TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60049 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (189ms).
 | – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (73ms), PolynomialLinearRange4iUR (2214ms), DependencyGraph (69ms), PolynomialLinearRange8NegiUR (5200ms), DependencyGraph (60ms), ReductionPairSAT (52156ms)].

The following open problems remain:



Open Dependency Pair Problem 2

Dependency Pairs

app#(app(times, app(s, x)), y)app#(app(plus, app(app(times, x), y)), y)app#(app(app(fold, f), x), app(app(cons, y), z))app#(app(fold, f), x)
app#(app(app(fold, f), x), app(app(cons, y), z))app#(app(f, y), app(app(app(fold, f), x), z))app#(app(plus, app(s, x)), y)app#(plus, x)
app#(app(app(fold, f), x), app(app(cons, y), z))app#(app(app(fold, f), x), z)app#(app(app(fold, f), x), app(app(cons, y), z))app#(f, y)
app#(app(times, app(s, x)), y)app#(plus, app(app(times, x), y))app#(app(times, app(s, x)), y)app#(times, x)
app#(app(plus, app(s, x)), y)app#(s, app(app(plus, x), y))app#(app(times, app(s, x)), y)app#(app(times, x), y)
app#(app(app(fold, f), x), app(app(cons, y), z))app#(fold, f)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)

Rewrite Rules

app(app(app(fold, f), x), nil)xapp(app(app(fold, f), x), app(app(cons, y), z))app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(times, 0), y)0app(app(times, app(s, x)), y)app(app(plus, app(app(times, x), y)), y)
sumapp(app(fold, add), 0)prodapp(app(fold, mul), app(s, 0))

Original Signature

Termination of terms over the following signature is verified: plus, app, 0, s, times, sum, mul, add, fold, prod, cons, nil


Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

app#(app(times, app(s, x)), y)app#(app(plus, app(app(times, x), y)), y)app#(app(app(fold, f), x), app(app(cons, y), z))app#(app(fold, f), x)
app#(app(app(fold, f), x), app(app(cons, y), z))app#(app(f, y), app(app(app(fold, f), x), z))app#(app(app(fold, f), x), app(app(cons, y), z))app#(app(app(fold, f), x), z)
app#(app(plus, app(s, x)), y)app#(plus, x)prod#app#(app(fold, mul), app(s, 0))
app#(app(app(fold, f), x), app(app(cons, y), z))app#(f, y)app#(app(times, app(s, x)), y)app#(plus, app(app(times, x), y))
app#(app(times, app(s, x)), y)app#(times, x)app#(app(plus, app(s, x)), y)app#(s, app(app(plus, x), y))
sum#app#(app(fold, add), 0)app#(app(times, app(s, x)), y)app#(app(times, x), y)
prod#app#(fold, mul)prod#app#(s, 0)
app#(app(app(fold, f), x), app(app(cons, y), z))app#(fold, f)sum#app#(fold, add)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)

Rewrite Rules

app(app(app(fold, f), x), nil)xapp(app(app(fold, f), x), app(app(cons, y), z))app(app(f, y), app(app(app(fold, f), x), z))
app(app(plus, 0), y)yapp(app(plus, app(s, x)), y)app(s, app(app(plus, x), y))
app(app(times, 0), y)0app(app(times, app(s, x)), y)app(app(plus, app(app(times, x), y)), y)
sumapp(app(fold, add), 0)prodapp(app(fold, mul), app(s, 0))

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, times, sum, fold, add, mul, nil, cons, prod

Strategy


The following SCCs where found

app#(app(times, app(s, x)), y) → app#(plus, app(app(times, x), y))app#(app(times, app(s, x)), y) → app#(app(plus, app(app(times, x), y)), y)
app#(app(app(fold, f), x), app(app(cons, y), z)) → app#(app(fold, f), x)app#(app(times, app(s, x)), y) → app#(times, x)
app#(app(plus, app(s, x)), y) → app#(s, app(app(plus, x), y))app#(app(app(fold, f), x), app(app(cons, y), z)) → app#(app(f, y), app(app(app(fold, f), x), z))
app#(app(times, app(s, x)), y) → app#(app(times, x), y)app#(app(app(fold, f), x), app(app(cons, y), z)) → app#(app(app(fold, f), x), z)
app#(app(app(fold, f), x), app(app(cons, y), z)) → app#(fold, f)app#(app(plus, app(s, x)), y) → app#(plus, x)
app#(app(plus, app(s, x)), y) → app#(app(plus, x), y)app#(app(app(fold, f), x), app(app(cons, y), z)) → app#(f, y)