YES
The TRS could be proven terminating. The proof took 43351 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (695ms).
| – Problem 2 was processed with processor SubtermCriterion (0ms).
| | – Problem 9 was processed with processor SubtermCriterion (0ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor PolynomialLinearRange4 (642ms).
| | – Problem 12 was processed with processor PolynomialLinearRange4 (261ms).
| | | – Problem 13 was processed with processor PolynomialLinearRange4 (269ms).
| | | | – Problem 14 was processed with processor PolynomialLinearRange4 (154ms).
| | | | | – Problem 15 was processed with processor PolynomialLinearRange4 (184ms).
| | | | | | – Problem 16 was processed with processor PolynomialLinearRange4 (148ms).
| | | | | | | – Problem 17 was processed with processor PolynomialLinearRange4 (138ms).
| | | | | | | | – Problem 18 was processed with processor PolynomialLinearRange4 (134ms).
| | | | | | | | | – Problem 19 was processed with processor PolynomialLinearRange4 (131ms).
| | | | | | | | | | – Problem 20 was processed with processor PolynomialLinearRange4 (128ms).
| – Problem 5 was processed with processor SubtermCriterion (0ms).
| | – Problem 10 was processed with processor SubtermCriterion (0ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| | – Problem 11 was processed with processor SubtermCriterion (1ms).
| – Problem 7 was processed with processor SubtermCriterion (0ms).
| – Problem 8 was processed with processor SubtermCriterion (0ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) | | active#(fst(0, Z)) | → | mark#(nil) |
| add#(X1, mark(X2)) | → | add#(X1, X2) | | active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) |
| mark#(fst(X1, X2)) | → | mark#(X2) | | active#(fst(s(X), cons(Y, Z))) | → | fst#(X, Z) |
| mark#(fst(X1, X2)) | → | active#(fst(mark(X1), mark(X2))) | | mark#(add(X1, X2)) | → | mark#(X2) |
| active#(len(cons(X, Z))) | → | len#(Z) | | mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) |
| active#(fst(s(X), cons(Y, Z))) | → | mark#(cons(Y, fst(X, Z))) | | mark#(add(X1, X2)) | → | mark#(X1) |
| active#(len(nil)) | → | mark#(0) | | fst#(active(X1), X2) | → | fst#(X1, X2) |
| mark#(from(X)) | → | mark#(X) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(len(X)) | → | len#(mark(X)) | | mark#(len(X)) | → | active#(len(mark(X))) |
| active#(add(0, X)) | → | mark#(X) | | add#(mark(X1), X2) | → | add#(X1, X2) |
| active#(from(X)) | → | mark#(cons(X, from(s(X)))) | | mark#(fst(X1, X2)) | → | mark#(X1) |
| mark#(fst(X1, X2)) | → | fst#(mark(X1), mark(X2)) | | mark#(s(X)) | → | active#(s(X)) |
| active#(from(X)) | → | s#(X) | | fst#(X1, mark(X2)) | → | fst#(X1, X2) |
| cons#(X1, active(X2)) | → | cons#(X1, X2) | | active#(add(s(X), Y)) | → | add#(X, Y) |
| mark#(from(X)) | → | from#(mark(X)) | | len#(mark(X)) | → | len#(X) |
| mark#(from(X)) | → | active#(from(mark(X))) | | mark#(add(X1, X2)) | → | add#(mark(X1), mark(X2)) |
| from#(active(X)) | → | from#(X) | | add#(active(X1), X2) | → | add#(X1, X2) |
| mark#(len(X)) | → | mark#(X) | | active#(len(cons(X, Z))) | → | s#(len(Z)) |
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | from#(mark(X)) | → | from#(X) |
| mark#(nil) | → | active#(nil) | | fst#(X1, active(X2)) | → | fst#(X1, X2) |
| active#(add(s(X), Y)) | → | s#(add(X, Y)) | | active#(from(X)) | → | cons#(X, from(s(X))) |
| add#(X1, active(X2)) | → | add#(X1, X2) | | cons#(X1, mark(X2)) | → | cons#(X1, X2) |
| mark#(cons(X1, X2)) | → | cons#(mark(X1), X2) | | mark#(0) | → | active#(0) |
| cons#(active(X1), X2) | → | cons#(X1, X2) | | active#(len(cons(X, Z))) | → | mark#(s(len(Z))) |
| s#(mark(X)) | → | s#(X) | | mark#(s(X)) | → | s#(X) |
| s#(active(X)) | → | s#(X) | | active#(fst(s(X), cons(Y, Z))) | → | cons#(Y, fst(X, Z)) |
| fst#(mark(X1), X2) | → | fst#(X1, X2) | | len#(active(X)) | → | len#(X) |
| active#(from(X)) | → | from#(s(X)) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, from, len, add, nil, cons
Strategy
The following SCCs where found
| from#(active(X)) → from#(X) | from#(mark(X)) → from#(X) |
| s#(mark(X)) → s#(X) | s#(active(X)) → s#(X) |
| cons#(X1, active(X2)) → cons#(X1, X2) | cons#(mark(X1), X2) → cons#(X1, X2) |
| cons#(active(X1), X2) → cons#(X1, X2) | cons#(X1, mark(X2)) → cons#(X1, X2) |
| fst#(X1, mark(X2)) → fst#(X1, X2) | fst#(X1, active(X2)) → fst#(X1, X2) |
| fst#(active(X1), X2) → fst#(X1, X2) | fst#(mark(X1), X2) → fst#(X1, X2) |
| len#(mark(X)) → len#(X) | len#(active(X)) → len#(X) |
| mark#(cons(X1, X2)) → active#(cons(mark(X1), X2)) | mark#(from(X)) → mark#(X) |
| mark#(cons(X1, X2)) → mark#(X1) | active#(add(s(X), Y)) → mark#(s(add(X, Y))) |
| active#(add(0, X)) → mark#(X) | mark#(len(X)) → active#(len(mark(X))) |
| active#(len(cons(X, Z))) → mark#(s(len(Z))) | mark#(fst(X1, X2)) → mark#(X1) |
| active#(from(X)) → mark#(cons(X, from(s(X)))) | mark#(fst(X1, X2)) → mark#(X2) |
| mark#(s(X)) → active#(s(X)) | mark#(from(X)) → active#(from(mark(X))) |
| mark#(fst(X1, X2)) → active#(fst(mark(X1), mark(X2))) | mark#(len(X)) → mark#(X) |
| mark#(add(X1, X2)) → mark#(X2) | mark#(add(X1, X2)) → active#(add(mark(X1), mark(X2))) |
| mark#(add(X1, X2)) → mark#(X1) | active#(fst(s(X), cons(Y, Z))) → mark#(cons(Y, fst(X, Z))) |
| add#(active(X1), X2) → add#(X1, X2) | add#(X1, mark(X2)) → add#(X1, X2) |
| add#(X1, active(X2)) → add#(X1, X2) | add#(mark(X1), X2) → add#(X1, X2) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| add#(active(X1), X2) | → | add#(X1, X2) | | add#(X1, mark(X2)) | → | add#(X1, X2) |
| add#(X1, active(X2)) | → | add#(X1, X2) | | add#(mark(X1), X2) | → | add#(X1, X2) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, from, len, add, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| add#(active(X1), X2) | → | add#(X1, X2) | | add#(mark(X1), X2) | → | add#(X1, X2) |
Problem 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| add#(X1, mark(X2)) | → | add#(X1, X2) | | add#(X1, active(X2)) | → | add#(X1, X2) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, len, from, add, cons, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| add#(X1, mark(X2)) | → | add#(X1, X2) | | add#(X1, active(X2)) | → | add#(X1, X2) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(active(X)) | → | s#(X) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, from, len, add, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| s#(mark(X)) | → | s#(X) | | s#(active(X)) | → | s#(X) |
Problem 4: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) | | mark#(from(X)) | → | mark#(X) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) |
| mark#(len(X)) | → | active#(len(mark(X))) | | active#(add(0, X)) | → | mark#(X) |
| active#(from(X)) | → | mark#(cons(X, from(s(X)))) | | mark#(fst(X1, X2)) | → | mark#(X1) |
| active#(len(cons(X, Z))) | → | mark#(s(len(Z))) | | mark#(s(X)) | → | active#(s(X)) |
| mark#(fst(X1, X2)) | → | mark#(X2) | | mark#(from(X)) | → | active#(from(mark(X))) |
| mark#(fst(X1, X2)) | → | active#(fst(mark(X1), mark(X2))) | | mark#(len(X)) | → | mark#(X) |
| mark#(add(X1, X2)) | → | mark#(X2) | | mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) |
| mark#(add(X1, X2)) | → | mark#(X1) | | active#(fst(s(X), cons(Y, Z))) | → | mark#(cons(Y, fst(X, Z))) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, from, len, add, nil, cons
Strategy
Polynomial Interpretation
- 0: 3
- active(x): 1
- active#(x): x
- add(x,y): 2
- cons(x,y): 0
