Term Rewriting System R:
[x, y]
merge(x, nil) -> x
merge(nil, y) -> y
merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

MERGE(++(x, y), ++(u, v)) -> MERGE(y, ++(u, v))
MERGE(++(x, y), ++(u, v)) -> MERGE(++(x, y), v)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pair:

MERGE(++(x, y), ++(u, v)) -> MERGE(y, ++(u, v))


Rules:


merge(x, nil) -> x
merge(nil, y) -> y
merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))


Strategy:

innermost




On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

MERGE(++(x, y), ++(u, v)) -> MERGE(y, ++(u, v))
one new Dependency Pair is created:

MERGE(++(x, ++(x'', y'')), ++(u, v)) -> MERGE(++(x'', y''), ++(u, v))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Forward Instantiation Transformation


Dependency Pair:

MERGE(++(x, ++(x'', y'')), ++(u, v)) -> MERGE(++(x'', y''), ++(u, v))


Rules:


merge(x, nil) -> x
merge(nil, y) -> y
merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))


Strategy:

innermost




On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

MERGE(++(x, ++(x'', y'')), ++(u, v)) -> MERGE(++(x'', y''), ++(u, v))
one new Dependency Pair is created:

MERGE(++(x, ++(x'''', ++(x''''', y''''))), ++(u, v)) -> MERGE(++(x'''', ++(x''''', y'''')), ++(u, v))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Polynomial Ordering


Dependency Pair:

MERGE(++(x, ++(x'''', ++(x''''', y''''))), ++(u, v)) -> MERGE(++(x'''', ++(x''''', y'''')), ++(u, v))


Rules:


merge(x, nil) -> x
merge(nil, y) -> y
merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))


Strategy:

innermost




The following dependency pair can be strictly oriented:

MERGE(++(x, ++(x'''', ++(x''''', y''''))), ++(u, v)) -> MERGE(++(x'''', ++(x''''', y'''')), ++(u, v))


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(v)=  0  
  POL(++(x1, x2))=  1 + x2  
  POL(MERGE(x1, x2))=  1 + x1  
  POL(u)=  0  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 4
                 ↳Dependency Graph


Dependency Pair:


Rules:


merge(x, nil) -> x
merge(nil, y) -> y
merge(++(x, y), ++(u, v)) -> ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) -> ++(u, merge(++(x, y), v))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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