The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(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(dbl1(0)) → mark(01)
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0, cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0)) → mark(01)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01) → ok(01)
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
0' :: 0':mark:nil:01':ok
mark :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
s :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbls :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
nil :: 0':mark:nil:01':ok
cons :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
indx :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
from :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
01' :: 0':mark:nil:01':ok
s1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
quote :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
proper :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
ok :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
top :: 0':mark:nil:01':ok → top
hole_0':mark:nil:01':ok1_0 :: 0':mark:nil:01':ok
hole_top2_0 :: top
gen_0':mark:nil:01':ok3_0 :: Nat → 0':mark:nil:01':ok

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
active, s, dbl, cons, dbls, sel, indx, from, s1, dbl1, sel1, quote, 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
s1 < active
dbl1 < active
sel1 < active
quote < active
active < top
s < proper
dbl < proper
cons < proper
dbls < proper
sel < proper
indx < proper
from < proper
s1 < proper
dbl1 < proper
sel1 < proper
quote < 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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
0' :: 0':mark:nil:01':ok
mark :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
s :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbls :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
nil :: 0':mark:nil:01':ok
cons :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
indx :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
from :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
01' :: 0':mark:nil:01':ok
s1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
quote :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
proper :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
ok :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
top :: 0':mark:nil:01':ok → top
hole_0':mark:nil:01':ok1_0 :: 0':mark:nil:01':ok
hole_top2_0 :: top
gen_0':mark:nil:01':ok3_0 :: Nat → 0':mark:nil:01':ok

Generator Equations:
gen_0':mark:nil:01':ok3_0(0) ⇔ 0'
gen_0':mark:nil:01':ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:01':ok3_0(x))

The following defined symbols remain to be analysed:
s, active, dbl, cons, dbls, sel, indx, from, s1, dbl1, sel1, quote, 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
s1 < active
dbl1 < active
sel1 < active
quote < active
active < top
s < proper
dbl < proper
cons < proper
dbls < proper
sel < proper
indx < proper
from < proper
s1 < proper
dbl1 < proper
sel1 < proper
quote < 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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
0' :: 0':mark:nil:01':ok
mark :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
s :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbls :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
nil :: 0':mark:nil:01':ok
cons :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
indx :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
from :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
01' :: 0':mark:nil:01':ok
s1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
quote :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
proper :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
ok :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
top :: 0':mark:nil:01':ok → top
hole_0':mark:nil:01':ok1_0 :: 0':mark:nil:01':ok
hole_top2_0 :: top
gen_0':mark:nil:01':ok3_0 :: Nat → 0':mark:nil:01':ok

Generator Equations:
gen_0':mark:nil:01':ok3_0(0) ⇔ 0'
gen_0':mark:nil:01':ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:01':ok3_0(x))

The following defined symbols remain to be analysed:
dbl, active, cons, dbls, sel, indx, from, s1, dbl1, sel1, quote, proper, top

They will be analysed ascendingly in the following order:
dbl < active
cons < active
dbls < active
sel < active
indx < active
from < active
s1 < active
dbl1 < active
sel1 < active
quote < active
active < top
dbl < proper
cons < proper
dbls < proper
sel < proper
indx < proper
from < proper
s1 < proper
dbl1 < proper
sel1 < proper
quote < proper
proper < top

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol dbl.

(12) 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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
0' :: 0':mark:nil:01':ok
mark :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
s :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbls :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
nil :: 0':mark:nil:01':ok
cons :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
indx :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
from :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
01' :: 0':mark:nil:01':ok
s1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
quote :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
proper :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
ok :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
top :: 0':mark:nil:01':ok → top
hole_0':mark:nil:01':ok1_0 :: 0':mark:nil:01':ok
hole_top2_0 :: top
gen_0':mark:nil:01':ok3_0 :: Nat → 0':mark:nil:01':ok

Generator Equations:
gen_0':mark:nil:01':ok3_0(0) ⇔ 0'
gen_0':mark:nil:01':ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:01':ok3_0(x))

The following defined symbols remain to be analysed:
cons, active, dbls, sel, indx, from, s1, dbl1, sel1, quote, proper, top

They will be analysed ascendingly in the following order:
cons < active
dbls < active
sel < active
indx < active
from < active
s1 < active
dbl1 < active
sel1 < active
quote < active
active < top
cons < proper
dbls < proper
sel < proper
indx < proper
from < proper
s1 < proper
dbl1 < proper
sel1 < proper
quote < proper
proper < top

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol cons.

(14) 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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
0' :: 0':mark:nil:01':ok
mark :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
s :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbls :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
nil :: 0':mark:nil:01':ok
cons :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
indx :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
from :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
01' :: 0':mark:nil:01':ok
s1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
quote :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
proper :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
ok :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
top :: 0':mark:nil:01':ok → top
hole_0':mark:nil:01':ok1_0 :: 0':mark:nil:01':ok
hole_top2_0 :: top
gen_0':mark:nil:01':ok3_0 :: Nat → 0':mark:nil:01':ok

Generator Equations:
gen_0':mark:nil:01':ok3_0(0) ⇔ 0'
gen_0':mark:nil:01':ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:01':ok3_0(x))

The following defined symbols remain to be analysed:
dbls, active, sel, indx, from, s1, dbl1, sel1, quote, proper, top

They will be analysed ascendingly in the following order:
dbls < active
sel < active
indx < active
from < active
s1 < active
dbl1 < active
sel1 < active
quote < active
active < top
dbls < proper
sel < proper
indx < proper
from < proper
s1 < proper
dbl1 < proper
sel1 < proper
quote < proper
proper < top

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol dbls.

(16) 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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
0' :: 0':mark:nil:01':ok
mark :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
s :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbls :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
nil :: 0':mark:nil:01':ok
cons :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
indx :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
from :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
01' :: 0':mark:nil:01':ok
s1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
quote :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
proper :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
ok :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
top :: 0':mark:nil:01':ok → top
hole_0':mark:nil:01':ok1_0 :: 0':mark:nil:01':ok
hole_top2_0 :: top
gen_0':mark:nil:01':ok3_0 :: Nat → 0':mark:nil:01':ok

Generator Equations:
gen_0':mark:nil:01':ok3_0(0) ⇔ 0'
gen_0':mark:nil:01':ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:01':ok3_0(x))

The following defined symbols remain to be analysed:
sel, active, indx, from, s1, dbl1, sel1, quote, proper, top

They will be analysed ascendingly in the following order:
sel < active
indx < active
from < active
s1 < active
dbl1 < active
sel1 < active
quote < active
active < top
sel < proper
indx < proper
from < proper
s1 < proper
dbl1 < proper
sel1 < proper
quote < proper
proper < top

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol sel.

(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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
0' :: 0':mark:nil:01':ok
mark :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
s :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbls :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
nil :: 0':mark:nil:01':ok
cons :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
indx :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
from :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
01' :: 0':mark:nil:01':ok
s1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
quote :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
proper :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
ok :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
top :: 0':mark:nil:01':ok → top
hole_0':mark:nil:01':ok1_0 :: 0':mark:nil:01':ok
hole_top2_0 :: top
gen_0':mark:nil:01':ok3_0 :: Nat → 0':mark:nil:01':ok

Generator Equations:
gen_0':mark:nil:01':ok3_0(0) ⇔ 0'
gen_0':mark:nil:01':ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:01':ok3_0(x))

The following defined symbols remain to be analysed:
indx, active, from, s1, dbl1, sel1, quote, proper, top

They will be analysed ascendingly in the following order:
indx < active
from < active
s1 < active
dbl1 < active
sel1 < active
quote < active
active < top
indx < proper
from < proper
s1 < proper
dbl1 < proper
sel1 < proper
quote < proper
proper < top

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol indx.

(20) 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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
0' :: 0':mark:nil:01':ok
mark :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
s :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbls :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
nil :: 0':mark:nil:01':ok
cons :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
indx :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
from :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
01' :: 0':mark:nil:01':ok
s1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
quote :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
proper :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
ok :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
top :: 0':mark:nil:01':ok → top
hole_0':mark:nil:01':ok1_0 :: 0':mark:nil:01':ok
hole_top2_0 :: top
gen_0':mark:nil:01':ok3_0 :: Nat → 0':mark:nil:01':ok

Generator Equations:
gen_0':mark:nil:01':ok3_0(0) ⇔ 0'
gen_0':mark:nil:01':ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:01':ok3_0(x))

The following defined symbols remain to be analysed:
from, active, s1, dbl1, sel1, quote, proper, top

They will be analysed ascendingly in the following order:
from < active
s1 < active
dbl1 < active
sel1 < active
quote < active
active < top
from < proper
s1 < proper
dbl1 < proper
sel1 < proper
quote < proper
proper < top

(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol from.

(22) 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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
0' :: 0':mark:nil:01':ok
mark :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
s :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbls :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
nil :: 0':mark:nil:01':ok
cons :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
indx :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
from :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
01' :: 0':mark:nil:01':ok
s1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
quote :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
proper :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
ok :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
top :: 0':mark:nil:01':ok → top
hole_0':mark:nil:01':ok1_0 :: 0':mark:nil:01':ok
hole_top2_0 :: top
gen_0':mark:nil:01':ok3_0 :: Nat → 0':mark:nil:01':ok

Generator Equations:
gen_0':mark:nil:01':ok3_0(0) ⇔ 0'
gen_0':mark:nil:01':ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:01':ok3_0(x))

The following defined symbols remain to be analysed:
s1, active, dbl1, sel1, quote, proper, top

They will be analysed ascendingly in the following order:
s1 < active
dbl1 < active
sel1 < active
quote < active
active < top
s1 < proper
dbl1 < proper
sel1 < proper
quote < proper
proper < top

(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol s1.

(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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
0' :: 0':mark:nil:01':ok
mark :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
s :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbls :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
nil :: 0':mark:nil:01':ok
cons :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
indx :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
from :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
01' :: 0':mark:nil:01':ok
s1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
quote :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
proper :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
ok :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
top :: 0':mark:nil:01':ok → top
hole_0':mark:nil:01':ok1_0 :: 0':mark:nil:01':ok
hole_top2_0 :: top
gen_0':mark:nil:01':ok3_0 :: Nat → 0':mark:nil:01':ok

Generator Equations:
gen_0':mark:nil:01':ok3_0(0) ⇔ 0'
gen_0':mark:nil:01':ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:01':ok3_0(x))

The following defined symbols remain to be analysed:
dbl1, active, sel1, quote, proper, top

They will be analysed ascendingly in the following order:
dbl1 < active
sel1 < active
quote < active
active < top
dbl1 < proper
sel1 < proper
quote < proper
proper < top

(25) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol dbl1.

(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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
0' :: 0':mark:nil:01':ok
mark :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
s :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbls :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
nil :: 0':mark:nil:01':ok
cons :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
indx :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
from :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
01' :: 0':mark:nil:01':ok
s1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
quote :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
proper :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
ok :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
top :: 0':mark:nil:01':ok → top
hole_0':mark:nil:01':ok1_0 :: 0':mark:nil:01':ok
hole_top2_0 :: top
gen_0':mark:nil:01':ok3_0 :: Nat → 0':mark:nil:01':ok

Generator Equations:
gen_0':mark:nil:01':ok3_0(0) ⇔ 0'
gen_0':mark:nil:01':ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:01':ok3_0(x))

The following defined symbols remain to be analysed:
sel1, active, quote, proper, top

They will be analysed ascendingly in the following order:
sel1 < active
quote < active
active < top
sel1 < proper
quote < proper
proper < top

(27) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol sel1.

(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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
0' :: 0':mark:nil:01':ok
mark :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
s :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbls :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
nil :: 0':mark:nil:01':ok
cons :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
indx :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
from :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
01' :: 0':mark:nil:01':ok
s1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
quote :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
proper :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
ok :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
top :: 0':mark:nil:01':ok → top
hole_0':mark:nil:01':ok1_0 :: 0':mark:nil:01':ok
hole_top2_0 :: top
gen_0':mark:nil:01':ok3_0 :: Nat → 0':mark:nil:01':ok

Generator Equations:
gen_0':mark:nil:01':ok3_0(0) ⇔ 0'
gen_0':mark:nil:01':ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:01':ok3_0(x))

The following defined symbols remain to be analysed:
quote, active, proper, top

They will be analysed ascendingly in the following order:
quote < active
active < top
quote < proper
proper < top

(29) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol quote.

