WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + 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 WORST_CASE(?,O(n^3)) + 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 WORST_CASE(?,O(n^3)) + 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 WORST_CASE(?,O(n^3)) + 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 WORST_CASE(?,O(n^3)) + 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: Decompose WORST_CASE(?,O(n^3)) + 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: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: +#(s(x),y) -> c_2(+#(x,y)) - Weak DPs: ++#(:(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} Problem (S) - Strict DPs: ++#(:(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 DPs: +#(s(x),y) -> c_2(+#(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} ** Step 6.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: +#(s(x),y) -> c_2(+#(x,y)) - Weak DPs: ++#(:(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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:+#(s(x),y) -> c_2(+#(x,y)) -->_1 +#(s(x),y) -> c_2(+#(x,y)):1 2:W:++#(:(x,xs),ys) -> c_3(++#(xs,ys)) -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):2 3:W:-#(s(x),s(y)) -> c_7(-#(x,y)) -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):3 4:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8 -->_2 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 -->_3 length#(:(x,xs)) -> c_10(length#(xs)):5 5:W:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):5 6:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):3 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):6 7:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))) -->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):2 -->_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 8:W:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) -->_2 +#(s(x),y) -> c_2(+#(x,y)):1 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: length#(:(x,xs)) -> c_10(length#(xs)) 6: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) 3: -#(s(x),s(y)) -> c_7(-#(x,y)) 2: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) ** Step 6.a:2: SimplifyRHS WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: +#(s(x),y) -> c_2(+#(x,y)) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) 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: SimplifyRHS + Details: Consider the dependency graph 1:S:+#(s(x),y) -> c_2(+#(x,y)) -->_1 +#(s(x),y) -> c_2(+#(x,y)):1 4:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8 -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):7 7:W: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 8:W:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) -->_2 +#(s(x),y) -> c_2(+#(x,y)):1 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):8 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) ** Step 6.a:3: UsableRules WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: +#(s(x),y) -> c_2(+#(x,y)) - Weak DPs: avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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 sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs)) sum(:(x,nil())) -> :(x,nil()) +#(s(x),y) -> c_2(+#(x,y)) avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) ** Step 6.a:4: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: +#(s(x),y) -> c_2(+#(x,y)) - Weak DPs: avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(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 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) and a lower component +#(s(x),y) -> c_2(+#(x,y)) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) Further, following extension rules are added to the lower component. avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))) *** Step 6.a:4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) Consider the set of all dependency pairs 1: avg#(xs) -> c_8(sum#(xs)) 2: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {2} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 6.a:4.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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 = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_8) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {++,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#} TcT has computed the following interpretation: p(+) = [8] x1 + [0] p(++) = [6] x1 + [2] x2 + [2] p(-) = [1] x1 + [2] x2 + [0] p(0) = [0] p(:) = [1] x2 + [4] p(avg) = [1] x1 + [1] p(hd) = [4] x1 + [0] p(length) = [0] p(nil) = [3] p(quot) = [1] x1 + [8] x2 + [1] p(s) = [1] x1 + [2] p(sum) = [7] p(+#) = [1] x1 + [0] p(++#) = [1] p(-#) = [1] x1 + [1] x2 + [1] p(avg#) = [2] x1 + [2] p(hd#) = [2] x1 + [0] p(length#) = [1] p(quot#) = [1] x1 + [1] x2 + [2] p(sum#) = [1] x1 + [1] p(c_1) = [1] p(c_2) = [0] p(c_3) = [2] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] p(c_7) = [1] x1 + [1] p(c_8) = [2] x1 + [0] p(c_9) = [1] p(c_10) = [2] p(c_11) = [1] p(c_12) = [1] p(c_13) = [2] p(c_14) = [1] x1 + [0] p(c_15) = [2] x1 + [1] x2 + [0] p(c_16) = [0] Following rules are strictly oriented: sum#(++(xs,:(x,:(y,ys)))) = [6] xs + [2] ys + [19] > [6] xs + [17] = c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) Following rules are (at-least) weakly oriented: avg#(xs) = [2] xs + [2] >= [2] xs + [2] = c_8(sum#(xs)) ++(:(x,xs),ys) = [6] xs + [2] ys + [26] >= [6] xs + [2] ys + [6] = :(x,++(xs,ys)) ++(nil(),ys) = [2] ys + [20] >= [1] ys + [0] = ys sum(:(x,:(y,xs))) = [7] >= [7] = sum(:(+(x,y),xs)) sum(:(x,nil())) = [7] >= [7] = :(x,nil()) **** Step 6.a:4.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: avg#(xs) -> c_8(sum#(xs)) - Weak DPs: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.a:4.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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:W:avg#(xs) -> c_8(sum#(xs)) -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):2 2:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: avg#(xs) -> c_8(sum#(xs)) 2: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) **** Step 6.a:4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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 already closed. The intended complexity is O(1). *** Step 6.a:4.b:1: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: +#(s(x),y) -> c_2(+#(x,y)) - Weak DPs: avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) sum#(++(xs,:(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 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) and a lower component +#(s(x),y) -> c_2(+#(x,y)) Further, following extension rules are added to the lower component. avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))) sum#(:(x,:(y,xs))) -> +#(x,y) sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs)) **** Step 6.a:4.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) - Weak DPs: avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) Consider the set of all dependency pairs 1: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) 2: avg#(xs) -> sum#(xs) 3: sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) 4: sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 6.a:4.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) - Weak DPs: avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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 = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_15) = {1} Following symbols are considered usable: {+,++,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#} TcT has computed the following interpretation: p(+) = [1] x2 + [1] p(++) = [1] x1 + [1] x2 + [0] p(-) = [1] x1 + [2] x2 + [0] p(0) = [4] p(:) = [1] x1 + [1] x2 + [3] p(avg) = [1] p(hd) = [0] p(length) = [0] p(nil) = [2] p(quot) = [1] x1 + [0] p(s) = [1] x1 + [0] p(sum) = [1] x1 + [0] p(+#) = [0] p(++#) = [1] x2 + [2] p(-#) = [2] x1 + [1] p(avg#) = [5] x1 + [0] p(hd#) = [1] x1 + [4] p(length#) = [0] p(quot#) = [2] x2 + [2] p(sum#) = [4] x1 + [0] p(c_1) = [0] p(c_2) = [2] x1 + [0] p(c_3) = [1] x1 + [1] p(c_4) = [2] p(c_5) = [2] p(c_6) = [1] p(c_7) = [2] x1 + [0] p(c_8) = [1] x1 + [0] p(c_9) = [0] p(c_10) = [1] p(c_11) = [2] p(c_12) = [0] p(c_13) = [1] x1 + [1] x2 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] x1 + [8] x2 + [0] p(c_16) = [1] Following rules are strictly oriented: sum#(:(x,:(y,xs))) = [4] x + [4] xs + [4] y + [24] > [4] xs + [4] y + [16] = c_15(sum#(:(+(x,y),xs)),+#(x,y)) Following rules are (at-least) weakly oriented: avg#(xs) = [5] xs + [0] >= [4] xs + [0] = sum#(xs) sum#(++(xs,:(x,:(y,ys)))) = [4] x + [4] xs + [4] y + [4] ys + [24] >= [4] x + [4] xs + [4] y + [4] ys + [24] = sum#(++(xs,sum(:(x,:(y,ys))))) sum#(++(xs,:(x,:(y,ys)))) = [4] x + [4] xs + [4] y + [4] ys + [24] >= [4] x + [4] y + [4] ys + [24] = sum#(:(x,:(y,ys))) +(0(),y) = [1] y + [1] >= [1] y + [0] = y +(s(x),y) = [1] y + [1] >= [1] y + [1] = s(+(x,y)) ++(:(x,xs),ys) = [1] x + [1] xs + [1] ys + [3] >= [1] x + [1] xs + [1] ys + [3] = :(x,++(xs,ys)) ++(nil(),ys) = [1] ys + [2] >= [1] ys + [0] = ys sum(:(x,:(y,xs))) = [1] x + [1] xs + [1] y + [6] >= [1] xs + [1] y + [4] = sum(:(+(x,y),xs)) sum(:(x,nil())) = [1] x + [5] >= [1] x + [5] = :(x,nil()) ***** Step 6.a:4.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) sum#(++(xs,:(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 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 6.a:4.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) sum#(++(xs,:(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 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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:W:avg#(xs) -> sum#(xs) -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):4 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))):3 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))):2 2:W:sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):4 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))):3 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))):2 3:W:sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))) -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):4 4:W: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)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: avg#(xs) -> sum#(xs) 2: sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) 3: sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))) 4: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) ***** Step 6.a:4.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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 already closed. The intended complexity is O(1). **** Step 6.a:4.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: +#(s(x),y) -> c_2(+#(x,y)) - Weak DPs: avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))) sum#(:(x,:(y,xs))) -> +#(x,y) sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs)) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: +#(s(x),y) -> c_2(+#(x,y)) Consider the set of all dependency pairs 1: +#(s(x),y) -> c_2(+#(x,y)) 2: avg#(xs) -> sum#(xs) 3: sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) 4: sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))) 5: sum#(:(x,:(y,xs))) -> +#(x,y) 6: sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 6.a:4.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: +#(s(x),y) -> c_2(+#(x,y)) - Weak DPs: avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))) sum#(:(x,:(y,xs))) -> +#(x,y) sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs)) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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 = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1} Following symbols are considered usable: {+,++,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#} TcT has computed the following interpretation: p(+) = [1] x1 + [1] x2 + [0] p(++) = [1] x1 + [2] x2 + [0] p(-) = [0] p(0) = [7] p(:) = [1] x1 + [1] x2 + [0] p(avg) = [0] p(hd) = [0] p(length) = [0] p(nil) = [0] p(quot) = [0] p(s) = [1] x1 + [4] p(sum) = [1] x1 + [0] p(+#) = [2] x1 + [1] x2 + [4] p(++#) = [1] x2 + [0] p(-#) = [1] x1 + [0] p(avg#) = [4] x1 + [4] p(hd#) = [0] p(length#) = [0] p(quot#) = [0] p(sum#) = [4] x1 + [4] p(c_1) = [0] p(c_2) = [1] x1 + [6] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] Following rules are strictly oriented: +#(s(x),y) = [2] x + [1] y + [12] > [2] x + [1] y + [10] = c_2(+#(x,y)) Following rules are (at-least) weakly oriented: avg#(xs) = [4] xs + [4] >= [4] xs + [4] = sum#(xs) sum#(++(xs,:(x,:(y,ys)))) = [8] x + [4] xs + [8] y + [8] ys + [4] >= [8] x + [4] xs + [8] y + [8] ys + [4] = sum#(++(xs,sum(:(x,:(y,ys))))) sum#(++(xs,:(x,:(y,ys)))) = [8] x + [4] xs + [8] y + [8] ys + [4] >= [4] x + [4] y + [4] ys + [4] = sum#(:(x,:(y,ys))) sum#(:(x,:(y,xs))) = [4] x + [4] xs + [4] y + [4] >= [2] x + [1] y + [4] = +#(x,y) sum#(:(x,:(y,xs))) = [4] x + [4] xs + [4] y + [4] >= [4] x + [4] xs + [4] y + [4] = sum#(:(+(x,y),xs)) +(0(),y) = [1] y + [7] >= [1] y + [0] = y +(s(x),y) = [1] x + [1] y + [4] >= [1] x + [1] y + [4] = s(+(x,y)) ++(:(x,xs),ys) = [1] x + [1] xs + [2] ys + [0] >= [1] x + [1] xs + [2] ys + [0] = :(x,++(xs,ys)) ++(nil(),ys) = [2] ys + [0] >= [1] ys + [0] = ys sum(:(x,:(y,xs))) = [1] x + [1] xs + [1] y + [0] >= [1] x + [1] xs + [1] y + [0] = sum(:(+(x,y),xs)) sum(:(x,nil())) = [1] x + [0] >= [1] x + [0] = :(x,nil()) ***** Step 6.a:4.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: +#(s(x),y) -> c_2(+#(x,y)) avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))) sum#(:(x,:(y,xs))) -> +#(x,y) sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs)) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 6.a:4.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: +#(s(x),y) -> c_2(+#(x,y)) avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))) sum#(:(x,:(y,xs))) -> +#(x,y) sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs)) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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:W:+#(s(x),y) -> c_2(+#(x,y)) -->_1 +#(s(x),y) -> c_2(+#(x,y)):1 2:W:avg#(xs) -> sum#(xs) -->_1 sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs)):6 -->_1 sum#(:(x,:(y,xs))) -> +#(x,y):5 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))):4 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))):3 3:W:sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) -->_1 sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs)):6 -->_1 sum#(:(x,:(y,xs))) -> +#(x,y):5 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))):4 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))):3 4:W:sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))) -->_1 sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs)):6 -->_1 sum#(:(x,:(y,xs))) -> +#(x,y):5 5:W:sum#(:(x,:(y,xs))) -> +#(x,y) -->_1 +#(s(x),y) -> c_2(+#(x,y)):1 6:W:sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs)) -->_1 sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs)):6 -->_1 sum#(:(x,:(y,xs))) -> +#(x,y):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: avg#(xs) -> sum#(xs) 3: sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) 4: sum#(++(xs,:(x,:(y,ys)))) -> sum#(:(x,:(y,ys))) 6: sum#(:(x,:(y,xs))) -> sum#(:(+(x,y),xs)) 5: sum#(:(x,:(y,xs))) -> +#(x,y) 1: +#(s(x),y) -> c_2(+#(x,y)) ***** Step 6.a:4.