(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(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: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
activate(n__natsFrom(X)) →+ cons(activate(X), n__natsFrom(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__natsFrom(X)].
The result substitution is [ ].
The rewrite sequence
activate(n__natsFrom(X)) →+ cons(activate(X), n__natsFrom(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__natsFrom(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:
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(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: FULL
(5) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
u/1
(6) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), X, activate(XS))
u(pair(YS, ZS), X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(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: FULL
(7) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(8) Obligation:
TRS:
Rules:
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), X, activate(XS))
u(pair(YS, ZS), X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(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:
natsFrom :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
cons :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
n__natsFrom :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
n__s :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
fst :: pair → n__s:n__natsFrom:cons:0':nil
pair :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
snd :: pair → n__s:n__natsFrom:cons:0':nil
splitAt :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
0' :: n__s:n__natsFrom:cons:0':nil
nil :: n__s:n__natsFrom:cons:0':nil
s :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
u :: pair → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
activate :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
head :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
tail :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
sel :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
afterNth :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
take :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
hole_n__s:n__natsFrom:cons:0':nil1_0 :: n__s:n__natsFrom:cons:0':nil
hole_pair2_0 :: pair
gen_n__s:n__natsFrom:cons:0':nil3_0 :: Nat → n__s:n__natsFrom:cons:0':nil
(9) 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
(10) Obligation:
TRS:
Rules:
natsFrom(
N) →
cons(
N,
n__natsFrom(
n__s(
N)))
fst(
pair(
XS,
YS)) →
XSsnd(
pair(
XS,
YS)) →
YSsplitAt(
0',
XS) →
pair(
nil,
XS)
splitAt(
s(
N),
cons(
X,
XS)) →
u(
splitAt(
N,
activate(
XS)),
X,
activate(
XS))
u(
pair(
YS,
ZS),
X,
XS) →
pair(
cons(
activate(
X),
YS),
ZS)
head(
cons(
N,
XS)) →
Ntail(
cons(
N,
XS)) →
activate(
XS)
sel(
N,
XS) →
head(
afterNth(
N,
XS))
take(
N,
XS) →
fst(
splitAt(
N,
XS))
afterNth(
N,
XS) →
snd(
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:
natsFrom :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
cons :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
n__natsFrom :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
n__s :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
fst :: pair → n__s:n__natsFrom:cons:0':nil
pair :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
snd :: pair → n__s:n__natsFrom:cons:0':nil
splitAt :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
0' :: n__s:n__natsFrom:cons:0':nil
nil :: n__s:n__natsFrom:cons:0':nil
s :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
u :: pair → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
activate :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
head :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
tail :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
sel :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
afterNth :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
take :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
hole_n__s:n__natsFrom:cons:0':nil1_0 :: n__s:n__natsFrom:cons:0':nil
hole_pair2_0 :: pair
gen_n__s:n__natsFrom:cons:0':nil3_0 :: Nat → n__s:n__natsFrom:cons:0':nil
Generator Equations:
gen_n__s:n__natsFrom:cons:0':nil3_0(0) ⇔ 0'
gen_n__s:n__natsFrom:cons:0':nil3_0(+(x, 1)) ⇔ cons(0', gen_n__s:n__natsFrom:cons:0':nil3_0(x))
The following defined symbols remain to be analysed:
activate, splitAt
They will be analysed ascendingly in the following order:
activate < splitAt
(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol activate.
(12) Obligation:
TRS:
Rules:
natsFrom(
N) →
cons(
N,
n__natsFrom(
n__s(
N)))
fst(
pair(
XS,
YS)) →
XSsnd(
pair(
XS,
YS)) →
YSsplitAt(
0',
XS) →
pair(
nil,
XS)
splitAt(
s(
N),
cons(
X,
XS)) →
u(
splitAt(
N,
activate(
XS)),
X,
activate(
XS))
u(
pair(
YS,
ZS),
X,
XS) →
pair(
cons(
activate(
X),
YS),
ZS)
head(
cons(
N,
XS)) →
Ntail(
cons(
N,
XS)) →
activate(
XS)
sel(
N,
XS) →
head(
afterNth(
N,
XS))
take(
N,
XS) →
fst(
splitAt(
N,
XS))
afterNth(
N,
XS) →
snd(
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:
natsFrom :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
cons :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
n__natsFrom :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
n__s :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
fst :: pair → n__s:n__natsFrom:cons:0':nil
pair :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
snd :: pair → n__s:n__natsFrom:cons:0':nil
splitAt :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
0' :: n__s:n__natsFrom:cons:0':nil
nil :: n__s:n__natsFrom:cons:0':nil
s :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
u :: pair → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
activate :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
head :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
tail :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
sel :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
afterNth :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
take :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
hole_n__s:n__natsFrom:cons:0':nil1_0 :: n__s:n__natsFrom:cons:0':nil
hole_pair2_0 :: pair
gen_n__s:n__natsFrom:cons:0':nil3_0 :: Nat → n__s:n__natsFrom:cons:0':nil
Generator Equations:
gen_n__s:n__natsFrom:cons:0':nil3_0(0) ⇔ 0'
gen_n__s:n__natsFrom:cons:0':nil3_0(+(x, 1)) ⇔ cons(0', gen_n__s:n__natsFrom:cons:0':nil3_0(x))
The following defined symbols remain to be analysed:
splitAt
(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol splitAt.
(14) Obligation:
TRS:
Rules:
natsFrom(
N) →
cons(
N,
n__natsFrom(
n__s(
N)))
fst(
pair(
XS,
YS)) →
XSsnd(
pair(
XS,
YS)) →
YSsplitAt(
0',
XS) →
pair(
nil,
XS)
splitAt(
s(
N),
cons(
X,
XS)) →
u(
splitAt(
N,
activate(
XS)),
X,
activate(
XS))
u(
pair(
YS,
ZS),
X,
XS) →
pair(
cons(
activate(
X),
YS),
ZS)
head(
cons(
N,
XS)) →
Ntail(
cons(
N,
XS)) →
activate(
XS)
sel(
N,
XS) →
head(
afterNth(
N,
XS))
take(
N,
XS) →
fst(
splitAt(
N,
XS))
afterNth(
N,
XS) →
snd(
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:
natsFrom :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
cons :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
n__natsFrom :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
n__s :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
fst :: pair → n__s:n__natsFrom:cons:0':nil
pair :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
snd :: pair → n__s:n__natsFrom:cons:0':nil
splitAt :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
0' :: n__s:n__natsFrom:cons:0':nil
nil :: n__s:n__natsFrom:cons:0':nil
s :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
u :: pair → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → pair
activate :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
head :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
tail :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
sel :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
afterNth :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
take :: n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil → n__s:n__natsFrom:cons:0':nil
hole_n__s:n__natsFrom:cons:0':nil1_0 :: n__s:n__natsFrom:cons:0':nil
hole_pair2_0 :: pair
gen_n__s:n__natsFrom:cons:0':nil3_0 :: Nat → n__s:n__natsFrom:cons:0':nil
Generator Equations:
gen_n__s:n__natsFrom:cons:0':nil3_0(0) ⇔ 0'
gen_n__s:n__natsFrom:cons:0':nil3_0(+(x, 1)) ⇔ cons(0', gen_n__s:n__natsFrom:cons:0':nil3_0(x))
No more defined symbols left to analyse.