We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict Trs:
{ sqr(0()) -> 0()
, sqr(s(x)) -> s(+(sqr(x), double(x)))
, sqr(s(x)) -> +(sqr(x), s(double(x)))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We add the following dependency tuples:
Strict DPs:
{ sqr^#(0()) -> c_1()
, sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x))
, sqr^#(s(x)) ->
c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x))
, +^#(x, 0()) -> c_4()
, +^#(x, s(y)) -> c_5(+^#(x, y))
, double^#(0()) -> c_6()
, double^#(s(x)) -> c_7(double^#(x)) }
and mark the set of starting terms.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ sqr^#(0()) -> c_1()
, sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x))
, sqr^#(s(x)) ->
c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x))
, +^#(x, 0()) -> c_4()
, +^#(x, s(y)) -> c_5(+^#(x, y))
, double^#(0()) -> c_6()
, double^#(s(x)) -> c_7(double^#(x)) }
Weak Trs:
{ sqr(0()) -> 0()
, sqr(s(x)) -> s(+(sqr(x), double(x)))
, sqr(s(x)) -> +(sqr(x), s(double(x)))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We estimate the number of application of {1,4,6} by applications of
Pre({1,4,6}) = {2,3,5,7}. Here rules are labeled as follows:
DPs:
{ 1: sqr^#(0()) -> c_1()
, 2: sqr^#(s(x)) ->
c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x))
, 3: sqr^#(s(x)) ->
c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x))
, 4: +^#(x, 0()) -> c_4()
, 5: +^#(x, s(y)) -> c_5(+^#(x, y))
, 6: double^#(0()) -> c_6()
, 7: double^#(s(x)) -> c_7(double^#(x)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x))
, sqr^#(s(x)) ->
c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x))
, +^#(x, s(y)) -> c_5(+^#(x, y))
, double^#(s(x)) -> c_7(double^#(x)) }
Weak DPs:
{ sqr^#(0()) -> c_1()
, +^#(x, 0()) -> c_4()
, double^#(0()) -> c_6() }
Weak Trs:
{ sqr(0()) -> 0()
, sqr(s(x)) -> s(+(sqr(x), double(x)))
, sqr(s(x)) -> +(sqr(x), s(double(x)))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ sqr^#(0()) -> c_1()
, +^#(x, 0()) -> c_4()
, double^#(0()) -> c_6() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x))
, sqr^#(s(x)) ->
c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x))
, +^#(x, s(y)) -> c_5(+^#(x, y))
, double^#(s(x)) -> c_7(double^#(x)) }
Weak Trs:
{ sqr(0()) -> 0()
, sqr(s(x)) -> s(+(sqr(x), double(x)))
, sqr(s(x)) -> +(sqr(x), s(double(x)))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^2))
We decompose the input problem according to the dependency graph
into the upper component
{ sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x))
, sqr^#(s(x)) ->
c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x)) }
and lower component
{ +^#(x, s(y)) -> c_5(+^#(x, y))
, double^#(s(x)) -> c_7(double^#(x)) }
Further, following extension rules are added to the lower
component.
{ sqr^#(s(x)) -> sqr^#(x)
, sqr^#(s(x)) -> +^#(sqr(x), s(double(x)))
, sqr^#(s(x)) -> +^#(sqr(x), double(x))
, sqr^#(s(x)) -> double^#(x) }
TcT solves the upper component with certificate YES(O(1),O(n^1)).
Sub-proof:
----------
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x))
, sqr^#(s(x)) ->
c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x)) }
Weak Trs:
{ sqr(0()) -> 0()
, sqr(s(x)) -> s(+(sqr(x), double(x)))
, sqr(s(x)) -> +(sqr(x), s(double(x)))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'Small Polynomial Path Order (PS,1-bounded)'
to orient following rules strictly.
DPs:
{ 1: sqr^#(s(x)) ->
c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x))
, 2: sqr^#(s(x)) ->
c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x)) }
Sub-proof:
----------
The input was oriented with the instance of 'Small Polynomial Path
Order (PS,1-bounded)' as induced by the safe mapping
safe(sqr) = {}, safe(0) = {}, safe(s) = {1}, safe(+) = {1, 2},
safe(double) = {}, safe(sqr^#) = {}, safe(c_2) = {},
safe(+^#) = {}, safe(double^#) = {}, safe(c_3) = {}
and precedence
empty .
Following symbols are considered recursive:
{sqr^#}
The recursion depth is 1.
Further, following argument filtering is employed:
pi(sqr) = [], pi(0) = [], pi(s) = [1], pi(+) = [1, 2],
pi(double) = [], pi(sqr^#) = [1], pi(c_2) = [1, 2, 3],
pi(+^#) = [], pi(double^#) = [], pi(c_3) = [1, 2, 3]
Usable defined function symbols are a subset of:
{sqr^#, +^#, double^#}
For your convenience, here are the satisfied ordering constraints:
pi(sqr^#(s(x))) = sqr^#(s(; x);)
> c_2(+^#(), sqr^#(x;), double^#();)
= pi(c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x)))
pi(sqr^#(s(x))) = sqr^#(s(; x);)
> c_3(+^#(), sqr^#(x;), double^#();)
= pi(c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x)))
The strictly oriented rules are moved into the weak component.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak DPs:
{ sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x))
, sqr^#(s(x)) ->
c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x)) }
Weak Trs:
{ sqr(0()) -> 0()
, sqr(s(x)) -> s(+(sqr(x), double(x)))
, sqr(s(x)) -> +(sqr(x), s(double(x)))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ sqr^#(s(x)) -> c_2(+^#(sqr(x), double(x)), sqr^#(x), double^#(x))
, sqr^#(s(x)) ->
c_3(+^#(sqr(x), s(double(x))), sqr^#(x), double^#(x)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak Trs:
{ sqr(0()) -> 0()
, sqr(s(x)) -> s(+(sqr(x), double(x)))
, sqr(s(x)) -> +(sqr(x), s(double(x)))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
No rule is usable, rules are removed from the input problem.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Rules: Empty
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
We return to the main proof.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ +^#(x, s(y)) -> c_5(+^#(x, y))
, double^#(s(x)) -> c_7(double^#(x)) }
Weak DPs:
{ sqr^#(s(x)) -> sqr^#(x)
, sqr^#(s(x)) -> +^#(sqr(x), s(double(x)))
, sqr^#(s(x)) -> +^#(sqr(x), double(x))
, sqr^#(s(x)) -> double^#(x) }
Weak Trs:
{ sqr(0()) -> 0()
, sqr(s(x)) -> s(+(sqr(x), double(x)))
, sqr(s(x)) -> +(sqr(x), s(double(x)))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.
DPs:
{ 2: double^#(s(x)) -> c_7(double^#(x))
, 3: sqr^#(s(x)) -> sqr^#(x)
, 4: sqr^#(s(x)) -> +^#(sqr(x), s(double(x)))
, 5: sqr^#(s(x)) -> +^#(sqr(x), double(x))
, 6: sqr^#(s(x)) -> double^#(x) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_5) = {1}, Uargs(c_7) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[sqr](x1) = [0]
[0] = [0]
[s](x1) = [1] x1 + [4]
[+](x1, x2) = [0]
[double](x1) = [0]
[sqr^#](x1) = [2] x1 + [0]
[+^#](x1, x2) = [0]
[double^#](x1) = [2] x1 + [0]
[c_5](x1) = [1] x1 + [0]
[c_7](x1) = [1] x1 + [1]
The order satisfies the following ordering constraints:
[sqr(0())] = [0]
>= [0]
= [0()]
[sqr(s(x))] = [0]
? [4]
= [s(+(sqr(x), double(x)))]
[sqr(s(x))] = [0]
>= [0]
= [+(sqr(x), s(double(x)))]
[+(x, 0())] = [0]
? [1] x + [0]
= [x]
[+(x, s(y))] = [0]
? [4]
= [s(+(x, y))]
[double(0())] = [0]
>= [0]
= [0()]
[double(s(x))] = [0]
? [8]
= [s(s(double(x)))]
[sqr^#(s(x))] = [2] x + [8]
> [2] x + [0]
= [sqr^#(x)]
[sqr^#(s(x))] = [2] x + [8]
> [0]
= [+^#(sqr(x), s(double(x)))]
[sqr^#(s(x))] = [2] x + [8]
> [0]
= [+^#(sqr(x), double(x))]
[sqr^#(s(x))] = [2] x + [8]
> [2] x + [0]
= [double^#(x)]
[+^#(x, s(y))] = [0]