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,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 already closed. The intended complexity is O(1). ** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ++#(:(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 DPs: +#(s(x),y) -> c_2(+#(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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:++#(:(x,xs),ys) -> c_3(++#(xs,ys)) -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1 2:S:-#(s(x),s(y)) -> c_7(-#(x,y)) -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):2 3:S:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):7 -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):6 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):5 -->_3 length#(:(x,xs)) -> c_10(length#(xs)):4 4:S:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):4 5: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)):5 -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):2 6: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)):7 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):7 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):6 -->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1 7:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)) -->_2 +#(s(x),y) -> c_2(+#(x,y)):8 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):7 8:W:+#(s(x),y) -> c_2(+#(x,y)) -->_1 +#(s(x),y) -> c_2(+#(x,y)):8 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: +#(s(x),y) -> c_2(+#(x,y)) ** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ++#(:(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: SimplifyRHS + Details: Consider the dependency graph 1:S:++#(:(x,xs),ys) -> c_3(++#(xs,ys)) -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1 2:S:-#(s(x),s(y)) -> c_7(-#(x,y)) -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):2 3:S:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):7 -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):6 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):5 -->_3 length#(:(x,xs)) -> c_10(length#(xs)):4 4:S:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):4 5: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)):5 -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):2 6: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)):7 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y)):7 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):6 -->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1 7: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)):7 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) ** Step 6.b:3: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ++#(:(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))) - 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_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) - Weak DPs: -#(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))) - 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_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} Problem (S) - Strict DPs: -#(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))) - Weak DPs: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) - 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_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} *** Step 6.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) - Weak DPs: -#(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))) - 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_15/1,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:++#(:(x,xs),ys) -> c_3(++#(xs,ys)) -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1 2:W:-#(s(x),s(y)) -> c_7(-#(x,y)) -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):2 3:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):7 -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):6 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):5 -->_3 length#(:(x,xs)) -> c_10(length#(xs)):4 4:W:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):4 5:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):2 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):5 6:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))) -->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1 -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):7 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):7 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):6 7:W:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: length#(:(x,xs)) -> c_10(length#(xs)) 5: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) 7: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) 2: -#(s(x),s(y)) -> c_7(-#(x,y)) *** Step 6.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))) - 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_15/1,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:++#(:(x,xs),ys) -> c_3(++#(xs,ys)) -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1 3:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):6 6:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))) -->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):6 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))) *** Step 6.b:3.a:3: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) - Weak DPs: avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))) - 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,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 sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs)) sum(:(x,nil())) -> :(x,nil()) ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))) *** Step 6.b:3.a:4: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) - Weak DPs: avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))) and a lower component ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) Further, following extension rules are added to the lower component. avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys)))) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) **** Step 6.b:3.a:4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))) - Weak DPs: avg#(xs) -> c_8(sum#(xs)) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))) Consider the set of all dependency pairs 1: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))) 2: avg#(xs) -> c_8(sum#(xs)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 6.b:3.a:4.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))) - Weak DPs: avg#(xs) -> c_8(sum#(xs)) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,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 = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_8) = {1}, uargs(c_14) = {1,2} Following symbols are considered usable: {++,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#} TcT has computed the following interpretation: p(+) = [1] x1 + [6] p(++) = [4] x1 + [2] x2 + [2] p(-) = [4] x2 + [2] p(0) = [0] p(:) = [1] x2 + [4] p(avg) = [1] p(hd) = [8] p(length) = [2] x1 + [2] p(nil) = [3] p(quot) = [1] x2 + [0] p(s) = [1] x1 + [2] p(sum) = [7] p(+#) = [2] x1 + [1] x2 + [4] p(++#) = [1] p(-#) = [1] x2 + [2] p(avg#) = [4] x1 + [6] p(hd#) = [1] x1 + [1] p(length#) = [2] x1 + [8] p(quot#) = [1] x2 + [0] p(sum#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [2] x1 + [1] p(c_3) = [8] x1 + [0] p(c_4) = [1] p(c_5) = [1] p(c_6) = [4] p(c_7) = [2] x1 + [1] p(c_8) = [1] x1 + [6] p(c_9) = [1] p(c_10) = [1] x1 + [0] p(c_11) = [2] p(c_12) = [0] p(c_13) = [1] x1 + [1] x2 + [0] p(c_14) = [1] x1 + [1] x2 + [0] p(c_15) = [1] p(c_16) = [1] Following rules are strictly oriented: sum#(++(xs,:(x,:(y,ys)))) = [4] xs + [2] ys + [18] > [4] xs + [17] = c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))) Following rules are (at-least) weakly oriented: avg#(xs) = [4] xs + [6] >= [1] xs + [6] = c_8(sum#(xs)) ++(:(x,xs),ys) = [4] xs + [2] ys + [18] >= [4] xs + [2] ys + [6] = :(x,++(xs,ys)) ++(nil(),ys) = [2] ys + [14] >= [1] ys + [0] = ys sum(:(x,:(y,xs))) = [7] >= [7] = sum(:(+(x,y),xs)) sum(:(x,nil())) = [7] >= [7] = :(x,nil()) ***** Step 6.b:3.a:4.