(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
from(X) → cons(X, n__from(s(X)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, activate(XS))
from(X) → n__from(X)
activate(n__from(X)) → from(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
after(s(N), cons(X, XS)) →+ after(N, XS)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [N / s(N), XS / cons(X, XS)].
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:
from(X) → cons(X, n__from(s(X)))
after(0', XS) → XS
after(s(N), cons(X, XS)) → after(N, activate(XS))
from(X) → n__from(X)
activate(n__from(X)) → from(X)
activate(X) → X
S is empty.
Rewrite Strategy: INNERMOST
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
Innermost TRS:
Rules:
from(X) → cons(X, n__from(s(X)))
after(0', XS) → XS
after(s(N), cons(X, XS)) → after(N, activate(XS))
from(X) → n__from(X)
activate(n__from(X)) → from(X)
activate(X) → X
Types:
from :: s:0' → n__from:cons
cons :: s:0' → n__from:cons → n__from:cons
n__from :: s:0' → n__from:cons
s :: s:0' → s:0'
after :: s:0' → n__from:cons → n__from:cons
0' :: s:0'
activate :: n__from:cons → n__from:cons
hole_n__from:cons1_0 :: n__from:cons
hole_s:0'2_0 :: s:0'
gen_n__from:cons3_0 :: Nat → n__from:cons
gen_s:0'4_0 :: Nat → s:0'
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
after
(8) Obligation:
Innermost TRS:
Rules:
from(
X) →
cons(
X,
n__from(
s(
X)))
after(
0',
XS) →
XSafter(
s(
N),
cons(
X,
XS)) →
after(
N,
activate(
XS))
from(
X) →
n__from(
X)
activate(
n__from(
X)) →
from(
X)
activate(
X) →
XTypes:
from :: s:0' → n__from:cons
cons :: s:0' → n__from:cons → n__from:cons
n__from :: s:0' → n__from:cons
s :: s:0' → s:0'
after :: s:0' → n__from:cons → n__from:cons
0' :: s:0'
activate :: n__from:cons → n__from:cons
hole_n__from:cons1_0 :: n__from:cons
hole_s:0'2_0 :: s:0'
gen_n__from:cons3_0 :: Nat → n__from:cons
gen_s:0'4_0 :: Nat → s:0'
Generator Equations:
gen_n__from:cons3_0(0) ⇔ n__from(0')
gen_n__from:cons3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))
The following defined symbols remain to be analysed:
after
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol after.
(10) Obligation:
Innermost TRS:
Rules:
from(
X) →
cons(
X,
n__from(
s(
X)))
after(
0',
XS) →
XSafter(
s(
N),
cons(
X,
XS)) →
after(
N,
activate(
XS))
from(
X) →
n__from(
X)
activate(
n__from(
X)) →
from(
X)
activate(
X) →
XTypes:
from :: s:0' → n__from:cons
cons :: s:0' → n__from:cons → n__from:cons
n__from :: s:0' → n__from:cons
s :: s:0' → s:0'
after :: s:0' → n__from:cons → n__from:cons
0' :: s:0'
activate :: n__from:cons → n__from:cons
hole_n__from:cons1_0 :: n__from:cons
hole_s:0'2_0 :: s:0'
gen_n__from:cons3_0 :: Nat → n__from:cons
gen_s:0'4_0 :: Nat → s:0'
Generator Equations:
gen_n__from:cons3_0(0) ⇔ n__from(0')
gen_n__from:cons3_0(+(x, 1)) ⇔ cons(0', gen_n__from:cons3_0(x))
gen_s:0'4_0(0) ⇔ 0'
gen_s:0'4_0(+(x, 1)) ⇔ s(gen_s:0'4_0(x))
No more defined symbols left to analyse.