Term Rewriting System R:
[x, y, z]
implies(not(x), y) -> or(x, y)
implies(not(x), or(y, z)) -> implies(y, or(x, z))
implies(x, or(y, z)) -> or(y, implies(x, z))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

IMPLIES(not(x), or(y, z)) -> IMPLIES(y, or(x, z))
IMPLIES(x, or(y, z)) -> IMPLIES(x, z)

Furthermore, R contains one SCC.

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

Dependency Pairs:

IMPLIES(x, or(y, z)) -> IMPLIES(x, z)
IMPLIES(not(x), or(y, z)) -> IMPLIES(y, or(x, z))

Rules:

implies(not(x), y) -> or(x, y)
implies(not(x), or(y, z)) -> implies(y, or(x, z))
implies(x, or(y, z)) -> or(y, implies(x, z))

Strategy:

innermost

The following dependency pair can be strictly oriented:

IMPLIES(not(x), or(y, z)) -> IMPLIES(y, or(x, z))

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(or(x1, x2)) =  x1 + x2 POL(not(x1)) =  1 + x1 POL(IMPLIES(x1, x2)) =  1 + x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
IMPLIES(x1, x2) -> IMPLIES(x1, x2)
not(x1) -> not(x1)
or(x1, x2) -> or(x1, x2)

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

Dependency Pair:

IMPLIES(x, or(y, z)) -> IMPLIES(x, z)

Rules:

implies(not(x), y) -> or(x, y)
implies(not(x), or(y, z)) -> implies(y, or(x, z))
implies(x, or(y, z)) -> or(y, implies(x, z))

Strategy:

innermost

The following dependency pair can be strictly oriented:

IMPLIES(x, or(y, z)) -> IMPLIES(x, z)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(or(x1, x2)) =  1 + x1 + x2 POL(IMPLIES(x1, x2)) =  x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
IMPLIES(x1, x2) -> IMPLIES(x1, x2)
or(x1, x2) -> or(x1, x2)

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

Dependency Pair:

Rules:

implies(not(x), y) -> or(x, y)
implies(not(x), or(y, z)) -> implies(y, or(x, z))
implies(x, or(y, z)) -> or(y, implies(x, z))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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