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 6.b:3.a:4.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,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:W:avg#(xs) -> c_8(sum#(xs)) -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))):2 2:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))) -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: avg#(xs) -> c_8(sum#(xs)) 2: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys))))) ***** Step 6.b:3.a:4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,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 already closed. The intended complexity is O(1). **** Step 6.b:3.a:4.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) - Weak DPs: avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys)))) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) Consider the set of all dependency pairs 1: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) 2: avg#(xs) -> sum#(xs) 3: sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys)))) 4: sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 6.b:3.a:4.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) - Weak DPs: avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys)))) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,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 = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1} Following symbols are considered usable: {+,++,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#} TcT has computed the following interpretation: p(+) = [1] x1 + [1] x2 + [2] p(++) = [1] x1 + [1] x2 + [0] p(-) = [1] x1 + [1] x2 + [0] p(0) = [8] p(:) = [1] x1 + [1] x2 + [2] p(avg) = [0] p(hd) = [1] x1 + [0] p(length) = [1] x1 + [0] p(nil) = [0] p(quot) = [2] x1 + [4] p(s) = [13] p(sum) = [1] x1 + [0] p(+#) = [2] x1 + [0] p(++#) = [4] x1 + [0] p(-#) = [1] x1 + [2] x2 + [4] p(avg#) = [5] x1 + [0] p(hd#) = [1] x1 + [2] p(length#) = [0] p(quot#) = [1] x1 + [1] p(sum#) = [4] x1 + [0] p(c_1) = [2] p(c_2) = [1] p(c_3) = [1] x1 + [4] p(c_4) = [4] p(c_5) = [2] p(c_6) = [1] p(c_7) = [0] p(c_8) = [0] p(c_9) = [2] p(c_10) = [2] x1 + [0] p(c_11) = [0] p(c_12) = [2] p(c_13) = [1] x1 + [2] p(c_14) = [2] x1 + [1] x2 + [0] p(c_15) = [8] p(c_16) = [1] Following rules are strictly oriented: ++#(:(x,xs),ys) = [4] x + [4] xs + [8] > [4] xs + [4] = c_3(++#(xs,ys)) Following rules are (at-least) weakly oriented: avg#(xs) = [5] xs + [0] >= [4] xs + [0] = sum#(xs) sum#(++(xs,:(x,:(y,ys)))) = [4] x + [4] xs + [4] y + [4] ys + [16] >= [4] xs + [0] = ++#(xs,sum(:(x,:(y,ys)))) sum#(++(xs,:(x,:(y,ys)))) = [4] x + [4] xs + [4] y + [4] ys + [16] >= [4] x + [4] xs + [4] y + [4] ys + [16] = sum#(++(xs,sum(:(x,:(y,ys))))) +(0(),y) = [1] y + [10] >= [1] y + [0] = y +(s(x),y) = [1] y + [15] >= [13] = s(+(x,y)) ++(:(x,xs),ys) = [1] x + [1] xs + [1] ys + [2] >= [1] x + [1] xs + [1] ys + [2] = :(x,++(xs,ys)) ++(nil(),ys) = [1] ys + [0] >= [1] ys + [0] = ys sum(:(x,:(y,xs))) = [1] x + [1] xs + [1] y + [4] >= [1] x + [1] xs + [1] y + [4] = sum(:(+(x,y),xs)) sum(:(x,nil())) = [1] x + [2] >= [1] x + [2] = :(x,nil()) ***** Step 6.b:3.a:4.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys)))) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 6.b:3.a:4.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) avg#(xs) -> sum#(xs) sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys)))) sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,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:W:++#(:(x,xs),ys) -> c_3(++#(xs,ys)) -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1 2:W:avg#(xs) -> sum#(xs) -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))):4 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys)))):3 3:W:sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys)))) -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1 4:W:sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) -->_1 sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))):4 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys)))):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: avg#(xs) -> sum#(xs) 4: sum#(++(xs,:(x,:(y,ys)))) -> sum#(++(xs,sum(:(x,:(y,ys))))) 3: sum#(++(xs,:(x,:(y,ys)))) -> ++#(xs,sum(:(x,:(y,ys)))) 1: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) ***** Step 6.b:3.a:4.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,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 already closed. The intended complexity is O(1). *** Step 6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(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))) - Weak DPs: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) - 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_15/1,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),s(y)) -> c_7(-#(x,y)) -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):1 2:S:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6 -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):5 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):4 -->_3 length#(:(x,xs)) -> c_10(length#(xs)):3 3:S:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):3 4: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)):4 -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):1 5:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))) -->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):7 -->_3 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):5 6:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6 7:W:++#(:(x,xs),ys) -> c_3(++#(xs,ys)) -->_1 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: ++#(:(x,xs),ys) -> c_3(++#(xs,ys)) *** Step 6.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(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))) - 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_15/1,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),s(y)) -> c_7(-#(x,y)) -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):1 2:S:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6 -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):5 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):4 -->_3 length#(:(x,xs)) -> c_10(length#(xs)):3 3:S:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):3 4: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)):4 -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):1 5: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))):6 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))) ,++#(xs,sum(:(x,:(y,ys)))) ,sum#(:(x,:(y,ys)))):5 6:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) *** Step 6.b:3.b:3: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(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))))),sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),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_14/2 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: -#(s(x),s(y)) -> c_7(-#(x,y)) - Weak DPs: 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))))),sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),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_14/2,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} Problem (S) - Strict DPs: 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))))),sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) - Weak DPs: -#(s(x),s(y)) -> c_7(-#(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_14/2,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} **** Step 6.b:3.