Problem SK90 2.41

Tool Bounds

Execution Time	60.030903ms
Answer	TIMEOUT
Input	SK90 2.41

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  rem(g(x, y), s(z)) -> rem(x, z)
     , rem(g(x, y), 0()) -> g(x, y)
     , rem(nil(), y) -> nil()
     , f(x, g(y, z)) -> g(f(x, y), z)
     , f(x, nil()) -> g(nil(), x)
     , norm(g(x, y)) -> s(norm(x))
     , norm(nil()) -> 0()}
  StartTerms: all
  Strategy: none

Certificate: TIMEOUT

Proof:
  Computation stopped due to timeout after 60.0 seconds.

Arrrr..

Tool CDI

Execution Time	0.46180487ms
Answer	YES(?,O(n^2))
Input	SK90 2.41

stdout:

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

This TRS is terminating using the deltarestricted interpretation
rem(delta, X1, X0) =  + 0*X0 + 1*X1 + 0 + 1*X0*delta + 0*X1*delta + 2*delta
f(delta, X1, X0) =  + 1*X0 + 0*X1 + 1 + 2*X0*delta + 1*X1*delta + 2*delta
g(delta, X1, X0) =  + 0*X0 + 1*X1 + 1 + 1*X0*delta + 0*X1*delta + 0*delta
s(delta, X0) =  + 0*X0 + 0 + 1*X0*delta + 1*delta
nil(delta) =  + 0 + 0*delta
norm(delta, X0) =  + 0*X0 + 0 + 2*X0*delta + 2*delta
0(delta) =  + 0 + 0*delta
rem_tau_1(delta) = delta/(1 + 0 * delta)
rem_tau_2(delta) = delta/(0 + 1 * delta)
f_tau_1(delta) = delta/(0 + 1 * delta)
f_tau_2(delta) = delta/(1 + 2 * delta)
g_tau_1(delta) = delta/(1 + 0 * delta)
g_tau_2(delta) = delta/(0 + 1 * delta)
s_tau_1(delta) = delta/(0 + 1 * delta)
norm_tau_1(delta) = delta/(0 + 2 * delta)

Time: 0.420463 seconds
Statistics:
Number of monomials: 376
Last formula building started for bound 3
Last SAT solving started for bound 3

Tool EDA

Execution Time	0.5373192ms
Answer	YES(?,O(n^2))
Input	SK90 2.41

stdout:

YES(?,O(n^2))

We consider the following Problem:

  Strict Trs:
    {  rem(g(x, y), s(z)) -> rem(x, z)
     , rem(g(x, y), 0()) -> g(x, y)
     , rem(nil(), y) -> nil()
     , f(x, g(y, z)) -> g(f(x, y), z)
     , f(x, nil()) -> g(nil(), x)
     , norm(g(x, y)) -> s(norm(x))
     , norm(nil()) -> 0()}
  StartTerms: all
  Strategy: none

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

Proof:
  We have the following EDA-non-satisfying matrix interpretation:
  Interpretation Functions:
   nil() = [3]
           [2]
   norm(x1) = [1 0] x1 + [0]
              [0 0]      [0]
   0() = [0]
         [0]
   g(x1, x2) = [1 1] x1 + [1 0] x2 + [3]
               [0 1]      [0 0]      [2]
   s(x1) = [1 0] x1 + [0]
           [0 0]      [0]
   f(x1, x2) = [1 0] x1 + [1 3] x2 + [0]
               [0 0]      [0 1]      [2]
   rem(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                 [0 1]      [0 0]      [2]

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

Tool IDA

Execution Time	0.8114481ms
Answer	YES(?,O(n^2))
Input	SK90 2.41

stdout:

YES(?,O(n^2))

We consider the following Problem:

  Strict Trs:
    {  rem(g(x, y), s(z)) -> rem(x, z)
     , rem(g(x, y), 0()) -> g(x, y)
     , rem(nil(), y) -> nil()
     , f(x, g(y, z)) -> g(f(x, y), z)
     , f(x, nil()) -> g(nil(), x)
     , norm(g(x, y)) -> s(norm(x))
     , norm(nil()) -> 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:
   nil() = [2]
           [2]
   norm(x1) = [1 1] x1 + [1]
              [0 0]      [0]
   0() = [0]
         [0]
   g(x1, x2) = [1 0] x1 + [1 0] x2 + [0]
               [0 1]      [0 0]      [3]
   s(x1) = [1 0] x1 + [0]
           [0 0]      [0]
   f(x1, x2) = [1 0] x1 + [1 3] x2 + [0]
               [0 0]      [0 1]      [3]
   rem(x1, x2) = [1 3] x1 + [1 0] x2 + [0]
                 [0 1]      [0 0]      [2]

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

Tool TRI

Execution Time	0.22038603ms
Answer	YES(?,O(n^2))
Input	SK90 2.41

stdout:

YES(?,O(n^2))

We consider the following Problem:

  Strict Trs:
    {  rem(g(x, y), s(z)) -> rem(x, z)
     , rem(g(x, y), 0()) -> g(x, y)
     , rem(nil(), y) -> nil()
     , f(x, g(y, z)) -> g(f(x, y), z)
     , f(x, nil()) -> g(nil(), x)
     , norm(g(x, y)) -> s(norm(x))
     , norm(nil()) -> 0()}
  StartTerms: all
  Strategy: none

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

Proof:
  We have the following triangular matrix interpretation:
  Interpretation Functions:
   nil() = [0]
           [1]
   norm(x1) = [1 1] x1 + [0]
              [0 1]      [3]
   0() = [0]
         [3]
   g(x1, x2) = [1 0] x1 + [1 1] x2 + [0]
               [0 1]      [0 1]      [1]
   s(x1) = [1 0] x1 + [0]
           [0 1]      [1]
   f(x1, x2) = [1 3] x1 + [1 1] x2 + [0]
               [0 1]      [0 1]      [3]
   rem(x1, x2) = [1 1] x1 + [1 0] x2 + [3]
                 [0 1]      [0 1]      [0]

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

Tool TRI2

Execution Time	0.18457294ms
Answer	YES(?,O(n^2))
Input	SK90 2.41

stdout:

YES(?,O(n^2))

We consider the following Problem:

  Strict Trs:
    {  rem(g(x, y), s(z)) -> rem(x, z)
     , rem(g(x, y), 0()) -> g(x, y)
     , rem(nil(), y) -> nil()
     , f(x, g(y, z)) -> g(f(x, y), z)
     , f(x, nil()) -> g(nil(), x)
     , norm(g(x, y)) -> s(norm(x))
     , norm(nil()) -> 0()}
  StartTerms: all
  Strategy: none

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

Proof:
  We have the following triangular matrix interpretation:
  Interpretation Functions:
   nil() = [2]
           [1]
   norm(x1) = [1 2] x1 + [0]
              [0 0]      [0]
   0() = [0]
         [0]
   g(x1, x2) = [1 0] x1 + [1 3] x2 + [0]
               [0 1]      [0 1]      [2]
   s(x1) = [1 0] x1 + [1]
           [0 0]      [0]
   f(x1, x2) = [1 3] x1 + [1 2] x2 + [0]
               [0 1]      [0 1]      [2]
   rem(x1, x2) = [1 0] x1 + [1 0] x2 + [3]
                 [0 1]      [0 0]      [0]

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