(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: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
activate(n__cons(n__from(X128_0), X2)) →+ cons(cons(activate(X128_0), n__from(n__s(activate(X128_0)))), X2)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X128_0 / n__cons(n__from(X128_0), X2)].
The result substitution is [ ].
The rewrite sequence
activate(n__cons(n__from(X128_0), X2)) →+ cons(cons(activate(X128_0), n__from(n__s(activate(X128_0)))), X2)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0,0].
The pumping substitution is [X128_0 / n__cons(n__from(X128_0), X2)].
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(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: FULL
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
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:
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) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
activate(
gen_n__cons:n__s:n__from2_0(
+(
1,
n4_0))) →
*3_0, rt ∈ Ω(n4
0)
Induction Base:
activate(gen_n__cons:n__s:n__from2_0(+(1, 0)))
Induction Step:
activate(gen_n__cons:n__s:n__from2_0(+(1, +(n4_0, 1)))) →RΩ(1)
cons(activate(gen_n__cons:n__s:n__from2_0(+(1, n4_0))), hole_n__cons:n__s:n__from1_0) →IH
cons(*3_0, hole_n__cons:n__s:n__from1_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(10) Complex Obligation (BEST)
(11) Obligation:
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
Lemmas:
activate(gen_n__cons:n__s:n__from2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
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.
(12) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
activate(gen_n__cons:n__s:n__from2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
(13) BOUNDS(n^1, INF)
(14) Obligation:
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
Lemmas:
activate(gen_n__cons:n__s:n__from2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
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.
(15) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
activate(gen_n__cons:n__s:n__from2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
(16) BOUNDS(n^1, INF)