Term Rewriting System R:
[x, y, f, g]
app(app(const, x), y) -> x
app(app(app(subst, f), g), x) -> app(app(f, x), app(g, x))
app(app(fix, f), x) -> app(app(f, app(fix, f)), x)

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

APP(app(app(subst, f), g), x) -> APP(app(f, x), app(g, x))
APP(app(app(subst, f), g), x) -> APP(f, x)
APP(app(app(subst, f), g), x) -> APP(g, x)
APP(app(fix, f), x) -> APP(app(f, app(fix, f)), x)
APP(app(fix, f), x) -> APP(f, app(fix, f))

Furthermore, R contains one SCC.

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

Dependency Pairs:

APP(app(fix, f), x) -> APP(f, app(fix, f))
APP(app(fix, f), x) -> APP(app(f, app(fix, f)), x)
APP(app(app(subst, f), g), x) -> APP(g, x)
APP(app(app(subst, f), g), x) -> APP(f, x)
APP(app(app(subst, f), g), x) -> APP(app(f, x), app(g, x))

Rules:

app(app(const, x), y) -> x
app(app(app(subst, f), g), x) -> app(app(f, x), app(g, x))
app(app(fix, f), x) -> app(app(f, app(fix, f)), x)

Strategy:

innermost

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

APP(app(app(subst, f), g), x) -> APP(f, x)
two new Dependency Pairs are created:

APP(app(app(subst, app(app(subst, f''), g'')), g), x'') -> APP(app(app(subst, f''), g''), x'')
APP(app(app(subst, app(fix, f'')), g), x'') -> APP(app(fix, f''), x'')

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

APP(app(app(subst, app(fix, f'')), g), x'') -> APP(app(fix, f''), x'')
APP(app(app(subst, app(app(subst, f''), g'')), g), x'') -> APP(app(app(subst, f''), g''), x'')
APP(app(fix, f), x) -> APP(app(f, app(fix, f)), x)
APP(app(app(subst, f), g), x) -> APP(g, x)
APP(app(app(subst, f), g), x) -> APP(app(f, x), app(g, x))
APP(app(fix, f), x) -> APP(f, app(fix, f))

Rules:

app(app(const, x), y) -> x
app(app(app(subst, f), g), x) -> app(app(f, x), app(g, x))
app(app(fix, f), x) -> app(app(f, app(fix, f)), x)

Strategy:

innermost

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

APP(app(app(subst, f), g), x) -> APP(g, x)
four new Dependency Pairs are created:

APP(app(app(subst, f), app(app(subst, f''), g'')), x'') -> APP(app(app(subst, f''), g''), x'')
APP(app(app(subst, f), app(fix, f'')), x'') -> APP(app(fix, f''), x'')
APP(app(app(subst, f), app(app(subst, app(app(subst, f''''), g'''')), g'')), x') -> APP(app(app(subst, app(app(subst, f''''), g'''')), g''), x')
APP(app(app(subst, f), app(app(subst, app(fix, f'''')), g'')), x') -> APP(app(app(subst, app(fix, f'''')), g''), x')

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:

APP(app(app(subst, f), app(app(subst, app(fix, f'''')), g'')), x') -> APP(app(app(subst, app(fix, f'''')), g''), x')
APP(app(app(subst, f), app(app(subst, app(app(subst, f''''), g'''')), g'')), x') -> APP(app(app(subst, app(app(subst, f''''), g'''')), g''), x')
APP(app(app(subst, f), app(fix, f'')), x'') -> APP(app(fix, f''), x'')
APP(app(app(subst, f), app(app(subst, f''), g'')), x'') -> APP(app(app(subst, f''), g''), x'')
APP(app(app(subst, app(app(subst, f''), g'')), g), x'') -> APP(app(app(subst, f''), g''), x'')
APP(app(fix, f), x) -> APP(f, app(fix, f))
APP(app(app(subst, f), g), x) -> APP(app(f, x), app(g, x))
APP(app(fix, f), x) -> APP(app(f, app(fix, f)), x)
APP(app(app(subst, app(fix, f'')), g), x'') -> APP(app(fix, f''), x'')

Rules:

app(app(const, x), y) -> x
app(app(app(subst, f), g), x) -> app(app(f, x), app(g, x))
app(app(fix, f), x) -> app(app(f, app(fix, f)), x)

Strategy:

innermost

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