MAYBE
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict Trs:
{ from(X) -> cons(X, from(s(X)))
, length(cons(X, Y)) -> s(length1(Y))
, length(nil()) -> 0()
, length1(X) -> length(X) }
Obligation:
runtime complexity
Answer:
MAYBE
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'WithProblem (timeout of 60 seconds)' failed due to the
following reason:
We add the following weak dependency pairs:
Strict DPs:
{ from^#(X) -> c_1(X, from^#(s(X)))
, length^#(cons(X, Y)) -> c_2(length1^#(Y))
, length^#(nil()) -> c_3()
, length1^#(X) -> c_4(length^#(X)) }
and mark the set of starting terms.
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ from^#(X) -> c_1(X, from^#(s(X)))
, length^#(cons(X, Y)) -> c_2(length1^#(Y))
, length^#(nil()) -> c_3()
, length1^#(X) -> c_4(length^#(X)) }
Strict Trs:
{ from(X) -> cons(X, from(s(X)))
, length(cons(X, Y)) -> s(length1(Y))
, length(nil()) -> 0()
, length1(X) -> length(X) }
Obligation:
runtime complexity
Answer:
MAYBE
No rule is usable, rules are removed from the input problem.
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ from^#(X) -> c_1(X, from^#(s(X)))
, length^#(cons(X, Y)) -> c_2(length1^#(Y))
, length^#(nil()) -> c_3()
, length1^#(X) -> c_4(length^#(X)) }
Obligation:
runtime complexity
Answer:
MAYBE
The weightgap principle applies (using the following constant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(c_1) = {2}, Uargs(c_2) = {1}, Uargs(c_4) = {1}
TcT has computed the following constructor-restricted matrix
interpretation.
[cons](x1, x2) = [1 1] x2 + [2]
[0 1] [1]
[s](x1) = [1 1] x1 + [2]
[0 0] [2]
[nil] = [2]
[1]
[from^#](x1) = [1 1] x1 + [2]
[1 1] [2]
[c_1](x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[1 2] [0 1] [1]
[length^#](x1) = [1 2] x1 + [2]
[1 2] [2]
[c_2](x1) = [1 0] x1 + [2]
[0 1] [0]
[length1^#](x1) = [1 2] x1 + [2]
[1 2] [2]
[c_3] = [1]
[1]
[c_4](x1) = [1 0] x1 + [2]
[0 1] [1]
The following symbols are considered usable
{from^#, length^#, length1^#}
The order satisfies the following ordering constraints:
[from^#(X)] = [1 1] X + [2]
[1 1] [2]
? [1 1] X + [7]
[2 3] [7]
= [c_1(X, from^#(s(X)))]
[length^#(cons(X, Y))] = [1 3] Y + [6]
[1 3] [6]
> [1 2] Y + [4]
[1 2] [2]
= [c_2(length1^#(Y))]
[length^#(nil())] = [6]
[6]
> [1]
[1]
= [c_3()]
[length1^#(X)] = [1 2] X + [2]
[1 2] [2]
? [1 2] X + [4]
[1 2] [3]
= [c_4(length^#(X))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ from^#(X) -> c_1(X, from^#(s(X)))
, length1^#(X) -> c_4(length^#(X)) }
Weak DPs:
{ length^#(cons(X, Y)) -> c_2(length1^#(Y))
, length^#(nil()) -> c_3() }
Obligation:
runtime complexity
Answer:
MAYBE
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ length^#(nil()) -> c_3() }
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ from^#(X) -> c_1(X, from^#(s(X)))
, length1^#(X) -> c_4(length^#(X)) }
Weak DPs: { length^#(cons(X, Y)) -> c_2(length1^#(Y)) }
Obligation:
runtime complexity
Answer:
MAYBE
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'empty' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'Inspecting Problem...' failed due to the following reason:
We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.
