(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
proper(dbl(X)) → dbl(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(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
dbl(mark(X)) →+ mark(dbl(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(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0', cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
proper(dbl(X)) → dbl(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(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(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0', cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(dbl(X)) → dbl(active(X))
active(dbls(X)) → dbls(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(indx(X1, X2)) → indx(active(X1), X2)
dbl(mark(X)) → mark(dbl(X))
dbls(mark(X)) → mark(dbls(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
indx(mark(X1), X2) → mark(indx(X1, X2))
proper(dbl(X)) → dbl(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(dbls(X)) → dbls(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(indx(X1, X2)) → indx(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
dbl(ok(X)) → ok(dbl(X))
s(ok(X)) → ok(s(X))
dbls(ok(X)) → ok(dbls(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
indx(ok(X1), ok(X2)) → ok(indx(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: 0':mark:nil:ok → 0':mark:nil:ok
dbl :: 0':mark:nil:ok → 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok → 0':mark:nil:ok
s :: 0':mark:nil:ok → 0':mark:nil:ok
dbls :: 0':mark:nil:ok → 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
sel :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
indx :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
from :: 0':mark:nil:ok → 0':mark:nil:ok
proper :: 0':mark:nil:ok → 0':mark:nil:ok
ok :: 0':mark:nil:ok → 0':mark:nil:ok
top :: 0':mark:nil:ok → top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat → 0':mark:nil:ok
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active,
s,
dbl,
cons,
dbls,
sel,
indx,
from,
proper,
topThey will be analysed ascendingly in the following order:
s < active
dbl < active
cons < active
dbls < active
sel < active
indx < active
from < active
active < top
s < proper
dbl < proper
cons < proper
dbls < proper
sel < proper
indx < proper
from < proper
proper < top
(8) Obligation:
TRS:
Rules:
active(
dbl(
0')) →
mark(
0')
active(
dbl(
s(
X))) →
mark(
s(
s(
dbl(
X))))
active(
dbls(
nil)) →
mark(
nil)
active(
dbls(
cons(
X,
Y))) →
mark(
cons(
dbl(
X),
dbls(
Y)))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
indx(
nil,
X)) →
mark(
nil)
active(
indx(
cons(
X,
Y),
Z)) →
mark(
cons(
sel(
X,
Z),
indx(
Y,
Z)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
dbl(
X)) →
dbl(
active(
X))
active(
dbls(
X)) →
dbls(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
active(
indx(
X1,
X2)) →
indx(
active(
X1),
X2)
dbl(
mark(
X)) →
mark(
dbl(
X))
dbls(
mark(
X)) →
mark(
dbls(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
indx(
mark(
X1),
X2) →
mark(
indx(
X1,
X2))
proper(
dbl(
X)) →
dbl(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
dbls(
X)) →
dbls(
proper(
X))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
proper(
indx(
X1,
X2)) →
indx(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
dbl(
ok(
X)) →
ok(
dbl(
X))
s(
ok(
X)) →
ok(
s(
X))
dbls(
ok(
X)) →
ok(
dbls(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
indx(
ok(
X1),
ok(
X2)) →
ok(
indx(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:nil:ok → 0':mark:nil:ok
dbl :: 0':mark:nil:ok → 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok → 0':mark:nil:ok
s :: 0':mark:nil:ok → 0':mark:nil:ok
dbls :: 0':mark:nil:ok → 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
sel :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
indx :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
from :: 0':mark:nil:ok → 0':mark:nil:ok
proper :: 0':mark:nil:ok → 0':mark:nil:ok
ok :: 0':mark:nil:ok → 0':mark:nil:ok
top :: 0':mark:nil:ok → top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat → 0':mark:nil:ok
Generator Equations:
gen_0':mark:nil:ok3_0(0) ⇔ 0'
gen_0':mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
s, active, dbl, cons, dbls, sel, indx, from, proper, top
They will be analysed ascendingly in the following order:
s < active
dbl < active
cons < active
dbls < active
sel < active
indx < active
from < active
active < top
s < proper
dbl < proper
cons < proper
dbls < proper
sel < proper
indx < proper
from < proper
proper < top
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol s.
(10) Obligation:
TRS:
Rules:
active(
dbl(
0')) →
mark(
0')
active(
dbl(
s(
X))) →
mark(
s(
s(
dbl(
X))))
active(
dbls(
nil)) →
mark(
nil)
active(
dbls(
cons(
X,
Y))) →
mark(
cons(
dbl(
X),
dbls(
Y)))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
indx(
nil,
X)) →
mark(
nil)
active(
indx(
cons(
X,
Y),
Z)) →
mark(
cons(
sel(
X,
Z),
indx(
Y,
Z)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
dbl(
X)) →
dbl(
active(
X))
active(
dbls(
X)) →
dbls(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
active(
indx(
X1,
X2)) →
indx(
active(
X1),
X2)
dbl(
mark(
X)) →
mark(
dbl(
X))
dbls(
mark(
X)) →
mark(
dbls(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
indx(
mark(
X1),
X2) →
mark(
indx(
X1,
X2))
proper(
dbl(
X)) →
dbl(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
dbls(
X)) →
dbls(
proper(
X))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
proper(
indx(
X1,
X2)) →
indx(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
dbl(
ok(
X)) →
ok(
dbl(
X))
s(
ok(
X)) →
ok(
s(
X))
dbls(
ok(
X)) →
ok(
dbls(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
indx(
ok(
X1),
ok(
X2)) →
ok(
indx(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:nil:ok → 0':mark:nil:ok
dbl :: 0':mark:nil:ok → 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok → 0':mark:nil:ok
s :: 0':mark:nil:ok → 0':mark:nil:ok
dbls :: 0':mark:nil:ok → 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
sel :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
indx :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
from :: 0':mark:nil:ok → 0':mark:nil:ok
proper :: 0':mark:nil:ok → 0':mark:nil:ok
ok :: 0':mark:nil:ok → 0':mark:nil:ok
top :: 0':mark:nil:ok → top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat → 0':mark:nil:ok
Generator Equations:
gen_0':mark:nil:ok3_0(0) ⇔ 0'
gen_0':mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
dbl, active, cons, dbls, sel, indx, from, proper, top
They will be analysed ascendingly in the following order:
dbl < active
cons < active
dbls < active
sel < active
indx < active
from < active
active < top
dbl < proper
cons < proper
dbls < proper
sel < proper
indx < proper
from < proper
proper < top
(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
dbl(
gen_0':mark:nil:ok3_0(
+(
1,
n9_0))) →
*4_0, rt ∈ Ω(n9
0)
Induction Base:
dbl(gen_0':mark:nil:ok3_0(+(1, 0)))
Induction Step:
dbl(gen_0':mark:nil:ok3_0(+(1, +(n9_0, 1)))) →RΩ(1)
mark(dbl(gen_0':mark:nil:ok3_0(+(1, n9_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(12) Complex Obligation (BEST)
(13) Obligation:
TRS:
Rules:
active(
dbl(
0')) →
mark(
0')
active(
dbl(
s(
X))) →
mark(
s(
s(
dbl(
X))))
active(
dbls(
nil)) →
mark(
nil)
active(
dbls(
cons(
X,
Y))) →
mark(
cons(
dbl(
X),
dbls(
Y)))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
indx(
nil,
X)) →
mark(
nil)
active(
indx(
cons(
X,
Y),
Z)) →
mark(
cons(
sel(
X,
Z),
indx(
Y,
Z)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
dbl(
X)) →
dbl(
active(
X))
active(
dbls(
X)) →
dbls(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
active(
indx(
X1,
X2)) →
indx(
active(
X1),
X2)
dbl(
mark(
X)) →
mark(
dbl(
X))
dbls(
mark(
X)) →
mark(
dbls(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
indx(
mark(
X1),
X2) →
mark(
indx(
X1,
X2))
proper(
dbl(
X)) →
dbl(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
dbls(
X)) →
dbls(
proper(
X))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
proper(
indx(
X1,
X2)) →
indx(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
dbl(
ok(
X)) →
ok(
dbl(
X))
s(
ok(
X)) →
ok(
s(
X))
dbls(
ok(
X)) →
ok(
dbls(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
indx(
ok(
X1),
ok(
X2)) →
ok(
indx(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:nil:ok → 0':mark:nil:ok
dbl :: 0':mark:nil:ok → 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok → 0':mark:nil:ok
s :: 0':mark:nil:ok → 0':mark:nil:ok
dbls :: 0':mark:nil:ok → 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
sel :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
indx :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
from :: 0':mark:nil:ok → 0':mark:nil:ok
proper :: 0':mark:nil:ok → 0':mark:nil:ok
ok :: 0':mark:nil:ok → 0':mark:nil:ok
top :: 0':mark:nil:ok → top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat → 0':mark:nil:ok
Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
Generator Equations:
gen_0':mark:nil:ok3_0(0) ⇔ 0'
gen_0':mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
cons, active, dbls, sel, indx, from, proper, top
They will be analysed ascendingly in the following order:
cons < active
dbls < active
sel < active
indx < active
from < active
active < top
cons < proper
dbls < proper
sel < proper
indx < proper
from < proper
proper < top
(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol cons.
(15) Obligation:
TRS:
Rules:
active(
dbl(
0')) →
mark(
0')
active(
dbl(
s(
X))) →
mark(
s(
s(
dbl(
X))))
active(
dbls(
nil)) →
mark(
nil)
active(
dbls(
cons(
X,
Y))) →
mark(
cons(
dbl(
X),
dbls(
Y)))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
indx(
nil,
X)) →
mark(
nil)
active(
indx(
cons(
X,
Y),
Z)) →
mark(
cons(
sel(
X,
Z),
indx(
Y,
Z)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
dbl(
X)) →
dbl(
active(
X))
active(
dbls(
X)) →
dbls(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
active(
indx(
X1,
X2)) →
indx(
active(
X1),
X2)
dbl(
mark(
X)) →
mark(
dbl(
X))
dbls(
mark(
X)) →
mark(
dbls(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
indx(
mark(
X1),
X2) →
mark(
indx(
X1,
X2))
proper(
dbl(
X)) →
dbl(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
dbls(
X)) →
dbls(
proper(
X))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
proper(
indx(
X1,
X2)) →
indx(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
dbl(
ok(
X)) →
ok(
dbl(
X))
s(
ok(
X)) →
ok(
s(
X))
dbls(
ok(
X)) →
ok(
dbls(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
indx(
ok(
X1),
ok(
X2)) →
ok(
indx(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:nil:ok → 0':mark:nil:ok
dbl :: 0':mark:nil:ok → 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok → 0':mark:nil:ok
s :: 0':mark:nil:ok → 0':mark:nil:ok
dbls :: 0':mark:nil:ok → 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
sel :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
indx :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
from :: 0':mark:nil:ok → 0':mark:nil:ok
proper :: 0':mark:nil:ok → 0':mark:nil:ok
ok :: 0':mark:nil:ok → 0':mark:nil:ok
top :: 0':mark:nil:ok → top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat → 0':mark:nil:ok
Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
Generator Equations:
gen_0':mark:nil:ok3_0(0) ⇔ 0'
gen_0':mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
dbls, active, sel, indx, from, proper, top
They will be analysed ascendingly in the following order:
dbls < active
sel < active
indx < active
from < active
active < top
dbls < proper
sel < proper
indx < proper
from < proper
proper < top
(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
dbls(
gen_0':mark:nil:ok3_0(
+(
1,
n399_0))) →
*4_0, rt ∈ Ω(n399
0)
Induction Base:
dbls(gen_0':mark:nil:ok3_0(+(1, 0)))
Induction Step:
dbls(gen_0':mark:nil:ok3_0(+(1, +(n399_0, 1)))) →RΩ(1)
mark(dbls(gen_0':mark:nil:ok3_0(+(1, n399_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(17) Complex Obligation (BEST)
(18) Obligation:
TRS:
Rules:
active(
dbl(
0')) →
mark(
0')
active(
dbl(
s(
X))) →
mark(
s(
s(
dbl(
X))))
active(
dbls(
nil)) →
mark(
nil)
active(
dbls(
cons(
X,
Y))) →
mark(
cons(
dbl(
X),
dbls(
Y)))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
indx(
nil,
X)) →
mark(
nil)
active(
indx(
cons(
X,
Y),
Z)) →
mark(
cons(
sel(
X,
Z),
indx(
Y,
Z)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
dbl(
X)) →
dbl(
active(
X))
active(
dbls(
X)) →
dbls(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
active(
indx(
X1,
X2)) →
indx(
active(
X1),
X2)
dbl(
mark(
X)) →
mark(
dbl(
X))
dbls(
mark(
X)) →
mark(
dbls(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
indx(
mark(
X1),
X2) →
mark(
indx(
X1,
X2))
proper(
dbl(
X)) →
dbl(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
dbls(
X)) →
dbls(
proper(
X))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
proper(
indx(
X1,
X2)) →
indx(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
dbl(
ok(
X)) →
ok(
dbl(
X))
s(
ok(
X)) →
ok(
s(
X))
dbls(
ok(
X)) →
ok(
dbls(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
indx(
ok(
X1),
ok(
X2)) →
ok(
indx(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:nil:ok → 0':mark:nil:ok
dbl :: 0':mark:nil:ok → 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok → 0':mark:nil:ok
s :: 0':mark:nil:ok → 0':mark:nil:ok
dbls :: 0':mark:nil:ok → 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
sel :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
indx :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
from :: 0':mark:nil:ok → 0':mark:nil:ok
proper :: 0':mark:nil:ok → 0':mark:nil:ok
ok :: 0':mark:nil:ok → 0':mark:nil:ok
top :: 0':mark:nil:ok → top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat → 0':mark:nil:ok
Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
dbls(gen_0':mark:nil:ok3_0(+(1, n399_0))) → *4_0, rt ∈ Ω(n3990)
Generator Equations:
gen_0':mark:nil:ok3_0(0) ⇔ 0'
gen_0':mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
sel, active, indx, from, proper, top
They will be analysed ascendingly in the following order:
sel < active
indx < active
from < active
active < top
sel < proper
indx < proper
from < proper
proper < top
(19) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
sel(
gen_0':mark:nil:ok3_0(
+(
1,
n878_0)),
gen_0':mark:nil:ok3_0(
b)) →
*4_0, rt ∈ Ω(n878
0)
Induction Base:
sel(gen_0':mark:nil:ok3_0(+(1, 0)), gen_0':mark:nil:ok3_0(b))
Induction Step:
sel(gen_0':mark:nil:ok3_0(+(1, +(n878_0, 1))), gen_0':mark:nil:ok3_0(b)) →RΩ(1)
mark(sel(gen_0':mark:nil:ok3_0(+(1, n878_0)), gen_0':mark:nil:ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(20) Complex Obligation (BEST)
(21) Obligation:
TRS:
Rules:
active(
dbl(
0')) →
mark(
0')
active(
dbl(
s(
X))) →
mark(
s(
s(
dbl(
X))))
active(
dbls(
nil)) →
mark(
nil)
active(
dbls(
cons(
X,
Y))) →
mark(
cons(
dbl(
X),
dbls(
Y)))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
indx(
nil,
X)) →
mark(
nil)
active(
indx(
cons(
X,
Y),
Z)) →
mark(
cons(
sel(
X,
Z),
indx(
Y,
Z)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
dbl(
X)) →
dbl(
active(
X))
active(
dbls(
X)) →
dbls(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
active(
indx(
X1,
X2)) →
indx(
active(
X1),
X2)
dbl(
mark(
X)) →
mark(
dbl(
X))
dbls(
mark(
X)) →
mark(
dbls(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
indx(
mark(
X1),
X2) →
mark(
indx(
X1,
X2))
proper(
dbl(
X)) →
dbl(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
dbls(
X)) →
dbls(
proper(
X))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
proper(
indx(
X1,
X2)) →
indx(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
dbl(
ok(
X)) →
ok(
dbl(
X))
s(
ok(
X)) →
ok(
s(
X))
dbls(
ok(
X)) →
ok(
dbls(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
indx(
ok(
X1),
ok(
X2)) →
ok(
indx(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:nil:ok → 0':mark:nil:ok
dbl :: 0':mark:nil:ok → 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok → 0':mark:nil:ok
s :: 0':mark:nil:ok → 0':mark:nil:ok
dbls :: 0':mark:nil:ok → 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
sel :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
indx :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
from :: 0':mark:nil:ok → 0':mark:nil:ok
proper :: 0':mark:nil:ok → 0':mark:nil:ok
ok :: 0':mark:nil:ok → 0':mark:nil:ok
top :: 0':mark:nil:ok → top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat → 0':mark:nil:ok
Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
dbls(gen_0':mark:nil:ok3_0(+(1, n399_0))) → *4_0, rt ∈ Ω(n3990)
sel(gen_0':mark:nil:ok3_0(+(1, n878_0)), gen_0':mark:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8780)
Generator Equations:
gen_0':mark:nil:ok3_0(0) ⇔ 0'
gen_0':mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
indx, active, from, proper, top
They will be analysed ascendingly in the following order:
indx < active
from < active
active < top
indx < proper
from < proper
proper < top
(22) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
indx(
gen_0':mark:nil:ok3_0(
+(
1,
n2470_0)),
gen_0':mark:nil:ok3_0(
b)) →
*4_0, rt ∈ Ω(n2470
0)
Induction Base:
indx(gen_0':mark:nil:ok3_0(+(1, 0)), gen_0':mark:nil:ok3_0(b))
Induction Step:
indx(gen_0':mark:nil:ok3_0(+(1, +(n2470_0, 1))), gen_0':mark:nil:ok3_0(b)) →RΩ(1)
mark(indx(gen_0':mark:nil:ok3_0(+(1, n2470_0)), gen_0':mark:nil:ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(23) Complex Obligation (BEST)
(24) Obligation:
TRS:
Rules:
active(
dbl(
0')) →
mark(
0')
active(
dbl(
s(
X))) →
mark(
s(
s(
dbl(
X))))
active(
dbls(
nil)) →
mark(
nil)
active(
dbls(
cons(
X,
Y))) →
mark(
cons(
dbl(
X),
dbls(
Y)))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
indx(
nil,
X)) →
mark(
nil)
active(
indx(
cons(
X,
Y),
Z)) →
mark(
cons(
sel(
X,
Z),
indx(
Y,
Z)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
dbl(
X)) →
dbl(
active(
X))
active(
dbls(
X)) →
dbls(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
active(
indx(
X1,
X2)) →
indx(
active(
X1),
X2)
dbl(
mark(
X)) →
mark(
dbl(
X))
dbls(
mark(
X)) →
mark(
dbls(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
indx(
mark(
X1),
X2) →
mark(
indx(
X1,
X2))
proper(
dbl(
X)) →
dbl(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
dbls(
X)) →
dbls(
proper(
X))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
proper(
indx(
X1,
X2)) →
indx(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
dbl(
ok(
X)) →
ok(
dbl(
X))
s(
ok(
X)) →
ok(
s(
X))
dbls(
ok(
X)) →
ok(
dbls(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
indx(
ok(
X1),
ok(
X2)) →
ok(
indx(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:nil:ok → 0':mark:nil:ok
dbl :: 0':mark:nil:ok → 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok → 0':mark:nil:ok
s :: 0':mark:nil:ok → 0':mark:nil:ok
dbls :: 0':mark:nil:ok → 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
sel :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
indx :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
from :: 0':mark:nil:ok → 0':mark:nil:ok
proper :: 0':mark:nil:ok → 0':mark:nil:ok
ok :: 0':mark:nil:ok → 0':mark:nil:ok
top :: 0':mark:nil:ok → top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat → 0':mark:nil:ok
Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
dbls(gen_0':mark:nil:ok3_0(+(1, n399_0))) → *4_0, rt ∈ Ω(n3990)
sel(gen_0':mark:nil:ok3_0(+(1, n878_0)), gen_0':mark:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8780)
indx(gen_0':mark:nil:ok3_0(+(1, n2470_0)), gen_0':mark:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24700)
Generator Equations:
gen_0':mark:nil:ok3_0(0) ⇔ 0'
gen_0':mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
from, active, proper, top
They will be analysed ascendingly in the following order:
from < active
active < top
from < proper
proper < top
(25) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol from.
(26) Obligation:
TRS:
Rules:
active(
dbl(
0')) →
mark(
0')
active(
dbl(
s(
X))) →
mark(
s(
s(
dbl(
X))))
active(
dbls(
nil)) →
mark(
nil)
active(
dbls(
cons(
X,
Y))) →
mark(
cons(
dbl(
X),
dbls(
Y)))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
indx(
nil,
X)) →
mark(
nil)
active(
indx(
cons(
X,
Y),
Z)) →
mark(
cons(
sel(
X,
Z),
indx(
Y,
Z)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
dbl(
X)) →
dbl(
active(
X))
active(
dbls(
X)) →
dbls(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
active(
indx(
X1,
X2)) →
indx(
active(
X1),
X2)
dbl(
mark(
X)) →
mark(
dbl(
X))
dbls(
mark(
X)) →
mark(
dbls(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
indx(
mark(
X1),
X2) →
mark(
indx(
X1,
X2))
proper(
dbl(
X)) →
dbl(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
dbls(
X)) →
dbls(
proper(
X))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
proper(
indx(
X1,
X2)) →
indx(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
dbl(
ok(
X)) →
ok(
dbl(
X))
s(
ok(
X)) →
ok(
s(
X))
dbls(
ok(
X)) →
ok(
dbls(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
indx(
ok(
X1),
ok(
X2)) →
ok(
indx(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:nil:ok → 0':mark:nil:ok
dbl :: 0':mark:nil:ok → 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok → 0':mark:nil:ok
s :: 0':mark:nil:ok → 0':mark:nil:ok
dbls :: 0':mark:nil:ok → 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
sel :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
indx :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
from :: 0':mark:nil:ok → 0':mark:nil:ok
proper :: 0':mark:nil:ok → 0':mark:nil:ok
ok :: 0':mark:nil:ok → 0':mark:nil:ok
top :: 0':mark:nil:ok → top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat → 0':mark:nil:ok
Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
dbls(gen_0':mark:nil:ok3_0(+(1, n399_0))) → *4_0, rt ∈ Ω(n3990)
sel(gen_0':mark:nil:ok3_0(+(1, n878_0)), gen_0':mark:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8780)
indx(gen_0':mark:nil:ok3_0(+(1, n2470_0)), gen_0':mark:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24700)
Generator Equations:
gen_0':mark:nil:ok3_0(0) ⇔ 0'
gen_0':mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':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
(27) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol active.
(28) Obligation:
TRS:
Rules:
active(
dbl(
0')) →
mark(
0')
active(
dbl(
s(
X))) →
mark(
s(
s(
dbl(
X))))
active(
dbls(
nil)) →
mark(
nil)
active(
dbls(
cons(
X,
Y))) →
mark(
cons(
dbl(
X),
dbls(
Y)))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
indx(
nil,
X)) →
mark(
nil)
active(
indx(
cons(
X,
Y),
Z)) →
mark(
cons(
sel(
X,
Z),
indx(
Y,
Z)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
dbl(
X)) →
dbl(
active(
X))
active(
dbls(
X)) →
dbls(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
active(
indx(
X1,
X2)) →
indx(
active(
X1),
X2)
dbl(
mark(
X)) →
mark(
dbl(
X))
dbls(
mark(
X)) →
mark(
dbls(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
indx(
mark(
X1),
X2) →
mark(
indx(
X1,
X2))
proper(
dbl(
X)) →
dbl(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
dbls(
X)) →
dbls(
proper(
X))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
proper(
indx(
X1,
X2)) →
indx(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
dbl(
ok(
X)) →
ok(
dbl(
X))
s(
ok(
X)) →
ok(
s(
X))
dbls(
ok(
X)) →
ok(
dbls(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
indx(
ok(
X1),
ok(
X2)) →
ok(
indx(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:nil:ok → 0':mark:nil:ok
dbl :: 0':mark:nil:ok → 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok → 0':mark:nil:ok
s :: 0':mark:nil:ok → 0':mark:nil:ok
dbls :: 0':mark:nil:ok → 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
sel :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
indx :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
from :: 0':mark:nil:ok → 0':mark:nil:ok
proper :: 0':mark:nil:ok → 0':mark:nil:ok
ok :: 0':mark:nil:ok → 0':mark:nil:ok
top :: 0':mark:nil:ok → top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat → 0':mark:nil:ok
Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
dbls(gen_0':mark:nil:ok3_0(+(1, n399_0))) → *4_0, rt ∈ Ω(n3990)
sel(gen_0':mark:nil:ok3_0(+(1, n878_0)), gen_0':mark:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8780)
indx(gen_0':mark:nil:ok3_0(+(1, n2470_0)), gen_0':mark:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24700)
Generator Equations:
gen_0':mark:nil:ok3_0(0) ⇔ 0'
gen_0':mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':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
(29) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol proper.
(30) Obligation:
TRS:
Rules:
active(
dbl(
0')) →
mark(
0')
active(
dbl(
s(
X))) →
mark(
s(
s(
dbl(
X))))
active(
dbls(
nil)) →
mark(
nil)
active(
dbls(
cons(
X,
Y))) →
mark(
cons(
dbl(
X),
dbls(
Y)))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
indx(
nil,
X)) →
mark(
nil)
active(
indx(
cons(
X,
Y),
Z)) →
mark(
cons(
sel(
X,
Z),
indx(
Y,
Z)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
dbl(
X)) →
dbl(
active(
X))
active(
dbls(
X)) →
dbls(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
active(
indx(
X1,
X2)) →
indx(
active(
X1),
X2)
dbl(
mark(
X)) →
mark(
dbl(
X))
dbls(
mark(
X)) →
mark(
dbls(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
indx(
mark(
X1),
X2) →
mark(
indx(
X1,
X2))
proper(
dbl(
X)) →
dbl(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
dbls(
X)) →
dbls(
proper(
X))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
proper(
indx(
X1,
X2)) →
indx(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
dbl(
ok(
X)) →
ok(
dbl(
X))
s(
ok(
X)) →
ok(
s(
X))
dbls(
ok(
X)) →
ok(
dbls(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
indx(
ok(
X1),
ok(
X2)) →
ok(
indx(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:nil:ok → 0':mark:nil:ok
dbl :: 0':mark:nil:ok → 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok → 0':mark:nil:ok
s :: 0':mark:nil:ok → 0':mark:nil:ok
dbls :: 0':mark:nil:ok → 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
sel :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
indx :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
from :: 0':mark:nil:ok → 0':mark:nil:ok
proper :: 0':mark:nil:ok → 0':mark:nil:ok
ok :: 0':mark:nil:ok → 0':mark:nil:ok
top :: 0':mark:nil:ok → top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat → 0':mark:nil:ok
Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
dbls(gen_0':mark:nil:ok3_0(+(1, n399_0))) → *4_0, rt ∈ Ω(n3990)
sel(gen_0':mark:nil:ok3_0(+(1, n878_0)), gen_0':mark:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8780)
indx(gen_0':mark:nil:ok3_0(+(1, n2470_0)), gen_0':mark:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24700)
Generator Equations:
gen_0':mark:nil:ok3_0(0) ⇔ 0'
gen_0':mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:ok3_0(x))
The following defined symbols remain to be analysed:
top
(31) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol top.
(32) Obligation:
TRS:
Rules:
active(
dbl(
0')) →
mark(
0')
active(
dbl(
s(
X))) →
mark(
s(
s(
dbl(
X))))
active(
dbls(
nil)) →
mark(
nil)
active(
dbls(
cons(
X,
Y))) →
mark(
cons(
dbl(
X),
dbls(
Y)))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
indx(
nil,
X)) →
mark(
nil)
active(
indx(
cons(
X,
Y),
Z)) →
mark(
cons(
sel(
X,
Z),
indx(
Y,
Z)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
dbl(
X)) →
dbl(
active(
X))
active(
dbls(
X)) →
dbls(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
active(
indx(
X1,
X2)) →
indx(
active(
X1),
X2)
dbl(
mark(
X)) →
mark(
dbl(
X))
dbls(
mark(
X)) →
mark(
dbls(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
indx(
mark(
X1),
X2) →
mark(
indx(
X1,
X2))
proper(
dbl(
X)) →
dbl(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
dbls(
X)) →
dbls(
proper(
X))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
proper(
indx(
X1,
X2)) →
indx(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
dbl(
ok(
X)) →
ok(
dbl(
X))
s(
ok(
X)) →
ok(
s(
X))
dbls(
ok(
X)) →
ok(
dbls(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
indx(
ok(
X1),
ok(
X2)) →
ok(
indx(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:nil:ok → 0':mark:nil:ok
dbl :: 0':mark:nil:ok → 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok → 0':mark:nil:ok
s :: 0':mark:nil:ok → 0':mark:nil:ok
dbls :: 0':mark:nil:ok → 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
sel :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
indx :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
from :: 0':mark:nil:ok → 0':mark:nil:ok
proper :: 0':mark:nil:ok → 0':mark:nil:ok
ok :: 0':mark:nil:ok → 0':mark:nil:ok
top :: 0':mark:nil:ok → top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat → 0':mark:nil:ok
Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
dbls(gen_0':mark:nil:ok3_0(+(1, n399_0))) → *4_0, rt ∈ Ω(n3990)
sel(gen_0':mark:nil:ok3_0(+(1, n878_0)), gen_0':mark:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8780)
indx(gen_0':mark:nil:ok3_0(+(1, n2470_0)), gen_0':mark:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24700)
Generator Equations:
gen_0':mark:nil:ok3_0(0) ⇔ 0'
gen_0':mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:ok3_0(x))
No more defined symbols left to analyse.
(33) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
(34) BOUNDS(n^1, INF)
(35) Obligation:
TRS:
Rules:
active(
dbl(
0')) →
mark(
0')
active(
dbl(
s(
X))) →
mark(
s(
s(
dbl(
X))))
active(
dbls(
nil)) →
mark(
nil)
active(
dbls(
cons(
X,
Y))) →
mark(
cons(
dbl(
X),
dbls(
Y)))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
indx(
nil,
X)) →
mark(
nil)
active(
indx(
cons(
X,
Y),
Z)) →
mark(
cons(
sel(
X,
Z),
indx(
Y,
Z)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
dbl(
X)) →
dbl(
active(
X))
active(
dbls(
X)) →
dbls(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
active(
indx(
X1,
X2)) →
indx(
active(
X1),
X2)
dbl(
mark(
X)) →
mark(
dbl(
X))
dbls(
mark(
X)) →
mark(
dbls(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
indx(
mark(
X1),
X2) →
mark(
indx(
X1,
X2))
proper(
dbl(
X)) →
dbl(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
dbls(
X)) →
dbls(
proper(
X))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
proper(
indx(
X1,
X2)) →
indx(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
dbl(
ok(
X)) →
ok(
dbl(
X))
s(
ok(
X)) →
ok(
s(
X))
dbls(
ok(
X)) →
ok(
dbls(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
indx(
ok(
X1),
ok(
X2)) →
ok(
indx(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:nil:ok → 0':mark:nil:ok
dbl :: 0':mark:nil:ok → 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok → 0':mark:nil:ok
s :: 0':mark:nil:ok → 0':mark:nil:ok
dbls :: 0':mark:nil:ok → 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
sel :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
indx :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
from :: 0':mark:nil:ok → 0':mark:nil:ok
proper :: 0':mark:nil:ok → 0':mark:nil:ok
ok :: 0':mark:nil:ok → 0':mark:nil:ok
top :: 0':mark:nil:ok → top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat → 0':mark:nil:ok
Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
dbls(gen_0':mark:nil:ok3_0(+(1, n399_0))) → *4_0, rt ∈ Ω(n3990)
sel(gen_0':mark:nil:ok3_0(+(1, n878_0)), gen_0':mark:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8780)
indx(gen_0':mark:nil:ok3_0(+(1, n2470_0)), gen_0':mark:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24700)
Generator Equations:
gen_0':mark:nil:ok3_0(0) ⇔ 0'
gen_0':mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:ok3_0(x))
No more defined symbols left to analyse.
(36) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
(37) BOUNDS(n^1, INF)
(38) Obligation:
TRS:
Rules:
active(
dbl(
0')) →
mark(
0')
active(
dbl(
s(
X))) →
mark(
s(
s(
dbl(
X))))
active(
dbls(
nil)) →
mark(
nil)
active(
dbls(
cons(
X,
Y))) →
mark(
cons(
dbl(
X),
dbls(
Y)))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
indx(
nil,
X)) →
mark(
nil)
active(
indx(
cons(
X,
Y),
Z)) →
mark(
cons(
sel(
X,
Z),
indx(
Y,
Z)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
dbl(
X)) →
dbl(
active(
X))
active(
dbls(
X)) →
dbls(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
active(
indx(
X1,
X2)) →
indx(
active(
X1),
X2)
dbl(
mark(
X)) →
mark(
dbl(
X))
dbls(
mark(
X)) →
mark(
dbls(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
indx(
mark(
X1),
X2) →
mark(
indx(
X1,
X2))
proper(
dbl(
X)) →
dbl(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
dbls(
X)) →
dbls(
proper(
X))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
proper(
indx(
X1,
X2)) →
indx(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
dbl(
ok(
X)) →
ok(
dbl(
X))
s(
ok(
X)) →
ok(
s(
X))
dbls(
ok(
X)) →
ok(
dbls(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
indx(
ok(
X1),
ok(
X2)) →
ok(
indx(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:nil:ok → 0':mark:nil:ok
dbl :: 0':mark:nil:ok → 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok → 0':mark:nil:ok
s :: 0':mark:nil:ok → 0':mark:nil:ok
dbls :: 0':mark:nil:ok → 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
sel :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
indx :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
from :: 0':mark:nil:ok → 0':mark:nil:ok
proper :: 0':mark:nil:ok → 0':mark:nil:ok
ok :: 0':mark:nil:ok → 0':mark:nil:ok
top :: 0':mark:nil:ok → top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat → 0':mark:nil:ok
Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
dbls(gen_0':mark:nil:ok3_0(+(1, n399_0))) → *4_0, rt ∈ Ω(n3990)
sel(gen_0':mark:nil:ok3_0(+(1, n878_0)), gen_0':mark:nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8780)
Generator Equations:
gen_0':mark:nil:ok3_0(0) ⇔ 0'
gen_0':mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:ok3_0(x))
No more defined symbols left to analyse.
