Term Rewriting System R:
[x, y]
not(true) -> false
not(false) -> true
odd(0) -> false
odd(s(x)) -> not(odd(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(s(x), y) -> s(+(x, y))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

ODD(s(x)) -> NOT(odd(x))
ODD(s(x)) -> ODD(x)
+'(x, s(y)) -> +'(x, y)
+'(s(x), y) -> +'(x, y)

Furthermore, R contains two SCCs.

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

Dependency Pair:

ODD(s(x)) -> ODD(x)

Rules:

not(true) -> false
not(false) -> true
odd(0) -> false
odd(s(x)) -> not(odd(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(s(x), y) -> s(+(x, y))

Strategy:

innermost

The following dependency pair can be strictly oriented:

ODD(s(x)) -> ODD(x)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
ODD(x1) -> ODD(x1)
s(x1) -> s(x1)

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

Dependency Pair:

Rules:

not(true) -> false
not(false) -> true
odd(0) -> false
odd(s(x)) -> not(odd(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(s(x), y) -> s(+(x, y))

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 Pairs:

+'(s(x), y) -> +'(x, y)
+'(x, s(y)) -> +'(x, y)

Rules:

not(true) -> false
not(false) -> true
odd(0) -> false
odd(s(x)) -> not(odd(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(s(x), y) -> s(+(x, y))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

+'(s(x), y) -> +'(x, y)
+'(x, s(y)) -> +'(x, y)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
+'(x1, x2) -> +'(x1, x2)
s(x1) -> s(x1)

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

Dependency Pair:

Rules:

not(true) -> false
not(false) -> true
odd(0) -> false
odd(s(x)) -> not(odd(x))
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(s(x), y) -> s(+(x, y))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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