Problem AProVE 10 ex5

Tool Bounds

Execution Time	2.5157928e-2ms
Answer	YES(?,O(n^1))
Input	AProVE 10 ex5

stdout:

YES(?,O(n^1))

We consider the following Problem:

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

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

Proof:
  The problem is match-bounded by 2.
  The enriched problem is compatible with the following automaton:
  {  0_0() -> 1
   , 0_1() -> 1
   , 0_1() -> 2
   , 0_2() -> 1
   , 0_2() -> 2
   , g_0(1) -> 1
   , g_1(1) -> 2
   , s_0(1) -> 1
   , f_0(1) -> 1
   , f_1(2) -> 1
   , f_1(2) -> 2}

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

Tool CDI

Execution Time	0.111490965ms
Answer	YES(?,O(n^2))
Input	AProVE 10 ex5

stdout:

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

This TRS is terminating using the deltarestricted interpretation
s(delta, X0) =  + 1*X0 + 0 + 1*X0*delta + 1*delta
f(delta, X0) =  + 1*X0 + 0 + 1*X0*delta + 0*delta
g(delta, X0) =  + 1*X0 + 0 + 0*X0*delta + 1*delta
0(delta) =  + 3 + 0*delta
s_tau_1(delta) = delta/(1 + 1 * delta)
f_tau_1(delta) = delta/(1 + 1 * delta)
g_tau_1(delta) = delta/(1 + 0 * delta)

Time: 0.074302 seconds
Statistics:
Number of monomials: 89
Last formula building started for bound 3
Last SAT solving started for bound 3

Tool EDA

Execution Time	0.14420795ms
Answer	YES(?,O(n^1))
Input	AProVE 10 ex5

stdout:

YES(?,O(n^1))

We consider the following Problem:

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

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

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

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

Tool IDA

Execution Time	0.2628281ms
Answer	YES(?,O(n^1))
Input	AProVE 10 ex5

stdout:

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  f(0()) -> 0()
     , g(s(x)) -> f(g(x))
     , g(0()) -> 0()}
  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:
   0() = [0]
   g(x1) = [1] x1 + [1]
   s(x1) = [1] x1 + [2]
   f(x1) = [1] x1 + [1]

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

Tool TRI

Execution Time	5.7497025e-2ms
Answer	YES(?,O(n^1))
Input	AProVE 10 ex5

stdout:

YES(?,O(n^1))

We consider the following Problem:

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

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

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

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

Tool TRI2

Execution Time	7.50339e-2ms
Answer	YES(?,O(n^2))
Input	AProVE 10 ex5

stdout:

YES(?,O(n^2))

We consider the following Problem:

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

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

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

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