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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(x, c(y)) -> F(x, s(f(y, y)))
F(x, c(y)) -> F(y, y)
F(s(x), y) -> F(x, s(c(y)))

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Size-Change Principle
       →DP Problem 2
         ↳SCP


Dependency Pair:

F(s(x), y) -> F(x, s(c(y)))


Rules:


f(x, c(y)) -> f(x, s(f(y, y)))
f(s(x), y) -> f(x, s(c(y)))





We number the DPs as follows:
  1. F(s(x), y) -> F(x, s(c(y)))
and get the following Size-Change Graph(s):
{1} , {1}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
c(x1) -> c(x1)
s(x1) -> s(x1)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳Size-Change Principle


Dependency Pair:

F(x, c(y)) -> F(y, y)


Rules:


f(x, c(y)) -> f(x, s(f(y, y)))
f(s(x), y) -> f(x, s(c(y)))





We number the DPs as follows:
  1. F(x, c(y)) -> F(y, y)
and get the following Size-Change Graph(s):
{1} , {1}
2>1
2>2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
2>1
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
c(x1) -> c(x1)

We obtain no new DP problems.

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