Term Rewriting System R:
[x, y, z, u, v]
if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

IF(if(x, y, z), u, v) -> IF(x, if(y, u, v), if(z, u, v))
IF(if(x, y, z), u, v) -> IF(y, u, v)
IF(if(x, y, z), u, v) -> IF(z, u, v)

Furthermore, R contains one SCC.

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

Dependency Pairs:

IF(if(x, y, z), u, v) -> IF(z, u, v)
IF(if(x, y, z), u, v) -> IF(y, u, v)

Rule:

if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))

Strategy:

innermost

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

IF(if(x, y, z), u, v) -> IF(y, u, v)
one new Dependency Pair is created:

IF(if(x, if(x'', y'', z''), z), u'', v'') -> IF(if(x'', y'', z''), u'', v'')

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

IF(if(x, if(x'', y'', z''), z), u'', v'') -> IF(if(x'', y'', z''), u'', v'')
IF(if(x, y, z), u, v) -> IF(z, u, v)

Rule:

if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))

Strategy:

innermost

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

IF(if(x, y, z), u, v) -> IF(z, u, v)
two new Dependency Pairs are created:

IF(if(x, y, if(x'', y'', z'')), u'', v'') -> IF(if(x'', y'', z''), u'', v'')
IF(if(x, y, if(x'', if(x'''', y'''', z''''), z'')), u', v') -> IF(if(x'', if(x'''', y'''', z''''), z''), u', v')

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 3`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

IF(if(x, y, if(x'', if(x'''', y'''', z''''), z'')), u', v') -> IF(if(x'', if(x'''', y'''', z''''), z''), u', v')
IF(if(x, y, if(x'', y'', z'')), u'', v'') -> IF(if(x'', y'', z''), u'', v'')
IF(if(x, if(x'', y'', z''), z), u'', v'') -> IF(if(x'', y'', z''), u'', v'')

Rule:

if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:00 minutes