Term Rewriting System R:
[x, y, z]
f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(x, g(y, z)) -> F(x, y)
++'(x, g(y, z)) -> ++'(x, y)
MEM(g(x, y), z) -> MEM(x, z)
MEM(x, max(x)) -> NULL(x)
MAX(g(g(g(x, y), z), u)) -> MAX(g(g(x, y), z))

Furthermore, R contains four SCCs.


   R
DPs
       →DP Problem 1
Forward Instantiation Transformation
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst


Dependency Pair:

F(x, g(y, z)) -> F(x, y)


Rules:


f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)


Strategy:

innermost




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

F(x, g(y, z)) -> F(x, y)
one new Dependency Pair is created:

F(x'', g(g(y'', z''), z)) -> F(x'', g(y'', z''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 5
Forward Instantiation Transformation
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst


Dependency Pair:

F(x'', g(g(y'', z''), z)) -> F(x'', g(y'', z''))


Rules:


f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)


Strategy:

innermost




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

F(x'', g(g(y'', z''), z)) -> F(x'', g(y'', z''))
one new Dependency Pair is created:

F(x'''', g(g(g(y'''', z''''), z''0), z)) -> F(x'''', g(g(y'''', z''''), z''0))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 5
FwdInst
             ...
               →DP Problem 6
Polynomial Ordering
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst


Dependency Pair:

F(x'''', g(g(g(y'''', z''''), z''0), z)) -> F(x'''', g(g(y'''', z''''), z''0))


Rules:


f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(x'''', g(g(g(y'''', z''''), z''0), z)) -> F(x'''', g(g(y'''', z''''), z''0))


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 5
FwdInst
             ...
               →DP Problem 7
Dependency Graph
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst


Dependency Pair:


Rules:


f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
Forward Instantiation Transformation
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst


Dependency Pair:

++'(x, g(y, z)) -> ++'(x, y)


Rules:


f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)


Strategy:

innermost




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

++'(x, g(y, z)) -> ++'(x, y)
one new Dependency Pair is created:

++'(x'', g(g(y'', z''), z)) -> ++'(x'', g(y'', z''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
           →DP Problem 8
Forward Instantiation Transformation
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst


Dependency Pair:

++'(x'', g(g(y'', z''), z)) -> ++'(x'', g(y'', z''))


Rules:


f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)


Strategy:

innermost




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

++'(x'', g(g(y'', z''), z)) -> ++'(x'', g(y'', z''))
one new Dependency Pair is created:

++'(x'''', g(g(g(y'''', z''''), z''0), z)) -> ++'(x'''', g(g(y'''', z''''), z''0))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
           →DP Problem 8
FwdInst
             ...
               →DP Problem 9
Polynomial Ordering
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst


Dependency Pair:

++'(x'''', g(g(g(y'''', z''''), z''0), z)) -> ++'(x'''', g(g(y'''', z''''), z''0))


Rules:


f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)


Strategy:

innermost




The following dependency pair can be strictly oriented:

++'(x'''', g(g(g(y'''', z''''), z''0), z)) -> ++'(x'''', g(g(y'''', z''''), z''0))


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
           →DP Problem 8
FwdInst
             ...
               →DP Problem 10
Dependency Graph
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst


Dependency Pair:


Rules:


f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
Forward Instantiation Transformation
       →DP Problem 4
FwdInst


Dependency Pair:

MEM(g(x, y), z) -> MEM(x, z)


Rules:


f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)


Strategy:

innermost




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

MEM(g(x, y), z) -> MEM(x, z)
one new Dependency Pair is created:

MEM(g(g(x'', y''), y), z'') -> MEM(g(x'', y''), z'')

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
           →DP Problem 11
Forward Instantiation Transformation
       →DP Problem 4
FwdInst


Dependency Pair:

MEM(g(g(x'', y''), y), z'') -> MEM(g(x'', y''), z'')


Rules:


f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)


Strategy:

innermost




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

MEM(g(g(x'', y''), y), z'') -> MEM(g(x'', y''), z'')
one new Dependency Pair is created:

MEM(g(g(g(x'''', y''''), y''0), y), z'''') -> MEM(g(g(x'''', y''''), y''0), z'''')

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
           →DP Problem 11
FwdInst
             ...
               →DP Problem 12
Polynomial Ordering
       →DP Problem 4
FwdInst


Dependency Pair:

MEM(g(g(g(x'''', y''''), y''0), y), z'''') -> MEM(g(g(x'''', y''''), y''0), z'''')


Rules:


f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)


Strategy:

innermost




The following dependency pair can be strictly oriented:

MEM(g(g(g(x'''', y''''), y''0), y), z'''') -> MEM(g(g(x'''', y''''), y''0), z'''')


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
           →DP Problem 11
FwdInst
             ...
               →DP Problem 13
Dependency Graph
       →DP Problem 4
FwdInst


Dependency Pair:


Rules:


f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
Forward Instantiation Transformation


Dependency Pair:

MAX(g(g(g(x, y), z), u)) -> MAX(g(g(x, y), z))


Rules:


f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)


Strategy:

innermost




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

MAX(g(g(g(x, y), z), u)) -> MAX(g(g(x, y), z))
one new Dependency Pair is created:

MAX(g(g(g(g(x'', y''), y0), u), u)) -> MAX(g(g(g(x'', y''), y0), u))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
           →DP Problem 14
Forward Instantiation Transformation


Dependency Pair:

MAX(g(g(g(g(x'', y''), y0), u), u)) -> MAX(g(g(g(x'', y''), y0), u))


Rules:


f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)


Strategy:

innermost




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

MAX(g(g(g(g(x'', y''), y0), u), u)) -> MAX(g(g(g(x'', y''), y0), u))
one new Dependency Pair is created:

MAX(g(g(g(g(g(x'''', y''''), y''0), u), u), u)) -> MAX(g(g(g(g(x'''', y''''), y''0), u), u))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
           →DP Problem 14
FwdInst
             ...
               →DP Problem 15
Polynomial Ordering


Dependency Pair:

MAX(g(g(g(g(g(x'''', y''''), y''0), u), u), u)) -> MAX(g(g(g(g(x'''', y''''), y''0), u), u))


Rules:


f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)


Strategy:

innermost




The following dependency pair can be strictly oriented:

MAX(g(g(g(g(g(x'''', y''''), y''0), u), u), u)) -> MAX(g(g(g(g(x'''', y''''), y''0), u), u))


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(MAX(x1))=  1 + x1  
  POL(g(x1, x2))=  1 + x1  
  POL(u)=  0  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
       →DP Problem 2
FwdInst
       →DP Problem 3
FwdInst
       →DP Problem 4
FwdInst
           →DP Problem 14
FwdInst
             ...
               →DP Problem 16
Dependency Graph


Dependency Pair:


Rules:


f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
++(x, nil) -> x
++(x, g(y, z)) -> g(++(x, y), z)
null(nil) -> true
null(g(x, y)) -> false
mem(nil, y) -> false
mem(g(x, y), z) -> or(=(y, z), mem(x, z))
mem(x, max(x)) -> not(null(x))
max(g(g(nil, x), y)) -> max'(x, y)
max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes