Term Rewriting System R:
[x, y]
active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVE(f(x)) -> F(active(x))
ACTIVE(f(x)) -> ACTIVE(x)
TOP(active(c)) -> TOP(mark(c))
TOP(mark(x)) -> TOP(check(x))
TOP(mark(x)) -> CHECK(x)
TOP(found(x)) -> TOP(active(x))
TOP(found(x)) -> ACTIVE(x)
CHECK(f(x)) -> F(check(x))
CHECK(f(x)) -> CHECK(x)
CHECK(x) -> START(match(f(X), x))
CHECK(x) -> MATCH(f(X), x)
CHECK(x) -> F(X)
MATCH(f(x), f(y)) -> F(match(x, y))
MATCH(f(x), f(y)) -> MATCH(x, y)
MATCH(X, x) -> PROPER(x)
PROPER(f(x)) -> F(proper(x))
PROPER(f(x)) -> PROPER(x)
F(ok(x)) -> F(x)
F(found(x)) -> F(x)
F(mark(x)) -> F(x)

Furthermore, R contains six SCCs.


   R
DPs
       →DP Problem 1
Argument Filtering and Ordering
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS


Dependency Pairs:

F(mark(x)) -> F(x)
F(found(x)) -> F(x)
F(ok(x)) -> F(x)


Rules:


active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(ok(x)) -> F(x)


There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(found(x1))=  x1  
  POL(mark(x1))=  x1  
  POL(ok(x1))=  1 + x1  
  POL(F(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
ok(x1) -> ok(x1)
mark(x1) -> mark(x1)
found(x1) -> found(x1)


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 7
Argument Filtering and Ordering
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS


Dependency Pairs:

F(mark(x)) -> F(x)
F(found(x)) -> F(x)


Rules:


active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(mark(x)) -> F(x)


There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(found(x1))=  x1  
  POL(mark(x1))=  1 + x1  
  POL(F(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
mark(x1) -> mark(x1)
found(x1) -> found(x1)


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 7
AFS
             ...
               →DP Problem 8
Argument Filtering and Ordering
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS


Dependency Pair:

F(found(x)) -> F(x)


Rules:


active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(found(x)) -> F(x)


There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(found(x1))=  1 + x1  
  POL(F(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
found(x1) -> found(x1)


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 7
AFS
             ...
               →DP Problem 9
Dependency Graph
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS


Dependency Pair:


Rules:


active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
Argument Filtering and Ordering
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS


Dependency Pair:

ACTIVE(f(x)) -> ACTIVE(x)


Rules:


active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

ACTIVE(f(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(f(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
ACTIVE(x1) -> ACTIVE(x1)
f(x1) -> f(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
           →DP Problem 10
Dependency Graph
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS


Dependency Pair:


Rules:


active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
Argument Filtering and Ordering
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS


Dependency Pair:

PROPER(f(x)) -> PROPER(x)


Rules:


active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

PROPER(f(x)) -> PROPER(x)


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))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
PROPER(x1) -> PROPER(x1)
f(x1) -> f(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
           →DP Problem 11
Dependency Graph
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS


Dependency Pair:


Rules:


active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
Argument Filtering and Ordering
       →DP Problem 5
AFS
       →DP Problem 6
AFS


Dependency Pair:

MATCH(f(x), f(y)) -> MATCH(x, y)


Rules:


active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

MATCH(f(x), f(y)) -> MATCH(x, y)


There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(MATCH(x1, x2))=  x1 + x2  
  POL(f(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
MATCH(x1, x2) -> MATCH(x1, x2)
f(x1) -> f(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
           →DP Problem 12
Dependency Graph
       →DP Problem 5
AFS
       →DP Problem 6
AFS


Dependency Pair:


Rules:


active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
Argument Filtering and Ordering
       →DP Problem 6
AFS


Dependency Pair:

CHECK(f(x)) -> CHECK(x)


Rules:


active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

CHECK(f(x)) -> CHECK(x)


