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))

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



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

if(true, x, y) -> x

where the Polynomial interpretation:
  POL(if(x1, x2, x3))=  x1 + x2 + x3  
  POL(v)=  0  
  POL(false)=  0  
  POL(true)=  1  
  POL(u)=  0  
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:

if(false, x, y) -> y

where the Polynomial interpretation:
  POL(if(x1, x2, x3))=  x1 + x2 + x3  
  POL(v)=  0  
  POL(false)=  1  
  POL(u)=  0  
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:

if(x, y, y) -> y

where the Polynomial interpretation:
  POL(if(x1, x2, x3))=  1 + 2·x1 + x2 + x3  
  POL(v)=  0  
  POL(u)=  0  
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
                 ↳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
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳RRRPolo
             ...
               →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)


Rule:


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





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):
{2, 1} , {2, 1}
1>1
2=2
3=3

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

We obtain no new DP problems.

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