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:

eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
lt(0, s(y)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
bin2s(nil) → 0
bin2s(cons(x, xs)) → bin2ss(x, xs)
bin2ss(x, nil) → x
bin2ss(x, cons(0, xs)) → bin2ss(double(x), xs)
bin2ss(x, cons(1, xs)) → bin2ss(s(double(x)), xs)
half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
log(0) → 0
log(s(0)) → 0
log(s(s(x))) → s(log(half(s(s(x)))))
more(nil) → nil
more(cons(xs, ys)) → cons(cons(0, xs), cons(cons(1, xs), cons(xs, ys)))
s2bin(x) → s2bin1(x, 0, cons(nil, nil))
s2bin1(x, y, lists) → if1(lt(y, log(x)), x, y, lists)
if1(true, x, y, lists) → s2bin1(x, s(y), more(lists))
if1(false, x, y, lists) → s2bin2(x, lists)
s2bin2(x, nil) → bug_list_not
s2bin2(x, cons(xs, ys)) → if2(eq(x, bin2s(xs)), x, xs, ys)
if2(true, x, xs, ys) → xs
if2(false, x, xs, ys) → s2bin2(x, ys)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
bin2s(nil) → 0'
bin2s(cons(x, xs)) → bin2ss(x, xs)
bin2ss(x, nil) → x
bin2ss(x, cons(0', xs)) → bin2ss(double(x), xs)
bin2ss(x, cons(1', xs)) → bin2ss(s(double(x)), xs)
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
log(0') → 0'
log(s(0')) → 0'
log(s(s(x))) → s(log(half(s(s(x)))))
more(nil) → nil
more(cons(xs, ys)) → cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys)))
s2bin(x) → s2bin1(x, 0', cons(nil, nil))
s2bin1(x, y, lists) → if1(lt(y, log(x)), x, y, lists)
if1(true, x, y, lists) → s2bin1(x, s(y), more(lists))
if1(false, x, y, lists) → s2bin2(x, lists)
s2bin2(x, nil) → bug_list_not
s2bin2(x, cons(xs, ys)) → if2(eq(x, bin2s(xs)), x, xs, ys)
if2(true, x, xs, ys) → xs
if2(false, x, xs, ys) → s2bin2(x, ys)

S is empty.
Rewrite Strategy: FULL

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

Sliced the following arguments:
double/0

(6) Obligation:

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

eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
bin2s(nil) → 0'
bin2s(cons(x, xs)) → bin2ss(x, xs)
bin2ss(x, nil) → x
bin2ss(x, cons(0', xs)) → bin2ss(double, xs)
bin2ss(x, cons(1', xs)) → bin2ss(s(double), xs)
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
log(0') → 0'
log(s(0')) → 0'
log(s(s(x))) → s(log(half(s(s(x)))))
more(nil) → nil
more(cons(xs, ys)) → cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys)))
s2bin(x) → s2bin1(x, 0', cons(nil, nil))
s2bin1(x, y, lists) → if1(lt(y, log(x)), x, y, lists)
if1(true, x, y, lists) → s2bin1(x, s(y), more(lists))
if1(false, x, y, lists) → s2bin2(x, lists)
s2bin2(x, nil) → bug_list_not
s2bin2(x, cons(xs, ys)) → if2(eq(x, bin2s(xs)), x, xs, ys)
if2(true, x, xs, ys) → xs
if2(false, x, xs, ys) → s2bin2(x, ys)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(8) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
bin2s(nil) → 0'
bin2s(cons(x, xs)) → bin2ss(x, xs)
bin2ss(x, nil) → x
bin2ss(x, cons(0', xs)) → bin2ss(double, xs)
bin2ss(x, cons(1', xs)) → bin2ss(s(double), xs)
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
log(0') → 0'
log(s(0')) → 0'
log(s(s(x))) → s(log(half(s(s(x)))))
more(nil) → nil
more(cons(xs, ys)) → cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys)))
s2bin(x) → s2bin1(x, 0', cons(nil, nil))
s2bin1(x, y, lists) → if1(lt(y, log(x)), x, y, lists)
if1(true, x, y, lists) → s2bin1(x, s(y), more(lists))
if1(false, x, y, lists) → s2bin2(x, lists)
s2bin2(x, nil) → bug_list_not
s2bin2(x, cons(xs, ys)) → if2(eq(x, bin2s(xs)), x, xs, ys)
if2(true, x, xs, ys) → xs
if2(false, x, xs, ys) → s2bin2(x, ys)

Types:
eq :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
0' :: 0':s:nil:cons:double:1':bug_list_not
true :: true:false
s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
false :: true:false
lt :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
bin2s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
nil :: 0':s:nil:cons:double:1':bug_list_not
cons :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bin2ss :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
double :: 0':s:nil:cons:double:1':bug_list_not
1' :: 0':s:nil:cons:double:1':bug_list_not
half :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
log :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
more :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin1 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
if1 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin2 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bug_list_not :: 0':s:nil:cons:double:1':bug_list_not
if2 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
hole_true:false1_0 :: true:false
hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not
gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat → 0':s:nil:cons:double:1':bug_list_not

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

Heuristically decided to analyse the following defined symbols:
eq, lt, bin2ss, half, log, s2bin1, s2bin2

They will be analysed ascendingly in the following order:
eq < s2bin2
lt < s2bin1
half < log
log < s2bin1
s2bin2 < s2bin1

(10) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
bin2s(nil) → 0'
bin2s(cons(x, xs)) → bin2ss(x, xs)
bin2ss(x, nil) → x
bin2ss(x, cons(0', xs)) → bin2ss(double, xs)
bin2ss(x, cons(1', xs)) → bin2ss(s(double), xs)
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
log(0') → 0'
log(s(0')) → 0'
log(s(s(x))) → s(log(half(s(s(x)))))
more(nil) → nil
more(cons(xs, ys)) → cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys)))
s2bin(x) → s2bin1(x, 0', cons(nil, nil))
s2bin1(x, y, lists) → if1(lt(y, log(x)), x, y, lists)
if1(true, x, y, lists) → s2bin1(x, s(y), more(lists))
if1(false, x, y, lists) → s2bin2(x, lists)
s2bin2(x, nil) → bug_list_not
s2bin2(x, cons(xs, ys)) → if2(eq(x, bin2s(xs)), x, xs, ys)
if2(true, x, xs, ys) → xs
if2(false, x, xs, ys) → s2bin2(x, ys)

Types:
eq :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
0' :: 0':s:nil:cons:double:1':bug_list_not
true :: true:false
s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
false :: true:false
lt :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
bin2s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
nil :: 0':s:nil:cons:double:1':bug_list_not
cons :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bin2ss :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
double :: 0':s:nil:cons:double:1':bug_list_not
1' :: 0':s:nil:cons:double:1':bug_list_not
half :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
log :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
more :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin1 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
if1 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin2 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bug_list_not :: 0':s:nil:cons:double:1':bug_list_not
if2 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
hole_true:false1_0 :: true:false
hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not
gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat → 0':s:nil:cons:double:1':bug_list_not

Generator Equations:
gen_0':s:nil:cons:double:1':bug_list_not3_0(0) ⇔ 0'
gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) ⇔ s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x))

The following defined symbols remain to be analysed:
eq, lt, bin2ss, half, log, s2bin1, s2bin2

They will be analysed ascendingly in the following order:
eq < s2bin2
lt < s2bin1
half < log
log < s2bin1
s2bin2 < s2bin1

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

Proved the following rewrite lemma:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Induction Base:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(0), gen_0':s:nil:cons:double:1':bug_list_not3_0(0)) →_R^Ω(1)
true

Induction Step:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(+(n5_0, 1)), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(n5_0, 1))) →_R^Ω(1)
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) →_IH
true

