(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(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) DependencyGraphProof (BOTH BOUNDS(ID, ID) transformation)
The following rules are not reachable from basic terms in the dependency graph and can be removed:
2nd(cons(X, n__cons(Y, Z))) → activate(Y)
(2) 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:
from(X) → cons(X, n__from(s(X)))
from(X) → n__from(X)
activate(n__from(X)) → from(X)
activate(X) → X
cons(X1, X2) → n__cons(X1, X2)
activate(n__cons(X1, X2)) → cons(X1, X2)
Rewrite Strategy: INNERMOST
(3) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
activate(n__from(z0)) → from(z0)
activate(z0) → z0
activate(n__cons(z0, z1)) → cons(z0, z1)
cons(z0, z1) → n__cons(z0, z1)
Tuples:
FROM(z0) → c(CONS(z0, n__from(s(z0))))
FROM(z0) → c1
ACTIVATE(n__from(z0)) → c2(FROM(z0))
ACTIVATE(z0) → c3
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1))
CONS(z0, z1) → c5
S tuples:
FROM(z0) → c(CONS(z0, n__from(s(z0))))
FROM(z0) → c1
ACTIVATE(n__from(z0)) → c2(FROM(z0))
ACTIVATE(z0) → c3
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1))
CONS(z0, z1) → c5
K tuples:none
Defined Rule Symbols:
from, activate, cons
Defined Pair Symbols:
FROM, ACTIVATE, CONS
Compound Symbols:
c, c1, c2, c3, c4, c5
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 6 trailing nodes:
FROM(z0) → c1
CONS(z0, z1) → c5
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1))
ACTIVATE(n__from(z0)) → c2(FROM(z0))
FROM(z0) → c(CONS(z0, n__from(s(z0))))
ACTIVATE(z0) → c3
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
activate(n__from(z0)) → from(z0)
activate(z0) → z0
activate(n__cons(z0, z1)) → cons(z0, z1)
cons(z0, z1) → n__cons(z0, z1)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
from, activate, cons
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(8) BOUNDS(1, 1)