MAYBE
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict Trs:
{ fst(0(), Z) -> nil()
, fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
, from(X) -> cons(X, from(s(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, len(nil()) -> 0()
, len(cons(X, Z)) -> s(len(Z)) }
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:
{ fst^#(0(), Z) -> c_1()
, fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z))
, from^#(X) -> c_3(X, from^#(s(X)))
, add^#(0(), X) -> c_4(X)
, add^#(s(X), Y) -> c_5(add^#(X, Y))
, len^#(nil()) -> c_6()
, len^#(cons(X, Z)) -> c_7(len^#(Z)) }
and mark the set of starting terms.
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ fst^#(0(), Z) -> c_1()
, fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z))
, from^#(X) -> c_3(X, from^#(s(X)))
, add^#(0(), X) -> c_4(X)
, add^#(s(X), Y) -> c_5(add^#(X, Y))
, len^#(nil()) -> c_6()
, len^#(cons(X, Z)) -> c_7(len^#(Z)) }
Strict Trs:
{ fst(0(), Z) -> nil()
, fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
, from(X) -> cons(X, from(s(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, len(nil()) -> 0()
, len(cons(X, Z)) -> s(len(Z)) }
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:
{ fst^#(0(), Z) -> c_1()
, fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z))
, from^#(X) -> c_3(X, from^#(s(X)))
, add^#(0(), X) -> c_4(X)
, add^#(s(X), Y) -> c_5(add^#(X, Y))
, len^#(nil()) -> c_6()
, len^#(cons(X, Z)) -> c_7(len^#(Z)) }
Obligation:
runtime complexity
Answer:
MAYBE
The weightgap principle applies (using the following constant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(c_2) = {2}, Uargs(c_3) = {2}, Uargs(c_5) = {1},
Uargs(c_7) = {1}
TcT has computed the following constructor-restricted matrix
interpretation.
[0] = [0]
[0]
[nil] = [1]
[1]
[s](x1) = [1 0] x1 + [0]
[0 0] [0]
[cons](x1, x2) = [1 2] x1 + [1 0] x2 + [0]
[0 1] [0 0] [0]
[fst^#](x1, x2) = [0]
[2]
[c_1] = [1]
[1]
[c_2](x1, x2) = [1 0] x2 + [2]
[0 1] [0]
[from^#](x1) = [0 0] x1 + [2]
[1 1] [2]
[c_3](x1, x2) = [0 0] x1 + [1 0] x2 + [2]
[1 2] [0 1] [2]
[add^#](x1, x2) = [0 0] x1 + [2 1] x2 + [0]
[2 2] [1 2] [1]
[c_4](x1) = [1 1] x1 + [1]
[1 1] [1]
[c_5](x1) = [1 0] x1 + [2]
[0 1] [0]
[len^#](x1) = [0 0] x1 + [1]
[2 2] [1]
[c_6] = [0]
[1]
[c_7](x1) = [1 0] x1 + [1]
[0 1] [0]
The following symbols are considered usable
{fst^#, from^#, add^#, len^#}
The order satisfies the following ordering constraints:
[fst^#(0(), Z)] = [0]
[2]
? [1]
[1]
= [c_1()]
[fst^#(s(X), cons(Y, Z))] = [0]
[2]
? [2]
[2]
= [c_2(Y, fst^#(X, Z))]
[from^#(X)] = [0 0] X + [2]
[1 1] [2]
? [0 0] X + [4]
[2 2] [4]
= [c_3(X, from^#(s(X)))]
[add^#(0(), X)] = [2 1] X + [0]
[1 2] [1]
? [1 1] X + [1]
[1 1] [1]
= [c_4(X)]
[add^#(s(X), Y)] = [0 0] X + [2 1] Y + [0]
[2 0] [1 2] [1]
? [0 0] X + [2 1] Y + [2]
[2 2] [1 2] [1]
= [c_5(add^#(X, Y))]
[len^#(nil())] = [1]
[5]
> [0]
[1]
= [c_6()]
[len^#(cons(X, Z))] = [0 0] Z + [0 0] X + [1]
[2 0] [2 6] [1]
? [0 0] Z + [2]
[2 2] [1]
= [c_7(len^#(Z))]
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:
{ fst^#(0(), Z) -> c_1()
, fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z))
, from^#(X) -> c_3(X, from^#(s(X)))
, add^#(0(), X) -> c_4(X)
, add^#(s(X), Y) -> c_5(add^#(X, Y))
, len^#(cons(X, Z)) -> c_7(len^#(Z)) }
Weak DPs: { len^#(nil()) -> c_6() }
Obligation:
runtime complexity
Answer:
MAYBE
We estimate the number of application of {1} by applications of
Pre({1}) = {2,3,4}. Here rules are labeled as follows:
DPs:
{ 1: fst^#(0(), Z) -> c_1()
, 2: fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z))
, 3: from^#(X) -> c_3(X, from^#(s(X)))
, 4: add^#(0(), X) -> c_4(X)
, 5: add^#(s(X), Y) -> c_5(add^#(X, Y))
, 6: len^#(cons(X, Z)) -> c_7(len^#(Z))
, 7: len^#(nil()) -> c_6() }
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z))
, from^#(X) -> c_3(X, from^#(s(X)))
, add^#(0(), X) -> c_4(X)
, add^#(s(X), Y) -> c_5(add^#(X, Y))
, len^#(cons(X, Z)) -> c_7(len^#(Z)) }
Weak DPs:
{ fst^#(0(), Z) -> c_1()
, len^#(nil()) -> c_6() }
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.
{ fst^#(0(), Z) -> c_1()
, len^#(nil()) -> c_6() }
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z))
, from^#(X) -> c_3(X, from^#(s(X)))
, add^#(0(), X) -> c_4(X)
, add^#(s(X), Y) -> c_5(add^#(X, Y))
, len^#(cons(X, Z)) -> c_7(len^#(Z)) }
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:
{ 1: fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z))
, 5: len^#(cons(X, Z)) -> c_7(len^#(Z)) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {2}, Uargs(c_3) = {2}, Uargs(c_5) = {1},
Uargs(c_7) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[fst](x1, x2) = [7] x1 + [7] x2 + [0]
[0] = [7]
[nil] = [0]
[s](x1) = [0]
[cons](x1, x2) = [1] x1 + [1] x2 + [4]
[from](x1) = [7] x1 + [0]
[add](x1, x2) = [7] x1 + [7] x2 + [0]
[len](x1) = [7] x1 + [0]
[fst^#](x1, x2) = [2] x2 + [0]
[c_1] = [0]
[c_2](x1, x2) = [1] x1 + [1] x2 + [5]
[from^#](x1) = [4] x1 + [0]
[c_3](x1, x2) = [3] x1 + [2] x2 + [0]
[add^#](x1, x2) = [0]
[c_4](x1) = [0]
[c_5](x1) = [2] x1 + [0]
[len^#](x1) = [2] x1 + [0]
[c_6] = [0]
[c_7](x1) = [1] x1 + [5]
The following symbols are considered usable
{fst^#, from^#, add^#, len^#}
The order satisfies the following ordering constraints:
[fst^#(s(X), cons(Y, Z))] = [2] Z + [2] Y + [8]
> [2] Z + [1] Y + [5]
= [c_2(Y, fst^#(X, Z))]
[from^#(X)] = [4] X + [0]
>= [3] X + [0]
= [c_3(X, from^#(s(X)))]
[add^#(0(), X)] = [0]
>= [0]
= [c_4(X)]
[add^#(s(X), Y)] = [0]
>= [0]
= [c_5(add^#(X, Y))]
[len^#(cons(X, Z))] = [2] Z + [2] X + [8]
> [2] Z + [5]
= [c_7(len^#(Z))]
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_3(X, from^#(s(X)))
, add^#(0(), X) -> c_4(X)
, add^#(s(X), Y) -> c_5(add^#(X, Y)) }
Weak DPs:
{ fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z))
, len^#(cons(X, Z)) -> c_7(len^#(Z)) }
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.
{ len^#(cons(X, Z)) -> c_7(len^#(Z)) }
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ from^#(X) -> c_3(X, from^#(s(X)))
, add^#(0(), X) -> c_4(X)
, add^#(s(X), Y) -> c_5(add^#(X, Y)) }
Weak DPs: { fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z)) }
Obligation:
runtime complexity
Answer:
MAYBE
We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.
DPs:
{ 2: add^#(0(), X) -> c_4(X)
, 4: fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z)) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {2}, Uargs(c_3) = {2}, Uargs(c_5) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[fst](x1, x2) = [7] x1 + [7] x2 + [0]
[0] = [4]
[nil] = [0]
[s](x1) = [1] x1 + [0]
[cons](x1, x2) = [1] x1 + [1] x2 + [4]
[from](x1) = [7] x1 + [0]
[add](x1, x2) = [7] x1 + [7] x2 + [0]
[len](x1) = [7] x1 + [0]
[fst^#](x1, x2) = [1] x2 + [0]
[c_1] = [0]
[c_2](x1, x2) = [1] x1 + [1] x2 + [1]
[from^#](x1) = [0]
[c_3](x1, x2) = [1] x2 + [0]
[add^#](x1, x2) = [2] x1 + [7] x2 + [0]
[c_4](x1) = [7] x1 + [7]
[c_5](x1) = [1] x1 + [0]
[len^#](x1) = [7] x1 + [0]
[c_6] = [0]
[c_7](x1) = [7] x1 + [0]
The following symbols are considered usable
{fst^#, from^#, add^#}
The order satisfies the following ordering constraints:
[fst^#(s(X), cons(Y, Z))] = [1] Z + [1] Y + [4]
> [1] Z + [1] Y + [1]
= [c_2(Y, fst^#(X, Z))]
[from^#(X)] = [0]
>= [0]
= [c_3(X, from^#(s(X)))]
[add^#(0(), X)] = [7] X + [8]
> [7] X + [7]
= [c_4(X)]
[add^#(s(X), Y)] = [2] X + [7] Y + [0]
>= [2] X + [7] Y + [0]
= [c_5(add^#(X, Y))]
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_3(X, from^#(s(X)))
, add^#(s(X), Y) -> c_5(add^#(X, Y)) }
Weak DPs:
{ fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z))
, add^#(0(), X) -> c_4(X) }
Obligation:
runtime complexity
Answer:
MAYBE
We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.
DPs:
{ 2: add^#(s(X), Y) -> c_5(add^#(X, Y))
, 3: fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z))
, 4: add^#(0(), X) -> c_4(X) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {2}, Uargs(c_3) = {2}, Uargs(c_5) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[fst](x1, x2) = [7] x1 + [7] x2 + [0]
[0] = [2]
[nil] = [0]
[s](x1) = [1] x1 + [2]
[cons](x1, x2) = [1] x1 + [1] x2 + [7]
[from](x1) = [7] x1 + [0]
[add](x1, x2) = [7] x1 + [7] x2 + [0]
[len](x1) = [7] x1 + [0]
[fst^#](x1, x2) = [4] x1 + [0]
[c_1] = [0]
[c_2](x1, x2) = [1] x2 + [1]
[from^#](x1) = [0]
[c_3](x1, x2) = [1] x2 + [0]
[add^#](x1, x2) = [4] x1 + [7] x2 + [0]
[c_4](x1) = [7] x1 + [7]
[c_5](x1) = [1] x1 + [1]
[len^#](x1) = [7] x1 + [0]
[c_6] = [0]
[c_7](x1) = [7] x1 + [0]
The following symbols are considered usable
{fst^#, from^#, add^#}
The order satisfies the following ordering constraints:
[fst^#(s(X), cons(Y, Z))] = [4] X + [8]
> [4] X + [1]
= [c_2(Y, fst^#(X, Z))]
[from^#(X)] = [0]
>= [0]
= [c_3(X, from^#(s(X)))]
[add^#(0(), X)] = [7] X + [8]
> [7] X + [7]
= [c_4(X)]
[add^#(s(X), Y)] = [4] X + [7] Y + [8]
> [4] X + [7] Y + [1]
= [c_5(add^#(X, Y))]
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_3(X, from^#(s(X))) }
Weak DPs:
{ fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z))
, add^#(0(), X) -> c_4(X)
, add^#(s(X), Y) -> c_5(add^#(X, 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) '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(s) = {1}, Uargs(cons) = {2}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[fst](x1, x2) = [1] x1 + [1] x2 + [0]
[0] = [4]
[nil] = [3]
[s](x1) = [1] x1 + [0]
[cons](x1, x2) = [1] x2 + [0]
[from](x1) = [1] x1 + [0]
[add](x1, x2) = [1] x2 + [0]
[len](x1) = [0]
The following symbols are considered usable
{fst, from, add, len}
The order satisfies the following ordering constraints:
[fst(0(), Z)] = [1] Z + [4]
> [3]
= [nil()]
[fst(s(X), cons(Y, Z))] = [1] Z + [1] X + [0]
>= [1] Z + [1] X + [0]
= [cons(Y, fst(X, Z))]
[from(X)] = [1] X + [0]
>= [1] X + [0]
= [cons(X, from(s(X)))]
[add(0(), X)] = [1] X + [0]
>= [1] X + [0]
= [X]
[add(s(X), Y)] = [1] Y + [0]
>= [1] Y + [0]
= [s(add(X, Y))]
[len(nil())] = [0]
? [4]
= [0()]
[len(cons(X, Z))] = [0]
>= [0]
= [s(len(Z))]
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:
{ fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
, from(X) -> cons(X, from(s(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, len(nil()) -> 0()
, len(cons(X, Z)) -> s(len(Z)) }
Weak Trs: { fst(0(), Z) -> nil() }
Obligation:
runtime complexity
Answer:
MAYBE
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(cons) = {2}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[fst](x1, x2) = [1] x2 + [0]
[0] = [4]
[nil] = [0]
[s](x1) = [1] x1 + [0]
[cons](x1, x2) = [1] x2 + [0]
[from](x1) = [1] x1 + [0]
[add](x1, x2) = [1] x1 + [1] x2 + [0]
[len](x1) = [0]
The following symbols are considered usable
{fst, from, add, len}
The order satisfies the following ordering constraints:
[fst(0(), Z)] = [1] Z + [0]
>= [0]
= [nil()]
[fst(s(X), cons(Y, Z))] = [1] Z + [0]
>= [1] Z + [0]
= [cons(Y, fst(X, Z))]
[from(X)] = [1] X + [0]
>= [1] X + [0]
= [cons(X, from(s(X)))]
[add(0(), X)] = [1] X + [4]
> [1] X + [0]
= [X]
[add(s(X), Y)] = [1] X + [1] Y + [0]
>= [1] X + [1] Y + [0]
= [s(add(X, Y))]
[len(nil())] = [0]
? [4]
= [0()]
[len(cons(X, Z))] = [0]
>= [0]
= [s(len(Z))]
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:
{ fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
, from(X) -> cons(X, from(s(X)))
, add(s(X), Y) -> s(add(X, Y))
, len(nil()) -> 0()
, len(cons(X, Z)) -> s(len(Z)) }
Weak Trs:
{ fst(0(), Z) -> nil()
, add(0(), X) -> X }
Obligation:
runtime complexity
Answer:
MAYBE
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(cons) = {2}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[fst](x1, x2) = [1] x2 + [1]
[0] = [0]
[nil] = [1]
[s](x1) = [1] x1 + [0]
[cons](x1, x2) = [1] x2 + [0]
[from](x1) = [1] x1 + [0]
[add](x1, x2) = [1] x1 + [1] x2 + [0]
[len](x1) = [1] x1 + [0]
The following symbols are considered usable
{fst, from, add, len}
The order satisfies the following ordering constraints:
[fst(0(), Z)] = [1] Z + [1]
>= [1]
= [nil()]
[fst(s(X), cons(Y, Z))] = [1] Z + [1]
>= [1] Z + [1]
= [cons(Y, fst(X, Z))]
[from(X)] = [1] X + [0]
>= [1] X + [0]
= [cons(X, from(s(X)))]
[add(0(), X)] = [1] X + [0]
>= [1] X + [0]
= [X]
[add(s(X), Y)] = [1] X + [1] Y + [0]
>= [1] X + [1] Y + [0]
= [s(add(X, Y))]
[len(nil())] = [1]
> [0]
= [0()]
[len(cons(X, Z))] = [1] Z + [0]
>= [1] Z + [0]
= [s(len(Z))]
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:
{ fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
, from(X) -> cons(X, from(s(X)))
, add(s(X), Y) -> s(add(X, Y))
, len(cons(X, Z)) -> s(len(Z)) }
Weak Trs:
{ fst(0(), Z) -> nil()
, add(0(), X) -> X
, len(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(s) = {1}, Uargs(cons) = {2}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[fst](x1, x2) = [1] x1 + [1] x2 + [1]
[0] = [3]
[nil] = [4]
[s](x1) = [1] x1 + [4]
[cons](x1, x2) = [1] x2 + [0]
[from](x1) = [0]
[add](x1, x2) = [1] x2 + [4]
[len](x1) = [1] x1 + [0]
The following symbols are considered usable
{fst, from, add, len}
The order satisfies the following ordering constraints:
[fst(0(), Z)] = [1] Z + [4]
>= [4]
= [nil()]
[fst(s(X), cons(Y, Z))] = [1] Z + [1] X + [5]
> [1] Z + [1] X + [1]
= [cons(Y, fst(X, Z))]
[from(X)] = [0]
>= [0]
= [cons(X, from(s(X)))]
[add(0(), X)] = [1] X + [4]
> [1] X + [0]
= [X]
[add(s(X), Y)] = [1] Y + [4]
? [1] Y + [8]
= [s(add(X, Y))]
[len(nil())] = [4]
> [3]
= [0()]
[len(cons(X, Z))] = [1] Z + [0]
? [1] Z + [4]
= [s(len(Z))]
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)))
, add(s(X), Y) -> s(add(X, Y))
, len(cons(X, Z)) -> s(len(Z)) }
Weak Trs:
{ fst(0(), Z) -> nil()
, fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
, add(0(), X) -> X
, len(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(s) = {1}, Uargs(cons) = {2}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[fst](x1, x2) = [1] x1 + [1] x2 + [0]
[0] = [0]
[nil] = [0]
[s](x1) = [1] x1 + [4]
[cons](x1, x2) = [1] x2 + [5]
[from](x1) = [0]
[add](x1, x2) = [1] x2 + [0]
[len](x1) = [1] x1 + [4]
The following symbols are considered usable
{fst, from, add, len}
The order satisfies the following ordering constraints:
[fst(0(), Z)] = [1] Z + [0]
>= [0]
= [nil()]
[fst(s(X), cons(Y, Z))] = [1] Z + [1] X + [9]
> [1] Z + [1] X + [5]
= [cons(Y, fst(X, Z))]
[from(X)] = [0]
? [5]
= [cons(X, from(s(X)))]
[add(0(), X)] = [1] X + [0]
>= [1] X + [0]
= [X]
[add(s(X), Y)] = [1] Y + [0]
? [1] Y + [4]
= [s(add(X, Y))]
[len(nil())] = [4]
> [0]
= [0()]
[len(cons(X, Z))] = [1] Z + [9]
> [1] Z + [8]
= [s(len(Z))]
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)))
, add(s(X), Y) -> s(add(X, Y)) }
Weak Trs:
{ fst(0(), Z) -> nil()
, fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
, add(0(), X) -> X
, len(nil()) -> 0()
, len(cons(X, Z)) -> s(len(Z)) }
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:
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(cons) = {2}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[fst](x1, x2) = [0 1] x2 + [0]
[0 1] [0]
[0] = [0]
[0]
[nil] = [0]
[0]
[s](x1) = [1 0] x1 + [0]
[0 1] [1]
[cons](x1, x2) = [1 0] x2 + [0]
[0 1] [1]
[from](x1) = [0]
[0]
[add](x1, x2) = [0 4] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[len](x1) = [0 1] x1 + [0]
[0 1] [0]
The following symbols are considered usable
{fst, from, add, len}
The order satisfies the following ordering constraints:
[fst(0(), Z)] = [0 1] Z + [0]
[0 1] [0]
>= [0]
[0]
= [nil()]
[fst(s(X), cons(Y, Z))] = [0 1] Z + [1]
[0 1] [1]
> [0 1] Z + [0]
[0 1] [1]
= [cons(Y, fst(X, Z))]
[from(X)] = [0]
[0]
? [0]
[1]
= [cons(X, from(s(X)))]
[add(0(), X)] = [1 0] X + [0]
[0 1] [0]
>= [1 0] X + [0]
[0 1] [0]
= [X]
[add(s(X), Y)] = [0 4] X + [1 0] Y + [4]
[0 1] [0 1] [1]
> [0 4] X + [1 0] Y + [0]
[0 1] [0 1] [1]
= [s(add(X, Y))]
[len(nil())] = [0]
[0]
>= [0]
[0]
= [0()]
[len(cons(X, Z))] = [0 1] Z + [1]
[0 1] [1]
> [0 1] Z + [0]
[0 1] [1]
= [s(len(Z))]
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:
{ fst(0(), Z) -> nil()
, fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, len(nil()) -> 0()
, len(cons(X, Z)) -> s(len(Z)) }
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) '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 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) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed
due to the following reason:
We add the following weak dependency pairs:
Strict DPs:
{ fst^#(0(), Z) -> c_1()
, fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z))
, from^#(X) -> c_3(X, from^#(s(X)))
, add^#(0(), X) -> c_4(X)
, add^#(s(X), Y) -> c_5(add^#(X, Y))
, len^#(nil()) -> c_6()
, len^#(cons(X, Z)) -> c_7(len^#(Z)) }
and mark the set of starting terms.
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ fst^#(0(), Z) -> c_1()
, fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z))
, from^#(X) -> c_3(X, from^#(s(X)))
, add^#(0(), X) -> c_4(X)
, add^#(s(X), Y) -> c_5(add^#(X, Y))
, len^#(nil()) -> c_6()
, len^#(cons(X, Z)) -> c_7(len^#(Z)) }
Strict Trs:
{ fst(0(), Z) -> nil()
, fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
, from(X) -> cons(X, from(s(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, len(nil()) -> 0()
, len(cons(X, Z)) -> s(len(Z)) }
Obligation:
runtime complexity
Answer:
MAYBE
We estimate the number of application of {1,6} by applications of
Pre({1,6}) = {2,3,4,7}. Here rules are labeled as follows:
DPs:
{ 1: fst^#(0(), Z) -> c_1()
, 2: fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z))
, 3: from^#(X) -> c_3(X, from^#(s(X)))
, 4: add^#(0(), X) -> c_4(X)
, 5: add^#(s(X), Y) -> c_5(add^#(X, Y))
, 6: len^#(nil()) -> c_6()
, 7: len^#(cons(X, Z)) -> c_7(len^#(Z)) }
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict DPs:
{ fst^#(s(X), cons(Y, Z)) -> c_2(Y, fst^#(X, Z))
, from^#(X) -> c_3(X, from^#(s(X)))
, add^#(0(), X) -> c_4(X)
, add^#(s(X), Y) -> c_5(add^#(X, Y))
, len^#(cons(X, Z)) -> c_7(len^#(Z)) }
Strict Trs:
{ fst(0(), Z) -> nil()
, fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z))
, from(X) -> cons(X, from(s(X)))
, add(0(), X) -> X
, add(s(X), Y) -> s(add(X, Y))
, len(nil()) -> 0()
, len(cons(X, Z)) -> s(len(Z)) }
Weak DPs:
{ fst^#(0(), Z) -> c_1()
, len^#(nil()) -> c_6() }
Obligation:
runtime complexity
Answer:
MAYBE
Empty strict component of the problem is NOT empty.
Arrrr..