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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(0, 1, x) -> F(s(x), x, x)
F(x, y, s(z)) -> F(0, 1, z)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Usable Rules (Innermost)


Dependency Pairs:

F(x, y, s(z)) -> F(0, 1, z)
F(0, 1, x) -> F(s(x), x, x)


Rules:


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


Strategy:

innermost




As we are in the innermost case, we can delete all 4 non-usable-rules.


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


Dependency Pairs:

F(x, y, s(z)) -> F(0, 1, z)
F(0, 1, x) -> F(s(x), x, x)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. F(x, y, s(z)) -> F(0, 1, z)
  2. F(0, 1, x) -> F(s(x), x, x)
and get the following Size-Change Graph(s):
{1} , {1}
3>3
{2} , {2}
3=2
3=3

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
3>3
{1} , {2}
3>2
3>3
{2} , {1}
3>3

DP: empty set
Oriented Rules: none

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

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

We obtain no new DP problems.

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