(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
U11(tt, N, X, XS) → U12(splitAt(activate(N), activate(XS)), activate(X))
U12(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
afterNth(N, XS) → snd(splitAt(N, XS))
and(tt, X) → activate(X)
fst(pair(X, Y)) → X
head(cons(N, XS)) → N
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → head(afterNth(N, XS))
snd(pair(X, Y)) → Y
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U11(tt, N, X, activate(XS))
tail(cons(N, XS)) → activate(XS)
take(N, XS) → fst(splitAt(N, XS))
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(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__natsFrom(X)) →+ natsFrom(activate(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / n__natsFrom(X)].
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:
U11(tt, N, X, XS) → U12(splitAt(activate(N), activate(XS)), activate(X))
U12(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
afterNth(N, XS) → snd(splitAt(N, XS))
and(tt, X) → activate(X)
fst(pair(X, Y)) → X
head(cons(N, XS)) → N
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → head(afterNth(N, XS))
snd(pair(X, Y)) → Y
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U11(tt, N, X, activate(XS))
tail(cons(N, XS)) → activate(XS)
take(N, XS) → fst(splitAt(N, XS))
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(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:
U11(tt, N, X, XS) → U12(splitAt(activate(N), activate(XS)), activate(X))
U12(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
afterNth(N, XS) → snd(splitAt(N, XS))
and(tt, X) → activate(X)
fst(pair(X, Y)) → X
head(cons(N, XS)) → N
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → head(afterNth(N, XS))
snd(pair(X, Y)) → Y
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U11(tt, N, X, activate(XS))
tail(cons(N, XS)) → activate(XS)
take(N, XS) → fst(splitAt(N, XS))
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X
Types:
U11 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
tt :: tt
U12 :: pair → cons:n__s:n__natsFrom:0':nil → pair
splitAt :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
activate :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
pair :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
cons :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
afterNth :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
snd :: pair → cons:n__s:n__natsFrom:0':nil
and :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
fst :: pair → cons:n__s:n__natsFrom:0':nil
head :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
natsFrom :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
n__natsFrom :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
n__s :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
sel :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
0' :: cons:n__s:n__natsFrom:0':nil
nil :: cons:n__s:n__natsFrom:0':nil
s :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
tail :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
take :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
hole_pair1_0 :: pair
hole_tt2_0 :: tt
hole_cons:n__s:n__natsFrom:0':nil3_0 :: cons:n__s:n__natsFrom:0':nil
gen_cons:n__s:n__natsFrom:0':nil4_0 :: Nat → cons:n__s:n__natsFrom:0':nil
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
splitAt,
activateThey will be analysed ascendingly in the following order:
activate < splitAt
(8) Obligation:
Innermost TRS:
Rules:
U11(
tt,
N,
X,
XS) →
U12(
splitAt(
activate(
N),
activate(
XS)),
activate(
X))
U12(
pair(
YS,
ZS),
X) →
pair(
cons(
activate(
X),
YS),
ZS)
afterNth(
N,
XS) →
snd(
splitAt(
N,
XS))
and(
tt,
X) →
activate(
X)
fst(
pair(
X,
Y)) →
Xhead(
cons(
N,
XS)) →
NnatsFrom(
N) →
cons(
N,
n__natsFrom(
n__s(
N)))
sel(
N,
XS) →
head(
afterNth(
N,
XS))
snd(
pair(
X,
Y)) →
YsplitAt(
0',
XS) →
pair(
nil,
XS)
splitAt(
s(
N),
cons(
X,
XS)) →
U11(
tt,
N,
X,
activate(
XS))
tail(
cons(
N,
XS)) →
activate(
XS)
take(
N,
XS) →
fst(
splitAt(
N,
XS))
natsFrom(
X) →
n__natsFrom(
X)
s(
X) →
n__s(
X)
activate(
n__natsFrom(
X)) →
natsFrom(
activate(
X))
activate(
n__s(
X)) →
s(
activate(
X))
activate(
X) →
XTypes:
U11 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
tt :: tt
U12 :: pair → cons:n__s:n__natsFrom:0':nil → pair
splitAt :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
activate :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
pair :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
cons :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
afterNth :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
snd :: pair → cons:n__s:n__natsFrom:0':nil
and :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
fst :: pair → cons:n__s:n__natsFrom:0':nil
head :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
natsFrom :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
n__natsFrom :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
n__s :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
sel :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
0' :: cons:n__s:n__natsFrom:0':nil
nil :: cons:n__s:n__natsFrom:0':nil
s :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
tail :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
take :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
hole_pair1_0 :: pair
hole_tt2_0 :: tt
hole_cons:n__s:n__natsFrom:0':nil3_0 :: cons:n__s:n__natsFrom:0':nil
gen_cons:n__s:n__natsFrom:0':nil4_0 :: Nat → cons:n__s:n__natsFrom:0':nil
Generator Equations:
gen_cons:n__s:n__natsFrom:0':nil4_0(0) ⇔ 0'
gen_cons:n__s:n__natsFrom:0':nil4_0(+(x, 1)) ⇔ cons(0', gen_cons:n__s:n__natsFrom:0':nil4_0(x))
The following defined symbols remain to be analysed:
activate, splitAt
They will be analysed ascendingly in the following order:
activate < splitAt
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol activate.
