YES(O(1), O(n^1)) 0.00/0.79 YES(O(1), O(n^1)) 0.00/0.82 0.00/0.82 0.00/0.82
0.00/0.82 0.00/0.820 CpxTRS0.00/0.82
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))0.00/0.82
↳2 CdtProblem0.00/0.82
↳3 CdtUnreachableProof (⇔)0.00/0.82
↳4 CdtProblem0.00/0.82
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))0.00/0.82
↳6 CdtProblem0.00/0.82
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))0.00/0.82
↳8 CdtProblem0.00/0.82
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))0.00/0.82
↳10 CdtProblem0.00/0.82
↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID))0.00/0.82
↳12 BOUNDS(O(1), O(1))0.00/0.82
first(0, X) → nil 0.00/0.82
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z))) 0.00/0.82
from(X) → cons(X, n__from(n__s(X))) 0.00/0.82
first(X1, X2) → n__first(X1, X2) 0.00/0.82
from(X) → n__from(X) 0.00/0.82
s(X) → n__s(X) 0.00/0.82
activate(n__first(X1, X2)) → first(activate(X1), activate(X2)) 0.00/0.82
activate(n__from(X)) → from(activate(X)) 0.00/0.82
activate(n__s(X)) → s(activate(X)) 0.00/0.82
activate(X) → X
Tuples:
first(0, z0) → nil 0.00/0.82
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 0.00/0.82
first(z0, z1) → n__first(z0, z1) 0.00/0.82
from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.82
from(z0) → n__from(z0) 0.00/0.82
s(z0) → n__s(z0) 0.00/0.82
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 0.00/0.82
activate(n__from(z0)) → from(activate(z0)) 0.00/0.82
activate(n__s(z0)) → s(activate(z0)) 0.00/0.82
activate(z0) → z0
S tuples:
FIRST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z2)) 0.00/0.82
ACTIVATE(n__first(z0, z1)) → c6(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.82
ACTIVATE(n__from(z0)) → c7(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.82
ACTIVATE(n__s(z0)) → c8(S(activate(z0)), ACTIVATE(z0))
K tuples:none
FIRST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z2)) 0.00/0.82
ACTIVATE(n__first(z0, z1)) → c6(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.82
ACTIVATE(n__from(z0)) → c7(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.82
ACTIVATE(n__s(z0)) → c8(S(activate(z0)), ACTIVATE(z0))
first, from, s, activate
FIRST, ACTIVATE
c1, c6, c7, c8
FIRST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z2))
Tuples:
first(0, z0) → nil 0.00/0.82
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 0.00/0.82
first(z0, z1) → n__first(z0, z1) 0.00/0.82
from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.82
from(z0) → n__from(z0) 0.00/0.82
s(z0) → n__s(z0) 0.00/0.82
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 0.00/0.82
activate(n__from(z0)) → from(activate(z0)) 0.00/0.82
activate(n__s(z0)) → s(activate(z0)) 0.00/0.82
activate(z0) → z0
S tuples:
ACTIVATE(n__first(z0, z1)) → c6(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.82
ACTIVATE(n__from(z0)) → c7(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.82
ACTIVATE(n__s(z0)) → c8(S(activate(z0)), ACTIVATE(z0))
K tuples:none
ACTIVATE(n__first(z0, z1)) → c6(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.82
ACTIVATE(n__from(z0)) → c7(FROM(activate(z0)), ACTIVATE(z0)) 0.00/0.82
ACTIVATE(n__s(z0)) → c8(S(activate(z0)), ACTIVATE(z0))
first, from, s, activate
ACTIVATE
c6, c7, c8
Tuples:
first(0, z0) → nil 0.00/0.82
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 0.00/0.82
first(z0, z1) → n__first(z0, z1) 0.00/0.82
from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.82
from(z0) → n__from(z0) 0.00/0.82
s(z0) → n__s(z0) 0.00/0.82
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 0.00/0.82
activate(n__from(z0)) → from(activate(z0)) 0.00/0.82
activate(n__s(z0)) → s(activate(z0)) 0.00/0.82
activate(z0) → z0
S tuples:
ACTIVATE(n__first(z0, z1)) → c6(ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.82
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0)) 0.00/0.82
ACTIVATE(n__s(z0)) → c8(ACTIVATE(z0))
K tuples:none
ACTIVATE(n__first(z0, z1)) → c6(ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.82
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0)) 0.00/0.82
ACTIVATE(n__s(z0)) → c8(ACTIVATE(z0))
first, from, s, activate
ACTIVATE
c6, c7, c8
We considered the (Usable) Rules:none
ACTIVATE(n__first(z0, z1)) → c6(ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.82
ACTIVATE(n__s(z0)) → c8(ACTIVATE(z0))
The order we found is given by the following interpretation:
ACTIVATE(n__first(z0, z1)) → c6(ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.82
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0)) 0.00/0.82
ACTIVATE(n__s(z0)) → c8(ACTIVATE(z0))
POL(ACTIVATE(x1)) = [1] + [2]x1 0.00/0.82
POL(c6(x1, x2)) = x1 + x2 0.00/0.82
POL(c7(x1)) = x1 0.00/0.82
POL(c8(x1)) = x1 0.00/0.82
POL(n__first(x1, x2)) = [1] + x1 + x2 0.00/0.82
POL(n__from(x1)) = x1 0.00/0.82
POL(n__s(x1)) = [1] + x1
Tuples:
first(0, z0) → nil 0.00/0.82
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 0.00/0.82
first(z0, z1) → n__first(z0, z1) 0.00/0.82
from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.82
from(z0) → n__from(z0) 0.00/0.82
s(z0) → n__s(z0) 0.00/0.82
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 0.00/0.82
activate(n__from(z0)) → from(activate(z0)) 0.00/0.82
activate(n__s(z0)) → s(activate(z0)) 0.00/0.82
activate(z0) → z0
S tuples:
ACTIVATE(n__first(z0, z1)) → c6(ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.82
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0)) 0.00/0.82
ACTIVATE(n__s(z0)) → c8(ACTIVATE(z0))
K tuples:
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
Defined Rule Symbols:
ACTIVATE(n__first(z0, z1)) → c6(ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.82
ACTIVATE(n__s(z0)) → c8(ACTIVATE(z0))
first, from, s, activate
ACTIVATE
c6, c7, c8
We considered the (Usable) Rules:none
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
The order we found is given by the following interpretation:
ACTIVATE(n__first(z0, z1)) → c6(ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.82
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0)) 0.00/0.82
ACTIVATE(n__s(z0)) → c8(ACTIVATE(z0))
POL(ACTIVATE(x1)) = [4] + x1 0.00/0.82
POL(c6(x1, x2)) = x1 + x2 0.00/0.82
POL(c7(x1)) = x1 0.00/0.82
POL(c8(x1)) = x1 0.00/0.82
POL(n__first(x1, x2)) = [5] + x1 + x2 0.00/0.82
POL(n__from(x1)) = [1] + x1 0.00/0.82
POL(n__s(x1)) = x1
Tuples:
first(0, z0) → nil 0.00/0.82
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2))) 0.00/0.82
first(z0, z1) → n__first(z0, z1) 0.00/0.82
from(z0) → cons(z0, n__from(n__s(z0))) 0.00/0.82
from(z0) → n__from(z0) 0.00/0.82
s(z0) → n__s(z0) 0.00/0.82
activate(n__first(z0, z1)) → first(activate(z0), activate(z1)) 0.00/0.82
activate(n__from(z0)) → from(activate(z0)) 0.00/0.82
activate(n__s(z0)) → s(activate(z0)) 0.00/0.82
activate(z0) → z0
S tuples:none
ACTIVATE(n__first(z0, z1)) → c6(ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.82
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0)) 0.00/0.82
ACTIVATE(n__s(z0)) → c8(ACTIVATE(z0))
Defined Rule Symbols:
ACTIVATE(n__first(z0, z1)) → c6(ACTIVATE(z0), ACTIVATE(z1)) 0.00/0.82
ACTIVATE(n__s(z0)) → c8(ACTIVATE(z0)) 0.00/0.82
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
first, from, s, activate
ACTIVATE
c6, c7, c8