We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ p(s(x)) -> x
, fac(s(x)) -> times(s(x), fac(p(s(x))))
, fac(0()) -> s(0()) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ p^#(s(x)) -> c_1(x)
, fac^#(s(x)) -> c_2(x, fac^#(p(s(x))))
, fac^#(0()) -> c_3() }
and mark the set of starting terms.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ p^#(s(x)) -> c_1(x)
, fac^#(s(x)) -> c_2(x, fac^#(p(s(x))))
, fac^#(0()) -> c_3() }
Strict Trs:
{ p(s(x)) -> x
, fac(s(x)) -> times(s(x), fac(p(s(x))))
, fac(0()) -> s(0()) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We replace rewrite rules by usable rules:
Strict Usable Rules: { p(s(x)) -> x }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ p^#(s(x)) -> c_1(x)
, fac^#(s(x)) -> c_2(x, fac^#(p(s(x))))
, fac^#(0()) -> c_3() }
Strict Trs: { p(s(x)) -> x }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following constant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(fac^#) = {1}, Uargs(c_2) = {2}
TcT has computed the following constructor-restricted matrix
interpretation.
[p](x1) = [1 0] x1 + [0]
[0 1] [0]
[s](x1) = [1 0] x1 + [2]
[0 1] [0]
[0] = [0]
[0]
[p^#](x1) = [0]
[0]
[c_1](x1) = [0]
[0]
[fac^#](x1) = [2 0] x1 + [0]
[0 0] [0]
[c_2](x1, x2) = [1 0] x2 + [0]
[0 1] [0]
[c_3] = [0]
[0]
The order satisfies the following ordering constraints:
[p(s(x))] = [1 0] x + [2]
[0 1] [0]
> [1 0] x + [0]
[0 1] [0]
= [x]
[p^#(s(x))] = [0]
[0]
>= [0]
[0]
= [c_1(x)]
[fac^#(s(x))] = [2 0] x + [4]
[0 0] [0]
>= [2 0] x + [4]
[0 0] [0]
= [c_2(x, fac^#(p(s(x))))]
[fac^#(0())] = [0]
[0]
>= [0]
[0]
= [c_3()]
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 YES(O(1),O(n^1)).
Strict DPs:
{ p^#(s(x)) -> c_1(x)
, fac^#(s(x)) -> c_2(x, fac^#(p(s(x))))
, fac^#(0()) -> c_3() }
Weak Trs: { p(s(x)) -> x }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We estimate the number of application of {3} by applications of
Pre({3}) = {1,2}. Here rules are labeled as follows:
DPs:
{ 1: p^#(s(x)) -> c_1(x)
, 2: fac^#(s(x)) -> c_2(x, fac^#(p(s(x))))
, 3: fac^#(0()) -> c_3() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ p^#(s(x)) -> c_1(x)
, fac^#(s(x)) -> c_2(x, fac^#(p(s(x)))) }
Weak DPs: { fac^#(0()) -> c_3() }
Weak Trs: { p(s(x)) -> x }
Obligation:
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.
{ fac^#(0()) -> c_3() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ p^#(s(x)) -> c_1(x)
, fac^#(s(x)) -> c_2(x, fac^#(p(s(x)))) }
Weak Trs: { p(s(x)) -> x }
Obligation:
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: p^#(s(x)) -> c_1(x) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {2}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[p](x1) = [0]
[s](x1) = [0]
[p^#](x1) = [1] x1 + [6]
[c_1](x1) = [1]
[fac^#](x1) = [0]
[c_2](x1, x2) = [4] x2 + [0]
The order satisfies the following ordering constraints:
[p(s(x))] = [0]
? [1] x + [0]
= [x]
[p^#(s(x))] = [6]
> [1]
= [c_1(x)]
[fac^#(s(x))] = [0]
>= [0]
= [c_2(x, fac^#(p(s(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: { fac^#(s(x)) -> c_2(x, fac^#(p(s(x)))) }
Weak DPs: { p^#(s(x)) -> c_1(x) }
Weak Trs: { p(s(x)) -> x }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'matrix interpretation of dimension 3' to
orient following rules strictly.
DPs:
{ 1: fac^#(s(x)) -> c_2(x, fac^#(p(s(x)))) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {2}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA) and not(IDA(1)).
[0 0 1] [0]
[p](x1) = [0 1 0] x1 + [0]
[0 3 1] [0]
[1 0 0] [1]
[s](x1) = [0 1 5] x1 + [1]
[1 0 0] [0]
[0]
[p^#](x1) = [0]
[0]
[0]
[c_1](x1) = [0]
[0]
[5 0 0] [4]
[fac^#](x1) = [0 1 1] x1 + [0]
[0 0 0] [3]
[1 0 1] [0]
[c_2](x1, x2) = [0 0 0] x2 + [0]
[0 0 0] [0]
The order satisfies the following ordering constraints:
[p(s(x))] = [1 0 0] [0]
[0 1 5] x + [1]
[1 3 15] [3]
>= [1 0 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= [x]
[p^#(s(x))] = [0]
[0]
[0]
>= [0]
[0]
[0]
= [c_1(x)]
[fac^#(s(x))] = [5 0 0] [9]
[1 1 5] x + [1]
[0 0 0] [3]
> [5 0 0] [7]
[0 0 0] x + [0]
[0 0 0] [0]
= [c_2(x, fac^#(p(s(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:
{ p^#(s(x)) -> c_1(x)
, fac^#(s(x)) -> c_2(x, fac^#(p(s(x)))) }
Weak Trs: { p(s(x)) -> x }
Obligation:
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.
{ p^#(s(x)) -> c_1(x)
, fac^#(s(x)) -> c_2(x, fac^#(p(s(x)))) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak Trs: { p(s(x)) -> x }
Obligation:
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:
runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^1))