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(fst(0, Z)) → mark(nil)
active(fst(s(X), cons(Y, Z))) → mark(cons(Y, fst(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(len(nil)) → mark(0)
active(len(cons(X, Z))) → mark(s(len(Z)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(fst(X1, X2)) → fst(active(X1), X2)
active(fst(X1, X2)) → fst(X1, active(X2))
active(from(X)) → from(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(len(X)) → len(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
fst(mark(X1), X2) → mark(fst(X1, X2))
fst(X1, mark(X2)) → mark(fst(X1, X2))
from(mark(X)) → mark(from(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
len(mark(X)) → mark(len(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(fst(X1, X2)) → fst(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(len(X)) → len(proper(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
fst(ok(X1), ok(X2)) → ok(fst(X1, X2))
from(ok(X)) → ok(from(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
len(ok(X)) → ok(len(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
cons(mark(X1), X2) →⁺ mark(cons(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(fst(0', Z)) → mark(nil)
active(fst(s(X), cons(Y, Z))) → mark(cons(Y, fst(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(len(nil)) → mark(0')
active(len(cons(X, Z))) → mark(s(len(Z)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(fst(X1, X2)) → fst(active(X1), X2)
active(fst(X1, X2)) → fst(X1, active(X2))
active(from(X)) → from(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(len(X)) → len(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
fst(mark(X1), X2) → mark(fst(X1, X2))
fst(X1, mark(X2)) → mark(fst(X1, X2))
from(mark(X)) → mark(from(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
len(mark(X)) → mark(len(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(fst(X1, X2)) → fst(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(len(X)) → len(proper(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
fst(ok(X1), ok(X2)) → ok(fst(X1, X2))
from(ok(X)) → ok(from(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
len(ok(X)) → ok(len(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
active(fst(0', Z)) → mark(nil)
active(fst(s(X), cons(Y, Z))) → mark(cons(Y, fst(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(len(nil)) → mark(0')
active(len(cons(X, Z))) → mark(s(len(Z)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(fst(X1, X2)) → fst(active(X1), X2)
active(fst(X1, X2)) → fst(X1, active(X2))
active(from(X)) → from(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(len(X)) → len(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
fst(mark(X1), X2) → mark(fst(X1, X2))
fst(X1, mark(X2)) → mark(fst(X1, X2))
from(mark(X)) → mark(from(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
len(mark(X)) → mark(len(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(fst(X1, X2)) → fst(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(len(X)) → len(proper(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
fst(ok(X1), ok(X2)) → ok(fst(X1, X2))
from(ok(X)) → ok(from(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
len(ok(X)) → ok(len(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
fst :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
add :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
len :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

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

Heuristically decided to analyse the following defined symbols:
active, cons, fst, from, s, add, len, proper, top

They will be analysed ascendingly in the following order:
cons < active
fst < active
from < active
s < active
add < active
len < active
active < top
cons < proper
fst < proper
from < proper
s < proper
add < proper
len < proper
proper < top

(8) Obligation:

TRS:
Rules:
active(fst(0', Z)) → mark(nil)
active(fst(s(X), cons(Y, Z))) → mark(cons(Y, fst(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(len(nil)) → mark(0')
active(len(cons(X, Z))) → mark(s(len(Z)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(fst(X1, X2)) → fst(active(X1), X2)
active(fst(X1, X2)) → fst(X1, active(X2))
active(from(X)) → from(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(len(X)) → len(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
fst(mark(X1), X2) → mark(fst(X1, X2))
fst(X1, mark(X2)) → mark(fst(X1, X2))
from(mark(X)) → mark(from(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
len(mark(X)) → mark(len(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(fst(X1, X2)) → fst(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(len(X)) → len(proper(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
fst(ok(X1), ok(X2)) → ok(fst(X1, X2))
from(ok(X)) → ok(from(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
len(ok(X)) → ok(len(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
fst :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
add :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
len :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

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

The following defined symbols remain to be analysed:
cons, active, fst, from, s, add, len, proper, top

They will be analysed ascendingly in the following order:
cons < active
fst < active
from < active
s < active
add < active
len < active
active < top
cons < proper
fst < proper
from < proper
s < proper
add < proper
len < proper
proper < top

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
Rules:
active(fst(0', Z)) → mark(nil)
active(fst(s(X), cons(Y, Z))) → mark(cons(Y, fst(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(len(nil)) → mark(0')
active(len(cons(X, Z))) → mark(s(len(Z)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(fst(X1, X2)) → fst(active(X1), X2)
active(fst(X1, X2)) → fst(X1, active(X2))
active(from(X)) → from(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(len(X)) → len(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
fst(mark(X1), X2) → mark(fst(X1, X2))
fst(X1, mark(X2)) → mark(fst(X1, X2))
from(mark(X)) → mark(from(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
len(mark(X)) → mark(len(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(fst(X1, X2)) → fst(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(len(X)) → len(proper(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
fst(ok(X1), ok(X2)) → ok(fst(X1, X2))
from(ok(X)) → ok(from(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
len(ok(X)) → ok(len(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
fst :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
add :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
len :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

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

The following defined symbols remain to be analysed:
fst, active, from, s, add, len, proper, top

They will be analysed ascendingly in the following order:
fst < active
from < active
s < active
add < active
len < active
active < top
fst < proper
from < proper
s < proper
add < proper
len < proper
proper < top

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

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

(12) Obligation:

TRS:
Rules:
active(fst(0', Z)) → mark(nil)
active(fst(s(X), cons(Y, Z))) → mark(cons(Y, fst(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(len(nil)) → mark(0')
active(len(cons(X, Z))) → mark(s(len(Z)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(fst(X1, X2)) → fst(active(X1), X2)
active(fst(X1, X2)) → fst(X1, active(X2))
active(from(X)) → from(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(len(X)) → len(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
fst(mark(X1), X2) → mark(fst(X1, X2))
fst(X1, mark(X2)) → mark(fst(X1, X2))
from(mark(X)) → mark(from(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
len(mark(X)) → mark(len(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(fst(X1, X2)) → fst(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(len(X)) → len(proper(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
fst(ok(X1), ok(X2)) → ok(fst(X1, X2))
from(ok(X)) → ok(from(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
len(ok(X)) → ok(len(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
fst :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
add :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
len :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

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

The following defined symbols remain to be analysed:
from, active, s, add, len, proper, top

They will be analysed ascendingly in the following order:
from < active
s < active
add < active
len < active
active < top
from < proper
s < proper
add < proper
len < proper
proper < top

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

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

(14) Obligation:

TRS:
Rules:
active(fst(0', Z)) → mark(nil)
active(fst(s(X), cons(Y, Z))) → mark(cons(Y, fst(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(len(nil)) → mark(0')
active(len(cons(X, Z))) → mark(s(len(Z)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(fst(X1, X2)) → fst(active(X1), X2)
active(fst(X1, X2)) → fst(X1, active(X2))
active(from(X)) → from(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(len(X)) → len(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
fst(mark(X1), X2) → mark(fst(X1, X2))
fst(X1, mark(X2)) → mark(fst(X1, X2))
from(mark(X)) → mark(from(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
len(mark(X)) → mark(len(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(fst(X1, X2)) → fst(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(len(X)) → len(proper(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
fst(ok(X1), ok(X2)) → ok(fst(X1, X2))
from(ok(X)) → ok(from(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
len(ok(X)) → ok(len(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
fst :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
add :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
len :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

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

The following defined symbols remain to be analysed:
s, active, add, len, proper, top

They will be analysed ascendingly in the following order:
s < active
add < active
len < active
active < top
s < proper
add < proper
len < proper
proper < top

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

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

(16) Obligation:

TRS:
Rules:
active(fst(0', Z)) → mark(nil)
active(fst(s(X), cons(Y, Z))) → mark(cons(Y, fst(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(len(nil)) → mark(0')
active(len(cons(X, Z))) → mark(s(len(Z)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(fst(X1, X2)) → fst(active(X1), X2)
active(fst(X1, X2)) → fst(X1, active(X2))
active(from(X)) → from(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(len(X)) → len(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
fst(mark(X1), X2) → mark(fst(X1, X2))
fst(X1, mark(X2)) → mark(fst(X1, X2))
from(mark(X)) → mark(from(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
len(mark(X)) → mark(len(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(fst(X1, X2)) → fst(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(len(X)) → len(proper(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
fst(ok(X1), ok(X2)) → ok(fst(X1, X2))
from(ok(X)) → ok(from(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
len(ok(X)) → ok(len(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
fst :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
add :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
len :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

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

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

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

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

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

(18) Obligation:

TRS:
Rules:
active(fst(0', Z)) → mark(nil)
active(fst(s(X), cons(Y, Z))) → mark(cons(Y, fst(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(len(nil)) → mark(0')
active(len(cons(X, Z))) → mark(s(len(Z)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(fst(X1, X2)) → fst(active(X1), X2)
active(fst(X1, X2)) → fst(X1, active(X2))
active(from(X)) → from(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(len(X)) → len(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
fst(mark(X1), X2) → mark(fst(X1, X2))
fst(X1, mark(X2)) → mark(fst(X1, X2))
from(mark(X)) → mark(from(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
len(mark(X)) → mark(len(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(fst(X1, X2)) → fst(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(len(X)) → len(proper(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
fst(ok(X1), ok(X2)) → ok(fst(X1, X2))
from(ok(X)) → ok(from(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
len(ok(X)) → ok(len(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
fst :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
add :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
len :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

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

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

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

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

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

(20) Obligation:

TRS:
Rules:
active(fst(0', Z)) → mark(nil)
active(fst(s(X), cons(Y, Z))) → mark(cons(Y, fst(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(len(nil)) → mark(0')
active(len(cons(X, Z))) → mark(s(len(Z)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(fst(X1, X2)) → fst(active(X1), X2)
active(fst(X1, X2)) → fst(X1, active(X2))
active(from(X)) → from(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(len(X)) → len(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
fst(mark(X1), X2) → mark(fst(X1, X2))
fst(X1, mark(X2)) → mark(fst(X1, X2))
from(mark(X)) → mark(from(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
len(mark(X)) → mark(len(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(fst(X1, X2)) → fst(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(len(X)) → len(proper(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
fst(ok(X1), ok(X2)) → ok(fst(X1, X2))
from(ok(X)) → ok(from(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
len(ok(X)) → ok(len(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
fst :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
add :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
len :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

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

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

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

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

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

(22) Obligation:

TRS:
Rules:
active(fst(0', Z)) → mark(nil)
active(fst(s(X), cons(Y, Z))) → mark(cons(Y, fst(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(len(nil)) → mark(0')
active(len(cons(X, Z))) → mark(s(len(Z)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(fst(X1, X2)) → fst(active(X1), X2)
active(fst(X1, X2)) → fst(X1, active(X2))
active(from(X)) → from(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(len(X)) → len(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
fst(mark(X1), X2) → mark(fst(X1, X2))
fst(X1, mark(X2)) → mark(fst(X1, X2))
from(mark(X)) → mark(from(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
len(mark(X)) → mark(len(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(fst(X1, X2)) → fst(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(len(X)) → len(proper(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
fst(ok(X1), ok(X2)) → ok(fst(X1, X2))
from(ok(X)) → ok(from(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
len(ok(X)) → ok(len(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
fst :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
add :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
len :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

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

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

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

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

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

(24) Obligation:

TRS:
Rules:
active(fst(0', Z)) → mark(nil)
active(fst(s(X), cons(Y, Z))) → mark(cons(Y, fst(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(len(nil)) → mark(0')
active(len(cons(X, Z))) → mark(s(len(Z)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(fst(X1, X2)) → fst(active(X1), X2)
active(fst(X1, X2)) → fst(X1, active(X2))
active(from(X)) → from(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(len(X)) → len(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
fst(mark(X1), X2) → mark(fst(X1, X2))
fst(X1, mark(X2)) → mark(fst(X1, X2))
from(mark(X)) → mark(from(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
len(mark(X)) → mark(len(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(fst(X1, X2)) → fst(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(len(X)) → len(proper(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
fst(ok(X1), ok(X2)) → ok(fst(X1, X2))
from(ok(X)) → ok(from(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
len(ok(X)) → ok(len(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
fst :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
add :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
len :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

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

The following defined symbols remain to be analysed:
top

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

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

(26) Obligation:

TRS:
Rules:
active(fst(0', Z)) → mark(nil)
active(fst(s(X), cons(Y, Z))) → mark(cons(Y, fst(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(len(nil)) → mark(0')
active(len(cons(X, Z))) → mark(s(len(Z)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(fst(X1, X2)) → fst(active(X1), X2)
active(fst(X1, X2)) → fst(X1, active(X2))
active(from(X)) → from(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(len(X)) → len(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
fst(mark(X1), X2) → mark(fst(X1, X2))
fst(X1, mark(X2)) → mark(fst(X1, X2))
from(mark(X)) → mark(from(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
len(mark(X)) → mark(len(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(fst(X1, X2)) → fst(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(len(X)) → len(proper(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
fst(ok(X1), ok(X2)) → ok(fst(X1, X2))
from(ok(X)) → ok(from(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
len(ok(X)) → ok(len(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':nil:mark:ok → 0':nil:mark:ok
fst :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
0' :: 0':nil:mark:ok
mark :: 0':nil:mark:ok → 0':nil:mark:ok
nil :: 0':nil:mark:ok
s :: 0':nil:mark:ok → 0':nil:mark:ok
cons :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
from :: 0':nil:mark:ok → 0':nil:mark:ok
add :: 0':nil:mark:ok → 0':nil:mark:ok → 0':nil:mark:ok
len :: 0':nil:mark:ok → 0':nil:mark:ok
proper :: 0':nil:mark:ok → 0':nil:mark:ok
ok :: 0':nil:mark:ok → 0':nil:mark:ok
top :: 0':nil:mark:ok → top
hole_0':nil:mark:ok1_0 :: 0':nil:mark:ok
hole_top2_0 :: top
gen_0':nil:mark:ok3_0 :: Nat → 0':nil:mark:ok

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

No more defined symbols left to analyse.