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

(12) Complex Obligation (BEST)

(13) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
bin2s(nil) → 0'
bin2s(cons(x, xs)) → bin2ss(x, xs)
bin2ss(x, nil) → x
bin2ss(x, cons(0', xs)) → bin2ss(double, xs)
bin2ss(x, cons(1', xs)) → bin2ss(s(double), xs)
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
log(0') → 0'
log(s(0')) → 0'
log(s(s(x))) → s(log(half(s(s(x)))))
more(nil) → nil
more(cons(xs, ys)) → cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys)))
s2bin(x) → s2bin1(x, 0', cons(nil, nil))
s2bin1(x, y, lists) → if1(lt(y, log(x)), x, y, lists)
if1(true, x, y, lists) → s2bin1(x, s(y), more(lists))
if1(false, x, y, lists) → s2bin2(x, lists)
s2bin2(x, nil) → bug_list_not
s2bin2(x, cons(xs, ys)) → if2(eq(x, bin2s(xs)), x, xs, ys)
if2(true, x, xs, ys) → xs
if2(false, x, xs, ys) → s2bin2(x, ys)

Types:
eq :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
0' :: 0':s:nil:cons:double:1':bug_list_not
true :: true:false
s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
false :: true:false
lt :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
bin2s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
nil :: 0':s:nil:cons:double:1':bug_list_not
cons :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bin2ss :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
double :: 0':s:nil:cons:double:1':bug_list_not
1' :: 0':s:nil:cons:double:1':bug_list_not
half :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
log :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
more :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin1 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
if1 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin2 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bug_list_not :: 0':s:nil:cons:double:1':bug_list_not
if2 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
hole_true:false1_0 :: true:false
hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not
gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat → 0':s:nil:cons:double:1':bug_list_not

Lemmas:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s:nil:cons:double:1':bug_list_not3_0(0) ⇔ 0'
gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) ⇔ s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x))

The following defined symbols remain to be analysed:
lt, bin2ss, half, log, s2bin1, s2bin2

They will be analysed ascendingly in the following order:
lt < s2bin1
half < log
log < s2bin1
s2bin2 < s2bin1

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

Proved the following rewrite lemma:
lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n618_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n618_0))) → true, rt ∈ Ω(1 + n618₀)

Induction Base:
lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, 0))) →_R^Ω(1)
true

Induction Step:
lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(+(n618_0, 1)), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, +(n618_0, 1)))) →_R^Ω(1)
lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n618_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n618_0))) →_IH
true

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

(15) Complex Obligation (BEST)

(16) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
bin2s(nil) → 0'
bin2s(cons(x, xs)) → bin2ss(x, xs)
bin2ss(x, nil) → x
bin2ss(x, cons(0', xs)) → bin2ss(double, xs)
bin2ss(x, cons(1', xs)) → bin2ss(s(double), xs)
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
log(0') → 0'
log(s(0')) → 0'
log(s(s(x))) → s(log(half(s(s(x)))))
more(nil) → nil
more(cons(xs, ys)) → cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys)))
s2bin(x) → s2bin1(x, 0', cons(nil, nil))
s2bin1(x, y, lists) → if1(lt(y, log(x)), x, y, lists)
if1(true, x, y, lists) → s2bin1(x, s(y), more(lists))
if1(false, x, y, lists) → s2bin2(x, lists)
s2bin2(x, nil) → bug_list_not
s2bin2(x, cons(xs, ys)) → if2(eq(x, bin2s(xs)), x, xs, ys)
if2(true, x, xs, ys) → xs
if2(false, x, xs, ys) → s2bin2(x, ys)

Types:
eq :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
0' :: 0':s:nil:cons:double:1':bug_list_not
true :: true:false
s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
false :: true:false
lt :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
bin2s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
nil :: 0':s:nil:cons:double:1':bug_list_not
cons :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bin2ss :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
double :: 0':s:nil:cons:double:1':bug_list_not
1' :: 0':s:nil:cons:double:1':bug_list_not
half :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
log :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
more :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin1 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
if1 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin2 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bug_list_not :: 0':s:nil:cons:double:1':bug_list_not
if2 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
hole_true:false1_0 :: true:false
hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not
gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat → 0':s:nil:cons:double:1':bug_list_not

