Term Rewriting System R:
[y, x, f]
app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(app(curry, f), x), y) -> app(app(f, x), y)
add -> app(curry, plus)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(plus, app(s, x)), y) -> APP(s, app(app(plus, x), y))
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(plus, app(s, x)), y) -> APP(plus, x)
APP(app(app(curry, f), x), y) -> APP(app(f, x), y)
APP(app(app(curry, f), x), y) -> APP(f, x)
ADD -> APP(curry, plus)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

APP(app(app(curry, f), x), y) -> APP(f, x)
APP(app(app(curry, f), x), y) -> APP(app(f, x), y)
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)


Rules:


app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(app(curry, f), x), y) -> app(app(f, x), y)
add -> app(curry, plus)


Strategy:

innermost




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

APP(app(app(curry, f), x), y) -> APP(f, x)
two new Dependency Pairs are created:

APP(app(app(curry, app(app(curry, f''), x'')), x0), y) -> APP(app(app(curry, f''), x''), x0)
APP(app(app(curry, app(plus, app(s, x''))), x0), y) -> APP(app(plus, app(s, x'')), x0)

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

APP(app(app(curry, app(plus, app(s, x''))), x0), y) -> APP(app(plus, app(s, x'')), x0)
APP(app(app(curry, app(app(curry, f''), x'')), x0), y) -> APP(app(app(curry, f''), x''), x0)
APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)
APP(app(app(curry, f), x), y) -> APP(app(f, x), y)


Rules:


app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(app(curry, f), x), y) -> app(app(f, x), y)
add -> app(curry, plus)


Strategy:

innermost




The following dependency pairs can be strictly oriented:

APP(app(app(curry, app(plus, app(s, x''))), x0), y) -> APP(app(plus, app(s, x'')), x0)
APP(app(app(curry, app(app(curry, f''), x'')), x0), y) -> APP(app(app(curry, f''), x''), x0)
APP(app(app(curry, f), x), y) -> APP(app(f, x), y)


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

app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(app(curry, f), x), y) -> app(app(f, x), y)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(plus)=  0  
  POL(0)=  1  
  POL(curry)=  1  
  POL(s)=  0  
  POL(app(x1, x2))=  x1 + x2  
  POL(APP(x1, x2))=  1 + x1  

resulting in one new DP problem. 



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


Dependency Pair:

APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)


Rules:


app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(app(curry, f), x), y) -> app(app(f, x), y)
add -> app(curry, plus)


Strategy:

innermost




The following dependency pair can be strictly oriented:

APP(app(plus, app(s, x)), y) -> APP(app(plus, x), y)


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

app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(app(curry, f), x), y) -> app(app(f, x), y)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(plus)=  0  
  POL(0)=  1  
  POL(curry)=  1  
  POL(s)=  0  
  POL(app(x1, x2))=  1 + x1 + x2  
  POL(APP(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(app(curry, f), x), y) -> app(app(f, x), y)
add -> app(curry, plus)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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