*** 1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
top(sent(x)) -> top(check(rest(x)))
Weak DP Rules:
Weak TRS Rules:
Signature:
{check/1,rest/1,top/1} / {cons/2,nil/0,sent/1}
Obligation:
Innermost
basic terms: {check,rest,top}/{cons,nil,sent}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
check#(cons(x,y)) -> c_1()
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
check#(rest(x)) -> c_4(rest#(check(x)),check#(x))
check#(sent(x)) -> c_5(check#(x))
rest#(cons(x,y)) -> c_6()
rest#(nil()) -> c_7()
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x))
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
check#(cons(x,y)) -> c_1()
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
check#(rest(x)) -> c_4(rest#(check(x)),check#(x))
check#(sent(x)) -> c_5(check#(x))
rest#(cons(x,y)) -> c_6()
rest#(nil()) -> c_7()
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
top(sent(x)) -> top(check(rest(x)))
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0,c_8/3}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
check#(cons(x,y)) -> c_1()
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
check#(rest(x)) -> c_4(rest#(check(x)),check#(x))
check#(sent(x)) -> c_5(check#(x))
rest#(cons(x,y)) -> c_6()
rest#(nil()) -> c_7()
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x))
*** 1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
check#(cons(x,y)) -> c_1()
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
check#(rest(x)) -> c_4(rest#(check(x)),check#(x))
check#(sent(x)) -> c_5(check#(x))
rest#(cons(x,y)) -> c_6()
rest#(nil()) -> c_7()
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0,c_8/3}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{1,6,7}
by application of
Pre({1,6,7}) = {2,3,4,5,8}.
Here rules are labelled as follows:
1: check#(cons(x,y)) -> c_1()
2: check#(cons(x,y)) ->
c_2(check#(y))
3: check#(cons(x,y)) ->
c_3(check#(x))
4: check#(rest(x)) ->
c_4(rest#(check(x)),check#(x))
5: check#(sent(x)) ->
c_5(check#(x))
6: rest#(cons(x,y)) -> c_6()
7: rest#(nil()) -> c_7()
8: top#(sent(x)) ->
c_8(top#(check(rest(x)))
,check#(rest(x))
,rest#(x))
*** 1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
check#(rest(x)) -> c_4(rest#(check(x)),check#(x))
check#(sent(x)) -> c_5(check#(x))
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x))
Strict TRS Rules:
Weak DP Rules:
check#(cons(x,y)) -> c_1()
rest#(cons(x,y)) -> c_6()
rest#(nil()) -> c_7()
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0,c_8/3}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:check#(cons(x,y)) -> c_2(check#(y))
-->_1 check#(sent(x)) -> c_5(check#(x)):4
-->_1 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3
-->_1 check#(cons(x,y)) -> c_3(check#(x)):2
-->_1 check#(cons(x,y)) -> c_1():6
-->_1 check#(cons(x,y)) -> c_2(check#(y)):1
2:S:check#(cons(x,y)) -> c_3(check#(x))
-->_1 check#(sent(x)) -> c_5(check#(x)):4
-->_1 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3
-->_1 check#(cons(x,y)) -> c_1():6
-->_1 check#(cons(x,y)) -> c_3(check#(x)):2
-->_1 check#(cons(x,y)) -> c_2(check#(y)):1
3:S:check#(rest(x)) -> c_4(rest#(check(x)),check#(x))
-->_2 check#(sent(x)) -> c_5(check#(x)):4
-->_1 rest#(nil()) -> c_7():8
-->_1 rest#(cons(x,y)) -> c_6():7
-->_2 check#(cons(x,y)) -> c_1():6
-->_2 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3