Lemmas:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n618_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n618_0))) → true, rt ∈ Ω(1 + n618₀)

Generator Equations:
gen_0':s:nil:cons:double:1':bug_list_not3_0(0) ⇔ 0'
gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) ⇔ s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x))

The following defined symbols remain to be analysed:
bin2ss, half, log, s2bin1, s2bin2

They will be analysed ascendingly in the following order:
half < log
log < s2bin1
s2bin2 < s2bin1

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

Could not prove a rewrite lemma for the defined symbol bin2ss.

(18) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
bin2s(nil) → 0'
bin2s(cons(x, xs)) → bin2ss(x, xs)
bin2ss(x, nil) → x
bin2ss(x, cons(0', xs)) → bin2ss(double, xs)
bin2ss(x, cons(1', xs)) → bin2ss(s(double), xs)
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
log(0') → 0'
log(s(0')) → 0'
log(s(s(x))) → s(log(half(s(s(x)))))
more(nil) → nil
more(cons(xs, ys)) → cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys)))
s2bin(x) → s2bin1(x, 0', cons(nil, nil))
s2bin1(x, y, lists) → if1(lt(y, log(x)), x, y, lists)
if1(true, x, y, lists) → s2bin1(x, s(y), more(lists))
if1(false, x, y, lists) → s2bin2(x, lists)
s2bin2(x, nil) → bug_list_not
s2bin2(x, cons(xs, ys)) → if2(eq(x, bin2s(xs)), x, xs, ys)
if2(true, x, xs, ys) → xs
if2(false, x, xs, ys) → s2bin2(x, ys)

Types:
eq :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
0' :: 0':s:nil:cons:double:1':bug_list_not
true :: true:false
s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
false :: true:false
lt :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
bin2s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
nil :: 0':s:nil:cons:double:1':bug_list_not
cons :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bin2ss :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
double :: 0':s:nil:cons:double:1':bug_list_not
1' :: 0':s:nil:cons:double:1':bug_list_not
half :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
log :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
more :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin1 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
if1 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin2 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bug_list_not :: 0':s:nil:cons:double:1':bug_list_not
if2 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
hole_true:false1_0 :: true:false
hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not
gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat → 0':s:nil:cons:double:1':bug_list_not

Lemmas:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n618_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n618_0))) → true, rt ∈ Ω(1 + n618₀)

Generator Equations:
gen_0':s:nil:cons:double:1':bug_list_not3_0(0) ⇔ 0'
gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) ⇔ s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x))

The following defined symbols remain to be analysed:
half, log, s2bin1, s2bin2

They will be analysed ascendingly in the following order:
half < log
log < s2bin1
s2bin2 < s2bin1

(19) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, n1023_0))) → gen_0':s:nil:cons:double:1':bug_list_not3_0(n1023_0), rt ∈ Ω(1 + n1023₀)

Induction Base:
half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, 0))) →_R^Ω(1)
0'

Induction Step:
half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, +(n1023_0, 1)))) →_R^Ω(1)
s(half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, n1023_0)))) →_IH
s(gen_0':s:nil:cons:double:1':bug_list_not3_0(c1024_0))

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

(20) Complex Obligation (BEST)

(21) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
bin2s(nil) → 0'
bin2s(cons(x, xs)) → bin2ss(x, xs)
bin2ss(x, nil) → x
bin2ss(x, cons(0', xs)) → bin2ss(double, xs)
bin2ss(x, cons(1', xs)) → bin2ss(s(double), xs)
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
log(0') → 0'
log(s(0')) → 0'
log(s(s(x))) → s(log(half(s(s(x)))))
more(nil) → nil
more(cons(xs, ys)) → cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys)))
s2bin(x) → s2bin1(x, 0', cons(nil, nil))
s2bin1(x, y, lists) → if1(lt(y, log(x)), x, y, lists)
if1(true, x, y, lists) → s2bin1(x, s(y), more(lists))
if1(false, x, y, lists) → s2bin2(x, lists)
s2bin2(x, nil) → bug_list_not
s2bin2(x, cons(xs, ys)) → if2(eq(x, bin2s(xs)), x, xs, ys)
if2(true, x, xs, ys) → xs
if2(false, x, xs, ys) → s2bin2(x, ys)

Types:
eq :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
0' :: 0':s:nil:cons:double:1':bug_list_not
true :: true:false
s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
false :: true:false
lt :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
bin2s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
nil :: 0':s:nil:cons:double:1':bug_list_not
cons :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bin2ss :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
double :: 0':s:nil:cons:double:1':bug_list_not
1' :: 0':s:nil:cons:double:1':bug_list_not
half :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
log :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
more :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin1 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
if1 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin2 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bug_list_not :: 0':s:nil:cons:double:1':bug_list_not
if2 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
hole_true:false1_0 :: true:false
hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not
gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat → 0':s:nil:cons:double:1':bug_list_not

Lemmas:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n618_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n618_0))) → true, rt ∈ Ω(1 + n618₀)
half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, n1023_0))) → gen_0':s:nil:cons:double:1':bug_list_not3_0(n1023_0), rt ∈ Ω(1 + n1023₀)

Generator Equations:
gen_0':s:nil:cons:double:1':bug_list_not3_0(0) ⇔ 0'
gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) ⇔ s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x))

The following defined symbols remain to be analysed:
log, s2bin1, s2bin2

They will be analysed ascendingly in the following order:
log < s2bin1
s2bin2 < s2bin1

(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(23) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
bin2s(nil) → 0'
bin2s(cons(x, xs)) → bin2ss(x, xs)
bin2ss(x, nil) → x
bin2ss(x, cons(0', xs)) → bin2ss(double, xs)
bin2ss(x, cons(1', xs)) → bin2ss(s(double), xs)
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
log(0') → 0'
log(s(0')) → 0'
log(s(s(x))) → s(log(half(s(s(x)))))
more(nil) → nil
more(cons(xs, ys)) → cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys)))
s2bin(x) → s2bin1(x, 0', cons(nil, nil))
s2bin1(x, y, lists) → if1(lt(y, log(x)), x, y, lists)
if1(true, x, y, lists) → s2bin1(x, s(y), more(lists))
if1(false, x, y, lists) → s2bin2(x, lists)
s2bin2(x, nil) → bug_list_not
s2bin2(x, cons(xs, ys)) → if2(eq(x, bin2s(xs)), x, xs, ys)
if2(true, x, xs, ys) → xs
if2(false, x, xs, ys) → s2bin2(x, ys)

Types:
eq :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
0' :: 0':s:nil:cons:double:1':bug_list_not
true :: true:false
s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
false :: true:false
lt :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
bin2s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
nil :: 0':s:nil:cons:double:1':bug_list_not
cons :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bin2ss :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
double :: 0':s:nil:cons:double:1':bug_list_not
1' :: 0':s:nil:cons:double:1':bug_list_not
half :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
log :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
more :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin1 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
if1 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin2 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bug_list_not :: 0':s:nil:cons:double:1':bug_list_not
if2 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
hole_true:false1_0 :: true:false
hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not
gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat → 0':s:nil:cons:double:1':bug_list_not

Lemmas:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n618_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n618_0))) → true, rt ∈ Ω(1 + n618₀)
half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, n1023_0))) → gen_0':s:nil:cons:double:1':bug_list_not3_0(n1023_0), rt ∈ Ω(1 + n1023₀)

Generator Equations:
gen_0':s:nil:cons:double:1':bug_list_not3_0(0) ⇔ 0'
gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) ⇔ s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x))

The following defined symbols remain to be analysed:
s2bin2, s2bin1

They will be analysed ascendingly in the following order:
s2bin2 < s2bin1

(24) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol s2bin2.

