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:

le(s(x), 0) → false
le(0, y) → true
le(s(x), s(y)) → le(x, y)
double(0) → 0
double(s(x)) → s(s(double(x)))
log(0) → logError
log(s(x)) → loop(s(x), s(0), 0)
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))
maplog(xs) → mapIter(xs, nil)
mapIter(xs, ys) → ifmap(isempty(xs), xs, ys)
ifmap(true, xs, ys) → ys
ifmap(false, xs, ys) → mapIter(droplast(xs), cons(log(last(xs)), ys))
isempty(nil) → true
isempty(cons(x, xs)) → false
last(nil) → error
last(cons(x, nil)) → x
last(cons(x, cons(y, xs))) → last(cons(y, xs))
droplast(nil) → nil
droplast(cons(x, nil)) → nil
droplast(cons(x, cons(y, xs))) → cons(x, droplast(cons(y, xs)))
a → b
a → c

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
le(s(x), s(y)) →⁺ le(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:

le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))
maplog(xs) → mapIter(xs, nil)
mapIter(xs, ys) → ifmap(isempty(xs), xs, ys)
ifmap(true, xs, ys) → ys
ifmap(false, xs, ys) → mapIter(droplast(xs), cons(log(last(xs)), ys))
isempty(nil) → true
isempty(cons(x, xs)) → false
last(nil) → error
last(cons(x, nil)) → x
last(cons(x, cons(y, xs))) → last(cons(y, xs))
droplast(nil) → nil
droplast(cons(x, nil)) → nil
droplast(cons(x, cons(y, xs))) → cons(x, droplast(cons(y, xs)))
a → b
a → c

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))
maplog(xs) → mapIter(xs, nil)
mapIter(xs, ys) → ifmap(isempty(xs), xs, ys)
ifmap(true, xs, ys) → ys
ifmap(false, xs, ys) → mapIter(droplast(xs), cons(log(last(xs)), ys))
isempty(nil) → true
isempty(cons(x, xs)) → false
last(nil) → error
last(cons(x, nil)) → x
last(cons(x, cons(y, xs))) → last(cons(y, xs))
droplast(nil) → nil
droplast(cons(x, nil)) → nil
droplast(cons(x, cons(y, xs))) → cons(x, droplast(cons(y, xs)))
a → b
a → c

Types:
le :: s:0':logError:error → s:0':logError:error → false:true
s :: s:0':logError:error → s:0':logError:error
0' :: s:0':logError:error
false :: false:true
true :: false:true
double :: s:0':logError:error → s:0':logError:error
log :: s:0':logError:error → s:0':logError:error
logError :: s:0':logError:error
loop :: s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
if :: false:true → s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
maplog :: nil:cons → nil:cons
mapIter :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
ifmap :: false:true → nil:cons → nil:cons → nil:cons
isempty :: nil:cons → false:true
droplast :: nil:cons → nil:cons
cons :: s:0':logError:error → nil:cons → nil:cons
last :: nil:cons → s:0':logError:error
error :: s:0':logError:error
a :: b:c
b :: b:c
c :: b:c
hole_false:true1_0 :: false:true
hole_s:0':logError:error2_0 :: s:0':logError:error
hole_nil:cons3_0 :: nil:cons
hole_b:c4_0 :: b:c
gen_s:0':logError:error5_0 :: Nat → s:0':logError:error
gen_nil:cons6_0 :: Nat → nil:cons

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

Heuristically decided to analyse the following defined symbols:
le, double, loop, mapIter, droplast, last

They will be analysed ascendingly in the following order:
le < loop
double < loop
droplast < mapIter
last < mapIter

(8) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))
maplog(xs) → mapIter(xs, nil)
mapIter(xs, ys) → ifmap(isempty(xs), xs, ys)
ifmap(true, xs, ys) → ys
ifmap(false, xs, ys) → mapIter(droplast(xs), cons(log(last(xs)), ys))
isempty(nil) → true
isempty(cons(x, xs)) → false
last(nil) → error
last(cons(x, nil)) → x
last(cons(x, cons(y, xs))) → last(cons(y, xs))
droplast(nil) → nil
droplast(cons(x, nil)) → nil
droplast(cons(x, cons(y, xs))) → cons(x, droplast(cons(y, xs)))
a → b
a → c

