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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

+'(x, s(y)) -> S(+(x, y))
+'(x, s(y)) -> +'(x, y)
S(+(0, y)) -> S(y)

Furthermore, R contains two SCCs. 


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


Dependency Pair:

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


Rules:


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


Strategy:

innermost




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

S(+(0, y)) -> S(y)
one new Dependency Pair is created:

S(+(0, +(0, y''))) -> S(+(0, y''))

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

S(+(0, +(0, y''))) -> S(+(0, y''))


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

S(+(0, +(0, y''))) -> S(+(0, y''))


Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  1  
  POL(s(x1))=  0  
  POL(S(x1))=  1 + x1  
  POL(+(x1, x2))=  x1 + x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 3
             ↳Polo
             ...
               →DP Problem 4
                 ↳Dependency Graph
       →DP Problem 2
         ↳FwdInst


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pair:

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


Rules:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 5
             ↳Polynomial Ordering


Dependency Pair:

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


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


Additionally, the following usable rule for innermost w.r.t. to the implicit AFS can be oriented:

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(s(x1))=  1 + x1  
  POL(+(x1, x2))=  1 + x2  
  POL(+'(x1, x2))=  x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 5
             ↳Polo
             ...
               →DP Problem 6
                 ↳Dependency Graph


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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