Term Rewriting System R:
[x, y, z]
if(true, x, y) -> x
if(false, x, y) -> y
if(x, y, y) -> y
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`
`         ↳Size-Change Principle`

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

We number the DPs as follows:
1. IF(if(x, y, z), u, v) -> IF(z, u, v)
2. IF(if(x, y, z), u, v) -> IF(y, u, v)
3. IF(if(x, y, z), u, v) -> IF(x, if(y, u, v), if(z, u, v))
and get the following Size-Change Graph(s):
{3, 2, 1} , {3, 2, 1}
1>1
2=2
3=3
{3, 2, 1} , {3, 2, 1}
1>1

which lead(s) to this/these maximal multigraph(s):
{3, 2, 1} , {3, 2, 1}
1>1
{3, 2, 1} , {3, 2, 1}
1>1
2=2
3=3

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
trivial

We obtain no new DP problems.

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