-->_2 check#(cons(x,y)) -> c_3(check#(x)):2
-->_2 check#(cons(x,y)) -> c_2(check#(y)):1
4:S:check#(sent(x)) -> c_5(check#(x))
-->_1 check#(cons(x,y)) -> c_1():6
-->_1 check#(sent(x)) -> c_5(check#(x)):4
-->_1 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3
-->_1 check#(cons(x,y)) -> c_3(check#(x)):2
-->_1 check#(cons(x,y)) -> c_2(check#(y)):1
5:S:top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x))
-->_3 rest#(nil()) -> c_7():8
-->_3 rest#(cons(x,y)) -> c_6():7
-->_2 check#(cons(x,y)) -> c_1():6
-->_1 top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x)):5
-->_2 check#(sent(x)) -> c_5(check#(x)):4
-->_2 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3
-->_2 check#(cons(x,y)) -> c_3(check#(x)):2
-->_2 check#(cons(x,y)) -> c_2(check#(y)):1
6:W:check#(cons(x,y)) -> c_1()
7:W:rest#(cons(x,y)) -> c_6()
8:W:rest#(nil()) -> c_7()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
7: rest#(cons(x,y)) -> c_6()
8: rest#(nil()) -> c_7()
6: check#(cons(x,y)) -> c_1()
*** 1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
check#(rest(x)) -> c_4(rest#(check(x)),check#(x))
check#(sent(x)) -> c_5(check#(x))
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0,c_8/3}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:check#(cons(x,y)) -> c_2(check#(y))
-->_1 check#(sent(x)) -> c_5(check#(x)):4
-->_1 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3
-->_1 check#(cons(x,y)) -> c_3(check#(x)):2
-->_1 check#(cons(x,y)) -> c_2(check#(y)):1
2:S:check#(cons(x,y)) -> c_3(check#(x))
-->_1 check#(sent(x)) -> c_5(check#(x)):4
-->_1 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3
-->_1 check#(cons(x,y)) -> c_3(check#(x)):2
-->_1 check#(cons(x,y)) -> c_2(check#(y)):1
3:S:check#(rest(x)) -> c_4(rest#(check(x)),check#(x))
-->_2 check#(sent(x)) -> c_5(check#(x)):4
-->_2 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3
-->_2 check#(cons(x,y)) -> c_3(check#(x)):2
-->_2 check#(cons(x,y)) -> c_2(check#(y)):1
4:S:check#(sent(x)) -> c_5(check#(x))
-->_1 check#(sent(x)) -> c_5(check#(x)):4
-->_1 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3
-->_1 check#(cons(x,y)) -> c_3(check#(x)):2
-->_1 check#(cons(x,y)) -> c_2(check#(y)):1
5:S:top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x))
-->_1 top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)),rest#(x)):5
-->_2 check#(sent(x)) -> c_5(check#(x)):4
-->_2 check#(rest(x)) -> c_4(rest#(check(x)),check#(x)):3
-->_2 check#(cons(x,y)) -> c_3(check#(x)):2
-->_2 check#(cons(x,y)) -> c_2(check#(y)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
check#(rest(x)) -> c_4(check#(x))
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
*** 1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
check#(rest(x)) -> c_4(check#(x))
check#(sent(x)) -> c_5(check#(x))
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
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:
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
check#(rest(x)) -> c_4(check#(x))
check#(sent(x)) -> c_5(check#(x))
Strict TRS Rules:
Weak DP Rules:
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Problem (S)
Strict DP Rules:
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
Strict TRS Rules:
Weak DP Rules:
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
check#(rest(x)) -> c_4(check#(x))
