(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
2nd(cons(X, n__cons(Y, Z))) → activate(Y)
from(X) → cons(X, n__from(s(X)))
cons(X1, X2) → n__cons(X1, X2)
from(X) → n__from(X)
activate(n__cons(X1, X2)) → cons(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
2nd(cons(z0, n__cons(z1, z2))) → activate(z1)
from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
cons(z0, z1) → n__cons(z0, z1)
activate(n__cons(z0, z1)) → cons(z0, z1)
activate(n__from(z0)) → from(z0)
activate(z0) → z0
Tuples:
2ND(cons(z0, n__cons(z1, z2))) → c(ACTIVATE(z1))
FROM(z0) → c1(CONS(z0, n__from(s(z0))))
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1))
ACTIVATE(n__from(z0)) → c5(FROM(z0))
S tuples:
2ND(cons(z0, n__cons(z1, z2))) → c(ACTIVATE(z1))
FROM(z0) → c1(CONS(z0, n__from(s(z0))))
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1))
ACTIVATE(n__from(z0)) → c5(FROM(z0))
K tuples:none
Defined Rule Symbols:
2nd, from, cons, activate
Defined Pair Symbols:
2ND, FROM, ACTIVATE
Compound Symbols:
c, c1, c4, c5
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing nodes:
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1))
FROM(z0) → c1(CONS(z0, n__from(s(z0))))
2ND(cons(z0, n__cons(z1, z2))) → c(ACTIVATE(z1))
ACTIVATE(n__from(z0)) → c5(FROM(z0))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
2nd(cons(z0, n__cons(z1, z2))) → activate(z1)
from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
cons(z0, z1) → n__cons(z0, z1)
activate(n__cons(z0, z1)) → cons(z0, z1)
activate(n__from(z0)) → from(z0)
activate(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
2nd, from, cons, activate
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(6) BOUNDS(O(1), O(1))