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(s(x),s(y)) -> c_7(-#(x,y)) - Weak DPs: 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))))),sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),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_14/2 ,c_15/1,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),s(y)) -> c_7(-#(x,y)) -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):1 2:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6 -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):5 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):4 -->_3 length#(:(x,xs)) -> c_10(length#(xs)):3 3:W:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):3 4:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):1 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):4 5:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):5 6:W:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):6 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: length#(:(x,xs)) -> c_10(length#(xs)) 5: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) 6: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) **** Step 6.b:3.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(s(x),s(y)) -> c_7(-#(x,y)) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(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_14/2 ,c_15/1,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),s(y)) -> c_7(-#(x,y)) -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):1 2:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):4 4:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):1 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):4 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))) **** Step 6.b:3.b:3.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(s(x),s(y)) -> c_7(-#(x,y)) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs))) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: -#(s(x),s(y)) -> c_7(-#(x,y)) Consider the set of all dependency pairs 1: -#(s(x),s(y)) -> c_7(-#(x,y)) 2: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs))) 3: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ***** Step 6.b:3.b:3.a:3.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(s(x),s(y)) -> c_7(-#(x,y)) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs))) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_7) = {1}, uargs(c_8) = {1}, uargs(c_13) = {1,2} Following symbols are considered usable: {+,++,-,hd,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#} TcT has computed the following interpretation: p(+) = x1 + x2 p(++) = x1 + x2 p(-) = x1 p(0) = 0 p(:) = x1 + x2 p(avg) = 0 p(hd) = x1 p(length) = 0 p(nil) = 0 p(quot) = 0 p(s) = 1 + x1 p(sum) = x1 p(+#) = 0 p(++#) = 0 p(-#) = x1 p(avg#) = x1^2 p(hd#) = 0 p(length#) = 0 p(quot#) = x1^2 p(sum#) = 0 p(c_1) = 0 p(c_2) = 0 p(c_3) = 0 p(c_4) = 0 p(c_5) = 0 p(c_6) = 0 p(c_7) = x1 p(c_8) = x1 p(c_9) = 0 p(c_10) = 0 p(c_11) = 0 p(c_12) = 0 p(c_13) = x1 + x2 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 Following rules are strictly oriented: -#(s(x),s(y)) = 1 + x > x = c_7(-#(x,y)) Following rules are (at-least) weakly oriented: avg#(xs) = xs^2 >= xs^2 = c_8(quot#(hd(sum(xs)),length(xs))) quot#(s(x),s(y)) = 1 + 2*x + x^2 >= x + x^2 = c_13(quot#(-(x,y),s(y)),-#(x,y)) +(0(),y) = y >= y = y +(s(x),y) = 1 + x + y >= 1 + x + y = s(+(x,y)) ++(:(x,xs),ys) = x + xs + ys >= x + xs + ys = :(x,++(xs,ys)) ++(nil(),ys) = ys >= ys = ys -(x,0()) = x >= x = x -(0(),s(y)) = 0 >= 0 = 0() -(s(x),s(y)) = 1 + x >= x = -(x,y) hd(:(x,xs)) = x + xs >= x = x sum(++(xs,:(x,:(y,ys)))) = x + xs + y + ys >= x + xs + y + ys = sum(++(xs,sum(:(x,:(y,ys))))) sum(:(x,:(y,xs))) = x + xs + y >= x + xs + y = sum(:(+(x,y),xs)) sum(:(x,nil())) = x >= x = :(x,nil()) ***** Step 6.b:3.b:3.a:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: -#(s(x),s(y)) -> c_7(-#(x,y)) avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs))) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 6.b:3.b:3.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: -#(s(x),s(y)) -> c_7(-#(x,y)) avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs))) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,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:W:-#(s(x),s(y)) -> c_7(-#(x,y)) -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):1 2:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs))) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):3 3:W: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)):3 -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs))) 3: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) 1: -#(s(x),s(y)) -> c_7(-#(x,y)) ***** Step 6.b:3.b:3.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/2,c_14/2 ,c_15/1,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 already closed. The intended complexity is O(1). **** Step 6.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 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))))),sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) - Weak DPs: -#(s(x),s(y)) -> c_7(-#(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_14/2 ,c_15/1,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:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):3 -->_3 length#(:(x,xs)) -> c_10(length#(xs)):2 2:S:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):2 3:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)) -->_2 -#(s(x),s(y)) -> c_7(-#(x,y)):6 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):3 4:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4 5:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 6:W:-#(s(x),s(y)) -> c_7(-#(x,y)) -->_1 -#(s(x),s(y)) -> c_7(-#(x,y)):6 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: -#(s(x),s(y)) -> c_7(-#(x,y)) **** Step 6.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 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))))),sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),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_14/2 ,c_15/1,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:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)),-#(x,y)):3 -->_3 length#(:(x,xs)) -> c_10(length#(xs)):2 2:S:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):2 3: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)):3 4:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4 5:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) **** Step 6.b:3.b:3.b:3: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 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))) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),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_13/1,c_14/2 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) - Weak DPs: length#(:(x,xs)) -> c_10(length#(xs)) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) - 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_13/1,c_14/2,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} Problem (S) - Strict DPs: length#(:(x,xs)) -> c_10(length#(xs)) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),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_13/1,c_14/2,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} ***** Step 6.b:3.b:3.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) - Weak DPs: length#(:(x,xs)) -> c_10(length#(xs)) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) - 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_13/1,c_14/2 ,c_15/1,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:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):3 -->_3 length#(:(x,xs)) -> c_10(length#(xs)):2 2:W:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):2 3:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):3 4:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4 5:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: length#(:(x,xs)) -> c_10(length#(xs)) 3: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) ***** Step 6.b:3.b:3.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) - Weak DPs: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) - 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_13/1,c_14/2 ,c_15/1,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:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4 4:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4 5:S:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: avg#(xs) -> c_8(sum#(xs)) ***** Step 6.b:3.b:3.b:3.a:3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: avg#(xs) -> c_8(sum#(xs)) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) - Weak DPs: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) - 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2 ,c_15/1,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 sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs)) sum(:(x,nil())) -> :(x,nil()) avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) ***** Step 6.b:3.b:3.b:3.a:4: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: avg#(xs) -> c_8(sum#(xs)) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) - Weak DPs: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: avg#(xs) -> c_8(sum#(xs)) 2: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) The strictly oriented rules are moved into the weak component. ****** Step 6.b:3.b:3.b:3.a:4.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: avg#(xs) -> c_8(sum#(xs)) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) - Weak DPs: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2 ,c_15/1,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 = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_8) = {1}, uargs(c_14) = {1,2}, uargs(c_15) = {1} Following symbols are considered usable: {++,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#} TcT has computed the following interpretation: p(+) = [8] p(++) = [1] x1 + [2] x2 + [0] p(-) = [1] x1 + [8] x2 + [1] p(0) = [4] p(:) = [1] x2 + [1] p(avg) = [1] x1 + [0] p(hd) = [1] x1 + [1] p(length) = [2] x1 + [2] p(nil) = [0] p(quot) = [2] x2 + [1] p(s) = [1] x1 + [14] p(sum) = [1] p(+#) = [1] x1 + [1] x2 + [1] p(++#) = [1] x1 + [1] x2 + [0] p(-#) = [1] p(avg#) = [8] x1 + [9] p(hd#) = [1] x1 + [4] p(length#) = [2] x1 + [1] p(quot#) = [1] x1 + [2] p(sum#) = [2] x1 + [0] p(c_1) = [2] p(c_2) = [0] p(c_3) = [1] x1 + [1] p(c_4) = [1] p(c_5) = [2] p(c_6) = [2] p(c_7) = [1] x1 + [1] p(c_8) = [4] x1 + [2] p(c_9) = [0] p(c_10) = [1] x1 + [8] p(c_11) = [0] p(c_12) = [0] p(c_13) = [2] x1 + [2] p(c_14) = [1] x1 + [1] x2 + [0] p(c_15) = [1] x1 + [0] p(c_16) = [0] Following rules are strictly oriented: avg#(xs) = [8] xs + [9] > [8] xs + [2] = c_8(sum#(xs)) sum#(:(x,:(y,xs))) = [2] xs + [4] > [2] xs + [2] = c_15(sum#(:(+(x,y),xs))) Following rules are (at-least) weakly oriented: sum#(++(xs,:(x,:(y,ys)))) = [2] xs + [4] ys + [8] >= [2] xs + [2] ys + [8] = c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) ++(:(x,xs),ys) = [1] xs + [2] ys + [1] >= [1] xs + [2] ys + [1] = :(x,++(xs,ys)) ++(nil(),ys) = [2] ys + [0] >= [1] ys + [0] = ys sum(:(x,:(y,xs))) = [1] >= [1] = sum(:(+(x,y),xs)) sum(:(x,nil())) = [1] >= [1] = :(x,nil()) ****** Step 6.b:3.b:3.b:3.a:4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 6.b:3.b:3.b:3.a:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2 ,c_15/1,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:W:avg#(xs) -> c_8(sum#(xs)) -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):3 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):2 2:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):3 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):3 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):2 3:W:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: avg#(xs) -> c_8(sum#(xs)) 2: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) 3: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) ****** Step 6.b:3.b:3.b:3.a:4.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2 ,c_15/1,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 already closed. The intended complexity is O(1). ***** Step 6.b:3.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: length#(:(x,xs)) -> c_10(length#(xs)) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),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_13/1,c_14/2 ,c_15/1,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:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):1 2:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):2 3:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):3 4:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):3 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):2 -->_3 length#(:(x,xs)) -> c_10(length#(xs)):1 5:W:sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) -->_1 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))) ***** Step 6.b:3.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: length#(:(x,xs)) -> c_10(length#(xs)) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) - 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_13/1,c_14/2 ,c_15/1,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:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):1 2:S:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):2 3:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))) -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):3 4:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):3 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):2 -->_3 length#(:(x,xs)) -> c_10(length#(xs)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) ***** Step 6.b:3.b:3.b:3.b:3: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: length#(:(x,xs)) -> c_10(length#(xs)) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - 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_13/1,c_14/1 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: length#(:(x,xs)) -> c_10(length#(xs)) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - 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_13/1,c_14/1,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} Problem (S) - Strict DPs: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) length#(:(x,xs)) -> c_10(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_13/1,c_14/1,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} ****** Step 6.b:3.b:3.b:3.