Problem Der95 02

Tool Bounds

Execution Time	3.1186104e-2ms
Answer	YES(?,O(n^1))
Input	Der95 02

stdout:

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  .(y, .(i(y), z)) -> z
     , .(i(y), .(y, z)) -> z
     , i(i(x)) -> x
     , i(1()) -> 1()
     , .(x, i(x)) -> 1()
     , .(i(x), x) -> 1()
     , .(x, 1()) -> x
     , .(1(), x) -> x}
  StartTerms: all
  Strategy: none

Certificate: YES(?,O(n^1))

Proof:
  The problem is match-bounded by 1.
  The enriched problem is compatible with the following automaton:
  {  1_0() -> 1
   , 1_1() -> 1
   , ._0(1, 1) -> 1
   , i_0(1) -> 1}

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

Tool CDI

Execution Time	0.335114ms
Answer	YES(?,O(n^2))
Input	Der95 02

stdout:

YES(?,O(n^2))
QUADRATIC upper bound on the derivational complexity

This TRS is terminating using the deltarestricted interpretation
i(delta, X0) =  + 1*X0 + 0 + 0*X0*delta + 2*delta
1(delta) =  + 0 + 1*delta
.(delta, X1, X0) =  + 1*X0 + 1*X1 + 0 + 0*X0*delta + 0*X1*delta + 0*delta
i_tau_1(delta) = delta/(1 + 0 * delta)
._tau_1(delta) = delta/(1 + 0 * delta)
._tau_2(delta) = delta/(1 + 0 * delta)

Time: 0.295943 seconds
Statistics:
Number of monomials: 297
Last formula building started for bound 3
Last SAT solving started for bound 3

Tool EDA

Execution Time	0.16859007ms
Answer	YES(?,O(n^1))
Input	Der95 02

stdout:

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  .(y, .(i(y), z)) -> z
     , .(i(y), .(y, z)) -> z
     , i(i(x)) -> x
     , i(1()) -> 1()
     , .(x, i(x)) -> 1()
     , .(i(x), x) -> 1()
     , .(x, 1()) -> x
     , .(1(), x) -> x}
  StartTerms: all
  Strategy: none

Certificate: YES(?,O(n^1))

Proof:
  We have the following EDA-non-satisfying matrix interpretation:
  Interpretation Functions:
   1() = [0]
   .(x1, x2) = [1] x1 + [1] x2 + [2]
   i(x1) = [1] x1 + [1]

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

Tool IDA

Execution Time	0.27212882ms
Answer	YES(?,O(n^1))
Input	Der95 02

stdout:

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  .(y, .(i(y), z)) -> z
     , .(i(y), .(y, z)) -> z
     , i(i(x)) -> x
     , i(1()) -> 1()
     , .(x, i(x)) -> 1()
     , .(i(x), x) -> 1()
     , .(x, 1()) -> x
     , .(1(), x) -> x}
  StartTerms: all
  Strategy: none

Certificate: YES(?,O(n^1))

Proof:
  We have the following EDA-non-satisfying and IDA(1)-non-satisfying matrix interpretation:
  Interpretation Functions:
   1() = [0]
   .(x1, x2) = [1] x1 + [1] x2 + [1]
   i(x1) = [1] x1 + [1]

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

Tool TRI

Execution Time	6.708217e-2ms
Answer	YES(?,O(n^1))
Input	Der95 02

stdout:

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  .(y, .(i(y), z)) -> z
     , .(i(y), .(y, z)) -> z
     , i(i(x)) -> x
     , i(1()) -> 1()
     , .(x, i(x)) -> 1()
     , .(i(x), x) -> 1()
     , .(x, 1()) -> x
     , .(1(), x) -> x}
  StartTerms: all
  Strategy: none

Certificate: YES(?,O(n^1))

Proof:
  We have the following triangular matrix interpretation:
  Interpretation Functions:
   1() = [0]
   .(x1, x2) = [1] x1 + [1] x2 + [2]
   i(x1) = [1] x1 + [1]

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

Tool TRI2

Execution Time	0.12351012ms
Answer	YES(?,O(n^2))
Input	Der95 02

stdout:

YES(?,O(n^2))

We consider the following Problem:

  Strict Trs:
    {  .(y, .(i(y), z)) -> z
     , .(i(y), .(y, z)) -> z
     , i(i(x)) -> x
     , i(1()) -> 1()
     , .(x, i(x)) -> 1()
     , .(i(x), x) -> 1()
     , .(x, 1()) -> x
     , .(1(), x) -> x}
  StartTerms: all
  Strategy: none

Certificate: YES(?,O(n^2))

Proof:
  We have the following triangular matrix interpretation:
  Interpretation Functions:
   1() = [2]
         [0]
   .(x1, x2) = [1 0] x1 + [1 2] x2 + [2]
               [0 1]      [0 1]      [1]
   i(x1) = [1 0] x1 + [2]
           [0 1]      [0]

Hurray, we answered YES(?,O(n^2))