Term Rewriting System R:
[X, Y, Z, X1, X2, X3, X4]
plus(s(X), plus(Y, Z)) -> plus(X, plus(s(s(Y)), Z))
plus(s(X1), plus(X2, plus(X3, X4))) -> plus(X1, plus(X3, plus(X2, X4)))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

PLUS(s(X), plus(Y, Z)) -> PLUS(X, plus(s(s(Y)), Z))
PLUS(s(X), plus(Y, Z)) -> PLUS(s(s(Y)), Z)
PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X1, plus(X3, plus(X2, X4)))
PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X3, plus(X2, X4))
PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X2, X4)

Furthermore, R contains one SCC.

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

Dependency Pairs:

PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X3, plus(X2, X4))
PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X1, plus(X3, plus(X2, X4)))
PLUS(s(X), plus(Y, Z)) -> PLUS(s(s(Y)), Z)
PLUS(s(X), plus(Y, Z)) -> PLUS(X, plus(s(s(Y)), Z))

Rules:

plus(s(X), plus(Y, Z)) -> plus(X, plus(s(s(Y)), Z))
plus(s(X1), plus(X2, plus(X3, X4))) -> plus(X1, plus(X3, plus(X2, X4)))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X3, plus(X2, X4))
PLUS(s(X), plus(Y, Z)) -> PLUS(s(s(Y)), Z)

The following usable rules for innermost w.r.t. to the AFS can be oriented:

plus(s(X), plus(Y, Z)) -> plus(X, plus(s(s(Y)), Z))
plus(s(X1), plus(X2, plus(X3, X4))) -> plus(X1, plus(X3, plus(X2, X4)))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(PLUS(x1, x2)) =  1 + x1 + x2 POL(plus(x1, x2)) =  1 + x1 + x2 POL(s(x1)) =  x1

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

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

Dependency Pairs:

PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X1, plus(X3, plus(X2, X4)))
PLUS(s(X), plus(Y, Z)) -> PLUS(X, plus(s(s(Y)), Z))

Rules:

plus(s(X), plus(Y, Z)) -> plus(X, plus(s(s(Y)), Z))
plus(s(X1), plus(X2, plus(X3, X4))) -> plus(X1, plus(X3, plus(X2, X4)))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

PLUS(s(X1), plus(X2, plus(X3, X4))) -> PLUS(X1, plus(X3, plus(X2, X4)))
PLUS(s(X), plus(Y, Z)) -> PLUS(X, plus(s(s(Y)), Z))

The following usable rules for innermost w.r.t. to the AFS can be oriented:

plus(s(X), plus(Y, Z)) -> plus(X, plus(s(s(Y)), Z))
plus(s(X1), plus(X2, plus(X3, X4))) -> plus(X1, plus(X3, plus(X2, X4)))

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

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

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

Dependency Pair:

Rules:

plus(s(X), plus(Y, Z)) -> plus(X, plus(s(s(Y)), Z))
plus(s(X1), plus(X2, plus(X3, X4))) -> plus(X1, plus(X3, plus(X2, X4)))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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