DPs:
{ 2: length1^#(X) -> c_4(length^#(X))
, 3: length^#(cons(X, Y)) -> c_2(length1^#(Y)) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_1) = {2}, Uargs(c_2) = {1}, Uargs(c_4) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[from](x1) = [7] x1 + [0]
[cons](x1, x2) = [1] x1 + [1] x2 + [4]
[s](x1) = [0]
[length](x1) = [7] x1 + [0]
[nil] = [0]
[0] = [0]
[length1](x1) = [7] x1 + [0]
[from^#](x1) = [4] x1 + [0]
[c_1](x1, x2) = [3] x1 + [2] x2 + [0]
[length^#](x1) = [2] x1 + [0]
[c_2](x1) = [1] x1 + [3]
[length1^#](x1) = [2] x1 + [4]
[c_3] = [0]
[c_4](x1) = [1] x1 + [1]
The following symbols are considered usable
{from^#, length^#, length1^#}
The order satisfies the following ordering constraints:
[from^#(X)] = [4] X + [0]
>= [3] X + [0]
= [c_1(X, from^#(s(X)))]
[length^#(cons(X, Y))] = [2] X + [2] Y + [8]
> [2] Y + [7]
= [c_2(length1^#(Y))]
[length1^#(X)] = [2] X + [4]
> [2] X + [1]
= [c_4(length^#(X))]
The strictly oriented rules are moved into the weak component.
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs: { from^#(X) -> c_1(X, from^#(s(X))) }
Weak DPs:
{ length^#(cons(X, Y)) -> c_2(length1^#(Y))
, length1^#(X) -> c_4(length^#(X)) }
Obligation:
runtime complexity
Answer:
MAYBE
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ length^#(cons(X, Y)) -> c_2(length1^#(Y))
, length1^#(X) -> c_4(length^#(X)) }
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs: { from^#(X) -> c_1(X, from^#(s(X))) }
Obligation:
runtime complexity
Answer:
MAYBE
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'empty' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'Fastest' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'WithProblem' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'empty' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'Polynomial Path Order (PS)' failed due to the following reason:
The processor is inapplicable, reason:
Processor only applicable for innermost runtime complexity analysis
2) 'Fastest (timeout of 5 seconds)' failed due to the following
reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'Bounds with perSymbol-enrichment and initial automaton 'match''
failed due to the following reason:
match-boundness of the problem could not be verified.
2) 'Bounds with minimal-enrichment and initial automaton 'match''
failed due to the following reason:
match-boundness of the problem could not be verified.
3) 'Polynomial Path Order (PS)' failed due to the following reason:
The processor is inapplicable, reason:
Processor only applicable for innermost runtime complexity analysis
2) 'Best' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)'
failed due to the following reason:
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(cons) = {2}, Uargs(s) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[from](x1) = [1] x1 + [0]
[cons](x1, x2) = [1] x2 + [0]
[s](x1) = [1] x1 + [0]
[length](x1) = [1] x1 + [0]
[nil] = [1]
[0] = [0]
[length1](x1) = [1] x1 + [0]
The following symbols are considered usable
{from, length, length1}
The order satisfies the following ordering constraints:
[from(X)] = [1] X + [0]
>= [1] X + [0]
= [cons(X, from(s(X)))]
[length(cons(X, Y))] = [1] Y + [0]
>= [1] Y + [0]
= [s(length1(Y))]
[length(nil())] = [1]
> [0]
= [0()]
[length1(X)] = [1] X + [0]
>= [1] X + [0]
= [length(X)]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict Trs:
{ from(X) -> cons(X, from(s(X)))
, length(cons(X, Y)) -> s(length1(Y))
, length1(X) -> length(X) }
Weak Trs: { length(nil()) -> 0() }
Obligation:
runtime complexity
Answer:
MAYBE
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(cons) = {2}, Uargs(s) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[from](x1) = [1] x1 + [0]
[cons](x1, x2) = [1] x2 + [0]
[s](x1) = [1] x1 + [0]
[length](x1) = [1] x1 + [1]
[nil] = [7]
[0] = [0]
[length1](x1) = [1] x1 + [0]
The following symbols are considered usable
{from, length, length1}
The order satisfies the following ordering constraints:
[from(X)] = [1] X + [0]
>= [1] X + [0]
= [cons(X, from(s(X)))]
[length(cons(X, Y))] = [1] Y + [1]
> [1] Y + [0]
= [s(length1(Y))]
[length(nil())] = [8]
> [0]
= [0()]
[length1(X)] = [1] X + [0]
? [1] X + [1]
= [length(X)]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict Trs:
{ from(X) -> cons(X, from(s(X)))