check#(sent(x)) -> c_5(check#(x))
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
*** 1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
check#(rest(x)) -> c_4(check#(x))
check#(sent(x)) -> c_5(check#(x))
Strict TRS Rules:
Weak DP Rules:
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: check#(cons(x,y)) ->
c_2(check#(y))
2: check#(cons(x,y)) ->
c_3(check#(x))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
check#(rest(x)) -> c_4(check#(x))
check#(sent(x)) -> c_5(check#(x))
Strict TRS Rules:
Weak DP Rules:
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, 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},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1},
uargs(c_8) = {1,2}
Following symbols are considered usable:
{check,rest,check#,rest#,top#}
TcT has computed the following interpretation:
p(check) = [1 0] x1 + [0]
[0 1] [0]
p(cons) = [0 0] x1 + [1 4] x2 + [0]
[0 1] [0 1] [2]
p(nil) = [0]
[0]
p(rest) = [1 0] x1 + [1]
[0 4] [0]
p(sent) = [1 4] x1 + [1]
[0 1] [0]
p(top) = [0 2] x1 + [1]
[1 0] [0]
p(check#) = [0 1] x1 + [0]
[0 2] [0]
p(rest#) = [2 0] x1 + [2]
[0 0] [1]
p(top#) = [1 0] x1 + [3]
[2 2] [6]
p(c_1) = [4]
[1]
p(c_2) = [1 0] x1 + [0]
[0 1] [4]
p(c_3) = [1 0] x1 + [0]
[0 0] [0]
p(c_4) = [1 1] x1 + [0]
[6 0] [0]
p(c_5) = [1 0] x1 + [0]
[2 0] [0]
p(c_6) = [1]
[0]
p(c_7) = [0]
[0]
p(c_8) = [1 0] x1 + [1 0] x2 + [0]
[2 0] [0 0] [0]
Following rules are strictly oriented:
check#(cons(x,y)) = [0 1] x + [0 1] y + [2]
[0 2] [0 2] [4]
> [0 1] y + [0]
[0 2] [4]
= c_2(check#(y))
check#(cons(x,y)) = [0 1] x + [0 1] y + [2]
[0 2] [0 2] [4]
> [0 1] x + [0]
[0 0] [0]
= c_3(check#(x))
Following rules are (at-least) weakly oriented:
check#(rest(x)) = [0 4] x + [0]
[0 8] [0]
>= [0 3] x + [0]
[0 6] [0]
= c_4(check#(x))
check#(sent(x)) = [0 1] x + [0]
[0 2] [0]
>= [0 1] x + [0]
[0 2] [0]
= c_5(check#(x))
top#(sent(x)) = [1 4] x + [4]
[2 10] [8]
>= [1 4] x + [4]
[2 0] [8]
= c_8(top#(check(rest(x)))
,check#(rest(x)))
check(cons(x,y)) = [0 0] x + [1 4] y + [0]
[0 1] [0 1] [2]
>= [0 0] x + [1 4] y + [0]
[0 1] [0 1] [2]
= cons(x,y)
check(cons(x,y)) = [0 0] x + [1 4] y + [0]
[0 1] [0 1] [2]
>= [0 0] x + [1 4] y + [0]
[0 1] [0 1] [2]
= cons(x,check(y))
check(cons(x,y)) = [0 0] x + [1 4] y + [0]
[0 1] [0 1] [2]
>= [0 0] x + [1 4] y + [0]
[0 1] [0 1] [2]
= cons(check(x),y)
check(rest(x)) = [1 0] x + [1]
[0 4] [0]
>= [1 0] x + [1]
[0 4] [0]
= rest(check(x))
check(sent(x)) = [1 4] x + [1]
[0 1] [0]
>= [1 4] x + [1]
[0 1] [0]
= sent(check(x))
rest(cons(x,y)) = [0 0] x + [1 4] y + [1]
[0 4] [0 4] [8]
>= [1 4] y + [1]
[0 1] [0]
= sent(y)
rest(nil()) = [1]
[0]
>= [1]
[0]
= sent(nil())
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
check#(rest(x)) -> c_4(check#(x))
check#(sent(x)) -> c_5(check#(x))
Strict TRS Rules:
Weak DP Rules:
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
check#(rest(x)) -> c_4(check#(x))
check#(sent(x)) -> c_5(check#(x))
Strict TRS Rules:
Weak DP Rules:
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
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
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
and a lower component
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
check#(rest(x)) -> c_4(check#(x))
check#(sent(x)) -> c_5(check#(x))
Further, following extension rules are added to the lower component.
