Term Rewriting System R:
[y, x, z]
f(nil) -> nil
f(.(nil, y)) -> .(nil, f(y))
f(.(.(x, y), z)) -> f(.(x, .(y, z)))
g(nil) -> nil
g(.(x, nil)) -> .(g(x), nil)
g(.(x, .(y, z))) -> g(.(.(x, y), z))

Innermost Termination of R to be shown.

`   R`
`     ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

f(nil) -> nil

where the Polynomial interpretation:
 POL(g(x1)) =  x1 POL(nil) =  0 POL(.(x1, x2)) =  x1 + x2 POL(f(x1)) =  1 + x1
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

g(nil) -> nil

where the Polynomial interpretation:
 POL(g(x1)) =  1 + x1 POL(nil) =  0 POL(.(x1, x2)) =  x1 + x2 POL(f(x1)) =  x1
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

f(.(nil, y)) -> .(nil, f(y))

where the Polynomial interpretation:
 POL(g(x1)) =  x1 POL(nil) =  1 POL(.(x1, x2)) =  x1 + x2 POL(f(x1)) =  2·x1
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS4`
`                 ↳Removing Redundant Rules`

Removing the following rules from R which fullfill a polynomial ordering:

g(.(x, nil)) -> .(g(x), nil)

where the Polynomial interpretation:
 POL(g(x1)) =  2·x1 POL(nil) =  1 POL(.(x1, x2)) =  x1 + x2 POL(f(x1)) =  2 + x1
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →TRS5`
`                 ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(.(.(x, y), z)) -> F(.(x, .(y, z)))
G(.(x, .(y, z))) -> G(.(.(x, y), z))

Furthermore, R contains two SCCs.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →DP Problem 1`
`                 ↳Usable Rules (Innermost)`

Dependency Pair:

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

Rules:

f(.(.(x, y), z)) -> f(.(x, .(y, z)))
g(.(x, .(y, z))) -> g(.(.(x, y), z))

Strategy:

innermost

As we are in the innermost case, we can delete all 2 non-usable-rules.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →DP Problem 3`
`                 ↳Size-Change Principle`

Dependency Pair:

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

Rule:

none

Strategy:

innermost

We number the DPs as follows:
1. F(.(.(x, y), z)) -> F(.(x, .(y, z)))
and get the following Size-Change Graph(s):
{1} , {1}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
.(x1, x2) -> .(x1)

We obtain no new DP problems.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →DP Problem 2`
`                 ↳Usable Rules (Innermost)`

Dependency Pair:

G(.(x, .(y, z))) -> G(.(.(x, y), z))

Rules:

f(.(.(x, y), z)) -> f(.(x, .(y, z)))
g(.(x, .(y, z))) -> g(.(.(x, y), z))

Strategy:

innermost

As we are in the innermost case, we can delete all 2 non-usable-rules.

`   R`
`     ↳RRRPolo`
`       →TRS2`
`         ↳RRRPolo`
`           →TRS3`
`             ↳RRRPolo`
`             ...`
`               →DP Problem 4`
`                 ↳Size-Change Principle`

Dependency Pair:

G(.(x, .(y, z))) -> G(.(.(x, y), z))

Rule:

none

Strategy:

innermost

We number the DPs as follows:
1. G(.(x, .(y, z))) -> G(.(.(x, y), z))
and get the following Size-Change Graph(s):
{1} , {1}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
.(x1, x2) -> .(x2)

We obtain no new DP problems.

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