Term Rewriting System R:
[x, y]
g(0, f(x, x)) -> x
g(x, s(y)) -> g(f(x, y), 0)
g(s(x), y) -> g(f(x, y), 0)
g(f(x, y), 0) -> f(g(x, 0), g(y, 0))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

G(x, s(y)) -> G(f(x, y), 0)
G(s(x), y) -> G(f(x, y), 0)
G(f(x, y), 0) -> G(x, 0)
G(f(x, y), 0) -> G(y, 0)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Instantiation Transformation


Dependency Pairs:

G(f(x, y), 0) -> G(y, 0)
G(s(x), y) -> G(f(x, y), 0)
G(f(x, y), 0) -> G(x, 0)


Rules:


g(0, f(x, x)) -> x
g(x, s(y)) -> g(f(x, y), 0)
g(s(x), y) -> g(f(x, y), 0)
g(f(x, y), 0) -> f(g(x, 0), g(y, 0))


Strategy:

innermost




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

G(s(x), y) -> G(f(x, y), 0)
two new Dependency Pairs are created:

G(s(x''), 0) -> G(f(x'', 0), 0)
G(s(x'), 0) -> G(f(x', 0), 0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

G(s(x'), 0) -> G(f(x', 0), 0)
G(s(x''), 0) -> G(f(x'', 0), 0)
G(f(x, y), 0) -> G(x, 0)
G(f(x, y), 0) -> G(y, 0)


Rules:


g(0, f(x, x)) -> x
g(x, s(y)) -> g(f(x, y), 0)
g(s(x), y) -> g(f(x, y), 0)
g(f(x, y), 0) -> f(g(x, 0), g(y, 0))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:00 minutes