TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60000 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (5119ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| | – Problem 12 was processed with processor ReductionPairSAT (85ms).
| – Problem 3 was processed with processor SubtermCriterion (4ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| | – Problem 13 was processed with processor ReductionPairSAT (47ms).
| – Problem 5 was processed with processor SubtermCriterion (2ms).
| | – Problem 14 was processed with processor ReductionPairSAT (82ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| | – Problem 15 was processed with processor ReductionPairSAT (44ms).
| – Problem 7 was processed with processor SubtermCriterion (1ms).
| – Problem 8 was processed with processor SubtermCriterion (3ms).
| – Problem 9 was processed with processor SubtermCriterion (2ms).
| – Problem 10 remains open; application of the following processors failed [SubtermCriterion (0ms), DependencyGraph (7ms), PolynomialLinearRange4iUR (2000ms), DependencyGraph (5ms), PolynomialLinearRange8NegiUR (6001ms), DependencyGraph (5ms), ReductionPairSAT (14361ms), DependencyGraph (5ms), ReductionPairSAT (14159ms), DependencyGraph (5ms), SizeChangePrinciple (timeout)].
| – Problem 11 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 10
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(minus(X, 0)) | → | mark(0) |
| active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) | | active(quot(0, s(Y))) | → | mark(0) |
| active(quot(s(X), s(Y))) | → | mark(s(quot(minus(X, Y), s(Y)))) | | active(zWquot(XS, nil)) | → | mark(nil) |
| active(zWquot(nil, XS)) | → | mark(nil) | | active(zWquot(cons(X, XS), cons(Y, YS))) | → | mark(cons(quot(X, Y), zWquot(XS, YS))) |
| active(from(X)) | → | from(active(X)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(minus(X1, X2)) | → | minus(active(X1), X2) |
| active(minus(X1, X2)) | → | minus(X1, active(X2)) | | active(quot(X1, X2)) | → | quot(active(X1), X2) |
| active(quot(X1, X2)) | → | quot(X1, active(X2)) | | active(zWquot(X1, X2)) | → | zWquot(active(X1), X2) |
| active(zWquot(X1, X2)) | → | zWquot(X1, active(X2)) | | from(mark(X)) | → | mark(from(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| minus(mark(X1), X2) | → | mark(minus(X1, X2)) | | minus(X1, mark(X2)) | → | mark(minus(X1, X2)) |
| quot(mark(X1), X2) | → | mark(quot(X1, X2)) | | quot(X1, mark(X2)) | → | mark(quot(X1, X2)) |
| zWquot(mark(X1), X2) | → | mark(zWquot(X1, X2)) | | zWquot(X1, mark(X2)) | → | mark(zWquot(X1, X2)) |
| proper(from(X)) | → | from(proper(X)) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(s(X)) | → | s(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(quot(X1, X2)) | → | quot(proper(X1), proper(X2)) | | proper(zWquot(X1, X2)) | → | zWquot(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | from(ok(X)) | → | ok(from(X)) |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| quot(ok(X1), ok(X2)) | → | ok(quot(X1, X2)) | | zWquot(ok(X1), ok(X2)) | → | ok(zWquot(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, mark, from, 0, s, zWquot, active, ok, proper, sel, quot, top, cons, nil
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| minus#(ok(X1), ok(X2)) | → | minus#(X1, X2) | | proper#(cons(X1, X2)) | → | proper#(X1) |
| top#(ok(X)) | → | top#(active(X)) | | active#(quot(s(X), s(Y))) | → | quot#(minus(X, Y), s(Y)) |
| active#(quot(X1, X2)) | → | active#(X2) | | minus#(X1, mark(X2)) | → | minus#(X1, X2) |
| proper#(minus(X1, X2)) | → | proper#(X1) | | active#(zWquot(cons(X, XS), cons(Y, YS))) | → | cons#(quot(X, Y), zWquot(XS, YS)) |
| cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) | | active#(quot(X1, X2)) | → | quot#(active(X1), X2) |
| from#(ok(X)) | → | from#(X) | | active#(cons(X1, X2)) | → | cons#(active(X1), X2) |
| active#(minus(X1, X2)) | → | minus#(X1, active(X2)) | | active#(sel(X1, X2)) | → | active#(X2) |
| proper#(quot(X1, X2)) | → | quot#(proper(X1), proper(X2)) | | top#(mark(X)) | → | proper#(X) |
| proper#(from(X)) | → | proper#(X) | | top#(mark(X)) | → | top#(proper(X)) |
| proper#(cons(X1, X2)) | → | proper#(X2) | | zWquot#(mark(X1), X2) | → | zWquot#(X1, X2) |
| proper#(zWquot(X1, X2)) | → | proper#(X1) | | zWquot#(X1, mark(X2)) | → | zWquot#(X1, X2) |
| active#(zWquot(X1, X2)) | → | active#(X2) | | active#(quot(s(X), s(Y))) | → | s#(Y) |
| active#(minus(X1, X2)) | → | minus#(active(X1), X2) | | sel#(X1, mark(X2)) | → | sel#(X1, X2) |
| active#(quot(s(X), s(Y))) | → | s#(quot(minus(X, Y), s(Y))) | | proper#(zWquot(X1, X2)) | → | zWquot#(proper(X1), proper(X2)) |
| quot#(X1, mark(X2)) | → | quot#(X1, X2) | | active#(from(X)) | → | s#(X) |
| proper#(s(X)) | → | proper#(X) | | active#(minus(s(X), s(Y))) | → | minus#(X, Y) |
| active#(zWquot(cons(X, XS), cons(Y, YS))) | → | quot#(X, Y) | | sel#(ok(X1), ok(X2)) | → | sel#(X1, X2) |
| active#(zWquot(X1, X2)) | → | active#(X1) | | quot#(ok(X1), ok(X2)) | → | quot#(X1, X2) |
| active#(cons(X1, X2)) | → | active#(X1) | | active#(zWquot(X1, X2)) | → | zWquot#(X1, active(X2)) |
| sel#(mark(X1), X2) | → | sel#(X1, X2) | | proper#(quot(X1, X2)) | → | proper#(X1) |
| active#(minus(X1, X2)) | → | active#(X1) | | cons#(mark(X1), X2) | → | cons#(X1, X2) |
| active#(from(X)) | → | from#(active(X)) | | active#(quot(X1, X2)) | → | quot#(X1, active(X2)) |
