(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: INNERMOST
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
cons(mark(X1), X2) →+ mark(cons(X1, X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
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:
active(pairNs) → mark(cons(0', incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0', XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
S is empty.
Rewrite Strategy: INNERMOST
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
Innermost TRS:
Rules:
active(pairNs) → mark(cons(0', incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0', XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active,
cons,
incr,
s,
take,
pair,
zip,
repItems,
tail,
proper,
topThey will be analysed ascendingly in the following order:
cons < active
incr < active
s < active
take < active
pair < active
zip < active
repItems < active
tail < active
active < top
cons < proper
incr < proper
s < proper
take < proper
pair < proper
zip < proper
repItems < proper
tail < proper
proper < top
(8) Obligation:
Innermost TRS:
Rules:
active(
pairNs) →
mark(
cons(
0',
incr(
oddNs)))
active(
oddNs) →
mark(
incr(
pairNs))
active(
incr(
cons(
X,
XS))) →
mark(
cons(
s(
X),
incr(
XS)))
active(
take(
0',
XS)) →
mark(
nil)
active(
take(
s(
N),
cons(
X,
XS))) →
mark(
cons(
X,
take(
N,
XS)))
active(
zip(
nil,
XS)) →
mark(
nil)
active(
zip(
X,
nil)) →
mark(
nil)
active(
zip(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
pair(
X,
Y),
zip(
XS,
YS)))
active(
tail(
cons(
X,
XS))) →
mark(
XS)
active(
repItems(
nil)) →
mark(
nil)
active(
repItems(
cons(
X,
XS))) →
mark(
cons(
X,
cons(
X,
repItems(
XS))))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
incr(
X)) →
incr(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
take(
X1,
X2)) →
take(
active(
X1),
X2)
active(
take(
X1,
X2)) →
take(
X1,
active(
X2))
active(
zip(
X1,
X2)) →
zip(
active(
X1),
X2)
active(
zip(
X1,
X2)) →
zip(
X1,
active(
X2))
active(
pair(
X1,
X2)) →
pair(
active(
X1),
X2)
active(
pair(
X1,
X2)) →
pair(
X1,
active(
X2))
active(
tail(
X)) →
tail(
active(
X))
active(
repItems(
X)) →
repItems(
active(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
incr(
mark(
X)) →
mark(
incr(
X))
s(
mark(
X)) →
mark(
s(
X))
take(
mark(
X1),
X2) →
mark(
take(
X1,
X2))
take(
X1,
mark(
X2)) →
mark(
take(
X1,
X2))
zip(
mark(
X1),
X2) →
mark(
zip(
X1,
X2))
zip(
X1,
mark(
X2)) →
mark(
zip(
X1,
X2))
pair(
mark(
X1),
X2) →
mark(
pair(
X1,
X2))
pair(
X1,
mark(
X2)) →
mark(
pair(
X1,
X2))
tail(
mark(
X)) →
mark(
tail(
X))
repItems(
mark(
X)) →
mark(
repItems(
X))
proper(
pairNs) →
ok(
pairNs)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
incr(
X)) →
incr(
proper(
X))
proper(
oddNs) →
ok(
oddNs)
proper(
s(
X)) →
s(
proper(
X))
proper(
take(
X1,
X2)) →
take(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
zip(
X1,
X2)) →
zip(
proper(
X1),
proper(
X2))
proper(
pair(
X1,
X2)) →
pair(
proper(
X1),
proper(
X2))
proper(
tail(
X)) →
tail(
proper(
X))
proper(
repItems(
X)) →
repItems(
proper(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
incr(
ok(
X)) →
ok(
incr(
X))
s(
ok(
X)) →
ok(
s(
X))
take(
ok(
X1),
ok(
X2)) →
ok(
take(
X1,
X2))
zip(
ok(
X1),
ok(
X2)) →
ok(
zip(
X1,
X2))
pair(
ok(
X1),
ok(
X2)) →
ok(
pair(
X1,
X2))
tail(
ok(
X)) →
ok(
tail(
X))
repItems(
ok(
X)) →
ok(
repItems(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok
Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
cons, active, incr, s, take, pair, zip, repItems, tail, proper, top
They will be analysed ascendingly in the following order:
cons < active
incr < active
s < active
take < active
pair < active
zip < active
repItems < active
tail < active
active < top
cons < proper
incr < proper
s < proper
take < proper
pair < proper
zip < proper
repItems < proper
tail < proper
proper < top
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol cons.
(10) Obligation:
Innermost TRS:
Rules:
active(
pairNs) →
mark(
cons(
0',
incr(
oddNs)))
active(
oddNs) →
mark(
incr(
pairNs))
active(
incr(
cons(
X,
XS))) →
mark(
cons(
s(
X),
incr(
XS)))
active(
take(
0',
XS)) →
mark(
nil)
active(
take(
s(
N),
cons(
X,
XS))) →
mark(
cons(
X,
take(
N,
XS)))
active(
zip(
nil,
XS)) →
mark(
nil)
active(
zip(
X,
nil)) →
mark(
nil)
active(
zip(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
pair(
X,
Y),
zip(
XS,
YS)))
active(
tail(
cons(
X,
XS))) →
mark(
XS)
active(
repItems(
nil)) →
mark(
nil)
active(
repItems(
cons(
X,
XS))) →
mark(
cons(
X,
cons(
X,
repItems(
XS))))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
incr(
X)) →
incr(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
take(
X1,
X2)) →
take(
active(
X1),
X2)
active(
take(
X1,
X2)) →
take(
X1,
active(
X2))
active(
zip(
X1,
X2)) →
zip(
active(
X1),
X2)
active(
zip(
X1,
X2)) →
zip(
X1,
active(
X2))
active(
pair(
X1,
X2)) →
pair(
active(
X1),
X2)
active(
pair(
X1,
X2)) →
pair(
X1,
active(
X2))
active(
tail(
X)) →
tail(
active(
X))
active(
repItems(
X)) →
repItems(
active(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
incr(
mark(
X)) →
mark(
incr(
X))
s(
mark(
X)) →
mark(
s(
X))
take(
mark(
X1),
X2) →
mark(
take(
X1,
X2))
take(
X1,
mark(
X2)) →
mark(
take(
X1,
X2))
zip(
mark(
X1),
X2) →
mark(
zip(
X1,
X2))
zip(
X1,
mark(
X2)) →
mark(
zip(
X1,
X2))
pair(
mark(
X1),
X2) →
mark(
pair(
X1,
X2))
pair(
X1,
mark(
X2)) →
mark(
pair(
X1,
X2))
tail(
mark(
X)) →
mark(
tail(
X))
repItems(
mark(
X)) →
mark(
repItems(
X))
proper(
pairNs) →
ok(
pairNs)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
incr(
X)) →
incr(
proper(
X))
proper(
oddNs) →
ok(
oddNs)
proper(
s(
X)) →
s(
proper(
X))
proper(
take(
X1,
X2)) →
take(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
zip(
X1,
X2)) →
zip(
proper(
X1),
proper(
X2))
proper(
pair(
X1,
X2)) →
pair(
proper(
X1),
proper(
X2))
proper(
tail(
X)) →
tail(
proper(
X))
proper(
repItems(
X)) →
repItems(
proper(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
incr(
ok(
X)) →
ok(
incr(
X))
s(
ok(
X)) →
ok(
s(
X))
take(
ok(
X1),
ok(
X2)) →
ok(
take(
X1,
X2))
zip(
ok(
X1),
ok(
X2)) →
ok(
zip(
X1,
X2))
pair(
ok(
X1),
ok(
X2)) →
ok(
pair(
X1,
X2))
tail(
ok(
X)) →
ok(
tail(
X))
repItems(
ok(
X)) →
ok(
repItems(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok
Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
incr, active, s, take, pair, zip, repItems, tail, proper, top
They will be analysed ascendingly in the following order:
incr < active
s < active
take < active
pair < active
zip < active
repItems < active
tail < active
active < top
incr < proper
s < proper
take < proper
pair < proper
zip < proper
repItems < proper
tail < proper
proper < top
(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol incr.
(12) Obligation:
Innermost TRS:
Rules:
active(
pairNs) →
mark(
cons(
0',
incr(
oddNs)))
active(
oddNs) →
mark(
incr(
pairNs))
active(
incr(
cons(
X,
XS))) →
mark(
cons(
s(
X),
incr(
XS)))
active(
take(
0',
XS)) →
mark(
nil)
active(
take(
s(
N),
cons(
X,
XS))) →
mark(
cons(
X,
take(
N,
XS)))
active(
zip(
nil,
XS)) →
mark(
nil)
active(
zip(
X,
nil)) →
mark(
nil)
active(
zip(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
pair(
X,
Y),
zip(
XS,
YS)))
active(
tail(
cons(
X,
XS))) →
mark(
XS)
active(
repItems(
nil)) →
mark(
nil)
active(
repItems(
cons(
X,
XS))) →
mark(
cons(
X,
cons(
X,
repItems(
XS))))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
incr(
X)) →
incr(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
take(
X1,
X2)) →
take(
active(
X1),
X2)
active(
take(
X1,
X2)) →
take(
X1,
active(
X2))
active(
zip(
X1,
X2)) →
zip(
active(
X1),
X2)
active(
zip(
X1,
X2)) →
zip(
X1,
active(
X2))
active(
pair(
X1,
X2)) →
pair(
active(
X1),
X2)
active(
pair(
X1,
X2)) →
pair(
X1,
active(
X2))
active(
tail(
X)) →
tail(
active(
X))
active(
repItems(
X)) →
repItems(
active(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
incr(
mark(
X)) →
mark(
incr(
X))
s(
mark(
X)) →
mark(
s(
X))
take(
mark(
X1),
X2) →
mark(
take(
X1,
X2))
take(
X1,
mark(
X2)) →
mark(
take(
X1,
X2))
zip(
mark(
X1),
X2) →
mark(
zip(
X1,
X2))
zip(
X1,
mark(
X2)) →
mark(
zip(
X1,
X2))
pair(
mark(
X1),
X2) →
mark(
pair(
X1,
X2))
pair(
X1,
mark(
X2)) →
mark(
pair(
X1,
X2))
tail(
mark(
X)) →
mark(
tail(
X))
repItems(
mark(
X)) →
mark(
repItems(
X))
proper(
pairNs) →
ok(
pairNs)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
incr(
X)) →
incr(
proper(
X))
proper(
oddNs) →
ok(
oddNs)
proper(
s(
X)) →
s(
proper(
X))
proper(
take(
X1,
X2)) →
take(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
zip(
X1,
X2)) →
zip(
proper(
X1),
proper(
X2))
proper(
pair(
X1,
X2)) →
pair(
proper(
X1),
proper(
X2))
proper(
tail(
X)) →
tail(
proper(
X))
proper(
repItems(
X)) →
repItems(
proper(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
incr(
ok(
X)) →
ok(
incr(
X))
s(
ok(
X)) →
ok(
s(
X))
take(
ok(
X1),
ok(
X2)) →
ok(
take(
X1,
X2))
zip(
ok(
X1),
ok(
X2)) →
ok(
zip(
X1,
X2))
pair(
ok(
X1),
ok(
X2)) →
ok(
pair(
X1,
X2))
tail(
ok(
X)) →
ok(
tail(
X))
repItems(
ok(
X)) →
ok(
repItems(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok
Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
s, active, take, pair, zip, repItems, tail, proper, top
They will be analysed ascendingly in the following order:
s < active
take < active
pair < active
zip < active
repItems < active
tail < active
active < top
s < proper
take < proper
pair < proper
zip < proper
repItems < proper
tail < proper
proper < top
(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol s.
(14) Obligation:
Innermost TRS:
Rules:
active(
pairNs) →
mark(
cons(
0',
incr(
oddNs)))
active(
oddNs) →
mark(
incr(
pairNs))
active(
incr(
cons(
X,
XS))) →
mark(
cons(
s(
X),
incr(
XS)))
active(
take(
0',
XS)) →
mark(
nil)
active(
take(
s(
N),
cons(
X,
XS))) →
mark(
cons(
X,
take(
N,
XS)))
active(
zip(
nil,
XS)) →
mark(
nil)
active(
zip(
X,
nil)) →
mark(
nil)
active(
zip(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
pair(
X,
Y),
zip(
XS,
YS)))
active(
tail(
cons(
X,
XS))) →
mark(
XS)
active(
repItems(
nil)) →
mark(
nil)
active(
repItems(
cons(
X,
XS))) →
mark(
cons(
X,
cons(
X,
repItems(
XS))))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
incr(
X)) →
incr(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
take(
X1,
X2)) →
take(
active(
X1),
X2)
active(
take(
X1,
X2)) →
take(
X1,
active(
X2))
active(
zip(
X1,
X2)) →
zip(
active(
X1),
X2)
active(
zip(
X1,
X2)) →
zip(
X1,
active(
X2))
active(
pair(
X1,
X2)) →
pair(
active(
X1),
X2)
active(
pair(
X1,
X2)) →
pair(
X1,
active(
X2))
active(
tail(
X)) →
tail(
active(
X))
active(
repItems(
X)) →
repItems(
active(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
incr(
mark(
X)) →
mark(
incr(
X))
s(
mark(
X)) →
mark(
s(
X))
take(
mark(
X1),
X2) →
mark(
take(
X1,
X2))
take(
X1,
mark(
X2)) →
mark(
take(
X1,
X2))
zip(
mark(
X1),
X2) →
mark(
zip(
X1,
X2))
zip(
X1,
mark(
X2)) →
mark(
zip(
X1,
X2))
pair(
mark(
X1),
X2) →
mark(
pair(
X1,
X2))
pair(
X1,
mark(
X2)) →
mark(
pair(
X1,
X2))
tail(
mark(
X)) →
mark(
tail(
X))
repItems(
mark(
X)) →
mark(
repItems(
X))
proper(
pairNs) →
ok(
pairNs)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
incr(
X)) →
incr(
proper(
X))
proper(
oddNs) →
ok(
oddNs)
proper(
s(
X)) →
s(
proper(
X))
proper(
take(
X1,
X2)) →
take(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
zip(
X1,
X2)) →
zip(
proper(
X1),
proper(
X2))
proper(
pair(
X1,
X2)) →
pair(
proper(
X1),
proper(
X2))
proper(
tail(
X)) →
tail(
proper(
X))
proper(
repItems(
X)) →
repItems(
proper(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
incr(
ok(
X)) →
ok(
incr(
X))
s(
ok(
X)) →
ok(
s(
X))
take(
ok(
X1),
ok(
X2)) →
ok(
take(
X1,
X2))
zip(
ok(
X1),
ok(
X2)) →
ok(
zip(
X1,
X2))
pair(
ok(
X1),
ok(
X2)) →
ok(
pair(
X1,
X2))
tail(
ok(
X)) →
ok(
tail(
X))
repItems(
ok(
X)) →
ok(
repItems(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok
Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
take, active, pair, zip, repItems, tail, proper, top
They will be analysed ascendingly in the following order:
take < active
pair < active
zip < active
repItems < active
tail < active
active < top
take < proper
pair < proper
zip < proper
repItems < proper
tail < proper
proper < top
(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol take.
(16) Obligation:
Innermost TRS:
Rules:
active(
pairNs) →
mark(
cons(
0',
incr(
oddNs)))
active(
oddNs) →
mark(
incr(
pairNs))
active(
incr(
cons(
X,
XS))) →
mark(
cons(
s(
X),
incr(
XS)))
active(
take(
0',
XS)) →
mark(
nil)
active(
take(
s(
N),
cons(
X,
XS))) →
mark(
cons(
X,
take(
N,
XS)))
active(
zip(
nil,
XS)) →
mark(
nil)
active(
zip(
X,
nil)) →
mark(
nil)
active(
zip(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
pair(
X,
Y),
zip(
XS,
YS)))
active(
tail(
cons(
X,
XS))) →
mark(
XS)
active(
repItems(
nil)) →
mark(
nil)
active(
repItems(
cons(
X,
XS))) →
mark(
cons(
X,
cons(
X,
repItems(
XS))))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
incr(
X)) →
incr(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
take(
X1,
X2)) →
take(
active(
X1),
X2)
active(
take(
X1,
X2)) →
take(
X1,
active(
X2))
active(
zip(
X1,
X2)) →
zip(
active(
X1),
X2)
active(
zip(
X1,
X2)) →
zip(
X1,
active(
X2))
active(
pair(
X1,
X2)) →
pair(
active(
X1),
X2)
active(
pair(
X1,
X2)) →
pair(
X1,
active(
X2))
active(
tail(
X)) →
tail(
active(
X))
active(
repItems(
X)) →
repItems(
active(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
incr(
mark(
X)) →
mark(
incr(
X))
s(
mark(
X)) →
mark(
s(
X))
take(
mark(
X1),
X2) →
mark(
take(
X1,
X2))
take(
X1,
mark(
X2)) →
mark(
take(
X1,
X2))
zip(
mark(
X1),
X2) →
mark(
zip(
X1,
X2))
zip(
X1,
mark(
X2)) →
mark(
zip(
X1,
X2))
pair(
mark(
X1),
X2) →
mark(
pair(
X1,
X2))
pair(
X1,
mark(
X2)) →
mark(
pair(
X1,
X2))
tail(
mark(
X)) →
mark(
tail(
X))
repItems(
mark(
X)) →
mark(
repItems(
X))
proper(
pairNs) →
ok(
pairNs)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
incr(
X)) →
incr(
proper(
X))
proper(
oddNs) →
ok(
oddNs)
proper(
s(
X)) →
s(
proper(
X))
proper(
take(
X1,
X2)) →
take(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
zip(
X1,
X2)) →
zip(
proper(
X1),
proper(
X2))
proper(
pair(
X1,
X2)) →
pair(
proper(
X1),
proper(
X2))
proper(
tail(
X)) →
tail(
proper(
X))
proper(
repItems(
X)) →
repItems(
proper(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
incr(
ok(
X)) →
ok(
incr(
X))
s(
ok(
X)) →
ok(
s(
X))
take(
ok(
X1),
ok(
X2)) →
ok(
take(
X1,
X2))
zip(
ok(
X1),
ok(
X2)) →
ok(
zip(
X1,
X2))
pair(
ok(
X1),
ok(
X2)) →
ok(
pair(
X1,
X2))
tail(
ok(
X)) →
ok(
tail(
X))
repItems(
ok(
X)) →
ok(
repItems(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok
Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
pair, active, zip, repItems, tail, proper, top
They will be analysed ascendingly in the following order:
pair < active
zip < active
repItems < active
tail < active
active < top
pair < proper
zip < proper
repItems < proper
tail < proper
proper < top
(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol pair.
(18) Obligation:
Innermost TRS:
Rules:
active(
pairNs) →
mark(
cons(
0',
incr(
oddNs)))
active(
oddNs) →
mark(
incr(
pairNs))
active(
incr(
cons(
X,
XS))) →
mark(
cons(
s(
X),
incr(
XS)))
active(
take(
0',
XS)) →
mark(
nil)
active(
take(
s(
N),
cons(
X,
XS))) →
mark(
cons(
X,
take(
N,
XS)))
active(
zip(
nil,
XS)) →
mark(
nil)
active(
zip(
X,
nil)) →
mark(
nil)
active(
zip(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
pair(
X,
Y),
zip(
XS,
YS)))
active(
tail(
cons(
X,
XS))) →
mark(
XS)
active(
repItems(
nil)) →
mark(
nil)
active(
repItems(
cons(
X,
XS))) →
mark(
cons(
X,
cons(
X,
repItems(
XS))))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
incr(
X)) →
incr(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
take(
X1,
X2)) →
take(
active(
X1),
X2)
active(
take(
X1,
X2)) →
take(
X1,
active(
X2))
active(
zip(
X1,
X2)) →
zip(
active(
X1),
X2)
active(
zip(
X1,
X2)) →
zip(
X1,
active(
X2))
active(
pair(
X1,
X2)) →
pair(
active(
X1),
X2)
active(
pair(
X1,
X2)) →
pair(
X1,
active(
X2))
active(
tail(
X)) →
tail(
active(
X))
active(
repItems(
X)) →
repItems(
active(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
incr(
mark(
X)) →
mark(
incr(
X))
s(
mark(
X)) →
mark(
s(
X))
take(
mark(
X1),
X2) →
mark(
take(
X1,
X2))
take(
X1,
mark(
X2)) →
mark(
take(
X1,
X2))
zip(
mark(
X1),
X2) →
mark(
zip(
X1,
X2))
zip(
X1,
mark(
X2)) →
mark(
zip(
X1,
X2))
pair(
mark(
X1),
X2) →
mark(
pair(
X1,
X2))
pair(
X1,
mark(
X2)) →
mark(
pair(
X1,
X2))
tail(
mark(
X)) →
mark(
tail(
X))
repItems(
mark(
X)) →
mark(
repItems(
X))
proper(
pairNs) →
ok(
pairNs)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
incr(
X)) →
incr(
proper(
X))
proper(
oddNs) →
ok(
oddNs)
proper(
s(
X)) →
s(
proper(
X))
proper(
take(
X1,
X2)) →
take(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
zip(
X1,
X2)) →
zip(
proper(
X1),
proper(
X2))
proper(
pair(
X1,
X2)) →
pair(
proper(
X1),
proper(
X2))
proper(
tail(
X)) →
tail(
proper(
X))
proper(
repItems(
X)) →
repItems(
proper(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
incr(
ok(
X)) →
ok(
incr(
X))
s(
ok(
X)) →
ok(
s(
X))
take(
ok(
X1),
ok(
X2)) →
ok(
take(
X1,
X2))
zip(
ok(
X1),
ok(
X2)) →
ok(
zip(
X1,
X2))
pair(
ok(
X1),
ok(
X2)) →
ok(
pair(
X1,
X2))
tail(
ok(
X)) →
ok(
tail(
X))
repItems(
ok(
X)) →
ok(
repItems(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok
Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
zip, active, repItems, tail, proper, top
They will be analysed ascendingly in the following order:
zip < active
repItems < active
tail < active
active < top
zip < proper
repItems < proper
tail < proper
proper < top
(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol zip.
