(0) Obligation:
Runtime Complexity TRS:
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(n__s(X)))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
activate(n__from(X)) →+ cons(activate(X), n__from(n__s(activate(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / n__from(X)].
The result substitution is [ ].
The rewrite sequence
activate(n__from(X)) →+ cons(activate(X), n__from(n__s(activate(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
The pumping substitution is [X / n__from(X)].
The result substitution is [ ].
(2) BOUNDS(2^n, 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(cons1(X, cons(Y, Z))) → Y
2nd(cons(X, X1)) → 2nd(cons1(X, activate(X1)))
from(X) → cons(X, n__from(n__s(X)))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X
S is empty.
Rewrite Strategy: FULL
(5) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
cons1/0
(6) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
2nd(cons1(cons(Y, Z))) → Y
2nd(cons(X, X1)) → 2nd(cons1(activate(X1)))
from(X) → cons(X, n__from(n__s(X)))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X
S is empty.
Rewrite Strategy: FULL
(7) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(8) Obligation:
TRS:
Rules:
2nd(cons1(cons(Y, Z))) → Y
2nd(cons(X, X1)) → 2nd(cons1(activate(X1)))
from(X) → cons(X, n__from(n__s(X)))
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X
Types:
2nd :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
cons1 :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
cons :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
activate :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
from :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
n__from :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
n__s :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
s :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
hole_cons:cons1:n__s:n__from1_0 :: cons:cons1:n__s:n__from
gen_cons:cons1:n__s:n__from2_0 :: Nat → cons:cons1:n__s:n__from
(9) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
2nd,
activateThey will be analysed ascendingly in the following order:
activate < 2nd
(10) Obligation:
TRS:
Rules:
2nd(
cons1(
cons(
Y,
Z))) →
Y2nd(
cons(
X,
X1)) →
2nd(
cons1(
activate(
X1)))
from(
X) →
cons(
X,
n__from(
n__s(
X)))
from(
X) →
n__from(
X)
s(
X) →
n__s(
X)
activate(
n__from(
X)) →
from(
activate(
X))
activate(
n__s(
X)) →
s(
activate(
X))
activate(
X) →
XTypes:
2nd :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
cons1 :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
cons :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
activate :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
from :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
n__from :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
n__s :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
s :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
hole_cons:cons1:n__s:n__from1_0 :: cons:cons1:n__s:n__from
gen_cons:cons1:n__s:n__from2_0 :: Nat → cons:cons1:n__s:n__from
Generator Equations:
gen_cons:cons1:n__s:n__from2_0(0) ⇔ hole_cons:cons1:n__s:n__from1_0
gen_cons:cons1:n__s:n__from2_0(+(x, 1)) ⇔ cons(hole_cons:cons1:n__s:n__from1_0, gen_cons:cons1:n__s:n__from2_0(x))
The following defined symbols remain to be analysed:
activate, 2nd
They will be analysed ascendingly in the following order:
activate < 2nd
(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol activate.
(12) Obligation:
TRS:
Rules:
2nd(
cons1(
cons(
Y,
Z))) →
Y2nd(
cons(
X,
X1)) →
2nd(
cons1(
activate(
X1)))
from(
X) →
cons(
X,
n__from(
n__s(
X)))
from(
X) →
n__from(
X)
s(
X) →
n__s(
X)
activate(
n__from(
X)) →
from(
activate(
X))
activate(
n__s(
X)) →
s(
activate(
X))
activate(
X) →
XTypes:
2nd :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
cons1 :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
cons :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
activate :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
from :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
n__from :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
n__s :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
s :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
hole_cons:cons1:n__s:n__from1_0 :: cons:cons1:n__s:n__from
gen_cons:cons1:n__s:n__from2_0 :: Nat → cons:cons1:n__s:n__from
Generator Equations:
gen_cons:cons1:n__s:n__from2_0(0) ⇔ hole_cons:cons1:n__s:n__from1_0
gen_cons:cons1:n__s:n__from2_0(+(x, 1)) ⇔ cons(hole_cons:cons1:n__s:n__from1_0, gen_cons:cons1:n__s:n__from2_0(x))
The following defined symbols remain to be analysed:
2nd
(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol 2nd.
(14) Obligation:
TRS:
Rules:
2nd(
cons1(
cons(
Y,
Z))) →
Y2nd(
cons(
X,
X1)) →
2nd(
cons1(
activate(
X1)))
from(
X) →
cons(
X,
n__from(
n__s(
X)))
from(
X) →
n__from(
X)
s(
X) →
n__s(
X)
activate(
n__from(
X)) →
from(
activate(
X))
activate(
n__s(
X)) →
s(
activate(
X))
activate(
X) →
XTypes:
2nd :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
cons1 :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
cons :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
activate :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
from :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
n__from :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
n__s :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
s :: cons:cons1:n__s:n__from → cons:cons1:n__s:n__from
hole_cons:cons1:n__s:n__from1_0 :: cons:cons1:n__s:n__from
gen_cons:cons1:n__s:n__from2_0 :: Nat → cons:cons1:n__s:n__from
Generator Equations:
gen_cons:cons1:n__s:n__from2_0(0) ⇔ hole_cons:cons1:n__s:n__from1_0
gen_cons:cons1:n__s:n__from2_0(+(x, 1)) ⇔ cons(hole_cons:cons1:n__s:n__from1_0, gen_cons:cons1:n__s:n__from2_0(x))
No more defined symbols left to analyse.