Term Rewriting System R:
[x, y, z]
del(.(x, .(y, z))) -> f(=(x, y), x, y, z)
f(true, x, y, z) -> del(.(y, z))
f(false, x, y, z) -> .(x, del(.(y, z)))
=(nil, nil) -> true
=(.(x, y), nil) -> false
=(nil, .(y, z)) -> false
=(.(x, y), .(u, v)) -> and(=(x, u), =(y, v))

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

DEL(.(x, .(y, z))) -> F(=(x, y), x, y, z)
DEL(.(x, .(y, z))) -> ='(x, y)
F(true, x, y, z) -> DEL(.(y, z))
F(false, x, y, z) -> DEL(.(y, z))
='(.(x, y), .(u, v)) -> ='(x, u)
='(.(x, y), .(u, v)) -> ='(y, v)

Furthermore, R contains one SCC.

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

Dependency Pairs:

F(false, x, y, z) -> DEL(.(y, z))
F(true, x, y, z) -> DEL(.(y, z))
DEL(.(x, .(y, z))) -> F(=(x, y), x, y, z)

Rules:

del(.(x, .(y, z))) -> f(=(x, y), x, y, z)
f(true, x, y, z) -> del(.(y, z))
f(false, x, y, z) -> .(x, del(.(y, z)))
=(nil, nil) -> true
=(.(x, y), nil) -> false
=(nil, .(y, z)) -> false
=(.(x, y), .(u, v)) -> and(=(x, u), =(y, v))

The following dependency pair can be strictly oriented:

F(true, x, y, z) -> DEL(.(y, z))

The following usable rules using the Ce-refinement can be oriented:

=(nil, nil) -> true
=(.(x, y), nil) -> false
=(nil, .(y, z)) -> false
=(.(x, y), .(u, v)) -> and(=(x, u), =(y, v))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(DEL(x1)) =  x1 POL(false) =  0 POL(=) =  1 POL(true) =  1 POL(.(x1, x2)) =  1 + x1 + x2 POL(F(x1, x2, x3, x4)) =  1 + x1 + x2 + x3 + x4

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2, x3, x4) -> F(x1, x2, x3, x4)
DEL(x1) -> DEL(x1)
.(x1, x2) -> .(x1, x2)
=(x1, x2) -> =
and(x1, x2) -> x1

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

Dependency Pairs:

F(false, x, y, z) -> DEL(.(y, z))
DEL(.(x, .(y, z))) -> F(=(x, y), x, y, z)

Rules:

del(.(x, .(y, z))) -> f(=(x, y), x, y, z)
f(true, x, y, z) -> del(.(y, z))
f(false, x, y, z) -> .(x, del(.(y, z)))
=(nil, nil) -> true
=(.(x, y), nil) -> false
=(nil, .(y, z)) -> false
=(.(x, y), .(u, v)) -> and(=(x, u), =(y, v))

The following dependency pair can be strictly oriented:

DEL(.(x, .(y, z))) -> F(=(x, y), x, y, z)

The following usable rules using the Ce-refinement can be oriented:

=(nil, nil) -> true
=(.(x, y), nil) -> false
=(nil, .(y, z)) -> false
=(.(x, y), .(u, v)) -> and(=(x, u), =(y, v))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(and(x1, x2)) =  x1 + x2 POL(DEL(x1)) =  x1 POL(false) =  0 POL(=) =  0 POL(true) =  0 POL(.(x1, x2)) =  1 + x1 + x2 POL(F(x1, x2, x3, x4)) =  1 + x1 + x2 + x3 + x4

resulting in one new DP problem.
Used Argument Filtering System:
DEL(x1) -> DEL(x1)
F(x1, x2, x3, x4) -> F(x1, x2, x3, x4)
.(x1, x2) -> .(x1, x2)
=(x1, x2) -> =
and(x1, x2) -> and(x1, x2)

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

Dependency Pair:

F(false, x, y, z) -> DEL(.(y, z))

Rules:

del(.(x, .(y, z))) -> f(=(x, y), x, y, z)
f(true, x, y, z) -> del(.(y, z))
f(false, x, y, z) -> .(x, del(.(y, z)))
=(nil, nil) -> true
=(.(x, y), nil) -> false
=(nil, .(y, z)) -> false
=(.(x, y), .(u, v)) -> and(=(x, u), =(y, v))

Using the Dependency Graph resulted in no new DP problems.

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