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(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
sel(mark(X1), X2) →⁺ mark(sel(X1, X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

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

Heuristically decided to analyse the following defined symbols:
active, sel, cons, first, from, s, sel1, quote, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, top

They will be analysed ascendingly in the following order:
sel < active
cons < active
first < active
from < active
s < active
sel1 < active
quote < active
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
sel < proper
cons < proper
first < proper
from < proper
s < proper
sel1 < proper
quote < proper
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top

(8) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

The following defined symbols remain to be analysed:
sel, active, cons, first, from, s, sel1, quote, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, top

They will be analysed ascendingly in the following order:
sel < active
cons < active
first < active
from < active
s < active
sel1 < active
quote < active
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
sel < proper
cons < proper
first < proper
from < proper
s < proper
sel1 < proper
quote < proper
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)), gen_mark:0':nil:nil1:01':ok3_0(b))

Induction Step:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n5_0, 1))), gen_mark:0':nil:nil1:01':ok3_0(b)) →_R^Ω(1)
mark(sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

The following defined symbols remain to be analysed:
cons, active, first, from, s, sel1, quote, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, top

They will be analysed ascendingly in the following order:
cons < active
first < active
from < active
s < active
sel1 < active
quote < active
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
cons < proper
first < proper
from < proper
s < proper
sel1 < proper
quote < proper
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)

Induction Base:
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)), gen_mark:0':nil:nil1:01':ok3_0(b))

Induction Step:
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n1821_0, 1))), gen_mark:0':nil:nil1:01':ok3_0(b)) →_R^Ω(1)
mark(cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

The following defined symbols remain to be analysed:
first, active, from, s, sel1, quote, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, top

They will be analysed ascendingly in the following order:
first < active
from < active
s < active
sel1 < active
quote < active
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
first < proper
from < proper
s < proper
sel1 < proper
quote < proper
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)

Induction Base:
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)), gen_mark:0':nil:nil1:01':ok3_0(b))

Induction Step:
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n3742_0, 1))), gen_mark:0':nil:nil1:01':ok3_0(b)) →_R^Ω(1)
mark(first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

The following defined symbols remain to be analysed:
from, active, s, sel1, quote, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, top

They will be analysed ascendingly in the following order:
from < active
s < active
sel1 < active
quote < active
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
from < proper
s < proper
sel1 < proper
quote < proper
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)

Induction Base:
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)))

Induction Step:
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n6166_0, 1)))) →_R^Ω(1)
mark(from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(19) Complex Obligation (BEST)

(20) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

The following defined symbols remain to be analysed:
s, active, sel1, quote, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, top

They will be analysed ascendingly in the following order:
s < active
sel1 < active
quote < active
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
s < proper
sel1 < proper
quote < proper
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top

(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)

Induction Base:
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)))

Induction Step:
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n7153_0, 1)))) →_R^Ω(1)
mark(s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(22) Complex Obligation (BEST)

(23) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

The following defined symbols remain to be analysed:
sel1, active, quote, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, top

They will be analysed ascendingly in the following order:
sel1 < active
quote < active
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
sel1 < proper
quote < proper
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top

(24) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)

Induction Base:
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)), gen_mark:0':nil:nil1:01':ok3_0(b))

Induction Step:
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n8241_0, 1))), gen_mark:0':nil:nil1:01':ok3_0(b)) →_R^Ω(1)
mark(sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(25) Complex Obligation (BEST)

(26) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

The following defined symbols remain to be analysed:
quote, active, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, top

They will be analysed ascendingly in the following order:
quote < active
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
quote < proper
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top

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

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

(28) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

The following defined symbols remain to be analysed:
cons1, active, first1, quote1, s1, unquote, fcons, unquote1, proper, top

They will be analysed ascendingly in the following order:
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top

(29) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n11411₀)

Induction Base:
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)), gen_mark:0':nil:nil1:01':ok3_0(b))

Induction Step:
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n11411_0, 1))), gen_mark:0':nil:nil1:01':ok3_0(b)) →_R^Ω(1)
mark(cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(30) Complex Obligation (BEST)

(31) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n11411₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

The following defined symbols remain to be analysed:
first1, active, quote1, s1, unquote, fcons, unquote1, proper, top

They will be analysed ascendingly in the following order:
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top

(32) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n14855₀)

Induction Base:
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)), gen_mark:0':nil:nil1:01':ok3_0(b))

Induction Step:
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n14855_0, 1))), gen_mark:0':nil:nil1:01':ok3_0(b)) →_R^Ω(1)
mark(first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(33) Complex Obligation (BEST)

(34) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n11411₀)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n14855₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

The following defined symbols remain to be analysed:
quote1, active, s1, unquote, fcons, unquote1, proper, top

They will be analysed ascendingly in the following order:
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top

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

Could not prove a rewrite lemma for the defined symbol quote1.

(36) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n11411₀)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n14855₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

The following defined symbols remain to be analysed:
s1, active, unquote, fcons, unquote1, proper, top

They will be analysed ascendingly in the following order:
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top

(37) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n18643₀)

Induction Base:
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)))

Induction Step:
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n18643_0, 1)))) →_R^Ω(1)
mark(s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(38) Complex Obligation (BEST)

(39) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n11411₀)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n14855₀)
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n18643₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

The following defined symbols remain to be analysed:
unquote, active, fcons, unquote1, proper, top

They will be analysed ascendingly in the following order:
unquote < active
fcons < active
unquote1 < active
active < top
unquote < proper
fcons < proper
unquote1 < proper
proper < top

(40) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, n20282_0))) → *4_0, rt ∈ Ω(n20282₀)

Induction Base:
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)))

Induction Step:
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n20282_0, 1)))) →_R^Ω(1)
mark(unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, n20282_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(41) Complex Obligation (BEST)

(42) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n11411₀)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n14855₀)
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n18643₀)
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, n20282_0))) → *4_0, rt ∈ Ω(n20282₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

The following defined symbols remain to be analysed:
fcons, active, unquote1, proper, top

They will be analysed ascendingly in the following order:
fcons < active
unquote1 < active
active < top
fcons < proper
unquote1 < proper
proper < top

(43) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
fcons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n22022_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n22022₀)

Induction Base:
fcons(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)), gen_mark:0':nil:nil1:01':ok3_0(b))

Induction Step:
fcons(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n22022_0, 1))), gen_mark:0':nil:nil1:01':ok3_0(b)) →_R^Ω(1)
mark(fcons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n22022_0)), gen_mark:0':nil:nil1:01':ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(44) Complex Obligation (BEST)

(45) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n11411₀)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n14855₀)
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n18643₀)
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, n20282_0))) → *4_0, rt ∈ Ω(n20282₀)
fcons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n22022_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n22022₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

The following defined symbols remain to be analysed:
unquote1, active, proper, top

They will be analysed ascendingly in the following order:
unquote1 < active
active < top
unquote1 < proper
proper < top

(46) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
unquote1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n26486_0))) → *4_0, rt ∈ Ω(n26486₀)

Induction Base:
unquote1(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)))

Induction Step:
unquote1(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n26486_0, 1)))) →_R^Ω(1)
mark(unquote1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n26486_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(47) Complex Obligation (BEST)

(48) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n11411₀)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n14855₀)
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n18643₀)
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, n20282_0))) → *4_0, rt ∈ Ω(n20282₀)
fcons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n22022_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n22022₀)
unquote1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n26486_0))) → *4_0, rt ∈ Ω(n26486₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':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

(49) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n11411₀)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n14855₀)
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n18643₀)
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, n20282_0))) → *4_0, rt ∈ Ω(n20282₀)
fcons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n22022_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n22022₀)
unquote1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n26486_0))) → *4_0, rt ∈ Ω(n26486₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

No more defined symbols left to analyse.

(50) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n11411₀)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n14855₀)
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n18643₀)
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, n20282_0))) → *4_0, rt ∈ Ω(n20282₀)
fcons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n22022_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n22022₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

No more defined symbols left to analyse.

(51) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n11411₀)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n14855₀)
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n18643₀)
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, n20282_0))) → *4_0, rt ∈ Ω(n20282₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

No more defined symbols left to analyse.

(52) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n11411₀)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n14855₀)
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n18643₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

No more defined symbols left to analyse.

(53) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n11411₀)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n14855₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

No more defined symbols left to analyse.

(54) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n11411₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

No more defined symbols left to analyse.

(55) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n8241₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

No more defined symbols left to analyse.

(56) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n7153₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

No more defined symbols left to analyse.

(57) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n6166₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

No more defined symbols left to analyse.

(58) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n3742₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

No more defined symbols left to analyse.

(59) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n1821₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

No more defined symbols left to analyse.

(60) Obligation:

TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok

Lemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))

No more defined symbols left to analyse.