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
         ↳Polynomial Ordering
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo


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




The following dependency pair can be strictly oriented:

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


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

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

resulting in one new DP problem. 



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


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
         ↳Polo
       →DP Problem 2
         ↳Polynomial Ordering
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo


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




The following dependency pair can be strictly oriented:

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


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

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

resulting in one new DP problem. 



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


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
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polynomial Ordering
       →DP Problem 4
         ↳Polo


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




The following dependency pair can be strictly oriented:

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


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

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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
           →DP Problem 7
             ↳Dependency Graph
       →DP Problem 4
         ↳Polo


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
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polynomial Ordering


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




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost 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
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Polo
       →DP Problem 4
         ↳Polo
           →DP Problem 8
             ↳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