(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
from(X) → cons(X, n__from(n__s(X)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, activate(XS))
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:
from(X) → cons(X, n__from(n__s(X)))
after(0', XS) → XS
after(s(N), cons(X, XS)) → after(N, activate(XS))
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:
cons/0
(6) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
from(X) → cons(n__from(n__s(X)))
after(0', XS) → XS
after(s(N), cons(XS)) → after(N, activate(XS))
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:
from(X) → cons(n__from(n__s(X)))
after(0', XS) → XS
after(s(N), cons(XS)) → after(N, activate(XS))
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:
from :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
cons :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
n__from :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
n__s :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
after :: n__s:n__from:cons:0' → n__s:n__from:cons:0' → n__s:n__from:cons:0'
0' :: n__s:n__from:cons:0'
s :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
activate :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
hole_n__s:n__from:cons:0'1_0 :: n__s:n__from:cons:0'
gen_n__s:n__from:cons:0'2_0 :: Nat → n__s:n__from:cons:0'
(9) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
after,
activateThey will be analysed ascendingly in the following order:
activate < after
(10) Obligation:
TRS:
Rules:
from(
X) →
cons(
n__from(
n__s(
X)))
after(
0',
XS) →
XSafter(
s(
N),
cons(
XS)) →
after(
N,
activate(
XS))
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:
from :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
cons :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
n__from :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
n__s :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
after :: n__s:n__from:cons:0' → n__s:n__from:cons:0' → n__s:n__from:cons:0'
0' :: n__s:n__from:cons:0'
s :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
activate :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
hole_n__s:n__from:cons:0'1_0 :: n__s:n__from:cons:0'
gen_n__s:n__from:cons:0'2_0 :: Nat → n__s:n__from:cons:0'
Generator Equations:
gen_n__s:n__from:cons:0'2_0(0) ⇔ 0'
gen_n__s:n__from:cons:0'2_0(+(x, 1)) ⇔ cons(gen_n__s:n__from:cons:0'2_0(x))
The following defined symbols remain to be analysed:
activate, after
They will be analysed ascendingly in the following order:
activate < after
(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol activate.
(12) Obligation:
TRS:
Rules:
from(
X) →
cons(
n__from(
n__s(
X)))
after(
0',
XS) →
XSafter(
s(
N),
cons(
XS)) →
after(
N,
activate(
XS))
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:
from :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
cons :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
n__from :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
n__s :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
after :: n__s:n__from:cons:0' → n__s:n__from:cons:0' → n__s:n__from:cons:0'
0' :: n__s:n__from:cons:0'
s :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
activate :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
hole_n__s:n__from:cons:0'1_0 :: n__s:n__from:cons:0'
gen_n__s:n__from:cons:0'2_0 :: Nat → n__s:n__from:cons:0'
Generator Equations:
gen_n__s:n__from:cons:0'2_0(0) ⇔ 0'
gen_n__s:n__from:cons:0'2_0(+(x, 1)) ⇔ cons(gen_n__s:n__from:cons:0'2_0(x))
The following defined symbols remain to be analysed:
after
(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol after.
(14) Obligation:
TRS:
Rules:
from(
X) →
cons(
n__from(
n__s(
X)))
after(
0',
XS) →
XSafter(
s(
N),
cons(
XS)) →
after(
N,
activate(
XS))
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:
from :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
cons :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
n__from :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
n__s :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
after :: n__s:n__from:cons:0' → n__s:n__from:cons:0' → n__s:n__from:cons:0'
0' :: n__s:n__from:cons:0'
s :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
activate :: n__s:n__from:cons:0' → n__s:n__from:cons:0'
hole_n__s:n__from:cons:0'1_0 :: n__s:n__from:cons:0'
gen_n__s:n__from:cons:0'2_0 :: Nat → n__s:n__from:cons:0'
Generator Equations:
gen_n__s:n__from:cons:0'2_0(0) ⇔ 0'
gen_n__s:n__from:cons:0'2_0(+(x, 1)) ⇔ cons(gen_n__s:n__from:cons:0'2_0(x))
No more defined symbols left to analyse.