Term Rewriting System R:
[x, y, z, u, v]
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))
if(x, if(x, y, z), z) -> if(x, y, z)
if(x, y, if(x, y, z)) -> if(x, y, z)

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
         ↳Usable Rules (Innermost)


Dependency Pairs:

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


Strategy:

innermost




As we are in the innermost case, we can delete all 6 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 2
             ↳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)


Rule:

none


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)
and get the following Size-Change Graph(s):
{1, 2} , {1, 2}
1>1
2=2
3=3

which lead(s) to this/these maximal multigraph(s):
{1, 2} , {1, 2}
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:
if(x1, x2, x3) -> if(x1, x2, x3)

We obtain no new DP problems.

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