*** 1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS 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)
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())
Weak DP Rules:
Weak TRS Rules:
Signature:
{+/2,++/2,-/2,avg/1,hd/1,length/1,quot/2,sum/1} / {0/0,:/2,nil/0,s/1}
Obligation:
Innermost
basic terms: {+,++,-,avg,hd,length,quot,sum}/{0,:,nil,s}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
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.
*** 1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(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()
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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)
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
UsableRules
Proof:
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()
*** 1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(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()
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
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()
*** 1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(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))
Strict TRS Rules:
Weak DP Rules:
+#(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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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()
*** 1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
SimplifyRHS
Proof:
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))
*** 1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
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 DP Rules:
+#(s(x),y) -> c_2(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
++#(:(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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Problem (S)
Strict DP Rules:
++#(:(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))
Strict TRS Rules:
Weak DP Rules:
+#(s(x),y) -> c_2(+#(x,y))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
*** 1.1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(s(x),y) -> c_2(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
++#(:(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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(s(x),y) -> c_2(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
SimplifyRHS
Proof:
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))))
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(s(x),y) -> c_2(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
UsableRules
Proof:
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))
*** 1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
+#(s(x),y) -> c_2(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
Proof:
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)))
*** 1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
avg#(xs) -> c_8(sum#(xs))
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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, greedy = NoGreedy}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.
*** 1.1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
avg#(xs) -> c_8(sum#(xs))
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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(+) = [12] x1 + [3]
p(++) = [1] x1 + [1] x2 + [0]
p(-) = [1]
p(0) = [0]
p(:) = [1] x2 + [8]
p(avg) = [2] x1 + [0]
p(hd) = [1] x1 + [1]
p(length) = [2]
p(nil) = [0]
p(quot) = [1] x2 + [0]
p(s) = [1] x1 + [0]
p(sum) = [8]
p(+#) = [1]
p(++#) = [2] x1 + [2] x2 + [2]
p(-#) = [2] x2 + [1]
p(avg#) = [1] x1 + [8]
p(hd#) = [0]
p(length#) = [0]
p(quot#) = [2]
p(sum#) = [1] x1 + [8]
p(c_1) = [1]
p(c_2) = [1] x1 + [1]
p(c_3) = [1]
p(c_4) = [2]
p(c_5) = [1]
p(c_6) = [1]
p(c_7) = [1] x1 + [1]
p(c_8) = [1] x1 + [0]
p(c_9) = [0]
p(c_10) = [1] x1 + [1]
p(c_11) = [2]
p(c_12) = [2]
p(c_13) = [1] x2 + [2]
p(c_14) = [1] x1 + [6]
p(c_15) = [8] x1 + [1]
p(c_16) = [1]
Following rules are strictly oriented:
sum#(++(xs,:(x,:(y,ys)))) = [1] xs + [1] ys + [24]
> [1] xs + [22]
= c_14(sum#(++(xs
,sum(:(x,:(y,ys)))))
,sum#(:(x,:(y,ys))))
Following rules are (at-least) weakly oriented:
avg#(xs) = [1] xs + [8]
>= [1] xs + [8]
= c_8(sum#(xs))
++(:(x,xs),ys) = [1] xs + [1] ys + [8]
>= [1] xs + [1] ys + [8]
= :(x,++(xs,ys))
++(nil(),ys) = [1] ys + [0]
>= [1] ys + [0]
= ys
sum(:(x,:(y,xs))) = [8]
>= [8]
= sum(:(+(x,y),xs))
sum(:(x,nil())) = [8]
>= [8]
= :(x,nil())
*** 1.1.1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
avg#(xs) -> c_8(sum#(xs))
Strict TRS Rules:
Weak DP Rules:
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))))
*** 1.1.1.1.1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.1.1.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
+#(s(x),y) -> c_2(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
Proof:
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))
*** 1.1.1.1.1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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, greedy = NoGreedy}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.
*** 1.1.1.1.1.1.1.1.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)),+#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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(+) = [13]
p(++) = [2] x1 + [1] x2 + [5]
p(-) = [2] x1 + [0]
p(0) = [0]
p(:) = [1] x2 + [8]
p(avg) = [0]
p(hd) = [0]
p(length) = [0]
p(nil) = [0]
p(quot) = [0]
p(s) = [1] x1 + [0]
p(sum) = [11]
p(+#) = [1]
p(++#) = [1] x2 + [0]
p(-#) = [1] x1 + [0]
p(avg#) = [1] x1 + [9]
p(hd#) = [2]
p(length#) = [4] x1 + [1]
p(quot#) = [1] x2 + [1]
p(sum#) = [1] x1 + [0]
p(c_1) = [1]
p(c_2) = [1] x1 + [1]
p(c_3) = [0]
p(c_4) = [1]
p(c_5) = [1]
p(c_6) = [1]
p(c_7) = [1] x1 + [1]
p(c_8) = [0]
p(c_9) = [1]
p(c_10) = [1] x1 + [0]
p(c_11) = [2]
p(c_12) = [0]
p(c_13) = [1] x2 + [0]
p(c_14) = [1] x1 + [1] x2 + [4]
p(c_15) = [1] x1 + [6] x2 + [0]
p(c_16) = [0]
Following rules are strictly oriented:
sum#(:(x,:(y,xs))) = [1] xs + [16]
> [1] xs + [14]
= c_15(sum#(:(+(x,y),xs)),+#(x,y))
Following rules are (at-least) weakly oriented:
avg#(xs) = [1] xs + [9]
>= [1] xs + [0]
= sum#(xs)
sum#(++(xs,:(x,:(y,ys)))) = [2] xs + [1] ys + [21]
>= [2] xs + [16]
= sum#(++(xs,sum(:(x,:(y,ys)))))
sum#(++(xs,:(x,:(y,ys)))) = [2] xs + [1] ys + [21]
>= [1] ys + [16]
= sum#(:(x,:(y,ys)))
++(:(x,xs),ys) = [2] xs + [1] ys + [21]
>= [2] xs + [1] ys + [13]
= :(x,++(xs,ys))
++(nil(),ys) = [1] ys + [5]
>= [1] ys + [0]
= ys
sum(:(x,:(y,xs))) = [11]
>= [11]
= sum(:(+(x,y),xs))
sum(:(x,nil())) = [11]
>= [8]
= :(x,nil())
*** 1.1.1.1.1.1.1.1.1.1.2.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.1.2.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.1.1.1.1.1.2.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.1.1.1.1.2.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
+#(s(x),y) -> c_2(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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, greedy = NoGreedy}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.
*** 1.1.1.1.1.1.1.1.1.1.2.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
+#(s(x),y) -> c_2(+#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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 + [1] x2 + [0]
p(-) = [0]
p(0) = [0]
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 + [3]
p(sum) = [1] x1 + [0]
p(+#) = [2] x1 + [2] x2 + [2]
p(++#) = [0]
p(-#) = [0]
p(avg#) = [2] x1 + [3]
p(hd#) = [0]
p(length#) = [0]
p(quot#) = [4] x1 + [1] x2 + [0]
p(sum#) = [2] x1 + [2]
p(c_1) = [0]
p(c_2) = [1] x1 + [2]
p(c_3) = [1] x1 + [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [2] x1 + [0]
p(c_8) = [0]
p(c_9) = [0]
p(c_10) = [1] x1 + [0]
p(c_11) = [0]
p(c_12) = [0]
p(c_13) = [4] x2 + [0]
p(c_14) = [1] x1 + [0]
p(c_15) = [1] x1 + [2] x2 + [0]
p(c_16) = [0]
Following rules are strictly oriented:
+#(s(x),y) = [2] x + [2] y + [8]
> [2] x + [2] y + [4]
= c_2(+#(x,y))
Following rules are (at-least) weakly oriented:
avg#(xs) = [2] xs + [3]
>= [2] xs + [2]
= sum#(xs)
sum#(++(xs,:(x,:(y,ys)))) = [2] x + [2] xs + [2] y + [2] ys + [2]
>= [2] x + [2] xs + [2] y + [2] ys + [2]
= sum#(++(xs,sum(:(x,:(y,ys)))))
sum#(++(xs,:(x,:(y,ys)))) = [2] x + [2] xs + [2] y + [2] ys + [2]
>= [2] x + [2] y + [2] ys + [2]
= sum#(:(x,:(y,ys)))
sum#(:(x,:(y,xs))) = [2] x + [2] xs + [2] y + [2]
>= [2] x + [2] y + [2]
= +#(x,y)
sum#(:(x,:(y,xs))) = [2] x + [2] xs + [2] y + [2]
>= [2] x + [2] xs + [2] y + [2]
= sum#(:(+(x,y),xs))
+(0(),y) = [1] y + [0]
>= [1] y + [0]
= y
+(s(x),y) = [1] x + [1] y + [3]
>= [1] x + [1] y + [3]
= s(+(x,y))
++(:(x,xs),ys) = [1] x + [1] xs + [1] ys + [0]
>= [1] x + [1] xs + [1] ys + [0]
= :(x,++(xs,ys))
++(nil(),ys) = [1] 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())
*** 1.1.1.1.1.1.1.1.1.1.2.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
+#(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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.1.2.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
+#(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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.1.1.1.1.1.2.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
++#(:(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))
Strict TRS Rules:
Weak DP Rules:
+#(s(x),y) -> c_2(+#(x,y))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
