(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(2nd(cons(X, cons(Y, Z)))) → mark(Y)
active(from(X)) → mark(cons(X, from(s(X))))
active(2nd(X)) → 2nd(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
2nd(mark(X)) → mark(2nd(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
proper(2nd(X)) → 2nd(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
2nd(ok(X)) → ok(2nd(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
2nd(mark(X)) →+ mark(2nd(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / mark(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:
active(2nd(cons(X, cons(Y, Z)))) → mark(Y)
active(from(X)) → mark(cons(X, from(s(X))))
active(2nd(X)) → 2nd(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
2nd(mark(X)) → mark(2nd(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
proper(2nd(X)) → 2nd(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
2nd(ok(X)) → ok(2nd(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
S is empty.
Rewrite Strategy: FULL
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
TRS:
Rules:
active(2nd(cons(X, cons(Y, Z)))) → mark(Y)
active(from(X)) → mark(cons(X, from(s(X))))
active(2nd(X)) → 2nd(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
2nd(mark(X)) → mark(2nd(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
proper(2nd(X)) → 2nd(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
2nd(ok(X)) → ok(2nd(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:ok → mark:ok
2nd :: mark:ok → mark:ok
cons :: mark:ok → mark:ok → mark:ok
mark :: mark:ok → mark:ok
from :: mark:ok → mark:ok
s :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active,
cons,
from,
s,
2nd,
proper,
topThey will be analysed ascendingly in the following order:
cons < active
from < active
s < active
2nd < active
active < top
cons < proper
from < proper
s < proper
2nd < proper
proper < top
(8) Obligation:
TRS:
Rules:
active(
2nd(
cons(
X,
cons(
Y,
Z)))) →
mark(
Y)
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
2nd(
X)) →
2nd(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
2nd(
mark(
X)) →
mark(
2nd(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
proper(
2nd(
X)) →
2nd(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
2nd(
ok(
X)) →
ok(
2nd(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
2nd :: mark:ok → mark:ok
cons :: mark:ok → mark:ok → mark:ok
mark :: mark:ok → mark:ok
from :: mark:ok → mark:ok
s :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))
The following defined symbols remain to be analysed:
cons, active, from, s, 2nd, proper, top
They will be analysed ascendingly in the following order:
cons < active
from < active
s < active
2nd < active
active < top
cons < proper
from < proper
s < proper
2nd < proper
proper < top
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol cons.
(10) Obligation:
TRS:
Rules:
active(
2nd(
cons(
X,
cons(
Y,
Z)))) →
mark(
Y)
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
2nd(
X)) →
2nd(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
2nd(
mark(
X)) →
mark(
2nd(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
proper(
2nd(
X)) →
2nd(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
2nd(
ok(
X)) →
ok(
2nd(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
2nd :: mark:ok → mark:ok
cons :: mark:ok → mark:ok → mark:ok
mark :: mark:ok → mark:ok
from :: mark:ok → mark:ok
s :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))
The following defined symbols remain to be analysed:
from, active, s, 2nd, proper, top
They will be analysed ascendingly in the following order:
from < active
s < active
2nd < active
active < top
from < proper
s < proper
2nd < proper
proper < top
(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol from.
(12) Obligation:
TRS:
Rules:
active(
2nd(
cons(
X,
cons(
Y,
Z)))) →
mark(
Y)
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
2nd(
X)) →
2nd(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
2nd(
mark(
X)) →
mark(
2nd(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
proper(
2nd(
X)) →
2nd(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
2nd(
ok(
X)) →
ok(
2nd(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
2nd :: mark:ok → mark:ok
cons :: mark:ok → mark:ok → mark:ok
mark :: mark:ok → mark:ok
from :: mark:ok → mark:ok
s :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))
The following defined symbols remain to be analysed:
s, active, 2nd, proper, top
They will be analysed ascendingly in the following order:
s < active
2nd < active
active < top
s < proper
2nd < proper
proper < top
(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol s.
(14) Obligation:
TRS:
Rules:
active(
2nd(
cons(
X,
cons(
Y,
Z)))) →
mark(
Y)
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
2nd(
X)) →
2nd(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
2nd(
mark(
X)) →
mark(
2nd(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
proper(
2nd(
X)) →
2nd(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
2nd(
ok(
X)) →
ok(
2nd(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
2nd :: mark:ok → mark:ok
cons :: mark:ok → mark:ok → mark:ok
mark :: mark:ok → mark:ok
from :: mark:ok → mark:ok
s :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))
The following defined symbols remain to be analysed:
2nd, active, proper, top
They will be analysed ascendingly in the following order:
2nd < active
active < top
2nd < proper
proper < top
(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol 2nd.
(16) Obligation:
TRS:
Rules:
active(
2nd(
cons(
X,
cons(
Y,
Z)))) →
mark(
Y)
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
2nd(
X)) →
2nd(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
2nd(
mark(
X)) →
mark(
2nd(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
proper(
2nd(
X)) →
2nd(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
2nd(
ok(
X)) →
ok(
2nd(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
2nd :: mark:ok → mark:ok
cons :: mark:ok → mark:ok → mark:ok
mark :: mark:ok → mark:ok
from :: mark:ok → mark:ok
s :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_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
(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol active.
(18) Obligation:
TRS:
Rules:
active(
2nd(
cons(
X,
cons(
Y,
Z)))) →
mark(
Y)
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
2nd(
X)) →
2nd(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
2nd(
mark(
X)) →
mark(
2nd(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
proper(
2nd(
X)) →
2nd(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
2nd(
ok(
X)) →
ok(
2nd(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
2nd :: mark:ok → mark:ok
cons :: mark:ok → mark:ok → mark:ok
mark :: mark:ok → mark:ok
from :: mark:ok → mark:ok
s :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_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
(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol proper.
(20) Obligation:
TRS:
Rules:
active(
2nd(
cons(
X,
cons(
Y,
Z)))) →
mark(
Y)
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
2nd(
X)) →
2nd(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
2nd(
mark(
X)) →
mark(
2nd(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
proper(
2nd(
X)) →
2nd(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
2nd(
ok(
X)) →
ok(
2nd(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
2nd :: mark:ok → mark:ok
cons :: mark:ok → mark:ok → mark:ok
mark :: mark:ok → mark:ok
from :: mark:ok → mark:ok
s :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))
The following defined symbols remain to be analysed:
top
(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol top.
(22) Obligation:
TRS:
Rules:
active(
2nd(
cons(
X,
cons(
Y,
Z)))) →
mark(
Y)
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
2nd(
X)) →
2nd(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
from(
X)) →
from(
active(
X))
active(
s(
X)) →
s(
active(
X))
2nd(
mark(
X)) →
mark(
2nd(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
from(
mark(
X)) →
mark(
from(
X))
s(
mark(
X)) →
mark(
s(
X))
proper(
2nd(
X)) →
2nd(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
proper(
s(
X)) →
s(
proper(
X))
2nd(
ok(
X)) →
ok(
2nd(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
s(
ok(
X)) →
ok(
s(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:ok → mark:ok
2nd :: mark:ok → mark:ok
cons :: mark:ok → mark:ok → mark:ok
mark :: mark:ok → mark:ok
from :: mark:ok → mark:ok
s :: mark:ok → mark:ok
proper :: mark:ok → mark:ok
ok :: mark:ok → mark:ok
top :: mark:ok → top
hole_mark:ok1_0 :: mark:ok
hole_top2_0 :: top
gen_mark:ok3_0 :: Nat → mark:ok
Generator Equations:
gen_mark:ok3_0(0) ⇔ hole_mark:ok1_0
gen_mark:ok3_0(+(x, 1)) ⇔ mark(gen_mark:ok3_0(x))
No more defined symbols left to analyse.