Term Rewriting System R:
[y, x, z]
f(0, y) > y
f(x, 0) > x
f(i(x), y) > i(x)
f(f(x, y), z) > f(x, f(y, z))
f(g(x, y), z) > g(f(x, z), f(y, z))
f(1, g(x, y)) > x
f(2, g(x, y)) > y
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(f(x, y), z) > F(x, f(y, z))
F(f(x, y), z) > F(y, z)
F(g(x, y), z) > F(x, z)
F(g(x, y), z) > F(y, z)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Usable Rules (Innermost)
Dependency Pairs:
F(g(x, y), z) > F(y, z)
F(g(x, y), z) > F(x, z)
F(f(x, y), z) > F(y, z)
Rules:
f(0, y) > y
f(x, 0) > x
f(i(x), y) > i(x)
f(f(x, y), z) > f(x, f(y, z))
f(g(x, y), z) > g(f(x, z), f(y, z))
f(1, g(x, y)) > x
f(2, g(x, y)) > y
Strategy:
innermost
As we are in the innermost case, we can delete all 7 nonusablerules.
R
↳DPs
→DP Problem 1
↳UsableRules
→DP Problem 2
↳SizeChange Principle
Dependency Pairs:
F(g(x, y), z) > F(y, z)
F(g(x, y), z) > F(x, z)
F(f(x, y), z) > F(y, z)
Rule:
none
Strategy:
innermost
We number the DPs as follows:
 F(g(x, y), z) > F(y, z)
 F(g(x, y), z) > F(x, z)
 F(f(x, y), z) > F(y, z)
and get the following SizeChange Graph(s): {1, 2, 3}  ,  {1, 2, 3} 

1  >  1 
2  =  2 

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

1  >  1 
2  =  2 

D_{P}: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with NonStrict Precedence.
trivial
with Argument Filtering System:
g(x_{1}, x_{2}) > g(x_{1}, x_{2})
f(x_{1}, x_{2}) > f(x_{1}, x_{2})
We obtain no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes