Term Rewriting System R:
[X, Z, Y]
h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

H(X, Z) -> F(X, s(X), Z)
F(X, Y, g(X, Y)) -> H(0, g(X, Y))
G(X, s(Y)) -> G(X, Y)

Furthermore, R contains one SCC. 


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


Dependency Pair:

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


Rules:


h(X, Z) -> f(X, s(X), Z)
f(X, Y, g(X, Y)) -> h(0, g(X, Y))
g(0, Y) -> 0
g(X, s(Y)) -> g(X, Y)


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 Pair:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. G(X, s(Y)) -> G(X, Y)
and get the following Size-Change Graph(s):
{1} , {1}
1=1
2>2

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1=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:
s(x1) -> s(x1)

We obtain no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:01 minutes