Term Rewriting System R:
[x, y]
exp(x, 0) -> s(0)
exp(x, s(y)) -> *(x, exp(x, y))
*(0, y) -> 0
*(s(x), y) -> +(y, *(x, y))
-(0, y) -> 0
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

EXP(x, s(y)) -> *'(x, exp(x, y))
EXP(x, s(y)) -> EXP(x, y)
*'(s(x), y) -> *'(x, y)
-'(s(x), s(y)) -> -'(x, y)

Furthermore, R contains three SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`

Dependency Pair:

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

Rules:

exp(x, 0) -> s(0)
exp(x, s(y)) -> *(x, exp(x, y))
*(0, y) -> 0
*(s(x), y) -> +(y, *(x, y))
-(0, y) -> 0
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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 4`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`

Dependency Pair:

Rules:

exp(x, 0) -> s(0)
exp(x, s(y)) -> *(x, exp(x, y))
*(0, y) -> 0
*(s(x), y) -> +(y, *(x, y))
-(0, y) -> 0
-(x, 0) -> x
-(s(x), s(y)) -> -(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`
`       →DP Problem 3`
`         ↳AFS`

Dependency Pair:

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

Rules:

exp(x, 0) -> s(0)
exp(x, s(y)) -> *(x, exp(x, y))
*(0, y) -> 0
*(s(x), y) -> +(y, *(x, y))
-(0, y) -> 0
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)

Strategy:

innermost

The following dependency pair can be strictly oriented:

-'(s(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: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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 5`
`             ↳Dependency Graph`
`       →DP Problem 3`
`         ↳AFS`

Dependency Pair:

Rules:

exp(x, 0) -> s(0)
exp(x, s(y)) -> *(x, exp(x, y))
*(0, y) -> 0
*(s(x), y) -> +(y, *(x, y))
-(0, y) -> 0
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳Argument Filtering and Ordering`

Dependency Pair:

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

Rules:

exp(x, 0) -> s(0)
exp(x, s(y)) -> *(x, exp(x, y))
*(0, y) -> 0
*(s(x), y) -> +(y, *(x, y))
-(0, y) -> 0
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`       →DP Problem 3`
`         ↳AFS`
`           →DP Problem 6`
`             ↳Dependency Graph`

Dependency Pair:

Rules:

exp(x, 0) -> s(0)
exp(x, s(y)) -> *(x, exp(x, y))
*(0, y) -> 0
*(s(x), y) -> +(y, *(x, y))
-(0, y) -> 0
-(x, 0) -> x
-(s(x), s(y)) -> -(x, y)

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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