Problem AG01 3.24

Tool Bounds

Execution Time	2.846694e-2ms
Answer	YES(?,O(n^1))
Input	AG01 3.24

stdout:

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  f(s(s(x))) -> f(f(s(x)))
     , f(s(0())) -> s(0())
     , f(0()) -> s(0())}
  StartTerms: all
  Strategy: none

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

Proof:
  The problem is match-bounded by 3.
  The enriched problem is compatible with the following automaton:
  {  0_0() -> 1
   , 0_1() -> 4
   , 0_2() -> 7
   , 0_3() -> 8
   , f_0(1) -> 1
   , f_1(2) -> 1
   , f_1(2) -> 2
   , f_1(3) -> 2
   , f_2(2) -> 2
   , f_2(5) -> 2
   , f_2(6) -> 5
   , s_0(1) -> 1
   , s_1(1) -> 3
   , s_1(4) -> 1
   , s_1(4) -> 2
   , s_1(7) -> 2
   , s_2(4) -> 6
   , s_2(7) -> 1
   , s_2(7) -> 2
   , s_2(7) -> 5
   , s_3(8) -> 2}

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

Tool CDI

Execution Time	1.6217439ms
Answer	YES(?,O(n^2))
Input	AG01 3.24

stdout:

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

This TRS is terminating using the deltarestricted interpretation
f(delta, X0) =  + 0*X0 + 0 + 2*X0*delta + 1*delta
0(delta) =  + 2 + 0*delta
s(delta, X0) =  + 0*X0 + 0 + 1*X0*delta + 2*delta
f_tau_1(delta) = delta/(0 + 2 * delta)
s_tau_1(delta) = delta/(0 + 1 * delta)

Time: 1.584324 seconds
Statistics:
Number of monomials: 247
Last formula building started for bound 3
Last SAT solving started for bound 3

Tool EDA

Execution Time	0.41037393ms
Answer	YES(?,O(n^2))
Input	AG01 3.24

stdout:

YES(?,O(n^2))

We consider the following Problem:

  Strict Trs:
    {  f(s(s(x))) -> f(f(s(x)))
     , f(s(0())) -> s(0())
     , f(0()) -> s(0())}
  StartTerms: all
  Strategy: none

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

Proof:
  We have the following EDA-non-satisfying matrix interpretation:
  Interpretation Functions:
   0() = [0]
         [0]
   f(x1) = [1 2] x1 + [2]
           [0 0]      [1]
   s(x1) = [1 3] x1 + [0]
           [0 1]      [1]

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

Tool IDA

Execution Time	0.45045805ms
Answer	YES(?,O(n^2))
Input	AG01 3.24

stdout:

YES(?,O(n^2))

We consider the following Problem:

  Strict Trs:
    {  f(s(s(x))) -> f(f(s(x)))
     , f(s(0())) -> s(0())
     , f(0()) -> s(0())}
  StartTerms: all
  Strategy: none

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

Proof:
  We have the following EDA-non-satisfying and IDA(2)-non-satisfying matrix interpretation:
  Interpretation Functions:
   0() = [0]
         [0]
   f(x1) = [1 2] x1 + [2]
           [0 0]      [1]
   s(x1) = [1 3] x1 + [0]
           [0 1]      [1]

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

Tool TRI

Execution Time	0.17341995ms
Answer	YES(?,O(n^1))
Input	AG01 3.24

stdout:

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  f(s(s(x))) -> f(f(s(x)))
     , f(s(0())) -> s(0())
     , f(0()) -> s(0())}
  StartTerms: all
  Strategy: none

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

Proof:
  We have the following triangular matrix interpretation:
  Interpretation Functions:
   0() = [1]
         [1]
   f(x1) = [1 2] x1 + [1]
           [0 0]      [0]
   s(x1) = [1 0] x1 + [2]
           [0 0]      [0]

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

Tool TRI2

Execution Time	0.1520729ms
Answer	YES(?,O(n^2))
Input	AG01 3.24

stdout:

YES(?,O(n^2))

We consider the following Problem:

  Strict Trs:
    {  f(s(s(x))) -> f(f(s(x)))
     , f(s(0())) -> s(0())
     , f(0()) -> s(0())}
  StartTerms: all
  Strategy: none

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

Proof:
  We have the following triangular matrix interpretation:
  Interpretation Functions:
   0() = [3]
         [0]
   f(x1) = [1 1] x1 + [1]
           [0 0]      [1]
   s(x1) = [1 2] x1 + [0]
           [0 1]      [1]

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