Problem Transformed CSR 04 Ex26 Luc03b L

Tool Bounds

Execution Time	8.368111e-2ms
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex26 Luc03b L

stdout:

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  first(s(), cons(Y)) -> cons(Y)
     , first(0(), X) -> nil()
     , add(s(), Y) -> s()
     , add(0(), X) -> X
     , dbl(s()) -> s()
     , dbl(0()) -> 0()
     , sqr(s()) -> s()
     , sqr(0()) -> 0()
     , terms(N) -> cons(recip(sqr(N)))}
  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:
  {  terms_0(1) -> 1
   , sqr_0(1) -> 1
   , sqr_1(1) -> 2
   , recip_0(1) -> 1
   , recip_1(2) -> 1
   , cons_0(1) -> 1
   , cons_1(1) -> 1
   , 0_0() -> 1
   , 0_1() -> 1
   , 0_1() -> 2
   , 0_2() -> 2
   , s_0() -> 1
   , s_1() -> 1
   , s_1() -> 2
   , s_2() -> 2
   , dbl_0(1) -> 1
   , add_0(1, 1) -> 1
   , first_0(1, 1) -> 1
   , nil_0() -> 1
   , nil_1() -> 1}

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

Tool CDI

Execution Time	0.378268ms
Answer	YES(?,O(n^2))
Input	Transformed CSR 04 Ex26 Luc03b L

stdout:

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

This TRS is terminating using the deltarestricted interpretation
first(delta, X1, X0) =  + 1*X0 + 0*X1 + 0 + 0*X0*delta + 1*X1*delta + 1*delta
nil(delta) =  + 0 + 0*delta
add(delta, X1, X0) =  + 1*X0 + 0*X1 + 3 + 0*X0*delta + 2*X1*delta + 3*delta
dbl(delta, X0) =  + 1*X0 + 3 + 2*X0*delta + 2*delta
s(delta) =  + 2 + 0*delta
0(delta) =  + 2 + 0*delta
terms(delta, X0) =  + 0*X0 + 2 + 2*X0*delta + 2*delta
sqr(delta, X0) =  + 1*X0 + 1 + 1*X0*delta + 0*delta
recip(delta, X0) =  + 0*X0 + 0 + 1*X0*delta + 0*delta
cons(delta, X0) =  + 1*X0 + 0 + 0*X0*delta + 0*delta
first_tau_1(delta) = delta/(0 + 1 * delta)
first_tau_2(delta) = delta/(1 + 0 * delta)
add_tau_1(delta) = delta/(0 + 2 * delta)
add_tau_2(delta) = delta/(1 + 0 * delta)
dbl_tau_1(delta) = delta/(1 + 2 * delta)
terms_tau_1(delta) = delta/(0 + 2 * delta)
sqr_tau_1(delta) = delta/(1 + 1 * delta)
recip_tau_1(delta) = delta/(0 + 1 * delta)
cons_tau_1(delta) = delta/(1 + 0 * delta)

Time: 0.338175 seconds
Statistics:
Number of monomials: 251
Last formula building started for bound 3
Last SAT solving started for bound 3

Tool EDA

Execution Time	0.12351203ms
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex26 Luc03b L

stdout:

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  first(s(), cons(Y)) -> cons(Y)
     , first(0(), X) -> nil()
     , add(s(), Y) -> s()
     , add(0(), X) -> X
     , dbl(s()) -> s()
     , dbl(0()) -> 0()
     , sqr(s()) -> s()
     , sqr(0()) -> 0()
     , terms(N) -> cons(recip(sqr(N)))}
  StartTerms: all
  Strategy: none

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

Proof:
  We have the following EDA-non-satisfying matrix interpretation:
  Interpretation Functions:
   terms(x1) = [1] x1 + [3]
   sqr(x1) = [1] x1 + [1]
   recip(x1) = [1] x1 + [1]
   cons(x1) = [1] x1 + [0]
   0() = [0]
   s() = [0]
   dbl(x1) = [1] x1 + [1]
   add(x1, x2) = [1] x1 + [1] x2 + [3]
   first(x1, x2) = [1] x1 + [1] x2 + [1]
   nil() = [0]

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

Tool IDA

Execution Time	0.65058994ms
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex26 Luc03b L

stdout:

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  first(s(), cons(Y)) -> cons(Y)
     , first(0(), X) -> nil()
     , add(s(), Y) -> s()
     , add(0(), X) -> X
     , dbl(s()) -> s()
     , dbl(0()) -> 0()
     , sqr(s()) -> s()
     , sqr(0()) -> 0()
     , terms(N) -> cons(recip(sqr(N)))}
  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:
   terms(x1) = [1] x1 + [3]
   sqr(x1) = [1] x1 + [1]
   recip(x1) = [1] x1 + [1]
   cons(x1) = [1] x1 + [0]
   0() = [0]
   s() = [0]
   dbl(x1) = [1] x1 + [1]
   add(x1, x2) = [1] x1 + [1] x2 + [3]
   first(x1, x2) = [1] x1 + [1] x2 + [1]
   nil() = [0]

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

Tool TRI

Execution Time	9.879899e-2ms
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex26 Luc03b L

stdout:

YES(?,O(n^1))

We consider the following Problem:

  Strict Trs:
    {  first(s(), cons(Y)) -> cons(Y)
     , first(0(), X) -> nil()
     , add(s(), Y) -> s()
     , add(0(), X) -> X
     , dbl(s()) -> s()
     , dbl(0()) -> 0()
     , sqr(s()) -> s()
     , sqr(0()) -> 0()
     , terms(N) -> cons(recip(sqr(N)))}
  StartTerms: all
  Strategy: none

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

Proof:
  We have the following triangular matrix interpretation:
  Interpretation Functions:
   terms(x1) = [1] x1 + [3]
   sqr(x1) = [1] x1 + [1]
   recip(x1) = [1] x1 + [1]
   cons(x1) = [1] x1 + [0]
   0() = [0]
   s() = [0]
   dbl(x1) = [1] x1 + [1]
   add(x1, x2) = [1] x1 + [1] x2 + [3]
   first(x1, x2) = [1] x1 + [1] x2 + [1]
   nil() = [0]

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

Tool TRI2

Execution Time	0.15700221ms
Answer	YES(?,O(n^2))
Input	Transformed CSR 04 Ex26 Luc03b L

stdout:

YES(?,O(n^2))

We consider the following Problem:

  Strict Trs:
    {  first(s(), cons(Y)) -> cons(Y)
     , first(0(), X) -> nil()
     , add(s(), Y) -> s()
     , add(0(), X) -> X
     , dbl(s()) -> s()
     , dbl(0()) -> 0()
     , sqr(s()) -> s()
     , sqr(0()) -> 0()
     , terms(N) -> cons(recip(sqr(N)))}
  StartTerms: all
  Strategy: none

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

Proof:
  We have the following triangular matrix interpretation:
  Interpretation Functions:
   terms(x1) = [1 3] x1 + [3]
               [0 1]      [3]
   sqr(x1) = [1 2] x1 + [2]
             [0 0]      [1]
   recip(x1) = [1 0] x1 + [0]
               [0 0]      [3]
   cons(x1) = [1 0] x1 + [0]
              [0 0]      [3]
   0() = [1]
         [1]
   s() = [2]
         [0]
   dbl(x1) = [1 0] x1 + [2]
             [0 1]      [3]
   add(x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                 [0 1]      [0 1]      [3]
   first(x1, x2) = [1 0] x1 + [1 2] x2 + [0]
                   [0 1]      [0 1]      [3]
   nil() = [0]
           [0]

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