Term Rewriting System R:
[x, y, z]
max(L(x)) -> x
max(N(L(0), L(y))) -> y
max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

MAX(N(L(s(x)), L(s(y)))) -> MAX(N(L(x), L(y)))
MAX(N(L(x), N(y, z))) -> MAX(N(L(x), L(max(N(y, z)))))
MAX(N(L(x), N(y, z))) -> MAX(N(y, z))

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`

Dependency Pair:

MAX(N(L(s(x)), L(s(y)))) -> MAX(N(L(x), L(y)))

Rules:

max(L(x)) -> x
max(N(L(0), L(y))) -> y
max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

MAX(N(L(s(x)), L(s(y)))) -> MAX(N(L(x), L(y)))
one new Dependency Pair is created:

MAX(N(L(s(s(x''))), L(s(s(y''))))) -> MAX(N(L(s(x'')), L(s(y''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 3`
`             ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳FwdInst`

Dependency Pair:

MAX(N(L(s(s(x''))), L(s(s(y''))))) -> MAX(N(L(s(x'')), L(s(y''))))

Rules:

max(L(x)) -> x
max(N(L(0), L(y))) -> y
max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

MAX(N(L(s(s(x''))), L(s(s(y''))))) -> MAX(N(L(s(x'')), L(s(y''))))
one new Dependency Pair is created:

MAX(N(L(s(s(s(x'''')))), L(s(s(s(y'''')))))) -> MAX(N(L(s(s(x''''))), L(s(s(y'''')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 3`
`             ↳FwdInst`
`             ...`
`               →DP Problem 4`
`                 ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳FwdInst`

Dependency Pair:

MAX(N(L(s(s(s(x'''')))), L(s(s(s(y'''')))))) -> MAX(N(L(s(s(x''''))), L(s(s(y'''')))))

Rules:

max(L(x)) -> x
max(N(L(0), L(y))) -> y
max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

MAX(N(L(s(s(s(x'''')))), L(s(s(s(y'''')))))) -> MAX(N(L(s(s(x''''))), L(s(s(y'''')))))

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 3`
`             ↳FwdInst`
`             ...`
`               →DP Problem 5`
`                 ↳Dependency Graph`
`       →DP Problem 2`
`         ↳FwdInst`

Dependency Pair:

Rules:

max(L(x)) -> x
max(N(L(0), L(y))) -> y
max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳Forward Instantiation Transformation`

Dependency Pair:

MAX(N(L(x), N(y, z))) -> MAX(N(y, z))

Rules:

max(L(x)) -> x
max(N(L(0), L(y))) -> y
max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

MAX(N(L(x), N(y, z))) -> MAX(N(y, z))
one new Dependency Pair is created:

MAX(N(L(x), N(L(x''), N(y'', z'')))) -> MAX(N(L(x''), N(y'', z'')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 6`
`             ↳Forward Instantiation Transformation`

Dependency Pair:

MAX(N(L(x), N(L(x''), N(y'', z'')))) -> MAX(N(L(x''), N(y'', z'')))

Rules:

max(L(x)) -> x
max(N(L(0), L(y))) -> y
max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

MAX(N(L(x), N(L(x''), N(y'', z'')))) -> MAX(N(L(x''), N(y'', z'')))
one new Dependency Pair is created:

MAX(N(L(x), N(L(x''''), N(L(x'''''), N(y'''', z''''))))) -> MAX(N(L(x''''), N(L(x'''''), N(y'''', z''''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 6`
`             ↳FwdInst`
`             ...`
`               →DP Problem 7`
`                 ↳Polynomial Ordering`

Dependency Pair:

MAX(N(L(x), N(L(x''''), N(L(x'''''), N(y'''', z''''))))) -> MAX(N(L(x''''), N(L(x'''''), N(y'''', z''''))))

Rules:

max(L(x)) -> x
max(N(L(0), L(y))) -> y
max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

MAX(N(L(x), N(L(x''''), N(L(x'''''), N(y'''', z''''))))) -> MAX(N(L(x''''), N(L(x'''''), N(y'''', z''''))))

There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(MAX(x1)) =  1 + x1 POL(L(x1)) =  0 POL(N(x1, x2)) =  1 + x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`       →DP Problem 2`
`         ↳FwdInst`
`           →DP Problem 6`
`             ↳FwdInst`
`             ...`
`               →DP Problem 8`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

max(L(x)) -> x
max(N(L(0), L(y))) -> y
max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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