Term Rewriting System R:
[f, x, h, t, l]
app(app(app(fold, f), nil), x) -> x
app(app(app(fold, f), app(app(cons, h), t)), x) -> app(app(app(fold, f), t), app(app(f, x), h))
app(sum, l) -> app(app(app(fold, add), l), 0)
app(app(app(fold, mul), l), 1) -> app(prod, l)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(app(fold, f), app(app(cons, h), t)), x) -> APP(app(app(fold, f), t), app(app(f, x), h))
APP(app(app(fold, f), app(app(cons, h), t)), x) -> APP(app(fold, f), t)
APP(app(app(fold, f), app(app(cons, h), t)), x) -> APP(app(f, x), h)
APP(app(app(fold, f), app(app(cons, h), t)), x) -> APP(f, x)
APP(sum, l) -> APP(app(app(fold, add), l), 0)
APP(sum, l) -> APP(app(fold, add), l)
APP(sum, l) -> APP(fold, add)
APP(app(app(fold, mul), l), 1) -> APP(prod, l)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(sum, l) -> APP(app(fold, add), l)
APP(sum, l) -> APP(app(app(fold, add), l), 0)
APP(app(app(fold, f), app(app(cons, h), t)), x) -> APP(f, x)
APP(app(app(fold, f), app(app(cons, h), t)), x) -> APP(app(f, x), h)
APP(app(app(fold, f), app(app(cons, h), t)), x) -> APP(app(app(fold, f), t), app(app(f, x), h))


Rules:


app(app(app(fold, f), nil), x) -> x
app(app(app(fold, f), app(app(cons, h), t)), x) -> app(app(app(fold, f), t), app(app(f, x), h))
app(sum, l) -> app(app(app(fold, add), l), 0)
app(app(app(fold, mul), l), 1) -> app(prod, l)


Strategy:

innermost




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

APP(sum, l) -> APP(app(fold, add), l)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

APP(app(app(fold, f), app(app(cons, h), t)), x) -> APP(f, x)
APP(app(app(fold, f), app(app(cons, h), t)), x) -> APP(app(f, x), h)
APP(app(app(fold, f), app(app(cons, h), t)), x) -> APP(app(app(fold, f), t), app(app(f, x), h))
APP(sum, l) -> APP(app(app(fold, add), l), 0)


Rules:


app(app(app(fold, f), nil), x) -> x
app(app(app(fold, f), app(app(cons, h), t)), x) -> app(app(app(fold, f), t), app(app(f, x), h))
app(sum, l) -> app(app(app(fold, add), l), 0)
app(app(app(fold, mul), l), 1) -> app(prod, l)


Strategy:

innermost




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

APP(app(app(fold, f), app(app(cons, h), t)), x) -> APP(f, x)
two new Dependency Pairs are created:

APP(app(app(fold, app(app(fold, f''), app(app(cons, h''), t''))), app(app(cons, h), t)), x'') -> APP(app(app(fold, f''), app(app(cons, h''), t'')), x'')
APP(app(app(fold, sum), app(app(cons, h), t)), x') -> APP(sum, x')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(app(fold, sum), app(app(cons, h), t)), x') -> APP(sum, x')
APP(app(app(fold, app(app(fold, f''), app(app(cons, h''), t''))), app(app(cons, h), t)), x'') -> APP(app(app(fold, f''), app(app(cons, h''), t'')), x'')
APP(sum, l) -> APP(app(app(fold, add), l), 0)
APP(app(app(fold, f), app(app(cons, h), t)), x) -> APP(app(app(fold, f), t), app(app(f, x), h))
APP(app(app(fold, f), app(app(cons, h), t)), x) -> APP(app(f, x), h)


Rules:


app(app(app(fold, f), nil), x) -> x
app(app(app(fold, f), app(app(cons, h), t)), x) -> app(app(app(fold, f), t), app(app(f, x), h))
app(sum, l) -> app(app(app(fold, add), l), 0)
app(app(app(fold, mul), l), 1) -> app(prod, l)


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:01 minutes