Term Rewriting System R:
[x, y, z]
f(0, 1, g(x, y), z) -> f(g(x, y), g(x, y), g(x, y), h(x))
g(0, 1) -> 0
g(0, 1) -> 1
h(g(x, y)) -> h(x)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(0, 1, g(x, y), z) -> F(g(x, y), g(x, y), g(x, y), h(x))
F(0, 1, g(x, y), z) -> H(x)
H(g(x, y)) -> H(x)

Furthermore, R contains two SCCs. 


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


Dependency Pair:

H(g(x, y)) -> H(x)


Rules:


f(0, 1, g(x, y), z) -> f(g(x, y), g(x, y), g(x, y), h(x))
g(0, 1) -> 0
g(0, 1) -> 1
h(g(x, y)) -> h(x)





We number the DPs as follows:
  1. H(g(x, y)) -> H(x)
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:
g(x1, x2) -> g(x1, x2)

We obtain no new DP problems. 


   R
     ↳DPs
       →DP Problem 1
         ↳SCP
       →DP Problem 2
         ↳Non Termination


Dependency Pair:

F(0, 1, g(x, y), z) -> F(g(x, y), g(x, y), g(x, y), h(x))


Rules:


f(0, 1, g(x, y), z) -> f(g(x, y), g(x, y), g(x, y), h(x))
g(0, 1) -> 0
g(0, 1) -> 1
h(g(x, y)) -> h(x)





Found an infinite P-chain over R:
P =

F(0, 1, g(x, y), z) -> F(g(x, y), g(x, y), g(x, y), h(x))

R =

f(0, 1, g(x, y), z) -> f(g(x, y), g(x, y), g(x, y), h(x))
g(0, 1) -> 0
g(0, 1) -> 1
h(g(x, y)) -> h(x)

s = F(g(0, 1), g(0, 1), g(0, 1), h(0))
evaluates to t =F(g(0, 1), g(0, 1), g(0, 1), h(0))

Thus, s starts an infinite chain.

Non-Termination of R could be shown.
Duration:
0:00 minutes