Types:
le :: s:0':logError:error → s:0':logError:error → false:true
s :: s:0':logError:error → s:0':logError:error
0' :: s:0':logError:error
false :: false:true
true :: false:true
double :: s:0':logError:error → s:0':logError:error
log :: s:0':logError:error → s:0':logError:error
logError :: s:0':logError:error
loop :: s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
if :: false:true → s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
maplog :: nil:cons → nil:cons
mapIter :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
ifmap :: false:true → nil:cons → nil:cons → nil:cons
isempty :: nil:cons → false:true
droplast :: nil:cons → nil:cons
cons :: s:0':logError:error → nil:cons → nil:cons
last :: nil:cons → s:0':logError:error
error :: s:0':logError:error
a :: b:c
b :: b:c
c :: b:c
hole_false:true1_0 :: false:true
hole_s:0':logError:error2_0 :: s:0':logError:error
hole_nil:cons3_0 :: nil:cons
hole_b:c4_0 :: b:c
gen_s:0':logError:error5_0 :: Nat → s:0':logError:error
gen_nil:cons6_0 :: Nat → nil:cons

Generator Equations:
gen_s:0':logError:error5_0(0) ⇔ 0'
gen_s:0':logError:error5_0(+(x, 1)) ⇔ s(gen_s:0':logError:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
le, double, loop, mapIter, droplast, last

They will be analysed ascendingly in the following order:
le < loop
double < loop
droplast < mapIter
last < mapIter

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

Proved the following rewrite lemma:
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) → false, rt ∈ Ω(1 + n8₀)

Induction Base:
le(gen_s:0':logError:error5_0(+(1, 0)), gen_s:0':logError:error5_0(0)) →_R^Ω(1)
false

Induction Step:
le(gen_s:0':logError:error5_0(+(1, +(n8_0, 1))), gen_s:0':logError:error5_0(+(n8_0, 1))) →_R^Ω(1)
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) →_IH
false

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))
maplog(xs) → mapIter(xs, nil)
mapIter(xs, ys) → ifmap(isempty(xs), xs, ys)
ifmap(true, xs, ys) → ys
ifmap(false, xs, ys) → mapIter(droplast(xs), cons(log(last(xs)), ys))
isempty(nil) → true
isempty(cons(x, xs)) → false
last(nil) → error
last(cons(x, nil)) → x
last(cons(x, cons(y, xs))) → last(cons(y, xs))
droplast(nil) → nil
droplast(cons(x, nil)) → nil
droplast(cons(x, cons(y, xs))) → cons(x, droplast(cons(y, xs)))
a → b
a → c

Types:
le :: s:0':logError:error → s:0':logError:error → false:true
s :: s:0':logError:error → s:0':logError:error
0' :: s:0':logError:error
false :: false:true
true :: false:true
double :: s:0':logError:error → s:0':logError:error
log :: s:0':logError:error → s:0':logError:error
logError :: s:0':logError:error
loop :: s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
if :: false:true → s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
maplog :: nil:cons → nil:cons
mapIter :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
ifmap :: false:true → nil:cons → nil:cons → nil:cons
isempty :: nil:cons → false:true
droplast :: nil:cons → nil:cons
cons :: s:0':logError:error → nil:cons → nil:cons
last :: nil:cons → s:0':logError:error
error :: s:0':logError:error
a :: b:c
b :: b:c
c :: b:c
hole_false:true1_0 :: false:true
hole_s:0':logError:error2_0 :: s:0':logError:error
hole_nil:cons3_0 :: nil:cons
hole_b:c4_0 :: b:c
gen_s:0':logError:error5_0 :: Nat → s:0':logError:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) → false, rt ∈ Ω(1 + n8₀)

Generator Equations:
gen_s:0':logError:error5_0(0) ⇔ 0'
gen_s:0':logError:error5_0(+(x, 1)) ⇔ s(gen_s:0':logError:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
double, loop, mapIter, droplast, last

They will be analysed ascendingly in the following order:
double < loop
droplast < mapIter
last < mapIter

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

Proved the following rewrite lemma:
double(gen_s:0':logError:error5_0(n365_0)) → gen_s:0':logError:error5_0(*(2, n365_0)), rt ∈ Ω(1 + n365₀)

Induction Base:
double(gen_s:0':logError:error5_0(0)) →_R^Ω(1)
0'

Induction Step:
double(gen_s:0':logError:error5_0(+(n365_0, 1))) →_R^Ω(1)
s(s(double(gen_s:0':logError:error5_0(n365_0)))) →_IH
s(s(gen_s:0':logError:error5_0(*(2, c366_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:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))
maplog(xs) → mapIter(xs, nil)
mapIter(xs, ys) → ifmap(isempty(xs), xs, ys)
ifmap(true, xs, ys) → ys
ifmap(false, xs, ys) → mapIter(droplast(xs), cons(log(last(xs)), ys))
isempty(nil) → true
isempty(cons(x, xs)) → false
last(nil) → error
last(cons(x, nil)) → x
last(cons(x, cons(y, xs))) → last(cons(y, xs))
droplast(nil) → nil
droplast(cons(x, nil)) → nil
droplast(cons(x, cons(y, xs))) → cons(x, droplast(cons(y, xs)))
a → b
a → c

Types:
le :: s:0':logError:error → s:0':logError:error → false:true
s :: s:0':logError:error → s:0':logError:error
0' :: s:0':logError:error
false :: false:true
true :: false:true
double :: s:0':logError:error → s:0':logError:error
log :: s:0':logError:error → s:0':logError:error
logError :: s:0':logError:error
loop :: s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
if :: false:true → s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
maplog :: nil:cons → nil:cons
mapIter :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
ifmap :: false:true → nil:cons → nil:cons → nil:cons
isempty :: nil:cons → false:true
droplast :: nil:cons → nil:cons
cons :: s:0':logError:error → nil:cons → nil:cons
last :: nil:cons → s:0':logError:error
error :: s:0':logError:error
a :: b:c
b :: b:c
c :: b:c
hole_false:true1_0 :: false:true
hole_s:0':logError:error2_0 :: s:0':logError:error
hole_nil:cons3_0 :: nil:cons
hole_b:c4_0 :: b:c
gen_s:0':logError:error5_0 :: Nat → s:0':logError:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) → false, rt ∈ Ω(1 + n8₀)
double(gen_s:0':logError:error5_0(n365_0)) → gen_s:0':logError:error5_0(*(2, n365_0)), rt ∈ Ω(1 + n365₀)

Generator Equations:
gen_s:0':logError:error5_0(0) ⇔ 0'
gen_s:0':logError:error5_0(+(x, 1)) ⇔ s(gen_s:0':logError:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
loop, mapIter, droplast, last

They will be analysed ascendingly in the following order:
droplast < mapIter
last < mapIter

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

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

(16) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))
maplog(xs) → mapIter(xs, nil)
mapIter(xs, ys) → ifmap(isempty(xs), xs, ys)
ifmap(true, xs, ys) → ys
ifmap(false, xs, ys) → mapIter(droplast(xs), cons(log(last(xs)), ys))
isempty(nil) → true
isempty(cons(x, xs)) → false
last(nil) → error
last(cons(x, nil)) → x
last(cons(x, cons(y, xs))) → last(cons(y, xs))
droplast(nil) → nil
droplast(cons(x, nil)) → nil
droplast(cons(x, cons(y, xs))) → cons(x, droplast(cons(y, xs)))
a → b
a → c