top#(sent(x)) -> check#(rest(x))
top#(sent(x)) -> top#(check(rest(x)))
*** 1.1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: top#(sent(x)) ->
c_8(top#(check(rest(x)))
,check#(rest(x)))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
NaturalMI {miDimension = 3, miDegree = 2, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_8) = {1,2}
Following symbols are considered usable:
{check,rest,check#,rest#,top#}
TcT has computed the following interpretation:
p(check) = [0 2 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(cons) = [0 0 0] [0 2 0] [0]
[0 0 3] x1 + [0 1 2] x2 + [2]
[0 0 0] [0 0 0] [0]
p(nil) = [0]
[0]
[2]
p(rest) = [0 2 3] [0]
[0 1 1] x1 + [0]
[0 0 0] [0]
p(sent) = [0 2 2] [1]
[0 1 0] x1 + [2]
[0 0 0] [0]
p(top) = [0 2 0] [1]
[2 1 0] x1 + [1]
[0 1 0] [0]
p(check#) = [0 0 0] [0]
[0 2 0] x1 + [1]
[0 0 0] [0]
p(rest#) = [0]
[0]
[2]
p(top#) = [2 0 0] [0]
[0 0 0] x1 + [0]
[2 0 0] [0]
p(c_1) = [1]
[2]
[1]
p(c_2) = [0]
[1]
[2]
p(c_3) = [0 0 0] [0]
[0 0 0] x1 + [0]
[0 2 2] [2]
p(c_4) = [0]
[0]
[1]
p(c_5) = [0 0 0] [0]
[0 0 0] x1 + [2]
[0 0 1] [0]
p(c_6) = [1]
[0]
[1]
p(c_7) = [0]
[0]
[2]
p(c_8) = [1 0 0] [1 0 0] [1]
[0 0 0] x1 + [0 0 1] x2 + [0]
[0 0 0] [1 2 2] [0]
Following rules are strictly oriented:
top#(sent(x)) = [0 4 4] [2]
[0 0 0] x + [0]
[0 4 4] [2]
> [0 4 4] [1]
[0 0 0] x + [0]
[0 4 4] [2]
= c_8(top#(check(rest(x)))
,check#(rest(x)))
Following rules are (at-least) weakly oriented:
check(cons(x,y)) = [0 0 6] [0 2 4] [4]
[0 0 3] x + [0 1 2] y + [2]
[0 0 0] [0 0 0] [0]
>= [0 0 0] [0 2 0] [0]
[0 0 3] x + [0 1 2] y + [2]
[0 0 0] [0 0 0] [0]
= cons(x,y)
check(cons(x,y)) = [0 0 6] [0 2 4] [4]
[0 0 3] x + [0 1 2] y + [2]
[0 0 0] [0 0 0] [0]
>= [0 0 0] [0 2 0] [0]
[0 0 3] x + [0 1 0] y + [2]
[0 0 0] [0 0 0] [0]
= cons(x,check(y))
check(cons(x,y)) = [0 0 6] [0 2 4] [4]
[0 0 3] x + [0 1 2] y + [2]
[0 0 0] [0 0 0] [0]
>= [0 2 0] [0]
[0 1 2] y + [2]
[0 0 0] [0]
= cons(check(x),y)
check(rest(x)) = [0 2 2] [0]
[0 1 1] x + [0]
[0 0 0] [0]
>= [0 2 0] [0]
[0 1 0] x + [0]
[0 0 0] [0]
= rest(check(x))
check(sent(x)) = [0 2 0] [4]
[0 1 0] x + [2]
[0 0 0] [0]
>= [0 2 0] [1]
[0 1 0] x + [2]
[0 0 0] [0]
= sent(check(x))
rest(cons(x,y)) = [0 0 6] [0 2 4] [4]
[0 0 3] x + [0 1 2] y + [2]
[0 0 0] [0 0 0] [0]
>= [0 2 2] [1]
[0 1 0] y + [2]
[0 0 0] [0]
= sent(y)
rest(nil()) = [6]
[2]
[0]
>= [5]
[2]
[0]
= sent(nil())
*** 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:
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
Assumption
Proof:
()
*** 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:
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
-->_1 top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: top#(sent(x)) ->
c_8(top#(check(rest(x)))
,check#(rest(x)))
*** 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:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.1.2.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
check#(rest(x)) -> c_4(check#(x))
check#(sent(x)) -> c_5(check#(x))
Strict TRS Rules:
Weak DP Rules:
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
top#(sent(x)) -> check#(rest(x))
top#(sent(x)) -> top#(check(rest(x)))
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
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: check#(rest(x)) ->
c_4(check#(x))
2: check#(sent(x)) ->
c_5(check#(x))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.2.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
check#(rest(x)) -> c_4(check#(x))
check#(sent(x)) -> c_5(check#(x))
Strict TRS Rules:
Weak DP Rules:
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
top#(sent(x)) -> check#(rest(x))
top#(sent(x)) -> top#(check(rest(x)))
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
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},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1}
Following symbols are considered usable:
{check,rest,check#,rest#,top#}
TcT has computed the following interpretation:
p(check) = [1] x1 + [0]
p(cons) = [1] x1 + [1] x2 + [0]
p(nil) = [1]
p(rest) = [1] x1 + [2]
p(sent) = [1] x1 + [2]
p(top) = [0]
p(check#) = [8] x1 + [0]
p(rest#) = [1] x1 + [2]
p(top#) = [8] x1 + [7]
p(c_1) = [4]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [1] x1 + [6]
p(c_5) = [1] x1 + [9]
p(c_6) = [2]
p(c_7) = [0]
p(c_8) = [2] x1 + [1] x2 + [2]
Following rules are strictly oriented:
check#(rest(x)) = [8] x + [16]
> [8] x + [6]
= c_4(check#(x))
check#(sent(x)) = [8] x + [16]
> [8] x + [9]
= c_5(check#(x))
Following rules are (at-least) weakly oriented:
check#(cons(x,y)) = [8] x + [8] y + [0]
>= [8] y + [0]
= c_2(check#(y))
check#(cons(x,y)) = [8] x + [8] y + [0]
>= [8] x + [0]
= c_3(check#(x))
top#(sent(x)) = [8] x + [23]
>= [8] x + [16]
= check#(rest(x))
top#(sent(x)) = [8] x + [23]
>= [8] x + [23]
= top#(check(rest(x)))
check(cons(x,y)) = [1] x + [1] y + [0]
>= [1] x + [1] y + [0]
= cons(x,y)
check(cons(x,y)) = [1] x + [1] y + [0]
>= [1] x + [1] y + [0]
= cons(x,check(y))
check(cons(x,y)) = [1] x + [1] y + [0]
>= [1] x + [1] y + [0]
= cons(check(x),y)
check(rest(x)) = [1] x + [2]
>= [1] x + [2]
= rest(check(x))
check(sent(x)) = [1] x + [2]
>= [1] x + [2]
= sent(check(x))
rest(cons(x,y)) = [1] x + [1] y + [2]
>= [1] y + [2]
= sent(y)
rest(nil()) = [3]
>= [3]
= sent(nil())
*** 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:
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
check#(rest(x)) -> c_4(check#(x))
check#(sent(x)) -> c_5(check#(x))
top#(sent(x)) -> check#(rest(x))
top#(sent(x)) -> top#(check(rest(x)))
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
Assumption
Proof:
()
*** 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:
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
check#(rest(x)) -> c_4(check#(x))
check#(sent(x)) -> c_5(check#(x))
top#(sent(x)) -> check#(rest(x))
top#(sent(x)) -> top#(check(rest(x)))
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:check#(cons(x,y)) -> c_2(check#(y))
-->_1 check#(sent(x)) -> c_5(check#(x)):4
-->_1 check#(rest(x)) -> c_4(check#(x)):3
-->_1 check#(cons(x,y)) -> c_3(check#(x)):2
-->_1 check#(cons(x,y)) -> c_2(check#(y)):1
2:W:check#(cons(x,y)) -> c_3(check#(x))
-->_1 check#(sent(x)) -> c_5(check#(x)):4
-->_1 check#(rest(x)) -> c_4(check#(x)):3
-->_1 check#(cons(x,y)) -> c_3(check#(x)):2
-->_1 check#(cons(x,y)) -> c_2(check#(y)):1
3:W:check#(rest(x)) -> c_4(check#(x))
-->_1 check#(sent(x)) -> c_5(check#(x)):4
-->_1 check#(rest(x)) -> c_4(check#(x)):3
-->_1 check#(cons(x,y)) -> c_3(check#(x)):2
-->_1 check#(cons(x,y)) -> c_2(check#(y)):1
4:W:check#(sent(x)) -> c_5(check#(x))
-->_1 check#(sent(x)) -> c_5(check#(x)):4
-->_1 check#(rest(x)) -> c_4(check#(x)):3
-->_1 check#(cons(x,y)) -> c_3(check#(x)):2
-->_1 check#(cons(x,y)) -> c_2(check#(y)):1
5:W:top#(sent(x)) -> check#(rest(x))
-->_1 check#(sent(x)) -> c_5(check#(x)):4
-->_1 check#(rest(x)) -> c_4(check#(x)):3
-->_1 check#(cons(x,y)) -> c_3(check#(x)):2
-->_1 check#(cons(x,y)) -> c_2(check#(y)):1
6:W:top#(sent(x)) -> top#(check(rest(x)))
-->_1 top#(sent(x)) -> top#(check(rest(x))):6
-->_1 top#(sent(x)) -> check#(rest(x)):5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: top#(sent(x)) ->
top#(check(rest(x)))
5: top#(sent(x)) -> check#(rest(x))
1: check#(cons(x,y)) ->
c_2(check#(y))
4: check#(sent(x)) ->
c_5(check#(x))
3: check#(rest(x)) ->
c_4(check#(x))
2: check#(cons(x,y)) ->
c_3(check#(x))
*** 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:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
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^1))] ***
Considered Problem:
Strict DP Rules:
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
Strict TRS Rules:
Weak DP Rules:
check#(cons(x,y)) -> c_2(check#(y))
check#(cons(x,y)) -> c_3(check#(x))
check#(rest(x)) -> c_4(check#(x))
check#(sent(x)) -> c_5(check#(x))
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
-->_2 check#(sent(x)) -> c_5(check#(x)):5
-->_2 check#(rest(x)) -> c_4(check#(x)):4
-->_2 check#(cons(x,y)) -> c_3(check#(x)):3
-->_2 check#(cons(x,y)) -> c_2(check#(y)):2
-->_1 top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x))):1
2:W:check#(cons(x,y)) -> c_2(check#(y))
-->_1 check#(sent(x)) -> c_5(check#(x)):5
-->_1 check#(rest(x)) -> c_4(check#(x)):4
-->_1 check#(cons(x,y)) -> c_3(check#(x)):3
-->_1 check#(cons(x,y)) -> c_2(check#(y)):2
3:W:check#(cons(x,y)) -> c_3(check#(x))
-->_1 check#(sent(x)) -> c_5(check#(x)):5
-->_1 check#(rest(x)) -> c_4(check#(x)):4
-->_1 check#(cons(x,y)) -> c_3(check#(x)):3
-->_1 check#(cons(x,y)) -> c_2(check#(y)):2
4:W:check#(rest(x)) -> c_4(check#(x))
-->_1 check#(sent(x)) -> c_5(check#(x)):5
-->_1 check#(rest(x)) -> c_4(check#(x)):4
-->_1 check#(cons(x,y)) -> c_3(check#(x)):3
-->_1 check#(cons(x,y)) -> c_2(check#(y)):2
5:W:check#(sent(x)) -> c_5(check#(x))
-->_1 check#(sent(x)) -> c_5(check#(x)):5
-->_1 check#(rest(x)) -> c_4(check#(x)):4
-->_1 check#(cons(x,y)) -> c_3(check#(x)):3
-->_1 check#(cons(x,y)) -> c_2(check#(y)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: check#(sent(x)) ->
c_5(check#(x))
4: check#(rest(x)) ->
c_4(check#(x))
3: check#(cons(x,y)) ->
c_3(check#(x))
2: check#(cons(x,y)) ->
c_2(check#(y))
*** 1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x)))
-->_1 top#(sent(x)) -> c_8(top#(check(rest(x))),check#(rest(x))):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
top#(sent(x)) -> c_8(top#(check(rest(x))))
*** 1.1.1.1.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
top#(sent(x)) -> c_8(top#(check(rest(x))))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: top#(sent(x)) ->
c_8(top#(check(rest(x))))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
top#(sent(x)) -> c_8(top#(check(rest(x))))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
