Term Rewriting System R:
[X, Y, X1, X2]
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(hd(cons(X, Y))) -> mark(X)
active(tl(cons(X, Y))) -> mark(Y)
active(adx(X)) -> adx(active(X))
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
active(tl(X)) -> tl(active(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
tl(mark(X)) -> mark(tl(X))
tl(ok(X)) -> ok(tl(X))
proper(nats) -> ok(nats)
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0) -> ok(0)
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

active(nats) -> mark(adx(zeros))

where the Polynomial interpretation:
  POL(proper(x1))=  x1  
  POL(adx(x1))=  x1  
  POL(incr(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(tl(x1))=  x1  
  POL(ok(x1))=  x1  
  POL(top(x1))=  1 + x1  
  POL(active(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(hd(x1))=  x1  
  POL(nats)=  1  
  POL(s(x1))=  x1  
  POL(zeros)=  0  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R. 


   R
     ↳RRRPolo
       →TRS2
         ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

active(tl(cons(X, Y))) -> mark(Y)

where the Polynomial interpretation:
  POL(proper(x1))=  x1  
  POL(adx(x1))=  x1  
  POL(incr(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(tl(x1))=  1 + x1  
  POL(ok(x1))=  x1  
  POL(top(x1))=  1 + x1  
  POL(active(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(hd(x1))=  x1  
  POL(nats)=  0  
  POL(s(x1))=  x1  
  POL(zeros)=  0  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVE(adx(cons(X, Y))) -> INCR(cons(X, adx(Y)))
ACTIVE(adx(cons(X, Y))) -> CONS(X, adx(Y))
ACTIVE(adx(cons(X, Y))) -> ADX(Y)
ACTIVE(tl(X)) -> TL(active(X))
ACTIVE(tl(X)) -> ACTIVE(X)
ACTIVE(adx(X)) -> ADX(active(X))
ACTIVE(adx(X)) -> ACTIVE(X)
ACTIVE(zeros) -> CONS(0, zeros)
ACTIVE(incr(cons(X, Y))) -> CONS(s(X), incr(Y))
ACTIVE(incr(cons(X, Y))) -> S(X)
ACTIVE(incr(cons(X, Y))) -> INCR(Y)
ACTIVE(incr(X)) -> INCR(active(X))
ACTIVE(incr(X)) -> ACTIVE(X)
ACTIVE(hd(X)) -> HD(active(X))
ACTIVE(hd(X)) -> ACTIVE(X)
ADX(mark(X)) -> ADX(X)
ADX(ok(X)) -> ADX(X)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
INCR(mark(X)) -> INCR(X)
INCR(ok(X)) -> INCR(X)
HD(mark(X)) -> HD(X)
HD(ok(X)) -> HD(X)
PROPER(hd(X)) -> HD(proper(X))
PROPER(hd(X)) -> PROPER(X)
PROPER(tl(X)) -> TL(proper(X))
PROPER(tl(X)) -> PROPER(X)
PROPER(adx(X)) -> ADX(proper(X))
PROPER(adx(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(incr(X)) -> INCR(proper(X))
PROPER(incr(X)) -> PROPER(X)
PROPER(s(X)) -> S(proper(X))
PROPER(s(X)) -> PROPER(X)
TOP(mark(X)) -> TOP(proper(X))
TOP(mark(X)) -> PROPER(X)
TOP(ok(X)) -> TOP(active(X))
TOP(ok(X)) -> ACTIVE(X)
TL(ok(X)) -> TL(X)
TL(mark(X)) -> TL(X)
S(ok(X)) -> S(X)

Furthermore, R contains nine SCCs. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 1
                 ↳Size-Change Principle


Dependency Pairs:

INCR(ok(X)) -> INCR(X)
INCR(mark(X)) -> INCR(X)


Rules:


active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(cons(X, Y))) -> mark(X)
active(hd(X)) -> hd(active(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
proper(nats) -> ok(nats)
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
s(ok(X)) -> ok(s(X))





We number the DPs as follows:
  1. INCR(ok(X)) -> INCR(X)
  2. INCR(mark(X)) -> INCR(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
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 2
                 ↳Size-Change Principle


Dependency Pair:

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


Rules:


active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(cons(X, Y))) -> mark(X)
active(hd(X)) -> hd(active(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
proper(nats) -> ok(nats)
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
s(ok(X)) -> ok(s(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
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 3
                 ↳Size-Change Principle


Dependency Pairs:

ADX(ok(X)) -> ADX(X)
ADX(mark(X)) -> ADX(X)


