(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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
app(mark(X1), X2) →+ mark(app(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(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(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(app(nil, YS)) → mark(YS)
active(app(cons(X, XS), YS)) → mark(cons(X, app(XS, YS)))
active(from(X)) → mark(cons(X, from(s(X))))
active(zWadr(nil, YS)) → mark(nil)
active(zWadr(XS, nil)) → mark(nil)
active(zWadr(cons(X, XS), cons(Y, YS))) → mark(cons(app(Y, cons(X, nil)), zWadr(XS, YS)))
active(prefix(L)) → mark(cons(nil, zWadr(L, prefix(L))))
active(app(X1, X2)) → app(active(X1), X2)
active(app(X1, X2)) → app(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(zWadr(X1, X2)) → zWadr(active(X1), X2)
active(zWadr(X1, X2)) → zWadr(X1, active(X2))
active(prefix(X)) → prefix(active(X))
app(mark(X1), X2) → mark(app(X1, X2))
app(X1, mark(X2)) → mark(app(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
zWadr(mark(X1), X2) → mark(zWadr(X1, X2))
zWadr(X1, mark(X2)) → mark(zWadr(X1, X2))
prefix(mark(X)) → mark(prefix(X))
proper(app(X1, X2)) → app(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(zWadr(X1, X2)) → zWadr(proper(X1), proper(X2))
proper(prefix(X)) → prefix(proper(X))
app(ok(X1), ok(X2)) → ok(app(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
zWadr(ok(X1), ok(X2)) → ok(zWadr(X1, X2))
prefix(ok(X)) → ok(prefix(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: nil:mark:ok → nil:mark:ok
app :: nil:mark:ok → nil:mark:ok → nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok → nil:mark:ok
cons :: nil:mark:ok → nil:mark:ok → nil:mark:ok
from :: nil:mark:ok → nil:mark:ok
s :: nil:mark:ok → nil:mark:ok
zWadr :: nil:mark:ok → nil:mark:ok → nil:mark:ok
prefix :: nil:mark:ok → nil:mark:ok
proper :: nil:mark:ok → nil:mark:ok
ok :: nil:mark:ok → nil:mark:ok
top :: nil:mark:ok → top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active,
cons,
app,
from,
s,
zWadr,
prefix,
proper,
topThey will be analysed ascendingly in the following order:
cons < active
app < active
from < active
s < active
zWadr < active
prefix < active
active < top
cons < proper
app < proper
from < proper
s < proper
zWadr < proper
prefix < proper
proper < top
(8) Obligation:
Innermost TRS:
Rules:
active(
app(
nil,
YS)) →
mark(
YS)
active(
app(
cons(
X,
XS),
YS)) →
mark(
cons(
X,
app(
XS,
YS)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
zWadr(
nil,
YS)) →
mark(
nil)
active(
zWadr(
XS,
nil)) →
mark(
nil)
active(
zWadr(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
app(
Y,
cons(
X,
nil)),
zWadr(
XS,
YS)))
active(
prefix(
L)) →
mark(
cons(
nil,
zWadr(
L,
prefix(
L))))
active(
app(
X1,
X2)) →
app(
active(
X1),
X2)
active(
app(
X1,
X2)) →
app(
X1,
active(
X2))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
zWadr(
X1,
X2)) →
zWadr(
active(
X1),
X2)
active(
zWadr(
X1,
X2)) →
zWadr(
X1,
active(
X2))
active(
prefix(
X)) →
prefix(
active(
X))
app(
mark(
X1),
X2) →
mark(
app(
X1,
X2))
app(
X1,
mark(
X2)) →
mark(
app(
X1,
X2))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
zWadr(
mark(
X1),
X2) →
mark(
zWadr(
X1,
X2))
zWadr(
X1,
mark(
X2)) →
mark(
zWadr(
X1,
X2))
prefix(
mark(
X)) →
mark(
prefix(
X))
proper(
app(
X1,
X2)) →
app(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
proper(
zWadr(
X1,
X2)) →
zWadr(
proper(
X1),
proper(
X2))
proper(
prefix(
X)) →
prefix(
proper(
X))
app(
ok(
X1),
ok(
X2)) →
ok(
app(
X1,
X2))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
zWadr(
ok(
X1),
ok(
X2)) →
ok(
zWadr(
X1,
X2))
prefix(
ok(
X)) →
ok(
prefix(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: nil:mark:ok → nil:mark:ok
app :: nil:mark:ok → nil:mark:ok → nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok → nil:mark:ok
cons :: nil:mark:ok → nil:mark:ok → nil:mark:ok
from :: nil:mark:ok → nil:mark:ok
s :: nil:mark:ok → nil:mark:ok
zWadr :: nil:mark:ok → nil:mark:ok → nil:mark:ok
prefix :: nil:mark:ok → nil:mark:ok
proper :: nil:mark:ok → nil:mark:ok
ok :: nil:mark:ok → nil:mark:ok
top :: nil:mark:ok → top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok
Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))
The following defined symbols remain to be analysed:
cons, active, app, from, s, zWadr, prefix, proper, top
They will be analysed ascendingly in the following order:
cons < active
app < active
from < active
s < active
zWadr < active
prefix < active
active < top
cons < proper
app < proper
from < proper
s < proper
zWadr < proper
prefix < 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(
app(
nil,
YS)) →
mark(
YS)
active(
app(
cons(
X,
XS),
YS)) →
mark(
cons(
X,
app(
XS,
YS)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
zWadr(
nil,
YS)) →
mark(
nil)
active(
zWadr(
XS,
nil)) →
mark(
nil)
active(
zWadr(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
app(
Y,
cons(
X,
nil)),
zWadr(
XS,
YS)))
active(
prefix(
L)) →
mark(
cons(
nil,
zWadr(
L,
prefix(
L))))
active(
app(
X1,
X2)) →
app(
active(
X1),
X2)
active(
app(
X1,
X2)) →
app(
X1,
active(
X2))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
zWadr(
X1,
X2)) →
zWadr(
active(
X1),
X2)
active(
zWadr(
X1,
X2)) →
zWadr(
X1,
active(
X2))
active(
prefix(
X)) →
prefix(
active(
X))
app(
mark(
X1),
X2) →
mark(
app(
X1,
X2))
app(
X1,
mark(
X2)) →
mark(
app(
X1,
X2))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
zWadr(
mark(
X1),
X2) →
mark(
zWadr(
X1,
X2))
zWadr(
X1,
mark(
X2)) →
mark(
zWadr(
X1,
X2))
prefix(
mark(
X)) →
mark(
prefix(
X))
proper(
app(
X1,
X2)) →
app(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
proper(
zWadr(
X1,
X2)) →
zWadr(
proper(
X1),
proper(
X2))
proper(
prefix(
X)) →
prefix(
proper(
X))
app(
ok(
X1),
ok(
X2)) →
ok(
app(
X1,
X2))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
zWadr(
ok(
X1),
ok(
X2)) →
ok(
zWadr(
X1,
X2))
prefix(
ok(
X)) →
ok(
prefix(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: nil:mark:ok → nil:mark:ok
app :: nil:mark:ok → nil:mark:ok → nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok → nil:mark:ok
cons :: nil:mark:ok → nil:mark:ok → nil:mark:ok
from :: nil:mark:ok → nil:mark:ok
s :: nil:mark:ok → nil:mark:ok
zWadr :: nil:mark:ok → nil:mark:ok → nil:mark:ok
prefix :: nil:mark:ok → nil:mark:ok
proper :: nil:mark:ok → nil:mark:ok
ok :: nil:mark:ok → nil:mark:ok
top :: nil:mark:ok → top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok
Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))
The following defined symbols remain to be analysed:
app, active, from, s, zWadr, prefix, proper, top
They will be analysed ascendingly in the following order:
app < active
from < active
s < active
zWadr < active
prefix < active
active < top
app < proper
from < proper
s < proper
zWadr < proper
prefix < proper
proper < top
(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol app.
(12) Obligation:
Innermost TRS:
Rules:
active(
app(
nil,
YS)) →
mark(
YS)
active(
app(
cons(
X,
XS),
YS)) →
mark(
cons(
X,
app(
XS,
YS)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
zWadr(
nil,
YS)) →
mark(
nil)
active(
zWadr(
XS,
nil)) →
mark(
nil)
active(
zWadr(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
app(
Y,
cons(
X,
nil)),
zWadr(
XS,
YS)))
active(
prefix(
L)) →
mark(
cons(
nil,
zWadr(
L,
prefix(
L))))
active(
app(
X1,
X2)) →
app(
active(
X1),
X2)
active(
app(
X1,
X2)) →
app(
X1,
active(
X2))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
zWadr(
X1,
X2)) →
zWadr(
active(
X1),
X2)
active(
zWadr(
X1,
X2)) →
zWadr(
X1,
active(
X2))
active(
prefix(
X)) →
prefix(
active(
X))
app(
mark(
X1),
X2) →
mark(
app(
X1,
X2))
app(
X1,
mark(
X2)) →
mark(
app(
X1,
X2))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
zWadr(
mark(
X1),
X2) →
mark(
zWadr(
X1,
X2))
zWadr(
X1,
mark(
X2)) →
mark(
zWadr(
X1,
X2))
prefix(
mark(
X)) →
mark(
prefix(
X))
proper(
app(
X1,
X2)) →
app(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
proper(
zWadr(
X1,
X2)) →
zWadr(
proper(
X1),
proper(
X2))
proper(
prefix(
X)) →
prefix(
proper(
X))
app(
ok(
X1),
ok(
X2)) →
ok(
app(
X1,
X2))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
zWadr(
ok(
X1),
ok(
X2)) →
ok(
zWadr(
X1,
X2))
prefix(
ok(
X)) →
ok(
prefix(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: nil:mark:ok → nil:mark:ok
app :: nil:mark:ok → nil:mark:ok → nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok → nil:mark:ok
cons :: nil:mark:ok → nil:mark:ok → nil:mark:ok
from :: nil:mark:ok → nil:mark:ok
s :: nil:mark:ok → nil:mark:ok
zWadr :: nil:mark:ok → nil:mark:ok → nil:mark:ok
prefix :: nil:mark:ok → nil:mark:ok
proper :: nil:mark:ok → nil:mark:ok
ok :: nil:mark:ok → nil:mark:ok
top :: nil:mark:ok → top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok
Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))
The following defined symbols remain to be analysed:
from, active, s, zWadr, prefix, proper, top
They will be analysed ascendingly in the following order:
from < active
s < active
zWadr < active
prefix < active
active < top
from < proper
s < proper
zWadr < proper
prefix < proper
proper < top
(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol from.
(14) Obligation:
Innermost TRS:
Rules:
active(
app(
nil,
YS)) →
mark(
YS)
active(
app(
cons(
X,
XS),
YS)) →
mark(
cons(
X,
app(
XS,
YS)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
zWadr(
nil,
YS)) →
mark(
nil)
active(
zWadr(
XS,
nil)) →
mark(
nil)
active(
zWadr(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
app(
Y,
cons(
X,
nil)),
zWadr(
XS,
YS)))
active(
prefix(
L)) →
mark(
cons(
nil,
zWadr(
L,
prefix(
L))))
active(
app(
X1,
X2)) →
app(
active(
X1),
X2)
active(
app(
X1,
X2)) →
app(
X1,
active(
X2))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
zWadr(
X1,
X2)) →
zWadr(
active(
X1),
X2)
active(
zWadr(
X1,
X2)) →
zWadr(
X1,
active(
X2))
active(
prefix(
X)) →
prefix(
active(
X))
app(
mark(
X1),
X2) →
mark(
app(
X1,
X2))
app(
X1,
mark(
X2)) →
mark(
app(
X1,
X2))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
zWadr(
mark(
X1),
X2) →
mark(
zWadr(
X1,
X2))
zWadr(
X1,
mark(
X2)) →
mark(
zWadr(
X1,
X2))
prefix(
mark(
X)) →
mark(
prefix(
X))
proper(
app(
X1,
X2)) →
app(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
proper(
zWadr(
X1,
X2)) →
zWadr(
proper(
X1),
proper(
X2))
proper(
prefix(
X)) →
prefix(
proper(
X))
app(
ok(
X1),
ok(
X2)) →
ok(
app(
X1,
X2))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
zWadr(
ok(
X1),
ok(
X2)) →
ok(
zWadr(
X1,
X2))
prefix(
ok(
X)) →
ok(
prefix(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: nil:mark:ok → nil:mark:ok
app :: nil:mark:ok → nil:mark:ok → nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok → nil:mark:ok
cons :: nil:mark:ok → nil:mark:ok → nil:mark:ok
from :: nil:mark:ok → nil:mark:ok
s :: nil:mark:ok → nil:mark:ok
zWadr :: nil:mark:ok → nil:mark:ok → nil:mark:ok
prefix :: nil:mark:ok → nil:mark:ok
proper :: nil:mark:ok → nil:mark:ok
ok :: nil:mark:ok → nil:mark:ok
top :: nil:mark:ok → top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok
Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))
The following defined symbols remain to be analysed:
s, active, zWadr, prefix, proper, top
They will be analysed ascendingly in the following order:
s < active
zWadr < active
prefix < active
active < top
s < proper
zWadr < proper
prefix < proper
proper < top
(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol s.
(16) Obligation:
Innermost TRS:
Rules:
active(
app(
nil,
YS)) →
mark(
YS)
active(
app(
cons(
X,
XS),
YS)) →
mark(
cons(
X,
app(
XS,
YS)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
zWadr(
nil,
YS)) →
mark(
nil)
active(
zWadr(
XS,
nil)) →
mark(
nil)
active(
zWadr(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
app(
Y,
cons(
X,
nil)),
zWadr(
XS,
YS)))
active(
prefix(
L)) →
mark(
cons(
nil,
zWadr(
L,
prefix(
L))))
active(
app(
X1,
X2)) →
app(
active(
X1),
X2)
active(
app(
X1,
X2)) →
app(
X1,
active(
X2))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
zWadr(
X1,
X2)) →
zWadr(
active(
X1),
X2)
active(
zWadr(
X1,
X2)) →
zWadr(
X1,
active(
X2))
active(
prefix(
X)) →
prefix(
active(
X))
app(
mark(
X1),
X2) →
mark(
app(
X1,
X2))
app(
X1,
mark(
X2)) →
mark(
app(
X1,
X2))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
zWadr(
mark(
X1),
X2) →
mark(
zWadr(
X1,
X2))
zWadr(
X1,
mark(
X2)) →
mark(
zWadr(
X1,
X2))
prefix(
mark(
X)) →
mark(
prefix(
X))
proper(
app(
X1,
X2)) →
app(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
proper(
zWadr(
X1,
X2)) →
zWadr(
proper(
X1),
proper(
X2))
proper(
prefix(
X)) →
prefix(
proper(
X))
app(
ok(
X1),
ok(
X2)) →
ok(
app(
X1,
X2))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
zWadr(
ok(
X1),
ok(
X2)) →
ok(
zWadr(
X1,
X2))
prefix(
ok(
X)) →
ok(
prefix(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: nil:mark:ok → nil:mark:ok
app :: nil:mark:ok → nil:mark:ok → nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok → nil:mark:ok
cons :: nil:mark:ok → nil:mark:ok → nil:mark:ok
from :: nil:mark:ok → nil:mark:ok
s :: nil:mark:ok → nil:mark:ok
zWadr :: nil:mark:ok → nil:mark:ok → nil:mark:ok
prefix :: nil:mark:ok → nil:mark:ok
proper :: nil:mark:ok → nil:mark:ok
ok :: nil:mark:ok → nil:mark:ok
top :: nil:mark:ok → top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok
Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))
The following defined symbols remain to be analysed:
zWadr, active, prefix, proper, top
They will be analysed ascendingly in the following order:
zWadr < active
prefix < active
active < top
zWadr < proper
prefix < proper
proper < top
(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol zWadr.
