Term Rewriting System R:
[X, Y]
f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(c(X, s(Y))) -> F(c(s(X), Y))
G(c(s(X), Y)) -> F(c(X, s(Y)))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Instantiation Transformation


Dependency Pair:

F(c(X, s(Y))) -> F(c(s(X), Y))


Rules:


f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))


Strategy:

innermost




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

F(c(X, s(Y))) -> F(c(s(X), Y))
one new Dependency Pair is created:

F(c(s(X''), s(Y''))) -> F(c(s(s(X'')), Y''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Instantiation Transformation


Dependency Pair:

F(c(s(X''), s(Y''))) -> F(c(s(s(X'')), Y''))


Rules:


f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))


Strategy:

innermost




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

F(c(s(X''), s(Y''))) -> F(c(s(s(X'')), Y''))
one new Dependency Pair is created:

F(c(s(s(X'''')), s(Y''''))) -> F(c(s(s(s(X''''))), Y''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 3
                 ↳Argument Filtering and Ordering


Dependency Pair:

F(c(s(s(X'''')), s(Y''''))) -> F(c(s(s(s(X''''))), Y''''))


Rules:


f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(c(s(s(X'''')), s(Y''''))) -> F(c(s(s(s(X''''))), Y''''))


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:
trivial

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


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


Dependency Pair:


Rules:


f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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