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

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

F(s(x), s(y)) -> F(x, y)
G(0, x) -> G(f(x, x), x)
G(0, x) -> F(x, x)

Furthermore, R contains two SCCs.

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

Dependency Pair:

F(s(x), s(y)) -> F(x, y)

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

F(s(x), s(y)) -> F(x, y)

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

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

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

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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

Dependency Pair:

G(0, x) -> G(f(x, x), x)

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

G(0, x) -> G(f(x, x), x)

The following usable rules for innermost can be oriented:

f(x, 0) -> s(0)
f(s(x), s(y)) -> s(f(x, y))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(0) =  1 POL(G(x1, x2)) =  x1 + x2 POL(s) =  0 POL(f) =  0

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

R
DPs
→DP Problem 1
AFS
→DP Problem 2
AFS
→DP Problem 4
Dependency Graph

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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