Types:
le :: s:0':logError:error → s:0':logError:error → false:true
s :: s:0':logError:error → s:0':logError:error
0' :: s:0':logError:error
false :: false:true
true :: false:true
double :: s:0':logError:error → s:0':logError:error
log :: s:0':logError:error → s:0':logError:error
logError :: s:0':logError:error
loop :: s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
if :: false:true → s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
maplog :: nil:cons → nil:cons
mapIter :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
ifmap :: false:true → nil:cons → nil:cons → nil:cons
isempty :: nil:cons → false:true
droplast :: nil:cons → nil:cons
cons :: s:0':logError:error → nil:cons → nil:cons
last :: nil:cons → s:0':logError:error
error :: s:0':logError:error
a :: b:c
b :: b:c
c :: b:c
hole_false:true1_0 :: false:true
hole_s:0':logError:error2_0 :: s:0':logError:error
hole_nil:cons3_0 :: nil:cons
hole_b:c4_0 :: b:c
gen_s:0':logError:error5_0 :: Nat → s:0':logError:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) → false, rt ∈ Ω(1 + n8₀)
double(gen_s:0':logError:error5_0(n365_0)) → gen_s:0':logError:error5_0(*(2, n365_0)), rt ∈ Ω(1 + n365₀)

Generator Equations:
gen_s:0':logError:error5_0(0) ⇔ 0'
gen_s:0':logError:error5_0(+(x, 1)) ⇔ s(gen_s:0':logError:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
droplast, mapIter, last

They will be analysed ascendingly in the following order:
droplast < mapIter
last < mapIter

(17) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
droplast(gen_nil:cons6_0(+(1, n1157_0))) → gen_nil:cons6_0(n1157_0), rt ∈ Ω(1 + n1157₀)

Induction Base:
droplast(gen_nil:cons6_0(+(1, 0))) →_R^Ω(1)
nil

Induction Step:
droplast(gen_nil:cons6_0(+(1, +(n1157_0, 1)))) →_R^Ω(1)
cons(0', droplast(cons(0', gen_nil:cons6_0(n1157_0)))) →_IH
cons(0', gen_nil:cons6_0(c1158_0))

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

(18) Complex Obligation (BEST)

(19) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))
maplog(xs) → mapIter(xs, nil)
mapIter(xs, ys) → ifmap(isempty(xs), xs, ys)
ifmap(true, xs, ys) → ys
ifmap(false, xs, ys) → mapIter(droplast(xs), cons(log(last(xs)), ys))
isempty(nil) → true
isempty(cons(x, xs)) → false
last(nil) → error
last(cons(x, nil)) → x
last(cons(x, cons(y, xs))) → last(cons(y, xs))
droplast(nil) → nil
droplast(cons(x, nil)) → nil
droplast(cons(x, cons(y, xs))) → cons(x, droplast(cons(y, xs)))
a → b
a → c

Types:
le :: s:0':logError:error → s:0':logError:error → false:true
s :: s:0':logError:error → s:0':logError:error
0' :: s:0':logError:error
false :: false:true
true :: false:true
double :: s:0':logError:error → s:0':logError:error
log :: s:0':logError:error → s:0':logError:error
logError :: s:0':logError:error
loop :: s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
if :: false:true → s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
maplog :: nil:cons → nil:cons
mapIter :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
ifmap :: false:true → nil:cons → nil:cons → nil:cons
isempty :: nil:cons → false:true
droplast :: nil:cons → nil:cons
cons :: s:0':logError:error → nil:cons → nil:cons
last :: nil:cons → s:0':logError:error
error :: s:0':logError:error
a :: b:c
b :: b:c
c :: b:c
hole_false:true1_0 :: false:true
hole_s:0':logError:error2_0 :: s:0':logError:error
hole_nil:cons3_0 :: nil:cons
hole_b:c4_0 :: b:c
gen_s:0':logError:error5_0 :: Nat → s:0':logError:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) → false, rt ∈ Ω(1 + n8₀)
double(gen_s:0':logError:error5_0(n365_0)) → gen_s:0':logError:error5_0(*(2, n365_0)), rt ∈ Ω(1 + n365₀)
droplast(gen_nil:cons6_0(+(1, n1157_0))) → gen_nil:cons6_0(n1157_0), rt ∈ Ω(1 + n1157₀)

