Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS
↳1 AAECC Innermost (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
  ↳7 QDP
    ↳8 QDPOrderProof (⇔)
    ↳9 QDP
    ↳10 PisEmptyProof (⇔)
    ↳11 TRUE
  ↳12 QDP
    ↳13 QDPOrderProof (⇔)
    ↳14 QDP
    ↳15 PisEmptyProof (⇔)
    ↳16 TRUE
  ↳17 QDP
    ↳18 QDPOrderProof (⇔)
    ↳19 QDP
    ↳20 PisEmptyProof (⇔)
    ↳21 TRUE
  ↳22 QDP
    ↳23 QDPOrderProof (⇔)
    ↳24 QDP
    ↳25 PisEmptyProof (⇔)
    ↳26 TRUE
  ↳27 QDP
  ↳28 QDP

(0) Obligation:

Q restricted rewrite system:
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

Q is empty.

(1) AAECC Innermost (EQUIVALENT transformation)

We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is

double(0) → 0
double(s(x)) → s(s(double(x)))
if(true, x, y, z) → z
if(false, x, y, z) → loop(x, double(y), s(z))
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)))
le(s(x), 0) → false
le(0, y) → true
le(s(x), s(y)) → le(x, y)
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)
maplog(xs) → mapIter(xs, nil)
mapIter(xs, ys) → ifmap(isempty(xs), xs, ys)

The TRS R 2 is

a → b
a → c

The signature Sigma is {a, b, c}

(2) Obligation:

Q restricted rewrite system:
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

The set Q consists of the following terms:

le(s(x0), 0)
le(0, x0)
le(s(x0), s(x1))
double(0)
double(s(x0))
log(0)
log(s(x0))
loop(x0, s(x1), x2)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
maplog(x0)
mapIter(x0, x1)
ifmap(true, x0, x1)
ifmap(false, x0, x1)
isempty(nil)
isempty(cons(x0, x1))
last(nil)
last(cons(x0, nil))
last(cons(x0, cons(x1, x2)))
droplast(nil)
droplast(cons(x0, nil))
droplast(cons(x0, cons(x1, x2)))
a

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LE(s(x), s(y)) → LE(x, y)
DOUBLE(s(x)) → DOUBLE(x)
LOG(s(x)) → LOOP(s(x), s(0), 0)
LOOP(x, s(y), z) → IF(le(x, s(y)), x, s(y), z)
LOOP(x, s(y), z) → LE(x, s(y))
IF(false, x, y, z) → LOOP(x, double(y), s(z))
IF(false, x, y, z) → DOUBLE(y)
MAPLOG(xs) → MAPITER(xs, nil)
MAPITER(xs, ys) → IFMAP(isempty(xs), xs, ys)
MAPITER(xs, ys) → ISEMPTY(xs)
IFMAP(false, xs, ys) → MAPITER(droplast(xs), cons(log(last(xs)), ys))
IFMAP(false, xs, ys) → DROPLAST(xs)
IFMAP(false, xs, ys) → LOG(last(xs))
IFMAP(false, xs, ys) → LAST(xs)
LAST(cons(x, cons(y, xs))) → LAST(cons(y, xs))
DROPLAST(cons(x, cons(y, xs))) → DROPLAST(cons(y, xs))

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

The set Q consists of the following terms:

le(s(x0), 0)
le(0, x0)
le(s(x0), s(x1))
double(0)
double(s(x0))
log(0)
log(s(x0))
loop(x0, s(x1), x2)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
maplog(x0)
mapIter(x0, x1)
ifmap(true, x0, x1)
ifmap(false, x0, x1)
isempty(nil)
isempty(cons(x0, x1))
last(nil)
last(cons(x0, nil))
last(cons(x0, cons(x1, x2)))
droplast(nil)
droplast(cons(x0, nil))
droplast(cons(x0, cons(x1, x2)))
a

We have to consider all minimal (P,Q,R)-chains.