- from(x): 2
- fst(x,y): 2
- len(x): 2
- mark(x): 1
- mark#(x): 2
- nil: 2
- s(x): 2
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| fst(active(X1), X2) | → | fst(X1, X2) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(s(X)) | → | active(s(X)) | | add(X1, mark(X2)) | → | add(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | active(len(cons(X, Z))) | → | mark(s(len(Z))) |
| mark(from(X)) | → | active(from(mark(X))) | | fst(X1, mark(X2)) | → | fst(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | add(mark(X1), X2) | → | add(X1, X2) |
| mark(nil) | → | active(nil) | | mark(0) | → | active(0) |
| s(active(X)) | → | s(X) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(len(X)) | → | active(len(mark(X))) |
| len(active(X)) | → | len(X) | | active(fst(0, Z)) | → | mark(nil) |
| add(active(X1), X2) | → | add(X1, X2) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| len(mark(X)) | → | len(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| active(add(0, X)) | → | mark(X) | | s(mark(X)) | → | s(X) |
| active(len(nil)) | → | mark(0) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) |
Problem 12: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(from(X)) | → | mark#(X) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) | | mark#(len(X)) | → | active#(len(mark(X))) |
| active#(add(0, X)) | → | mark#(X) | | active#(from(X)) | → | mark#(cons(X, from(s(X)))) |
| mark#(fst(X1, X2)) | → | mark#(X1) | | active#(len(cons(X, Z))) | → | mark#(s(len(Z))) |
| mark#(s(X)) | → | active#(s(X)) | | mark#(fst(X1, X2)) | → | mark#(X2) |
| mark#(from(X)) | → | active#(from(mark(X))) | | mark#(fst(X1, X2)) | → | active#(fst(mark(X1), mark(X2))) |
| mark#(add(X1, X2)) | → | mark#(X2) | | mark#(len(X)) | → | mark#(X) |
| mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) | | active#(fst(s(X), cons(Y, Z))) | → | mark#(cons(Y, fst(X, Z))) |
| mark#(add(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, len, from, add, cons, nil
Strategy
Polynomial Interpretation
- 0: 3
- active(x): 1
- active#(x): 2x
- add(x,y): 1
- cons(x,y): 1
- from(x): 1
- fst(x,y): 1
- len(x): 1
- mark(x): 1
- mark#(x): 2
- nil: 0
- s(x): 0
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| fst(active(X1), X2) | → | fst(X1, X2) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(s(X)) | → | active(s(X)) | | add(X1, mark(X2)) | → | add(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | active(len(cons(X, Z))) | → | mark(s(len(Z))) |
| mark(from(X)) | → | active(from(mark(X))) | | fst(X1, mark(X2)) | → | fst(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | add(mark(X1), X2) | → | add(X1, X2) |
| mark(nil) | → | active(nil) | | mark(0) | → | active(0) |
| s(active(X)) | → | s(X) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(len(X)) | → | active(len(mark(X))) |
| len(active(X)) | → | len(X) | | active(fst(0, Z)) | → | mark(nil) |
| add(active(X1), X2) | → | add(X1, X2) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| len(mark(X)) | → | len(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| active(add(0, X)) | → | mark(X) | | s(mark(X)) | → | s(X) |
| active(len(nil)) | → | mark(0) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(s(X)) | → | active#(s(X)) |
Problem 13: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(from(X)) | → | mark#(X) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) | | mark#(len(X)) | → | active#(len(mark(X))) |
| active#(add(0, X)) | → | mark#(X) | | active#(from(X)) | → | mark#(cons(X, from(s(X)))) |
| mark#(fst(X1, X2)) | → | mark#(X1) | | active#(len(cons(X, Z))) | → | mark#(s(len(Z))) |
| mark#(fst(X1, X2)) | → | mark#(X2) | | mark#(from(X)) | → | active#(from(mark(X))) |