Rules:


active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(cons(X, Y))) -> mark(X)
active(hd(X)) -> hd(active(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
proper(nats) -> ok(nats)
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
s(ok(X)) -> ok(s(X))





We number the DPs as follows:
  1. ADX(ok(X)) -> ADX(X)
  2. ADX(mark(X)) -> ADX(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
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 4
                 ↳Size-Change Principle


Dependency Pairs:

TL(mark(X)) -> TL(X)
TL(ok(X)) -> TL(X)


Rules:


active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(cons(X, Y))) -> mark(X)
active(hd(X)) -> hd(active(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
proper(nats) -> ok(nats)
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
s(ok(X)) -> ok(s(X))





We number the DPs as follows:
  1. TL(mark(X)) -> TL(X)
  2. TL(ok(X)) -> TL(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
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 5
                 ↳Size-Change Principle


Dependency Pair:

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


Rules:


active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(cons(X, Y))) -> mark(X)
active(hd(X)) -> hd(active(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
proper(nats) -> ok(nats)
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
s(ok(X)) -> ok(s(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
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 6
                 ↳Size-Change Principle


Dependency Pairs:

HD(ok(X)) -> HD(X)
HD(mark(X)) -> HD(X)


Rules:


active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(cons(X, Y))) -> mark(X)
active(hd(X)) -> hd(active(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
proper(nats) -> ok(nats)
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
s(ok(X)) -> ok(s(X))





We number the DPs as follows:
  1. HD(ok(X)) -> HD(X)
  2. HD(mark(X)) -> HD(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
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 7
                 ↳Size-Change Principle


Dependency Pairs:

ACTIVE(hd(X)) -> ACTIVE(X)
ACTIVE(incr(X)) -> ACTIVE(X)
ACTIVE(adx(X)) -> ACTIVE(X)
ACTIVE(tl(X)) -> ACTIVE(X)


Rules:


active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(cons(X, Y))) -> mark(X)
active(hd(X)) -> hd(active(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
proper(nats) -> ok(nats)
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
s(ok(X)) -> ok(s(X))





We number the DPs as follows:
  1. ACTIVE(hd(X)) -> ACTIVE(X)
  2. ACTIVE(incr(X)) -> ACTIVE(X)
  3. ACTIVE(adx(X)) -> ACTIVE(X)
  4. ACTIVE(tl(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:
adx(x1) -> adx(x1)
hd(x1) -> hd(x1)
incr(x1) -> incr(x1)
tl(x1) -> tl(x1)

We obtain no new DP problems. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 8
                 ↳Size-Change Principle


Dependency Pairs:

PROPER(s(X)) -> PROPER(X)
PROPER(incr(X)) -> PROPER(X)
PROPER(cons(X1, X2)) -> PROPER(X2)
PROPER(cons(X1, X2)) -> PROPER(X1)
PROPER(adx(X)) -> PROPER(X)
PROPER(tl(X)) -> PROPER(X)
PROPER(hd(X)) -> PROPER(X)


Rules:


active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(cons(X, Y))) -> mark(X)
active(hd(X)) -> hd(active(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
proper(nats) -> ok(nats)
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
s(ok(X)) -> ok(s(X))





We number the DPs as follows:
  1. PROPER(s(X)) -> PROPER(X)
  2. PROPER(incr(X)) -> PROPER(X)
  3. PROPER(cons(X1, X2)) -> PROPER(X2)
  4. PROPER(cons(X1, X2)) -> PROPER(X1)
  5. PROPER(adx(X)) -> PROPER(X)
  6. PROPER(tl(X)) -> PROPER(X)
  7. PROPER(hd(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:
adx(x1) -> adx(x1)
cons(x1, x2) -> cons(x1, x2)
hd(x1) -> hd(x1)
incr(x1) -> incr(x1)
s(x1) -> s(x1)
tl(x1) -> tl(x1)

We obtain no new DP problems. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 9
                 ↳Modular Removal of Rules


Dependency Pairs:

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


Rules:


active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(cons(X, Y))) -> mark(X)
active(hd(X)) -> hd(active(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
proper(nats) -> ok(nats)
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
s(ok(X)) -> ok(s(X))





We have the following set of usable rules:

active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(cons(X, Y))) -> mark(X)
active(hd(X)) -> hd(active(X))
proper(nats) -> ok(nats)
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
  POL(proper(x1))=  x1  
  POL(adx(x1))=  x1  
  POL(incr(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(tl(x1))=  x1  
  POL(TOP(x1))=  1 + x1  
  POL(ok(x1))=  x1  
  POL(active(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(hd(x1))=  x1  
  POL(nats)=  0  
  POL(s(x1))=  x1  
  POL(zeros)=  0  

We have the following set D of usable symbols: {proper, adx, incr, mark, tl, TOP, ok, active, 0, cons, hd, nats, s, zeros}
No Dependency Pairs can be deleted.
2 non usable rules have been deleted.