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 6 SCCs with 8 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

Q DP problem:
The TRS P consists of the following rules:

DROPLAST(cons(x, cons(y, xs))) → DROPLAST(cons(y, xs))

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

The set Q consists of the following terms:

le(s(x0), 0)
le(0, x0)
le(s(x0), s(x1))
double(0)
double(s(x0))
log(0)
log(s(x0))
loop(x0, s(x1), x2)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
maplog(x0)
mapIter(x0, x1)
ifmap(true, x0, x1)
ifmap(false, x0, x1)
isempty(nil)
isempty(cons(x0, x1))
last(nil)
last(cons(x0, nil))
last(cons(x0, cons(x1, x2)))
droplast(nil)
droplast(cons(x0, nil))
droplast(cons(x0, cons(x1, x2)))
a

We have to consider all minimal (P,Q,R)-chains.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


DROPLAST(cons(x, cons(y, xs))) → DROPLAST(cons(y, xs))
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:
cons2 > DROPLAST1

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

Q DP problem:
P is empty.
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

The set Q consists of the following terms:

le(s(x0), 0)
le(0, x0)
le(s(x0), s(x1))
double(0)
double(s(x0))
log(0)
log(s(x0))
loop(x0, s(x1), x2)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
maplog(x0)
mapIter(x0, x1)
ifmap(true, x0, x1)
ifmap(false, x0, x1)
isempty(nil)
isempty(cons(x0, x1))
last(nil)
last(cons(x0, nil))
last(cons(x0, cons(x1, x2)))
droplast(nil)
droplast(cons(x0, nil))
droplast(cons(x0, cons(x1, x2)))
a

We have to consider all minimal (P,Q,R)-chains.

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LAST(cons(x, cons(y, xs))) → LAST(cons(y, xs))

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

The set Q consists of the following terms:

le(s(x0), 0)
le(0, x0)
le(s(x0), s(x1))
double(0)
double(s(x0))
log(0)
log(s(x0))
loop(x0, s(x1), x2)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
maplog(x0)
mapIter(x0, x1)
ifmap(true, x0, x1)
ifmap(false, x0, x1)
isempty(nil)
isempty(cons(x0, x1))
last(nil)
last(cons(x0, nil))
last(cons(x0, cons(x1, x2)))
droplast(nil)
droplast(cons(x0, nil))
droplast(cons(x0, cons(x1, x2)))
a

We have to consider all minimal (P,Q,R)-chains.

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


LAST(cons(x, cons(y, xs))) → LAST(cons(y, xs))
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:
cons2 > LAST1

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

Q DP problem:
P is empty.
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

The set Q consists of the following terms:

le(s(x0), 0)
le(0, x0)
le(s(x0), s(x1))
double(0)
double(s(x0))
log(0)
log(s(x0))
loop(x0, s(x1), x2)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
maplog(x0)
mapIter(x0, x1)
ifmap(true, x0, x1)
ifmap(false, x0, x1)
isempty(nil)
isempty(cons(x0, x1))
last(nil)
last(cons(x0, nil))
last(cons(x0, cons(x1, x2)))
droplast(nil)
droplast(cons(x0, nil))
droplast(cons(x0, cons(x1, x2)))
a

We have to consider all minimal (P,Q,R)-chains.

(15) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(16) TRUE

(17) Obligation:

Q DP problem:
The TRS P consists of the following rules:

DOUBLE(s(x)) → DOUBLE(x)

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

The set Q consists of the following terms:

le(s(x0), 0)
le(0, x0)
le(s(x0), s(x1))
double(0)
double(s(x0))
log(0)
log(s(x0))
loop(x0, s(x1), x2)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
maplog(x0)
mapIter(x0, x1)
ifmap(true, x0, x1)
ifmap(false, x0, x1)
isempty(nil)
isempty(cons(x0, x1))
last(nil)
last(cons(x0, nil))
last(cons(x0, cons(x1, x2)))
droplast(nil)
droplast(cons(x0, nil))
droplast(cons(x0, cons(x1, x2)))
a

We have to consider all minimal (P,Q,R)-chains.

(18) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


DOUBLE(s(x)) → DOUBLE(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive Path Order [RPO].
Precedence:
s1 > DOUBLE1

The following usable rules [FROCOS05] were oriented: none

(19) Obligation:

Q DP problem:
P is empty.
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

The set Q consists of the following terms:

le(s(x0), 0)
le(0, x0)
le(s(x0), s(x1))
double(0)
double(s(x0))
log(0)
log(s(x0))
loop(x0, s(x1), x2)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
maplog(x0)
mapIter(x0, x1)
ifmap(true, x0, x1)
ifmap(false, x0, x1)
isempty(nil)
isempty(cons(x0, x1))
last(nil)
last(cons(x0, nil))
last(cons(x0, cons(x1, x2)))
droplast(nil)
droplast(cons(x0, nil))
droplast(cons(x0, cons(x1, x2)))
a

We have to consider all minimal (P,Q,R)-chains.

(20) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(21) TRUE

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LE(s(x), s(y)) → LE(x, y)

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

The set Q consists of the following terms:

le(s(x0), 0)
le(0, x0)
le(s(x0), s(x1))
double(0)
double(s(x0))
log(0)
log(s(x0))
loop(x0, s(x1), x2)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
maplog(x0)
mapIter(x0, x1)
ifmap(true, x0, x1)
ifmap(false, x0, x1)
isempty(nil)
isempty(cons(x0, x1))
last(nil)
last(cons(x0, nil))
last(cons(x0, cons(x1, x2)))
droplast(nil)
droplast(cons(x0, nil))
droplast(cons(x0, cons(x1, x2)))
a

We have to consider all minimal (P,Q,R)-chains.

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


LE(s(x), s(y)) → LE(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LE(x1, x2)  =  LE(x2)
s(x1)  =  s(x1)

Recursive Path Order [RPO].
Precedence:
s1 > LE1

The following usable rules [FROCOS05] were oriented: none

(24) Obligation:

Q DP problem:
P is empty.
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

The set Q consists of the following terms:

le(s(x0), 0)
le(0, x0)
le(s(x0), s(x1))
double(0)
double(s(x0))
log(0)
log(s(x0))
loop(x0, s(x1), x2)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
maplog(x0)
mapIter(x0, x1)
ifmap(true, x0, x1)
ifmap(false, x0, x1)
isempty(nil)
isempty(cons(x0, x1))
last(nil)
last(cons(x0, nil))
last(cons(x0, cons(x1, x2)))
droplast(nil)
droplast(cons(x0, nil))
droplast(cons(x0, cons(x1, x2)))
a

We have to consider all minimal (P,Q,R)-chains.

(25) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(26) TRUE

(27) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LOOP(x, s(y), z) → IF(le(x, s(y)), x, s(y), z)
IF(false, x, y, z) → LOOP(x, double(y), s(z))

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

The set Q consists of the following terms:

le(s(x0), 0)
le(0, x0)
le(s(x0), s(x1))
double(0)
double(s(x0))
log(0)
log(s(x0))
loop(x0, s(x1), x2)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
maplog(x0)
mapIter(x0, x1)
ifmap(true, x0, x1)
ifmap(false, x0, x1)
isempty(nil)
isempty(cons(x0, x1))
last(nil)
last(cons(x0, nil))
last(cons(x0, cons(x1, x2)))
droplast(nil)
droplast(cons(x0, nil))
droplast(cons(x0, cons(x1, x2)))
a

We have to consider all minimal (P,Q,R)-chains.

(28) Obligation:

Q DP problem:
The TRS P consists of the following rules:

IFMAP(false, xs, ys) → MAPITER(droplast(xs), cons(log(last(xs)), ys))
MAPITER(xs, ys) → IFMAP(isempty(xs), xs, ys)

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

The set Q consists of the following terms:

le(s(x0), 0)
le(0, x0)
le(s(x0), s(x1))
double(0)
double(s(x0))
log(0)
log(s(x0))
loop(x0, s(x1), x2)
if(true, x0, x1, x2)
if(false, x0, x1, x2)
maplog(x0)
mapIter(x0, x1)
ifmap(true, x0, x1)
ifmap(false, x0, x1)
isempty(nil)
isempty(cons(x0, x1))
last(nil)
last(cons(x0, nil))
last(cons(x0, cons(x1, x2)))
droplast(nil)
droplast(cons(x0, nil))
droplast(cons(x0, cons(x1, x2)))
a

We have to consider all minimal (P,Q,R)-chains.