++#(:(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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
SimplifyRHS
Proof:
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)))
*** 1.1.1.1.1.1.2.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
++#(:(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)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
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 DP Rules:
++#(:(x,xs),ys) -> c_3(++#(xs,ys))
Strict TRS Rules:
Weak DP Rules:
-#(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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Problem (S)
Strict DP Rules:
-#(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)))
Strict TRS Rules:
Weak DP Rules:
++#(:(x,xs),ys) -> c_3(++#(xs,ys))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
*** 1.1.1.1.1.1.2.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
++#(:(x,xs),ys) -> c_3(++#(xs,ys))
Strict TRS Rules:
Weak DP Rules:
-#(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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
-->_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: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))):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
-->_2 ++#(:(x,xs),ys) -> c_3(++#(xs,ys)):1
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))
*** 1.1.1.1.1.1.2.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
++#(:(x,xs),ys) -> c_3(++#(xs,ys))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
SimplifyRHS
Proof:
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))))
-->_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
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)))))
*** 1.1.1.1.1.1.2.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
++#(:(x,xs),ys) -> c_3(++#(xs,ys))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
UsableRules
Proof:
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)))))
*** 1.1.1.1.1.1.2.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
++#(:(x,xs),ys) -> c_3(++#(xs,ys))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
Proof:
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)))))
*** 1.1.1.1.1.1.2.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(sum#(xs))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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, greedy = NoGreedy}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.
*** 1.1.1.1.1.1.2.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),++#(xs,sum(:(x,:(y,ys)))))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(sum#(xs))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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(+) = [2] x1 + [1] x2 + [6]
p(++) = [10] x1 + [4] x2 + [2]
p(-) = [1] x2 + [2]
p(0) = [1]
p(:) = [1] x2 + [1]
p(avg) = [1] x1 + [1]
p(hd) = [1] x1 + [1]
p(length) = [1] x1 + [0]
p(nil) = [0]
p(quot) = [0]
p(s) = [0]
p(sum) = [1]
p(+#) = [1]
p(++#) = [1] x2 + [0]
p(-#) = [1] x1 + [1]
p(avg#) = [5] x1 + [0]
p(hd#) = [1] x1 + [8]
p(length#) = [1] x1 + [2]
p(quot#) = [2] x2 + [1]
p(sum#) = [2] x1 + [0]
p(c_1) = [0]
p(c_2) = [2]
p(c_3) = [4]
p(c_4) = [1]
p(c_5) = [1]
p(c_6) = [0]
p(c_7) = [1] x1 + [1]
p(c_8) = [2] x1 + [0]
p(c_9) = [1]
p(c_10) = [1]
p(c_11) = [2]
p(c_12) = [2]
p(c_13) = [2] x1 + [1]
p(c_14) = [1] x1 + [1] x2 + [1]
p(c_15) = [1]
p(c_16) = [0]
Following rules are strictly oriented:
sum#(++(xs,:(x,:(y,ys)))) = [20] xs + [8] ys + [20]
> [20] xs + [14]
= c_14(sum#(++(xs
,sum(:(x,:(y,ys)))))
,++#(xs,sum(:(x,:(y,ys)))))
Following rules are (at-least) weakly oriented:
avg#(xs) = [5] xs + [0]
>= [4] xs + [0]
= c_8(sum#(xs))
++(:(x,xs),ys) = [10] xs + [4] ys + [12]
>= [10] xs + [4] ys + [3]
= :(x,++(xs,ys))
++(nil(),ys) = [4] ys + [2]
>= [1] ys + [0]
= ys
sum(:(x,:(y,xs))) = [1]
>= [1]
= sum(:(+(x,y),xs))
sum(:(x,nil())) = [1]
>= [1]
= :(x,nil())
*** 1.1.1.1.1.1.2.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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)))))
*** 1.1.1.1.1.1.2.1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.2.1.1.1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
++#(:(x,xs),ys) -> c_3(++#(xs,ys))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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, greedy = NoGreedy}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.