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: length#(:(x,xs)) -> c_10(length#(xs)) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - 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_13/1,c_14/1 ,c_15/1,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:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):1 2:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):2 3:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):3 4:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_3 length#(:(x,xs)) -> c_10(length#(xs)):1 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):2 -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) 2: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) ****** Step 6.b:3.b:3.b:3.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: length#(:(x,xs)) -> c_10(length#(xs)) - 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_13/1,c_14/1 ,c_15/1,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:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):1 4:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_3 length#(:(x,xs)) -> c_10(length#(xs)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: avg#(xs) -> c_8(length#(xs)) ****** Step 6.b:3.b:3.b:3.b:3.a:3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: length#(:(x,xs)) -> c_10(length#(xs)) - Weak DPs: avg#(xs) -> c_8(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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,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: avg#(xs) -> c_8(length#(xs)) length#(:(x,xs)) -> c_10(length#(xs)) ****** Step 6.b:3.b:3.b:3.b:3.a:4: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: length#(:(x,xs)) -> c_10(length#(xs)) - Weak DPs: avg#(xs) -> c_8(length#(xs)) - 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: length#(:(x,xs)) -> c_10(length#(xs)) Consider the set of all dependency pairs 1: length#(:(x,xs)) -> c_10(length#(xs)) 2: avg#(xs) -> c_8(length#(xs)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ******* Step 6.b:3.b:3.b:3.b:3.a:4.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: length#(:(x,xs)) -> c_10(length#(xs)) - Weak DPs: avg#(xs) -> c_8(length#(xs)) - 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,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 = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_8) = {1}, uargs(c_10) = {1} Following symbols are considered usable: {+#,++#,-#,avg#,hd#,length#,quot#,sum#} TcT has computed the following interpretation: p(+) = [0] p(++) = [0] p(-) = [0] p(0) = [0] p(:) = [1] x1 + [1] x2 + [4] p(avg) = [0] p(hd) = [0] p(length) = [0] p(nil) = [0] p(quot) = [0] p(s) = [1] x1 + [0] p(sum) = [0] p(+#) = [0] p(++#) = [0] p(-#) = [0] p(avg#) = [8] x1 + [0] p(hd#) = [0] p(length#) = [1] x1 + [0] p(quot#) = [0] p(sum#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [8] x1 + [0] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] Following rules are strictly oriented: length#(:(x,xs)) = [1] x + [1] xs + [4] > [1] xs + [0] = c_10(length#(xs)) Following rules are (at-least) weakly oriented: avg#(xs) = [8] xs + [0] >= [8] xs + [0] = c_8(length#(xs)) ******* Step 6.b:3.b:3.b:3.b:3.a:4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: avg#(xs) -> c_8(length#(xs)) length#(:(x,xs)) -> c_10(length#(xs)) - 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 6.b:3.b:3.b:3.b:3.a:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: avg#(xs) -> c_8(length#(xs)) length#(:(x,xs)) -> c_10(length#(xs)) - 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,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:W:avg#(xs) -> c_8(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):2 2:W:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: avg#(xs) -> c_8(length#(xs)) 2: length#(:(x,xs)) -> c_10(length#(xs)) ******* Step 6.b:3.b:3.b:3.b:3.a:4.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,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 already closed. The intended complexity is O(1). ****** Step 6.b:3.b:3.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) length#(:(x,xs)) -> c_10(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_13/1,c_14/1 ,c_15/1,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:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):1 2:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):2 3:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_3 length#(:(x,xs)) -> c_10(length#(xs)):4 -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):2 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):1 4:W:length#(:(x,xs)) -> c_10(length#(xs)) -->_1 length#(:(x,xs)) -> c_10(length#(xs)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: length#(:(x,xs)) -> c_10(length#(xs)) ****** Step 6.b:3.b:3.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - 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_13/1,c_14/1 ,c_15/1,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:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):1 2:S:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):2 3:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs)) -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):2 -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(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)) ****** Step 6.b:3.b:3.b:3.b:3.b:3: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(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_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - 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_8/2,c_9/0,c_10/1,c_11/0,c_12/0 ,c_13/1,c_14/1,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} Problem (S) - Strict DPs: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs)) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(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_8/2,c_9/0,c_10/1,c_11/0,c_12/0 ,c_13/1,c_14/1,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} ******* Step 6.b:3.b:3.b:3.b:3.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - 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_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,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:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):1 2:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):2 3:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs)) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):1 -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) ******* Step 6.b:3.b:3.b:3.b:3.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(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_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,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:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):1 3:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs)) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(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))) ******* Step 6.b:3.b:3.b:3.b:3.b:3.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) - Weak DPs: avg#(xs) -> c_8(quot#(hd(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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) Consider the set of all dependency pairs 1: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) 2: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs))) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ******** Step 6.b:3.b:3.b:3.b:3.b:3.