NaturalMI {miDimension = 3, miDegree = 2, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_8) = {1}
Following symbols are considered usable:
{check,rest,check#,rest#,top#}
TcT has computed the following interpretation:
p(check) = [0 2 0] [0]
[0 1 0] x1 + [0]
[0 0 0] [0]
p(cons) = [0 2 2] [0 2 3] [0]
[0 1 1] x1 + [0 1 2] x2 + [2]
[0 0 0] [0 0 0] [0]
p(nil) = [2]
[0]
[2]
p(rest) = [1 0 2] [0]
[0 1 1] x1 + [0]
[0 0 0] [0]
p(sent) = [0 2 3] [0]
[0 1 0] x1 + [2]
[0 0 0] [0]
p(top) = [0 1 0] [1]
[1 0 0] x1 + [0]
[1 1 0] [1]
p(check#) = [0 0 2] [0]
[0 0 0] x1 + [1]
[0 2 2] [0]
p(rest#) = [0 2 0] [2]
[0 2 0] x1 + [2]
[1 0 2] [0]
p(top#) = [1 1 0] [0]
[0 1 2] x1 + [1]
[2 0 1] [3]
p(c_1) = [0]
[1]
[1]
p(c_2) = [1 0 2] [0]
[2 1 1] x1 + [2]
[1 1 0] [1]
p(c_3) = [1 0 0] [0]
[0 2 2] x1 + [2]
[2 0 0] [0]
p(c_4) = [0 0 0] [2]
[0 0 0] x1 + [2]
[0 0 1] [1]
p(c_5) = [0 0 0] [2]
[0 0 0] x1 + [2]
[0 1 0] [0]
p(c_6) = [2]
[0]
[2]
p(c_7) = [1]
[0]
[0]
p(c_8) = [1 0 0] [0]
[0 0 0] x1 + [1]
[1 1 0] [2]
Following rules are strictly oriented:
top#(sent(x)) = [0 3 3] [2]
[0 1 0] x + [3]
[0 4 6] [3]
> [0 3 3] [0]
[0 0 0] x + [1]
[0 4 4] [3]
= c_8(top#(check(rest(x))))
Following rules are (at-least) weakly oriented:
check(cons(x,y)) = [0 2 2] [0 2 4] [4]
[0 1 1] x + [0 1 2] y + [2]
[0 0 0] [0 0 0] [0]
>= [0 2 2] [0 2 3] [0]
[0 1 1] x + [0 1 2] y + [2]
[0 0 0] [0 0 0] [0]
= cons(x,y)
check(cons(x,y)) = [0 2 2] [0 2 4] [4]
[0 1 1] x + [0 1 2] y + [2]
[0 0 0] [0 0 0] [0]
>= [0 2 2] [0 2 0] [0]
[0 1 1] x + [0 1 0] y + [2]
[0 0 0] [0 0 0] [0]
= cons(x,check(y))
check(cons(x,y)) = [0 2 2] [0 2 4] [4]
[0 1 1] x + [0 1 2] y + [2]
[0 0 0] [0 0 0] [0]
>= [0 2 0] [0 2 3] [0]
[0 1 0] x + [0 1 2] y + [2]
[0 0 0] [0 0 0] [0]
= cons(check(x),y)
check(rest(x)) = [0 2 2] [0]
[0 1 1] x + [0]
[0 0 0] [0]
>= [0 2 0] [0]
[0 1 0] x + [0]
[0 0 0] [0]
= rest(check(x))
check(sent(x)) = [0 2 0] [4]
[0 1 0] x + [2]
[0 0 0] [0]
>= [0 2 0] [0]
[0 1 0] x + [2]
[0 0 0] [0]
= sent(check(x))
rest(cons(x,y)) = [0 2 2] [0 2 3] [0]
[0 1 1] x + [0 1 2] y + [2]
[0 0 0] [0 0 0] [0]
>= [0 2 3] [0]
[0 1 0] y + [2]
[0 0 0] [0]
= sent(y)
rest(nil()) = [6]
[2]
[0]
>= [6]
[2]
[0]
= sent(nil())
*** 1.1.1.1.1.1.2.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
top#(sent(x)) -> c_8(top#(check(rest(x))))
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
top#(sent(x)) -> c_8(top#(check(rest(x))))
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:top#(sent(x)) -> c_8(top#(check(rest(x))))
-->_1 top#(sent(x)) -> c_8(top#(check(rest(x)))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: top#(sent(x)) ->
c_8(top#(check(rest(x))))
*** 1.1.1.1.1.1.2.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
check(cons(x,y)) -> cons(x,y)
check(cons(x,y)) -> cons(x,check(y))
check(cons(x,y)) -> cons(check(x),y)
check(rest(x)) -> rest(check(x))
check(sent(x)) -> sent(check(x))
rest(cons(x,y)) -> sent(y)
rest(nil()) -> sent(nil())
Signature:
{check/1,rest/1,top/1,check#/1,rest#/1,top#/1} / {cons/2,nil/0,sent/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1}
Obligation:
Innermost
basic terms: {check#,rest#,top#}/{cons,nil,sent}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).