(25) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
bin2s(nil) → 0'
bin2s(cons(x, xs)) → bin2ss(x, xs)
bin2ss(x, nil) → x
bin2ss(x, cons(0', xs)) → bin2ss(double, xs)
bin2ss(x, cons(1', xs)) → bin2ss(s(double), xs)
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
log(0') → 0'
log(s(0')) → 0'
log(s(s(x))) → s(log(half(s(s(x)))))
more(nil) → nil
more(cons(xs, ys)) → cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys)))
s2bin(x) → s2bin1(x, 0', cons(nil, nil))
s2bin1(x, y, lists) → if1(lt(y, log(x)), x, y, lists)
if1(true, x, y, lists) → s2bin1(x, s(y), more(lists))
if1(false, x, y, lists) → s2bin2(x, lists)
s2bin2(x, nil) → bug_list_not
s2bin2(x, cons(xs, ys)) → if2(eq(x, bin2s(xs)), x, xs, ys)
if2(true, x, xs, ys) → xs
if2(false, x, xs, ys) → s2bin2(x, ys)

Types:
eq :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
0' :: 0':s:nil:cons:double:1':bug_list_not
true :: true:false
s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
false :: true:false
lt :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
bin2s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
nil :: 0':s:nil:cons:double:1':bug_list_not
cons :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bin2ss :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
double :: 0':s:nil:cons:double:1':bug_list_not
1' :: 0':s:nil:cons:double:1':bug_list_not
half :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
log :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
more :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin1 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
if1 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin2 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bug_list_not :: 0':s:nil:cons:double:1':bug_list_not
if2 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
hole_true:false1_0 :: true:false
hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not
gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat → 0':s:nil:cons:double:1':bug_list_not

Lemmas:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n618_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n618_0))) → true, rt ∈ Ω(1 + n618₀)
half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, n1023_0))) → gen_0':s:nil:cons:double:1':bug_list_not3_0(n1023_0), rt ∈ Ω(1 + n1023₀)

Generator Equations:
gen_0':s:nil:cons:double:1':bug_list_not3_0(0) ⇔ 0'
gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) ⇔ s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x))

The following defined symbols remain to be analysed:
s2bin1

(26) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol s2bin1.

(27) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
bin2s(nil) → 0'
bin2s(cons(x, xs)) → bin2ss(x, xs)
bin2ss(x, nil) → x
bin2ss(x, cons(0', xs)) → bin2ss(double, xs)
bin2ss(x, cons(1', xs)) → bin2ss(s(double), xs)
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
log(0') → 0'
log(s(0')) → 0'
log(s(s(x))) → s(log(half(s(s(x)))))
more(nil) → nil
more(cons(xs, ys)) → cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys)))
s2bin(x) → s2bin1(x, 0', cons(nil, nil))
s2bin1(x, y, lists) → if1(lt(y, log(x)), x, y, lists)
if1(true, x, y, lists) → s2bin1(x, s(y), more(lists))
if1(false, x, y, lists) → s2bin2(x, lists)
s2bin2(x, nil) → bug_list_not
s2bin2(x, cons(xs, ys)) → if2(eq(x, bin2s(xs)), x, xs, ys)
if2(true, x, xs, ys) → xs
if2(false, x, xs, ys) → s2bin2(x, ys)

Types:
eq :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
0' :: 0':s:nil:cons:double:1':bug_list_not
true :: true:false
s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
false :: true:false
lt :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
bin2s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
nil :: 0':s:nil:cons:double:1':bug_list_not
cons :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bin2ss :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
double :: 0':s:nil:cons:double:1':bug_list_not
1' :: 0':s:nil:cons:double:1':bug_list_not
half :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
log :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
more :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin1 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
if1 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin2 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bug_list_not :: 0':s:nil:cons:double:1':bug_list_not
if2 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
hole_true:false1_0 :: true:false
hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not
gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat → 0':s:nil:cons:double:1':bug_list_not

Lemmas:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n618_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n618_0))) → true, rt ∈ Ω(1 + n618₀)
half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, n1023_0))) → gen_0':s:nil:cons:double:1':bug_list_not3_0(n1023_0), rt ∈ Ω(1 + n1023₀)

Generator Equations:
gen_0':s:nil:cons:double:1':bug_list_not3_0(0) ⇔ 0'
gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) ⇔ s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x))

No more defined symbols left to analyse.

(28) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(29) BOUNDS(n^1, INF)

(30) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
bin2s(nil) → 0'
bin2s(cons(x, xs)) → bin2ss(x, xs)
bin2ss(x, nil) → x
bin2ss(x, cons(0', xs)) → bin2ss(double, xs)
bin2ss(x, cons(1', xs)) → bin2ss(s(double), xs)
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
log(0') → 0'
log(s(0')) → 0'
log(s(s(x))) → s(log(half(s(s(x)))))
more(nil) → nil
more(cons(xs, ys)) → cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys)))
s2bin(x) → s2bin1(x, 0', cons(nil, nil))
s2bin1(x, y, lists) → if1(lt(y, log(x)), x, y, lists)
if1(true, x, y, lists) → s2bin1(x, s(y), more(lists))
if1(false, x, y, lists) → s2bin2(x, lists)
s2bin2(x, nil) → bug_list_not
s2bin2(x, cons(xs, ys)) → if2(eq(x, bin2s(xs)), x, xs, ys)
if2(true, x, xs, ys) → xs
if2(false, x, xs, ys) → s2bin2(x, ys)

Types:
eq :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
0' :: 0':s:nil:cons:double:1':bug_list_not
true :: true:false
s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
false :: true:false
lt :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
bin2s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
nil :: 0':s:nil:cons:double:1':bug_list_not
cons :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bin2ss :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
double :: 0':s:nil:cons:double:1':bug_list_not
1' :: 0':s:nil:cons:double:1':bug_list_not
half :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
log :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
more :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin1 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
if1 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin2 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bug_list_not :: 0':s:nil:cons:double:1':bug_list_not
if2 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
hole_true:false1_0 :: true:false
hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not
gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat → 0':s:nil:cons:double:1':bug_list_not

Lemmas:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n618_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n618_0))) → true, rt ∈ Ω(1 + n618₀)
half(gen_0':s:nil:cons:double:1':bug_list_not3_0(*(2, n1023_0))) → gen_0':s:nil:cons:double:1':bug_list_not3_0(n1023_0), rt ∈ Ω(1 + n1023₀)

Generator Equations:
gen_0':s:nil:cons:double:1':bug_list_not3_0(0) ⇔ 0'
gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) ⇔ s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x))

No more defined symbols left to analyse.

(31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(32) BOUNDS(n^1, INF)

(33) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
bin2s(nil) → 0'
bin2s(cons(x, xs)) → bin2ss(x, xs)
bin2ss(x, nil) → x
bin2ss(x, cons(0', xs)) → bin2ss(double, xs)
bin2ss(x, cons(1', xs)) → bin2ss(s(double), xs)
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
log(0') → 0'
log(s(0')) → 0'
log(s(s(x))) → s(log(half(s(s(x)))))
more(nil) → nil
more(cons(xs, ys)) → cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys)))
s2bin(x) → s2bin1(x, 0', cons(nil, nil))
s2bin1(x, y, lists) → if1(lt(y, log(x)), x, y, lists)
if1(true, x, y, lists) → s2bin1(x, s(y), more(lists))
if1(false, x, y, lists) → s2bin2(x, lists)
s2bin2(x, nil) → bug_list_not
s2bin2(x, cons(xs, ys)) → if2(eq(x, bin2s(xs)), x, xs, ys)
if2(true, x, xs, ys) → xs
if2(false, x, xs, ys) → s2bin2(x, ys)

Types:
eq :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
0' :: 0':s:nil:cons:double:1':bug_list_not
true :: true:false
s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
false :: true:false
lt :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
bin2s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
nil :: 0':s:nil:cons:double:1':bug_list_not
cons :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bin2ss :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
double :: 0':s:nil:cons:double:1':bug_list_not
1' :: 0':s:nil:cons:double:1':bug_list_not
half :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
log :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
more :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin1 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
if1 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin2 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bug_list_not :: 0':s:nil:cons:double:1':bug_list_not
if2 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
hole_true:false1_0 :: true:false
hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not
gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat → 0':s:nil:cons:double:1':bug_list_not

Lemmas:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)
lt(gen_0':s:nil:cons:double:1':bug_list_not3_0(n618_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(+(1, n618_0))) → true, rt ∈ Ω(1 + n618₀)

Generator Equations:
gen_0':s:nil:cons:double:1':bug_list_not3_0(0) ⇔ 0'
gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) ⇔ s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x))

No more defined symbols left to analyse.

(34) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(35) BOUNDS(n^1, INF)

(36) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(y)) → false
eq(s(x), 0') → false
eq(s(x), s(y)) → eq(x, y)
lt(0', s(y)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
bin2s(nil) → 0'
bin2s(cons(x, xs)) → bin2ss(x, xs)
bin2ss(x, nil) → x
bin2ss(x, cons(0', xs)) → bin2ss(double, xs)
bin2ss(x, cons(1', xs)) → bin2ss(s(double), xs)
half(0') → 0'
half(s(0')) → 0'
half(s(s(x))) → s(half(x))
log(0') → 0'
log(s(0')) → 0'
log(s(s(x))) → s(log(half(s(s(x)))))
more(nil) → nil
more(cons(xs, ys)) → cons(cons(0', xs), cons(cons(1', xs), cons(xs, ys)))
s2bin(x) → s2bin1(x, 0', cons(nil, nil))
s2bin1(x, y, lists) → if1(lt(y, log(x)), x, y, lists)
if1(true, x, y, lists) → s2bin1(x, s(y), more(lists))
if1(false, x, y, lists) → s2bin2(x, lists)
s2bin2(x, nil) → bug_list_not
s2bin2(x, cons(xs, ys)) → if2(eq(x, bin2s(xs)), x, xs, ys)
if2(true, x, xs, ys) → xs
if2(false, x, xs, ys) → s2bin2(x, ys)

Types:
eq :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
0' :: 0':s:nil:cons:double:1':bug_list_not
true :: true:false
s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
false :: true:false
lt :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → true:false
bin2s :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
nil :: 0':s:nil:cons:double:1':bug_list_not
cons :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bin2ss :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
double :: 0':s:nil:cons:double:1':bug_list_not
1' :: 0':s:nil:cons:double:1':bug_list_not
half :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
log :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
more :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin1 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
if1 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
s2bin2 :: 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
bug_list_not :: 0':s:nil:cons:double:1':bug_list_not
if2 :: true:false → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not → 0':s:nil:cons:double:1':bug_list_not
hole_true:false1_0 :: true:false
hole_0':s:nil:cons:double:1':bug_list_not2_0 :: 0':s:nil:cons:double:1':bug_list_not
gen_0':s:nil:cons:double:1':bug_list_not3_0 :: Nat → 0':s:nil:cons:double:1':bug_list_not

Lemmas:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s:nil:cons:double:1':bug_list_not3_0(0) ⇔ 0'
gen_0':s:nil:cons:double:1':bug_list_not3_0(+(x, 1)) ⇔ s(gen_0':s:nil:cons:double:1':bug_list_not3_0(x))

No more defined symbols left to analyse.

(37) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
eq(gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0), gen_0':s:nil:cons:double:1':bug_list_not3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(38) BOUNDS(n^1, INF)