s(y)) -> c_6() -#(s(x),s(y)) -> c_7(-#(x,y)) avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs)) hd#(:(x,xs)) -> c_9() length#(:(x,xs)) -> c_10(length#(xs)) length#(nil()) -> c_11() quot#(0(),s(y)) -> c_12() quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) sum#(:(x,nil())) -> c_16() - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> ys -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) avg(xs) -> quot(hd(sum(xs)),length(xs)) hd(:(x,xs)) -> x length(:(x,xs)) -> s(length(xs)) length(nil()) -> 0() quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(-(x,y),s(y))) sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys))))) sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs)) sum(:(x,nil())) -> :(x,nil()) - Signature: {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0 ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/4,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3 ,c_15/2,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> ys -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) hd(:(x,xs)) -> x length(:(x,xs)) -> s(length(xs)) length(nil()) -> 0() sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys))))) sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs)) sum(:(x,nil())) -> :(x,nil()) +#(0(),y) -> c_1() +#(s(x),y) -> c_2(+#(x,y)) ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) ++#(nil(),ys) -> c_4() -#(x,0()) -> c_5() -#(0(),s(y)) -> c_6() -#(s(x),s(y)) -> c_7(-#(x,y)) avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs)) hd#(:(x,xs)) -> c_9() length#(:(x,xs)) -> c_10(length#(xs)) length#(nil()) -> c_11() quot#(0(),s(y)) -> c_12() quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) sum#(:(x,nil())) -> c_16() * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: +#(0(),y) -> c_1() +#(s(x),y) -> c_2(+#(x,y)) ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) ++#(nil(),ys) -> c_4() -#(x,0()) -> c_5() -#(0(),s(y)) -> c_6() -#(s(x),s(y)) -> c_7(-#(x,y)) avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs)) hd#(:(x,xs)) -> c_9() length#(:(x,xs)) -> c_10(length#(xs)) length#(nil()) -> c_11() quot#(0(),s(y)) -> c_12() quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) sum#(:(x,nil())) -> c_16() - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> ys -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) hd(:(x,xs)) -> x length(:(x,xs)) -> s(length(xs)) length(nil()) -> 0() sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys))))) sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs)) sum(:(x,nil())) -> :(x,nil()) - Signature: {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0 ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/4,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3 ,c_15/2,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,5,6,9,11,12,16} by application of Pre({1,4,5,6,9,11,12,16}) = {2,3,7,8,10,13,14,15}. Here rules are labelled as follows: 1: +#(0(),y) -> c_1() 2: +#(s(x),y) -> c_2(+#(x,y)) 3: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) 4: ++#(nil(),ys) -> c_4() 5: -#(x,0()) -> c_5() 6: -#(0(),s(y)) -> c_6() 7: -#(s(x),s(y)) -> c_7(-#(x,y)) 8: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs)) 9: hd#(:(x,xs)) -> c_9() 10: length#(:(x,xs)) -> c_10(length#(xs)) 11: length#(nil()) -> c_11() 12: quot#(0(),s(y)) -> c_12() 13: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) 14: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))) 15: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) 16: sum#(:(x,nil())) -> c_16() * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: +#(s(x),y) -> c_2(+#(x,y)) ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) -#(s(x),s(y)) -> c_7(-#(x,y)) avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs)) length#(:(x,xs)) -> c_10(length#(xs)) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) - Weak DPs: +#(0(),y) -> c_1() ++#(nil(),ys) -> c_4() -#(x,0()) -> c_5() -#(0(),s(y)) -> c_6() hd#(:(x,xs)) -> c_9() length#(nil()) -> c_11() quot#(0(),s(y)) -> c_12() sum#(:(x,nil())) -> c_16() - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> ys -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) hd(:(x,xs)) -> x length(:(x,xs)) -> s(length(xs)) length(nil()) -> 0() sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys))))) sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs)) sum(:(x,nil())) -> :(x,nil()) - Signature: {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0 ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/4,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3 ,c_15/2,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:+#(s(x),y) -> c_2(+#(x,y)) -->_1 +#(0(),y) -> c_1():9 -->_1 +#(s(x),y) -> c_2(+#(x,y)):1 2:S:++#(:(x,xs),ys) -> c_3(++#(xs,ys)) -->_1 ++#(nil(),ys) -> c_4():10 -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):2 3:S:-#(s(x),s(y)) -> c_7(-#(x,y)) -->_1 -#(0(),s(y)) -> c_6():12 -->_1 -#(x,0()) -> c_5():11 -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):3 4:S:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs)) -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8 -->_3 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):7 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):6 -->_4 length#(:(x,xs)) -> c_10(length#(xs)):5 -->_3 sum#(:(x,nil())) -> c_16():16 -->_1 quot#(0(),s(y)) -> c_12():15 -->_4 length#(nil()) -> c_11():14 -->_2 hd#(:(x,xs)) -> c_9():13 5:S:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(nil()) -> c_11():14 -->_1 length#(:(x,xs)) -> c_10(length#(xs)):5 6:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) -->_1 quot#(0(),s(y)) -> c_12():15 -->_2 -#(0(),s(y)) -> c_6():12 -->_2 -#(x,0()) -> c_5():11 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):6 -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):3 7:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))) -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8 -->_1 sum#(:(x,nil())) -> c_16():16 -->_2 ++#(nil(),ys) -> c_4():10 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):7 -->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):2 8:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) -->_1 sum#(:(x,nil())) -> c_16():16 -->_2 +#(0(),y) -> c_1():9 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8 -->_2 +#(s(x),y) -> c_2(+#(x,y)):1 9:W:+#(0(),y) -> c_1() 10:W:++#(nil(),ys) -> c_4() 11:W:-#(x,0()) -> c_5() 12:W:-#(0(),s(y)) -> c_6() 13:W:hd#(:(x,xs)) -> c_9() 14:W:length#(nil()) -> c_11() 15:W:quot#(0(),s(y)) -> c_12() 16:W:sum#(:(x,nil())) -> c_16() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 13: hd#(:(x,xs)) -> c_9() 14: length#(nil()) -> c_11() 15: quot#(0(),s(y)) -> c_12() 16: sum#(:(x,nil())) -> c_16() 11: -#(x,0()) -> c_5() 12: -#(0(),s(y)) -> c_6() 10: ++#(nil(),ys) -> c_4() 9: +#(0(),y) -> c_1() * Step 5: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: +#(s(x),y) -> c_2(+#(x,y)) ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) -#(s(x),s(y)) -> c_7(-#(x,y)) avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs)) length#(:(x,xs)) -> c_10(length#(xs)) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> ys -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) hd(:(x,xs)) -> x length(:(x,xs)) -> s(length(xs)) length(nil()) -> 0() sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys))))) sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs)) sum(:(x,nil())) -> :(x,nil()) - Signature: {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0 ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/4,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3 ,c_15/2,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:+#(s(x),y) -> c_2(+#(x,y)) -->_1 +#(s(x),y) -> c_2(+#(x,y)):1 2:S:++#(:(x,xs),ys) -> c_3(++#(xs,ys)) -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):2 3:S:-#(s(x),s(y)) -> c_7(-#(x,y)) -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):3 4:S:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),hd#(sum(xs)),sum#(xs),length#(xs)) -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8 -->_3 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):7 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):6 -->_4 length#(:(x,xs)) -> c_10(length#(xs)):5 5:S:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):5 6:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):6 -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):3 7:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))) -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):7 -->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):2 8:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8 -->_2 +#(s(x),y) -> c_2(+#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) * Step 6: NaturalMI MAYBE + Considered Problem: - Strict DPs: +#(s(x),y) -> c_2(+#(x,y)) ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) -#(s(x),s(y)) -> c_7(-#(x,y)) avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) length#(:(x,xs)) -> c_10(length#(xs)) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> ys -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) hd(:(x,xs)) -> x length(:(x,xs)) -> s(length(xs)) length(nil()) -> 0() sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys))))) sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs)) sum(:(x,nil())) -> :(x,nil()) - Signature: {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0 ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3 ,c_15/2,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_7) = {1}, uargs(c_8) = {1,2,3}, uargs(c_10) = {1}, uargs(c_13) = {1,2}, uargs(c_14) = {1,2,3}, uargs(c_15) = {1,2} Following symbols are considered usable: {+#,++#,-#,avg#,hd#,length#,quot#,sum#} TcT has computed the following interpretation: p(+) = [4] x1 + [1] x2 + [6] p(++) = [4] x2 + [0] p(-) = [3] x1 + [3] x2 + [1] p(0) = [1] p(:) = [0] p(avg) = [1] p(hd) = [2] p(length) = [0] p(nil) = [4] p(quot) = [2] x2 + [0] p(s) = [0] p(sum) = [1] x1 + [2] p(+#) = [0] p(++#) = [0] p(-#) = [0] p(avg#) = [4] x1 + [5] p(hd#) = [4] x1 + [0] p(length#) = [0] p(quot#) = [0] p(sum#) = [0] p(c_1) = [0] p(c_2) = [4] x1 + [0] p(c_3) = [4] x1 + [0] p(c_4) = [0] p(c_5) = [1] p(c_6) = [1] p(c_7) = [4] x1 + [0] p(c_8) = [2] x1 + [1] x2 + [4] x3 + [3] p(c_9) = [1] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [2] p(c_13) = [4] x1 + [1] x2 + [0] p(c_14) = [4] x1 + [4] x2 + [2] x3 + [0] p(c_15) = [4] x1 + [4] x2 + [0] p(c_16) = [1] Following rules are strictly oriented: avg#(xs) = [4] xs + [5] > [3] = c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) Following rules are (at-least) weakly oriented: +#(s(x),y) = [0] >= [0] = c_2(+#(x,y)) ++#(:(x,xs),ys) = [0] >= [0] = c_3(++#(xs,ys)) -#(s(x),s(y)) = [0] >= [0] = c_7(-#(x,y)) length#(:(x,xs)) = [0] >= [0] = c_10(length#(xs)) quot#(s(x),s(y)) = [0] >= [0] = c_13(quot#(-(x,y),s(y)),-#(x,y)) sum#(++(xs,:(x,:(y,ys)))) = [0] >= [0] = c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))),sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) = [0] >= [0] = c_15(sum#(:(+(x,y),xs)),+#(x,y)) * Step 7: Failure MAYBE + Considered Problem: - Strict DPs: +#(s(x),y) -> c_2(+#(x,y)) ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) -#(s(x),s(y)) -> c_7(-#(x,y)) length#(:(x,xs)) -> c_10(length#(xs)) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> ys -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) hd(:(x,xs)) -> x length(:(x,xs)) -> s(length(xs)) length(nil()) -> 0() sum(++(xs,:(x,:(y,ys)))) -> sum(++(xs,sum(:(x,:(y,ys))))) sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs)) sum(:(x,nil())) -> :(x,nil()) - Signature: {+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1,+#/2,++#/2,-#/2,avg#/1,hd#/1,length#/1,quot#/2,sum#/1} / {0/0 ,:/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/3,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/3 ,c_15/2,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE