(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(n__s(X)))
cons(X1, X2) → n__cons(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X
Rewrite Strategy: INNERMOST
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
activate(n__cons(X1, X2)) →+ cons(activate(X1), X2)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / n__cons(X1, X2)].
The result substitution is [ ].
(2) BOUNDS(n^1, INF)
(3) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
2nd(cons(X, n__cons(Y, Z))) → activate(Y)
from(X) → cons(X, n__from(n__s(X)))
cons(X1, X2) → n__cons(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X
S is empty.
Rewrite Strategy: INNERMOST
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
Innermost TRS:
Rules:
2nd(cons(X, n__cons(Y, Z))) → activate(Y)
from(X) → cons(X, n__from(n__s(X)))
cons(X1, X2) → n__cons(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X
Types:
2nd :: n__cons:n__s:n__from → n__cons:n__s:n__from
cons :: n__cons:n__s:n__from → n__cons:n__s:n__from → n__cons:n__s:n__from
n__cons :: n__cons:n__s:n__from → n__cons:n__s:n__from → n__cons:n__s:n__from
activate :: n__cons:n__s:n__from → n__cons:n__s:n__from
from :: n__cons:n__s:n__from → n__cons:n__s:n__from
n__from :: n__cons:n__s:n__from → n__cons:n__s:n__from
n__s :: n__cons:n__s:n__from → n__cons:n__s:n__from
s :: n__cons:n__s:n__from → n__cons:n__s:n__from
hole_n__cons:n__s:n__from1_0 :: n__cons:n__s:n__from
gen_n__cons:n__s:n__from2_0 :: Nat → n__cons:n__s:n__from
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
activate
(8) Obligation:
Innermost TRS:
Rules:
2nd(
cons(
X,
n__cons(
Y,
Z))) →
activate(
Y)
from(
X) →
cons(
X,
n__from(
n__s(
X)))
cons(
X1,
X2) →
n__cons(
X1,
X2)
from(
X) →
n__from(
X)
s(
X) →
n__s(
X)
activate(
n__cons(
X1,
X2)) →
cons(
activate(
X1),
X2)
activate(
n__from(
X)) →
from(
activate(
X))
activate(
n__s(
X)) →
s(
activate(
X))
activate(
X) →
XTypes:
2nd :: n__cons:n__s:n__from → n__cons:n__s:n__from
cons :: n__cons:n__s:n__from → n__cons:n__s:n__from → n__cons:n__s:n__from
n__cons :: n__cons:n__s:n__from → n__cons:n__s:n__from → n__cons:n__s:n__from
activate :: n__cons:n__s:n__from → n__cons:n__s:n__from
from :: n__cons:n__s:n__from → n__cons:n__s:n__from
n__from :: n__cons:n__s:n__from → n__cons:n__s:n__from
n__s :: n__cons:n__s:n__from → n__cons:n__s:n__from
s :: n__cons:n__s:n__from → n__cons:n__s:n__from
hole_n__cons:n__s:n__from1_0 :: n__cons:n__s:n__from
gen_n__cons:n__s:n__from2_0 :: Nat → n__cons:n__s:n__from
Generator Equations:
gen_n__cons:n__s:n__from2_0(0) ⇔ hole_n__cons:n__s:n__from1_0
gen_n__cons:n__s:n__from2_0(+(x, 1)) ⇔ n__cons(gen_n__cons:n__s:n__from2_0(x), hole_n__cons:n__s:n__from1_0)
The following defined symbols remain to be analysed:
activate
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol activate.
(10) Obligation:
Innermost TRS:
Rules:
2nd(
cons(
X,
n__cons(
Y,
Z))) →
activate(
Y)
from(
X) →
cons(
X,
n__from(
n__s(
X)))
cons(
X1,
X2) →
n__cons(
X1,
X2)
from(
X) →
n__from(
X)
s(
X) →
n__s(
X)
activate(
n__cons(
X1,
X2)) →
cons(
activate(
X1),
X2)
activate(
n__from(
X)) →
from(
activate(
X))
activate(
n__s(
X)) →
s(
activate(
X))
activate(
X) →
XTypes:
2nd :: n__cons:n__s:n__from → n__cons:n__s:n__from
cons :: n__cons:n__s:n__from → n__cons:n__s:n__from → n__cons:n__s:n__from
n__cons :: n__cons:n__s:n__from → n__cons:n__s:n__from → n__cons:n__s:n__from
activate :: n__cons:n__s:n__from → n__cons:n__s:n__from
from :: n__cons:n__s:n__from → n__cons:n__s:n__from
n__from :: n__cons:n__s:n__from → n__cons:n__s:n__from
n__s :: n__cons:n__s:n__from → n__cons:n__s:n__from
s :: n__cons:n__s:n__from → n__cons:n__s:n__from
hole_n__cons:n__s:n__from1_0 :: n__cons:n__s:n__from
gen_n__cons:n__s:n__from2_0 :: Nat → n__cons:n__s:n__from
Generator Equations:
gen_n__cons:n__s:n__from2_0(0) ⇔ hole_n__cons:n__s:n__from1_0
gen_n__cons:n__s:n__from2_0(+(x, 1)) ⇔ n__cons(gen_n__cons:n__s:n__from2_0(x), hole_n__cons:n__s:n__from1_0)
No more defined symbols left to analyse.