YES

The TRS could be proven terminating. The proof took 771 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (16ms).
 | – Problem 2 was processed with processor PolynomialLinearRange4iUR (288ms).
 |    | – Problem 3 was processed with processor PolynomialLinearRange4iUR (158ms).

Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

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)
app#(app(app(curry, f), x), y)app#(f, x)add#app#(curry, plus)
app#(app(plus, app(s, x)), y)app#(plus, x)app#(app(plus, app(s, x)), y)app#(app(plus, x), y)

Rewrite Rules

app(app(plus, 0), y)yapp(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)addapp(curry, plus)

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, add, curry

Strategy


The following SCCs where found

app#(app(app(curry, f), x), y) → app#(app(f, x), y)app#(app(app(curry, f), x), y) → app#(f, x)
app#(app(plus, app(s, x)), y) → app#(app(plus, x), y)

Problem 2: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

app#(app(app(curry, f), x), y)app#(app(f, x), y)app#(app(app(curry, f), x), y)app#(f, x)
app#(app(plus, app(s, x)), y)app#(app(plus, x), y)

Rewrite Rules

app(app(plus, 0), y)yapp(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)addapp(curry, plus)

Original Signature

Termination of terms over the following signature is verified: app, plus, 0, s, add, curry

Strategy


Polynomial Interpretation

Improved Usable rules

app(app(plus, 0), y)yapp(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)

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

app#(app(plus, app(s, x)), y)app#(app(plus, x), y)

Problem 3: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

app#(app(app(curry, f), x), y)app#(f, x)app#(app(app(curry, f), x), y)app#(app(f, x), y)

Rewrite Rules

app(app(plus, 0), y)yapp(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)addapp(curry, plus)

Original Signature

Termination of terms over the following signature is verified: plus, app, 0, s, add, curry

Strategy


Polynomial Interpretation

Improved Usable rules

app(app(plus, 0), y)yapp(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)

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

app#(app(app(curry, f), x), y)app#(app(f, x), y)app#(app(app(curry, f), x), y)app#(f, x)