The result of this processor delivers one new DP problem.



   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 10
                 ↳Modular Removal of Rules


Dependency Pairs:

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


Rules:


active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(cons(X, Y))) -> mark(X)
active(hd(X)) -> hd(active(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
s(ok(X)) -> ok(s(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
proper(nats) -> ok(nats)
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)





We have the following set of usable rules:

active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(cons(X, Y))) -> mark(X)
active(hd(X)) -> hd(active(X))
proper(nats) -> ok(nats)
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
  POL(proper(x1))=  x1  
  POL(adx(x1))=  x1  
  POL(incr(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(tl(x1))=  x1  
  POL(TOP(x1))=  1 + x1  
  POL(ok(x1))=  x1  
  POL(active(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(hd(x1))=  1 + x1  
  POL(nats)=  0  
  POL(s(x1))=  x1  
  POL(zeros)=  0  

We have the following set D of usable symbols: {proper, adx, incr, mark, tl, TOP, ok, active, 0, cons, hd, nats, s, zeros}
No Dependency Pairs can be deleted.
The following rules can be deleted as the lhs is strictly greater than the corresponding rhs:

active(hd(cons(X, Y))) -> mark(X)


The result of this processor delivers one new DP problem.



   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 11
                 ↳Argument Filtering and Ordering


Dependency Pairs:

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


Rules:


active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
s(ok(X)) -> ok(s(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
proper(nats) -> ok(nats)
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)





The following dependency pair can be strictly oriented:

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


The following usable rules w.r.t. the AFS can be oriented:

active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
proper(nats) -> ok(nats)
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))


Used ordering: Lexicographic Path Order with Precedence:
adx > incr > mark
zeros > mark
zeros > 0

resulting in one new DP problem.
Used Argument Filtering System:
TOP(x1) -> x1
ok(x1) -> x1
active(x1) -> x1
adx(x1) -> adx(x1)
incr(x1) -> incr(x1)
cons(x1, x2) -> x1
mark(x1) -> mark(x1)
tl(x1) -> x1
hd(x1) -> x1
s(x1) -> x1
proper(x1) -> x1


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 12
                 ↳Modular Removal of Rules


Dependency Pair:

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


Rules:


active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
s(ok(X)) -> ok(s(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
proper(nats) -> ok(nats)
proper(hd(X)) -> hd(proper(X))
proper(tl(X)) -> tl(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(incr(X)) -> incr(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)





We have the following set of usable rules:

active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
  POL(active(x1))=  x1  
  POL(adx(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(hd(x1))=  x1  
  POL(incr(x1))=  x1  
  POL(s(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(tl(x1))=  x1  
  POL(zeros)=  0  
  POL(TOP(x1))=  1 + x1  
  POL(ok(x1))=  x1  

We have the following set D of usable symbols: {active, 0, adx, cons, hd, incr, s, zeros, tl, mark, TOP, ok}
No Dependency Pairs can be deleted.
9 non usable rules have been deleted.

The result of this processor delivers one new DP problem.



   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳DPs
             ...
               →DP Problem 13
                 ↳Modular Removal of Rules


Dependency Pair:

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


Rules:


active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
s(ok(X)) -> ok(s(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))





We have the following set of usable rules:

active(adx(cons(X, Y))) -> mark(incr(cons(X, adx(Y))))
active(tl(X)) -> tl(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, Y))) -> mark(cons(s(X), incr(Y)))
active(incr(X)) -> incr(active(X))
active(hd(X)) -> hd(active(X))
tl(ok(X)) -> ok(tl(X))
tl(mark(X)) -> mark(tl(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
hd(mark(X)) -> mark(hd(X))
hd(ok(X)) -> ok(hd(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
  POL(active(x1))=  x1  
  POL(adx(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(hd(x1))=  x1  
  POL(incr(x1))=  x1  
  POL(s(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(tl(x1))=  x1  
  POL(zeros)=  0  
  POL(TOP(x1))=  1 + x1  
  POL(ok(x1))=  1 + x1  

We have the following set D of usable symbols: {active, 0, adx, cons, hd, incr, s, zeros, tl, mark, TOP, ok}
The following Dependency Pairs can be deleted as the lhs is strictly greater than the corresponding rhs:

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

No Rules can be deleted.

After the removal, there are no SCCs in the dependency graph which results in no DP problems which have to be solved.


Termination of R successfully shown.
Duration:
0:12 minutes