(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(f(X)) → mark(cons(X, f(g(X))))
active(g(0)) → mark(s(0))
active(g(s(X))) → mark(s(s(g(X))))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(g(X)) → g(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
g(mark(X)) → mark(g(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(f(X)) → f(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
g(ok(X)) → ok(g(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
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
f(mark(X)) →+ mark(f(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(f(X)) → mark(cons(X, f(g(X))))
active(g(0')) → mark(s(0'))
active(g(s(X))) → mark(s(s(g(X))))
active(sel(0', cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(g(X)) → g(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
g(mark(X)) → mark(g(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(f(X)) → f(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
g(ok(X)) → ok(g(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
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(f(X)) → mark(cons(X, f(g(X))))
active(g(0')) → mark(s(0'))
active(g(s(X))) → mark(s(s(g(X))))
active(sel(0', cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(g(X)) → g(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
g(mark(X)) → mark(g(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(f(X)) → f(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
g(ok(X)) → ok(g(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':ok → mark:0':ok
f :: mark:0':ok → mark:0':ok
mark :: mark:0':ok → mark:0':ok
cons :: mark:0':ok → mark:0':ok → mark:0':ok
g :: mark:0':ok → mark:0':ok
0' :: mark:0':ok
s :: mark:0':ok → mark:0':ok
sel :: mark:0':ok → mark:0':ok → mark:0':ok
proper :: mark:0':ok → mark:0':ok
ok :: mark:0':ok → mark:0':ok
top :: mark:0':ok → top
hole_mark:0':ok1_0 :: mark:0':ok
hole_top2_0 :: top
gen_mark:0':ok3_0 :: Nat → mark:0':ok
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active,
cons,
f,
g,
s,
sel,
proper,
topThey will be analysed ascendingly in the following order:
cons < active
f < active
g < active
s < active
sel < active
active < top
cons < proper
f < proper
g < proper
s < proper
sel < proper
proper < top
(8) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
cons(
X,
f(
g(
X))))
active(
g(
0')) →
mark(
s(
0'))
active(
g(
s(
X))) →
mark(
s(
s(
g(
X))))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
f(
X)) →
f(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
f(
mark(
X)) →
mark(
f(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
s(
mark(
X)) →
mark(
s(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
proper(
f(
X)) →
f(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
f(
ok(
X)) →
ok(
f(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
g(
ok(
X)) →
ok(
g(
X))
s(
ok(
X)) →
ok(
s(
X))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:0':ok → mark:0':ok
f :: mark:0':ok → mark:0':ok
mark :: mark:0':ok → mark:0':ok
cons :: mark:0':ok → mark:0':ok → mark:0':ok
g :: mark:0':ok → mark:0':ok
0' :: mark:0':ok
s :: mark:0':ok → mark:0':ok
sel :: mark:0':ok → mark:0':ok → mark:0':ok
proper :: mark:0':ok → mark:0':ok
ok :: mark:0':ok → mark:0':ok
top :: mark:0':ok → top
hole_mark:0':ok1_0 :: mark:0':ok
hole_top2_0 :: top
gen_mark:0':ok3_0 :: Nat → mark:0':ok
Generator Equations:
gen_mark:0':ok3_0(0) ⇔ 0'
gen_mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
cons, active, f, g, s, sel, proper, top
They will be analysed ascendingly in the following order:
cons < active
f < active
g < active
s < active
sel < active
active < top
cons < proper
f < proper
g < proper
s < proper
sel < proper
proper < top
(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
cons(
gen_mark:0':ok3_0(
+(
1,
n5_0)),
gen_mark:0':ok3_0(
b)) →
*4_0, rt ∈ Ω(n5
0)
Induction Base:
cons(gen_mark:0':ok3_0(+(1, 0)), gen_mark:0':ok3_0(b))
Induction Step:
cons(gen_mark:0':ok3_0(+(1, +(n5_0, 1))), gen_mark:0':ok3_0(b)) →RΩ(1)
mark(cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(10) Complex Obligation (BEST)
(11) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
cons(
X,
f(
g(
X))))
active(
g(
0')) →
mark(
s(
0'))
active(
g(
s(
X))) →
mark(
s(
s(
g(
X))))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
f(
X)) →
f(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
f(
mark(
X)) →
mark(
f(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
s(
mark(
X)) →
mark(
s(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
proper(
f(
X)) →
f(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
f(
ok(
X)) →
ok(
f(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
g(
ok(
X)) →
ok(
g(
X))
s(
ok(
X)) →
ok(
s(
X))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:0':ok → mark:0':ok
f :: mark:0':ok → mark:0':ok
mark :: mark:0':ok → mark:0':ok
cons :: mark:0':ok → mark:0':ok → mark:0':ok
g :: mark:0':ok → mark:0':ok
0' :: mark:0':ok
s :: mark:0':ok → mark:0':ok
sel :: mark:0':ok → mark:0':ok → mark:0':ok
proper :: mark:0':ok → mark:0':ok
ok :: mark:0':ok → mark:0':ok
top :: mark:0':ok → top
hole_mark:0':ok1_0 :: mark:0':ok
hole_top2_0 :: top
gen_mark:0':ok3_0 :: Nat → mark:0':ok
Lemmas:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_mark:0':ok3_0(0) ⇔ 0'
gen_mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
f, active, g, s, sel, proper, top
They will be analysed ascendingly in the following order:
f < active
g < active
s < active
sel < active
active < top
f < proper
g < proper
s < proper
sel < proper
proper < top
(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
f(
gen_mark:0':ok3_0(
+(
1,
n902_0))) →
*4_0, rt ∈ Ω(n902
0)
Induction Base:
f(gen_mark:0':ok3_0(+(1, 0)))
Induction Step:
f(gen_mark:0':ok3_0(+(1, +(n902_0, 1)))) →RΩ(1)
mark(f(gen_mark:0':ok3_0(+(1, n902_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(13) Complex Obligation (BEST)
(14) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
cons(
X,
f(
g(
X))))
active(
g(
0')) →
mark(
s(
0'))
active(
g(
s(
X))) →
mark(
s(
s(
g(
X))))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
f(
X)) →
f(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
f(
mark(
X)) →
mark(
f(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
s(
mark(
X)) →
mark(
s(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
proper(
f(
X)) →
f(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
f(
ok(
X)) →
ok(
f(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
g(
ok(
X)) →
ok(
g(
X))
s(
ok(
X)) →
ok(
s(
X))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:0':ok → mark:0':ok
f :: mark:0':ok → mark:0':ok
mark :: mark:0':ok → mark:0':ok
cons :: mark:0':ok → mark:0':ok → mark:0':ok
g :: mark:0':ok → mark:0':ok
0' :: mark:0':ok
s :: mark:0':ok → mark:0':ok
sel :: mark:0':ok → mark:0':ok → mark:0':ok
proper :: mark:0':ok → mark:0':ok
ok :: mark:0':ok → mark:0':ok
top :: mark:0':ok → top
hole_mark:0':ok1_0 :: mark:0':ok
hole_top2_0 :: top
gen_mark:0':ok3_0 :: Nat → mark:0':ok
Lemmas:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
f(gen_mark:0':ok3_0(+(1, n902_0))) → *4_0, rt ∈ Ω(n9020)
Generator Equations:
gen_mark:0':ok3_0(0) ⇔ 0'
gen_mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
g, active, s, sel, proper, top
They will be analysed ascendingly in the following order:
g < active
s < active
sel < active
active < top
g < proper
s < proper
sel < proper
proper < top
(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
g(
gen_mark:0':ok3_0(
+(
1,
n1409_0))) →
*4_0, rt ∈ Ω(n1409
0)
Induction Base:
g(gen_mark:0':ok3_0(+(1, 0)))
Induction Step:
g(gen_mark:0':ok3_0(+(1, +(n1409_0, 1)))) →RΩ(1)
mark(g(gen_mark:0':ok3_0(+(1, n1409_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(16) Complex Obligation (BEST)
(17) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
cons(
X,
f(
g(
X))))
active(
g(
0')) →
mark(
s(
0'))
active(
g(
s(
X))) →
mark(
s(
s(
g(
X))))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
f(
X)) →
f(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
f(
mark(
X)) →
mark(
f(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
s(
mark(
X)) →
mark(
s(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
proper(
f(
X)) →
f(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
f(
ok(
X)) →
ok(
f(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
g(
ok(
X)) →
ok(
g(
X))
s(
ok(
X)) →
ok(
s(
X))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:0':ok → mark:0':ok
f :: mark:0':ok → mark:0':ok
mark :: mark:0':ok → mark:0':ok
cons :: mark:0':ok → mark:0':ok → mark:0':ok
g :: mark:0':ok → mark:0':ok
0' :: mark:0':ok
s :: mark:0':ok → mark:0':ok
sel :: mark:0':ok → mark:0':ok → mark:0':ok
proper :: mark:0':ok → mark:0':ok
ok :: mark:0':ok → mark:0':ok
top :: mark:0':ok → top
hole_mark:0':ok1_0 :: mark:0':ok
hole_top2_0 :: top
gen_mark:0':ok3_0 :: Nat → mark:0':ok
Lemmas:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
f(gen_mark:0':ok3_0(+(1, n902_0))) → *4_0, rt ∈ Ω(n9020)
g(gen_mark:0':ok3_0(+(1, n1409_0))) → *4_0, rt ∈ Ω(n14090)
Generator Equations:
gen_mark:0':ok3_0(0) ⇔ 0'
gen_mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
s, active, sel, proper, top
They will be analysed ascendingly in the following order:
s < active
sel < active
active < top
s < proper
sel < proper
proper < top
(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
s(
gen_mark:0':ok3_0(
+(
1,
n2017_0))) →
*4_0, rt ∈ Ω(n2017
0)
Induction Base:
s(gen_mark:0':ok3_0(+(1, 0)))
Induction Step:
s(gen_mark:0':ok3_0(+(1, +(n2017_0, 1)))) →RΩ(1)
mark(s(gen_mark:0':ok3_0(+(1, n2017_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(19) Complex Obligation (BEST)
(20) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
cons(
X,
f(
g(
X))))
active(
g(
0')) →
mark(
s(
0'))
active(
g(
s(
X))) →
mark(
s(
s(
g(
X))))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
f(
X)) →
f(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
f(
mark(
X)) →
mark(
f(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
s(
mark(
X)) →
mark(
s(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
proper(
f(
X)) →
f(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
f(
ok(
X)) →
ok(
f(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
g(
ok(
X)) →
ok(
g(
X))
s(
ok(
X)) →
ok(
s(
X))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:0':ok → mark:0':ok
f :: mark:0':ok → mark:0':ok
mark :: mark:0':ok → mark:0':ok
cons :: mark:0':ok → mark:0':ok → mark:0':ok
g :: mark:0':ok → mark:0':ok
0' :: mark:0':ok
s :: mark:0':ok → mark:0':ok
sel :: mark:0':ok → mark:0':ok → mark:0':ok
proper :: mark:0':ok → mark:0':ok
ok :: mark:0':ok → mark:0':ok
top :: mark:0':ok → top
hole_mark:0':ok1_0 :: mark:0':ok
hole_top2_0 :: top
gen_mark:0':ok3_0 :: Nat → mark:0':ok
Lemmas:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
f(gen_mark:0':ok3_0(+(1, n902_0))) → *4_0, rt ∈ Ω(n9020)
g(gen_mark:0':ok3_0(+(1, n1409_0))) → *4_0, rt ∈ Ω(n14090)
s(gen_mark:0':ok3_0(+(1, n2017_0))) → *4_0, rt ∈ Ω(n20170)
Generator Equations:
gen_mark:0':ok3_0(0) ⇔ 0'
gen_mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
sel, active, proper, top
They will be analysed ascendingly in the following order:
sel < active
active < top
sel < proper
proper < top
(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
sel(
gen_mark:0':ok3_0(
+(
1,
n2726_0)),
gen_mark:0':ok3_0(
b)) →
*4_0, rt ∈ Ω(n2726
0)
Induction Base:
sel(gen_mark:0':ok3_0(+(1, 0)), gen_mark:0':ok3_0(b))
Induction Step:
sel(gen_mark:0':ok3_0(+(1, +(n2726_0, 1))), gen_mark:0':ok3_0(b)) →RΩ(1)
mark(sel(gen_mark:0':ok3_0(+(1, n2726_0)), gen_mark:0':ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(22) Complex Obligation (BEST)
(23) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
cons(
X,
f(
g(
X))))
active(
g(
0')) →
mark(
s(
0'))
active(
g(
s(
X))) →
mark(
s(
s(
g(
X))))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
f(
X)) →
f(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
f(
mark(
X)) →
mark(
f(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
s(
mark(
X)) →
mark(
s(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
proper(
f(
X)) →
f(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
f(
ok(
X)) →
ok(
f(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
g(
ok(
X)) →
ok(
g(
X))
s(
ok(
X)) →
ok(
s(
X))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:0':ok → mark:0':ok
f :: mark:0':ok → mark:0':ok
mark :: mark:0':ok → mark:0':ok
cons :: mark:0':ok → mark:0':ok → mark:0':ok
g :: mark:0':ok → mark:0':ok
0' :: mark:0':ok
s :: mark:0':ok → mark:0':ok
sel :: mark:0':ok → mark:0':ok → mark:0':ok
proper :: mark:0':ok → mark:0':ok
ok :: mark:0':ok → mark:0':ok
top :: mark:0':ok → top
hole_mark:0':ok1_0 :: mark:0':ok
hole_top2_0 :: top
gen_mark:0':ok3_0 :: Nat → mark:0':ok
Lemmas:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
f(gen_mark:0':ok3_0(+(1, n902_0))) → *4_0, rt ∈ Ω(n9020)
g(gen_mark:0':ok3_0(+(1, n1409_0))) → *4_0, rt ∈ Ω(n14090)
s(gen_mark:0':ok3_0(+(1, n2017_0))) → *4_0, rt ∈ Ω(n20170)
sel(gen_mark:0':ok3_0(+(1, n2726_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n27260)
Generator Equations:
gen_mark:0':ok3_0(0) ⇔ 0'
gen_mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':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
(24) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol active.
(25) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
cons(
X,
f(
g(
X))))
active(
g(
0')) →
mark(
s(
0'))
active(
g(
s(
X))) →
mark(
s(
s(
g(
X))))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
f(
X)) →
f(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
f(
mark(
X)) →
mark(
f(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
s(
mark(
X)) →
mark(
s(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
proper(
f(
X)) →
f(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
f(
ok(
X)) →
ok(
f(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
g(
ok(
X)) →
ok(
g(
X))
s(
ok(
X)) →
ok(
s(
X))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:0':ok → mark:0':ok
f :: mark:0':ok → mark:0':ok
mark :: mark:0':ok → mark:0':ok
cons :: mark:0':ok → mark:0':ok → mark:0':ok
g :: mark:0':ok → mark:0':ok
0' :: mark:0':ok
s :: mark:0':ok → mark:0':ok
sel :: mark:0':ok → mark:0':ok → mark:0':ok
proper :: mark:0':ok → mark:0':ok
ok :: mark:0':ok → mark:0':ok
top :: mark:0':ok → top
hole_mark:0':ok1_0 :: mark:0':ok
hole_top2_0 :: top
gen_mark:0':ok3_0 :: Nat → mark:0':ok
Lemmas:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
f(gen_mark:0':ok3_0(+(1, n902_0))) → *4_0, rt ∈ Ω(n9020)
g(gen_mark:0':ok3_0(+(1, n1409_0))) → *4_0, rt ∈ Ω(n14090)
s(gen_mark:0':ok3_0(+(1, n2017_0))) → *4_0, rt ∈ Ω(n20170)
sel(gen_mark:0':ok3_0(+(1, n2726_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n27260)
Generator Equations:
gen_mark:0':ok3_0(0) ⇔ 0'
gen_mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
proper, top
They will be analysed ascendingly in the following order:
proper < top
(26) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol proper.
(27) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
cons(
X,
f(
g(
X))))
active(
g(
0')) →
mark(
s(
0'))
active(
g(
s(
X))) →
mark(
s(
s(
g(
X))))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
f(
X)) →
f(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
f(
mark(
X)) →
mark(
f(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
s(
mark(
X)) →
mark(
s(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
proper(
f(
X)) →
f(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
f(
ok(
X)) →
ok(
f(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
g(
ok(
X)) →
ok(
g(
X))
s(
ok(
X)) →
ok(
s(
X))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:0':ok → mark:0':ok
f :: mark:0':ok → mark:0':ok
mark :: mark:0':ok → mark:0':ok
cons :: mark:0':ok → mark:0':ok → mark:0':ok
g :: mark:0':ok → mark:0':ok
0' :: mark:0':ok
s :: mark:0':ok → mark:0':ok
sel :: mark:0':ok → mark:0':ok → mark:0':ok
proper :: mark:0':ok → mark:0':ok
ok :: mark:0':ok → mark:0':ok
top :: mark:0':ok → top
hole_mark:0':ok1_0 :: mark:0':ok
hole_top2_0 :: top
gen_mark:0':ok3_0 :: Nat → mark:0':ok
Lemmas:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
f(gen_mark:0':ok3_0(+(1, n902_0))) → *4_0, rt ∈ Ω(n9020)
g(gen_mark:0':ok3_0(+(1, n1409_0))) → *4_0, rt ∈ Ω(n14090)
s(gen_mark:0':ok3_0(+(1, n2017_0))) → *4_0, rt ∈ Ω(n20170)
sel(gen_mark:0':ok3_0(+(1, n2726_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n27260)
Generator Equations:
gen_mark:0':ok3_0(0) ⇔ 0'
gen_mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
top
(28) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol top.
(29) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
cons(
X,
f(
g(
X))))
active(
g(
0')) →
mark(
s(
0'))
active(
g(
s(
X))) →
mark(
s(
s(
g(
X))))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
f(
X)) →
f(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
f(
mark(
X)) →
mark(
f(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
s(
mark(
X)) →
mark(
s(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
proper(
f(
X)) →
f(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
f(
ok(
X)) →
ok(
f(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
g(
ok(
X)) →
ok(
g(
X))
s(
ok(
X)) →
ok(
s(
X))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:0':ok → mark:0':ok
f :: mark:0':ok → mark:0':ok
mark :: mark:0':ok → mark:0':ok
cons :: mark:0':ok → mark:0':ok → mark:0':ok
g :: mark:0':ok → mark:0':ok
0' :: mark:0':ok
s :: mark:0':ok → mark:0':ok
sel :: mark:0':ok → mark:0':ok → mark:0':ok
proper :: mark:0':ok → mark:0':ok
ok :: mark:0':ok → mark:0':ok
top :: mark:0':ok → top
hole_mark:0':ok1_0 :: mark:0':ok
hole_top2_0 :: top
gen_mark:0':ok3_0 :: Nat → mark:0':ok
Lemmas:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
f(gen_mark:0':ok3_0(+(1, n902_0))) → *4_0, rt ∈ Ω(n9020)
g(gen_mark:0':ok3_0(+(1, n1409_0))) → *4_0, rt ∈ Ω(n14090)
s(gen_mark:0':ok3_0(+(1, n2017_0))) → *4_0, rt ∈ Ω(n20170)
sel(gen_mark:0':ok3_0(+(1, n2726_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n27260)
Generator Equations:
gen_mark:0':ok3_0(0) ⇔ 0'
gen_mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':ok3_0(x))
No more defined symbols left to analyse.
(30) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(31) BOUNDS(n^1, INF)
(32) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
cons(
X,
f(
g(
X))))
active(
g(
0')) →
mark(
s(
0'))
active(
g(
s(
X))) →
mark(
s(
s(
g(
X))))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
f(
X)) →
f(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
f(
mark(
X)) →
mark(
f(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
s(
mark(
X)) →
mark(
s(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
proper(
f(
X)) →
f(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
f(
ok(
X)) →
ok(
f(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
g(
ok(
X)) →
ok(
g(
X))
s(
ok(
X)) →
ok(
s(
X))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:0':ok → mark:0':ok
f :: mark:0':ok → mark:0':ok
mark :: mark:0':ok → mark:0':ok
cons :: mark:0':ok → mark:0':ok → mark:0':ok
g :: mark:0':ok → mark:0':ok
0' :: mark:0':ok
s :: mark:0':ok → mark:0':ok
sel :: mark:0':ok → mark:0':ok → mark:0':ok
proper :: mark:0':ok → mark:0':ok
ok :: mark:0':ok → mark:0':ok
top :: mark:0':ok → top
hole_mark:0':ok1_0 :: mark:0':ok
hole_top2_0 :: top
gen_mark:0':ok3_0 :: Nat → mark:0':ok
Lemmas:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
f(gen_mark:0':ok3_0(+(1, n902_0))) → *4_0, rt ∈ Ω(n9020)
g(gen_mark:0':ok3_0(+(1, n1409_0))) → *4_0, rt ∈ Ω(n14090)
s(gen_mark:0':ok3_0(+(1, n2017_0))) → *4_0, rt ∈ Ω(n20170)
sel(gen_mark:0':ok3_0(+(1, n2726_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n27260)
Generator Equations:
gen_mark:0':ok3_0(0) ⇔ 0'
gen_mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':ok3_0(x))
No more defined symbols left to analyse.
(33) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(34) BOUNDS(n^1, INF)
(35) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
cons(
X,
f(
g(
X))))
active(
g(
0')) →
mark(
s(
0'))
active(
g(
s(
X))) →
mark(
s(
s(
g(
X))))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
f(
X)) →
f(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
f(
mark(
X)) →
mark(
f(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
s(
mark(
X)) →
mark(
s(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
proper(
f(
X)) →
f(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
f(
ok(
X)) →
ok(
f(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
g(
ok(
X)) →
ok(
g(
X))
s(
ok(
X)) →
ok(
s(
X))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:0':ok → mark:0':ok
f :: mark:0':ok → mark:0':ok
mark :: mark:0':ok → mark:0':ok
cons :: mark:0':ok → mark:0':ok → mark:0':ok
g :: mark:0':ok → mark:0':ok
0' :: mark:0':ok
s :: mark:0':ok → mark:0':ok
sel :: mark:0':ok → mark:0':ok → mark:0':ok
proper :: mark:0':ok → mark:0':ok
ok :: mark:0':ok → mark:0':ok
top :: mark:0':ok → top
hole_mark:0':ok1_0 :: mark:0':ok
hole_top2_0 :: top
gen_mark:0':ok3_0 :: Nat → mark:0':ok
Lemmas:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
f(gen_mark:0':ok3_0(+(1, n902_0))) → *4_0, rt ∈ Ω(n9020)
g(gen_mark:0':ok3_0(+(1, n1409_0))) → *4_0, rt ∈ Ω(n14090)
s(gen_mark:0':ok3_0(+(1, n2017_0))) → *4_0, rt ∈ Ω(n20170)
Generator Equations:
gen_mark:0':ok3_0(0) ⇔ 0'
gen_mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':ok3_0(x))
No more defined symbols left to analyse.
(36) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(37) BOUNDS(n^1, INF)
(38) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
cons(
X,
f(
g(
X))))
active(
g(
0')) →
mark(
s(
0'))
active(
g(
s(
X))) →
mark(
s(
s(
g(
X))))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
f(
X)) →
f(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
f(
mark(
X)) →
mark(
f(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
s(
mark(
X)) →
mark(
s(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
proper(
f(
X)) →
f(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
f(
ok(
X)) →
ok(
f(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
g(
ok(
X)) →
ok(
g(
X))
s(
ok(
X)) →
ok(
s(
X))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:0':ok → mark:0':ok
f :: mark:0':ok → mark:0':ok
mark :: mark:0':ok → mark:0':ok
cons :: mark:0':ok → mark:0':ok → mark:0':ok
g :: mark:0':ok → mark:0':ok
0' :: mark:0':ok
s :: mark:0':ok → mark:0':ok
sel :: mark:0':ok → mark:0':ok → mark:0':ok
proper :: mark:0':ok → mark:0':ok
ok :: mark:0':ok → mark:0':ok
top :: mark:0':ok → top
hole_mark:0':ok1_0 :: mark:0':ok
hole_top2_0 :: top
gen_mark:0':ok3_0 :: Nat → mark:0':ok
Lemmas:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
f(gen_mark:0':ok3_0(+(1, n902_0))) → *4_0, rt ∈ Ω(n9020)
g(gen_mark:0':ok3_0(+(1, n1409_0))) → *4_0, rt ∈ Ω(n14090)
Generator Equations:
gen_mark:0':ok3_0(0) ⇔ 0'
gen_mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':ok3_0(x))
No more defined symbols left to analyse.
(39) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(40) BOUNDS(n^1, INF)
(41) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
cons(
X,
f(
g(
X))))
active(
g(
0')) →
mark(
s(
0'))
active(
g(
s(
X))) →
mark(
s(
s(
g(
X))))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
f(
X)) →
f(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
f(
mark(
X)) →
mark(
f(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
s(
mark(
X)) →
mark(
s(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
proper(
f(
X)) →
f(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
f(
ok(
X)) →
ok(
f(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
g(
ok(
X)) →
ok(
g(
X))
s(
ok(
X)) →
ok(
s(
X))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:0':ok → mark:0':ok
f :: mark:0':ok → mark:0':ok
mark :: mark:0':ok → mark:0':ok
cons :: mark:0':ok → mark:0':ok → mark:0':ok
g :: mark:0':ok → mark:0':ok
0' :: mark:0':ok
s :: mark:0':ok → mark:0':ok
sel :: mark:0':ok → mark:0':ok → mark:0':ok
proper :: mark:0':ok → mark:0':ok
ok :: mark:0':ok → mark:0':ok
top :: mark:0':ok → top
hole_mark:0':ok1_0 :: mark:0':ok
hole_top2_0 :: top
gen_mark:0':ok3_0 :: Nat → mark:0':ok
Lemmas:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
f(gen_mark:0':ok3_0(+(1, n902_0))) → *4_0, rt ∈ Ω(n9020)
Generator Equations:
gen_mark:0':ok3_0(0) ⇔ 0'
gen_mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':ok3_0(x))
No more defined symbols left to analyse.
(42) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(43) BOUNDS(n^1, INF)
(44) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
cons(
X,
f(
g(
X))))
active(
g(
0')) →
mark(
s(
0'))
active(
g(
s(
X))) →
mark(
s(
s(
g(
X))))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
f(
X)) →
f(
active(
X))
active(
cons(
X1,
X2)) →
cons(
active(
X1),
X2)
active(
g(
X)) →
g(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
f(
mark(
X)) →
mark(
f(
X))
cons(
mark(
X1),
X2) →
mark(
cons(
X1,
X2))
g(
mark(
X)) →
mark(
g(
X))
s(
mark(
X)) →
mark(
s(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
proper(
f(
X)) →
f(
proper(
X))
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
g(
X)) →
g(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
f(
ok(
X)) →
ok(
f(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
g(
ok(
X)) →
ok(
g(
X))
s(
ok(
X)) →
ok(
s(
X))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: mark:0':ok → mark:0':ok
f :: mark:0':ok → mark:0':ok
mark :: mark:0':ok → mark:0':ok
cons :: mark:0':ok → mark:0':ok → mark:0':ok
g :: mark:0':ok → mark:0':ok
0' :: mark:0':ok
s :: mark:0':ok → mark:0':ok
sel :: mark:0':ok → mark:0':ok → mark:0':ok
proper :: mark:0':ok → mark:0':ok
ok :: mark:0':ok → mark:0':ok
top :: mark:0':ok → top
hole_mark:0':ok1_0 :: mark:0':ok
hole_top2_0 :: top
gen_mark:0':ok3_0 :: Nat → mark:0':ok
Lemmas:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_mark:0':ok3_0(0) ⇔ 0'
gen_mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':ok3_0(x))
No more defined symbols left to analyse.
(45) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(46) BOUNDS(n^1, INF)