We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ f(x, c(y)) -> f(x, s(f(y, y)))
, f(s(x), s(y)) -> f(x, s(c(s(y)))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following dependency tuples:
Strict DPs:
{ f^#(x, c(y)) -> c_1(f^#(x, s(f(y, y))), f^#(y, y))
, f^#(s(x), s(y)) -> c_2(f^#(x, s(c(s(y))))) }
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:
{ f^#(x, c(y)) -> c_1(f^#(x, s(f(y, y))), f^#(y, y))
, f^#(s(x), s(y)) -> c_2(f^#(x, s(c(s(y))))) }
Weak Trs:
{ f(x, c(y)) -> f(x, s(f(y, y)))
, f(s(x), s(y)) -> f(x, s(c(s(y)))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'matrix interpretation of dimension 2' to
orient following rules strictly.
DPs:
{ 2: f^#(s(x), s(y)) -> c_2(f^#(x, s(c(s(y))))) }
Trs: { f(x, c(y)) -> f(x, s(f(y, y))) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_1) = {1, 2}, Uargs(c_2) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA) and not(IDA(1)).
[f](x1, x2) = [0 0] x1 + [1 0] x2 + [1]
[0 2] [0 0] [0]
[c](x1) = [1 7] x1 + [6]
[0 0] [0]
[s](x1) = [0 1] x1 + [2]
[0 1] [3]
[f^#](x1, x2) = [0 1] x1 + [2 2] x2 + [2]
[0 0] [1 0] [0]
[c_1](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 0] [0]
[c_2](x1) = [1 1] x1 + [0]
[0 0] [0]
The order satisfies the following ordering constraints:
[f(x, c(y))] = [0 0] x + [1 7] y + [7]
[0 2] [0 0] [0]
> [0 0] x + [0 2] y + [3]
[0 2] [0 0] [0]
= [f(x, s(f(y, y)))]
[f(s(x), s(y))] = [0 0] x + [0 1] y + [3]
[0 2] [0 0] [6]
>= [0 0] x + [3]
[0 2] [0]
= [f(x, s(c(s(y))))]
[f^#(x, c(y))] = [0 1] x + [2 14] y + [14]
[0 0] [1 7] [6]
>= [0 1] x + [2 11] y + [14]
[0 0] [0 0] [0]
= [c_1(f^#(x, s(f(y, y))), f^#(y, y))]
[f^#(s(x), s(y))] = [0 1] x + [0 4] y + [15]
[0 0] [0 1] [2]
> [0 1] x + [14]
[0 0] [0]
= [c_2(f^#(x, s(c(s(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(n^1)).
Strict DPs: { f^#(x, c(y)) -> c_1(f^#(x, s(f(y, y))), f^#(y, y)) }
Weak DPs: { f^#(s(x), s(y)) -> c_2(f^#(x, s(c(s(y))))) }
Weak Trs:
{ f(x, c(y)) -> f(x, s(f(y, y)))
, f(s(x), s(y)) -> f(x, s(c(s(y)))) }
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.
{ f^#(s(x), s(y)) -> c_2(f^#(x, s(c(s(y))))) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { f^#(x, c(y)) -> c_1(f^#(x, s(f(y, y))), f^#(y, y)) }
Weak Trs:
{ f(x, c(y)) -> f(x, s(f(y, y)))
, f(s(x), s(y)) -> f(x, s(c(s(y)))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:
{ f^#(x, c(y)) -> c_1(f^#(x, s(f(y, y))), f^#(y, y)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { f^#(x, c(y)) -> c_1(f^#(y, y)) }
Weak Trs:
{ f(x, c(y)) -> f(x, s(f(y, y)))
, f(s(x), s(y)) -> f(x, s(c(s(y)))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^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(n^1)).
Strict DPs: { f^#(x, c(y)) -> c_1(f^#(y, y)) }
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: f^#(x, c(y)) -> c_1(f^#(y, y)) }
Sub-proof:
----------
The input was oriented with the instance of 'Small Polynomial Path
Order (PS,1-bounded)' as induced by the safe mapping
safe(f) = {}, safe(c) = {1}, safe(s) = {1}, safe(f^#) = {1},
safe(c_1) = {}, safe(c_2) = {}, safe(c) = {}, safe(c_1) = {}
and precedence
empty .
Following symbols are considered recursive:
{f^#}
The recursion depth is 1.
Further, following argument filtering is employed:
pi(f) = [], pi(c) = [1], pi(s) = [], pi(f^#) = [2], pi(c_1) = [],
pi(c_2) = [], pi(c) = [], pi(c_1) = [1]
Usable defined function symbols are a subset of:
{f^#}
For your convenience, here are the satisfied ordering constraints:
pi(f^#(x, c(y))) = f^#(c(; y);)
> c_1(f^#(y;);)
= pi(c_1(f^#(y, 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: { f^#(x, c(y)) -> c_1(f^#(y, y)) }
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.
{ f^#(x, c(y)) -> c_1(f^#(y, y)) }
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^1))