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
Argument Filtering and Ordering

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

The following dependency pairs can be strictly oriented:

F(g(x, y), z) -> F(y, z)
F(g(x, y), z) -> F(x, z)

There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g(x1, x2)) =  1 + x1 + x2 POL(F(x1, x2)) =  x1 + x2 POL(f(x1, x2)) =  x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
g(x1, x2) -> g(x1, x2)
f(x1, x2) -> f(x1, x2)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
Argument Filtering and Ordering

Dependency Pair:

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

The following dependency pair can be strictly oriented:

F(f(x, y), z) -> F(y, z)

There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(F(x1, x2)) =  x1 + x2 POL(f(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
f(x1, x2) -> f(x1, x2)

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
...
→DP Problem 3
Dependency Graph

Dependency Pair:

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

Using the Dependency Graph resulted in no new DP problems.

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