(18) Obligation:
Innermost TRS:
Rules:
active(
app(
nil,
YS)) →
mark(
YS)
active(
app(
cons(
X,
XS),
YS)) →
mark(
cons(
X,
app(
XS,
YS)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
zWadr(
nil,
YS)) →
mark(
nil)
active(
zWadr(
XS,
nil)) →
mark(
nil)
active(
zWadr(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
app(
Y,
cons(
X,
nil)),
zWadr(
XS,
YS)))
active(
prefix(
L)) →
mark(
cons(
nil,
zWadr(
L,
prefix(
L))))
active(
app(
X1,
X2)) →
app(
active(
X1),
X2)
active(
app(
X1,
X2)) →
app(
X1,
active(
X2))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
zWadr(
X1,
X2)) →
zWadr(
active(
X1),
X2)
active(
zWadr(
X1,
X2)) →
zWadr(
X1,
active(
X2))
active(
prefix(
X)) →
prefix(
active(
X))
app(
mark(
X1),
X2) →
mark(
app(
X1,
X2))
app(
X1,
mark(
X2)) →
mark(
app(
X1,
X2))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
zWadr(
mark(
X1),
X2) →
mark(
zWadr(
X1,
X2))
zWadr(
X1,
mark(
X2)) →
mark(
zWadr(
X1,
X2))
prefix(
mark(
X)) →
mark(
prefix(
X))
proper(
app(
X1,
X2)) →
app(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
proper(
zWadr(
X1,
X2)) →
zWadr(
proper(
X1),
proper(
X2))
proper(
prefix(
X)) →
prefix(
proper(
X))
app(
ok(
X1),
ok(
X2)) →
ok(
app(
X1,
X2))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
zWadr(
ok(
X1),
ok(
X2)) →
ok(
zWadr(
X1,
X2))
prefix(
ok(
X)) →
ok(
prefix(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: nil:mark:ok → nil:mark:ok
app :: nil:mark:ok → nil:mark:ok → nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok → nil:mark:ok
cons :: nil:mark:ok → nil:mark:ok → nil:mark:ok
from :: nil:mark:ok → nil:mark:ok
s :: nil:mark:ok → nil:mark:ok
zWadr :: nil:mark:ok → nil:mark:ok → nil:mark:ok
prefix :: nil:mark:ok → nil:mark:ok
proper :: nil:mark:ok → nil:mark:ok
ok :: nil:mark:ok → nil:mark:ok
top :: nil:mark:ok → top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok
Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))
The following defined symbols remain to be analysed:
prefix, active, proper, top
They will be analysed ascendingly in the following order:
prefix < active
active < top
prefix < proper
proper < top
(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol prefix.
(20) Obligation:
Innermost TRS:
Rules:
active(
app(
nil,
YS)) →
mark(
YS)
active(
app(
cons(
X,
XS),
YS)) →
mark(
cons(
X,
app(
XS,
YS)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
zWadr(
nil,
YS)) →
mark(
nil)
active(
zWadr(
XS,
nil)) →
mark(
nil)
active(
zWadr(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
app(
Y,
cons(
X,
nil)),
zWadr(
XS,
YS)))
active(
prefix(
L)) →
mark(
cons(
nil,
zWadr(
L,
prefix(
L))))
active(
app(
X1,
X2)) →
app(
active(
X1),
X2)
active(
app(
X1,
X2)) →
app(
X1,
active(
X2))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
zWadr(
X1,
X2)) →
zWadr(
active(
X1),
X2)
active(
zWadr(
X1,
X2)) →
zWadr(
X1,
active(
X2))
active(
prefix(
X)) →
prefix(
active(
X))
app(
mark(
X1),
X2) →
mark(
app(
X1,
X2))
app(
X1,
mark(
X2)) →
mark(
app(
X1,
X2))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
zWadr(
mark(
X1),
X2) →
mark(
zWadr(
X1,
X2))
zWadr(
X1,
mark(
X2)) →
mark(
zWadr(
X1,
X2))
prefix(
mark(
X)) →
mark(
prefix(
X))
proper(
app(
X1,
X2)) →
app(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
proper(
zWadr(
X1,
X2)) →
zWadr(
proper(
X1),
proper(
X2))
proper(
prefix(
X)) →
prefix(
proper(
X))
app(
ok(
X1),
ok(
X2)) →
ok(
app(
X1,
X2))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
zWadr(
ok(
X1),
ok(
X2)) →
ok(
zWadr(
X1,
X2))
prefix(
ok(
X)) →
ok(
prefix(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: nil:mark:ok → nil:mark:ok
app :: nil:mark:ok → nil:mark:ok → nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok → nil:mark:ok
cons :: nil:mark:ok → nil:mark:ok → nil:mark:ok
from :: nil:mark:ok → nil:mark:ok
s :: nil:mark:ok → nil:mark:ok
zWadr :: nil:mark:ok → nil:mark:ok → nil:mark:ok
prefix :: nil:mark:ok → nil:mark:ok
proper :: nil:mark:ok → nil:mark:ok
ok :: nil:mark:ok → nil:mark:ok
top :: nil:mark:ok → top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok
Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark: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
(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol active.
(22) Obligation:
Innermost TRS:
Rules:
active(
app(
nil,
YS)) →
mark(
YS)
active(
app(
cons(
X,
XS),
YS)) →
mark(
cons(
X,
app(
XS,
YS)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
zWadr(
nil,
YS)) →
mark(
nil)
active(
zWadr(
XS,
nil)) →
mark(
nil)
active(
zWadr(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
app(
Y,
cons(
X,
nil)),
zWadr(
XS,
YS)))
active(
prefix(
L)) →
mark(
cons(
nil,
zWadr(
L,
prefix(
L))))
active(
app(
X1,
X2)) →
app(
active(
X1),
X2)
active(
app(
X1,
X2)) →
app(
X1,
active(
X2))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
zWadr(
X1,
X2)) →
zWadr(
active(
X1),
X2)
active(
zWadr(
X1,
X2)) →
zWadr(
X1,
active(
X2))
active(
prefix(
X)) →
prefix(
active(
X))
app(
mark(
X1),
X2) →
mark(
app(
X1,
X2))
app(
X1,
mark(
X2)) →
mark(
app(
X1,
X2))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
zWadr(
mark(
X1),
X2) →
mark(
zWadr(
X1,
X2))
zWadr(
X1,
mark(
X2)) →
mark(
zWadr(
X1,
X2))
prefix(
mark(
X)) →
mark(
prefix(
X))
proper(
app(
X1,
X2)) →
app(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
proper(
zWadr(
X1,
X2)) →
zWadr(
proper(
X1),
proper(
X2))
proper(
prefix(
X)) →
prefix(
proper(
X))
app(
ok(
X1),
ok(
X2)) →
ok(
app(
X1,
X2))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
zWadr(
ok(
X1),
ok(
X2)) →
ok(
zWadr(
X1,
X2))
prefix(
ok(
X)) →
ok(
prefix(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: nil:mark:ok → nil:mark:ok
app :: nil:mark:ok → nil:mark:ok → nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok → nil:mark:ok
cons :: nil:mark:ok → nil:mark:ok → nil:mark:ok
from :: nil:mark:ok → nil:mark:ok
s :: nil:mark:ok → nil:mark:ok
zWadr :: nil:mark:ok → nil:mark:ok → nil:mark:ok
prefix :: nil:mark:ok → nil:mark:ok
proper :: nil:mark:ok → nil:mark:ok
ok :: nil:mark:ok → nil:mark:ok
top :: nil:mark:ok → top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok
Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))
The following defined symbols remain to be analysed:
proper, top
They will be analysed ascendingly in the following order:
proper < top
(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol proper.
(24) Obligation:
Innermost TRS:
Rules:
active(
app(
nil,
YS)) →
mark(
YS)
active(
app(
cons(
X,
XS),
YS)) →
mark(
cons(
X,
app(
XS,
YS)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
zWadr(
nil,
YS)) →
mark(
nil)
active(
zWadr(
XS,
nil)) →
mark(
nil)
active(
zWadr(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
app(
Y,
cons(
X,
nil)),
zWadr(
XS,
YS)))
active(
prefix(
L)) →
mark(
cons(
nil,
zWadr(
L,
prefix(
L))))
active(
app(
X1,
X2)) →
app(
active(
X1),
X2)
active(
app(
X1,
X2)) →
app(
X1,
active(
X2))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
zWadr(
X1,
X2)) →
zWadr(
active(
X1),
X2)
active(
zWadr(
X1,
X2)) →
zWadr(
X1,
active(
X2))
active(
prefix(
X)) →
prefix(
active(
X))
app(
mark(
X1),
X2) →
mark(
app(
X1,
X2))
app(
X1,
mark(
X2)) →
mark(
app(
X1,
X2))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
zWadr(
mark(
X1),
X2) →
mark(
zWadr(
X1,
X2))
zWadr(
X1,
mark(
X2)) →
mark(
zWadr(
X1,
X2))
prefix(
mark(
X)) →
mark(
prefix(
X))
proper(
app(
X1,
X2)) →
app(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
proper(
zWadr(
X1,
X2)) →
zWadr(
proper(
X1),
proper(
X2))
proper(
prefix(
X)) →
prefix(
proper(
X))
app(
ok(
X1),
ok(
X2)) →
ok(
app(
X1,
X2))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
zWadr(
ok(
X1),
ok(
X2)) →
ok(
zWadr(
X1,
X2))
prefix(
ok(
X)) →
ok(
prefix(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: nil:mark:ok → nil:mark:ok
app :: nil:mark:ok → nil:mark:ok → nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok → nil:mark:ok
cons :: nil:mark:ok → nil:mark:ok → nil:mark:ok
from :: nil:mark:ok → nil:mark:ok
s :: nil:mark:ok → nil:mark:ok
zWadr :: nil:mark:ok → nil:mark:ok → nil:mark:ok
prefix :: nil:mark:ok → nil:mark:ok
proper :: nil:mark:ok → nil:mark:ok
ok :: nil:mark:ok → nil:mark:ok
top :: nil:mark:ok → top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok
Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))
The following defined symbols remain to be analysed:
top
(25) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol top.
(26) Obligation:
Innermost TRS:
Rules:
active(
app(
nil,
YS)) →
mark(
YS)
active(
app(
cons(
X,
XS),
YS)) →
mark(
cons(
X,
app(
XS,
YS)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
zWadr(
nil,
YS)) →
mark(
nil)
active(
zWadr(
XS,
nil)) →
mark(
nil)
active(
zWadr(
cons(
X,
XS),
cons(
Y,
YS))) →
mark(
cons(
app(
Y,
cons(
X,
nil)),
zWadr(
XS,
YS)))
active(
prefix(
L)) →
mark(
cons(
nil,
zWadr(
L,
prefix(
L))))
active(
app(
X1,
X2)) →
app(
active(
X1),
X2)
active(
app(
X1,
X2)) →
app(
X1,
active(
X2))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
zWadr(
X1,
X2)) →
zWadr(
active(
X1),
X2)
active(
zWadr(
X1,
X2)) →
zWadr(
X1,
active(
X2))
active(
prefix(
X)) →
prefix(
active(
X))
app(
mark(
X1),
X2) →
mark(
app(
X1,
X2))
app(
X1,
mark(
X2)) →
mark(
app(
X1,
X2))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
zWadr(
mark(
X1),
X2) →
mark(
zWadr(
X1,
X2))
zWadr(
X1,
mark(
X2)) →
mark(
zWadr(
X1,
X2))
prefix(
mark(
X)) →
mark(
prefix(
X))
proper(
app(
X1,
X2)) →
app(
proper(
X1),
proper(
X2))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
proper(
zWadr(
X1,
X2)) →
zWadr(
proper(
X1),
proper(
X2))
proper(
prefix(
X)) →
prefix(
proper(
X))
app(
ok(
X1),
ok(
X2)) →
ok(
app(
X1,
X2))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
zWadr(
ok(
X1),
ok(
X2)) →
ok(
zWadr(
X1,
X2))
prefix(
ok(
X)) →
ok(
prefix(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: nil:mark:ok → nil:mark:ok
app :: nil:mark:ok → nil:mark:ok → nil:mark:ok
nil :: nil:mark:ok
mark :: nil:mark:ok → nil:mark:ok
cons :: nil:mark:ok → nil:mark:ok → nil:mark:ok
from :: nil:mark:ok → nil:mark:ok
s :: nil:mark:ok → nil:mark:ok
zWadr :: nil:mark:ok → nil:mark:ok → nil:mark:ok
prefix :: nil:mark:ok → nil:mark:ok
proper :: nil:mark:ok → nil:mark:ok
ok :: nil:mark:ok → nil:mark:ok
top :: nil:mark:ok → top
hole_nil:mark:ok1_0 :: nil:mark:ok
hole_top2_0 :: top
gen_nil:mark:ok3_0 :: Nat → nil:mark:ok
Generator Equations:
gen_nil:mark:ok3_0(0) ⇔ nil
gen_nil:mark:ok3_0(+(x, 1)) ⇔ mark(gen_nil:mark:ok3_0(x))
No more defined symbols left to analyse.