| mark#(fst(X1, X2)) | → | active#(fst(mark(X1), mark(X2))) | | mark#(add(X1, X2)) | → | mark#(X2) |
| mark#(len(X)) | → | mark#(X) | | mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) |
| mark#(add(X1, X2)) | → | mark#(X1) | | active#(fst(s(X), cons(Y, Z))) | → | mark#(cons(Y, fst(X, Z))) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, from, len, add, nil, cons
Strategy
Polynomial Interpretation
- 0: 0
- active(x): x
- active#(x): x + 1
- add(x,y): y + x
- cons(x,y): 2x
- from(x): 2x
- fst(x,y): 2y + x
- len(x): 2x + 2
- mark(x): x
- mark#(x): x + 1
- nil: 0
- s(x): 1
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| fst(active(X1), X2) | → | fst(X1, X2) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(s(X)) | → | active(s(X)) | | add(X1, mark(X2)) | → | add(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | active(len(cons(X, Z))) | → | mark(s(len(Z))) |
| mark(from(X)) | → | active(from(mark(X))) | | fst(X1, mark(X2)) | → | fst(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | add(mark(X1), X2) | → | add(X1, X2) |
| mark(nil) | → | active(nil) | | mark(0) | → | active(0) |
| s(active(X)) | → | s(X) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(len(X)) | → | active(len(mark(X))) |
| len(active(X)) | → | len(X) | | active(fst(0, Z)) | → | mark(nil) |
| add(active(X1), X2) | → | add(X1, X2) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| len(mark(X)) | → | len(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| active(add(0, X)) | → | mark(X) | | s(mark(X)) | → | s(X) |
| active(len(nil)) | → | mark(0) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(len(cons(X, Z))) | → | mark#(s(len(Z))) | | mark#(len(X)) | → | mark#(X) |
| active#(fst(s(X), cons(Y, Z))) | → | mark#(cons(Y, fst(X, Z))) |
Problem 14: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(from(X)) | → | mark#(X) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) | | mark#(len(X)) | → | active#(len(mark(X))) |
| active#(add(0, X)) | → | mark#(X) | | active#(from(X)) | → | mark#(cons(X, from(s(X)))) |
| mark#(fst(X1, X2)) | → | mark#(X1) | | mark#(fst(X1, X2)) | → | mark#(X2) |
| mark#(from(X)) | → | active#(from(mark(X))) | | mark#(fst(X1, X2)) | → | active#(fst(mark(X1), mark(X2))) |
| mark#(add(X1, X2)) | → | mark#(X2) | | mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) |
| mark#(add(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, len, from, add, cons, nil
Strategy
Polynomial Interpretation
- 0: 3
- active(x): 1
- active#(x): x + 1
- add(x,y): 1
- cons(x,y): 1
- from(x): 1
- fst(x,y): 1
- len(x): 0
- mark(x): 1
- mark#(x): 2
- nil: 3
- s(x): 1
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| fst(active(X1), X2) | → | fst(X1, X2) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(s(X)) | → | active(s(X)) | | add(X1, mark(X2)) | → | add(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | active(len(cons(X, Z))) | → | mark(s(len(Z))) |
| mark(from(X)) | → | active(from(mark(X))) | | fst(X1, mark(X2)) | → | fst(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | add(mark(X1), X2) | → | add(X1, X2) |
| mark(nil) | → | active(nil) | | mark(0) | → | active(0) |
| s(active(X)) | → | s(X) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(len(X)) | → | active(len(mark(X))) |
| len(active(X)) | → | len(X) | | active(fst(0, Z)) | → | mark(nil) |
| add(active(X1), X2) | → | add(X1, X2) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| len(mark(X)) | → | len(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| active(add(0, X)) | → | mark(X) | | s(mark(X)) | → | s(X) |
| active(len(nil)) | → | mark(0) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(len(X)) | → | active#(len(mark(X))) |