Generator Equations:
gen_s:0':logError:error5_0(0) ⇔ 0'
gen_s:0':logError:error5_0(+(x, 1)) ⇔ s(gen_s:0':logError:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
last, mapIter

They will be analysed ascendingly in the following order:
last < mapIter

(20) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
last(gen_nil:cons6_0(+(1, n1594_0))) → gen_s:0':logError:error5_0(0), rt ∈ Ω(1 + n1594₀)

Induction Base:
last(gen_nil:cons6_0(+(1, 0))) →_R^Ω(1)
0'

Induction Step:
last(gen_nil:cons6_0(+(1, +(n1594_0, 1)))) →_R^Ω(1)
last(cons(0', gen_nil:cons6_0(n1594_0))) →_IH
gen_s:0':logError:error5_0(0)

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

(21) Complex Obligation (BEST)

(22) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))
maplog(xs) → mapIter(xs, nil)
mapIter(xs, ys) → ifmap(isempty(xs), xs, ys)
ifmap(true, xs, ys) → ys
ifmap(false, xs, ys) → mapIter(droplast(xs), cons(log(last(xs)), ys))
isempty(nil) → true
isempty(cons(x, xs)) → false
last(nil) → error
last(cons(x, nil)) → x
last(cons(x, cons(y, xs))) → last(cons(y, xs))
droplast(nil) → nil
droplast(cons(x, nil)) → nil
droplast(cons(x, cons(y, xs))) → cons(x, droplast(cons(y, xs)))
a → b
a → c

Types:
le :: s:0':logError:error → s:0':logError:error → false:true
s :: s:0':logError:error → s:0':logError:error
0' :: s:0':logError:error
false :: false:true
true :: false:true
double :: s:0':logError:error → s:0':logError:error
log :: s:0':logError:error → s:0':logError:error
logError :: s:0':logError:error
loop :: s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
if :: false:true → s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
maplog :: nil:cons → nil:cons
mapIter :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
ifmap :: false:true → nil:cons → nil:cons → nil:cons
isempty :: nil:cons → false:true
droplast :: nil:cons → nil:cons
cons :: s:0':logError:error → nil:cons → nil:cons
last :: nil:cons → s:0':logError:error
error :: s:0':logError:error
a :: b:c
b :: b:c
c :: b:c
hole_false:true1_0 :: false:true
hole_s:0':logError:error2_0 :: s:0':logError:error
hole_nil:cons3_0 :: nil:cons
hole_b:c4_0 :: b:c
gen_s:0':logError:error5_0 :: Nat → s:0':logError:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) → false, rt ∈ Ω(1 + n8₀)
double(gen_s:0':logError:error5_0(n365_0)) → gen_s:0':logError:error5_0(*(2, n365_0)), rt ∈ Ω(1 + n365₀)
droplast(gen_nil:cons6_0(+(1, n1157_0))) → gen_nil:cons6_0(n1157_0), rt ∈ Ω(1 + n1157₀)
last(gen_nil:cons6_0(+(1, n1594_0))) → gen_s:0':logError:error5_0(0), rt ∈ Ω(1 + n1594₀)

Generator Equations:
gen_s:0':logError:error5_0(0) ⇔ 0'
gen_s:0':logError:error5_0(+(x, 1)) ⇔ s(gen_s:0':logError:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

The following defined symbols remain to be analysed:
mapIter

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

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

(24) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))
maplog(xs) → mapIter(xs, nil)
mapIter(xs, ys) → ifmap(isempty(xs), xs, ys)
ifmap(true, xs, ys) → ys
ifmap(false, xs, ys) → mapIter(droplast(xs), cons(log(last(xs)), ys))
isempty(nil) → true
isempty(cons(x, xs)) → false
last(nil) → error
last(cons(x, nil)) → x
last(cons(x, cons(y, xs))) → last(cons(y, xs))
droplast(nil) → nil
droplast(cons(x, nil)) → nil
droplast(cons(x, cons(y, xs))) → cons(x, droplast(cons(y, xs)))
a → b
a → c

