Term Rewriting System R:
[z, x, y]
+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

+'(a, b) -> +'(b, a)
+'(a, +(b, z)) -> +'(b, +(a, z))
+'(a, +(b, z)) -> +'(a, z)
+'(+(x, y), z) -> +'(x, +(y, z))
+'(+(x, y), z) -> +'(y, z)
F(+(x, y), z) -> +'(f(x, z), f(y, z))
F(+(x, y), z) -> F(x, z)
F(+(x, y), z) -> F(y, z)

Furthermore, R contains three SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳FwdInst


Dependency Pair:

+'(a, +(b, z)) -> +'(a, z)


Rules:


+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))


Strategy:

innermost




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

+'(a, +(b, z)) -> +'(a, z)
one new Dependency Pair is created:

+'(a, +(b, +(b, z''))) -> +'(a, +(b, z''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 4
             ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳FwdInst


Dependency Pair:

+'(a, +(b, +(b, z''))) -> +'(a, +(b, z''))


Rules:


+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))


Strategy:

innermost




The following dependency pair can be strictly oriented:

+'(a, +(b, +(b, z''))) -> +'(a, +(b, z''))


The following usable rules for innermost can be oriented:

+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))


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

resulting in one new DP problem.
Used Argument Filtering System:
+'(x1, x2) -> +'(x1, x2)
+(x1, x2) -> +(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 4
             ↳AFS
             ...
               →DP Problem 5
                 ↳Dependency Graph
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳FwdInst


Dependency Pair:


Rules:


+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳Forward Instantiation Transformation
       →DP Problem 3
         ↳FwdInst


Dependency Pair:

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


Rules:


+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 6
             ↳Argument Filtering and Ordering
       →DP Problem 3
         ↳FwdInst


Dependency Pair:

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


Rules:


+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


The following usable rules for innermost can be oriented:

+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))


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

resulting in one new DP problem.
Used Argument Filtering System:
+'(x1, x2) -> +'(x1, x2)
+(x1, x2) -> +(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
           →DP Problem 6
             ↳AFS
             ...
               →DP Problem 7
                 ↳Dependency Graph
       →DP Problem 3
         ↳FwdInst


Dependency Pair:


Rules:


+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳Forward Instantiation Transformation


Dependency Pair:

F(+(x, y), z) -> F(y, z)


Rules:


+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))


Strategy:

innermost




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

F(+(x, y), z) -> F(y, z)
one new Dependency Pair is created:

F(+(x, +(x'', y'')), z'') -> F(+(x'', y''), z'')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳FwdInst
           →DP Problem 8
             ↳Argument Filtering and Ordering


Dependency Pair:

F(+(x, +(x'', y'')), z'') -> F(+(x'', y''), z'')


Rules:


+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(+(x, +(x'', y'')), z'') -> F(+(x'', y''), z'')


The following usable rules for innermost can be oriented:

+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(b)=  0  
  POL(a)=  0  
  POL(F(x1, x2))=  1 + x1 + x2  
  POL(+(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
+(x1, x2) -> +(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
       →DP Problem 2
         ↳FwdInst
       →DP Problem 3
         ↳FwdInst
           →DP Problem 8
             ↳AFS
             ...
               →DP Problem 9
                 ↳Dependency Graph


Dependency Pair:


Rules:


+(a, b) -> +(b, a)
+(a, +(b, z)) -> +(b, +(a, z))
+(+(x, y), z) -> +(x, +(y, z))
f(a, y) -> a
f(b, y) -> b
f(+(x, y), z) -> +(f(x, z), f(y, z))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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