(0) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, 1).
The TRS R consists of the following rules:
2nd(cons1(X, cons(Y, Z))) → Y
2nd(cons(X, X1)) → 2nd(cons1(X, activate(X1)))
from(X) → cons(X, n__from(s(X)))
from(X) → n__from(X)
activate(n__from(X)) → from(X)
activate(X) → X
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
2nd(cons1(z0, cons(z1, z2))) → z1
2nd(cons(z0, z1)) → 2nd(cons1(z0, activate(z1)))
from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
activate(n__from(z0)) → from(z0)
activate(z0) → z0
Tuples:
2ND(cons1(z0, cons(z1, z2))) → c
2ND(cons(z0, z1)) → c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
FROM(z0) → c2
FROM(z0) → c3
ACTIVATE(n__from(z0)) → c4(FROM(z0))
ACTIVATE(z0) → c5
S tuples:
2ND(cons1(z0, cons(z1, z2))) → c
2ND(cons(z0, z1)) → c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
FROM(z0) → c2
FROM(z0) → c3
ACTIVATE(n__from(z0)) → c4(FROM(z0))
ACTIVATE(z0) → c5
K tuples:none
Defined Rule Symbols:
2nd, from, activate
Defined Pair Symbols:
2ND, FROM, ACTIVATE
Compound Symbols:
c, c1, c2, c3, c4, c5
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 6 trailing nodes:
2ND(cons1(z0, cons(z1, z2))) → c
2ND(cons(z0, z1)) → c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
FROM(z0) → c2
ACTIVATE(n__from(z0)) → c4(FROM(z0))
ACTIVATE(z0) → c5
FROM(z0) → c3
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
2nd(cons1(z0, cons(z1, z2))) → z1
2nd(cons(z0, z1)) → 2nd(cons1(z0, activate(z1)))
from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
activate(n__from(z0)) → from(z0)
activate(z0) → z0
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
2nd, from, activate
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(6) BOUNDS(1, 1)