Term Rewriting System R:
[X, X1, X2, X3]
active(f(a, b, X)) -> mark(f(X, X, X))
active(c) -> mark(a)
active(c) -> mark(b)
active(f(X1, X2, X3)) -> f(active(X1), X2, X3)
active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
f(mark(X1), X2, X3) -> mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(a) -> ok(a)
proper(b) -> ok(b)
proper(c) -> ok(c)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVE(f(a, b, X)) -> F(X, X, X)
ACTIVE(f(X1, X2, X3)) -> F(active(X1), X2, X3)
ACTIVE(f(X1, X2, X3)) -> ACTIVE(X1)
ACTIVE(f(X1, X2, X3)) -> F(X1, X2, active(X3))
ACTIVE(f(X1, X2, X3)) -> ACTIVE(X3)
F(mark(X1), X2, X3) -> F(X1, X2, X3)
F(X1, X2, mark(X3)) -> F(X1, X2, X3)
F(ok(X1), ok(X2), ok(X3)) -> F(X1, X2, X3)
PROPER(f(X1, X2, X3)) -> F(proper(X1), proper(X2), proper(X3))
PROPER(f(X1, X2, X3)) -> PROPER(X1)
PROPER(f(X1, X2, X3)) -> PROPER(X2)
PROPER(f(X1, X2, X3)) -> PROPER(X3)
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 four SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Size-Change Principle
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳Nar


Dependency Pairs:

F(ok(X1), ok(X2), ok(X3)) -> F(X1, X2, X3)
F(X1, X2, mark(X3)) -> F(X1, X2, X3)
F(mark(X1), X2, X3) -> F(X1, X2, X3)


Rules:


active(f(a, b, X)) -> mark(f(X, X, X))
active(c) -> mark(a)
active(c) -> mark(b)
active(f(X1, X2, X3)) -> f(active(X1), X2, X3)
active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
f(mark(X1), X2, X3) -> mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(a) -> ok(a)
proper(b) -> ok(b)
proper(c) -> ok(c)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. F(ok(X1), ok(X2), ok(X3)) -> F(X1, X2, X3)
  2. F(X1, X2, mark(X3)) -> F(X1, X2, X3)
  3. F(mark(X1), X2, X3) -> F(X1, X2, X3)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2>2
3>3
{3, 2, 1} , {3, 2, 1}
1=1
2=2
3>3
{3, 2, 1} , {3, 2, 1}
1>1
2=2
3=3

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

DP: empty set
Oriented Rules: none

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

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

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳Size-Change Principle
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳Nar


Dependency Pairs:

ACTIVE(f(X1, X2, X3)) -> ACTIVE(X3)
ACTIVE(f(X1, X2, X3)) -> ACTIVE(X1)


Rules:


active(f(a, b, X)) -> mark(f(X, X, X))
active(c) -> mark(a)
active(c) -> mark(b)
active(f(X1, X2, X3)) -> f(active(X1), X2, X3)
active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
f(mark(X1), X2, X3) -> mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(a) -> ok(a)
proper(b) -> ok(b)
proper(c) -> ok(c)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. ACTIVE(f(X1, X2, X3)) -> ACTIVE(X3)
  2. ACTIVE(f(X1, X2, X3)) -> ACTIVE(X1)
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:
f(x1, x2, x3) -> f(x1, x2, x3)

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
         ↳Nar


Dependency Pairs:

PROPER(f(X1, X2, X3)) -> PROPER(X3)
PROPER(f(X1, X2, X3)) -> PROPER(X2)
PROPER(f(X1, X2, X3)) -> PROPER(X1)


Rules:


active(f(a, b, X)) -> mark(f(X, X, X))
active(c) -> mark(a)
active(c) -> mark(b)
active(f(X1, X2, X3)) -> f(active(X1), X2, X3)
active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
f(mark(X1), X2, X3) -> mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(a) -> ok(a)
proper(b) -> ok(b)
proper(c) -> ok(c)
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))





We number the DPs as follows:
  1. PROPER(f(X1, X2, X3)) -> PROPER(X3)
  2. PROPER(f(X1, X2, X3)) -> PROPER(X2)
  3. PROPER(f(X1, X2, X3)) -> PROPER(X1)
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {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:
f(x1, x2, x3) -> f(x1, x2, x3)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳SCP
       →DP Problem 3
         ↳SCP
       →DP Problem 4
         ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


active(f(a, b, X)) -> mark(f(X, X, X))
active(c) -> mark(a)
active(c) -> mark(b)
active(f(X1, X2, X3)) -> f(active(X1), X2, X3)
active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
f(mark(X1), X2, X3) -> mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(a) -> ok(a)
proper(b) -> ok(b)
proper(c) -> ok(c)
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))
four new Dependency Pairs are created:

TOP(mark(f(X1', X2', X3'))) -> TOP(f(proper(X1'), proper(X2'), proper(X3')))
TOP(mark(a)) -> TOP(ok(a))
TOP(mark(b)) -> TOP(ok(b))
TOP(mark(c)) -> TOP(ok(c))

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
         ↳Nar
           →DP Problem 5
             ↳Narrowing Transformation


Dependency Pairs:

TOP(mark(b)) -> TOP(ok(b))
TOP(mark(a)) -> TOP(ok(a))
TOP(mark(f(X1', X2', X3'))) -> TOP(f(proper(X1'), proper(X2'), proper(X3')))
TOP(ok(X)) -> TOP(active(X))


Rules:


active(f(a, b, X)) -> mark(f(X, X, X))
active(c) -> mark(a)
active(c) -> mark(b)
active(f(X1, X2, X3)) -> f(active(X1), X2, X3)
active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
f(mark(X1), X2, X3) -> mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(a) -> ok(a)
proper(b) -> ok(b)
proper(c) -> ok(c)
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))
five new Dependency Pairs are created:

TOP(ok(f(a, b, X''))) -> TOP(mark(f(X'', X'', X'')))
TOP(ok(c)) -> TOP(mark(a))
TOP(ok(c)) -> TOP(mark(b))
TOP(ok(f(X1', X2', X3'))) -> TOP(f(active(X1'), X2', X3'))
TOP(ok(f(X1', X2', X3'))) -> TOP(f(X1', X2', active(X3')))

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
         ↳Nar
           →DP Problem 5
             ↳Nar
             ...
               →DP Problem 6
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

TOP(ok(f(X1', X2', X3'))) -> TOP(f(X1', X2', active(X3')))
TOP(ok(f(X1', X2', X3'))) -> TOP(f(active(X1'), X2', X3'))
TOP(ok(f(a, b, X''))) -> TOP(mark(f(X'', X'', X'')))
TOP(mark(f(X1', X2', X3'))) -> TOP(f(proper(X1'), proper(X2'), proper(X3')))


Rules:


active(f(a, b, X)) -> mark(f(X, X, X))
active(c) -> mark(a)
active(c) -> mark(b)
active(f(X1, X2, X3)) -> f(active(X1), X2, X3)
active(f(X1, X2, X3)) -> f(X1, X2, active(X3))
f(mark(X1), X2, X3) -> mark(f(X1, X2, X3))
f(X1, X2, mark(X3)) -> mark(f(X1, X2, X3))
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(a) -> ok(a)
proper(b) -> ok(b)
proper(c) -> ok(c)
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