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:

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(x))) → lastbit(x)
zero(0) → true
zero(s(x)) → false
conv(x) → conviter(x, cons(0, nil))
conviter(x, l) → if(zero(x), x, l)
if(true, x, l) → l
if(false, x, l) → conviter(half(x), cons(lastbit(x), l))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
lastbit(0') → 0'
lastbit(s(0')) → s(0')
lastbit(s(s(x))) → lastbit(x)
zero(0') → true
zero(s(x)) → false
conv(x) → conviter(x, cons(0', nil))
conviter(x, l) → if(zero(x), x, l)
if(true, x, l) → l
if(false, x, l) → conviter(half(x), cons(lastbit(x), l))

S is empty.
Rewrite Strategy: FULL

(5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
cons/1

(6) Obligation:

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

half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
lastbit(0') → 0'
lastbit(s(0')) → s(0')
lastbit(s(s(x))) → lastbit(x)
zero(0') → true
zero(s(x)) → false
conv(x) → conviter(x, cons(0'))
conviter(x, l) → if(zero(x), x, l)
if(true, x, l) → l
if(false, x, l) → conviter(half(x), cons(lastbit(x)))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
lastbit(0') → 0'
lastbit(s(0')) → s(0')
lastbit(s(s(x))) → lastbit(x)
zero(0') → true
zero(s(x)) → false
conv(x) → conviter(x, cons(0'))
conviter(x, l) → if(zero(x), x, l)
if(true, x, l) → l
if(false, x, l) → conviter(half(x), cons(lastbit(x)))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
lastbit :: 0':s → 0':s
zero :: 0':s → true:false
true :: true:false
false :: true:false
conv :: 0':s → cons
conviter :: 0':s → cons → cons
cons :: 0':s → cons
if :: true:false → 0':s → cons → cons
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_cons3_0 :: cons
gen_0':s4_0 :: Nat → 0':s

(9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
half, lastbit, conviter

They will be analysed ascendingly in the following order:
half < conviter
lastbit < conviter

(10) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
lastbit(0') → 0'
lastbit(s(0')) → s(0')
lastbit(s(s(x))) → lastbit(x)
zero(0') → true
zero(s(x)) → false
conv(x) → conviter(x, cons(0'))
conviter(x, l) → if(zero(x), x, l)
if(true, x, l) → l
if(false, x, l) → conviter(half(x), cons(lastbit(x)))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
lastbit :: 0':s → 0':s
zero :: 0':s → true:false
true :: true:false
false :: true:false
conv :: 0':s → cons
conviter :: 0':s → cons → cons
cons :: 0':s → cons
if :: true:false → 0':s → cons → cons
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_cons3_0 :: cons
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
half, lastbit, conviter

They will be analysed ascendingly in the following order:
half < conviter
lastbit < conviter

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

Proved the following rewrite lemma:
half(gen_0':s4_0(*(2, n6_0))) → gen_0':s4_0(n6_0), rt ∈ Ω(1 + n6₀)

Induction Base:
half(gen_0':s4_0(*(2, 0))) →_R^Ω(1)
0'

Induction Step:
half(gen_0':s4_0(*(2, +(n6_0, 1)))) →_R^Ω(1)
s(half(gen_0':s4_0(*(2, n6_0)))) →_IH
s(gen_0':s4_0(c7_0))

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
lastbit(0') → 0'
lastbit(s(0')) → s(0')
lastbit(s(s(x))) → lastbit(x)
zero(0') → true
zero(s(x)) → false
conv(x) → conviter(x, cons(0'))
conviter(x, l) → if(zero(x), x, l)
if(true, x, l) → l
if(false, x, l) → conviter(half(x), cons(lastbit(x)))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
lastbit :: 0':s → 0':s
zero :: 0':s → true:false
true :: true:false
false :: true:false
conv :: 0':s → cons
conviter :: 0':s → cons → cons
cons :: 0':s → cons
if :: true:false → 0':s → cons → cons
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_cons3_0 :: cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
half(gen_0':s4_0(*(2, n6_0))) → gen_0':s4_0(n6_0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
lastbit, conviter

They will be analysed ascendingly in the following order:
lastbit < conviter

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
lastbit(gen_0':s4_0(*(2, n308_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n308₀)

Induction Base:
lastbit(gen_0':s4_0(*(2, 0))) →_R^Ω(1)
0'

Induction Step:
lastbit(gen_0':s4_0(*(2, +(n308_0, 1)))) →_R^Ω(1)
lastbit(gen_0':s4_0(*(2, n308_0))) →_IH
gen_0':s4_0(0)

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

(15) Complex Obligation (BEST)

(16) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
lastbit(0') → 0'
lastbit(s(0')) → s(0')
lastbit(s(s(x))) → lastbit(x)
zero(0') → true
zero(s(x)) → false
conv(x) → conviter(x, cons(0'))
conviter(x, l) → if(zero(x), x, l)
if(true, x, l) → l
if(false, x, l) → conviter(half(x), cons(lastbit(x)))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
lastbit :: 0':s → 0':s
zero :: 0':s → true:false
true :: true:false
false :: true:false
conv :: 0':s → cons
conviter :: 0':s → cons → cons
cons :: 0':s → cons
if :: true:false → 0':s → cons → cons
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_cons3_0 :: cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
half(gen_0':s4_0(*(2, n6_0))) → gen_0':s4_0(n6_0), rt ∈ Ω(1 + n6₀)
lastbit(gen_0':s4_0(*(2, n308_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n308₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
conviter

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

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

(18) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
lastbit(0') → 0'
lastbit(s(0')) → s(0')
lastbit(s(s(x))) → lastbit(x)
zero(0') → true
zero(s(x)) → false
conv(x) → conviter(x, cons(0'))
conviter(x, l) → if(zero(x), x, l)
if(true, x, l) → l
if(false, x, l) → conviter(half(x), cons(lastbit(x)))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
lastbit :: 0':s → 0':s
zero :: 0':s → true:false
true :: true:false
false :: true:false
conv :: 0':s → cons
conviter :: 0':s → cons → cons
cons :: 0':s → cons
if :: true:false → 0':s → cons → cons
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_cons3_0 :: cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
half(gen_0':s4_0(*(2, n6_0))) → gen_0':s4_0(n6_0), rt ∈ Ω(1 + n6₀)
lastbit(gen_0':s4_0(*(2, n308_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n308₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
half(gen_0':s4_0(*(2, n6_0))) → gen_0':s4_0(n6_0), rt ∈ Ω(1 + n6₀)

(20) BOUNDS(n^1, INF)

(21) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
lastbit(0') → 0'
lastbit(s(0')) → s(0')
lastbit(s(s(x))) → lastbit(x)
zero(0') → true
zero(s(x)) → false
conv(x) → conviter(x, cons(0'))
conviter(x, l) → if(zero(x), x, l)
if(true, x, l) → l
if(false, x, l) → conviter(half(x), cons(lastbit(x)))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
lastbit :: 0':s → 0':s
zero :: 0':s → true:false
true :: true:false
false :: true:false
conv :: 0':s → cons
conviter :: 0':s → cons → cons
cons :: 0':s → cons
if :: true:false → 0':s → cons → cons
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_cons3_0 :: cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
half(gen_0':s4_0(*(2, n6_0))) → gen_0':s4_0(n6_0), rt ∈ Ω(1 + n6₀)
lastbit(gen_0':s4_0(*(2, n308_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n308₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
half(gen_0':s4_0(*(2, n6_0))) → gen_0':s4_0(n6_0), rt ∈ Ω(1 + n6₀)

(23) BOUNDS(n^1, INF)

(24) Obligation:

TRS:
Rules:
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
lastbit(0') → 0'
lastbit(s(0')) → s(0')
lastbit(s(s(x))) → lastbit(x)
zero(0') → true
zero(s(x)) → false
conv(x) → conviter(x, cons(0'))
conviter(x, l) → if(zero(x), x, l)
if(true, x, l) → l
if(false, x, l) → conviter(half(x), cons(lastbit(x)))

Types:
half :: 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
lastbit :: 0':s → 0':s
zero :: 0':s → true:false
true :: true:false
false :: true:false
conv :: 0':s → cons
conviter :: 0':s → cons → cons
cons :: 0':s → cons
if :: true:false → 0':s → cons → cons
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_cons3_0 :: cons
gen_0':s4_0 :: Nat → 0':s

Lemmas:
half(gen_0':s4_0(*(2, n6_0))) → gen_0':s4_0(n6_0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
half(gen_0':s4_0(*(2, n6_0))) → gen_0':s4_0(n6_0), rt ∈ Ω(1 + n6₀)

(26) BOUNDS(n^1, INF)