Problem Transformed CSR 04 Ex2 Luc03b Z

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 Ex2 Luc03b Z

stdout:

MAYBE

Problem:
fst(0(),Z) -> nil()
fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
from(X) -> cons(X,n__from(s(X)))
add(0(),X) -> X
add(s(X),Y) -> s(n__add(activate(X),Y))
len(nil()) -> 0()
len(cons(X,Z)) -> s(n__len(activate(Z)))
fst(X1,X2) -> n__fst(X1,X2)
from(X) -> n__from(X)
add(X1,X2) -> n__add(X1,X2)
len(X) -> n__len(X)
activate(n__fst(X1,X2)) -> fst(X1,X2)
activate(n__from(X)) -> from(X)
activate(n__add(X1,X2)) -> add(X1,X2)
activate(n__len(X)) -> len(X)
activate(X) -> X

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex2 Luc03b Z

stdout:

YES(?,O(n^1))

Tool IRC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex2 Luc03b Z

stdout:

YES(?,O(n^1))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  fst(0(), Z) -> nil()
     , fst(s(X), cons(Y, Z)) ->
       cons(Y, n__fst(activate(X), activate(Z)))
     , from(X) -> cons(X, n__from(s(X)))
     , add(0(), X) -> X
     , add(s(X), Y) -> s(n__add(activate(X), Y))
     , len(nil()) -> 0()
     , len(cons(X, Z)) -> s(n__len(activate(Z)))
     , fst(X1, X2) -> n__fst(X1, X2)
     , from(X) -> n__from(X)
     , add(X1, X2) -> n__add(X1, X2)
     , len(X) -> n__len(X)
     , activate(n__fst(X1, X2)) -> fst(X1, X2)
     , activate(n__from(X)) -> from(X)
     , activate(n__add(X1, X2)) -> add(X1, X2)
     , activate(n__len(X)) -> len(X)
     , activate(X) -> X}

Proof Output:
  'matrix-interpretation of dimension 1' proved the best result:

  Details:
  --------
    'matrix-interpretation of dimension 1' succeeded with the following output:
     'matrix-interpretation of dimension 1'
     --------------------------------------
     Answer:           YES(?,O(n^1))
     Input Problem:    innermost runtime-complexity with respect to
       Rules:
         {  fst(0(), Z) -> nil()
          , fst(s(X), cons(Y, Z)) ->
            cons(Y, n__fst(activate(X), activate(Z)))
          , from(X) -> cons(X, n__from(s(X)))
          , add(0(), X) -> X
          , add(s(X), Y) -> s(n__add(activate(X), Y))
          , len(nil()) -> 0()
          , len(cons(X, Z)) -> s(n__len(activate(Z)))
          , fst(X1, X2) -> n__fst(X1, X2)
          , from(X) -> n__from(X)
          , add(X1, X2) -> n__add(X1, X2)
          , len(X) -> n__len(X)
          , activate(n__fst(X1, X2)) -> fst(X1, X2)
          , activate(n__from(X)) -> from(X)
          , activate(n__add(X1, X2)) -> add(X1, X2)
          , activate(n__len(X)) -> len(X)
          , activate(X) -> X}

     Proof Output:
       The following argument positions are usable:
         Uargs(fst) = {}, Uargs(s) = {1}, Uargs(cons) = {2},
         Uargs(n__fst) = {1, 2}, Uargs(activate) = {}, Uargs(from) = {},
         Uargs(n__from) = {}, Uargs(add) = {}, Uargs(n__add) = {1},
         Uargs(len) = {}, Uargs(n__len) = {1}
       We have the following constructor-restricted matrix interpretation:
       Interpretation Functions:
        fst(x1, x2) = [3] x1 + [3] x2 + [3]
        0() = [2]
        nil() = [2]
        s(x1) = [1] x1 + [2]
        cons(x1, x2) = [0] x1 + [1] x2 + [2]
        n__fst(x1, x2) = [1] x1 + [1] x2 + [2]
        activate(x1) = [3] x1 + [4]
        from(x1) = [0] x1 + [3]
        n__from(x1) = [0] x1 + [0]
        add(x1, x2) = [3] x1 + [1] x2 + [4]
        n__add(x1, x2) = [1] x1 + [1] x2 + [2]
        len(x1) = [3] x1 + [4]
        n__len(x1) = [1] x1 + [2]

Tool RC1

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex2 Luc03b Z

stdout:

YES(?,O(n^1))

Tool RC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Transformed CSR 04 Ex2 Luc03b Z

stdout:

YES(?,O(n^1))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    runtime-complexity with respect to
  Rules:
    {  fst(0(), Z) -> nil()
     , fst(s(X), cons(Y, Z)) ->
       cons(Y, n__fst(activate(X), activate(Z)))
     , from(X) -> cons(X, n__from(s(X)))
     , add(0(), X) -> X
     , add(s(X), Y) -> s(n__add(activate(X), Y))
     , len(nil()) -> 0()
     , len(cons(X, Z)) -> s(n__len(activate(Z)))
     , fst(X1, X2) -> n__fst(X1, X2)
     , from(X) -> n__from(X)
     , add(X1, X2) -> n__add(X1, X2)
     , len(X) -> n__len(X)
     , activate(n__fst(X1, X2)) -> fst(X1, X2)
     , activate(n__from(X)) -> from(X)
     , activate(n__add(X1, X2)) -> add(X1, X2)
     , activate(n__len(X)) -> len(X)
     , activate(X) -> X}

Proof Output:
  'matrix-interpretation of dimension 1' proved the best result:

  Details:
  --------
    'matrix-interpretation of dimension 1' succeeded with the following output:
     'matrix-interpretation of dimension 1'
     --------------------------------------
     Answer:           YES(?,O(n^1))
     Input Problem:    runtime-complexity with respect to
       Rules:
         {  fst(0(), Z) -> nil()
          , fst(s(X), cons(Y, Z)) ->
            cons(Y, n__fst(activate(X), activate(Z)))
          , from(X) -> cons(X, n__from(s(X)))
          , add(0(), X) -> X
          , add(s(X), Y) -> s(n__add(activate(X), Y))
          , len(nil()) -> 0()
          , len(cons(X, Z)) -> s(n__len(activate(Z)))
          , fst(X1, X2) -> n__fst(X1, X2)
          , from(X) -> n__from(X)
          , add(X1, X2) -> n__add(X1, X2)
          , len(X) -> n__len(X)
          , activate(n__fst(X1, X2)) -> fst(X1, X2)
          , activate(n__from(X)) -> from(X)
          , activate(n__add(X1, X2)) -> add(X1, X2)
          , activate(n__len(X)) -> len(X)
          , activate(X) -> X}

     Proof Output:
       The following argument positions are usable:
         Uargs(fst) = {}, Uargs(s) = {1}, Uargs(cons) = {2},
         Uargs(n__fst) = {1, 2}, Uargs(activate) = {}, Uargs(from) = {},
         Uargs(n__from) = {}, Uargs(add) = {}, Uargs(n__add) = {1},
         Uargs(len) = {}, Uargs(n__len) = {1}
       We have the following constructor-restricted matrix interpretation:
       Interpretation Functions:
        fst(x1, x2) = [3] x1 + [3] x2 + [3]
        0() = [2]
        nil() = [2]
        s(x1) = [1] x1 + [2]
        cons(x1, x2) = [0] x1 + [1] x2 + [2]
        n__fst(x1, x2) = [1] x1 + [1] x2 + [2]
        activate(x1) = [3] x1 + [4]
        from(x1) = [0] x1 + [3]
        n__from(x1) = [0] x1 + [0]
        add(x1, x2) = [3] x1 + [1] x2 + [4]
        n__add(x1, x2) = [1] x1 + [1] x2 + [2]
        len(x1) = [3] x1 + [4]
        n__len(x1) = [1] x1 + [2]