*** 1.1.1.1.1.1.2.1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
++#(:(x,xs),ys) -> c_3(++#(xs,ys))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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(+) = [0]
p(++) = [2] x1 + [1] x2 + [2]
p(-) = [0]
p(0) = [0]
p(:) = [1] x2 + [8]
p(avg) = [8]
p(hd) = [4] x1 + [0]
p(length) = [1]
p(nil) = [4]
p(quot) = [1]
p(s) = [1] x1 + [4]
p(sum) = [12]
p(+#) = [2] x1 + [0]
p(++#) = [2] x1 + [0]
p(-#) = [0]
p(avg#) = [4] x1 + [1]
p(hd#) = [1]
p(length#) = [0]
p(quot#) = [1] x2 + [2]
p(sum#) = [1] x1 + [0]
p(c_1) = [1]
p(c_2) = [2] x1 + [1]
p(c_3) = [1] x1 + [8]
p(c_4) = [0]
p(c_5) = [1]
p(c_6) = [0]
p(c_7) = [8] x1 + [1]
p(c_8) = [8] x1 + [1]
p(c_9) = [2]
p(c_10) = [1] x1 + [0]
p(c_11) = [0]
p(c_12) = [0]
p(c_13) = [1] x1 + [1]
p(c_14) = [1] x2 + [2]
p(c_15) = [8]
p(c_16) = [2]
Following rules are strictly oriented:
++#(:(x,xs),ys) = [2] xs + [16]
> [2] xs + [8]
= c_3(++#(xs,ys))
Following rules are (at-least) weakly oriented:
avg#(xs) = [4] xs + [1]
>= [1] xs + [0]
= sum#(xs)
sum#(++(xs,:(x,:(y,ys)))) = [2] xs + [1] ys + [18]
>= [2] xs + [0]
= ++#(xs,sum(:(x,:(y,ys))))
sum#(++(xs,:(x,:(y,ys)))) = [2] xs + [1] ys + [18]
>= [2] xs + [14]
= sum#(++(xs,sum(:(x,:(y,ys)))))
++(:(x,xs),ys) = [2] xs + [1] ys + [18]
>= [2] xs + [1] ys + [10]
= :(x,++(xs,ys))
++(nil(),ys) = [1] ys + [10]
>= [1] ys + [0]
= ys
sum(:(x,:(y,xs))) = [12]
>= [12]
= sum(:(+(x,y),xs))
sum(:(x,nil())) = [12]
>= [12]
= :(x,nil())
*** 1.1.1.1.1.1.2.1.1.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
++#(:(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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.1.1.1.1.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
++#(:(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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.1.2.1.1.1.1.1.1.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.2.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
-#(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)))
Strict TRS Rules:
Weak DP Rules:
++#(:(x,xs),ys) -> c_3(++#(xs,ys))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.1.2.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
-#(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)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
SimplifyRHS
Proof:
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))))
*** 1.1.1.1.1.1.2.1.1.2.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
-#(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)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
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 DP Rules:
-#(s(x),s(y)) -> c_7(-#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Problem (S)
Strict DP Rules:
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)))
Strict TRS Rules:
Weak DP Rules:
-#(s(x),s(y)) -> c_7(-#(x,y))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
*** 1.1.1.1.1.1.2.1.1.2.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
-#(s(x),s(y)) -> c_7(-#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
-->_3 length#(:(x,xs)) -> c_10(length#(xs)):3
-->_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: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))
-->_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: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)))
*** 1.1.1.1.1.1.2.1.1.2.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
-#(s(x),s(y)) -> c_7(-#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
SimplifyRHS
Proof:
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))
-->_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
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)))
*** 1.1.1.1.1.1.2.1.1.2.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
-#(s(x),s(y)) -> c_7(-#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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, greedy = NoGreedy}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.
*** 1.1.1.1.1.1.2.1.1.2.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
-#(s(x),s(y)) -> c_7(-#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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 + x1^2 + 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(-#) = 1 + 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)) = 2 + x
> 1 + 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
>= 1 + 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 + 2*x*xs + x^2 + xs + xs^2 + ys
>= x + xs + xs^2 + 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 + xs^2 + y + ys
>= x + xs + xs^2 + 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())
*** 1.1.1.1.1.1.2.1.1.2.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
-#(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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.2.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
-#(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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.1.2.1.1.2.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.2.1.1.2.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
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)))
Strict TRS Rules:
Weak DP Rules:
-#(s(x),s(y)) -> c_7(-#(x,y))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
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)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
SimplifyRHS
Proof:
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)))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
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)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
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 DP Rules:
avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Problem (S)
Strict DP Rules:
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))))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S: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))):3
-->_3 length#(:(x,xs)) -> c_10(length#(xs)):2
-->_2 sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys)))):4
-->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
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)))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
Strict TRS Rules:
Weak DP Rules:
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S: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)))):4
-->_2 sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs))):5
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))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
avg#(xs) -> c_8(sum#(xs))
sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
Strict TRS Rules:
Weak DP Rules:
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
UsableRules
Proof:
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)))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
avg#(xs) -> c_8(sum#(xs))
sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
Strict TRS Rules:
Weak DP Rules:
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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.
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
avg#(xs) -> c_8(sum#(xs))
sum#(:(x,:(y,xs))) -> c_15(sum#(:(+(x,y),xs)))
Strict TRS Rules:
Weak DP Rules:
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))),sum#(:(x,:(y,ys))))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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(+) = [4] x1 + [1]
p(++) = [8] x1 + [2] x2 + [0]
p(-) = [0]
p(0) = [4]
p(:) = [1] x2 + [2]
p(avg) = [1]
p(hd) = [1]
p(length) = [0]
p(nil) = [0]
p(quot) = [1] x1 + [1]
p(s) = [1] x1 + [4]
p(sum) = [2]
p(+#) = [1] x1 + [2]
p(++#) = [1] x2 + [0]
p(-#) = [2] x2 + [0]
p(avg#) = [4] x1 + [8]
p(hd#) = [1] x1 + [2]
p(length#) = [2] x1 + [4]
p(quot#) = [2] x1 + [0]
p(sum#) = [2] x1 + [0]
p(c_1) = [1]
p(c_2) = [8] x1 + [2]
p(c_3) = [1] x1 + [0]
p(c_4) = [1]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [2]
p(c_8) = [2] x1 + [0]
p(c_9) = [1]
p(c_10) = [1] x1 + [1]
p(c_11) = [1]
p(c_12) = [0]
p(c_13) = [1]
p(c_14) = [1] x1 + [1] x2 + [0]
p(c_15) = [1] x1 + [2]
p(c_16) = [0]
Following rules are strictly oriented:
avg#(xs) = [4] xs + [8]
> [4] xs + [0]
= c_8(sum#(xs))
sum#(:(x,:(y,xs))) = [2] xs + [8]
> [2] xs + [6]
= c_15(sum#(:(+(x,y),xs)))
Following rules are (at-least) weakly oriented:
sum#(++(xs,:(x,:(y,ys)))) = [16] xs + [4] ys + [16]
>= [16] xs + [2] ys + [16]
= c_14(sum#(++(xs
,sum(:(x,:(y,ys)))))
,sum#(:(x,:(y,ys))))
++(:(x,xs),ys) = [8] xs + [2] ys + [16]
>= [8] xs + [2] ys + [2]
= :(x,++(xs,ys))
++(nil(),ys) = [2] ys + [0]
>= [1] ys + [0]
= ys
sum(:(x,:(y,xs))) = [2]
>= [2]
= sum(:(+(x,y),xs))
sum(:(x,nil())) = [2]
>= [2]
= :(x,nil())
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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)))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
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))))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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)))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
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))))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
SimplifyRHS
Proof:
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))))))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
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))))))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
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 DP Rules:
length#(:(x,xs)) -> c_10(length#(xs))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Problem (S)
Strict DP Rules:
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))))))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
length#(:(x,xs)) -> c_10(length#(xs))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
length#(:(x,xs)) -> c_10(length#(xs))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
-->_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
-->_3 length#(:(x,xs)) -> c_10(length#(xs)):1
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)))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
length#(:(x,xs)) -> c_10(length#(xs))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
SimplifyRHS
Proof:
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))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
length#(:(x,xs)) -> c_10(length#(xs))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(length#(xs))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
avg#(xs) -> c_8(length#(xs))
length#(:(x,xs)) -> c_10(length#(xs))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
length#(:(x,xs)) -> c_10(length#(xs))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(length#(xs))
Weak TRS Rules:
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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, greedy = NoGreedy}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.