, length1(X) -> length(X) }
Weak Trs:
{ length(cons(X, Y)) -> s(length1(Y))
, length(nil()) -> 0() }
Obligation:
runtime complexity
Answer:
MAYBE
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(cons) = {2}, Uargs(s) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[from](x1) = [1] x1 + [0]
[cons](x1, x2) = [1] x2 + [4]
[s](x1) = [1] x1 + [0]
[length](x1) = [1] x1 + [0]
[nil] = [7]
[0] = [7]
[length1](x1) = [1] x1 + [1]
The following symbols are considered usable
{from, length, length1}
The order satisfies the following ordering constraints:
[from(X)] = [1] X + [0]
? [1] X + [4]
= [cons(X, from(s(X)))]
[length(cons(X, Y))] = [1] Y + [4]
> [1] Y + [1]
= [s(length1(Y))]
[length(nil())] = [7]
>= [7]
= [0()]
[length1(X)] = [1] X + [1]
> [1] X + [0]
= [length(X)]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict Trs: { from(X) -> cons(X, from(s(X))) }
Weak Trs:
{ length(cons(X, Y)) -> s(length1(Y))
, length(nil()) -> 0()
, length1(X) -> length(X) }
Obligation:
runtime complexity
Answer:
MAYBE
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'empty' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'WithProblem' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'empty' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'Fastest' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'WithProblem' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'empty' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'WithProblem' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'empty' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'WithProblem' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'empty' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'WithProblem' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'WithProblem' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'empty' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'WithProblem' failed due to the following reason:
Empty strict component of the problem is NOT empty.
2) 'Best' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'bsearch-popstar (timeout of 60 seconds)' failed due to the
following reason:
The processor is inapplicable, reason:
Processor only applicable for innermost runtime complexity analysis
2) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due
to the following reason:
The processor is inapplicable, reason:
Processor only applicable for innermost runtime complexity analysis
3) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed
due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'Bounds with minimal-enrichment and initial automaton 'match''
failed due to the following reason:
match-boundness of the problem could not be verified.
2) 'Bounds with perSymbol-enrichment and initial automaton 'match''
failed due to the following reason:
match-boundness of the problem could not be verified.
3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed
due to the following reason:
We add the following weak dependency pairs:
Strict DPs:
{ from^#(X) -> c_1(X, from^#(s(X)))
, length^#(cons(X, Y)) -> c_2(length1^#(Y))
, length^#(nil()) -> c_3()
, length1^#(X) -> c_4(length^#(X)) }
and mark the set of starting terms.
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ from^#(X) -> c_1(X, from^#(s(X)))
, length^#(cons(X, Y)) -> c_2(length1^#(Y))
, length^#(nil()) -> c_3()
, length1^#(X) -> c_4(length^#(X)) }
Strict Trs:
{ from(X) -> cons(X, from(s(X)))
, length(cons(X, Y)) -> s(length1(Y))
, length(nil()) -> 0()
, length1(X) -> length(X) }
Obligation:
runtime complexity
Answer:
MAYBE
We estimate the number of application of {3} by applications of
Pre({3}) = {1,4}. Here rules are labeled as follows:
DPs:
{ 1: from^#(X) -> c_1(X, from^#(s(X)))
, 2: length^#(cons(X, Y)) -> c_2(length1^#(Y))
, 3: length^#(nil()) -> c_3()
, 4: length1^#(X) -> c_4(length^#(X)) }
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ from^#(X) -> c_1(X, from^#(s(X)))
, length^#(cons(X, Y)) -> c_2(length1^#(Y))
, length1^#(X) -> c_4(length^#(X)) }
Strict Trs:
{ from(X) -> cons(X, from(s(X)))
, length(cons(X, Y)) -> s(length1(Y))
, length(nil()) -> 0()
, length1(X) -> length(X) }
Weak DPs: { length^#(nil()) -> c_3() }
Obligation:
runtime complexity
Answer:
MAYBE
Empty strict component of the problem is NOT empty.
Arrrr..