YES

The TRS could be proven terminating. The proof took 570 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (8ms).
 | – Problem 2 was processed with processor PolynomialLinearRange4 (102ms).
 |    | – Problem 4 was processed with processor PolynomialLinearRange4 (117ms).
 | – Problem 3 was processed with processor PolynomialLinearRange4 (44ms).

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

T(f(g(X), Y)) → f#(g(X), Y)T(f(x_1, x_2)) → T(x_1)
T(f(x_1, x_2)) → T(x_2)T(g(x_1)) → T(x_1)
f#(g(X), Y) → f#(X, f(g(X), Y))

Rewrite Rules

f(g(X), Y) → f(X, f(g(X), Y))

Original Signature

Termination of terms over the following signature is verified: f, g

Strategy

Context-sensitive strategy:
μ(T) = ∅
μ(f) = μ(g) = μ(f#) = {1}

The following SCCs where found

T(f(x_1, x_2)) → T(x_1)T(f(x_1, x_2)) → T(x_2)
T(g(x_1)) → T(x_1)
f#(g(X), Y) → f#(X, f(g(X), Y))

Problem 2: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

T(f(x_1, x_2)) → T(x_1)T(f(x_1, x_2)) → T(x_2)
T(g(x_1)) → T(x_1)

Rewrite Rules

f(g(X), Y) → f(X, f(g(X), Y))

Original Signature

Termination of terms over the following signature is verified: f, g

Strategy

Context-sensitive strategy:
μ(T) = ∅
μ(f) = μ(g) = μ(f#) = {1}

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(g(x_1)) → T(x_1)

Problem 4: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

T(f(x_1, x_2)) → T(x_2)T(f(x_1, x_2)) → T(x_1)

Rewrite Rules

f(g(X), Y) → f(X, f(g(X), Y))

Original Signature

Termination of terms over the following signature is verified: f, g

Strategy

Context-sensitive strategy:
μ(T) = ∅
μ(f) = μ(g) = μ(f#) = {1}

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(f(x_1, x_2)) → T(x_1)T(f(x_1, x_2)) → T(x_2)

Problem 3: PolynomialLinearRange4

Dependency Pair Problem

Dependency Pairs

f#(g(X), Y) → f#(X, f(g(X), Y))

Rewrite Rules

f(g(X), Y) → f(X, f(g(X), Y))

Original Signature

Termination of terms over the following signature is verified: f, g

Strategy

Context-sensitive strategy:
μ(T) = ∅
μ(f) = μ(g) = μ(f#) = {1}

Polynomial Interpretation

Standard Usable rules

f(g(X), Y) → f(X, f(g(X), Y))

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

f#(g(X), Y) → f#(X, f(g(X), Y))