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
length#(:(x,xs)) -> c_10(length#(xs))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(length#(xs))
Weak TRS Rules:
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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 + [2]
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 + [9]
p(hd#) = [0]
p(length#) = [8] x1 + [7]
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) = [1] x1 + [2]
p(c_9) = [0]
p(c_10) = [1] x1 + [14]
p(c_11) = [0]
p(c_12) = [0]
p(c_13) = [8]
p(c_14) = [1]
p(c_15) = [2]
p(c_16) = [0]
Following rules are strictly oriented:
length#(:(x,xs)) = [8] x + [8] xs + [23]
> [8] xs + [21]
= c_10(length#(xs))
Following rules are (at-least) weakly oriented:
avg#(xs) = [8] xs + [9]
>= [8] xs + [9]
= c_8(length#(xs))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(length#(xs))
length#(:(x,xs)) -> c_10(length#(xs))
Weak TRS Rules:
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(length#(xs))
length#(:(x,xs)) -> c_10(length#(xs))
Weak TRS Rules:
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
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))))))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
length#(:(x,xs)) -> c_10(length#(xs))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
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))))))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs),length#(xs))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
SimplifyRHS
Proof:
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))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
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))))))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
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 DP Rules:
quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Problem (S)
Strict DP Rules:
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
-->_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
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))))))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
SimplifyRHS
Proof:
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)))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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, greedy = NoGreedy}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.
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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,length,sum,+#,++#,-#,avg#,hd#,length#,quot#,sum#}
TcT has computed the following interpretation:
p(+) = [1] x1 + [1] x2 + [0]
p(++) = [1] x1 + [1] x2 + [15]
p(-) = [1] x1 + [0]
p(0) = [0]
p(:) = [1] x1 + [1] x2 + [2]
p(avg) = [1] x1 + [1]
p(hd) = [4] x1 + [3]
p(length) = [2] x1 + [1]
p(nil) = [1]
p(quot) = [1] x1 + [2]
p(s) = [1] x1 + [4]
p(sum) = [1] x1 + [0]
p(+#) = [1] x1 + [1] x2 + [0]
p(++#) = [1] x1 + [2] x2 + [0]
p(-#) = [0]
p(avg#) = [9] x1 + [11]
p(hd#) = [1] x1 + [0]
p(length#) = [1]
p(quot#) = [1] x1 + [2] x2 + [4]
p(sum#) = [1] x1 + [1]
p(c_1) = [0]
p(c_2) = [1]
p(c_3) = [1]
p(c_4) = [0]
p(c_5) = [1]
p(c_6) = [2]
p(c_7) = [1]
p(c_8) = [1] x1 + [1]
p(c_9) = [1]
p(c_10) = [1] x1 + [1]
p(c_11) = [0]
p(c_12) = [1]
p(c_13) = [1] x1 + [2]
p(c_14) = [1]
p(c_15) = [1] x1 + [0]
p(c_16) = [0]
Following rules are strictly oriented:
quot#(s(x),s(y)) = [1] x + [2] y + [16]
> [1] x + [2] y + [14]
= c_13(quot#(-(x,y),s(y)))
Following rules are (at-least) weakly oriented:
avg#(xs) = [9] xs + [11]
>= [8] xs + [10]
= c_8(quot#(hd(sum(xs))
,length(xs)))
+(0(),y) = [1] y + [0]
>= [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 + [1] ys + [17]
