Termination Proof using AProVE

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

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

TIMES(x, s(y)) -> PLUS(times(x, y), x)
TIMES(x, s(y)) -> TIMES(x, y)
PLUS(x, s(y)) -> PLUS(x, y)
PLUS(s(x), y) -> PLUS(x, y)

Furthermore, R contains two SCCs.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
PLUS(s(x), y) -> PLUS(x, y)
PLUS(x, s(y)) -> PLUS(x, y)

Rules:

times(x, 0) -> 0
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0) -> x
plus(0, x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))
Dependency Pair:
TIMES(x, s(y)) -> TIMES(x, y)

Rules:

times(x, 0) -> 0
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0) -> x
plus(0, x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
PLUS(s(x), y) -> PLUS(x, y)
PLUS(x, s(y)) -> PLUS(x, y)

Rules:

times(x, 0) -> 0
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0) -> x
plus(0, x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))
Dependency Pair:
TIMES(x, s(y)) -> TIMES(x, y)

Rules:

times(x, 0) -> 0
times(x, s(y)) -> plus(times(x, y), x)
plus(x, 0) -> x
plus(0, x) -> x
plus(x, s(y)) -> s(plus(x, y))
plus(s(x), y) -> s(plus(x, y))

Termination of R could not be shown.
Duration:
0:00 minutes