YES(O(1),O(n^1))

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict Trs:
  { 0(1(2(3(4(x1))))) -> 0(2(3(1(4(x1)))))
  , 0(5(1(2(3(4(x1)))))) -> 0(1(2(5(3(4(x1))))))
  , 0(5(1(2(3(4(x1)))))) -> 0(5(2(1(3(4(x1))))))
  , 0(5(1(2(3(4(x1)))))) -> 5(0(2(3(1(4(x1))))))
  , 0(5(2(3(1(4(x1)))))) -> 0(1(5(2(3(4(x1)))))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add following weak dependency pairs:

Strict DPs:
  { 0^#(1(2(3(4(x1))))) -> c_1(0^#(2(3(1(4(x1))))))
  , 0^#(5(1(2(3(4(x1)))))) -> c_2(0^#(1(2(5(3(4(x1)))))))
  , 0^#(5(1(2(3(4(x1)))))) -> c_3(0^#(5(2(1(3(4(x1)))))))
  , 0^#(5(1(2(3(4(x1)))))) -> c_4(0^#(2(3(1(4(x1))))))
  , 0^#(5(2(3(1(4(x1)))))) -> c_5(0^#(1(5(2(3(4(x1))))))) }

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:
  { 0^#(1(2(3(4(x1))))) -> c_1(0^#(2(3(1(4(x1))))))
  , 0^#(5(1(2(3(4(x1)))))) -> c_2(0^#(1(2(5(3(4(x1)))))))
  , 0^#(5(1(2(3(4(x1)))))) -> c_3(0^#(5(2(1(3(4(x1)))))))
  , 0^#(5(1(2(3(4(x1)))))) -> c_4(0^#(2(3(1(4(x1))))))
  , 0^#(5(2(3(1(4(x1)))))) -> c_5(0^#(1(5(2(3(4(x1))))))) }
Strict Trs:
  { 0(1(2(3(4(x1))))) -> 0(2(3(1(4(x1)))))
  , 0(5(1(2(3(4(x1)))))) -> 0(1(2(5(3(4(x1))))))
  , 0(5(1(2(3(4(x1)))))) -> 0(5(2(1(3(4(x1))))))
  , 0(5(1(2(3(4(x1)))))) -> 5(0(2(3(1(4(x1))))))
  , 0(5(2(3(1(4(x1)))))) -> 0(1(5(2(3(4(x1)))))) }
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:
  { 0^#(1(2(3(4(x1))))) -> c_1(0^#(2(3(1(4(x1))))))
  , 0^#(5(1(2(3(4(x1)))))) -> c_2(0^#(1(2(5(3(4(x1)))))))
  , 0^#(5(1(2(3(4(x1)))))) -> c_3(0^#(5(2(1(3(4(x1)))))))
  , 0^#(5(1(2(3(4(x1)))))) -> c_4(0^#(2(3(1(4(x1))))))
  , 0^#(5(2(3(1(4(x1)))))) -> c_5(0^#(1(5(2(3(4(x1))))))) }
Obligation:
  innermost 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:
  none

TcT has computed following constructor-restricted matrix
interpretation.

    [1](x1) = [1] x1 + [0]
                          
    [2](x1) = [1] x1 + [0]
                          
    [3](x1) = [1] x1 + [0]
                          
    [4](x1) = [1] x1 + [2]
                          
    [5](x1) = [0]         
                          
  [0^#](x1) = [2] x1 + [0]
                          
  [c_1](x1) = [1]         
                          
  [c_2](x1) = [1]         
                          
  [c_3](x1) = [1]         
                          
  [c_4](x1) = [1]         
                          
  [c_5](x1) = [1]         

This order satisfies following ordering constraints:


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)).

Strict DPs:
  { 0^#(5(1(2(3(4(x1)))))) -> c_2(0^#(1(2(5(3(4(x1)))))))
  , 0^#(5(1(2(3(4(x1)))))) -> c_3(0^#(5(2(1(3(4(x1)))))))
  , 0^#(5(1(2(3(4(x1)))))) -> c_4(0^#(2(3(1(4(x1))))))
  , 0^#(5(2(3(1(4(x1)))))) -> c_5(0^#(1(5(2(3(4(x1))))))) }
Weak DPs: { 0^#(1(2(3(4(x1))))) -> c_1(0^#(2(3(1(4(x1)))))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

We estimate the number of application of {1,2,3,4} by applications
of Pre({1,2,3,4}) = {}. Here rules are labeled as follows:

  DPs:
    { 1: 0^#(5(1(2(3(4(x1)))))) -> c_2(0^#(1(2(5(3(4(x1)))))))
    , 2: 0^#(5(1(2(3(4(x1)))))) -> c_3(0^#(5(2(1(3(4(x1)))))))
    , 3: 0^#(5(1(2(3(4(x1)))))) -> c_4(0^#(2(3(1(4(x1))))))
    , 4: 0^#(5(2(3(1(4(x1)))))) -> c_5(0^#(1(5(2(3(4(x1)))))))
    , 5: 0^#(1(2(3(4(x1))))) -> c_1(0^#(2(3(1(4(x1)))))) }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(1)).

Weak DPs:
  { 0^#(1(2(3(4(x1))))) -> c_1(0^#(2(3(1(4(x1))))))
  , 0^#(5(1(2(3(4(x1)))))) -> c_2(0^#(1(2(5(3(4(x1)))))))
  , 0^#(5(1(2(3(4(x1)))))) -> c_3(0^#(5(2(1(3(4(x1)))))))
  , 0^#(5(1(2(3(4(x1)))))) -> c_4(0^#(2(3(1(4(x1))))))
  , 0^#(5(2(3(1(4(x1)))))) -> c_5(0^#(1(5(2(3(4(x1))))))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ 0^#(1(2(3(4(x1))))) -> c_1(0^#(2(3(1(4(x1))))))
, 0^#(5(1(2(3(4(x1)))))) -> c_2(0^#(1(2(5(3(4(x1)))))))
, 0^#(5(1(2(3(4(x1)))))) -> c_3(0^#(5(2(1(3(4(x1)))))))
, 0^#(5(1(2(3(4(x1)))))) -> c_4(0^#(2(3(1(4(x1))))))
, 0^#(5(2(3(1(4(x1)))))) -> c_5(0^#(1(5(2(3(4(x1))))))) }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(1)).

Rules: Empty
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

The input was oriented with the instance of 'Small Polynomial Path
Order (PS)' as induced by the safe mapping


and precedence

 empty .

Following symbols are considered recursive:

 {}

The recursion depth is 0.

Further, following argument filtering is employed:

 empty

Usable defined function symbols are a subset of:

 {}

For your convenience, here are the satisfied ordering constraints:


Hurray, we answered YES(O(1),O(n^1))