>= [1] x + [1] xs + [1] ys + [17]
= :(x,++(xs,ys))
++(nil(),ys) = [1] ys + [16]
>= [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 + [4]
>= [1] x + [0]
= -(x,y)
hd(:(x,xs)) = [4] x + [4] xs + [11]
>= [1] x + [0]
= x
length(:(x,xs)) = [2] x + [2] xs + [5]
>= [2] xs + [5]
= s(length(xs))
length(nil()) = [3]
>= [0]
= 0()
sum(++(xs,:(x,:(y,ys)))) = [1] x + [1] xs + [1] y + [1] ys + [19]
>= [1] x + [1] xs + [1] y + [1] ys + [19]
= sum(++(xs,sum(:(x,:(y,ys)))))
sum(:(x,:(y,xs))) = [1] x + [1] xs + [1] y + [4]
>= [1] x + [1] xs + [1] y + [2]
= sum(:(+(x,y),xs))
sum(:(x,nil())) = [1] x + [3]
>= [1] x + [3]
= :(x,nil())
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)))
quot#(s(x),s(y)) -> c_13(quot#(-(x,y),s(y)))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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)))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
Strict TRS Rules:
Weak DP Rules:
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 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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)))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(quot#(hd(sum(xs)),length(xs)),sum#(xs))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
SimplifyRHS
Proof:
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))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(sum#(xs))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
UsableRules
Proof:
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))))))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(sum#(xs))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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, greedy = NoGreedy}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.
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(sum#(xs))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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 + [9]
p(++) = [13] x1 + [4] x2 + [4]
p(-) = [2] x1 + [1]
p(0) = [2]
p(:) = [1] x2 + [1]
p(avg) = [8]
p(hd) = [2]
p(length) = [1] x1 + [0]
p(nil) = [0]
p(quot) = [2] x1 + [2]
p(s) = [1]
p(sum) = [1]
p(+#) = [1] x1 + [1] x2 + [1]
p(++#) = [2] x1 + [2] x2 + [1]
p(-#) = [4] x1 + [8]
p(avg#) = [2] x1 + [10]
p(hd#) = [1] x1 + [1]
p(length#) = [0]
p(quot#) = [2]
p(sum#) = [2] x1 + [0]
p(c_1) = [1]
p(c_2) = [2] x1 + [1]
p(c_3) = [2]
p(c_4) = [1]
p(c_5) = [1]
p(c_6) = [0]
p(c_7) = [2]
p(c_8) = [1] x1 + [0]
p(c_9) = [1]
p(c_10) = [1] x1 + [0]
p(c_11) = [1]
p(c_12) = [0]
p(c_13) = [1]
p(c_14) = [1] x1 + [0]
p(c_15) = [8]
p(c_16) = [8]
Following rules are strictly oriented:
sum#(++(xs,:(x,:(y,ys)))) = [26] xs + [8] ys + [24]
> [26] xs + [16]
= c_14(sum#(++(xs
,sum(:(x,:(y,ys))))))
Following rules are (at-least) weakly oriented:
avg#(xs) = [2] xs + [10]
>= [2] xs + [0]
= c_8(sum#(xs))
++(:(x,xs),ys) = [13] xs + [4] ys + [17]
>= [13] xs + [4] ys + [5]
= :(x,++(xs,ys))
++(nil(),ys) = [4] ys + [4]
>= [1] ys + [0]
= ys
sum(:(x,:(y,xs))) = [1]
>= [1]
= sum(:(+(x,y),xs))
sum(:(x,nil())) = [1]
>= [1]
= :(x,nil())
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(sum#(xs))
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
avg#(xs) -> c_8(sum#(xs))
sum#(++(xs,:(x,:(y,ys)))) -> c_14(sum#(++(xs,sum(:(x,:(y,ys))))))
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))))))
*** 1.1.1.1.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.2.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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())
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
basic terms: {+#,++#,-#,avg#,hd#,length#,quot#,sum#}/{0,:,nil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).