| active#(zWquot(cons(X, XS), cons(Y, YS))) | → | zWquot#(XS, YS) | | from#(mark(X)) | → | from#(X) |
| quot#(mark(X1), X2) | → | quot#(X1, X2) | | top#(ok(X)) | → | active#(X) |
| proper#(sel(X1, X2)) | → | sel#(proper(X1), proper(X2)) | | minus#(mark(X1), X2) | → | minus#(X1, X2) |
| active#(minus(X1, X2)) | → | active#(X2) | | active#(zWquot(X1, X2)) | → | zWquot#(active(X1), X2) |
| active#(sel(X1, X2)) | → | active#(X1) | | active#(quot(s(X), s(Y))) | → | minus#(X, Y) |
| active#(sel(s(N), cons(X, XS))) | → | sel#(N, XS) | | active#(from(X)) | → | cons#(X, from(s(X))) |
| proper#(from(X)) | → | from#(proper(X)) | | proper#(sel(X1, X2)) | → | proper#(X2) |
| proper#(minus(X1, X2)) | → | proper#(X2) | | proper#(quot(X1, X2)) | → | proper#(X2) |
| active#(sel(X1, X2)) | → | sel#(active(X1), X2) | | active#(sel(X1, X2)) | → | sel#(X1, active(X2)) |
| active#(from(X)) | → | active#(X) | | zWquot#(ok(X1), ok(X2)) | → | zWquot#(X1, X2) |
| active#(s(X)) | → | s#(active(X)) | | active#(quot(X1, X2)) | → | active#(X1) |
| s#(ok(X)) | → | s#(X) | | s#(mark(X)) | → | s#(X) |
| proper#(sel(X1, X2)) | → | proper#(X1) | | proper#(cons(X1, X2)) | → | cons#(proper(X1), proper(X2)) |
| proper#(minus(X1, X2)) | → | minus#(proper(X1), proper(X2)) | | active#(s(X)) | → | active#(X) |
| proper#(s(X)) | → | s#(proper(X)) | | proper#(zWquot(X1, X2)) | → | proper#(X2) |
| active#(from(X)) | → | from#(s(X)) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(minus(X, 0)) | → | mark(0) |
| active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) | | active(quot(0, s(Y))) | → | mark(0) |
| active(quot(s(X), s(Y))) | → | mark(s(quot(minus(X, Y), s(Y)))) | | active(zWquot(XS, nil)) | → | mark(nil) |
| active(zWquot(nil, XS)) | → | mark(nil) | | active(zWquot(cons(X, XS), cons(Y, YS))) | → | mark(cons(quot(X, Y), zWquot(XS, YS))) |
| active(from(X)) | → | from(active(X)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(minus(X1, X2)) | → | minus(active(X1), X2) |
| active(minus(X1, X2)) | → | minus(X1, active(X2)) | | active(quot(X1, X2)) | → | quot(active(X1), X2) |
| active(quot(X1, X2)) | → | quot(X1, active(X2)) | | active(zWquot(X1, X2)) | → | zWquot(active(X1), X2) |
| active(zWquot(X1, X2)) | → | zWquot(X1, active(X2)) | | from(mark(X)) | → | mark(from(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| minus(mark(X1), X2) | → | mark(minus(X1, X2)) | | minus(X1, mark(X2)) | → | mark(minus(X1, X2)) |
| quot(mark(X1), X2) | → | mark(quot(X1, X2)) | | quot(X1, mark(X2)) | → | mark(quot(X1, X2)) |
| zWquot(mark(X1), X2) | → | mark(zWquot(X1, X2)) | | zWquot(X1, mark(X2)) | → | mark(zWquot(X1, X2)) |
| proper(from(X)) | → | from(proper(X)) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(s(X)) | → | s(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(quot(X1, X2)) | → | quot(proper(X1), proper(X2)) | | proper(zWquot(X1, X2)) | → | zWquot(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | from(ok(X)) | → | ok(from(X)) |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| quot(ok(X1), ok(X2)) | → | ok(quot(X1, X2)) | | zWquot(ok(X1), ok(X2)) | → | ok(zWquot(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, mark, from, 0, s, zWquot, active, ok, proper, sel, quot, nil, top, cons
Strategy
The following SCCs where found
| cons#(mark(X1), X2) → cons#(X1, X2) | cons#(ok(X1), ok(X2)) → cons#(X1, X2) |
| active#(minus(X1, X2)) → active#(X2) | active#(sel(X1, X2)) → active#(X2) |
| active#(minus(X1, X2)) → active#(X1) | active#(quot(X1, X2)) → active#(X2) |
| active#(s(X)) → active#(X) | active#(from(X)) → active#(X) |
| active#(sel(X1, X2)) → active#(X1) | active#(zWquot(X1, X2)) → active#(X2) |
| active#(zWquot(X1, X2)) → active#(X1) | active#(quot(X1, X2)) → active#(X1) |
| active#(cons(X1, X2)) → active#(X1) |
| sel#(mark(X1), X2) → sel#(X1, X2) | sel#(ok(X1), ok(X2)) → sel#(X1, X2) |
| sel#(X1, mark(X2)) → sel#(X1, X2) |
| proper#(sel(X1, X2)) → proper#(X1) | proper#(s(X)) → proper#(X) |
| proper#(quot(X1, X2)) → proper#(X1) | proper#(cons(X1, X2)) → proper#(X1) |
| proper#(cons(X1, X2)) → proper#(X2) | proper#(zWquot(X1, X2)) → proper#(X1) |
| proper#(minus(X1, X2)) → proper#(X1) | proper#(zWquot(X1, X2)) → proper#(X2) |
| proper#(minus(X1, X2)) → proper#(X2) | proper#(sel(X1, X2)) → proper#(X2) |
| proper#(quot(X1, X2)) → proper#(X2) | proper#(from(X)) → proper#(X) |
| zWquot#(mark(X1), X2) → zWquot#(X1, X2) | zWquot#(X1, mark(X2)) → zWquot#(X1, X2) |
| zWquot#(ok(X1), ok(X2)) → zWquot#(X1, X2) |
| quot#(ok(X1), ok(X2)) → quot#(X1, X2) | quot#(mark(X1), X2) → quot#(X1, X2) |
| quot#(X1, mark(X2)) → quot#(X1, X2) |
| minus#(mark(X1), X2) → minus#(X1, X2) | minus#(ok(X1), ok(X2)) → minus#(X1, X2) |
| minus#(X1, mark(X2)) → minus#(X1, X2) |
| from#(mark(X)) → from#(X) | from#(ok(X)) → from#(X) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
| s#(mark(X)) → s#(X) | s#(ok(X)) → s#(X) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| sel#(mark(X1), X2) | → | sel#(X1, X2) | | sel#(ok(X1), ok(X2)) | → | sel#(X1, X2) |
| sel#(X1, mark(X2)) | → | sel#(X1, X2) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(minus(X, 0)) | → | mark(0) |
| active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) | | active(quot(0, s(Y))) | → | mark(0) |
| active(quot(s(X), s(Y))) | → | mark(s(quot(minus(X, Y), s(Y)))) | | active(zWquot(XS, nil)) | → | mark(nil) |