(20) Obligation:
Innermost TRS:
Rules:
active(
pairNs) →
mark(
cons(
0',
incr(
oddNs)))
active(
oddNs) →
mark(
incr(
pairNs))
active(
incr(
cons(
X,
XS))) →
mark(
cons(
s(
X),
incr(
XS)))
active(
take(
0',
XS)) →
mark(
nil)
active(
take(
s(
N),
cons(
X,
XS))) →
mark(
cons(
X,
take(
N,
XS)))
active(
zip(
nil,
XS)) →
mark(
nil)
active(
zip(
X,
nil)) →
mark(
nil)
active(
zip(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
pair(
X,
Y),
zip(
XS,
YS)))
active(
tail(
cons(
X,
XS))) →
mark(
XS)
active(
repItems(
nil)) →
mark(
nil)
active(
repItems(
cons(
X,
XS))) →
mark(
cons(
X,
cons(
X,
repItems(
XS))))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
incr(
X)) →
incr(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
take(
X1,
X2)) →
take(
active(
X1),
X2)
active(
take(
X1,
X2)) →
take(
X1,
active(
X2))
active(
zip(
X1,
X2)) →
zip(
active(
X1),
X2)
active(
zip(
X1,
X2)) →
zip(
X1,
active(
X2))
active(
pair(
X1,
X2)) →
pair(
active(
X1),
X2)
active(
pair(
X1,
X2)) →
pair(
X1,
active(
X2))
active(
tail(
X)) →
tail(
active(
X))
active(
repItems(
X)) →
repItems(
active(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
incr(
mark(
X)) →
mark(
incr(
X))
s(
mark(
X)) →
mark(
s(
X))
take(
mark(
X1),
X2) →
mark(
take(
X1,
X2))
take(
X1,
mark(
X2)) →
mark(
take(
X1,
X2))
zip(
mark(
X1),
X2) →
mark(
zip(
X1,
X2))
zip(
X1,
mark(
X2)) →
mark(
zip(
X1,
X2))
pair(
mark(
X1),
X2) →
mark(
pair(
X1,
X2))
pair(
X1,
mark(
X2)) →
mark(
pair(
X1,
X2))
tail(
mark(
X)) →
mark(
tail(
X))
repItems(
mark(
X)) →
mark(
repItems(
X))
proper(
pairNs) →
ok(
pairNs)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
incr(
X)) →
incr(
proper(
X))
proper(
oddNs) →
ok(
oddNs)
proper(
s(
X)) →
s(
proper(
X))
proper(
take(
X1,
X2)) →
take(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
zip(
X1,
X2)) →
zip(
proper(
X1),
proper(
X2))
proper(
pair(
X1,
X2)) →
pair(
proper(
X1),
proper(
X2))
proper(
tail(
X)) →
tail(
proper(
X))
proper(
repItems(
X)) →
repItems(
proper(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
incr(
ok(
X)) →
ok(
incr(
X))
s(
ok(
X)) →
ok(
s(
X))
take(
ok(
X1),
ok(
X2)) →
ok(
take(
X1,
X2))
zip(
ok(
X1),
ok(
X2)) →
ok(
zip(
X1,
X2))
pair(
ok(
X1),
ok(
X2)) →
ok(
pair(
X1,
X2))
tail(
ok(
X)) →
ok(
tail(
X))
repItems(
ok(
X)) →
ok(
repItems(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok
Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
repItems, active, tail, proper, top
They will be analysed ascendingly in the following order:
repItems < active
tail < active
active < top
repItems < proper
tail < proper
proper < top
(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol repItems.
