YES
The TRS could be proven terminating. The proof took 1817 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (93ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor PolynomialOrderingProcessor (318ms).
| | – Problem 7 was processed with processor DependencyGraph (2ms).
| | | – Problem 8 was processed with processor PolynomialOrderingProcessor (13ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| – Problem 5 was processed with processor PolynomialLinearRange4iUR (794ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| gcd#(s(x), s(y)) | → | gcd#(minus(max(x, y), min(x, transform(y))), s(min(x, y))) | | gcd#(s(x), s(y)) | → | min#(x, transform(y)) |
| transform#(s(x)) | → | transform#(x) | | gcd#(s(x), s(y)) | → | max#(x, y) |
| transform#(cons(x, y)) | → | cons#(x, x) | | max#(s(x), s(y)) | → | max#(x, y) |
| cons#(cons(x, z), s(y)) | → | transform#(x) | | gcd#(s(x), s(y)) | → | minus#(max(x, y), min(x, transform(y))) |
| transform#(cons(x, y)) | → | cons#(cons(x, x), x) | | gcd#(s(x), s(y)) | → | transform#(y) |
| cons#(x, cons(y, s(z))) | → | cons#(y, x) | | min#(s(x), s(y)) | → | min#(x, y) |
| minus#(s(x), s(y)) | → | minus#(x, y) | | gcd#(s(x), s(y)) | → | min#(x, y) |
Rewrite Rules
| min(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(x, 0) | → | x |
| max(0, y) | → | y | | max(s(x), s(y)) | → | s(max(x, y)) |
| minus(x, 0) | → | x | | minus(s(x), s(y)) | → | s(minus(x, y)) |
| gcd(s(x), s(y)) | → | gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y))) | | transform(x) | → | s(s(x)) |
| transform(cons(x, y)) | → | cons(cons(x, x), x) | | transform(cons(x, y)) | → | y |
| transform(s(x)) | → | s(s(transform(x))) | | cons(x, y) | → | y |
| cons(x, cons(y, s(z))) | → | cons(y, x) | | cons(cons(x, z), s(y)) | → | transform(x) |
Original Signature
Termination of terms over the following signature is verified: min, transform, 0, max, minus, s, gcd, cons
Strategy
The following SCCs where found
| gcd#(s(x), s(y)) → gcd#(minus(max(x, y), min(x, transform(y))), s(min(x, y))) |
| transform#(cons(x, y)) → cons#(cons(x, x), x) | cons#(x, cons(y, s(z))) → cons#(y, x) |
| transform#(s(x)) → transform#(x) | transform#(cons(x, y)) → cons#(x, x) |
| cons#(cons(x, z), s(y)) → transform#(x) |
| min#(s(x), s(y)) → min#(x, y) |
| minus#(s(x), s(y)) → minus#(x, y) |
| max#(s(x), s(y)) → max#(x, y) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| max#(s(x), s(y)) | → | max#(x, y) |
Rewrite Rules
| min(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(x, 0) | → | x |
| max(0, y) | → | y | | max(s(x), s(y)) | → | s(max(x, y)) |
| minus(x, 0) | → | x | | minus(s(x), s(y)) | → | s(minus(x, y)) |
| gcd(s(x), s(y)) | → | gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y))) | | transform(x) | → | s(s(x)) |
| transform(cons(x, y)) | → | cons(cons(x, x), x) | | transform(cons(x, y)) | → | y |
| transform(s(x)) | → | s(s(transform(x))) | | cons(x, y) | → | y |
| cons(x, cons(y, s(z))) | → | cons(y, x) | | cons(cons(x, z), s(y)) | → | transform(x) |
Original Signature
Termination of terms over the following signature is verified: min, transform, 0, max, minus, s, gcd, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| max#(s(x), s(y)) | → | max#(x, y) |
Problem 3: PolynomialOrderingProcessor
Dependency Pair Problem
Dependency Pairs
| transform#(cons(x, y)) | → | cons#(cons(x, x), x) | | cons#(x, cons(y, s(z))) | → | cons#(y, x) |
| transform#(s(x)) | → | transform#(x) | | transform#(cons(x, y)) | → | cons#(x, x) |
| cons#(cons(x, z), s(y)) | → | transform#(x) |
Rewrite Rules
| min(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(x, 0) | → | x |
| max(0, y) | → | y | | max(s(x), s(y)) | → | s(max(x, y)) |
| minus(x, 0) | → | x | | minus(s(x), s(y)) | → | s(minus(x, y)) |
| gcd(s(x), s(y)) | → | gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y))) | | transform(x) | → | s(s(x)) |
| transform(cons(x, y)) | → | cons(cons(x, x), x) | | transform(cons(x, y)) | → | y |
| transform(s(x)) | → | s(s(transform(x))) | | cons(x, y) | → | y |
| cons(x, cons(y, s(z))) | → | cons(y, x) | | cons(cons(x, z), s(y)) | → | transform(x) |
Original Signature
Termination of terms over the following signature is verified: min, transform, 0, max, minus, s, gcd, cons
Strategy
Polynomial Interpretation
- 0: -2
- cons(x,y): y + 2x + 1
- cons#(x,y): y + x + 1
- gcd(x,y): -2
- max(x,y): -2
- min(x,y): -2
- minus(x,y): -2
- s(x): x
- transform(x): 4x
- transform#(x): 2x + 2
Improved Usable rules
| cons(x, cons(y, s(z))) | → | cons(y, x) | | cons(cons(x, z), s(y)) | → | transform(x) |
| transform(x) | → | s(s(x)) | | transform(cons(x, y)) | → | cons(cons(x, x), x) |
| transform(s(x)) | → | s(s(transform(x))) | | transform(cons(x, y)) | → | y |
| cons(x, y) | → | y |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| transform#(cons(x, y)) | → | cons#(cons(x, x), x) | | cons#(x, cons(y, s(z))) | → | cons#(y, x) |
| transform#(cons(x, y)) | → | cons#(x, x) |
Problem 7: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| transform#(s(x)) | → | transform#(x) | | cons#(cons(x, z), s(y)) | → | transform#(x) |
Rewrite Rules
| min(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(x, 0) | → | x |
| max(0, y) | → | y | | max(s(x), s(y)) | → | s(max(x, y)) |
| minus(x, 0) | → | x | | minus(s(x), s(y)) | → | s(minus(x, y)) |
| gcd(s(x), s(y)) | → | gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y))) | | transform(x) | → | s(s(x)) |
| transform(cons(x, y)) | → | cons(cons(x, x), x) | | transform(cons(x, y)) | → | y |
| transform(s(x)) | → | s(s(transform(x))) | | cons(x, y) | → | y |
| cons(x, cons(y, s(z))) | → | cons(y, x) | | cons(cons(x, z), s(y)) | → | transform(x) |
Original Signature
Termination of terms over the following signature is verified: min, transform, minus, max, 0, s, cons, gcd
Strategy
The following SCCs where found
| transform#(s(x)) → transform#(x) |
Problem 8: PolynomialOrderingProcessor
Dependency Pair Problem
Dependency Pairs
| transform#(s(x)) | → | transform#(x) |
Rewrite Rules
| min(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(x, 0) | → | x |
| max(0, y) | → | y | | max(s(x), s(y)) | → | s(max(x, y)) |
| minus(x, 0) | → | x | | minus(s(x), s(y)) | → | s(minus(x, y)) |
| gcd(s(x), s(y)) | → | gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y))) | | transform(x) | → | s(s(x)) |
| transform(cons(x, y)) | → | cons(cons(x, x), x) | | transform(cons(x, y)) | → | y |
| transform(s(x)) | → | s(s(transform(x))) | | cons(x, y) | → | y |
| cons(x, cons(y, s(z))) | → | cons(y, x) | | cons(cons(x, z), s(y)) | → | transform(x) |
Original Signature
Termination of terms over the following signature is verified: min, transform, minus, max, 0, s, cons, gcd
Strategy
Polynomial Interpretation
- 0: -2
- cons(x,y): -2
- gcd(x,y): -2
- max(x,y): -2
- min(x,y): -2
- minus(x,y): -2
- s(x): x + 1
- transform(x): -2
- transform#(x): 4x - 2
There are no usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| transform#(s(x)) | → | transform#(x) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| minus#(s(x), s(y)) | → | minus#(x, y) |
Rewrite Rules
| min(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(x, 0) | → | x |
| max(0, y) | → | y | | max(s(x), s(y)) | → | s(max(x, y)) |
| minus(x, 0) | → | x | | minus(s(x), s(y)) | → | s(minus(x, y)) |
| gcd(s(x), s(y)) | → | gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y))) | | transform(x) | → | s(s(x)) |
| transform(cons(x, y)) | → | cons(cons(x, x), x) | | transform(cons(x, y)) | → | y |
| transform(s(x)) | → | s(s(transform(x))) | | cons(x, y) | → | y |
| cons(x, cons(y, s(z))) | → | cons(y, x) | | cons(cons(x, z), s(y)) | → | transform(x) |
Original Signature
Termination of terms over the following signature is verified: min, transform, 0, max, minus, s, gcd, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| minus#(s(x), s(y)) | → | minus#(x, y) |
Problem 5: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| gcd#(s(x), s(y)) | → | gcd#(minus(max(x, y), min(x, transform(y))), s(min(x, y))) |
Rewrite Rules
| min(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(x, 0) | → | x |
| max(0, y) | → | y | | max(s(x), s(y)) | → | s(max(x, y)) |
| minus(x, 0) | → | x | | minus(s(x), s(y)) | → | s(minus(x, y)) |
| gcd(s(x), s(y)) | → | gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y))) | | transform(x) | → | s(s(x)) |
| transform(cons(x, y)) | → | cons(cons(x, x), x) | | transform(cons(x, y)) | → | y |
| transform(s(x)) | → | s(s(transform(x))) | | cons(x, y) | → | y |
| cons(x, cons(y, s(z))) | → | cons(y, x) | | cons(cons(x, z), s(y)) | → | transform(x) |
Original Signature
Termination of terms over the following signature is verified: min, transform, 0, max, minus, s, gcd, cons
Strategy
Polynomial Interpretation
- 0: 0
- cons(x,y): 2
- gcd(x,y): 0
- gcd#(x,y): y + 2x
- max(x,y): y + x
- min(x,y): x
- minus(x,y): x
- s(x): 2x + 1
- transform(x): 0
Improved Usable rules
| min(0, y) | → | 0 | | max(s(x), s(y)) | → | s(max(x, y)) |
| minus(s(x), s(y)) | → | s(minus(x, y)) | | min(s(x), s(y)) | → | s(min(x, y)) |
| max(0, y) | → | y | | minus(x, 0) | → | x |
| min(x, 0) | → | 0 | | max(x, 0) | → | x |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| gcd#(s(x), s(y)) | → | gcd#(minus(max(x, y), min(x, transform(y))), s(min(x, y))) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| min#(s(x), s(y)) | → | min#(x, y) |
Rewrite Rules
| min(x, 0) | → | 0 | | min(0, y) | → | 0 |
| min(s(x), s(y)) | → | s(min(x, y)) | | max(x, 0) | → | x |
| max(0, y) | → | y | | max(s(x), s(y)) | → | s(max(x, y)) |
| minus(x, 0) | → | x | | minus(s(x), s(y)) | → | s(minus(x, y)) |
| gcd(s(x), s(y)) | → | gcd(minus(max(x, y), min(x, transform(y))), s(min(x, y))) | | transform(x) | → | s(s(x)) |
| transform(cons(x, y)) | → | cons(cons(x, x), x) | | transform(cons(x, y)) | → | y |
| transform(s(x)) | → | s(s(transform(x))) | | cons(x, y) | → | y |
| cons(x, cons(y, s(z))) | → | cons(y, x) | | cons(cons(x, z), s(y)) | → | transform(x) |
Original Signature
Termination of terms over the following signature is verified: min, transform, 0, max, minus, s, gcd, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| min#(s(x), s(y)) | → | min#(x, y) |