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

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVE(f(X, g(X), Y)) -> F(Y, Y, Y)
ACTIVE(g(X)) -> G(active(X))
ACTIVE(g(X)) -> ACTIVE(X)
G(mark(X)) -> G(X)
G(ok(X)) -> G(X)
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)
PROPER(g(X)) -> G(proper(X))
PROPER(g(X)) -> PROPER(X)
F(ok(X1), ok(X2), ok(X3)) -> F(X1, X2, 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 five SCCs.


   R
DPs
       →DP Problem 1
Polynomial Ordering
       →DP Problem 2
Polo
       →DP Problem 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Nar


Dependency Pair:

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


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ok(x1))=  1 + x1  
  POL(F(x1, x2, x3))=  x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 6
Dependency Graph
       →DP Problem 2
Polo
       →DP Problem 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Nar


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polynomial Ordering
       →DP Problem 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Nar


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(G(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
           →DP Problem 7
Polynomial Ordering
       →DP Problem 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Nar


Dependency Pair:

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


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(G(x1))=  x1  
  POL(mark(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
           →DP Problem 7
Polo
             ...
               →DP Problem 8
Dependency Graph
       →DP Problem 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Nar


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Polynomial Ordering
       →DP Problem 4
Polo
       →DP Problem 5
Nar


Dependency Pair:

ACTIVE(g(X)) -> ACTIVE(X)


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVE(g(X)) -> ACTIVE(X)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(ACTIVE(x1))=  x1  
  POL(g(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Polo
           →DP Problem 9
Dependency Graph
       →DP Problem 4
Polo
       →DP Problem 5
Nar


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Polo
       →DP Problem 4
Polynomial Ordering
       →DP Problem 5
Nar


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

PROPER(g(X)) -> PROPER(X)


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g(x1))=  1 + x1  
  POL(PROPER(x1))=  x1  
  POL(f(x1, x2, x3))=  x1 + x2 + x3  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Polo
       →DP Problem 4
Polo
           →DP Problem 10
Polynomial Ordering
       →DP Problem 5
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(X, g(X), Y)) -> mark(f(Y, Y, Y))
active(g(b)) -> mark(c)
active(b) -> mark(c)
active(g(X)) -> g(active(X))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2, X3)) -> f(proper(X1), proper(X2), proper(X3))
proper(g(X)) -> g(proper(X))
proper(b) -> ok(b)
proper(c) -> ok(c)
f(ok(X1), ok(X2), ok(X3)) -> ok(f(X1, X2, X3))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(PROPER(x1))=  x1  
  POL(f(x1, x2, x3))=  1 + x1 + x2 + x3  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Polo
       →DP Problem 4
Polo
           →DP Problem 10
Polo
             ...
               →DP Problem 11
Dependency Graph
       →DP Problem 5
Nar


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Narrowing Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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(g(X''))) -> TOP(g(proper(X'')))
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
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Nar
           →DP Problem 12
Narrowing Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

TOP(ok(X)) -> TOP(active(X))
four new Dependency Pairs are created:

TOP(ok(f(X'', g(X''), Y'))) -> TOP(mark(f(Y', Y', Y')))
TOP(ok(g(b))) -> TOP(mark(c))
TOP(ok(b)) -> TOP(mark(c))
TOP(ok(g(X''))) -> TOP(g(active(X'')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Polo
       →DP Problem 3
Polo
       →DP Problem 4
Polo
       →DP Problem 5
Nar
           →DP Problem 12
Nar
             ...
               →DP Problem 13
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rules:


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


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:02 minutes