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) - Weak DPs: avg#(xs) -> c_8(quot#(hd(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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,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 = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_8) = {1}, uargs(c_13) = {1} Following symbols are considered usable: {+,++,-,hd,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#} TcT has computed the following interpretation: p(+) = [1] x1 + [1] x2 + [0] p(++) = [2] x1 + [1] x2 + [8] p(-) = [1] x1 + [0] p(0) = [0] p(:) = [1] x1 + [1] x2 + [0] p(avg) = [2] x1 + [2] p(hd) = [1] x1 + [0] p(length) = [8] p(nil) = [0] p(quot) = [4] p(s) = [1] x1 + [2] p(sum) = [1] x1 + [0] p(+#) = [2] x2 + [0] p(++#) = [2] x1 + [1] x2 + [0] p(-#) = [1] p(avg#) = [12] x1 + [3] p(hd#) = [1] p(length#) = [2] p(quot#) = [12] x1 + [2] p(sum#) = [1] p(c_1) = [0] p(c_2) = [1] p(c_3) = [2] p(c_4) = [1] p(c_5) = [1] p(c_6) = [1] p(c_7) = [0] p(c_8) = [1] x1 + [1] p(c_9) = [1] p(c_10) = [1] x1 + [1] p(c_11) = [1] p(c_12) = [1] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [4] p(c_15) = [0] p(c_16) = [1] Following rules are strictly oriented: quot#(s(x),s(y)) = [12] x + [26] > [12] x + [2] = c_13(quot#(-(x,y),s(y))) Following rules are (at-least) weakly oriented: avg#(xs) = [12] xs + [3] >= [12] xs + [3] = c_8(quot#(hd(sum(xs)),length(xs))) +(0(),y) = [1] y + [0] >= [1] y + [0] = y +(s(x),y) = [1] x + [1] y + [2] >= [1] x + [1] y + [2] = s(+(x,y)) ++(:(x,xs),ys) = [2] x + [2] xs + [1] ys + [8] >= [1] x + [2] xs + [1] ys + [8] = :(x,++(xs,ys)) ++(nil(),ys) = [1] ys + [8] >= [1] ys + [0] = ys -(x,0()) = [1] x + [0] >= [1] x + [0] = x -(0(),s(y)) = [0] >= [0] = 0() -(s(x),s(y)) = [1] x + [2] >= [1] x + [0] = -(x,y) hd(:(x,xs)) = [1] x + [1] xs + [0] >= [1] x + [0] = x sum(++(xs,:(x,:(y,ys)))) = [1] x + [2] xs + [1] y + [1] ys + [8] >= [1] x + [2] xs + [1] y + [1] ys + [8] = sum(++(xs,sum(:(x,:(y,ys))))) sum(:(x,:(y,xs))) = [1] x + [1] xs + [1] y + [0] >= [1] x + [1] xs + [1] y + [0] = sum(:(+(x,y),xs)) sum(:(x,nil())) = [1] x + [0] >= [1] x + [0] = :(x,nil()) ******** Step 6.b:3.b:3.b:3.b:3.b:3.a:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs))) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******** Step 6.b:3.b:3.b:3.b:3.b:3.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs))) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,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:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs))) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):2 2:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs))) 2: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) ******** Step 6.b:3.b:3.b:3.b:3.b:3.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,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 already closed. The intended complexity is O(1). ******* Step 6.b:3.b:3.b:3.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs)) quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(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_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,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:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):1 2:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs)) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):3 -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):1 3:W:quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) -->_1 quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y))) ******* Step 6.b:3.b:3.b:3.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - Weak DPs: avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(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_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,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:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):1 2:W:avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs)) -->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: avg#(xs) -> c_8(sum#(xs)) ******* Step 6.b:3.b:3.b:3.b:3.b:3.b:3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - Weak DPs: avg#(xs) -> c_8(sum#(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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,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 sum(:(x,:(y,xs))) -> sum(:(+(x,y),xs)) sum(:(x,nil())) -> :(x,nil()) avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) ******* Step 6.b:3.b:3.b:3.b:3.b:3.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - Weak DPs: avg#(xs) -> c_8(sum#(xs)) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) Consider the set of all dependency pairs 1: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) 2: avg#(xs) -> c_8(sum#(xs)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ******** Step 6.b:3.b:3.b:3.b:3.b:3.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - Weak DPs: avg#(xs) -> c_8(sum#(xs)) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,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 = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_8) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {++,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#} TcT has computed the following interpretation: p(+) = [7] x1 + [1] x2 + [1] p(++) = [4] x1 + [4] x2 + [0] p(-) = [4] x1 + [2] x2 + [1] p(0) = [2] p(:) = [1] x2 + [1] p(avg) = [2] x1 + [0] p(hd) = [0] p(length) = [2] x1 + [2] p(nil) = [0] p(quot) = [1] x1 + [2] x2 + [1] p(s) = [1] x1 + [2] p(sum) = [1] p(+#) = [1] x1 + [2] x2 + [1] p(++#) = [1] x1 + [1] x2 + [2] p(-#) = [1] x1 + [1] x2 + [0] p(avg#) = [8] x1 + [12] p(hd#) = [0] p(length#) = [4] x1 + [1] p(quot#) = [8] x1 + [1] x2 + [1] p(sum#) = [2] x1 + [8] p(c_1) = [2] p(c_2) = [8] x1 + [0] p(c_3) = [1] x1 + [1] p(c_4) = [2] p(c_5) = [0] p(c_6) = [1] p(c_7) = [8] p(c_8) = [1] x1 + [4] p(c_9) = [4] p(c_10) = [2] p(c_11) = [1] p(c_12) = [0] p(c_13) = [1] x1 + [1] p(c_14) = [1] x1 + [0] p(c_15) = [2] x1 + [1] p(c_16) = [2] Following rules are strictly oriented: sum#(++(xs,:(x,:(y,ys)))) = [8] xs + [8] ys + [24] > [8] xs + [16] = c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) Following rules are (at-least) weakly oriented: avg#(xs) = [8] xs + [12] >= [2] xs + [12] = c_8(sum#(xs)) ++(:(x,xs),ys) = [4] xs + [4] ys + [4] >= [4] xs + [4] ys + [1] = :(x,++(xs,ys)) ++(nil(),ys) = [4] ys + [0] >= [1] ys + [0] = ys sum(:(x,:(y,xs))) = [1] >= [1] = sum(:(+(x,y),xs)) sum(:(x,nil())) = [1] >= [1] = :(x,nil()) ******** Step 6.b:3.b:3.b:3.b:3.b:3.b:4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,c_16/0} - Obligation: innermost runtime complexity wrt. defined symbols {+#,++#,-#,avg#,hd#,length#,quot# ,sum#} and constructors {0,:,nil,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******** Step 6.b:3.b:3.b:3.b:3.b:3.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: avg#(xs) -> c_8(sum#(xs)) sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,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:W:avg#(xs) -> c_8(sum#(xs)) -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):2 2:W:sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) -->_1 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: avg#(xs) -> c_8(sum#(xs)) 2: sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys)))))) ******** Step 6.b:3.b:3.b:3.b:3.b:3.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(0(),y) -> y +(s(x),y) -> s(+(x,y)) ++(:(x,xs),ys) -> :(x,++(xs,ys)) ++(nil(),ys) -> 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_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1 ,c_15/1,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 already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))