### (0) Obligation:

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

active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0, X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
and(mark(X1), X2) →+ mark(and(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 [ ].

### (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(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok → top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat → true:mark:false:0':nil:ok

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

Heuristically decided to analyse the following defined symbols:
active, s, add, cons, first, from, and, if, proper, top

They will be analysed ascendingly in the following order:
s < active
cons < active
first < active
from < active
and < active
if < active
active < top
s < proper
cons < proper
first < proper
from < proper
and < proper
if < proper
proper < top

### (8) Obligation:

TRS:
Rules:
active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok → top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat → true:mark:false:0':nil:ok

Generator Equations:
gen_true:mark:false:0':nil:ok3_0(0) ⇔ true
gen_true:mark:false:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_true:mark:false:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
s, active, add, cons, first, from, and, if, proper, top

They will be analysed ascendingly in the following order:
s < active
cons < active
first < active
from < active
and < active
if < active
active < top
s < proper
cons < proper
first < proper
from < proper
and < proper
if < proper
proper < top

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

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

### (10) Obligation:

TRS:
Rules:
active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok → top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat → true:mark:false:0':nil:ok

Generator Equations:
gen_true:mark:false:0':nil:ok3_0(0) ⇔ true
gen_true:mark:false:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_true:mark:false:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
add, active, cons, first, from, and, if, proper, top

They will be analysed ascendingly in the following order:
cons < active
first < active
from < active
and < active
if < active
active < top
cons < proper
first < proper
from < proper
and < proper
if < proper
proper < top

### (11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)

Induction Base:

Induction Step:
mark(*4_0)

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

### (13) Obligation:

TRS:
Rules:
active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok → top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat → true:mark:false:0':nil:ok

Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)

Generator Equations:
gen_true:mark:false:0':nil:ok3_0(0) ⇔ true
gen_true:mark:false:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_true:mark:false:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
cons, active, first, from, and, if, proper, top

They will be analysed ascendingly in the following order:
cons < active
first < active
from < active
and < active
if < active
active < top
cons < proper
first < proper
from < proper
and < proper
if < proper
proper < top

### (14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (15) Obligation:

TRS:
Rules:
active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok → top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat → true:mark:false:0':nil:ok

Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)

Generator Equations:
gen_true:mark:false:0':nil:ok3_0(0) ⇔ true
gen_true:mark:false:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_true:mark:false:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
first, active, from, and, if, proper, top

They will be analysed ascendingly in the following order:
first < active
from < active
and < active
if < active
active < top
first < proper
from < proper
and < proper
if < proper
proper < top

### (16) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
first(gen_true:mark:false:0':nil:ok3_0(+(1, n1029_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n10290)

Induction Base:
first(gen_true:mark:false:0':nil:ok3_0(+(1, 0)), gen_true:mark:false:0':nil:ok3_0(b))

Induction Step:
first(gen_true:mark:false:0':nil:ok3_0(+(1, +(n1029_0, 1))), gen_true:mark:false:0':nil:ok3_0(b)) →RΩ(1)
mark(first(gen_true:mark:false:0':nil:ok3_0(+(1, n1029_0)), gen_true:mark:false:0':nil:ok3_0(b))) →IH
mark(*4_0)

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

### (18) Obligation:

TRS:
Rules:
active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok → top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat → true:mark:false:0':nil:ok

Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)
first(gen_true:mark:false:0':nil:ok3_0(+(1, n1029_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n10290)

Generator Equations:
gen_true:mark:false:0':nil:ok3_0(0) ⇔ true
gen_true:mark:false:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_true:mark:false:0':nil:ok3_0(x))

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

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

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

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

### (20) Obligation:

TRS:
Rules:
active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok → top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat → true:mark:false:0':nil:ok

Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)
first(gen_true:mark:false:0':nil:ok3_0(+(1, n1029_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n10290)

Generator Equations:
gen_true:mark:false:0':nil:ok3_0(0) ⇔ true
gen_true:mark:false:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_true:mark:false:0':nil:ok3_0(x))

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

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

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

Proved the following rewrite lemma:
and(gen_true:mark:false:0':nil:ok3_0(+(1, n2551_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n25510)

Induction Base:
and(gen_true:mark:false:0':nil:ok3_0(+(1, 0)), gen_true:mark:false:0':nil:ok3_0(b))

Induction Step:
and(gen_true:mark:false:0':nil:ok3_0(+(1, +(n2551_0, 1))), gen_true:mark:false:0':nil:ok3_0(b)) →RΩ(1)
mark(and(gen_true:mark:false:0':nil:ok3_0(+(1, n2551_0)), gen_true:mark:false:0':nil:ok3_0(b))) →IH
mark(*4_0)

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

### (23) Obligation:

TRS:
Rules:
active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok → top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat → true:mark:false:0':nil:ok

Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)
first(gen_true:mark:false:0':nil:ok3_0(+(1, n1029_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n10290)
and(gen_true:mark:false:0':nil:ok3_0(+(1, n2551_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n25510)

Generator Equations:
gen_true:mark:false:0':nil:ok3_0(0) ⇔ true
gen_true:mark:false:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_true:mark:false:0':nil:ok3_0(x))

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

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

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

Proved the following rewrite lemma:
if(gen_true:mark:false:0':nil:ok3_0(+(1, n4164_0)), gen_true:mark:false:0':nil:ok3_0(b), gen_true:mark:false:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n41640)

Induction Base:
if(gen_true:mark:false:0':nil:ok3_0(+(1, 0)), gen_true:mark:false:0':nil:ok3_0(b), gen_true:mark:false:0':nil:ok3_0(c))

Induction Step:
if(gen_true:mark:false:0':nil:ok3_0(+(1, +(n4164_0, 1))), gen_true:mark:false:0':nil:ok3_0(b), gen_true:mark:false:0':nil:ok3_0(c)) →RΩ(1)
mark(if(gen_true:mark:false:0':nil:ok3_0(+(1, n4164_0)), gen_true:mark:false:0':nil:ok3_0(b), gen_true:mark:false:0':nil:ok3_0(c))) →IH
mark(*4_0)

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

### (26) Obligation:

TRS:
Rules:
active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok → top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat → true:mark:false:0':nil:ok

Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)
first(gen_true:mark:false:0':nil:ok3_0(+(1, n1029_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n10290)
and(gen_true:mark:false:0':nil:ok3_0(+(1, n2551_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n25510)
if(gen_true:mark:false:0':nil:ok3_0(+(1, n4164_0)), gen_true:mark:false:0':nil:ok3_0(b), gen_true:mark:false:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n41640)

Generator Equations:
gen_true:mark:false:0':nil:ok3_0(0) ⇔ true
gen_true:mark:false:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_true:mark:false:0':nil:ok3_0(x))

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

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

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

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

### (28) Obligation:

TRS:
Rules:
active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok → top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat → true:mark:false:0':nil:ok

Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)
first(gen_true:mark:false:0':nil:ok3_0(+(1, n1029_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n10290)
and(gen_true:mark:false:0':nil:ok3_0(+(1, n2551_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n25510)
if(gen_true:mark:false:0':nil:ok3_0(+(1, n4164_0)), gen_true:mark:false:0':nil:ok3_0(b), gen_true:mark:false:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n41640)

Generator Equations:
gen_true:mark:false:0':nil:ok3_0(0) ⇔ true
gen_true:mark:false:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_true:mark:false:0':nil:ok3_0(x))

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

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

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

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

### (30) Obligation:

TRS:
Rules:
active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok → top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat → true:mark:false:0':nil:ok

Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)
first(gen_true:mark:false:0':nil:ok3_0(+(1, n1029_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n10290)
and(gen_true:mark:false:0':nil:ok3_0(+(1, n2551_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n25510)
if(gen_true:mark:false:0':nil:ok3_0(+(1, n4164_0)), gen_true:mark:false:0':nil:ok3_0(b), gen_true:mark:false:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n41640)

