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(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0)) → mark(0)
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0, X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0) → ok(0)
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(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
terms(mark(X)) →⁺ mark(terms(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / mark(X)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

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

active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(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(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

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

Heuristically decided to analyse the following defined symbols:
active, cons, recip, sqr, terms, s, add, dbl, first, proper, top

They will be analysed ascendingly in the following order:
cons < active
recip < active
sqr < active
terms < active
s < active
add < active
dbl < active
first < active
active < top
cons < proper
recip < proper
sqr < proper
terms < proper
s < proper
add < proper
dbl < proper
first < proper
proper < top

(8) Obligation:

TRS:
Rules:
active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

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

The following defined symbols remain to be analysed:
cons, active, recip, sqr, terms, s, add, dbl, first, proper, top

They will be analysed ascendingly in the following order:
cons < active
recip < active
sqr < active
terms < active
s < active
add < active
dbl < active
first < active
active < top
cons < proper
recip < proper
sqr < proper
terms < proper
s < proper
add < proper
dbl < proper
first < proper
proper < top

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

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

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

Induction Step:
cons(gen_mark:0':nil:ok3_0(+(1, +(n5_0, 1))), gen_mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil: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(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

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

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

The following defined symbols remain to be analysed:
recip, active, sqr, terms, s, add, dbl, first, proper, top

They will be analysed ascendingly in the following order:
recip < active
sqr < active
terms < active
s < active
add < active
dbl < active
first < active
active < top
recip < proper
sqr < proper
terms < proper
s < proper
add < proper
dbl < proper
first < proper
proper < top

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

Proved the following rewrite lemma:
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)

Induction Base:
recip(gen_mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
recip(gen_mark:0':nil:ok3_0(+(1, +(n1106_0, 1)))) →_R^Ω(1)
mark(recip(gen_mark:0':nil:ok3_0(+(1, n1106_0)))) →_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(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)

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

The following defined symbols remain to be analysed:
sqr, active, terms, s, add, dbl, first, proper, top

They will be analysed ascendingly in the following order:
sqr < active
terms < active
s < active
add < active
dbl < active
first < active
active < top
sqr < proper
terms < proper
s < proper
add < proper
dbl < proper
first < proper
proper < top

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

Proved the following rewrite lemma:
sqr(gen_mark:0':nil:ok3_0(+(1, n1664_0))) → *4_0, rt ∈ Ω(n1664₀)

Induction Base:
sqr(gen_mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
sqr(gen_mark:0':nil:ok3_0(+(1, +(n1664_0, 1)))) →_R^Ω(1)
mark(sqr(gen_mark:0':nil:ok3_0(+(1, n1664_0)))) →_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(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)
sqr(gen_mark:0':nil:ok3_0(+(1, n1664_0))) → *4_0, rt ∈ Ω(n1664₀)

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

The following defined symbols remain to be analysed:
terms, active, s, add, dbl, first, proper, top

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

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

Proved the following rewrite lemma:
terms(gen_mark:0':nil:ok3_0(+(1, n2323_0))) → *4_0, rt ∈ Ω(n2323₀)

Induction Base:
terms(gen_mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
terms(gen_mark:0':nil:ok3_0(+(1, +(n2323_0, 1)))) →_R^Ω(1)
mark(terms(gen_mark:0':nil:ok3_0(+(1, n2323_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(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)
sqr(gen_mark:0':nil:ok3_0(+(1, n1664_0))) → *4_0, rt ∈ Ω(n1664₀)
terms(gen_mark:0':nil:ok3_0(+(1, n2323_0))) → *4_0, rt ∈ Ω(n2323₀)

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

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

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

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

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

(22) Obligation:

TRS:
Rules:
active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)
sqr(gen_mark:0':nil:ok3_0(+(1, n1664_0))) → *4_0, rt ∈ Ω(n1664₀)
terms(gen_mark:0':nil:ok3_0(+(1, n2323_0))) → *4_0, rt ∈ Ω(n2323₀)

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

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

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

(23) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
add(gen_mark:0':nil:ok3_0(+(1, n3101_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n3101₀)

Induction Base:
add(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))

Induction Step:
add(gen_mark:0':nil:ok3_0(+(1, +(n3101_0, 1))), gen_mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(add(gen_mark:0':nil:ok3_0(+(1, n3101_0)), gen_mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

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

(24) Complex Obligation (BEST)

(25) Obligation:

TRS:
Rules:
active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)
sqr(gen_mark:0':nil:ok3_0(+(1, n1664_0))) → *4_0, rt ∈ Ω(n1664₀)
terms(gen_mark:0':nil:ok3_0(+(1, n2323_0))) → *4_0, rt ∈ Ω(n2323₀)
add(gen_mark:0':nil:ok3_0(+(1, n3101_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n3101₀)

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

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

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

(26) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
dbl(gen_mark:0':nil:ok3_0(+(1, n5323_0))) → *4_0, rt ∈ Ω(n5323₀)

Induction Base:
dbl(gen_mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
dbl(gen_mark:0':nil:ok3_0(+(1, +(n5323_0, 1)))) →_R^Ω(1)
mark(dbl(gen_mark:0':nil:ok3_0(+(1, n5323_0)))) →_IH
mark(*4_0)

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

(27) Complex Obligation (BEST)

(28) Obligation:

TRS:
Rules:
active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)
sqr(gen_mark:0':nil:ok3_0(+(1, n1664_0))) → *4_0, rt ∈ Ω(n1664₀)
terms(gen_mark:0':nil:ok3_0(+(1, n2323_0))) → *4_0, rt ∈ Ω(n2323₀)
add(gen_mark:0':nil:ok3_0(+(1, n3101_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n3101₀)
dbl(gen_mark:0':nil:ok3_0(+(1, n5323_0))) → *4_0, rt ∈ Ω(n5323₀)

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

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

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

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

Proved the following rewrite lemma:
first(gen_mark:0':nil:ok3_0(+(1, n6334_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n6334₀)

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

Induction Step:
first(gen_mark:0':nil:ok3_0(+(1, +(n6334_0, 1))), gen_mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(first(gen_mark:0':nil:ok3_0(+(1, n6334_0)), gen_mark:0':nil: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(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)
sqr(gen_mark:0':nil:ok3_0(+(1, n1664_0))) → *4_0, rt ∈ Ω(n1664₀)
terms(gen_mark:0':nil:ok3_0(+(1, n2323_0))) → *4_0, rt ∈ Ω(n2323₀)
add(gen_mark:0':nil:ok3_0(+(1, n3101_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n3101₀)
dbl(gen_mark:0':nil:ok3_0(+(1, n5323_0))) → *4_0, rt ∈ Ω(n5323₀)
first(gen_mark:0':nil:ok3_0(+(1, n6334_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n6334₀)

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

(32) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(33) Obligation:

TRS:
Rules:
active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)
sqr(gen_mark:0':nil:ok3_0(+(1, n1664_0))) → *4_0, rt ∈ Ω(n1664₀)
terms(gen_mark:0':nil:ok3_0(+(1, n2323_0))) → *4_0, rt ∈ Ω(n2323₀)
add(gen_mark:0':nil:ok3_0(+(1, n3101_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n3101₀)
dbl(gen_mark:0':nil:ok3_0(+(1, n5323_0))) → *4_0, rt ∈ Ω(n5323₀)
first(gen_mark:0':nil:ok3_0(+(1, n6334_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n6334₀)

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

(34) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(35) Obligation:

TRS:
Rules:
active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)
sqr(gen_mark:0':nil:ok3_0(+(1, n1664_0))) → *4_0, rt ∈ Ω(n1664₀)
terms(gen_mark:0':nil:ok3_0(+(1, n2323_0))) → *4_0, rt ∈ Ω(n2323₀)
add(gen_mark:0':nil:ok3_0(+(1, n3101_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n3101₀)
dbl(gen_mark:0':nil:ok3_0(+(1, n5323_0))) → *4_0, rt ∈ Ω(n5323₀)
first(gen_mark:0':nil:ok3_0(+(1, n6334_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n6334₀)

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

The following defined symbols remain to be analysed:
top

(36) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(37) Obligation:

TRS:
Rules:
active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)
sqr(gen_mark:0':nil:ok3_0(+(1, n1664_0))) → *4_0, rt ∈ Ω(n1664₀)
terms(gen_mark:0':nil:ok3_0(+(1, n2323_0))) → *4_0, rt ∈ Ω(n2323₀)
add(gen_mark:0':nil:ok3_0(+(1, n3101_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n3101₀)
dbl(gen_mark:0':nil:ok3_0(+(1, n5323_0))) → *4_0, rt ∈ Ω(n5323₀)
first(gen_mark:0':nil:ok3_0(+(1, n6334_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n6334₀)

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

No more defined symbols left to analyse.

(38) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(39) BOUNDS(n^1, INF)

(40) Obligation:

TRS:
Rules:
active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)
sqr(gen_mark:0':nil:ok3_0(+(1, n1664_0))) → *4_0, rt ∈ Ω(n1664₀)
terms(gen_mark:0':nil:ok3_0(+(1, n2323_0))) → *4_0, rt ∈ Ω(n2323₀)
add(gen_mark:0':nil:ok3_0(+(1, n3101_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n3101₀)
dbl(gen_mark:0':nil:ok3_0(+(1, n5323_0))) → *4_0, rt ∈ Ω(n5323₀)
first(gen_mark:0':nil:ok3_0(+(1, n6334_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n6334₀)

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

No more defined symbols left to analyse.

(41) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(42) BOUNDS(n^1, INF)

(43) Obligation:

TRS:
Rules:
active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)
sqr(gen_mark:0':nil:ok3_0(+(1, n1664_0))) → *4_0, rt ∈ Ω(n1664₀)
terms(gen_mark:0':nil:ok3_0(+(1, n2323_0))) → *4_0, rt ∈ Ω(n2323₀)
add(gen_mark:0':nil:ok3_0(+(1, n3101_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n3101₀)
dbl(gen_mark:0':nil:ok3_0(+(1, n5323_0))) → *4_0, rt ∈ Ω(n5323₀)

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

No more defined symbols left to analyse.

(44) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(45) BOUNDS(n^1, INF)

(46) Obligation:

TRS:
Rules:
active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)
sqr(gen_mark:0':nil:ok3_0(+(1, n1664_0))) → *4_0, rt ∈ Ω(n1664₀)
terms(gen_mark:0':nil:ok3_0(+(1, n2323_0))) → *4_0, rt ∈ Ω(n2323₀)
add(gen_mark:0':nil:ok3_0(+(1, n3101_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n3101₀)

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

No more defined symbols left to analyse.

(47) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(48) BOUNDS(n^1, INF)

(49) Obligation:

TRS:
Rules:
active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)
sqr(gen_mark:0':nil:ok3_0(+(1, n1664_0))) → *4_0, rt ∈ Ω(n1664₀)
terms(gen_mark:0':nil:ok3_0(+(1, n2323_0))) → *4_0, rt ∈ Ω(n2323₀)

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

No more defined symbols left to analyse.

(50) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(51) BOUNDS(n^1, INF)

(52) Obligation:

TRS:
Rules:
active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)
sqr(gen_mark:0':nil:ok3_0(+(1, n1664_0))) → *4_0, rt ∈ Ω(n1664₀)

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

No more defined symbols left to analyse.

(53) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(54) BOUNDS(n^1, INF)

(55) Obligation:

TRS:
Rules:
active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

Lemmas:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)
recip(gen_mark:0':nil:ok3_0(+(1, n1106_0))) → *4_0, rt ∈ Ω(n1106₀)

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

No more defined symbols left to analyse.

(56) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(57) BOUNDS(n^1, INF)

(58) Obligation:

TRS:
Rules:
active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N))))
active(sqr(0')) → mark(0')
active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X))))
active(dbl(0')) → mark(0')
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(add(0', X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(first(0', X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(terms(X)) → terms(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(recip(X)) → recip(active(X))
active(sqr(X)) → sqr(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(dbl(X)) → dbl(active(X))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
terms(mark(X)) → mark(terms(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
recip(mark(X)) → mark(recip(X))
sqr(mark(X)) → mark(sqr(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
dbl(mark(X)) → mark(dbl(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
proper(terms(X)) → terms(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(recip(X)) → recip(proper(X))
proper(sqr(X)) → sqr(proper(X))
proper(s(X)) → s(proper(X))
proper(0') → ok(0')
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(dbl(X)) → dbl(proper(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
terms(ok(X)) → ok(terms(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
recip(ok(X)) → ok(recip(X))
sqr(ok(X)) → ok(sqr(X))
s(ok(X)) → ok(s(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
dbl(ok(X)) → ok(dbl(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:0':nil:ok → mark:0':nil:ok
terms :: mark:0':nil:ok → mark:0':nil:ok
mark :: mark:0':nil:ok → mark:0':nil:ok
cons :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
recip :: mark:0':nil:ok → mark:0':nil:ok
sqr :: mark:0':nil:ok → mark:0':nil:ok
s :: mark:0':nil:ok → mark:0':nil:ok
0' :: mark:0':nil:ok
add :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
dbl :: mark:0':nil:ok → mark:0':nil:ok
first :: mark:0':nil:ok → mark:0':nil:ok → mark:0':nil:ok
nil :: mark:0':nil:ok
proper :: mark:0':nil:ok → mark:0':nil:ok
ok :: mark:0':nil:ok → mark:0':nil:ok
top :: mark:0':nil:ok → top
hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
hole_top2_0 :: top
gen_mark:0':nil:ok3_0 :: Nat → mark:0':nil:ok

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

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

No more defined symbols left to analyse.

(59) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5₀)

(60) BOUNDS(n^1, INF)