(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
U11(tt, N, XS) → U12(tt, activate(N), activate(XS))
U12(tt, N, XS) → snd(splitAt(activate(N), activate(XS)))
U21(tt, X) → U22(tt, activate(X))
U22(tt, X) → activate(X)
U31(tt, N) → U32(tt, activate(N))
U32(tt, N) → activate(N)
U41(tt, N, XS) → U42(tt, activate(N), activate(XS))
U42(tt, N, XS) → head(afterNth(activate(N), activate(XS)))
U51(tt, Y) → U52(tt, activate(Y))
U52(tt, Y) → activate(Y)
U61(tt, N, X, XS) → U62(tt, activate(N), activate(X), activate(XS))
U62(tt, N, X, XS) → U63(tt, activate(N), activate(X), activate(XS))
U63(tt, N, X, XS) → U64(splitAt(activate(N), activate(XS)), activate(X))
U64(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
U71(tt, XS) → U72(tt, activate(XS))
U72(tt, XS) → activate(XS)
U81(tt, N, XS) → U82(tt, activate(N), activate(XS))
U82(tt, N, XS) → fst(splitAt(activate(N), activate(XS)))
afterNth(N, XS) → U11(tt, N, XS)
fst(pair(X, Y)) → U21(tt, X)
head(cons(N, XS)) → U31(tt, N)
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → U41(tt, N, XS)
snd(pair(X, Y)) → U51(tt, Y)
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U61(tt, N, X, activate(XS))
tail(cons(N, XS)) → U71(tt, activate(XS))
take(N, XS) → U81(tt, 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, XS) → U12(tt, activate(N), activate(XS))
U12(tt, N, XS) → snd(splitAt(activate(N), activate(XS)))
U21(tt, X) → U22(tt, activate(X))
U22(tt, X) → activate(X)
U31(tt, N) → U32(tt, activate(N))
U32(tt, N) → activate(N)
U41(tt, N, XS) → U42(tt, activate(N), activate(XS))
U42(tt, N, XS) → head(afterNth(activate(N), activate(XS)))
U51(tt, Y) → U52(tt, activate(Y))
U52(tt, Y) → activate(Y)
U61(tt, N, X, XS) → U62(tt, activate(N), activate(X), activate(XS))
U62(tt, N, X, XS) → U63(tt, activate(N), activate(X), activate(XS))
U63(tt, N, X, XS) → U64(splitAt(activate(N), activate(XS)), activate(X))
U64(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
U71(tt, XS) → U72(tt, activate(XS))
U72(tt, XS) → activate(XS)
U81(tt, N, XS) → U82(tt, activate(N), activate(XS))
U82(tt, N, XS) → fst(splitAt(activate(N), activate(XS)))
afterNth(N, XS) → U11(tt, N, XS)
fst(pair(X, Y)) → U21(tt, X)
head(cons(N, XS)) → U31(tt, N)
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → U41(tt, N, XS)
snd(pair(X, Y)) → U51(tt, Y)
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U61(tt, N, X, activate(XS))
tail(cons(N, XS)) → U71(tt, activate(XS))
take(N, XS) → U81(tt, 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, XS) → U12(tt, activate(N), activate(XS))
U12(tt, N, XS) → snd(splitAt(activate(N), activate(XS)))
U21(tt, X) → U22(tt, activate(X))
U22(tt, X) → activate(X)
U31(tt, N) → U32(tt, activate(N))
U32(tt, N) → activate(N)
U41(tt, N, XS) → U42(tt, activate(N), activate(XS))
U42(tt, N, XS) → head(afterNth(activate(N), activate(XS)))
U51(tt, Y) → U52(tt, activate(Y))
U52(tt, Y) → activate(Y)
U61(tt, N, X, XS) → U62(tt, activate(N), activate(X), activate(XS))
U62(tt, N, X, XS) → U63(tt, activate(N), activate(X), activate(XS))
U63(tt, N, X, XS) → U64(splitAt(activate(N), activate(XS)), activate(X))
U64(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
U71(tt, XS) → U72(tt, activate(XS))
U72(tt, XS) → activate(XS)
U81(tt, N, XS) → U82(tt, activate(N), activate(XS))
U82(tt, N, XS) → fst(splitAt(activate(N), activate(XS)))
afterNth(N, XS) → U11(tt, N, XS)
fst(pair(X, Y)) → U21(tt, X)
head(cons(N, XS)) → U31(tt, N)
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → U41(tt, N, XS)
snd(pair(X, Y)) → U51(tt, Y)
splitAt(0', XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U61(tt, N, X, activate(XS))
tail(cons(N, XS)) → U71(tt, activate(XS))
take(N, XS) → U81(tt, 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
tt :: tt
U12 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
activate :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
snd :: pair → cons:n__s:n__natsFrom:0':nil
splitAt :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
U21 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U22 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U31 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U32 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U41 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U42 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
head :: 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
U51 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U52 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U61 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
U62 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
