Term Rewriting System R:
[X]
f(a) -> f(c(a))
f(c(X)) -> X
f(c(a)) -> f(d(b))
f(a) -> f(d(a))
f(d(X)) -> X
f(c(b)) -> f(d(a))
e(g(X)) -> e(X)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(a) -> F(c(a))
F(c(a)) -> F(d(b))
F(a) -> F(d(a))
F(c(b)) -> F(d(a))
E(g(X)) -> E(X)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pair:

E(g(X)) -> E(X)


Rules:


f(a) -> f(c(a))
f(c(X)) -> X
f(c(a)) -> f(d(b))
f(a) -> f(d(a))
f(d(X)) -> X
f(c(b)) -> f(d(a))
e(g(X)) -> e(X)


Strategy:

innermost




The following dependency pair can be strictly oriented:

E(g(X)) -> E(X)


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
E(x1) -> E(x1)
g(x1) -> g(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Dependency Graph


Dependency Pair:


Rules:


f(a) -> f(c(a))
f(c(X)) -> X
f(c(a)) -> f(d(b))
f(a) -> f(d(a))
f(d(X)) -> X
f(c(b)) -> f(d(a))
e(g(X)) -> e(X)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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