Types:
le :: s:0':logError:error → s:0':logError:error → false:true
s :: s:0':logError:error → s:0':logError:error
0' :: s:0':logError:error
false :: false:true
true :: false:true
double :: s:0':logError:error → s:0':logError:error
log :: s:0':logError:error → s:0':logError:error
logError :: s:0':logError:error
loop :: s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
if :: false:true → s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
maplog :: nil:cons → nil:cons
mapIter :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
ifmap :: false:true → nil:cons → nil:cons → nil:cons
isempty :: nil:cons → false:true
droplast :: nil:cons → nil:cons
cons :: s:0':logError:error → nil:cons → nil:cons
last :: nil:cons → s:0':logError:error
error :: s:0':logError:error
a :: b:c
b :: b:c
c :: b:c
hole_false:true1_0 :: false:true
hole_s:0':logError:error2_0 :: s:0':logError:error
hole_nil:cons3_0 :: nil:cons
hole_b:c4_0 :: b:c
gen_s:0':logError:error5_0 :: Nat → s:0':logError:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) → false, rt ∈ Ω(1 + n8₀)
double(gen_s:0':logError:error5_0(n365_0)) → gen_s:0':logError:error5_0(*(2, n365_0)), rt ∈ Ω(1 + n365₀)
droplast(gen_nil:cons6_0(+(1, n1157_0))) → gen_nil:cons6_0(n1157_0), rt ∈ Ω(1 + n1157₀)
last(gen_nil:cons6_0(+(1, n1594_0))) → gen_s:0':logError:error5_0(0), rt ∈ Ω(1 + n1594₀)

Generator Equations:
gen_s:0':logError:error5_0(0) ⇔ 0'
gen_s:0':logError:error5_0(+(x, 1)) ⇔ s(gen_s:0':logError:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) → false, rt ∈ Ω(1 + n8₀)

(26) BOUNDS(n^1, INF)

(27) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))
maplog(xs) → mapIter(xs, nil)
mapIter(xs, ys) → ifmap(isempty(xs), xs, ys)
ifmap(true, xs, ys) → ys
ifmap(false, xs, ys) → mapIter(droplast(xs), cons(log(last(xs)), ys))
isempty(nil) → true
isempty(cons(x, xs)) → false
last(nil) → error
last(cons(x, nil)) → x
last(cons(x, cons(y, xs))) → last(cons(y, xs))
droplast(nil) → nil
droplast(cons(x, nil)) → nil
droplast(cons(x, cons(y, xs))) → cons(x, droplast(cons(y, xs)))
a → b
a → c

Types:
le :: s:0':logError:error → s:0':logError:error → false:true
s :: s:0':logError:error → s:0':logError:error
0' :: s:0':logError:error
false :: false:true
true :: false:true
double :: s:0':logError:error → s:0':logError:error
log :: s:0':logError:error → s:0':logError:error
logError :: s:0':logError:error
loop :: s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
if :: false:true → s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
maplog :: nil:cons → nil:cons
mapIter :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
ifmap :: false:true → nil:cons → nil:cons → nil:cons
isempty :: nil:cons → false:true
droplast :: nil:cons → nil:cons
cons :: s:0':logError:error → nil:cons → nil:cons
last :: nil:cons → s:0':logError:error
error :: s:0':logError:error
a :: b:c
b :: b:c
c :: b:c
hole_false:true1_0 :: false:true
hole_s:0':logError:error2_0 :: s:0':logError:error
hole_nil:cons3_0 :: nil:cons
hole_b:c4_0 :: b:c
gen_s:0':logError:error5_0 :: Nat → s:0':logError:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) → false, rt ∈ Ω(1 + n8₀)
double(gen_s:0':logError:error5_0(n365_0)) → gen_s:0':logError:error5_0(*(2, n365_0)), rt ∈ Ω(1 + n365₀)
droplast(gen_nil:cons6_0(+(1, n1157_0))) → gen_nil:cons6_0(n1157_0), rt ∈ Ω(1 + n1157₀)
last(gen_nil:cons6_0(+(1, n1594_0))) → gen_s:0':logError:error5_0(0), rt ∈ Ω(1 + n1594₀)

Generator Equations:
gen_s:0':logError:error5_0(0) ⇔ 0'
gen_s:0':logError:error5_0(+(x, 1)) ⇔ s(gen_s:0':logError:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

No more defined symbols left to analyse.

(28) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) → false, rt ∈ Ω(1 + n8₀)

(29) BOUNDS(n^1, INF)

(30) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))
maplog(xs) → mapIter(xs, nil)
mapIter(xs, ys) → ifmap(isempty(xs), xs, ys)
ifmap(true, xs, ys) → ys
ifmap(false, xs, ys) → mapIter(droplast(xs), cons(log(last(xs)), ys))
isempty(nil) → true
isempty(cons(x, xs)) → false
last(nil) → error
last(cons(x, nil)) → x
last(cons(x, cons(y, xs))) → last(cons(y, xs))
droplast(nil) → nil
droplast(cons(x, nil)) → nil
droplast(cons(x, cons(y, xs))) → cons(x, droplast(cons(y, xs)))
a → b
a → c

Types:
le :: s:0':logError:error → s:0':logError:error → false:true
s :: s:0':logError:error → s:0':logError:error
0' :: s:0':logError:error
false :: false:true
true :: false:true
double :: s:0':logError:error → s:0':logError:error
log :: s:0':logError:error → s:0':logError:error
logError :: s:0':logError:error
loop :: s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
if :: false:true → s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
maplog :: nil:cons → nil:cons
mapIter :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
ifmap :: false:true → nil:cons → nil:cons → nil:cons
isempty :: nil:cons → false:true
droplast :: nil:cons → nil:cons
cons :: s:0':logError:error → nil:cons → nil:cons
last :: nil:cons → s:0':logError:error
error :: s:0':logError:error
a :: b:c
b :: b:c
c :: b:c
hole_false:true1_0 :: false:true
hole_s:0':logError:error2_0 :: s:0':logError:error
hole_nil:cons3_0 :: nil:cons
hole_b:c4_0 :: b:c
gen_s:0':logError:error5_0 :: Nat → s:0':logError:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) → false, rt ∈ Ω(1 + n8₀)
double(gen_s:0':logError:error5_0(n365_0)) → gen_s:0':logError:error5_0(*(2, n365_0)), rt ∈ Ω(1 + n365₀)
droplast(gen_nil:cons6_0(+(1, n1157_0))) → gen_nil:cons6_0(n1157_0), rt ∈ Ω(1 + n1157₀)

Generator Equations:
gen_s:0':logError:error5_0(0) ⇔ 0'
gen_s:0':logError:error5_0(+(x, 1)) ⇔ s(gen_s:0':logError:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

No more defined symbols left to analyse.

(31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) → false, rt ∈ Ω(1 + n8₀)

(32) BOUNDS(n^1, INF)

(33) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))
maplog(xs) → mapIter(xs, nil)
mapIter(xs, ys) → ifmap(isempty(xs), xs, ys)
ifmap(true, xs, ys) → ys
ifmap(false, xs, ys) → mapIter(droplast(xs), cons(log(last(xs)), ys))
isempty(nil) → true
isempty(cons(x, xs)) → false
last(nil) → error
last(cons(x, nil)) → x
last(cons(x, cons(y, xs))) → last(cons(y, xs))
droplast(nil) → nil
droplast(cons(x, nil)) → nil
droplast(cons(x, cons(y, xs))) → cons(x, droplast(cons(y, xs)))
a → b
a → c

Types:
le :: s:0':logError:error → s:0':logError:error → false:true
s :: s:0':logError:error → s:0':logError:error
0' :: s:0':logError:error
false :: false:true
true :: false:true
double :: s:0':logError:error → s:0':logError:error
log :: s:0':logError:error → s:0':logError:error
logError :: s:0':logError:error
loop :: s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
if :: false:true → s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
maplog :: nil:cons → nil:cons
mapIter :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
ifmap :: false:true → nil:cons → nil:cons → nil:cons
isempty :: nil:cons → false:true
droplast :: nil:cons → nil:cons
cons :: s:0':logError:error → nil:cons → nil:cons
last :: nil:cons → s:0':logError:error
error :: s:0':logError:error
a :: b:c
b :: b:c
c :: b:c
hole_false:true1_0 :: false:true
hole_s:0':logError:error2_0 :: s:0':logError:error
hole_nil:cons3_0 :: nil:cons
hole_b:c4_0 :: b:c
gen_s:0':logError:error5_0 :: Nat → s:0':logError:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) → false, rt ∈ Ω(1 + n8₀)
double(gen_s:0':logError:error5_0(n365_0)) → gen_s:0':logError:error5_0(*(2, n365_0)), rt ∈ Ω(1 + n365₀)

Generator Equations:
gen_s:0':logError:error5_0(0) ⇔ 0'
gen_s:0':logError:error5_0(+(x, 1)) ⇔ s(gen_s:0':logError:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

No more defined symbols left to analyse.

(34) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) → false, rt ∈ Ω(1 + n8₀)

(35) BOUNDS(n^1, INF)

(36) Obligation:

TRS:
Rules:
le(s(x), 0') → false
le(0', y) → true
le(s(x), s(y)) → le(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
log(0') → logError
log(s(x)) → loop(s(x), s(0'), 0')
loop(x, s(y), z) → if(le(x, s(y)), x, s(y), z)
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))
maplog(xs) → mapIter(xs, nil)
mapIter(xs, ys) → ifmap(isempty(xs), xs, ys)
ifmap(true, xs, ys) → ys
ifmap(false, xs, ys) → mapIter(droplast(xs), cons(log(last(xs)), ys))
isempty(nil) → true
isempty(cons(x, xs)) → false
last(nil) → error
last(cons(x, nil)) → x
last(cons(x, cons(y, xs))) → last(cons(y, xs))
droplast(nil) → nil
droplast(cons(x, nil)) → nil
droplast(cons(x, cons(y, xs))) → cons(x, droplast(cons(y, xs)))
a → b
a → c

Types:
le :: s:0':logError:error → s:0':logError:error → false:true
s :: s:0':logError:error → s:0':logError:error
0' :: s:0':logError:error
false :: false:true
true :: false:true
double :: s:0':logError:error → s:0':logError:error
log :: s:0':logError:error → s:0':logError:error
logError :: s:0':logError:error
loop :: s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
if :: false:true → s:0':logError:error → s:0':logError:error → s:0':logError:error → s:0':logError:error
maplog :: nil:cons → nil:cons
mapIter :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
ifmap :: false:true → nil:cons → nil:cons → nil:cons
isempty :: nil:cons → false:true
droplast :: nil:cons → nil:cons
cons :: s:0':logError:error → nil:cons → nil:cons
last :: nil:cons → s:0':logError:error
error :: s:0':logError:error
a :: b:c
b :: b:c
c :: b:c
hole_false:true1_0 :: false:true
hole_s:0':logError:error2_0 :: s:0':logError:error
hole_nil:cons3_0 :: nil:cons
hole_b:c4_0 :: b:c
gen_s:0':logError:error5_0 :: Nat → s:0':logError:error
gen_nil:cons6_0 :: Nat → nil:cons

Lemmas:
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) → false, rt ∈ Ω(1 + n8₀)

Generator Equations:
gen_s:0':logError:error5_0(0) ⇔ 0'
gen_s:0':logError:error5_0(+(x, 1)) ⇔ s(gen_s:0':logError:error5_0(x))
gen_nil:cons6_0(0) ⇔ nil
gen_nil:cons6_0(+(x, 1)) ⇔ cons(0', gen_nil:cons6_0(x))

No more defined symbols left to analyse.

(37) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) → false, rt ∈ Ω(1 + n8₀)

(38) BOUNDS(n^1, INF)