Term Rewriting System R:
[x, y, u, v]
s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

S(f(x, y)) -> F(s(y), s(x))
S(f(x, y)) -> S(y)
S(f(x, y)) -> S(x)
S(g(x, y)) -> G(s(x), s(y))
S(g(x, y)) -> S(x)
S(g(x, y)) -> S(y)
F(g(x, y), g(u, v)) -> G(f(x, u), f(y, v))
F(g(x, y), g(u, v)) -> F(x, u)
F(g(x, y), g(u, v)) -> F(y, v)

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering
       →DP Problem 2
         ↳Polo


Dependency Pairs:

F(g(x, y), g(u, v)) -> F(y, v)
F(g(x, y), g(u, v)) -> F(x, u)


Rules:


s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a


Strategy:

innermost




The following dependency pairs can be strictly oriented:

F(g(x, y), g(u, v)) -> F(y, v)
F(g(x, y), g(u, v)) -> F(x, u)


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 + x2  
  POL(F(x1, x2))=  x1  

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polynomial Ordering


Dependency Pairs:

S(g(x, y)) -> S(y)
S(g(x, y)) -> S(x)
S(f(x, y)) -> S(x)
S(f(x, y)) -> S(y)


Rules:


s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a


Strategy:

innermost




The following dependency pairs can be strictly oriented:

S(g(x, y)) -> S(y)
S(g(x, y)) -> S(x)


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 + x2  
  POL(S(x1))=  x1  
  POL(f(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 4
             ↳Polynomial Ordering


Dependency Pairs:

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


Rules:


s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


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

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

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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