Term Rewriting System R:
[X, Y, Z, X1, X2]
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVE(dbl(s(X))) -> S(s(dbl(X)))
ACTIVE(dbl(s(X))) -> S(dbl(X))
ACTIVE(dbl(s(X))) -> DBL(X)
ACTIVE(dbls(cons(X, Y))) -> CONS(dbl(X), dbls(Y))
ACTIVE(dbls(cons(X, Y))) -> DBL(X)
ACTIVE(dbls(cons(X, Y))) -> DBLS(Y)
ACTIVE(sel(s(X), cons(Y, Z))) -> SEL(X, Z)
ACTIVE(indx(cons(X, Y), Z)) -> CONS(sel(X, Z), indx(Y, Z))
ACTIVE(indx(cons(X, Y), Z)) -> SEL(X, Z)
ACTIVE(indx(cons(X, Y), Z)) -> INDX(Y, Z)
ACTIVE(from(X)) -> CONS(X, from(s(X)))
ACTIVE(from(X)) -> FROM(s(X))
ACTIVE(from(X)) -> S(X)
ACTIVE(dbl(X)) -> DBL(active(X))
ACTIVE(dbl(X)) -> ACTIVE(X)
ACTIVE(dbls(X)) -> DBLS(active(X))
ACTIVE(dbls(X)) -> ACTIVE(X)
ACTIVE(sel(X1, X2)) -> SEL(active(X1), X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel(X1, X2)) -> SEL(X1, active(X2))
ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(indx(X1, X2)) -> INDX(active(X1), X2)
ACTIVE(indx(X1, X2)) -> ACTIVE(X1)
DBL(mark(X)) -> DBL(X)
DBL(ok(X)) -> DBL(X)
DBLS(mark(X)) -> DBLS(X)
DBLS(ok(X)) -> DBLS(X)
SEL(mark(X1), X2) -> SEL(X1, X2)
SEL(X1, mark(X2)) -> SEL(X1, X2)
SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
INDX(mark(X1), X2) -> INDX(X1, X2)
INDX(ok(X1), ok(X2)) -> INDX(X1, X2)
PROPER(dbl(X)) -> DBL(proper(X))
PROPER(dbl(X)) -> PROPER(X)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(dbls(X)) -> DBLS(proper(X))
PROPER(dbls(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> SEL(proper(X1), proper(X2))
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(indx(X1, X2)) -> INDX(proper(X1), proper(X2))
PROPER(indx(X1, X2)) -> PROPER(X1)
PROPER(indx(X1, X2)) -> PROPER(X2)
PROPER(from(X)) -> FROM(proper(X))
PROPER(from(X)) -> PROPER(X)
S(ok(X)) -> S(X)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
FROM(ok(X)) -> FROM(X)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)

Furthermore, R contains 10 SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Size-Change Principle
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar


Dependency Pair:

S(ok(X)) -> S(X)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. S(ok(X)) -> S(X)
and get the following Size-Change Graph(s):
{1} , {1}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳Size-Change Principle
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar


Dependency Pairs:

DBL(ok(X)) -> DBL(X)
DBL(mark(X)) -> DBL(X)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. DBL(ok(X)) -> DBL(X)
  2. DBL(mark(X)) -> DBL(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳Size-Change Principle
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar


Dependency Pair:

CONS(ok(X1), ok(X2)) -> CONS(X1, X2)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2>2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳Size-Change Principle
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar


Dependency Pairs:

DBLS(ok(X)) -> DBLS(X)
DBLS(mark(X)) -> DBLS(X)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. DBLS(ok(X)) -> DBLS(X)
  2. DBLS(mark(X)) -> DBLS(X)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳Size-Change Principle
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar


Dependency Pairs:

SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
SEL(X1, mark(X2)) -> SEL(X1, X2)
SEL(mark(X1), X2) -> SEL(X1, X2)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
  2. SEL(X1, mark(X2)) -> SEL(X1, X2)
  3. SEL(mark(X1), X2) -> SEL(X1, X2)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2=2
{3, 2, 1} , {3, 2, 1}
1=1
2>2
{3, 2, 1} , {3, 2, 1}
1>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳Size-Change Principle
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar


Dependency Pairs:

INDX(ok(X1), ok(X2)) -> INDX(X1, X2)
INDX(mark(X1), X2) -> INDX(X1, X2)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. INDX(ok(X1), ok(X2)) -> INDX(X1, X2)
  2. INDX(mark(X1), X2) -> INDX(X1, X2)
and get the following Size-Change Graph(s):
{2, 1} , {2, 1}
1>1
2>2
{2, 1} , {2, 1}
1>1
2=2

which lead(s) to this/these maximal multigraph(s):
{2, 1} , {2, 1}
1>1
2>2
{2, 1} , {2, 1}
1>1
2=2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳Size-Change Principle
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar


Dependency Pair:

FROM(ok(X)) -> FROM(X)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. FROM(ok(X)) -> FROM(X)
and get the following Size-Change Graph(s):
{1} , {1}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
ok(x1) -> ok(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳Size-Change Principle
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar


Dependency Pairs:

ACTIVE(indx(X1, X2)) -> ACTIVE(X1)
ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(dbls(X)) -> ACTIVE(X)
ACTIVE(dbl(X)) -> ACTIVE(X)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. ACTIVE(indx(X1, X2)) -> ACTIVE(X1)
  2. ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
  3. ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
  4. ACTIVE(dbls(X)) -> ACTIVE(X)
  5. ACTIVE(dbl(X)) -> ACTIVE(X)
and get the following Size-Change Graph(s):
{5, 4, 3, 2, 1} , {5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{5, 4, 3, 2, 1} , {5, 4, 3, 2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
dbls(x1) -> dbls(x1)
indx(x1, x2) -> indx(x1, x2)
dbl(x1) -> dbl(x1)
sel(x1, x2) -> sel(x1, x2)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Size-Change Principle
       →DP Problem 10
         ↳Nar


Dependency Pairs:

PROPER(from(X)) -> PROPER(X)
PROPER(indx(X1, X2)) -> PROPER(X2)
PROPER(indx(X1, X2)) -> PROPER(X1)
PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(dbls(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(dbl(X)) -> PROPER(X)


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. PROPER(from(X)) -> PROPER(X)
  2. PROPER(indx(X1, X2)) -> PROPER(X2)
  3. PROPER(indx(X1, X2)) -> PROPER(X1)
  4. PROPER(sel(X1, X2)) -> PROPER(X2)
  5. PROPER(sel(X1, X2)) -> PROPER(X1)
  6. PROPER(cons(X1, X2)) -> PROPER(X2)
  7. PROPER(cons(X1, X2)) -> PROPER(X1)
  8. PROPER(dbls(X)) -> PROPER(X)
  9. PROPER(s(X)) -> PROPER(X)
  10. PROPER(dbl(X)) -> PROPER(X)
and get the following Size-Change Graph(s):
{10, 9, 8, 7, 6, 5, 4, 3, 2, 1} , {10, 9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{10, 9, 8, 7, 6, 5, 4, 3, 2, 1} , {10, 9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
from(x1) -> from(x1)
dbls(x1) -> dbls(x1)
indx(x1, x2) -> indx(x1, x2)
cons(x1, x2) -> cons(x1, x2)
dbl(x1) -> dbl(x1)
sel(x1, x2) -> sel(x1, x2)
s(x1) -> s(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Narrowing Transformation


Dependency Pairs:

TOP(ok(X)) -> TOP(active(X))
TOP(mark(X)) -> TOP(proper(X))


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

TOP(mark(X)) -> TOP(proper(X))
nine new Dependency Pairs are created:

TOP(mark(dbl(X''))) -> TOP(dbl(proper(X'')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(dbls(X''))) -> TOP(dbls(proper(X'')))
TOP(mark(nil)) -> TOP(ok(nil))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(X2')))
TOP(mark(indx(X1', X2'))) -> TOP(indx(proper(X1'), proper(X2')))
TOP(mark(from(X''))) -> TOP(from(proper(X'')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar
           →DP Problem 11
             ↳Narrowing Transformation


Dependency Pairs:

TOP(mark(from(X''))) -> TOP(from(proper(X'')))
TOP(mark(indx(X1', X2'))) -> TOP(indx(proper(X1'), proper(X2')))
TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(X2')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(nil)) -> TOP(ok(nil))
TOP(mark(dbls(X''))) -> TOP(dbls(proper(X'')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(dbl(X''))) -> TOP(dbl(proper(X'')))
TOP(ok(X)) -> TOP(active(X))


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

TOP(ok(X)) -> TOP(active(X))
14 new Dependency Pairs are created:

TOP(ok(dbl(0))) -> TOP(mark(0))
TOP(ok(dbl(s(X'')))) -> TOP(mark(s(s(dbl(X'')))))
TOP(ok(dbls(nil))) -> TOP(mark(nil))
TOP(ok(dbls(cons(X'', Y')))) -> TOP(mark(cons(dbl(X''), dbls(Y'))))
TOP(ok(sel(0, cons(X'', Y')))) -> TOP(mark(X''))
TOP(ok(sel(s(X''), cons(Y', Z')))) -> TOP(mark(sel(X'', Z')))
TOP(ok(indx(nil, X''))) -> TOP(mark(nil))
TOP(ok(indx(cons(X'', Y'), Z'))) -> TOP(mark(cons(sel(X'', Z'), indx(Y', Z'))))
TOP(ok(from(X''))) -> TOP(mark(cons(X'', from(s(X'')))))
TOP(ok(dbl(X''))) -> TOP(dbl(active(X'')))
TOP(ok(dbls(X''))) -> TOP(dbls(active(X'')))
TOP(ok(sel(X1', X2'))) -> TOP(sel(active(X1'), X2'))
TOP(ok(sel(X1', X2'))) -> TOP(sel(X1', active(X2')))
TOP(ok(indx(X1', X2'))) -> TOP(indx(active(X1'), X2'))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar
           →DP Problem 11
             ↳Nar
             ...
               →DP Problem 12
                 ↳Negative Polynomial Order