(10) Obligation:
Innermost TRS:
Rules:
U11(
tt,
N,
X,
XS) →
U12(
splitAt(
activate(
N),
activate(
XS)),
activate(
X))
U12(
pair(
YS,
ZS),
X) →
pair(
cons(
activate(
X),
YS),
ZS)
afterNth(
N,
XS) →
snd(
splitAt(
N,
XS))
and(
tt,
X) →
activate(
X)
fst(
pair(
X,
Y)) →
Xhead(
cons(
N,
XS)) →
NnatsFrom(
N) →
cons(
N,
n__natsFrom(
n__s(
N)))
sel(
N,
XS) →
head(
afterNth(
N,
XS))
snd(
pair(
X,
Y)) →
YsplitAt(
0',
XS) →
pair(
nil,
XS)
splitAt(
s(
N),
cons(
X,
XS)) →
U11(
tt,
N,
X,
activate(
XS))
tail(
cons(
N,
XS)) →
activate(
XS)
take(
N,
XS) →
fst(
splitAt(
N,
XS))
natsFrom(
X) →
n__natsFrom(
X)
s(
X) →
n__s(
X)
activate(
n__natsFrom(
X)) →
natsFrom(
activate(
X))
activate(
n__s(
X)) →
s(
activate(
X))
activate(
X) →
XTypes:
U11 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
tt :: tt
U12 :: pair → cons:n__s:n__natsFrom:0':nil → pair
splitAt :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
activate :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
pair :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
cons :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
afterNth :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
snd :: pair → cons:n__s:n__natsFrom:0':nil
and :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
fst :: pair → cons:n__s:n__natsFrom:0':nil
head :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
natsFrom :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
n__natsFrom :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
n__s :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
sel :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
0' :: cons:n__s:n__natsFrom:0':nil
nil :: cons:n__s:n__natsFrom:0':nil
s :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
tail :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
take :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
hole_pair1_0 :: pair
hole_tt2_0 :: tt
hole_cons:n__s:n__natsFrom:0':nil3_0 :: cons:n__s:n__natsFrom:0':nil
gen_cons:n__s:n__natsFrom:0':nil4_0 :: Nat → cons:n__s:n__natsFrom:0':nil
Generator Equations:
gen_cons:n__s:n__natsFrom:0':nil4_0(0) ⇔ 0'
gen_cons:n__s:n__natsFrom:0':nil4_0(+(x, 1)) ⇔ cons(0', gen_cons:n__s:n__natsFrom:0':nil4_0(x))
The following defined symbols remain to be analysed:
splitAt
(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol splitAt.
(12) Obligation:
Innermost TRS:
Rules:
U11(
tt,
N,
X,
XS) →
U12(
splitAt(
activate(
N),
activate(
XS)),
activate(
X))
U12(
pair(
YS,
ZS),
X) →
pair(
cons(
activate(
X),
YS),
ZS)
afterNth(
N,
XS) →
snd(
splitAt(
N,
XS))
and(
tt,
X) →
activate(
X)
fst(
pair(
X,
Y)) →
Xhead(
cons(
N,
XS)) →
NnatsFrom(
N) →
cons(
N,
n__natsFrom(
n__s(
N)))
sel(
N,
XS) →
head(
afterNth(
N,
XS))
snd(
pair(
X,
Y)) →
YsplitAt(
0',
XS) →
pair(
nil,
XS)
splitAt(
s(
N),
cons(
X,
XS)) →
U11(
tt,
N,
X,
activate(
XS))
tail(
cons(
N,
XS)) →
activate(
XS)
take(
N,
XS) →
fst(
splitAt(
N,
XS))
natsFrom(
X) →
n__natsFrom(
X)
s(
X) →
n__s(
X)
activate(
n__natsFrom(
X)) →
natsFrom(
activate(
X))
activate(
n__s(
X)) →
s(
activate(
X))
activate(
X) →
XTypes:
U11 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
tt :: tt
U12 :: pair → cons:n__s:n__natsFrom:0':nil → pair
splitAt :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
activate :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
pair :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
cons :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
afterNth :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
snd :: pair → cons:n__s:n__natsFrom:0':nil
and :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
fst :: pair → cons:n__s:n__natsFrom:0':nil
head :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
natsFrom :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
n__natsFrom :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
n__s :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
sel :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
0' :: cons:n__s:n__natsFrom:0':nil
nil :: cons:n__s:n__natsFrom:0':nil
s :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
tail :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
take :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
hole_pair1_0 :: pair
hole_tt2_0 :: tt
hole_cons:n__s:n__natsFrom:0':nil3_0 :: cons:n__s:n__natsFrom:0':nil
gen_cons:n__s:n__natsFrom:0':nil4_0 :: Nat → cons:n__s:n__natsFrom:0':nil
Generator Equations:
gen_cons:n__s:n__natsFrom:0':nil4_0(0) ⇔ 0'
gen_cons:n__s:n__natsFrom:0':nil4_0(+(x, 1)) ⇔ cons(0', gen_cons:n__s:n__natsFrom:0':nil4_0(x))
No more defined symbols left to analyse.