Term Rewriting System R:
[X, L, X1, X2]
active(incr(nil)) -> mark(nil)
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(adx(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0, zeros))
active(head(cons(X, L))) -> mark(X)
active(tail(cons(X, L))) -> mark(L)
active(incr(X)) -> incr(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(adx(X)) -> adx(active(X))
active(head(X)) -> head(active(X))
active(tail(X)) -> tail(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(mark(X)) -> mark(s(X))
s(ok(X)) -> ok(s(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(incr(X)) -> incr(proper(X))
proper(nil) -> ok(nil)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(adx(X)) -> adx(proper(X))
proper(nats) -> ok(nats)
proper(zeros) -> ok(zeros)
proper(0) -> ok(0)
proper(head(X)) -> head(proper(X))
proper(tail(X)) -> tail(proper(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(adx(nil)) -> mark(nil)

where the Polynomial interpretation:
  POL(proper(x1))=  x1  
  POL(adx(x1))=  2·x1  
  POL(tail(x1))=  x1  
  POL(incr(x1))=  x1  
  POL(mark(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(nats)=  0  
  POL(nil)=  1  
  POL(s(x1))=  x1  
  POL(head(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
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVE(cons(X1, X2)) -> CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)
ACTIVE(tail(X)) -> TAIL(active(X))
ACTIVE(tail(X)) -> ACTIVE(X)
ACTIVE(adx(cons(X, L))) -> INCR(cons(X, adx(L)))
ACTIVE(adx(cons(X, L))) -> CONS(X, adx(L))
ACTIVE(adx(cons(X, L))) -> ADX(L)
ACTIVE(head(X)) -> HEAD(active(X))
ACTIVE(head(X)) -> ACTIVE(X)
ACTIVE(adx(X)) -> ADX(active(X))
ACTIVE(adx(X)) -> ACTIVE(X)
ACTIVE(zeros) -> CONS(0, zeros)
ACTIVE(incr(cons(X, L))) -> CONS(s(X), incr(L))
ACTIVE(incr(cons(X, L))) -> S(X)
ACTIVE(incr(cons(X, L))) -> INCR(L)
ACTIVE(incr(X)) -> INCR(active(X))
ACTIVE(incr(X)) -> ACTIVE(X)
ACTIVE(s(X)) -> S(active(X))
ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(nats) -> ADX(zeros)
CONS(mark(X1), X2) -> CONS(X1, X2)
CONS(ok(X1), ok(X2)) -> CONS(X1, X2)
TAIL(mark(X)) -> TAIL(X)
TAIL(ok(X)) -> TAIL(X)
PROPER(tail(X)) -> TAIL(proper(X))
PROPER(tail(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(head(X)) -> HEAD(proper(X))
PROPER(head(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)
INCR(mark(X)) -> INCR(X)
INCR(ok(X)) -> INCR(X)
ADX(mark(X)) -> ADX(X)
ADX(ok(X)) -> ADX(X)
HEAD(mark(X)) -> HEAD(X)
HEAD(ok(X)) -> HEAD(X)
S(ok(X)) -> S(X)
S(mark(X)) -> S(X)

Furthermore, R contains nine SCCs. 


   R
     ↳RRRPolo
       →TRS2
         ↳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
             ↳MRR


Dependency Pairs:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(head(cons(X, L))) -> mark(X)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(tail(cons(X, L))) -> mark(L)
active(nats) -> mark(adx(zeros))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(s(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
     ↳RRRPolo
       →TRS2
         ↳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
             ↳MRR


Dependency Pairs:

TAIL(ok(X)) -> TAIL(X)
TAIL(mark(X)) -> TAIL(X)


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(head(cons(X, L))) -> mark(X)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(tail(cons(X, L))) -> mark(L)
active(nats) -> mark(adx(zeros))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(s(X))





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


Dependency Pairs:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(head(cons(X, L))) -> mark(X)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(tail(cons(X, L))) -> mark(L)
active(nats) -> mark(adx(zeros))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(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
         ↳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
             ↳MRR


Dependency Pairs:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(head(cons(X, L))) -> mark(X)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(tail(cons(X, L))) -> mark(L)
active(nats) -> mark(adx(zeros))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(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
         ↳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
             ↳MRR


Dependency Pairs:

HEAD(ok(X)) -> HEAD(X)
HEAD(mark(X)) -> HEAD(X)


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(head(cons(X, L))) -> mark(X)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(tail(cons(X, L))) -> mark(L)
active(nats) -> mark(adx(zeros))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(s(X))





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


Dependency Pairs:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(head(cons(X, L))) -> mark(X)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(tail(cons(X, L))) -> mark(L)
active(nats) -> mark(adx(zeros))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(s(X))





We number the DPs as follows:
  1. S(mark(X)) -> S(X)
  2. S(ok(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
     ↳RRRPolo
       →TRS2
         ↳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
             ↳MRR


Dependency Pairs:

ACTIVE(s(X)) -> ACTIVE(X)
ACTIVE(incr(X)) -> ACTIVE(X)
ACTIVE(adx(X)) -> ACTIVE(X)
ACTIVE(head(X)) -> ACTIVE(X)
ACTIVE(tail(X)) -> ACTIVE(X)
ACTIVE(cons(X1, X2)) -> ACTIVE(X1)


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(head(cons(X, L))) -> mark(X)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(tail(cons(X, L))) -> mark(L)
active(nats) -> mark(adx(zeros))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(s(X))





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

We obtain no new DP problems. 


   R
     ↳RRRPolo
       →TRS2
         ↳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
             ↳MRR


Dependency Pairs:

PROPER(s(X)) -> PROPER(X)
PROPER(head(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(tail(X)) -> PROPER(X)


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(head(cons(X, L))) -> mark(X)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(tail(cons(X, L))) -> mark(L)
active(nats) -> mark(adx(zeros))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(s(X))





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

We obtain no new DP problems. 


   R
     ↳RRRPolo
       →TRS2
         ↳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
             ↳Modular Removal of Rules


Dependency Pairs:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(head(cons(X, L))) -> mark(X)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(tail(cons(X, L))) -> mark(L)
active(nats) -> mark(adx(zeros))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(s(X))





We have the following set of usable rules:

active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(head(cons(X, L))) -> mark(X)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(tail(cons(X, L))) -> mark(L)
active(nats) -> mark(adx(zeros))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(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))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(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(tail(x1))=  x1  
  POL(incr(x1))=  x1  
  POL(mark(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(nats)=  0  
  POL(nil)=  0  
  POL(s(x1))=  x1  
  POL(zeros)=  0  
  POL(head(x1))=  x1  

We have the following set D of usable symbols: {proper, adx, tail, incr, mark, TOP, ok, active, 0, cons, nats, nil, s, head, 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
         ↳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
             ↳MRR
             ...
               →DP Problem 10
                 ↳Modular Removal of Rules


Dependency Pairs:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(head(cons(X, L))) -> mark(X)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(tail(cons(X, L))) -> mark(L)
active(nats) -> mark(adx(zeros))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(s(X))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)





We have the following set of usable rules:

active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(head(cons(X, L))) -> mark(X)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(tail(cons(X, L))) -> mark(L)
active(nats) -> mark(adx(zeros))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(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))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(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(tail(x1))=  x1  
  POL(incr(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(TOP(x1))=  x1  
  POL(ok(x1))=  x1  
  POL(active(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(nats)=  1  
  POL(nil)=  0  
  POL(s(x1))=  x1  
  POL(zeros)=  0  
  POL(head(x1))=  x1  

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

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


The result of this processor delivers one new DP problem.



   R
     ↳RRRPolo
       →TRS2
         ↳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
             ↳MRR
             ...
               →DP Problem 11
                 ↳Modular Removal of Rules


Dependency Pairs:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(head(cons(X, L))) -> mark(X)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(tail(cons(X, L))) -> mark(L)
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(s(X))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)





We have the following set of usable rules:

active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(head(cons(X, L))) -> mark(X)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
active(tail(cons(X, L))) -> mark(L)
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(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))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(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(tail(x1))=  1 + x1  
  POL(incr(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(TOP(x1))=  x1  
  POL(ok(x1))=  x1  
  POL(active(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(nats)=  0  
  POL(nil)=  0  
  POL(s(x1))=  x1  
  POL(zeros)=  0  
  POL(head(x1))=  x1  

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

active(tail(cons(X, L))) -> mark(L)


The result of this processor delivers one new DP problem.



