Term Rewriting System R:
[X, Y, L, X1, X2]
active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, 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(eq(s(X), s(Y))) -> EQ(X, Y)
ACTIVE(inf(X)) -> CONS(X, inf(s(X)))
ACTIVE(inf(X)) -> INF(s(X))
ACTIVE(inf(X)) -> S(X)
ACTIVE(take(s(X), cons(Y, L))) -> CONS(Y, take(X, L))
ACTIVE(take(s(X), cons(Y, L))) -> TAKE(X, L)
ACTIVE(length(cons(X, L))) -> S(length(L))
ACTIVE(length(cons(X, L))) -> LENGTH(L)
ACTIVE(inf(X)) -> INF(active(X))
ACTIVE(inf(X)) -> ACTIVE(X)
ACTIVE(take(X1, X2)) -> TAKE(active(X1), X2)
ACTIVE(take(X1, X2)) -> ACTIVE(X1)
ACTIVE(take(X1, X2)) -> TAKE(X1, active(X2))
ACTIVE(take(X1, X2)) -> ACTIVE(X2)
ACTIVE(length(X)) -> LENGTH(active(X))
ACTIVE(length(X)) -> ACTIVE(X)
INF(mark(X)) -> INF(X)
INF(ok(X)) -> INF(X)
TAKE(mark(X1), X2) -> TAKE(X1, X2)
TAKE(X1, mark(X2)) -> TAKE(X1, X2)
TAKE(ok(X1), ok(X2)) -> TAKE(X1, X2)
LENGTH(mark(X)) -> LENGTH(X)
LENGTH(ok(X)) -> LENGTH(X)
PROPER(eq(X1, X2)) -> EQ(proper(X1), proper(X2))
PROPER(eq(X1, X2)) -> PROPER(X1)
PROPER(eq(X1, X2)) -> PROPER(X2)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
PROPER(inf(X)) -> INF(proper(X))
PROPER(inf(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(take(X1, X2)) -> TAKE(proper(X1), proper(X2))
PROPER(take(X1, X2)) -> PROPER(X1)
PROPER(take(X1, X2)) -> PROPER(X2)
PROPER(length(X)) -> LENGTH(proper(X))
PROPER(length(X)) -> PROPER(X)
EQ(ok(X1), ok(X2)) -> EQ(X1, X2)
S(ok(X)) -> S(X)
CONS(ok(X1), ok(X2)) -> CONS(X1, 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 nine 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
         ↳Nar


Dependency Pair:

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


Rules:


active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. EQ(ok(X1), ok(X2)) -> EQ(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
         ↳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
         ↳Nar


Dependency Pair:

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


Rules:


active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, 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)
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
         ↳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
         ↳Nar


Dependency Pairs:

INF(ok(X)) -> INF(X)
INF(mark(X)) -> INF(X)


Rules:


active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. INF(ok(X)) -> INF(X)
  2. INF(mark(X)) -> INF(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
         ↳SCP
       →DP Problem 9
         ↳Nar


Dependency Pair:

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


Rules:


active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, 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)
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
         ↳Size-Change Principle
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳SCP
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Nar


Dependency Pairs:

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


Rules:


active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. TAKE(ok(X1), ok(X2)) -> TAKE(X1, X2)
  2. TAKE(X1, mark(X2)) -> TAKE(X1, X2)
  3. TAKE(mark(X1), X2) -> TAKE(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
         ↳Nar


Dependency Pairs:

LENGTH(ok(X)) -> LENGTH(X)
LENGTH(mark(X)) -> LENGTH(X)


Rules:


active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. LENGTH(ok(X)) -> LENGTH(X)
  2. LENGTH(mark(X)) -> LENGTH(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
         ↳SCP
       →DP Problem 6
         ↳SCP
       →DP Problem 7
         ↳Size-Change Principle
       →DP Problem 8
         ↳SCP
       →DP Problem 9
         ↳Nar


Dependency Pairs:

ACTIVE(length(X)) -> ACTIVE(X)
ACTIVE(take(X1, X2)) -> ACTIVE(X2)
ACTIVE(take(X1, X2)) -> ACTIVE(X1)
ACTIVE(inf(X)) -> ACTIVE(X)