| active(zWquot(nil, XS)) | → | mark(nil) | | active(zWquot(cons(X, XS), cons(Y, YS))) | → | mark(cons(quot(X, Y), zWquot(XS, YS))) |
| active(from(X)) | → | from(active(X)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(minus(X1, X2)) | → | minus(active(X1), X2) |
| active(minus(X1, X2)) | → | minus(X1, active(X2)) | | active(quot(X1, X2)) | → | quot(active(X1), X2) |
| active(quot(X1, X2)) | → | quot(X1, active(X2)) | | active(zWquot(X1, X2)) | → | zWquot(active(X1), X2) |
| active(zWquot(X1, X2)) | → | zWquot(X1, active(X2)) | | from(mark(X)) | → | mark(from(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| minus(mark(X1), X2) | → | mark(minus(X1, X2)) | | minus(X1, mark(X2)) | → | mark(minus(X1, X2)) |
| quot(mark(X1), X2) | → | mark(quot(X1, X2)) | | quot(X1, mark(X2)) | → | mark(quot(X1, X2)) |
| zWquot(mark(X1), X2) | → | mark(zWquot(X1, X2)) | | zWquot(X1, mark(X2)) | → | mark(zWquot(X1, X2)) |
| proper(from(X)) | → | from(proper(X)) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(s(X)) | → | s(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(quot(X1, X2)) | → | quot(proper(X1), proper(X2)) | | proper(zWquot(X1, X2)) | → | zWquot(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | from(ok(X)) | → | ok(from(X)) |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| quot(ok(X1), ok(X2)) | → | ok(quot(X1, X2)) | | zWquot(ok(X1), ok(X2)) | → | ok(zWquot(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, mark, from, 0, s, zWquot, active, ok, proper, sel, quot, nil, top, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| sel#(mark(X1), X2) | → | sel#(X1, X2) | | sel#(ok(X1), ok(X2)) | → | sel#(X1, X2) |
Problem 12: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| sel#(X1, mark(X2)) | → | sel#(X1, X2) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(minus(X, 0)) | → | mark(0) |
| active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) | | active(quot(0, s(Y))) | → | mark(0) |
| active(quot(s(X), s(Y))) | → | mark(s(quot(minus(X, Y), s(Y)))) | | active(zWquot(XS, nil)) | → | mark(nil) |
| active(zWquot(nil, XS)) | → | mark(nil) | | active(zWquot(cons(X, XS), cons(Y, YS))) | → | mark(cons(quot(X, Y), zWquot(XS, YS))) |
| active(from(X)) | → | from(active(X)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(minus(X1, X2)) | → | minus(active(X1), X2) |
| active(minus(X1, X2)) | → | minus(X1, active(X2)) | | active(quot(X1, X2)) | → | quot(active(X1), X2) |
| active(quot(X1, X2)) | → | quot(X1, active(X2)) | | active(zWquot(X1, X2)) | → | zWquot(active(X1), X2) |
| active(zWquot(X1, X2)) | → | zWquot(X1, active(X2)) | | from(mark(X)) | → | mark(from(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| minus(mark(X1), X2) | → | mark(minus(X1, X2)) | | minus(X1, mark(X2)) | → | mark(minus(X1, X2)) |
| quot(mark(X1), X2) | → | mark(quot(X1, X2)) | | quot(X1, mark(X2)) | → | mark(quot(X1, X2)) |
| zWquot(mark(X1), X2) | → | mark(zWquot(X1, X2)) | | zWquot(X1, mark(X2)) | → | mark(zWquot(X1, X2)) |
| proper(from(X)) | → | from(proper(X)) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(s(X)) | → | s(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(quot(X1, X2)) | → | quot(proper(X1), proper(X2)) | | proper(zWquot(X1, X2)) | → | zWquot(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | from(ok(X)) | → | ok(from(X)) |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| quot(ok(X1), ok(X2)) | → | ok(quot(X1, X2)) | | zWquot(ok(X1), ok(X2)) | → | ok(zWquot(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, mark, from, 0, s, zWquot, active, ok, proper, sel, quot, cons, nil, top
Strategy
Function Precedence
mark < sel# < minus = from = 0 = s = zWquot = active = ok = proper = quot = sel = top = cons = nil
Argument Filtering
minus: 1 2
mark: 1
sel#: collapses to 2
from: collapses to 1
0: all arguments are removed from 0
s: collapses to 1
zWquot: all arguments are removed from zWquot
active: all arguments are removed from active
ok: all arguments are removed from ok
proper: all arguments are removed from proper
quot: all arguments are removed from quot
sel: 1 2
top: all arguments are removed from top
cons: all arguments are removed from cons
nil: all arguments are removed from nil
Status
minus: lexicographic with permutation 1 → 1 2 → 2
mark: lexicographic with permutation 1 → 1
0: multiset
zWquot: multiset
active: multiset
ok: multiset
proper: multiset
quot: multiset
sel: lexicographic with permutation 1 → 1 2 → 2
top: multiset
cons: multiset
nil: multiset
Usable Rules
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
| sel#(X1, mark(X2)) → sel#(X1, X2) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| from#(mark(X)) | → | from#(X) | | from#(ok(X)) | → | from#(X) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(minus(X, 0)) | → | mark(0) |
| active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) | | active(quot(0, s(Y))) | → | mark(0) |
| active(quot(s(X), s(Y))) | → | mark(s(quot(minus(X, Y), s(Y)))) | | active(zWquot(XS, nil)) | → | mark(nil) |
| active(zWquot(nil, XS)) | → | mark(nil) | | active(zWquot(cons(X, XS), cons(Y, YS))) | → | mark(cons(quot(X, Y), zWquot(XS, YS))) |
| active(from(X)) | → | from(active(X)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(minus(X1, X2)) | → | minus(active(X1), X2) |
| active(minus(X1, X2)) | → | minus(X1, active(X2)) | | active(quot(X1, X2)) | → | quot(active(X1), X2) |
| active(quot(X1, X2)) | → | quot(X1, active(X2)) | | active(zWquot(X1, X2)) | → | zWquot(active(X1), X2) |
| active(zWquot(X1, X2)) | → | zWquot(X1, active(X2)) | | from(mark(X)) | → | mark(from(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| minus(mark(X1), X2) | → | mark(minus(X1, X2)) | | minus(X1, mark(X2)) | → | mark(minus(X1, X2)) |
| quot(mark(X1), X2) | → | mark(quot(X1, X2)) | | quot(X1, mark(X2)) | → | mark(quot(X1, X2)) |
| zWquot(mark(X1), X2) | → | mark(zWquot(X1, X2)) | | zWquot(X1, mark(X2)) | → | mark(zWquot(X1, X2)) |
| proper(from(X)) | → | from(proper(X)) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(s(X)) | → | s(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(quot(X1, X2)) | → | quot(proper(X1), proper(X2)) | | proper(zWquot(X1, X2)) | → | zWquot(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | from(ok(X)) | → | ok(from(X)) |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| quot(ok(X1), ok(X2)) | → | ok(quot(X1, X2)) | | zWquot(ok(X1), ok(X2)) | → | ok(zWquot(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, mark, from, 0, s, zWquot, active, ok, proper, sel, quot, nil, top, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| from#(mark(X)) | → | from#(X) | | from#(ok(X)) | → | from#(X) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| minus#(mark(X1), X2) | → | minus#(X1, X2) | | minus#(ok(X1), ok(X2)) | → | minus#(X1, X2) |
| minus#(X1, mark(X2)) | → | minus#(X1, X2) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(minus(X, 0)) | → | mark(0) |
| active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) | | active(quot(0, s(Y))) | → | mark(0) |
| active(quot(s(X), s(Y))) | → | mark(s(quot(minus(X, Y), s(Y)))) | | active(zWquot(XS, nil)) | → | mark(nil) |
| active(zWquot(nil, XS)) | → | mark(nil) | | active(zWquot(cons(X, XS), cons(Y, YS))) | → | mark(cons(quot(X, Y), zWquot(XS, YS))) |
| active(from(X)) | → | from(active(X)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(minus(X1, X2)) | → | minus(active(X1), X2) |
| active(minus(X1, X2)) | → | minus(X1, active(X2)) | | active(quot(X1, X2)) | → | quot(active(X1), X2) |
| active(quot(X1, X2)) | → | quot(X1, active(X2)) | | active(zWquot(X1, X2)) | → | zWquot(active(X1), X2) |
| active(zWquot(X1, X2)) | → | zWquot(X1, active(X2)) | | from(mark(X)) | → | mark(from(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| minus(mark(X1), X2) | → | mark(minus(X1, X2)) | | minus(X1, mark(X2)) | → | mark(minus(X1, X2)) |
| quot(mark(X1), X2) | → | mark(quot(X1, X2)) | | quot(X1, mark(X2)) | → | mark(quot(X1, X2)) |
| zWquot(mark(X1), X2) | → | mark(zWquot(X1, X2)) | | zWquot(X1, mark(X2)) | → | mark(zWquot(X1, X2)) |
| proper(from(X)) | → | from(proper(X)) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(s(X)) | → | s(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(quot(X1, X2)) | → | quot(proper(X1), proper(X2)) | | proper(zWquot(X1, X2)) | → | zWquot(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | from(ok(X)) | → | ok(from(X)) |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| quot(ok(X1), ok(X2)) | → | ok(quot(X1, X2)) | | zWquot(ok(X1), ok(X2)) | → | ok(zWquot(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, mark, from, 0, s, zWquot, active, ok, proper, sel, quot, nil, top, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| minus#(ok(X1), ok(X2)) | → | minus#(X1, X2) | | minus#(mark(X1), X2) | → | minus#(X1, X2) |
Problem 13: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| minus#(X1, mark(X2)) | → | minus#(X1, X2) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(minus(X, 0)) | → | mark(0) |
| active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) | | active(quot(0, s(Y))) | → | mark(0) |
| active(quot(s(X), s(Y))) | → | mark(s(quot(minus(X, Y), s(Y)))) | | active(zWquot(XS, nil)) | → | mark(nil) |
| active(zWquot(nil, XS)) | → | mark(nil) | | active(zWquot(cons(X, XS), cons(Y, YS))) | → | mark(cons(quot(X, Y), zWquot(XS, YS))) |
| active(from(X)) | → | from(active(X)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(minus(X1, X2)) | → | minus(active(X1), X2) |
| active(minus(X1, X2)) | → | minus(X1, active(X2)) | | active(quot(X1, X2)) | → | quot(active(X1), X2) |
| active(quot(X1, X2)) | → | quot(X1, active(X2)) | | active(zWquot(X1, X2)) | → | zWquot(active(X1), X2) |
| active(zWquot(X1, X2)) | → | zWquot(X1, active(X2)) | | from(mark(X)) | → | mark(from(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| minus(mark(X1), X2) | → | mark(minus(X1, X2)) | | minus(X1, mark(X2)) | → | mark(minus(X1, X2)) |
| quot(mark(X1), X2) | → | mark(quot(X1, X2)) | | quot(X1, mark(X2)) | → | mark(quot(X1, X2)) |
| zWquot(mark(X1), X2) | → | mark(zWquot(X1, X2)) | | zWquot(X1, mark(X2)) | → | mark(zWquot(X1, X2)) |
| proper(from(X)) | → | from(proper(X)) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(s(X)) | → | s(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(quot(X1, X2)) | → | quot(proper(X1), proper(X2)) | | proper(zWquot(X1, X2)) | → | zWquot(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | from(ok(X)) | → | ok(from(X)) |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| quot(ok(X1), ok(X2)) | → | ok(quot(X1, X2)) | | zWquot(ok(X1), ok(X2)) | → | ok(zWquot(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, mark, from, 0, s, zWquot, active, ok, proper, sel, quot, cons, nil, top