U63 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
U64 :: pair → cons:n__s:n__natsFrom:0':nil → pair
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
U71 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U72 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U81 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U82 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
fst :: pair → 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_cons:n__s:n__natsFrom:0':nil1_0 :: cons:n__s:n__natsFrom:0':nil
hole_tt2_0 :: tt
hole_pair3_0 :: pair
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:
activate,
splitAtThey will be analysed ascendingly in the following order:
activate < splitAt
(8) Obligation:
Innermost TRS:
Rules:
U11(
tt,
N,
XS) →
U12(
tt,
activate(
N),
activate(
XS))
U12(
tt,
N,
XS) →
snd(
splitAt(
activate(
N),
activate(
XS)))
U21(
tt,
X) →
U22(
tt,
activate(
X))
U22(
tt,
X) →
activate(
X)
U31(
tt,
N) →
U32(
tt,
activate(
N))
U32(
tt,
N) →
activate(
N)
U41(
tt,
N,
XS) →
U42(
tt,
activate(
N),
activate(
XS))
U42(
tt,
N,
XS) →
head(
afterNth(
activate(
N),
activate(
XS)))
U51(
tt,
Y) →
U52(
tt,
activate(
Y))
U52(
tt,
Y) →
activate(
Y)
U61(
tt,
N,
X,
XS) →
U62(
tt,
activate(
N),
activate(
X),
activate(
XS))
U62(
tt,
N,
X,
XS) →
U63(
tt,
activate(
N),
activate(
X),
activate(
XS))
U63(
tt,
N,
X,
XS) →
U64(
splitAt(
activate(
N),
activate(
XS)),
activate(
X))
U64(
pair(
YS,
ZS),
X) →
pair(
cons(
activate(
X),
YS),
ZS)
U71(
tt,
XS) →
U72(
tt,
activate(
XS))
U72(
tt,
XS) →
activate(
XS)
U81(
tt,
N,
XS) →
U82(
tt,
activate(
N),
activate(
XS))
U82(
tt,
N,
XS) →
fst(
splitAt(
activate(
N),
activate(
XS)))
afterNth(
N,
XS) →
U11(
tt,
N,
XS)
fst(
pair(
X,
Y)) →
U21(
tt,
X)
head(
cons(
N,
XS)) →
U31(
tt,
N)
natsFrom(
N) →
cons(
N,
n__natsFrom(
n__s(
N)))
sel(
N,
XS) →
U41(
tt,
N,
XS)
snd(
pair(
X,
Y)) →
U51(
tt,
Y)
splitAt(
0',
XS) →
pair(
nil,
XS)
splitAt(
s(
N),
cons(
X,
XS)) →
U61(
tt,
N,
X,
activate(
XS))
tail(
cons(
N,
XS)) →
U71(
tt,
activate(
XS))
take(
N,
XS) →
U81(
tt,
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
tt :: tt
U12 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
activate :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
snd :: pair → cons:n__s:n__natsFrom:0':nil
splitAt :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
U21 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U22 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U31 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U32 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U41 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U42 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
head :: 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
U51 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U52 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U61 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
U62 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
U63 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
U64 :: pair → cons:n__s:n__natsFrom:0':nil → pair
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
U71 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U72 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U81 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U82 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
fst :: pair → 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_cons:n__s:n__natsFrom:0':nil1_0 :: cons:n__s:n__natsFrom:0':nil
hole_tt2_0 :: tt
hole_pair3_0 :: pair
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,
XS) →
U12(
tt,
activate(
N),
activate(
XS))
U12(
tt,
N,
XS) →
snd(
splitAt(
activate(
N),
activate(
XS)))
U21(
tt,
X) →
U22(
tt,
activate(
X))
U22(
tt,
X) →
activate(
X)
U31(
tt,
N) →
U32(
tt,
activate(
N))
U32(
tt,
N) →
activate(
N)
U41(
tt,
N,
XS) →
U42(
tt,
activate(
N),
activate(
XS))
U42(
tt,
N,
XS) →
head(
afterNth(
activate(
N),
activate(
XS)))
U51(
tt,
Y) →
U52(
tt,
activate(
Y))
U52(
tt,
Y) →
activate(
Y)
U61(
tt,
N,
X,
XS) →
U62(
tt,
activate(
N),
activate(
X),
activate(
XS))
U62(
tt,
N,
X,
XS) →
U63(
tt,
activate(
N),
activate(
X),
activate(
XS))
U63(
tt,
N,
X,
XS) →
U64(
splitAt(
activate(
N),
activate(
XS)),
activate(
X))
U64(
pair(
YS,
ZS),
X) →
pair(
cons(
activate(
X),
YS),
ZS)
U71(
tt,
XS) →
U72(
tt,
activate(
XS))
U72(
tt,
XS) →
activate(
XS)
U81(
tt,
N,
XS) →
U82(
tt,
activate(
N),
activate(
XS))
U82(
tt,
N,
XS) →
fst(
splitAt(
activate(
N),
activate(
XS)))
afterNth(
N,
XS) →
U11(
tt,
N,
XS)
fst(
pair(
X,
Y)) →
U21(
tt,
X)
head(
cons(
N,
XS)) →
U31(
tt,
N)
natsFrom(
N) →
cons(
N,
n__natsFrom(
n__s(
N)))
sel(
N,
XS) →
