Term Rewriting System R:
[X, Y, Z, X1, X2]
active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(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))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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(f(X)) -> CONS(X, f(g(X)))
ACTIVE(f(X)) -> F(g(X))
ACTIVE(f(X)) -> G(X)
ACTIVE(g(0)) -> S(0)
ACTIVE(g(s(X))) -> S(s(g(X)))
ACTIVE(g(s(X))) -> S(g(X))
ACTIVE(g(s(X))) -> G(X)
ACTIVE(sel(s(X), cons(Y, Z))) -> SEL(X, Z)
ACTIVE(f(X)) -> F(active(X))
ACTIVE(f(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(g(X)) -> G(active(X))
ACTIVE(g(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(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)
F(mark(X)) -> F(X)
F(ok(X)) -> F(X)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
G(mark(X)) -> G(X)
G(ok(X)) -> G(X)
S(mark(X)) -> S(X)
S(ok(X)) -> S(X)
SEL(mark(X1), X2) -> SEL(X1, X2)
SEL(X1, mark(X2)) -> SEL(X1, X2)
SEL(ok(X1), ok(X2)) -> SEL(X1, X2)
PROPER(f(X)) -> F(proper(X))
PROPER(f(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(g(X)) -> G(proper(X))
PROPER(g(X)) -> PROPER(X)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(sel(X1, X2)) -> SEL(proper(X1), proper(X2))
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(sel(X1, X2)) -> PROPER(X2)
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 eight 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
         ↳Nar


Dependency Pairs:

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


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(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))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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)
  2. CONS(mark(X1), X2) -> CONS(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
         ↳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
         ↳Nar


Dependency Pairs:

F(ok(X)) -> F(X)
F(mark(X)) -> F(X)


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(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))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. F(ok(X)) -> F(X)
  2. F(mark(X)) -> F(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
         ↳Nar


Dependency Pairs:

G(ok(X)) -> G(X)
G(mark(X)) -> G(X)


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(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))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. G(ok(X)) -> G(X)
  2. G(mark(X)) -> G(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
         ↳Size-Change Principle
       →DP Problem 5
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳Nar


Dependency Pairs:

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


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(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))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. S(ok(X)) -> S(X)
  2. S(mark(X)) -> S(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
         ↳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(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(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))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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
         ↳Nar


Dependency Pairs:

ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(g(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(f(X)) -> ACTIVE(X)


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(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))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. ACTIVE(sel(X1, X2)) -> ACTIVE(X2)
  2. ACTIVE(sel(X1, X2)) -> ACTIVE(X1)
  3. ACTIVE(s(X)) -> ACTIVE(X)
  4. ACTIVE(g(X)) -> ACTIVE(X)
  5. ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
  6. ACTIVE(f(X)) -> ACTIVE(X)
and get the following Size-Change Graph(s):
{6, 5, 4, 3, 2, 1} , {6, 5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{6, 5, 4, 3, 2, 1} , {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:
g(x1) -> g(x1)
cons(x1, x2) -> cons(x1, x2)
s(x1) -> s(x1)
sel(x1, x2) -> sel(x1, x2)
f(x1) -> f(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
         ↳Nar


Dependency Pairs:

PROPER(sel(X1, X2)) -> PROPER(X2)
PROPER(sel(X1, X2)) -> PROPER(X1)
PROPER(s(X)) -> PROPER(X)
PROPER(g(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(f(X)) -> PROPER(X)


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(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))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. PROPER(sel(X1, X2)) -> PROPER(X2)
  2. PROPER(sel(X1, X2)) -> PROPER(X1)
  3. PROPER(s(X)) -> PROPER(X)
  4. PROPER(g(X)) -> PROPER(X)
  5. PROPER(cons(X1, X2)) -> PROPER(X2)
  6. PROPER(cons(X1, X2)) -> PROPER(X1)
  7. PROPER(f(X)) -> PROPER(X)
and get the following Size-Change Graph(s):
{7, 6, 5, 4, 3, 2, 1} , {7, 6, 5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{7, 6, 5, 4, 3, 2, 1} , {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:
g(x1) -> g(x1)
cons(x1, x2) -> cons(x1, x2)
sel(x1, x2) -> sel(x1, x2)
s(x1) -> s(x1)
f(x1) -> f(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
         ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(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))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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))
six new Dependency Pairs are created:

TOP(mark(f(X''))) -> TOP(f(proper(X'')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(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
         ↳Nar
           →DP Problem 9
             ↳Narrowing Transformation


Dependency Pairs:

TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(X2')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(f(X''))) -> TOP(f(proper(X'')))
TOP(ok(X)) -> TOP(active(X))


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(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))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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))
11 new Dependency Pairs are created:

TOP(ok(f(X''))) -> TOP(mark(cons(X'', f(g(X'')))))
TOP(ok(g(0))) -> TOP(mark(s(0)))
TOP(ok(g(s(X'')))) -> TOP(mark(s(s(g(X'')))))
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(f(X''))) -> TOP(f(active(X'')))
TOP(ok(cons(X1', X2'))) -> TOP(cons(active(X1'), X2'))
TOP(ok(g(X''))) -> TOP(g(active(X'')))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(sel(X1', X2'))) -> TOP(sel(active(X1'), X2'))
TOP(ok(sel(X1', X2'))) -> TOP(sel(X1', active(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
         ↳Nar
           →DP Problem 9
             ↳Nar
             ...
               →DP Problem 10
                 ↳Negative Polynomial Order


Dependency Pairs:

TOP(ok(sel(X1', X2'))) -> TOP(sel(X1', active(X2')))
TOP(ok(sel(X1', X2'))) -> TOP(sel(active(X1'), X2'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(g(X''))) -> TOP(g(active(X'')))
TOP(ok(cons(X1', X2'))) -> TOP(cons(active(X1'), X2'))
TOP(ok(f(X''))) -> TOP(f(active(X'')))
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(g(s(X'')))) -> TOP(mark(s(s(g(X'')))))
TOP(ok(g(0))) -> TOP(mark(s(0)))
TOP(ok(f(X''))) -> TOP(mark(cons(X'', f(g(X'')))))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(f(X''))) -> TOP(f(proper(X'')))
TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(X2')))


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(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))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





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

TOP(ok(sel(0, cons(X'', Y')))) -> TOP(mark(X''))


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

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))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))


Used ordering:
Polynomial Order with Interpretation:

POL( TOP(x1) ) = x1

POL( ok(x1) ) = x1

POL( sel(x1, x2) ) = x2 + 1

POL( cons(x1, x2) ) = x1 + x2

POL( mark(x1) ) = x1

POL( g(x1) ) = 0

POL( s(x1) ) = 0

POL( proper(x1) ) = x1

POL( f(x1) ) = x1

POL( active(x1) ) = x1

POL( 0 ) = 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
         ↳Nar
           →DP Problem 9
             ↳Nar
             ...
               →DP Problem 11
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

TOP(ok(sel(X1', X2'))) -> TOP(sel(X1', active(X2')))
TOP(ok(sel(X1', X2'))) -> TOP(sel(active(X1'), X2'))
TOP(ok(s(X''))) -> TOP(s(active(X'')))
TOP(ok(g(X''))) -> TOP(g(active(X'')))
TOP(ok(cons(X1', X2'))) -> TOP(cons(active(X1'), X2'))
TOP(ok(f(X''))) -> TOP(f(active(X'')))
TOP(ok(sel(s(X''), cons(Y', Z')))) -> TOP(mark(sel(X'', Z')))
TOP(ok(g(s(X'')))) -> TOP(mark(s(s(g(X'')))))
TOP(ok(g(0))) -> TOP(mark(s(0)))
TOP(ok(f(X''))) -> TOP(mark(cons(X'', f(g(X'')))))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(g(X''))) -> TOP(g(proper(X'')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(f(X''))) -> TOP(f(proper(X'')))
TOP(mark(sel(X1', X2'))) -> TOP(sel(proper(X1'), proper(X2')))


Rules:


active(f(X)) -> mark(cons(X, f(g(X))))
active(g(0)) -> mark(s(0))
active(g(s(X))) -> mark(s(s(g(X))))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(f(X)) -> f(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(g(X)) -> g(active(X))
active(s(X)) -> s(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(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))
proper(f(X)) -> f(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
proper(0) -> ok(0)
proper(s(X)) -> s(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
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