YES

The TRS could be proven terminating. The proof took 391 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (8ms).
 | – Problem 2 was processed with processor PolynomialLinearRange4 (69ms).
 |    | – Problem 4 was processed with processor PolynomialLinearRange4 (119ms).
 | – Problem 3 was processed with processor PolynomialLinearRange4 (85ms).

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

after#(s(N), cons(X, XS)) → T(XS)T(s(x_1)) → T(x_1)
T(from(x_1)) → T(x_1)T(from(s(X))) → from#(s(X))
after#(s(N), cons(X, XS)) → after#(N, XS)

Rewrite Rules

from(X) → cons(X, from(s(X)))after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)

Original Signature

Termination of terms over the following signature is verified: after, 0, s, from, cons

Strategy

Context-sensitive strategy:
μ(T) = μ(0) = ∅
μ(from#) = μ(s) = μ(from) = μ(cons) = {1}
μ(after) = μ(after#) = {1, 2}

The following SCCs where found

T(s(x_1)) → T(x_1)T(from(x_1)) → T(x_1)
after#(s(N), cons(X, XS)) → after#(N, XS)

Problem 2: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

T(s(x_1)) → T(x_1)T(from(x_1)) → T(x_1)

Rewrite Rules

from(X) → cons(X, from(s(X)))after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)

Original Signature

Termination of terms over the following signature is verified: after, 0, s, from, cons

Strategy

Context-sensitive strategy:
μ(T) = μ(0) = ∅
μ(s) = μ(from#) = μ(from) = μ(cons) = {1}
μ(after) = μ(after#) = {1, 2}

Polynomial Interpretation

There are no usable rules

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

T(s(x_1)) → T(x_1)

Problem 4: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

T(from(x_1)) → T(x_1)

Rewrite Rules

from(X) → cons(X, from(s(X)))after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)

Original Signature

Termination of terms over the following signature is verified: after, 0, s, from, cons

Strategy

Context-sensitive strategy:
μ(T) = μ(0) = ∅
μ(from#) = μ(s) = μ(from) = μ(cons) = {1}
μ(after) = μ(after#) = {1, 2}

Polynomial Interpretation

There are no usable rules

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

T(from(x_1)) → T(x_1)

Problem 3: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

after#(s(N), cons(X, XS)) → after#(N, XS)

Rewrite Rules

from(X) → cons(X, from(s(X)))after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, XS)

Original Signature

Termination of terms over the following signature is verified: after, 0, s, from, cons

Strategy

Context-sensitive strategy:
μ(T) = μ(0) = ∅
μ(s) = μ(from#) = μ(from) = μ(cons) = {1}
μ(after) = μ(after#) = {1, 2}

Polynomial Interpretation

Standard Usable rules

from(X) → cons(X, from(s(X)))after(s(N), cons(X, XS)) → after(N, XS)
after(0, XS) → XS

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

after#(s(N), cons(X, XS)) → after#(N, XS)