There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(CHECK(x1))=  x1  
  POL(f(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
CHECK(x1) -> CHECK(x1)
f(x1) -> f(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
           →DP Problem 13
Dependency Graph
       →DP Problem 6
AFS


Dependency Pair:


Rules:


active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
Argument Filtering and Ordering


Dependency Pairs:

TOP(found(x)) -> TOP(active(x))
TOP(mark(x)) -> TOP(check(x))
TOP(active(c)) -> TOP(mark(c))


Rules:


active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

TOP(active(c)) -> TOP(mark(c))


The following usable rules for innermost can be oriented:

check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(active(x1))=  x1  
  POL(proper)=  1  
  POL(c)=  1  
  POL(X)=  1  
  POL(check)=  0  
  POL(found(x1))=  x1  
  POL(mark)=  0  
  POL(TOP(x1))=  x1  
  POL(ok(x1))=  x1  
  POL(f)=  0  
  POL(start(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
TOP(x1) -> TOP(x1)
active(x1) -> active(x1)
mark(x1) -> mark
check(x1) -> check
found(x1) -> found(x1)
f(x1) -> f
start(x1) -> start(x1)
match(x1, x2) -> x1
ok(x1) -> ok(x1)
proper(x1) -> proper


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
           →DP Problem 14
Argument Filtering and Ordering


Dependency Pairs:

TOP(found(x)) -> TOP(active(x))
TOP(mark(x)) -> TOP(check(x))


Rules:


active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

TOP(mark(x)) -> TOP(check(x))


The following usable rules for innermost can be oriented:

check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(active(x1))=  x1  
  POL(proper(x1))=  x1  
  POL(c)=  0  
  POL(check(x1))=  x1  
  POL(found(x1))=  x1  
  POL(mark(x1))=  1 + x1  
  POL(TOP(x1))=  1 + x1  
  POL(ok(x1))=  x1  
  POL(f(x1))=  1 + x1  
  POL(start(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
TOP(x1) -> TOP(x1)
mark(x1) -> mark(x1)
check(x1) -> check(x1)
found(x1) -> found(x1)
active(x1) -> active(x1)
f(x1) -> f(x1)
start(x1) -> start(x1)
match(x1, x2) -> x2
ok(x1) -> ok(x1)
proper(x1) -> proper(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
           →DP Problem 14
AFS
             ...
               →DP Problem 15
Argument Filtering and Ordering


Dependency Pair:

TOP(found(x)) -> TOP(active(x))


Rules:


active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)


Strategy:

innermost




The following dependency pair can be strictly oriented:

TOP(found(x)) -> TOP(active(x))


The following usable rules for innermost can be oriented:

active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(active(x1))=  x1  
  POL(found(x1))=  1 + x1  
  POL(mark(x1))=  x1  
  POL(TOP(x1))=  x1  
  POL(ok(x1))=  x1  
  POL(f(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
TOP(x1) -> TOP(x1)
found(x1) -> found(x1)
active(x1) -> active(x1)
f(x1) -> f(x1)
mark(x1) -> mark(x1)
ok(x1) -> ok(x1)


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
AFS
       →DP Problem 3
AFS
       →DP Problem 4
AFS
       →DP Problem 5
AFS
       →DP Problem 6
AFS
           →DP Problem 14
AFS
             ...
               →DP Problem 16
Dependency Graph


Dependency Pair:


Rules:


active(f(x)) -> mark(x)
active(f(x)) -> f(active(x))
top(active(c)) -> top(mark(c))
top(mark(x)) -> top(check(x))
top(found(x)) -> top(active(x))
check(f(x)) -> f(check(x))
check(x) -> start(match(f(X), x))
match(f(x), f(y)) -> f(match(x, y))
match(X, x) -> proper(x)
proper(c) -> ok(c)
proper(f(x)) -> f(proper(x))
f(ok(x)) -> ok(f(x))
f(found(x)) -> found(f(x))
f(mark(x)) -> mark(f(x))
start(ok(x)) -> found(x)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:11 minutes