(39) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
(40) BOUNDS(n^1, INF)
(41) Obligation:
TRS:
Rules:
active(
dbl(
0')) →
mark(
0')
active(
dbl(
s(
X))) →
mark(
s(
s(
dbl(
X))))
active(
dbls(
nil)) →
mark(
nil)
active(
dbls(
cons(
X,
Y))) →
mark(
cons(
dbl(
X),
dbls(
Y)))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
indx(
nil,
X)) →
mark(
nil)
active(
indx(
cons(
X,
Y),
Z)) →
mark(
cons(
sel(
X,
Z),
indx(
Y,
Z)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
dbl(
X)) →
dbl(
active(
X))
active(
dbls(
X)) →
dbls(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
active(
indx(
X1,
X2)) →
indx(
active(
X1),
X2)
dbl(
mark(
X)) →
mark(
dbl(
X))
dbls(
mark(
X)) →
mark(
dbls(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
indx(
mark(
X1),
X2) →
mark(
indx(
X1,
X2))
proper(
dbl(
X)) →
dbl(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
dbls(
X)) →
dbls(
proper(
X))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
proper(
indx(
X1,
X2)) →
indx(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
dbl(
ok(
X)) →
ok(
dbl(
X))
s(
ok(
X)) →
ok(
s(
X))
dbls(
ok(
X)) →
ok(
dbls(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
indx(
ok(
X1),
ok(
X2)) →
ok(
indx(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:nil:ok → 0':mark:nil:ok
dbl :: 0':mark:nil:ok → 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok → 0':mark:nil:ok
s :: 0':mark:nil:ok → 0':mark:nil:ok
dbls :: 0':mark:nil:ok → 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
sel :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
indx :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
from :: 0':mark:nil:ok → 0':mark:nil:ok
proper :: 0':mark:nil:ok → 0':mark:nil:ok
ok :: 0':mark:nil:ok → 0':mark:nil:ok
top :: 0':mark:nil:ok → top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat → 0':mark:nil:ok
Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
dbls(gen_0':mark:nil:ok3_0(+(1, n399_0))) → *4_0, rt ∈ Ω(n3990)
Generator Equations:
gen_0':mark:nil:ok3_0(0) ⇔ 0'
gen_0':mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:ok3_0(x))
No more defined symbols left to analyse.
(42) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
(43) BOUNDS(n^1, INF)
(44) Obligation:
TRS:
Rules:
active(
dbl(
0')) →
mark(
0')
active(
dbl(
s(
X))) →
mark(
s(
s(
dbl(
X))))
active(
dbls(
nil)) →
mark(
nil)
active(
dbls(
cons(
X,
Y))) →
mark(
cons(
dbl(
X),
dbls(
Y)))
active(
sel(
0',
cons(
X,
Y))) →
mark(
X)
active(
sel(
s(
X),
cons(
Y,
Z))) →
mark(
sel(
X,
Z))
active(
indx(
nil,
X)) →
mark(
nil)
active(
indx(
cons(
X,
Y),
Z)) →
mark(
cons(
sel(
X,
Z),
indx(
Y,
Z)))
active(
from(
X)) →
mark(
cons(
X,
from(
s(
X))))
active(
dbl(
X)) →
dbl(
active(
X))
active(
dbls(
X)) →
dbls(
active(
X))
active(
sel(
X1,
X2)) →
sel(
active(
X1),
X2)
active(
sel(
X1,
X2)) →
sel(
X1,
active(
X2))
active(
indx(
X1,
X2)) →
indx(
active(
X1),
X2)
dbl(
mark(
X)) →
mark(
dbl(
X))
dbls(
mark(
X)) →
mark(
dbls(
X))
sel(
mark(
X1),
X2) →
mark(
sel(
X1,
X2))
sel(
X1,
mark(
X2)) →
mark(
sel(
X1,
X2))
indx(
mark(
X1),
X2) →
mark(
indx(
X1,
X2))
proper(
dbl(
X)) →
dbl(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
dbls(
X)) →
dbls(
proper(
X))
proper(
nil) →
ok(
nil)
proper(
cons(
X1,
X2)) →
cons(
proper(
X1),
proper(
X2))
proper(
sel(
X1,
X2)) →
sel(
proper(
X1),
proper(
X2))
proper(
indx(
X1,
X2)) →
indx(
proper(
X1),
proper(
X2))
proper(
from(
X)) →
from(
proper(
X))
dbl(
ok(
X)) →
ok(
dbl(
X))
s(
ok(
X)) →
ok(
s(
X))
dbls(
ok(
X)) →
ok(
dbls(
X))
cons(
ok(
X1),
ok(
X2)) →
ok(
cons(
X1,
X2))
sel(
ok(
X1),
ok(
X2)) →
ok(
sel(
X1,
X2))
indx(
ok(
X1),
ok(
X2)) →
ok(
indx(
X1,
X2))
from(
ok(
X)) →
ok(
from(
X))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:nil:ok → 0':mark:nil:ok
dbl :: 0':mark:nil:ok → 0':mark:nil:ok
0' :: 0':mark:nil:ok
mark :: 0':mark:nil:ok → 0':mark:nil:ok
s :: 0':mark:nil:ok → 0':mark:nil:ok
dbls :: 0':mark:nil:ok → 0':mark:nil:ok
nil :: 0':mark:nil:ok
cons :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
sel :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
indx :: 0':mark:nil:ok → 0':mark:nil:ok → 0':mark:nil:ok
from :: 0':mark:nil:ok → 0':mark:nil:ok
proper :: 0':mark:nil:ok → 0':mark:nil:ok
ok :: 0':mark:nil:ok → 0':mark:nil:ok
top :: 0':mark:nil:ok → top
hole_0':mark:nil:ok1_0 :: 0':mark:nil:ok
hole_top2_0 :: top
gen_0':mark:nil:ok3_0 :: Nat → 0':mark:nil:ok
Lemmas:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
Generator Equations:
gen_0':mark:nil:ok3_0(0) ⇔ 0'
gen_0':mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:ok3_0(x))
No more defined symbols left to analyse.
(45) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
dbl(gen_0':mark:nil:ok3_0(+(1, n9_0))) → *4_0, rt ∈ Ω(n90)
(46) BOUNDS(n^1, INF)