Rules:


active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. ACTIVE(length(X)) -> ACTIVE(X)
  2. ACTIVE(take(X1, X2)) -> ACTIVE(X2)
  3. ACTIVE(take(X1, X2)) -> ACTIVE(X1)
  4. ACTIVE(inf(X)) -> ACTIVE(X)
and get the following Size-Change Graph(s):
{4, 3, 2, 1} , {4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{4, 3, 2, 1} , {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:
take(x1, x2) -> take(x1, x2)
inf(x1) -> inf(x1)
length(x1) -> length(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
         ↳Nar


Dependency Pairs:

PROPER(length(X)) -> PROPER(X)
PROPER(take(X1, X2)) -> PROPER(X2)
PROPER(take(X1, X2)) -> PROPER(X1)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(inf(X)) -> PROPER(X)
PROPER(s(X)) -> PROPER(X)
PROPER(eq(X1, X2)) -> PROPER(X2)
PROPER(eq(X1, X2)) -> PROPER(X1)


Rules:


active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. PROPER(length(X)) -> PROPER(X)
  2. PROPER(take(X1, X2)) -> PROPER(X2)
  3. PROPER(take(X1, X2)) -> PROPER(X1)
  4. PROPER(cons(X1, X2)) -> PROPER(X2)
  5. PROPER(cons(X1, X2)) -> PROPER(X1)
  6. PROPER(inf(X)) -> PROPER(X)
  7. PROPER(s(X)) -> PROPER(X)
  8. PROPER(eq(X1, X2)) -> PROPER(X2)
  9. PROPER(eq(X1, X2)) -> PROPER(X1)
and get the following Size-Change Graph(s):
{9, 8, 7, 6, 5, 4, 3, 2, 1} , {9, 8, 7, 6, 5, 4, 3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{9, 8, 7, 6, 5, 4, 3, 2, 1} , {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:
eq(x1, x2) -> eq(x1, x2)
cons(x1, x2) -> cons(x1, x2)
take(x1, x2) -> take(x1, x2)
inf(x1) -> inf(x1)
s(x1) -> s(x1)
length(x1) -> length(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
         ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, 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))
10 new Dependency Pairs are created:

TOP(mark(eq(X1', X2'))) -> TOP(eq(proper(X1'), proper(X2')))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(true)) -> TOP(ok(true))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(false)) -> TOP(ok(false))
TOP(mark(inf(X''))) -> TOP(inf(proper(X'')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(take(X1', X2'))) -> TOP(take(proper(X1'), proper(X2')))
TOP(mark(nil)) -> TOP(ok(nil))
TOP(mark(length(X''))) -> TOP(length(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
         ↳Nar
           →DP Problem 10
             ↳Narrowing Transformation


Dependency Pairs:

TOP(mark(length(X''))) -> TOP(length(proper(X'')))
TOP(mark(nil)) -> TOP(ok(nil))
TOP(mark(take(X1', X2'))) -> TOP(take(proper(X1'), proper(X2')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(inf(X''))) -> TOP(inf(proper(X'')))
TOP(mark(false)) -> TOP(ok(false))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(true)) -> TOP(ok(true))
TOP(mark(0)) -> TOP(ok(0))
TOP(mark(eq(X1', X2'))) -> TOP(eq(proper(X1'), proper(X2')))
TOP(ok(X)) -> TOP(active(X))


Rules:


active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, 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))
12 new Dependency Pairs are created:

TOP(ok(eq(0, 0))) -> TOP(mark(true))
TOP(ok(eq(s(X''), s(Y')))) -> TOP(mark(eq(X'', Y')))
TOP(ok(eq(X'', Y'))) -> TOP(mark(false))
TOP(ok(inf(X''))) -> TOP(mark(cons(X'', inf(s(X'')))))
TOP(ok(take(0, X''))) -> TOP(mark(nil))
TOP(ok(take(s(X''), cons(Y', L')))) -> TOP(mark(cons(Y', take(X'', L'))))
TOP(ok(length(nil))) -> TOP(mark(0))
TOP(ok(length(cons(X'', L')))) -> TOP(mark(s(length(L'))))
TOP(ok(inf(X''))) -> TOP(inf(active(X'')))
TOP(ok(take(X1', X2'))) -> TOP(take(active(X1'), X2'))
TOP(ok(take(X1', X2'))) -> TOP(take(X1', active(X2')))
TOP(ok(length(X''))) -> TOP(length(active(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
         ↳Nar
           →DP Problem 10
             ↳Nar
             ...
               →DP Problem 11
                 ↳Negative Polynomial Order


Dependency Pairs:

TOP(ok(length(X''))) -> TOP(length(active(X'')))
TOP(ok(take(X1', X2'))) -> TOP(take(X1', active(X2')))
TOP(ok(take(X1', X2'))) -> TOP(take(active(X1'), X2'))
TOP(ok(inf(X''))) -> TOP(inf(active(X'')))
TOP(ok(length(cons(X'', L')))) -> TOP(mark(s(length(L'))))
TOP(ok(take(s(X''), cons(Y', L')))) -> TOP(mark(cons(Y', take(X'', L'))))
TOP(ok(inf(X''))) -> TOP(mark(cons(X'', inf(s(X'')))))
TOP(ok(eq(s(X''), s(Y')))) -> TOP(mark(eq(X'', Y')))
TOP(mark(take(X1', X2'))) -> TOP(take(proper(X1'), proper(X2')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(inf(X''))) -> TOP(inf(proper(X'')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(eq(X1', X2'))) -> TOP(eq(proper(X1'), proper(X2')))
TOP(mark(length(X''))) -> TOP(length(proper(X'')))


Rules:


active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, 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(length(cons(X'', L')))) -> TOP(mark(s(length(L'))))


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

length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))


Used ordering:
Polynomial Order with Interpretation:

POL( TOP(x1) ) = x1

POL( ok(x1) ) = x1

POL( length(x1) ) = 1

POL( mark(x1) ) = x1

POL( s(x1) ) = 0

POL( take(x1, x2) ) = 0

POL( cons(x1, x2) ) = 0

POL( inf(x1) ) = 0

POL( eq(x1, x2) ) = 0

POL( active(x1) ) = 1

POL( true ) = 0

POL( false ) = 0

POL( nil ) = 0

POL( 0 ) = 0

POL( proper(x1) ) = 1


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
         ↳Nar
           →DP Problem 10
             ↳Nar
             ...
               →DP Problem 12
                 ↳Negative Polynomial Order


Dependency Pairs:

TOP(ok(length(X''))) -> TOP(length(active(X'')))
TOP(ok(take(X1', X2'))) -> TOP(take(X1', active(X2')))
TOP(ok(take(X1', X2'))) -> TOP(take(active(X1'), X2'))
TOP(ok(inf(X''))) -> TOP(inf(active(X'')))
TOP(ok(take(s(X''), cons(Y', L')))) -> TOP(mark(cons(Y', take(X'', L'))))
TOP(ok(inf(X''))) -> TOP(mark(cons(X'', inf(s(X'')))))
TOP(ok(eq(s(X''), s(Y')))) -> TOP(mark(eq(X'', Y')))
TOP(mark(take(X1', X2'))) -> TOP(take(proper(X1'), proper(X2')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(inf(X''))) -> TOP(inf(proper(X'')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(eq(X1', X2'))) -> TOP(eq(proper(X1'), proper(X2')))
TOP(mark(length(X''))) -> TOP(length(proper(X'')))


Rules:


active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, 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(take(s(X''), cons(Y', L')))) -> TOP(mark(cons(Y', take(X'', L'))))


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

length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))


Used ordering:
Polynomial Order with Interpretation:

POL( TOP(x1) ) = x1

POL( ok(x1) ) = x1

POL( take(x1, x2) ) = 1

POL( mark(x1) ) = x1

POL( cons(x1, x2) ) = 0

POL( inf(x1) ) = 0

POL( s(x1) ) = 0

POL( length(x1) ) = 0

POL( eq(x1, x2) ) = 0

POL( active(x1) ) = 1

POL( true ) = 0

POL( false ) = 0

POL( nil ) = 0

POL( 0 ) = 0

POL( proper(x1) ) = 1


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
         ↳Nar
           →DP Problem 10
             ↳Nar
             ...
               →DP Problem 13
                 ↳Negative Polynomial Order


Dependency Pairs:

TOP(ok(length(X''))) -> TOP(length(active(X'')))
TOP(ok(take(X1', X2'))) -> TOP(take(X1', active(X2')))
TOP(ok(take(X1', X2'))) -> TOP(take(active(X1'), X2'))
TOP(ok(inf(X''))) -> TOP(inf(active(X'')))
TOP(ok(inf(X''))) -> TOP(mark(cons(X'', inf(s(X'')))))
TOP(ok(eq(s(X''), s(Y')))) -> TOP(mark(eq(X'', Y')))
TOP(mark(take(X1', X2'))) -> TOP(take(proper(X1'), proper(X2')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(inf(X''))) -> TOP(inf(proper(X'')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(eq(X1', X2'))) -> TOP(eq(proper(X1'), proper(X2')))
TOP(mark(length(X''))) -> TOP(length(proper(X'')))


