MAYBE
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict Trs:
{ +(0(), y) -> y
, +(s(x), y) -> s(+(x, y))
, +(p(x), y) -> p(+(x, y))
, minus(0()) -> 0()
, minus(s(x)) -> p(minus(x))
, minus(p(x)) -> s(minus(x))
, *(0(), y) -> 0()
, *(s(x), y) -> +(*(x, y), y)
, *(p(x), y) -> +(*(x, y), minus(y)) }
Obligation:
innermost 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) 'Best' failed due to the following reason:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'WithProblem (timeout of 60 seconds)' failed due to the
following reason:
Computation stopped due to timeout after 60.0 seconds.
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:
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:
We use the processor 'polynomial interpretation' to orient
following rules strictly.
Trs:
{ *(s(x), y) -> +(*(x, y), y)
, *(p(x), y) -> +(*(x, y), minus(y)) }
The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^2)) . These rules are moved into the corresponding weak
component(s).
Sub-proof:
----------
The following argument positions are considered usable:
Uargs(+) = {1, 2}, Uargs(s) = {1}, Uargs(p) = {1}
TcT has computed the following constructor-restricted polynomial
interpretation.
[+](x1, x2) = x1 + x2
[0]() = 0
[s](x1) = 1 + x1
[p](x1) = 1 + x1
[minus](x1) = x1
[*](x1, x2) = x1*x2 + x1^2
The following symbols are considered usable
{+, minus, *}
This order satisfies the following ordering constraints.
[+(0(), y)] = y
>= y
= [y]
[+(s(x), y)] = 1 + x + y
>= 1 + x + y
= [s(+(x, y))]
[+(p(x), y)] = 1 + x + y
>= 1 + x + y
= [p(+(x, y))]
[minus(0())] =
>=
= [0()]
[minus(s(x))] = 1 + x
>= 1 + x
= [p(minus(x))]
[minus(p(x))] = 1 + x
>= 1 + x
= [s(minus(x))]
[*(0(), y)] =
>=
= [0()]
[*(s(x), y)] = y + x*y + 1 + 2*x + x^2
> x*y + x^2 + y
= [+(*(x, y), y)]
[*(p(x), y)] = y + x*y + 1 + 2*x + x^2
> x*y + x^2 + y
= [+(*(x, y), minus(y))]
We return to the main proof.
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict Trs:
{ +(0(), y) -> y
, +(s(x), y) -> s(+(x, y))
, +(p(x), y) -> p(+(x, y))
, minus(0()) -> 0()
, minus(s(x)) -> p(minus(x))
, minus(p(x)) -> s(minus(x))
, *(0(), y) -> 0() }
Weak Trs:
{ *(s(x), y) -> +(*(x, y), y)
, *(p(x), y) -> +(*(x, y), minus(y)) }
Obligation:
innermost runtime complexity
Answer:
MAYBE
We use the processor 'polynomial interpretation' to orient
following rules strictly.
Trs: { *(0(), y) -> 0() }
The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^2)) . These rules are moved into the corresponding weak
component(s).
Sub-proof:
----------
The following argument positions are considered usable:
Uargs(+) = {1, 2}, Uargs(s) = {1}, Uargs(p) = {1}
TcT has computed the following constructor-restricted polynomial
interpretation.
[+](x1, x2) = x1 + x2
[0]() = 0
[s](x1) = 1 + x1
[p](x1) = 1 + x1
[minus](x1) = x1
[*](x1, x2) = 1 + x1*x2
The following symbols are considered usable
{+, minus, *}
This order satisfies the following ordering constraints.
[+(0(), y)] = y
>= y
= [y]
[+(s(x), y)] = 1 + x + y
>= 1 + x + y
= [s(+(x, y))]
[+(p(x), y)] = 1 + x + y
>= 1 + x + y
= [p(+(x, y))]
[minus(0())] =
>=
= [0()]
[minus(s(x))] = 1 + x
>= 1 + x
= [p(minus(x))]
[minus(p(x))] = 1 + x
>= 1 + x
= [s(minus(x))]
[*(0(), y)] = 1
>
= [0()]
[*(s(x), y)] = 1 + y + x*y
>= 1 + x*y + y
= [+(*(x, y), y)]
[*(p(x), y)] = 1 + y + x*y
>= 1 + x*y + y
= [+(*(x, y), minus(y))]
We return to the main proof.
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict Trs:
{ +(0(), y) -> y
, +(s(x), y) -> s(+(x, y))
, +(p(x), y) -> p(+(x, y))
, minus(0()) -> 0()
, minus(s(x)) -> p(minus(x))
, minus(p(x)) -> s(minus(x)) }
Weak Trs:
{ *(0(), y) -> 0()
, *(s(x), y) -> +(*(x, y), y)
, *(p(x), y) -> +(*(x, y), minus(y)) }
Obligation:
innermost runtime complexity
Answer:
MAYBE
We use the processor 'polynomial interpretation' to orient
following rules strictly.
Trs: { +(0(), y) -> y }
The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^2)) . These rules are moved into the corresponding weak
component(s).
Sub-proof:
----------
The following argument positions are considered usable:
Uargs(+) = {1, 2}, Uargs(s) = {1}, Uargs(p) = {1}
TcT has computed the following constructor-restricted polynomial
interpretation.
[+](x1, x2) = 3 + x1 + x2
[0]() = 0
[s](x1) = 1 + x1
[p](x1) = 1 + x1
[minus](x1) = x1
[*](x1, x2) = 2*x1 + x1*x2 + 2*x1^2
The following symbols are considered usable
{+, minus, *}
This order satisfies the following ordering constraints.
[+(0(), y)] = 3 + y
> y
= [y]
[+(s(x), y)] = 4 + x + y
>= 4 + x + y
= [s(+(x, y))]
[+(p(x), y)] = 4 + x + y
>= 4 + x + y
= [p(+(x, y))]
[minus(0())] =
>=
= [0()]
[minus(s(x))] = 1 + x
>= 1 + x
= [p(minus(x))]
[minus(p(x))] = 1 + x
>= 1 + x
= [s(minus(x))]
[*(0(), y)] =
>=
= [0()]
[*(s(x), y)] = 4 + 6*x + y + x*y + 2*x^2
> 3 + 2*x + x*y + 2*x^2 + y
= [+(*(x, y), y)]
[*(p(x), y)] = 4 + 6*x + y + x*y + 2*x^2
> 3 + 2*x + x*y + 2*x^2 + y
= [+(*(x, y), minus(y))]
We return to the main proof.
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict Trs:
{ +(s(x), y) -> s(+(x, y))
, +(p(x), y) -> p(+(x, y))
, minus(0()) -> 0()
, minus(s(x)) -> p(minus(x))
, minus(p(x)) -> s(minus(x)) }
Weak Trs:
{ +(0(), y) -> y
, *(0(), y) -> 0()
, *(s(x), y) -> +(*(x, y), y)
, *(p(x), y) -> +(*(x, y), minus(y)) }
Obligation:
innermost runtime complexity
Answer:
MAYBE
We use the processor 'polynomial interpretation' to orient
following rules strictly.
Trs:
{ minus(0()) -> 0()
, minus(p(x)) -> s(minus(x)) }
The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^2)) . These rules are moved into the corresponding weak
component(s).
Sub-proof:
----------
The following argument positions are considered usable:
Uargs(+) = {1, 2}, Uargs(s) = {1}, Uargs(p) = {1}
TcT has computed the following constructor-restricted polynomial
interpretation.
[+](x1, x2) = x1 + 2*x2
[0]() = 2
[s](x1) = 1 + x1
[p](x1) = 2 + x1
[minus](x1) = 2*x1
[*](x1, x2) = 2*x1 + 2*x1*x2 + 2*x2
The following symbols are considered usable
{+, minus, *}
This order satisfies the following ordering constraints.
[+(0(), y)] = 2 + 2*y
> y
= [y]
[+(s(x), y)] = 1 + x + 2*y
>= 1 + x + 2*y
= [s(+(x, y))]
[+(p(x), y)] = 2 + x + 2*y
>= 2 + x + 2*y
= [p(+(x, y))]
[minus(0())] = 4
> 2
= [0()]
[minus(s(x))] = 2 + 2*x
>= 2 + 2*x
= [p(minus(x))]
[minus(p(x))] = 4 + 2*x
> 1 + 2*x
= [s(minus(x))]
[*(0(), y)] = 4 + 6*y
> 2
= [0()]
[*(s(x), y)] = 2 + 2*x + 4*y + 2*x*y
> 2*x + 2*x*y + 4*y
= [+(*(x, y), y)]
[*(p(x), y)] = 4 + 2*x + 6*y + 2*x*y
> 2*x + 2*x*y + 6*y
= [+(*(x, y), minus(y))]
We return to the main proof.
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict Trs:
{ +(s(x), y) -> s(+(x, y))
, +(p(x), y) -> p(+(x, y))
, minus(s(x)) -> p(minus(x)) }
Weak Trs:
{ +(0(), y) -> y
, minus(0()) -> 0()
, minus(p(x)) -> s(minus(x))
, *(0(), y) -> 0()
, *(s(x), y) -> +(*(x, y), y)
, *(p(x), y) -> +(*(x, y), minus(y)) }
Obligation:
innermost runtime complexity
Answer:
MAYBE
We use the processor 'polynomial interpretation' to orient
following rules strictly.
Trs: { minus(s(x)) -> p(minus(x)) }
The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^2)) . These rules are moved into the corresponding weak
component(s).
Sub-proof:
----------
The following argument positions are considered usable:
Uargs(+) = {1, 2}, Uargs(s) = {1}, Uargs(p) = {1}
TcT has computed the following constructor-restricted polynomial
interpretation.
[+](x1, x2) = 2 + x1 + x2
[0]() = 0
[s](x1) = 2 + x1
[p](x1) = 2 + x1
[minus](x1) = 2*x1
[*](x1, x2) = x1 + 2*x1*x2
The following symbols are considered usable
{+, minus, *}
This order satisfies the following ordering constraints.
[+(0(), y)] = 2 + y
> y
= [y]
[+(s(x), y)] = 4 + x + y
>= 4 + x + y
= [s(+(x, y))]
[+(p(x), y)] = 4 + x + y
>= 4 + x + y
= [p(+(x, y))]
[minus(0())] =
>=
= [0()]
[minus(s(x))] = 4 + 2*x
> 2 + 2*x
= [p(minus(x))]
[minus(p(x))] = 4 + 2*x
> 2 + 2*x
= [s(minus(x))]
[*(0(), y)] =
>=
= [0()]
[*(s(x), y)] = 2 + x + 4*y + 2*x*y
>= 2 + x + 2*x*y + y
= [+(*(x, y), y)]
[*(p(x), y)] = 2 + x + 4*y + 2*x*y
>= 2 + x + 2*x*y + 2*y
= [+(*(x, y), minus(y))]
We return to the main proof.
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict Trs:
{ +(s(x), y) -> s(+(x, y))
, +(p(x), y) -> p(+(x, y)) }
Weak Trs:
{ +(0(), y) -> y
, minus(0()) -> 0()
, minus(s(x)) -> p(minus(x))
, minus(p(x)) -> s(minus(x))
, *(0(), y) -> 0()
, *(s(x), y) -> +(*(x, y), y)
, *(p(x), y) -> +(*(x, y), minus(y)) }
Obligation:
innermost 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:
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:
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) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due
to the following reason:
The input cannot be shown compatible
2) 'bsearch-popstar (timeout of 60 seconds)' failed due to the
following reason:
The input cannot be shown compatible
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.
Arrrr..