Dependency Pairs:

TOP(ok(indx(X1', X2'))) -> TOP(indx(active(X1'), X2'))
TOP(ok(sel(X1', X2'))) -> TOP(sel(X1', active(X2')))
TOP(ok(sel(X1', X2'))) -> TOP(sel(active(X1'), X2'))
TOP(ok(dbls(X''))) -> TOP(dbls(active(X'')))
TOP(ok(dbl(X''))) -> TOP(dbl(active(X'')))
TOP(ok(from(X''))) -> TOP(mark(cons(X'', from(s(X'')))))
TOP(ok(indx(cons(X'', Y'), Z'))) -> TOP(mark(cons(sel(X'', Z'), indx(Y', Z'))))
TOP(ok(sel(s(X''), cons(Y', Z')))) -> TOP(mark(sel(X'', Z')))
TOP(ok(sel(0, cons(X'', Y')))) -> TOP(mark(X''))
TOP(ok(dbls(cons(X'', Y')))) -> TOP(mark(cons(dbl(X''), dbls(Y'))))
TOP(ok(dbl(s(X'')))) -> TOP(mark(s(s(dbl(X'')))))
TOP(mark(indx(X1', X2'))) -> TOP(indx(proper(X1'), proper(X2')))
TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(X2')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(dbls(X''))) -> TOP(dbls(proper(X'')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(dbl(X''))) -> TOP(dbl(proper(X'')))
TOP(mark(from(X''))) -> TOP(from(proper(X'')))


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





The following Dependency Pairs can be strictly oriented using the given order.

TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))


Moreover, the following usable rules (regarding the implicit AFS) are oriented.

indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
from(ok(X)) -> ok(from(X))
active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))


Used ordering:
Polynomial Order with Interpretation:

POL( TOP(x1) ) = x1

POL( mark(x1) ) = 1

POL( cons(x1, x2) ) = 0

POL( ok(x1) ) = x1

POL( indx(x1, x2) ) = 1

POL( from(x1) ) = 1

POL( sel(x1, x2) ) = 1

POL( dbls(x1) ) = 1

POL( dbl(x1) ) = 1

POL( s(x1) ) = 0

POL( active(x1) ) = 1

POL( proper(x1) ) = 1

POL( 0 ) = 0

POL( nil ) = 0


This results in one new DP problem. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳SCP
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳SCP
       →DP Problem 10
         ↳Nar
           →DP Problem 11
             ↳Nar
             ...
               →DP Problem 13
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

TOP(ok(indx(X1', X2'))) -> TOP(indx(active(X1'), X2'))
TOP(ok(sel(X1', X2'))) -> TOP(sel(X1', active(X2')))
TOP(ok(sel(X1', X2'))) -> TOP(sel(active(X1'), X2'))
TOP(ok(dbls(X''))) -> TOP(dbls(active(X'')))
TOP(ok(dbl(X''))) -> TOP(dbl(active(X'')))
TOP(ok(from(X''))) -> TOP(mark(cons(X'', from(s(X'')))))
TOP(ok(indx(cons(X'', Y'), Z'))) -> TOP(mark(cons(sel(X'', Z'), indx(Y', Z'))))
TOP(ok(sel(s(X''), cons(Y', Z')))) -> TOP(mark(sel(X'', Z')))
TOP(ok(sel(0, cons(X'', Y')))) -> TOP(mark(X''))
TOP(ok(dbls(cons(X'', Y')))) -> TOP(mark(cons(dbl(X''), dbls(Y'))))
TOP(ok(dbl(s(X'')))) -> TOP(mark(s(s(dbl(X'')))))
TOP(mark(indx(X1', X2'))) -> TOP(indx(proper(X1'), proper(X2')))
TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(X2')))
TOP(mark(dbls(X''))) -> TOP(dbls(proper(X'')))
TOP(mark(dbl(X''))) -> TOP(dbl(proper(X'')))
TOP(mark(from(X''))) -> TOP(from(proper(X'')))


Rules:


active(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))
active(dbl(X)) -> dbl(active(X))
active(dbls(X)) -> dbls(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(indx(X1, X2)) -> indx(active(X1), X2)
dbl(mark(X)) -> mark(dbl(X))
dbl(ok(X)) -> ok(dbl(X))
dbls(mark(X)) -> mark(dbls(X))
dbls(ok(X)) -> ok(dbls(X))
sel(mark(X1), X2) -> mark(sel(X1, X2))
sel(X1, mark(X2)) -> mark(sel(X1, X2))
sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
indx(mark(X1), X2) -> mark(indx(X1, X2))
indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
proper(dbl(X)) -> dbl(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(dbls(X)) -> dbls(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
proper(from(X)) -> from(proper(X))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
from(ok(X)) -> ok(from(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))




The Proof could not be continued due to a Timeout.
Termination of R could not be shown.
Duration:
1:00 minutes