(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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
0' :: 0':mark:nil:01':ok
mark :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
s :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbls :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
nil :: 0':mark:nil:01':ok
cons :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
indx :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
from :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
01' :: 0':mark:nil:01':ok
s1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
quote :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
proper :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
ok :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
top :: 0':mark:nil:01':ok → top
hole_0':mark:nil:01':ok1_0 :: 0':mark:nil:01':ok
hole_top2_0 :: top
gen_0':mark:nil:01':ok3_0 :: Nat → 0':mark:nil:01':ok

Generator Equations:
gen_0':mark:nil:01':ok3_0(0) ⇔ 0'
gen_0':mark:nil:01':ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:01':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

(31) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol active.

(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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
0' :: 0':mark:nil:01':ok
mark :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
s :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbls :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
nil :: 0':mark:nil:01':ok
cons :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
indx :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
from :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
01' :: 0':mark:nil:01':ok
s1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
quote :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
proper :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
ok :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
top :: 0':mark:nil:01':ok → top
hole_0':mark:nil:01':ok1_0 :: 0':mark:nil:01':ok
hole_top2_0 :: top
gen_0':mark:nil:01':ok3_0 :: Nat → 0':mark:nil:01':ok

Generator Equations:
gen_0':mark:nil:01':ok3_0(0) ⇔ 0'
gen_0':mark:nil:01':ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:01':ok3_0(x))

The following defined symbols remain to be analysed:
proper, top

They will be analysed ascendingly in the following order:
proper < top

(33) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol proper.

(34) 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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
0' :: 0':mark:nil:01':ok
mark :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
s :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbls :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
nil :: 0':mark:nil:01':ok
cons :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
indx :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
from :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
01' :: 0':mark:nil:01':ok
s1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
quote :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
proper :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
ok :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
top :: 0':mark:nil:01':ok → top
hole_0':mark:nil:01':ok1_0 :: 0':mark:nil:01':ok
hole_top2_0 :: top
gen_0':mark:nil:01':ok3_0 :: Nat → 0':mark:nil:01':ok

Generator Equations:
gen_0':mark:nil:01':ok3_0(0) ⇔ 0'
gen_0':mark:nil:01':ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:01':ok3_0(x))

The following defined symbols remain to be analysed:
top

(35) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol top.

(36) 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(dbl1(0')) → mark(01')
active(dbl1(s(X))) → mark(s1(s1(dbl1(X))))
active(sel1(0', cons(X, Y))) → mark(X)
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(quote(0')) → mark(01')
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(dbl(X))) → mark(dbl1(X))
active(quote(sel(X, Y))) → mark(sel1(X, Y))
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)
active(dbl1(X)) → dbl1(active(X))
active(s1(X)) → s1(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(quote(X)) → quote(active(X))
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))
dbl1(mark(X)) → mark(dbl1(X))
s1(mark(X)) → mark(s1(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
quote(mark(X)) → mark(quote(X))
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))
proper(dbl1(X)) → dbl1(proper(X))
proper(01') → ok(01')
proper(s1(X)) → s1(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(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))
dbl1(ok(X)) → ok(dbl1(X))
s1(ok(X)) → ok(s1(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
0' :: 0':mark:nil:01':ok
mark :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
s :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbls :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
nil :: 0':mark:nil:01':ok
cons :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
indx :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
from :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
dbl1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
01' :: 0':mark:nil:01':ok
s1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
sel1 :: 0':mark:nil:01':ok → 0':mark:nil:01':ok → 0':mark:nil:01':ok
quote :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
proper :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
ok :: 0':mark:nil:01':ok → 0':mark:nil:01':ok
top :: 0':mark:nil:01':ok → top
hole_0':mark:nil:01':ok1_0 :: 0':mark:nil:01':ok
hole_top2_0 :: top
gen_0':mark:nil:01':ok3_0 :: Nat → 0':mark:nil:01':ok

Generator Equations:
gen_0':mark:nil:01':ok3_0(0) ⇔ 0'
gen_0':mark:nil:01':ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:nil:01':ok3_0(x))

No more defined symbols left to analyse.