>= [0]
= [c_5(+^#(x, y))]
[double^#(s(x))] = [2] x + [8]
> [2] x + [1]
= [c_7(double^#(x))]
The strictly oriented rules are moved into the weak component.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { +^#(x, s(y)) -> c_5(+^#(x, y)) }
Weak DPs:
{ sqr^#(s(x)) -> sqr^#(x)
, sqr^#(s(x)) -> +^#(sqr(x), s(double(x)))
, sqr^#(s(x)) -> +^#(sqr(x), double(x))
, sqr^#(s(x)) -> double^#(x)
, double^#(s(x)) -> c_7(double^#(x)) }
Weak Trs:
{ sqr(0()) -> 0()
, sqr(s(x)) -> s(+(sqr(x), double(x)))
, sqr(s(x)) -> +(sqr(x), s(double(x)))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ sqr^#(s(x)) -> double^#(x)
, double^#(s(x)) -> c_7(double^#(x)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { +^#(x, s(y)) -> c_5(+^#(x, y)) }
Weak DPs:
{ sqr^#(s(x)) -> sqr^#(x)
, sqr^#(s(x)) -> +^#(sqr(x), s(double(x)))
, sqr^#(s(x)) -> +^#(sqr(x), double(x)) }
Weak Trs:
{ sqr(0()) -> 0()
, sqr(s(x)) -> s(+(sqr(x), double(x)))
, sqr(s(x)) -> +(sqr(x), s(double(x)))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.
DPs:
{ 1: +^#(x, s(y)) -> c_5(+^#(x, y))
, 2: sqr^#(s(x)) -> sqr^#(x)
, 3: sqr^#(s(x)) -> +^#(sqr(x), s(double(x)))
, 4: sqr^#(s(x)) -> +^#(sqr(x), double(x)) }
Trs:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x))) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_5) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[sqr](x1) = [0]
[0] = [0]
[s](x1) = [1] x1 + [1]
[+](x1, x2) = [0]
[double](x1) = [3] x1 + [1]
[sqr^#](x1) = [3] x1 + [5]
[+^#](x1, x2) = [1] x2 + [5]
[double^#](x1) = [0]
[c_5](x1) = [1] x1 + [0]
[c_7](x1) = [0]
The order satisfies the following ordering constraints:
[sqr(0())] = [0]
>= [0]
= [0()]
[sqr(s(x))] = [0]
? [1]
= [s(+(sqr(x), double(x)))]
[sqr(s(x))] = [0]
>= [0]
= [+(sqr(x), s(double(x)))]
[+(x, 0())] = [0]
? [1] x + [0]
= [x]
[+(x, s(y))] = [0]
? [1]
= [s(+(x, y))]
[double(0())] = [1]
> [0]
= [0()]
[double(s(x))] = [3] x + [4]
> [3] x + [3]
= [s(s(double(x)))]
[sqr^#(s(x))] = [3] x + [8]
> [3] x + [5]
= [sqr^#(x)]
[sqr^#(s(x))] = [3] x + [8]
> [3] x + [7]
= [+^#(sqr(x), s(double(x)))]
[sqr^#(s(x))] = [3] x + [8]
> [3] x + [6]
= [+^#(sqr(x), double(x))]
[+^#(x, s(y))] = [1] y + [6]
> [1] y + [5]
= [c_5(+^#(x, y))]
The strictly oriented rules are moved into the weak component.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak DPs:
{ sqr^#(s(x)) -> sqr^#(x)
, sqr^#(s(x)) -> +^#(sqr(x), s(double(x)))
, sqr^#(s(x)) -> +^#(sqr(x), double(x))
, +^#(x, s(y)) -> c_5(+^#(x, y)) }
Weak Trs:
{ sqr(0()) -> 0()
, sqr(s(x)) -> s(+(sqr(x), double(x)))
, sqr(s(x)) -> +(sqr(x), s(double(x)))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ sqr^#(s(x)) -> sqr^#(x)
, sqr^#(s(x)) -> +^#(sqr(x), s(double(x)))
, sqr^#(s(x)) -> +^#(sqr(x), double(x))
, +^#(x, s(y)) -> c_5(+^#(x, y)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak Trs:
{ sqr(0()) -> 0()
, sqr(s(x)) -> s(+(sqr(x), double(x)))
, sqr(s(x)) -> +(sqr(x), s(double(x)))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, double(0()) -> 0()
, double(s(x)) -> s(s(double(x))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
No rule is usable, rules are removed from the input problem.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Rules: Empty
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^2))