Rules:


active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, 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(inf(X''))) -> TOP(mark(cons(X'', inf(s(X'')))))


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

length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))


Used ordering:
Polynomial Order with Interpretation:

POL( TOP(x1) ) = x1

POL( ok(x1) ) = x1

POL( inf(x1) ) = 1

POL( mark(x1) ) = x1

POL( cons(x1, x2) ) = 0

POL( take(x1, x2) ) = 0

POL( length(x1) ) = 0

POL( s(x1) ) = 0

POL( eq(x1, x2) ) = 0

POL( active(x1) ) = 1

POL( true ) = 0

POL( false ) = 0

POL( nil ) = 0

POL( 0 ) = 0

POL( proper(x1) ) = 1


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
         ↳Nar
           →DP Problem 10
             ↳Nar
             ...
               →DP Problem 14
                 ↳Negative Polynomial Order


Dependency Pairs:

TOP(ok(length(X''))) -> TOP(length(active(X'')))
TOP(ok(take(X1', X2'))) -> TOP(take(X1', active(X2')))
TOP(ok(take(X1', X2'))) -> TOP(take(active(X1'), X2'))
TOP(ok(inf(X''))) -> TOP(inf(active(X'')))
TOP(ok(eq(s(X''), s(Y')))) -> TOP(mark(eq(X'', Y')))
TOP(mark(take(X1', X2'))) -> TOP(take(proper(X1'), proper(X2')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(inf(X''))) -> TOP(inf(proper(X'')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(eq(X1', X2'))) -> TOP(eq(proper(X1'), proper(X2')))
TOP(mark(length(X''))) -> TOP(length(proper(X'')))


Rules:


active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, 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(eq(s(X''), s(Y')))) -> TOP(mark(eq(X'', Y')))


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

length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))


Used ordering:
Polynomial Order with Interpretation:

POL( TOP(x1) ) = x1

POL( ok(x1) ) = x1

POL( eq(x1, x2) ) = x2

POL( s(x1) ) = x1 + 1

POL( mark(x1) ) = x1

POL( take(x1, x2) ) = 0

POL( length(x1) ) = 0

POL( proper(x1) ) = x1

POL( cons(x1, x2) ) = 0

POL( inf(x1) ) = 0

POL( active(x1) ) = x1 + 1

POL( 0 ) = 0

POL( true ) = 0

POL( false ) = 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
         ↳Nar
           →DP Problem 10
             ↳Nar
             ...
               →DP Problem 15
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

TOP(ok(length(X''))) -> TOP(length(active(X'')))
TOP(ok(take(X1', X2'))) -> TOP(take(X1', active(X2')))
TOP(ok(take(X1', X2'))) -> TOP(take(active(X1'), X2'))
TOP(ok(inf(X''))) -> TOP(inf(active(X'')))
TOP(mark(take(X1', X2'))) -> TOP(take(proper(X1'), proper(X2')))
TOP(mark(cons(X1', X2'))) -> TOP(cons(proper(X1'), proper(X2')))
TOP(mark(inf(X''))) -> TOP(inf(proper(X'')))
TOP(mark(s(X''))) -> TOP(s(proper(X'')))
TOP(mark(eq(X1', X2'))) -> TOP(eq(proper(X1'), proper(X2')))
TOP(mark(length(X''))) -> TOP(length(proper(X'')))


Rules:


active(eq(0, 0)) -> mark(true)
active(eq(s(X), s(Y))) -> mark(eq(X, Y))
active(eq(X, Y)) -> mark(false)
active(inf(X)) -> mark(cons(X, inf(s(X))))
active(take(0, X)) -> mark(nil)
active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
active(length(nil)) -> mark(0)
active(length(cons(X, L))) -> mark(s(length(L)))
active(inf(X)) -> inf(active(X))
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(length(X)) -> length(active(X))
inf(mark(X)) -> mark(inf(X))
inf(ok(X)) -> ok(inf(X))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
length(mark(X)) -> mark(length(X))
length(ok(X)) -> ok(length(X))
proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(true) -> ok(true)
proper(s(X)) -> s(proper(X))
proper(false) -> ok(false)
proper(inf(X)) -> inf(proper(X))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
proper(length(X)) -> length(proper(X))
eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
s(ok(X)) -> ok(s(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, 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