Term Rewriting System R:
[x, u, v, z, y]
admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) -> y

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

ADMIT(x, .(u, .(v, .(w, z)))) -> COND(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Instantiation Transformation`

Dependency Pair:

Rules:

admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) -> y

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

one new Dependency Pair is created:

ADMIT(carry(x'', u''', v''), .(u'', .(v0, .(w, z'')))) -> ADMIT(carry(carry(x'', u''', v''), u'', v0), z'')

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Inst`
`           →DP Problem 2`
`             ↳Instantiation Transformation`

Dependency Pair:

ADMIT(carry(x'', u''', v''), .(u'', .(v0, .(w, z'')))) -> ADMIT(carry(carry(x'', u''', v''), u'', v0), z'')

Rules:

admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) -> y

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

ADMIT(carry(x'', u''', v''), .(u'', .(v0, .(w, z'')))) -> ADMIT(carry(carry(x'', u''', v''), u'', v0), z'')
one new Dependency Pair is created:

ADMIT(carry(carry(x'''', u'''''', v''''), u''''', v''0), .(u''1, .(v0'', .(w, z'''')))) -> ADMIT(carry(carry(carry(x'''', u'''''', v''''), u''''', v''0), u''1, v0''), z'''')

The transformation is resulting in one new DP problem:

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

Dependency Pair:

ADMIT(carry(carry(x'''', u'''''', v''''), u''''', v''0), .(u''1, .(v0'', .(w, z'''')))) -> ADMIT(carry(carry(carry(x'''', u'''''', v''''), u''''', v''0), u''1, v0''), z'''')

Rules:

admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
cond(true, y) -> y

The following dependency pair can be strictly oriented:

ADMIT(carry(carry(x'''', u'''''', v''''), u''''', v''0), .(u''1, .(v0'', .(w, z'''')))) -> ADMIT(carry(carry(carry(x'''', u'''''', v''''), u''''', v''0), u''1, v0''), z'''')

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(carry(x1, x2, x3)) =  0 POL(.(x1, x2)) =  x1 + x2 POL(w) =  1 POL(ADMIT(x1, x2)) =  x2

resulting in one new DP problem.

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

Dependency Pair:

Rules: