Term Rewriting System R:
[x, y, z]
minus(0) -> 0
minus(minus(x)) -> x
+(x, 0) -> x
+(0, y) -> y
+(minus(1), 1) -> 0
+(x, minus(y)) -> minus(+(minus(x), y))
+(x, +(y, z)) -> +(+(x, y), z)
+(minus(+(x, 1)), 1) -> minus(x)

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



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

minus(0) -> 0
minus(minus(x)) -> x
+(minus(1), 1) -> 0

where the Polynomial interpretation:
  POL(0)=  0  
  POL(1)=  0  
  POL(minus(x1))=  1 + x1  
  POL(+(x1, x2))=  x1 + 2·x2  
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:

+(x, 0) -> x
+(0, y) -> y

where the Polynomial interpretation:
  POL(0)=  1  
  POL(1)=  0  
  POL(minus(x1))=  x1  
  POL(+(x1, x2))=  x1 + x2  
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:

+(x, +(y, z)) -> +(+(x, y), z)
+(minus(+(x, 1)), 1) -> minus(x)

where the Polynomial interpretation:
  POL(1)=  0  
  POL(minus(x1))=  x1  
  POL(+(x1, x2))=  1 + x1 + 2·x2  
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
                 ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


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



R contains the following Dependency Pairs:

+'(x, minus(y)) -> +'(minus(x), y)

Furthermore, R contains one SCC. 


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


Dependency Pair:

+'(x, minus(y)) -> +'(minus(x), y)


Rule:


+(x, minus(y)) -> minus(+(minus(x), y))


Strategy:

innermost




As we are in the innermost case, we can delete all 1 non-usable-rules.


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳RRRPolo
             ...
               →DP Problem 2
                 ↳Size-Change Principle


Dependency Pair:

+'(x, minus(y)) -> +'(minus(x), y)


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. +'(x, minus(y)) -> +'(minus(x), y)
and get the following Size-Change Graph(s):
{1} , {1}
2>2

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

DP: empty set
Oriented Rules: none

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

with Argument Filtering System:
minus(x1) -> minus(x1)

We obtain no new DP problems.

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