Strategy
Function Precedence
mark < minus = minus# = from = 0 = s = zWquot = active = ok = proper = quot = sel = top = cons = nil
Argument Filtering
minus: all arguments are removed from minus
minus#: collapses to 2
mark: 1
from: all arguments are removed from from
0: all arguments are removed from 0
s: all arguments are removed from s
zWquot: all arguments are removed from zWquot
active: all arguments are removed from active
ok: all arguments are removed from ok
proper: all arguments are removed from proper
quot: all arguments are removed from quot
sel: all arguments are removed from sel
top: all arguments are removed from top
cons: 1 2
nil: all arguments are removed from nil
Status
minus: multiset
mark: multiset
from: multiset
0: multiset
s: multiset
zWquot: multiset
active: multiset
ok: multiset
proper: multiset
quot: multiset
sel: multiset
top: multiset
cons: lexicographic with permutation 1 → 1 2 → 2
nil: multiset
Usable Rules
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
| minus#(X1, mark(X2)) → minus#(X1, X2) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| quot#(ok(X1), ok(X2)) | → | quot#(X1, X2) | | quot#(mark(X1), X2) | → | quot#(X1, X2) |
| quot#(X1, mark(X2)) | → | quot#(X1, X2) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(minus(X, 0)) | → | mark(0) |
| active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) | | active(quot(0, s(Y))) | → | mark(0) |
| active(quot(s(X), s(Y))) | → | mark(s(quot(minus(X, Y), s(Y)))) | | active(zWquot(XS, nil)) | → | mark(nil) |
| active(zWquot(nil, XS)) | → | mark(nil) | | active(zWquot(cons(X, XS), cons(Y, YS))) | → | mark(cons(quot(X, Y), zWquot(XS, YS))) |
| active(from(X)) | → | from(active(X)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(minus(X1, X2)) | → | minus(active(X1), X2) |
| active(minus(X1, X2)) | → | minus(X1, active(X2)) | | active(quot(X1, X2)) | → | quot(active(X1), X2) |
| active(quot(X1, X2)) | → | quot(X1, active(X2)) | | active(zWquot(X1, X2)) | → | zWquot(active(X1), X2) |
| active(zWquot(X1, X2)) | → | zWquot(X1, active(X2)) | | from(mark(X)) | → | mark(from(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| minus(mark(X1), X2) | → | mark(minus(X1, X2)) | | minus(X1, mark(X2)) | → | mark(minus(X1, X2)) |
| quot(mark(X1), X2) | → | mark(quot(X1, X2)) | | quot(X1, mark(X2)) | → | mark(quot(X1, X2)) |
| zWquot(mark(X1), X2) | → | mark(zWquot(X1, X2)) | | zWquot(X1, mark(X2)) | → | mark(zWquot(X1, X2)) |
| proper(from(X)) | → | from(proper(X)) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(s(X)) | → | s(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(quot(X1, X2)) | → | quot(proper(X1), proper(X2)) | | proper(zWquot(X1, X2)) | → | zWquot(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | from(ok(X)) | → | ok(from(X)) |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| quot(ok(X1), ok(X2)) | → | ok(quot(X1, X2)) | | zWquot(ok(X1), ok(X2)) | → | ok(zWquot(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, mark, from, 0, s, zWquot, active, ok, proper, sel, quot, nil, top, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| quot#(ok(X1), ok(X2)) | → | quot#(X1, X2) | | quot#(mark(X1), X2) | → | quot#(X1, X2) |
Problem 14: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| quot#(X1, mark(X2)) | → | quot#(X1, X2) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(minus(X, 0)) | → | mark(0) |
| active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) | | active(quot(0, s(Y))) | → | mark(0) |
| active(quot(s(X), s(Y))) | → | mark(s(quot(minus(X, Y), s(Y)))) | | active(zWquot(XS, nil)) | → | mark(nil) |
| active(zWquot(nil, XS)) | → | mark(nil) | | active(zWquot(cons(X, XS), cons(Y, YS))) | → | mark(cons(quot(X, Y), zWquot(XS, YS))) |
| active(from(X)) | → | from(active(X)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(minus(X1, X2)) | → | minus(active(X1), X2) |
| active(minus(X1, X2)) | → | minus(X1, active(X2)) | | active(quot(X1, X2)) | → | quot(active(X1), X2) |
| active(quot(X1, X2)) | → | quot(X1, active(X2)) | | active(zWquot(X1, X2)) | → | zWquot(active(X1), X2) |
| active(zWquot(X1, X2)) | → | zWquot(X1, active(X2)) | | from(mark(X)) | → | mark(from(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| minus(mark(X1), X2) | → | mark(minus(X1, X2)) | | minus(X1, mark(X2)) | → | mark(minus(X1, X2)) |
| quot(mark(X1), X2) | → | mark(quot(X1, X2)) | | quot(X1, mark(X2)) | → | mark(quot(X1, X2)) |
| zWquot(mark(X1), X2) | → | mark(zWquot(X1, X2)) | | zWquot(X1, mark(X2)) | → | mark(zWquot(X1, X2)) |
| proper(from(X)) | → | from(proper(X)) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(s(X)) | → | s(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(quot(X1, X2)) | → | quot(proper(X1), proper(X2)) | | proper(zWquot(X1, X2)) | → | zWquot(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | from(ok(X)) | → | ok(from(X)) |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| quot(ok(X1), ok(X2)) | → | ok(quot(X1, X2)) | | zWquot(ok(X1), ok(X2)) | → | ok(zWquot(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, mark, from, 0, s, zWquot, active, ok, proper, sel, quot, cons, nil, top
Strategy
Function Precedence
minus = mark = from = 0 = s = quot# = zWquot = active = ok = proper = quot = sel = top = cons = nil
Argument Filtering
minus: all arguments are removed from minus
mark: 1
from: all arguments are removed from from
0: all arguments are removed from 0
s: all arguments are removed from s
quot#: 2
zWquot: all arguments are removed from zWquot
active: collapses to 1
ok: 1
proper: all arguments are removed from proper
quot: 1 2
sel: 1 2
top: all arguments are removed from top
cons: 1 2
nil: all arguments are removed from nil
Status
minus: multiset
mark: multiset
from: multiset
0: multiset
s: multiset
quot#: multiset
zWquot: multiset
ok: lexicographic with permutation 1 → 1
proper: multiset
quot: lexicographic with permutation 1 → 2 2 → 1
sel: lexicographic with permutation 1 → 2 2 → 1
top: multiset
cons: lexicographic with permutation 1 → 1 2 → 2
nil: multiset
Usable Rules
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
| quot#(X1, mark(X2)) → quot#(X1, X2) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| zWquot#(mark(X1), X2) | → | zWquot#(X1, X2) | | zWquot#(X1, mark(X2)) | → | zWquot#(X1, X2) |
| zWquot#(ok(X1), ok(X2)) | → | zWquot#(X1, X2) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(minus(X, 0)) | → | mark(0) |
| active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) | | active(quot(0, s(Y))) | → | mark(0) |
| active(quot(s(X), s(Y))) | → | mark(s(quot(minus(X, Y), s(Y)))) | | active(zWquot(XS, nil)) | → | mark(nil) |
| active(zWquot(nil, XS)) | → | mark(nil) | | active(zWquot(cons(X, XS), cons(Y, YS))) | → | mark(cons(quot(X, Y), zWquot(XS, YS))) |
| active(from(X)) | → | from(active(X)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(minus(X1, X2)) | → | minus(active(X1), X2) |
| active(minus(X1, X2)) | → | minus(X1, active(X2)) | | active(quot(X1, X2)) | → | quot(active(X1), X2) |
| active(quot(X1, X2)) | → | quot(X1, active(X2)) | | active(zWquot(X1, X2)) | → | zWquot(active(X1), X2) |
| active(zWquot(X1, X2)) | → | zWquot(X1, active(X2)) | | from(mark(X)) | → | mark(from(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| minus(mark(X1), X2) | → | mark(minus(X1, X2)) | | minus(X1, mark(X2)) | → | mark(minus(X1, X2)) |
| quot(mark(X1), X2) | → | mark(quot(X1, X2)) | | quot(X1, mark(X2)) | → | mark(quot(X1, X2)) |
| zWquot(mark(X1), X2) | → | mark(zWquot(X1, X2)) | | zWquot(X1, mark(X2)) | → | mark(zWquot(X1, X2)) |
| proper(from(X)) | → | from(proper(X)) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(s(X)) | → | s(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(quot(X1, X2)) | → | quot(proper(X1), proper(X2)) | | proper(zWquot(X1, X2)) | → | zWquot(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | from(ok(X)) | → | ok(from(X)) |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| quot(ok(X1), ok(X2)) | → | ok(quot(X1, X2)) | | zWquot(ok(X1), ok(X2)) | → | ok(zWquot(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, mark, from, 0, s, zWquot, active, ok, proper, sel, quot, nil, top, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| zWquot#(mark(X1), X2) | → | zWquot#(X1, X2) | | zWquot#(ok(X1), ok(X2)) | → | zWquot#(X1, X2) |
Problem 15: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| zWquot#(X1, mark(X2)) | → | zWquot#(X1, X2) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(minus(X, 0)) | → | mark(0) |
| active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) | | active(quot(0, s(Y))) | → | mark(0) |
| active(quot(s(X), s(Y))) | → | mark(s(quot(minus(X, Y), s(Y)))) | | active(zWquot(XS, nil)) | → | mark(nil) |
| active(zWquot(nil, XS)) | → | mark(nil) | | active(zWquot(cons(X, XS), cons(Y, YS))) | → | mark(cons(quot(X, Y), zWquot(XS, YS))) |
| active(from(X)) | → | from(active(X)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(minus(X1, X2)) | → | minus(active(X1), X2) |
| active(minus(X1, X2)) | → | minus(X1, active(X2)) | | active(quot(X1, X2)) | → | quot(active(X1), X2) |
| active(quot(X1, X2)) | → | quot(X1, active(X2)) | | active(zWquot(X1, X2)) | → | zWquot(active(X1), X2) |
| active(zWquot(X1, X2)) | → | zWquot(X1, active(X2)) | | from(mark(X)) | → | mark(from(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| minus(mark(X1), X2) | → | mark(minus(X1, X2)) | | minus(X1, mark(X2)) | → | mark(minus(X1, X2)) |
| quot(mark(X1), X2) | → | mark(quot(X1, X2)) | | quot(X1, mark(X2)) | → | mark(quot(X1, X2)) |
| zWquot(mark(X1), X2) | → | mark(zWquot(X1, X2)) | | zWquot(X1, mark(X2)) | → | mark(zWquot(X1, X2)) |
| proper(from(X)) | → | from(proper(X)) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(s(X)) | → | s(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(quot(X1, X2)) | → | quot(proper(X1), proper(X2)) | | proper(zWquot(X1, X2)) | → | zWquot(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | from(ok(X)) | → | ok(from(X)) |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| quot(ok(X1), ok(X2)) | → | ok(quot(X1, X2)) | | zWquot(ok(X1), ok(X2)) | → | ok(zWquot(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, mark, from, 0, s, zWquot, active, ok, proper, sel, quot, cons, nil, top
Strategy
Function Precedence
zWquot# = minus = mark = from = 0 = s = zWquot = active = ok = proper = quot = sel = top = cons = nil
Argument Filtering
zWquot#: collapses to 2
minus: 1 2
mark: 1
from: all arguments are removed from from
0: all arguments are removed from 0
s: collapses to 1
zWquot: all arguments are removed from zWquot
active: all arguments are removed from active
ok: all arguments are removed from ok
proper: collapses to 1
quot: all arguments are removed from quot
sel: all arguments are removed from sel
top: collapses to 1
cons: 1 2
nil: all arguments are removed from nil
Status
minus: lexicographic with permutation 1 → 1 2 → 2
mark: multiset
from: multiset
0: multiset
zWquot: multiset
active: multiset
ok: multiset
quot: multiset
sel: multiset
cons: lexicographic with permutation 1 → 1 2 → 2
nil: multiset
Usable Rules
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
| zWquot#(X1, mark(X2)) → zWquot#(X1, X2) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(minus(X, 0)) | → | mark(0) |
| active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) | | active(quot(0, s(Y))) | → | mark(0) |
| active(quot(s(X), s(Y))) | → | mark(s(quot(minus(X, Y), s(Y)))) | | active(zWquot(XS, nil)) | → | mark(nil) |
| active(zWquot(nil, XS)) | → | mark(nil) | | active(zWquot(cons(X, XS), cons(Y, YS))) | → | mark(cons(quot(X, Y), zWquot(XS, YS))) |
| active(from(X)) | → | from(active(X)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(minus(X1, X2)) | → | minus(active(X1), X2) |
| active(minus(X1, X2)) | → | minus(X1, active(X2)) | | active(quot(X1, X2)) | → | quot(active(X1), X2) |
| active(quot(X1, X2)) | → | quot(X1, active(X2)) | | active(zWquot(X1, X2)) | → | zWquot(active(X1), X2) |
| active(zWquot(X1, X2)) | → | zWquot(X1, active(X2)) | | from(mark(X)) | → | mark(from(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| minus(mark(X1), X2) | → | mark(minus(X1, X2)) | | minus(X1, mark(X2)) | → | mark(minus(X1, X2)) |
| quot(mark(X1), X2) | → | mark(quot(X1, X2)) | | quot(X1, mark(X2)) | → | mark(quot(X1, X2)) |
| zWquot(mark(X1), X2) | → | mark(zWquot(X1, X2)) | | zWquot(X1, mark(X2)) | → | mark(zWquot(X1, X2)) |
| proper(from(X)) | → | from(proper(X)) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(s(X)) | → | s(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(quot(X1, X2)) | → | quot(proper(X1), proper(X2)) | | proper(zWquot(X1, X2)) | → | zWquot(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | from(ok(X)) | → | ok(from(X)) |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| quot(ok(X1), ok(X2)) | → | ok(quot(X1, X2)) | | zWquot(ok(X1), ok(X2)) | → | ok(zWquot(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, mark, from, 0, s, zWquot, active, ok, proper, sel, quot, nil, top, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(minus(X, 0)) | → | mark(0) |
| active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) | | active(quot(0, s(Y))) | → | mark(0) |
| active(quot(s(X), s(Y))) | → | mark(s(quot(minus(X, Y), s(Y)))) | | active(zWquot(XS, nil)) | → | mark(nil) |
| active(zWquot(nil, XS)) | → | mark(nil) | | active(zWquot(cons(X, XS), cons(Y, YS))) | → | mark(cons(quot(X, Y), zWquot(XS, YS))) |
| active(from(X)) | → | from(active(X)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(minus(X1, X2)) | → | minus(active(X1), X2) |
| active(minus(X1, X2)) | → | minus(X1, active(X2)) | | active(quot(X1, X2)) | → | quot(active(X1), X2) |
| active(quot(X1, X2)) | → | quot(X1, active(X2)) | | active(zWquot(X1, X2)) | → | zWquot(active(X1), X2) |
| active(zWquot(X1, X2)) | → | zWquot(X1, active(X2)) | | from(mark(X)) | → | mark(from(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| minus(mark(X1), X2) | → | mark(minus(X1, X2)) | | minus(X1, mark(X2)) | → | mark(minus(X1, X2)) |
| quot(mark(X1), X2) | → | mark(quot(X1, X2)) | | quot(X1, mark(X2)) | → | mark(quot(X1, X2)) |
| zWquot(mark(X1), X2) | → | mark(zWquot(X1, X2)) | | zWquot(X1, mark(X2)) | → | mark(zWquot(X1, X2)) |
| proper(from(X)) | → | from(proper(X)) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(s(X)) | → | s(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(quot(X1, X2)) | → | quot(proper(X1), proper(X2)) | | proper(zWquot(X1, X2)) | → | zWquot(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | from(ok(X)) | → | ok(from(X)) |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| quot(ok(X1), ok(X2)) | → | ok(quot(X1, X2)) | | zWquot(ok(X1), ok(X2)) | → | ok(zWquot(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, mark, from, 0, s, zWquot, active, ok, proper, sel, quot, nil, top, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) |
Problem 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(minus(X1, X2)) | → | active#(X2) | | active#(sel(X1, X2)) | → | active#(X2) |
| active#(minus(X1, X2)) | → | active#(X1) | | active#(quot(X1, X2)) | → | active#(X2) |
| active#(s(X)) | → | active#(X) | | active#(from(X)) | → | active#(X) |
| active#(sel(X1, X2)) | → | active#(X1) | | active#(zWquot(X1, X2)) | → | active#(X2) |
| active#(zWquot(X1, X2)) | → | active#(X1) | | active#(quot(X1, X2)) | → | active#(X1) |
| active#(cons(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(minus(X, 0)) | → | mark(0) |
| active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) | | active(quot(0, s(Y))) | → | mark(0) |
| active(quot(s(X), s(Y))) | → | mark(s(quot(minus(X, Y), s(Y)))) | | active(zWquot(XS, nil)) | → | mark(nil) |
| active(zWquot(nil, XS)) | → | mark(nil) | | active(zWquot(cons(X, XS), cons(Y, YS))) | → | mark(cons(quot(X, Y), zWquot(XS, YS))) |
| active(from(X)) | → | from(active(X)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(minus(X1, X2)) | → | minus(active(X1), X2) |
| active(minus(X1, X2)) | → | minus(X1, active(X2)) | | active(quot(X1, X2)) | → | quot(active(X1), X2) |
| active(quot(X1, X2)) | → | quot(X1, active(X2)) | | active(zWquot(X1, X2)) | → | zWquot(active(X1), X2) |
| active(zWquot(X1, X2)) | → | zWquot(X1, active(X2)) | | from(mark(X)) | → | mark(from(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| minus(mark(X1), X2) | → | mark(minus(X1, X2)) | | minus(X1, mark(X2)) | → | mark(minus(X1, X2)) |
| quot(mark(X1), X2) | → | mark(quot(X1, X2)) | | quot(X1, mark(X2)) | → | mark(quot(X1, X2)) |
| zWquot(mark(X1), X2) | → | mark(zWquot(X1, X2)) | | zWquot(X1, mark(X2)) | → | mark(zWquot(X1, X2)) |
| proper(from(X)) | → | from(proper(X)) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(s(X)) | → | s(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(quot(X1, X2)) | → | quot(proper(X1), proper(X2)) | | proper(zWquot(X1, X2)) | → | zWquot(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | from(ok(X)) | → | ok(from(X)) |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| quot(ok(X1), ok(X2)) | → | ok(quot(X1, X2)) | | zWquot(ok(X1), ok(X2)) | → | ok(zWquot(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, mark, from, 0, s, zWquot, active, ok, proper, sel, quot, nil, top, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(minus(X1, X2)) | → | active#(X2) | | active#(sel(X1, X2)) | → | active#(X2) |
| active#(sel(X1, X2)) | → | active#(X1) | | active#(s(X)) | → | active#(X) |
| active#(minus(X1, X2)) | → | active#(X1) | | active#(quot(X1, X2)) | → | active#(X2) |
| active#(from(X)) | → | active#(X) | | active#(zWquot(X1, X2)) | → | active#(X2) |
| active#(zWquot(X1, X2)) | → | active#(X1) | | active#(quot(X1, X2)) | → | active#(X1) |
| active#(cons(X1, X2)) | → | active#(X1) |
Problem 11: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(sel(X1, X2)) | → | proper#(X1) | | proper#(s(X)) | → | proper#(X) |
| proper#(quot(X1, X2)) | → | proper#(X1) | | proper#(cons(X1, X2)) | → | proper#(X1) |
| proper#(cons(X1, X2)) | → | proper#(X2) | | proper#(zWquot(X1, X2)) | → | proper#(X1) |
| proper#(zWquot(X1, X2)) | → | proper#(X2) | | proper#(minus(X1, X2)) | → | proper#(X1) |
| proper#(minus(X1, X2)) | → | proper#(X2) | | proper#(sel(X1, X2)) | → | proper#(X2) |
| proper#(quot(X1, X2)) | → | proper#(X2) | | proper#(from(X)) | → | proper#(X) |
Rewrite Rules
| active(from(X)) | → | mark(cons(X, from(s(X)))) | | active(sel(0, cons(X, XS))) | → | mark(X) |
| active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) | | active(minus(X, 0)) | → | mark(0) |
| active(minus(s(X), s(Y))) | → | mark(minus(X, Y)) | | active(quot(0, s(Y))) | → | mark(0) |
| active(quot(s(X), s(Y))) | → | mark(s(quot(minus(X, Y), s(Y)))) | | active(zWquot(XS, nil)) | → | mark(nil) |
| active(zWquot(nil, XS)) | → | mark(nil) | | active(zWquot(cons(X, XS), cons(Y, YS))) | → | mark(cons(quot(X, Y), zWquot(XS, YS))) |
| active(from(X)) | → | from(active(X)) | | active(cons(X1, X2)) | → | cons(active(X1), X2) |
| active(s(X)) | → | s(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(minus(X1, X2)) | → | minus(active(X1), X2) |
| active(minus(X1, X2)) | → | minus(X1, active(X2)) | | active(quot(X1, X2)) | → | quot(active(X1), X2) |
| active(quot(X1, X2)) | → | quot(X1, active(X2)) | | active(zWquot(X1, X2)) | → | zWquot(active(X1), X2) |
| active(zWquot(X1, X2)) | → | zWquot(X1, active(X2)) | | from(mark(X)) | → | mark(from(X)) |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | | s(mark(X)) | → | mark(s(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| minus(mark(X1), X2) | → | mark(minus(X1, X2)) | | minus(X1, mark(X2)) | → | mark(minus(X1, X2)) |
| quot(mark(X1), X2) | → | mark(quot(X1, X2)) | | quot(X1, mark(X2)) | → | mark(quot(X1, X2)) |
| zWquot(mark(X1), X2) | → | mark(zWquot(X1, X2)) | | zWquot(X1, mark(X2)) | → | mark(zWquot(X1, X2)) |
| proper(from(X)) | → | from(proper(X)) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(s(X)) | → | s(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(minus(X1, X2)) | → | minus(proper(X1), proper(X2)) |
| proper(quot(X1, X2)) | → | quot(proper(X1), proper(X2)) | | proper(zWquot(X1, X2)) | → | zWquot(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | from(ok(X)) | → | ok(from(X)) |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | minus(ok(X1), ok(X2)) | → | ok(minus(X1, X2)) |
| quot(ok(X1), ok(X2)) | → | ok(quot(X1, X2)) | | zWquot(ok(X1), ok(X2)) | → | ok(zWquot(X1, X2)) |
| top(mark(X)) | → | top(proper(X)) | | top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, mark, from, 0, s, zWquot, active, ok, proper, sel, quot, nil, top, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(sel(X1, X2)) | → | proper#(X1) | | proper#(quot(X1, X2)) | → | proper#(X1) |
| proper#(s(X)) | → | proper#(X) | | proper#(cons(X1, X2)) | → | proper#(X1) |
| proper#(cons(X1, X2)) | → | proper#(X2) | | proper#(zWquot(X1, X2)) | → | proper#(X1) |
| proper#(minus(X1, X2)) | → | proper#(X1) | | proper#(zWquot(X1, X2)) | → | proper#(X2) |
| proper#(sel(X1, X2)) | → | proper#(X2) | | proper#(minus(X1, X2)) | → | proper#(X2) |
| proper#(quot(X1, X2)) | → | proper#(X2) | | proper#(from(X)) | → | proper#(X) |