Generator Equations:
gen_true:mark:false:0':nil:ok3_0(0) ⇔ true
gen_true:mark:false:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_true:mark:false:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
top

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

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

### (32) Obligation:

TRS:
Rules:
active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok → top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat → true:mark:false:0':nil:ok

Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)
first(gen_true:mark:false:0':nil:ok3_0(+(1, n1029_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n10290)
and(gen_true:mark:false:0':nil:ok3_0(+(1, n2551_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n25510)
if(gen_true:mark:false:0':nil:ok3_0(+(1, n4164_0)), gen_true:mark:false:0':nil:ok3_0(b), gen_true:mark:false:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n41640)

Generator Equations:
gen_true:mark:false:0':nil:ok3_0(0) ⇔ true
gen_true:mark:false:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_true:mark:false:0':nil:ok3_0(x))

No more defined symbols left to analyse.

### (33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)

### (35) Obligation:

TRS:
Rules:
active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok → top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat → true:mark:false:0':nil:ok

Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)
first(gen_true:mark:false:0':nil:ok3_0(+(1, n1029_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n10290)
and(gen_true:mark:false:0':nil:ok3_0(+(1, n2551_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n25510)
if(gen_true:mark:false:0':nil:ok3_0(+(1, n4164_0)), gen_true:mark:false:0':nil:ok3_0(b), gen_true:mark:false:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n41640)

Generator Equations:
gen_true:mark:false:0':nil:ok3_0(0) ⇔ true
gen_true:mark:false:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_true:mark:false:0':nil:ok3_0(x))

No more defined symbols left to analyse.

### (36) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)

### (38) Obligation:

TRS:
Rules:
active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok → top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat → true:mark:false:0':nil:ok

Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)
first(gen_true:mark:false:0':nil:ok3_0(+(1, n1029_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n10290)
and(gen_true:mark:false:0':nil:ok3_0(+(1, n2551_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n25510)

Generator Equations:
gen_true:mark:false:0':nil:ok3_0(0) ⇔ true
gen_true:mark:false:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_true:mark:false:0':nil:ok3_0(x))

No more defined symbols left to analyse.

### (39) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)

### (41) Obligation:

TRS:
Rules:
active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok → top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat → true:mark:false:0':nil:ok

Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)
first(gen_true:mark:false:0':nil:ok3_0(+(1, n1029_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n10290)

Generator Equations:
gen_true:mark:false:0':nil:ok3_0(0) ⇔ true
gen_true:mark:false:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_true:mark:false:0':nil:ok3_0(x))

No more defined symbols left to analyse.

### (42) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)

### (44) Obligation:

TRS:
Rules:
active(and(true, X)) → mark(X)
active(and(false, Y)) → mark(false)
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(first(0', X)) → 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(and(X1, X2)) → and(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
s(ok(X)) → ok(s(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
and :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
true :: true:mark:false:0':nil:ok
mark :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
false :: true:mark:false:0':nil:ok
if :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
add :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
0' :: true:mark:false:0':nil:ok
s :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
first :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
nil :: true:mark:false:0':nil:ok
cons :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
from :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
proper :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
ok :: true:mark:false:0':nil:ok → true:mark:false:0':nil:ok
top :: true:mark:false:0':nil:ok → top
hole_true:mark:false:0':nil:ok1_0 :: true:mark:false:0':nil:ok
hole_top2_0 :: top
gen_true:mark:false:0':nil:ok3_0 :: Nat → true:mark:false:0':nil:ok

Lemmas:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)

Generator Equations:
gen_true:mark:false:0':nil:ok3_0(0) ⇔ true
gen_true:mark:false:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_true:mark:false:0':nil:ok3_0(x))

No more defined symbols left to analyse.

### (45) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
add(gen_true:mark:false:0':nil:ok3_0(+(1, n9_0)), gen_true:mark:false:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n90)