Problem 15: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(from(X)) | → | active#(from(mark(X))) | | mark#(from(X)) | → | mark#(X) |
| mark#(fst(X1, X2)) | → | active#(fst(mark(X1), mark(X2))) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(add(X1, X2)) | → | mark#(X2) | | active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) |
| active#(add(0, X)) | → | mark#(X) | | mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) |
| mark#(fst(X1, X2)) | → | mark#(X1) | | active#(from(X)) | → | mark#(cons(X, from(s(X)))) |
| mark#(add(X1, X2)) | → | mark#(X1) | | mark#(fst(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, from, len, add, nil, cons
Strategy
Polynomial Interpretation
- 0: 3
- active(x): 1
- active#(x): 2x
- add(x,y): 1
- cons(x,y): 1
- from(x): 1
- fst(x,y): 0
- len(x): 3
- mark(x): 1
- mark#(x): 2
- nil: 3
- s(x): 1
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| fst(active(X1), X2) | → | fst(X1, X2) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(s(X)) | → | active(s(X)) | | add(X1, mark(X2)) | → | add(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | active(len(cons(X, Z))) | → | mark(s(len(Z))) |
| mark(from(X)) | → | active(from(mark(X))) | | fst(X1, mark(X2)) | → | fst(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | active(from(X)) | → | mark(cons(X, from(s(X)))) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | add(mark(X1), X2) | → | add(X1, X2) |
| mark(nil) | → | active(nil) | | mark(0) | → | active(0) |
| s(active(X)) | → | s(X) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(len(X)) | → | active(len(mark(X))) |
| len(active(X)) | → | len(X) | | active(fst(0, Z)) | → | mark(nil) |
| add(active(X1), X2) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| fst(mark(X1), X2) | → | fst(X1, X2) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| active(add(0, X)) | → | mark(X) | | s(mark(X)) | → | s(X) |
| active(len(nil)) | → | mark(0) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(fst(X1, X2)) | → | active#(fst(mark(X1), mark(X2))) |
Problem 16: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(from(X)) | → | active#(from(mark(X))) | | mark#(from(X)) | → | mark#(X) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) |
| mark#(add(X1, X2)) | → | mark#(X2) | | active#(add(0, X)) | → | mark#(X) |
| mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) | | active#(from(X)) | → | mark#(cons(X, from(s(X)))) |
| mark#(fst(X1, X2)) | → | mark#(X1) | | mark#(add(X1, X2)) | → | mark#(X1) |
| mark#(fst(X1, X2)) | → | mark#(X2) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, len, from, add, cons, nil
Strategy
Polynomial Interpretation
- 0: 0
- active(x): x
- active#(x): 2x
- add(x,y): y + 2x
- cons(x,y): x + 1
- from(x): x + 1
- fst(x,y): 2y + 2x + 1
- len(x): 2
- mark(x): x
- mark#(x): 2x
- nil: 1
- s(x): 0
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| fst(active(X1), X2) | → | fst(X1, X2) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(s(X)) | → | active(s(X)) | | add(X1, mark(X2)) | → | add(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | active(len(cons(X, Z))) | → | mark(s(len(Z))) |
| mark(from(X)) | → | active(from(mark(X))) | | fst(X1, mark(X2)) | → | fst(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | fst(X1, active(X2)) | → | fst(X1, X2) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | add(mark(X1), X2) | → | add(X1, X2) |
| mark(nil) | → | active(nil) | | s(active(X)) | → | s(X) |
| mark(0) | → | active(0) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(len(X)) | → | active(len(mark(X))) |
| len(active(X)) | → | len(X) | | active(fst(0, Z)) | → | mark(nil) |
| add(active(X1), X2) | → | add(X1, X2) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| len(mark(X)) | → | len(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| active(add(0, X)) | → | mark(X) | | s(mark(X)) | → | s(X) |
| active(len(nil)) | → | mark(0) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(from(X)) | → | mark#(X) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(fst(X1, X2)) | → | mark#(X1) | | mark#(fst(X1, X2)) | → | mark#(X2) |
Problem 17: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(from(X)) | → | active#(from(mark(X))) | | mark#(add(X1, X2)) | → | mark#(X2) |
| active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) | | active#(add(0, X)) | → | mark#(X) |
| mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) | | active#(from(X)) | → | mark#(cons(X, from(s(X)))) |
| mark#(add(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, from, len, add, nil, cons
Strategy
Polynomial Interpretation
- 0: 1
- active(x): x
- active#(x): x
- add(x,y): y + 2x
- cons(x,y): 2
- from(x): 2
- fst(x,y): x + 2
- len(x): x + 1
- mark(x): x
- mark#(x): x
- nil: 0
- s(x): 0
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| fst(active(X1), X2) | → | fst(X1, X2) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(s(X)) | → | active(s(X)) | | add(X1, mark(X2)) | → | add(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | active(len(cons(X, Z))) | → | mark(s(len(Z))) |
| mark(from(X)) | → | active(from(mark(X))) | | fst(X1, mark(X2)) | → | fst(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | fst(X1, active(X2)) | → | fst(X1, X2) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | add(mark(X1), X2) | → | add(X1, X2) |
| mark(nil) | → | active(nil) | | s(active(X)) | → | s(X) |
| mark(0) | → | active(0) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(len(X)) | → | active(len(mark(X))) |
| len(active(X)) | → | len(X) | | active(fst(0, Z)) | → | mark(nil) |
| add(active(X1), X2) | → | add(X1, X2) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| len(mark(X)) | → | len(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| active(add(0, X)) | → | mark(X) | | s(mark(X)) | → | s(X) |
| active(len(nil)) | → | mark(0) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(add(0, X)) | → | mark#(X) |
Problem 18: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(from(X)) | → | active#(from(mark(X))) | | active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) |
| mark#(add(X1, X2)) | → | mark#(X2) | | mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) |
| active#(from(X)) | → | mark#(cons(X, from(s(X)))) | | mark#(add(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, len, from, add, cons, nil
Strategy
Polynomial Interpretation
- 0: 1
- active(x): x
- active#(x): 2x
- add(x,y): 2y + x
- cons(x,y): 0
- from(x): 1
- fst(x,y): x
- len(x): 1
- mark(x): x
- mark#(x): 2x
- nil: 1
- s(x): 0
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| fst(active(X1), X2) | → | fst(X1, X2) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(s(X)) | → | active(s(X)) | | add(X1, mark(X2)) | → | add(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | active(len(cons(X, Z))) | → | mark(s(len(Z))) |
| mark(from(X)) | → | active(from(mark(X))) | | fst(X1, mark(X2)) | → | fst(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | fst(X1, active(X2)) | → | fst(X1, X2) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | add(mark(X1), X2) | → | add(X1, X2) |
| mark(nil) | → | active(nil) | | s(active(X)) | → | s(X) |
| mark(0) | → | active(0) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(len(X)) | → | active(len(mark(X))) |
| len(active(X)) | → | len(X) | | active(fst(0, Z)) | → | mark(nil) |
| add(active(X1), X2) | → | add(X1, X2) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| len(mark(X)) | → | len(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| active(add(0, X)) | → | mark(X) | | s(mark(X)) | → | s(X) |
| active(len(nil)) | → | mark(0) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(from(X)) | → | mark#(cons(X, from(s(X)))) |
Problem 19: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| mark#(from(X)) | → | active#(from(mark(X))) | | mark#(add(X1, X2)) | → | mark#(X2) |
| active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) | | mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) |
| mark#(add(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, from, len, add, nil, cons
Strategy
Polynomial Interpretation
- 0: 3
- active(x): 3
- active#(x): x
- add(x,y): 1
- cons(x,y): 1
- from(x): 0
- fst(x,y): 1
- len(x): 1
- mark(x): 3
- mark#(x): 1
- nil: 1
- s(x): 1
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| fst(active(X1), X2) | → | fst(X1, X2) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(s(X)) | → | active(s(X)) | | add(X1, mark(X2)) | → | add(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | active(len(cons(X, Z))) | → | mark(s(len(Z))) |
| mark(from(X)) | → | active(from(mark(X))) | | fst(X1, mark(X2)) | → | fst(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | fst(X1, active(X2)) | → | fst(X1, X2) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | add(mark(X1), X2) | → | add(X1, X2) |
| mark(nil) | → | active(nil) | | s(active(X)) | → | s(X) |
| mark(0) | → | active(0) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(len(X)) | → | active(len(mark(X))) |
| len(active(X)) | → | len(X) | | active(fst(0, Z)) | → | mark(nil) |
| add(active(X1), X2) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| fst(mark(X1), X2) | → | fst(X1, X2) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| active(add(0, X)) | → | mark(X) | | s(mark(X)) | → | s(X) |
| active(len(nil)) | → | mark(0) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(from(X)) | → | active#(from(mark(X))) |
Problem 20: PolynomialLinearRange4
Dependency Pair Problem
Dependency Pairs
| active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) | | mark#(add(X1, X2)) | → | mark#(X2) |
| mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) | | mark#(add(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, len, from, add, cons, nil
Strategy
Polynomial Interpretation
- 0: 0
- active(x): x
- active#(x): 1
- add(x,y): 2y + x + 1
- cons(x,y): y
- from(x): 0
- fst(x,y): 0
- len(x): 1
- mark(x): 2x
- mark#(x): 2x
- nil: 0
- s(x): 0
Standard Usable rules
| cons(active(X1), X2) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| fst(active(X1), X2) | → | fst(X1, X2) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(s(X)) | → | active(s(X)) | | add(X1, mark(X2)) | → | add(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | active(len(cons(X, Z))) | → | mark(s(len(Z))) |
| mark(from(X)) | → | active(from(mark(X))) | | fst(X1, mark(X2)) | → | fst(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | fst(X1, active(X2)) | → | fst(X1, X2) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | add(mark(X1), X2) | → | add(X1, X2) |
| mark(nil) | → | active(nil) | | s(active(X)) | → | s(X) |
| mark(0) | → | active(0) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| from(active(X)) | → | from(X) | | mark(len(X)) | → | active(len(mark(X))) |
| len(active(X)) | → | len(X) | | active(fst(0, Z)) | → | mark(nil) |
| add(active(X1), X2) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| fst(mark(X1), X2) | → | fst(X1, X2) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| active(add(0, X)) | → | mark(X) | | s(mark(X)) | → | s(X) |
| active(len(nil)) | → | mark(0) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(add(X1, X2)) | → | mark#(X2) | | active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) |
| mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) | | mark#(add(X1, X2)) | → | mark#(X1) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| cons#(X1, active(X2)) | → | cons#(X1, X2) | | cons#(mark(X1), X2) | → | cons#(X1, X2) |
| cons#(active(X1), X2) | → | cons#(X1, X2) | | cons#(X1, mark(X2)) | → | cons#(X1, X2) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, from, len, add, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | cons#(active(X1), X2) | → | cons#(X1, X2) |
Problem 10: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| cons#(X1, active(X2)) | → | cons#(X1, X2) | | cons#(X1, mark(X2)) | → | cons#(X1, X2) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, len, from, add, cons, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| cons#(X1, active(X2)) | → | cons#(X1, X2) | | cons#(X1, mark(X2)) | → | cons#(X1, X2) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| fst#(X1, mark(X2)) | → | fst#(X1, X2) | | fst#(X1, active(X2)) | → | fst#(X1, X2) |
| fst#(active(X1), X2) | → | fst#(X1, X2) | | fst#(mark(X1), X2) | → | fst#(X1, X2) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, from, len, add, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| fst#(active(X1), X2) | → | fst#(X1, X2) | | fst#(mark(X1), X2) | → | fst#(X1, X2) |
Problem 11: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| fst#(X1, mark(X2)) | → | fst#(X1, X2) | | fst#(X1, active(X2)) | → | fst#(X1, X2) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, len, from, add, cons, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| fst#(X1, mark(X2)) | → | fst#(X1, X2) | | fst#(X1, active(X2)) | → | fst#(X1, X2) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| len#(mark(X)) | → | len#(X) | | len#(active(X)) | → | len#(X) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, from, len, add, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| len#(mark(X)) | → | len#(X) | | len#(active(X)) | → | len#(X) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| from#(active(X)) | → | from#(X) | | from#(mark(X)) | → | from#(X) |
Rewrite Rules
| active(fst(0, Z)) | → | mark(nil) | | active(fst(s(X), cons(Y, Z))) | → | mark(cons(Y, fst(X, Z))) |
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(len(nil)) | → | mark(0) |
| active(len(cons(X, Z))) | → | mark(s(len(Z))) | | mark(fst(X1, X2)) | → | active(fst(mark(X1), mark(X2))) |
| mark(0) | → | active(0) | | mark(nil) | → | active(nil) |
| mark(s(X)) | → | active(s(X)) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(from(X)) | → | active(from(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(len(X)) | → | active(len(mark(X))) | | fst(mark(X1), X2) | → | fst(X1, X2) |
| fst(X1, mark(X2)) | → | fst(X1, X2) | | fst(active(X1), X2) | → | fst(X1, X2) |
| fst(X1, active(X2)) | → | fst(X1, X2) | | s(mark(X)) | → | s(X) |
| s(active(X)) | → | s(X) | | cons(mark(X1), X2) | → | cons(X1, X2) |
| cons(X1, mark(X2)) | → | cons(X1, X2) | | cons(active(X1), X2) | → | cons(X1, X2) |
| cons(X1, active(X2)) | → | cons(X1, X2) | | from(mark(X)) | → | from(X) |
| from(active(X)) | → | from(X) | | add(mark(X1), X2) | → | add(X1, X2) |
| add(X1, mark(X2)) | → | add(X1, X2) | | add(active(X1), X2) | → | add(X1, X2) |
| add(X1, active(X2)) | → | add(X1, X2) | | len(mark(X)) | → | len(X) |
| len(active(X)) | → | len(X) |
Original Signature
Termination of terms over the following signature is verified: fst, 0, s, active, mark, from, len, add, nil, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| from#(active(X)) | → | from#(X) | | from#(mark(X)) | → | from#(X) |