(22) Obligation:
Innermost TRS:
Rules:
active(
pairNs) →
mark(
cons(
0',
incr(
oddNs)))
active(
oddNs) →
mark(
incr(
pairNs))
active(
incr(
cons(
X,
XS))) →
mark(
cons(
s(
X),
incr(
XS)))
active(
take(
0',
XS)) →
mark(
nil)
active(
take(
s(
N),
cons(
X,
XS))) →
mark(
cons(
X,
take(
N,
XS)))
active(
zip(
nil,
XS)) →
mark(
nil)
active(
zip(
X,
nil)) →
mark(
nil)
active(
zip(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
pair(
X,
Y),
zip(
XS,
YS)))
active(
tail(
cons(
X,
XS))) →
mark(
XS)
active(
repItems(
nil)) →
mark(
nil)
active(
repItems(
cons(
X,
XS))) →
mark(
cons(
X,
cons(
X,
repItems(
XS))))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
incr(
X)) →
incr(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
take(
X1,
X2)) →
take(
active(
X1),
X2)
active(
take(
X1,
X2)) →
take(
X1,
active(
X2))
active(
zip(
X1,
X2)) →
zip(
active(
X1),
X2)
active(
zip(
X1,
X2)) →
zip(
X1,
active(
X2))
active(
pair(
X1,
X2)) →
pair(
active(
X1),
X2)
active(
pair(
X1,
X2)) →
pair(
X1,
active(
X2))
active(
tail(
X)) →
tail(
active(
X))
active(
repItems(
X)) →
repItems(
active(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
incr(
mark(
X)) →
mark(
incr(
X))
s(
mark(
X)) →
mark(
s(
X))
take(
mark(
X1),
X2) →
mark(
take(
X1,
X2))
take(
X1,
mark(
X2)) →
mark(
take(
X1,
X2))
zip(
mark(
X1),
X2) →
mark(
zip(
X1,
X2))
zip(
X1,
mark(
X2)) →
mark(
zip(
X1,
X2))
pair(
mark(
X1),
X2) →
mark(
pair(
X1,
X2))
pair(
X1,
mark(
X2)) →
mark(
pair(
X1,
X2))
tail(
mark(
X)) →
mark(
tail(
X))
repItems(
mark(
X)) →
mark(
repItems(
X))
proper(
pairNs) →
ok(
pairNs)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
incr(
X)) →
incr(
proper(
X))
proper(
oddNs) →
ok(
oddNs)
proper(
s(
X)) →
s(
proper(
X))
proper(
take(
X1,
X2)) →
take(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
zip(
X1,
X2)) →
zip(
proper(
X1),
proper(
X2))
proper(
pair(
X1,
X2)) →
pair(
proper(
X1),
proper(
X2))
proper(
tail(
X)) →
tail(
proper(
X))
proper(
repItems(
X)) →
repItems(
proper(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
incr(
ok(
X)) →
ok(
incr(
X))
s(
ok(
X)) →
ok(
s(
X))
take(
ok(
X1),
ok(
X2)) →
ok(
take(
X1,
X2))
zip(
ok(
X1),
ok(
X2)) →
ok(
zip(
X1,
X2))
pair(
ok(
X1),
ok(
X2)) →
ok(
pair(
X1,
X2))
tail(
ok(
X)) →
ok(
tail(
X))
repItems(
ok(
X)) →
ok(
repItems(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok
Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
tail, active, proper, top
They will be analysed ascendingly in the following order:
tail < active
active < top
tail < proper
proper < top
(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol tail.
(24) Obligation:
Innermost TRS:
Rules:
active(
pairNs) →
mark(
cons(
0',
incr(
oddNs)))
active(
oddNs) →
mark(
incr(
pairNs))
active(
incr(
cons(
X,
XS))) →
mark(
cons(
s(
X),
incr(
XS)))
active(
take(
0',
XS)) →
mark(
nil)
active(
take(
s(
N),
cons(
X,
XS))) →
mark(
cons(
X,
take(
N,
XS)))
active(
zip(
nil,
XS)) →
mark(
nil)
active(
zip(
X,
nil)) →
mark(
nil)
active(
zip(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
pair(
X,
Y),
zip(
XS,
YS)))
active(
tail(
cons(
X,
XS))) →
mark(
XS)
active(
repItems(
nil)) →
mark(
nil)
active(
repItems(
cons(
X,
XS))) →
mark(
cons(
X,
cons(
X,
repItems(
XS))))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
incr(
X)) →
incr(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
take(
X1,
X2)) →
take(
active(
X1),
X2)
active(
take(
X1,
X2)) →
take(
X1,
active(
X2))
active(
zip(
X1,
X2)) →
zip(
active(
X1),
X2)
active(
zip(
X1,
X2)) →
zip(
X1,
active(
X2))
active(
pair(
X1,
X2)) →
pair(
active(
X1),
X2)
active(
pair(
X1,
X2)) →
pair(
X1,
active(
X2))
active(
tail(
X)) →
tail(
active(
X))
active(
repItems(
X)) →
repItems(
active(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
incr(
mark(
X)) →
mark(
incr(
X))
s(
mark(
X)) →
mark(
s(
X))
take(
mark(
X1),
X2) →
mark(
take(
X1,
X2))
take(
X1,
mark(
X2)) →
mark(
take(
X1,
X2))
zip(
mark(
X1),
X2) →
mark(
zip(
X1,
X2))
zip(
X1,
mark(
X2)) →
mark(
zip(
X1,
X2))
pair(
mark(
X1),
X2) →
mark(
pair(
X1,
X2))
pair(
X1,
mark(
X2)) →
mark(
pair(
X1,
X2))
tail(
mark(
X)) →
mark(
tail(
X))
repItems(
mark(
X)) →
mark(
repItems(
X))
proper(
pairNs) →
ok(
pairNs)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
incr(
X)) →
incr(
proper(
X))
proper(
oddNs) →
ok(
oddNs)
proper(
s(
X)) →
s(
proper(
X))
proper(
take(
X1,
X2)) →
take(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
zip(
X1,
X2)) →
zip(
proper(
X1),
proper(
X2))
proper(
pair(
X1,
X2)) →
pair(
proper(
X1),
proper(
X2))
proper(
tail(
X)) →
tail(
proper(
X))
proper(
repItems(
X)) →
repItems(
proper(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
incr(
ok(
X)) →
ok(
incr(
X))
s(
ok(
X)) →
ok(
s(
X))
take(
ok(
X1),
ok(
X2)) →
ok(
take(
X1,
X2))
zip(
ok(
X1),
ok(
X2)) →
ok(
zip(
X1,
X2))
pair(
ok(
X1),
ok(
X2)) →
ok(
pair(
X1,
X2))
tail(
ok(
X)) →
ok(
tail(
X))
repItems(
ok(
X)) →
ok(
repItems(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok
Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
active, proper, top
They will be analysed ascendingly in the following order:
active < top
proper < top
(25) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol active.
(26) Obligation:
Innermost TRS:
Rules:
active(
pairNs) →
mark(
cons(
0',
incr(
oddNs)))
active(
oddNs) →
mark(
incr(
pairNs))
active(
incr(
cons(
X,
XS))) →
mark(
cons(
s(
X),
incr(
XS)))
active(
take(
0',
XS)) →
mark(
nil)
active(
take(
s(
N),
cons(
X,
XS))) →
mark(
cons(
X,
take(
N,
XS)))
active(
zip(
nil,
XS)) →
mark(
nil)
active(
zip(
X,
nil)) →
mark(
nil)
active(
zip(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
pair(
X,
Y),
zip(
XS,
YS)))
active(
tail(
cons(
X,
XS))) →
mark(
XS)
active(
repItems(
nil)) →
mark(
nil)
active(
repItems(
cons(
X,
XS))) →
mark(
cons(
X,
cons(
X,
repItems(
XS))))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
incr(
X)) →
incr(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
take(
X1,
X2)) →
take(
active(
X1),
X2)
active(
take(
X1,
X2)) →
take(
X1,
active(
X2))
active(
zip(
X1,
X2)) →
zip(
active(
X1),
X2)
active(
zip(
X1,
X2)) →
zip(
X1,
active(
X2))
active(
pair(
X1,
X2)) →
pair(
active(
X1),
X2)
active(
pair(
X1,
X2)) →
pair(
X1,
active(
X2))
active(
tail(
X)) →
tail(
active(
X))
active(
repItems(
X)) →
repItems(
active(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
incr(
mark(
X)) →
mark(
incr(
X))
s(
mark(
X)) →
mark(
s(
X))
take(
mark(
X1),
X2) →
mark(
take(
X1,
X2))
take(
X1,
mark(
X2)) →
mark(
take(
X1,
X2))
zip(
mark(
X1),
X2) →
mark(
zip(
X1,
X2))
zip(
X1,
mark(
X2)) →
mark(
zip(
X1,
X2))
pair(
mark(
X1),
X2) →
mark(
pair(
X1,
X2))
pair(
X1,
mark(
X2)) →
mark(
pair(
X1,
X2))
tail(
mark(
X)) →
mark(
tail(
X))
repItems(
mark(
X)) →
mark(
repItems(
X))
proper(
pairNs) →
ok(
pairNs)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
incr(
X)) →
incr(
proper(
X))
proper(
oddNs) →
ok(
oddNs)
proper(
s(
X)) →
s(
proper(
X))
proper(
take(
X1,
X2)) →
take(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
zip(
X1,
X2)) →
zip(
proper(
X1),
proper(
X2))
proper(
pair(
X1,
X2)) →
pair(
proper(
X1),
proper(
X2))
proper(
tail(
X)) →
tail(
proper(
X))
proper(
repItems(
X)) →
repItems(
proper(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
incr(
ok(
X)) →
ok(
incr(
X))
s(
ok(
X)) →
ok(
s(
X))
take(
ok(
X1),
ok(
X2)) →
ok(
take(
X1,
X2))
zip(
ok(
X1),
ok(
X2)) →
ok(
zip(
X1,
X2))
pair(
ok(
X1),
ok(
X2)) →
ok(
pair(
X1,
X2))
tail(
ok(
X)) →
ok(
tail(
X))
repItems(
ok(
X)) →
ok(
repItems(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok
Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
proper, top
They will be analysed ascendingly in the following order:
proper < top
(27) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol proper.
(28) Obligation:
Innermost TRS:
Rules:
active(
pairNs) →
mark(
cons(
0',
incr(
oddNs)))
active(
oddNs) →
mark(
incr(
pairNs))
active(
incr(
cons(
X,
XS))) →
mark(
cons(
s(
X),
incr(
XS)))
active(
take(
0',
XS)) →
mark(
nil)
active(
take(
s(
N),
cons(
X,
XS))) →
mark(
cons(
X,
take(
N,
XS)))
active(
zip(
nil,
XS)) →
mark(
nil)
active(
zip(
X,
nil)) →
mark(
nil)
active(
zip(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
pair(
X,
Y),
zip(
XS,
YS)))
active(
tail(
cons(
X,
XS))) →
mark(
XS)
active(
repItems(
nil)) →
mark(
nil)
active(
repItems(
cons(
X,
XS))) →
mark(
cons(
X,
cons(
X,
repItems(
XS))))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
incr(
X)) →
incr(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
take(
X1,
X2)) →
take(
active(
X1),
X2)
active(
take(
X1,
X2)) →
take(
X1,
active(
X2))
active(
zip(
X1,
X2)) →
zip(
active(
X1),
X2)
active(
zip(
X1,
X2)) →
zip(
X1,
active(
X2))
active(
pair(
X1,
X2)) →
pair(
active(
X1),
X2)
active(
pair(
X1,
X2)) →
pair(
X1,
active(
X2))
active(
tail(
X)) →
tail(
active(
X))
active(
repItems(
X)) →
repItems(
active(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
incr(
mark(
X)) →
mark(
incr(
X))
s(
mark(
X)) →
mark(
s(
X))
take(
mark(
X1),
X2) →
mark(
take(
X1,
X2))
take(
X1,
mark(
X2)) →
mark(
take(
X1,
X2))
zip(
mark(
X1),
X2) →
mark(
zip(
X1,
X2))
zip(
X1,
mark(
X2)) →
mark(
zip(
X1,
X2))
pair(
mark(
X1),
X2) →
mark(
pair(
X1,
X2))
pair(
X1,
mark(
X2)) →
mark(
pair(
X1,
X2))
tail(
mark(
X)) →
mark(
tail(
X))
repItems(
mark(
X)) →
mark(
repItems(
X))
proper(
pairNs) →
ok(
pairNs)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
incr(
X)) →
incr(
proper(
X))
proper(
oddNs) →
ok(
oddNs)
proper(
s(
X)) →
s(
proper(
X))
proper(
take(
X1,
X2)) →
take(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
zip(
X1,
X2)) →
zip(
proper(
X1),
proper(
X2))
proper(
pair(
X1,
X2)) →
pair(
proper(
X1),
proper(
X2))
proper(
tail(
X)) →
tail(
proper(
X))
proper(
repItems(
X)) →
repItems(
proper(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
incr(
ok(
X)) →
ok(
incr(
X))
s(
ok(
X)) →
ok(
s(
X))
take(
ok(
X1),
ok(
X2)) →
ok(
take(
X1,
X2))
zip(
ok(
X1),
ok(
X2)) →
ok(
zip(
X1,
X2))
pair(
ok(
X1),
ok(
X2)) →
ok(
pair(
X1,
X2))
tail(
ok(
X)) →
ok(
tail(
X))
repItems(
ok(
X)) →
ok(
repItems(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok
Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
top
(29) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol top.
(30) Obligation:
Innermost TRS:
Rules:
active(
pairNs) →
mark(
cons(
0',
incr(
oddNs)))
active(
oddNs) →
mark(
incr(
pairNs))
active(
incr(
cons(
X,
XS))) →
mark(
cons(
s(
X),
incr(
XS)))
active(
take(
0',
XS)) →
mark(
nil)
active(
take(
s(
N),
cons(
X,
XS))) →
mark(
cons(
X,
take(
N,
XS)))
active(
zip(
nil,
XS)) →
mark(
nil)
active(
zip(
X,
nil)) →
mark(
nil)
active(
zip(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
pair(
X,
Y),
zip(
XS,
YS)))
active(
tail(
cons(
X,
XS))) →
mark(
XS)
active(
repItems(
nil)) →
mark(
nil)
active(
repItems(
cons(
X,
XS))) →
mark(
cons(
X,
cons(
X,
repItems(
XS))))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
incr(
X)) →
incr(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
take(
X1,
X2)) →
take(
active(
X1),
X2)
active(
take(
X1,
X2)) →
take(
X1,
active(
X2))
active(
zip(
X1,
X2)) →
zip(
active(
X1),
X2)
active(
zip(
X1,
X2)) →
zip(
X1,
active(
X2))
active(
pair(
X1,
X2)) →
pair(
active(
X1),
X2)
active(
pair(
X1,
X2)) →
pair(
X1,
active(
X2))
active(
tail(
X)) →
tail(
active(
X))
active(
repItems(
X)) →
repItems(
active(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
incr(
mark(
X)) →
mark(
incr(
X))
s(
mark(
X)) →
mark(
s(
X))
take(
mark(
X1),
X2) →
mark(
take(
X1,
X2))
take(
X1,
mark(
X2)) →
mark(
take(
X1,
X2))
zip(
mark(
X1),
X2) →
mark(
zip(
X1,
X2))
zip(
X1,
mark(
X2)) →
mark(
zip(
X1,
X2))
pair(
mark(
X1),
X2) →
mark(
pair(
X1,
X2))
pair(
X1,
mark(
X2)) →
mark(
pair(
X1,
X2))
tail(
mark(
X)) →
mark(
tail(
X))
repItems(
mark(
X)) →
mark(
repItems(
X))
proper(
pairNs) →
ok(
pairNs)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
incr(
X)) →
incr(
proper(
X))
proper(
oddNs) →
ok(
oddNs)
proper(
s(
X)) →
s(
proper(
X))
proper(
take(
X1,
X2)) →
take(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
zip(
X1,
X2)) →
zip(
proper(
X1),
proper(
X2))
proper(
pair(
X1,
X2)) →
pair(
proper(
X1),
proper(
X2))
proper(
tail(
X)) →
tail(
proper(
X))
proper(
repItems(
X)) →
repItems(
proper(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
incr(
ok(
X)) →
ok(
incr(
X))
s(
ok(
X)) →
ok(
s(
X))
take(
ok(
X1),
ok(
X2)) →
ok(
take(
X1,
X2))
zip(
ok(
X1),
ok(
X2)) →
ok(
zip(
X1,
X2))
pair(
ok(
X1),
ok(
X2)) →
ok(
pair(
X1,
X2))
tail(
ok(
X)) →
ok(
tail(
X))
repItems(
ok(
X)) →
ok(
repItems(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok
Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))
No more defined symbols left to analyse.