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


Dependency Pairs:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(head(cons(X, L))) -> mark(X)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(s(X))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)





We have the following set of usable rules:

active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(head(cons(X, L))) -> mark(X)
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(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))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(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(tail(x1))=  x1  
  POL(incr(x1))=  x1  
  POL(mark(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(nats)=  0  
  POL(nil)=  0  
  POL(s(x1))=  x1  
  POL(zeros)=  0  
  POL(head(x1))=  1 + x1  

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

active(head(cons(X, L))) -> mark(X)


The result of this processor delivers one new DP problem.



   R
     ↳RRRPolo
       →TRS2
         ↳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
             ↳MRR
             ...
               →DP Problem 13
                 ↳Argument Filtering and Ordering


Dependency Pairs:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(s(X))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(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(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(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))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(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
head(x1) -> x1
cons(x1, x2) -> x1
adx(x1) -> adx(x1)
incr(x1) -> incr(x1)
mark(x1) -> mark(x1)
s(x1) -> x1
tail(x1) -> x1
proper(x1) -> x1


   R
     ↳RRRPolo
       →TRS2
         ↳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
             ↳MRR
             ...
               →DP Problem 14
                 ↳Modular Removal of Rules


Dependency Pair:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(s(X))
proper(nats) -> ok(nats)
proper(nil) -> ok(nil)
proper(tail(X)) -> tail(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(head(X)) -> head(proper(X))
proper(s(X)) -> s(proper(X))
proper(0) -> ok(0)





We have the following set of usable rules:

active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(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))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(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(adx(x1))=  x1  
  POL(tail(x1))=  x1  
  POL(incr(x1))=  x1  
  POL(mark(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(nil)=  0  
  POL(s(x1))=  x1  
  POL(head(x1))=  x1  
  POL(zeros)=  0  

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

The result of this processor delivers one new DP problem.



   R
     ↳RRRPolo
       →TRS2
         ↳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
             ↳MRR
             ...
               →DP Problem 15
                 ↳Modular Removal of Rules


Dependency Pair:

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


Rules:


active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
incr(mark(X)) -> mark(incr(X))
incr(ok(X)) -> ok(incr(X))
adx(mark(X)) -> mark(adx(X))
adx(ok(X)) -> ok(adx(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(X))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(s(X))





We have the following set of usable rules:

active(cons(X1, X2)) -> cons(active(X1), X2)
active(tail(X)) -> tail(active(X))
active(incr(nil)) -> mark(nil)
active(adx(cons(X, L))) -> mark(incr(cons(X, adx(L))))
active(head(X)) -> head(active(X))
active(adx(X)) -> adx(active(X))
active(zeros) -> mark(cons(0, zeros))
active(incr(cons(X, L))) -> mark(cons(s(X), incr(L)))
active(incr(X)) -> incr(active(X))
active(s(X)) -> s(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
tail(mark(X)) -> mark(tail(X))
tail(ok(X)) -> ok(tail(X))
head(mark(X)) -> mark(head(X))
head(ok(X)) -> ok(head(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))
s(ok(X)) -> ok(s(X))
s(mark(X)) -> mark(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(adx(x1))=  x1  
  POL(tail(x1))=  x1  
  POL(incr(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(TOP(x1))=  1 + x1  
  POL(ok(x1))=  1 + x1  
  POL(active(x1))=  x1  
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(nil)=  0  
  POL(s(x1))=  x1  
  POL(head(x1))=  x1  
  POL(zeros)=  0  

We have the following set D of usable symbols: {adx, tail, incr, mark, ok, TOP, active, 0, cons, nil, s, head, zeros}
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:
0:16 minutes