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
             ↳Argument Filtering and Ordering


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




The following dependency pairs can be strictly oriented:

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)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{0, s} > f

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


   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳AFS
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


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




Using the Dependency Graph resulted in no new DP problems.

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