/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 244 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 96 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 490 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 32 ms]
        (18) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
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

----------------------------------------

(5) OrderProof (LOWER BOUND(ID))
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

----------------------------------------

(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


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

----------------------------------------

(7) RewriteLemmaProof (LOWER BOUND(ID))
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 in Omega(1 + n8_0)

Induction Base:
le(gen_s:0':logError:error5_0(+(1, 0)), gen_s:0':logError:error5_0(0)) ->_R^Omega(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^Omega(1)
le(gen_s:0':logError:error5_0(+(1, n8_0)), gen_s:0':logError:error5_0(n8_0)) ->_IH
false

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(8)
Complex Obligation (BEST)

----------------------------------------

(9)
Obligation:
Proved the lower bound n^1 for the following 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

----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
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 in Omega(1 + n8_0)


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

----------------------------------------

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_s:0':logError:error5_0(n341_0)) -> gen_s:0':logError:error5_0(*(2, n341_0)), rt in Omega(1 + n341_0)

Induction Base:
double(gen_s:0':logError:error5_0(0)) ->_R^Omega(1)
0'

Induction Step:
double(gen_s:0':logError:error5_0(+(n341_0, 1))) ->_R^Omega(1)
s(s(double(gen_s:0':logError:error5_0(n341_0)))) ->_IH
s(s(gen_s:0':logError:error5_0(*(2, c342_0))))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(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 in Omega(1 + n8_0)
double(gen_s:0':logError:error5_0(n341_0)) -> gen_s:0':logError:error5_0(*(2, n341_0)), rt in Omega(1 + n341_0)


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) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
droplast(gen_nil:cons6_0(+(1, n3031_0))) -> gen_nil:cons6_0(n3031_0), rt in Omega(1 + n3031_0)

Induction Base:
droplast(gen_nil:cons6_0(+(1, 0))) ->_R^Omega(1)
nil

Induction Step:
droplast(gen_nil:cons6_0(+(1, +(n3031_0, 1)))) ->_R^Omega(1)
cons(0', droplast(cons(0', gen_nil:cons6_0(n3031_0)))) ->_IH
cons(0', gen_nil:cons6_0(c3032_0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(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 in Omega(1 + n8_0)
double(gen_s:0':logError:error5_0(n341_0)) -> gen_s:0':logError:error5_0(*(2, n341_0)), rt in Omega(1 + n341_0)
droplast(gen_nil:cons6_0(+(1, n3031_0))) -> gen_nil:cons6_0(n3031_0), rt in Omega(1 + n3031_0)


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

----------------------------------------

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
last(gen_nil:cons6_0(+(1, n3468_0))) -> gen_s:0':logError:error5_0(0), rt in Omega(1 + n3468_0)

Induction Base:
last(gen_nil:cons6_0(+(1, 0))) ->_R^Omega(1)
0'

Induction Step:
last(gen_nil:cons6_0(+(1, +(n3468_0, 1)))) ->_R^Omega(1)
last(cons(0', gen_nil:cons6_0(n3468_0))) ->_IH
gen_s:0':logError:error5_0(0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(18)
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 in Omega(1 + n8_0)
double(gen_s:0':logError:error5_0(n341_0)) -> gen_s:0':logError:error5_0(*(2, n341_0)), rt in Omega(1 + n341_0)
droplast(gen_nil:cons6_0(+(1, n3031_0))) -> gen_nil:cons6_0(n3031_0), rt in Omega(1 + n3031_0)
last(gen_nil:cons6_0(+(1, n3468_0))) -> gen_s:0':logError:error5_0(0), rt in Omega(1 + n3468_0)


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