U41(
tt,
N,
XS)
snd(
pair(
X,
Y)) →
U51(
tt,
Y)
splitAt(
0',
XS) →
pair(
nil,
XS)
splitAt(
s(
N),
cons(
X,
XS)) →
U61(
tt,
N,
X,
activate(
XS))
tail(
cons(
N,
XS)) →
U71(
tt,
activate(
XS))
take(
N,
XS) →
U81(
tt,
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
tt :: tt
U12 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
activate :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
snd :: pair → cons:n__s:n__natsFrom:0':nil
splitAt :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
U21 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U22 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U31 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U32 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U41 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U42 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
head :: 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
U51 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U52 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U61 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
U62 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
U63 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
U64 :: pair → cons:n__s:n__natsFrom:0':nil → pair
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
U71 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U72 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U81 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U82 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
fst :: pair → 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_cons:n__s:n__natsFrom:0':nil1_0 :: cons:n__s:n__natsFrom:0':nil
hole_tt2_0 :: tt
hole_pair3_0 :: pair
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,
XS) →
U12(
tt,
activate(
N),
activate(
XS))
U12(
tt,
N,
XS) →
snd(
splitAt(
activate(
N),
activate(
XS)))
U21(
tt,
X) →
U22(
tt,
activate(
X))
U22(
tt,
X) →
activate(
X)
U31(
tt,
N) →
U32(
tt,
activate(
N))
U32(
tt,
N) →
activate(
N)
U41(
tt,
N,
XS) →
U42(
tt,
activate(
N),
activate(
XS))
U42(
tt,
N,
XS) →
head(
afterNth(
activate(
N),
activate(
XS)))
U51(
tt,
Y) →
U52(
tt,
activate(
Y))
U52(
tt,
Y) →
activate(
Y)
U61(
tt,
N,
X,
XS) →
U62(
tt,
activate(
N),
activate(
X),
activate(
XS))
U62(
tt,
N,
X,
XS) →
U63(
tt,
activate(
N),
activate(
X),
activate(
XS))
U63(
tt,
N,
X,
XS) →
U64(
splitAt(
activate(
N),
activate(
XS)),
activate(
X))
U64(
pair(
YS,
ZS),
X) →
pair(
cons(
activate(
X),
YS),
ZS)
U71(
tt,
XS) →
U72(
tt,
activate(
XS))
U72(
tt,
XS) →
activate(
XS)
U81(
tt,
N,
XS) →
U82(
tt,
activate(
N),
activate(
XS))
U82(
tt,
N,
XS) →
fst(
splitAt(
activate(
N),
activate(
XS)))
afterNth(
N,
XS) →
U11(
tt,
N,
XS)
fst(
pair(
X,
Y)) →
U21(
tt,
X)
head(
cons(
N,
XS)) →
U31(
tt,
N)
natsFrom(
N) →
cons(
N,
n__natsFrom(
n__s(
N)))
sel(
N,
XS) →
U41(
tt,
N,
XS)
snd(
pair(
X,
Y)) →
U51(
tt,
Y)
splitAt(
0',
XS) →
pair(
nil,
XS)
splitAt(
s(
N),
cons(
X,
XS)) →
U61(
tt,
N,
X,
activate(
XS))
tail(
cons(
N,
XS)) →
U71(
tt,
activate(
XS))
take(
N,
XS) →
U81(
tt,
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
tt :: tt
U12 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
activate :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
snd :: pair → cons:n__s:n__natsFrom:0':nil
splitAt :: cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
U21 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U22 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U31 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U32 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U41 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U42 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
head :: 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
U51 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U52 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U61 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
U62 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
U63 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → pair
U64 :: pair → cons:n__s:n__natsFrom:0':nil → pair
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
U71 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U72 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U81 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
U82 :: tt → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil → cons:n__s:n__natsFrom:0':nil
fst :: pair → 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_cons:n__s:n__natsFrom:0':nil1_0 :: cons:n__s:n__natsFrom:0':nil
hole_tt2_0 :: tt
hole_pair3_0 :: pair
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.