TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60001 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (2824ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| – Problem 3 remains open; application of the following processors failed [SubtermCriterion (0ms), DependencyGraph (6ms), PolynomialLinearRange4iUR (2500ms), DependencyGraph (4ms), PolynomialLinearRange8NegiUR (7500ms), DependencyGraph (5ms), ReductionPairSAT (4075ms), DependencyGraph (4ms), ReductionPairSAT (4110ms), DependencyGraph (4ms), SizeChangePrinciple (timeout)].
| – Problem 4 was processed with processor SubtermCriterion (0ms).
| | – Problem 11 was processed with processor ReductionPairSAT (87ms).
| – Problem 5 was processed with processor SubtermCriterion (1ms).
| – Problem 6 was processed with processor SubtermCriterion (3ms).
| – Problem 7 was processed with processor SubtermCriterion (2ms).
| – Problem 8 was processed with processor SubtermCriterion (1ms).
| | – Problem 12 was processed with processor ReductionPairSAT (75ms).
| – Problem 9 was processed with processor SubtermCriterion (0ms).
| – Problem 10 was processed with processor SubtermCriterion (0ms).
| | – Problem 13 was processed with processor ReductionPairSAT (78ms).
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(fib(N)) | → | mark(sel(N, fib1(s(0), s(0)))) | | active(fib1(X, Y)) | → | mark(cons(X, fib1(Y, add(X, Y)))) |
| active(add(0, X)) | → | mark(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(sel(0, cons(X, XS))) | → | mark(X) | | active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) |
| active(fib(X)) | → | fib(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(fib1(X1, X2)) | → | fib1(active(X1), X2) |
| active(fib1(X1, X2)) | → | fib1(X1, active(X2)) | | active(s(X)) | → | s(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fib(mark(X)) | → | mark(fib(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| fib1(mark(X1), X2) | → | mark(fib1(X1, X2)) | | fib1(X1, mark(X2)) | → | mark(fib1(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fib(X)) | → | fib(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(fib1(X1, X2)) | → | fib1(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) | | fib(ok(X)) | → | ok(fib(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | fib1(ok(X1), ok(X2)) | → | ok(fib1(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| add(ok(X1), ok(X2)) | → | ok(add(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: 0, s, active, ok, mark, proper, add, fib1, sel, fib, top, cons
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| proper#(fib(X)) | → | proper#(X) | | proper#(cons(X1, X2)) | → | proper#(X1) |
| top#(ok(X)) | → | top#(active(X)) | | fib1#(ok(X1), ok(X2)) | → | fib1#(X1, X2) |
| add#(X1, mark(X2)) | → | add#(X1, X2) | | cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) |
| active#(cons(X1, X2)) | → | cons#(active(X1), X2) | | fib#(mark(X)) | → | fib#(X) |
| active#(fib(X)) | → | fib#(active(X)) | | active#(fib1(X, Y)) | → | add#(X, Y) |
| active#(add(X1, X2)) | → | add#(active(X1), X2) | | active#(sel(X1, X2)) | → | active#(X2) |
| top#(mark(X)) | → | proper#(X) | | proper#(add(X1, X2)) | → | proper#(X2) |
| top#(mark(X)) | → | top#(proper(X)) | | proper#(cons(X1, X2)) | → | proper#(X2) |
| active#(fib(X)) | → | active#(X) | | active#(fib1(X, Y)) | → | cons#(X, fib1(Y, add(X, Y))) |
| proper#(add(X1, X2)) | → | proper#(X1) | | add#(mark(X1), X2) | → | add#(X1, X2) |
| active#(fib1(X1, X2)) | → | active#(X1) | | sel#(X1, mark(X2)) | → | sel#(X1, X2) |
| fib1#(mark(X1), X2) | → | fib1#(X1, X2) | | active#(fib1(X1, X2)) | → | fib1#(X1, active(X2)) |
| proper#(fib1(X1, X2)) | → | fib1#(proper(X1), proper(X2)) | | proper#(s(X)) | → | proper#(X) |
| active#(add(s(X), Y)) | → | add#(X, Y) | | active#(add(X1, X2)) | → | active#(X2) |
| sel#(ok(X1), ok(X2)) | → | sel#(X1, X2) | | fib#(ok(X)) | → | fib#(X) |
| active#(fib1(X1, X2)) | → | active#(X2) | | active#(fib1(X, Y)) | → | fib1#(Y, add(X, Y)) |
| active#(cons(X1, X2)) | → | active#(X1) | | sel#(mark(X1), X2) | → | sel#(X1, X2) |
| active#(fib(N)) | → | fib1#(s(0), s(0)) | | proper#(add(X1, X2)) | → | add#(proper(X1), proper(X2)) |
| proper#(fib1(X1, X2)) | → | proper#(X1) | | cons#(mark(X1), X2) | → | cons#(X1, X2) |
| active#(add(X1, X2)) | → | add#(X1, active(X2)) | | add#(ok(X1), ok(X2)) | → | add#(X1, X2) |
| top#(ok(X)) | → | active#(X) | | proper#(sel(X1, X2)) | → | sel#(proper(X1), proper(X2)) |
| active#(add(s(X), Y)) | → | s#(add(X, Y)) | | active#(fib1(X1, X2)) | → | fib1#(active(X1), X2) |
| active#(sel(X1, X2)) | → | active#(X1) | | active#(sel(s(N), cons(X, XS))) | → | sel#(N, XS) |
| proper#(fib1(X1, X2)) | → | proper#(X2) | | proper#(sel(X1, X2)) | → | proper#(X2) |
| fib1#(X1, mark(X2)) | → | fib1#(X1, X2) | | active#(add(X1, X2)) | → | active#(X1) |
| active#(sel(X1, X2)) | → | sel#(active(X1), X2) | | active#(sel(X1, X2)) | → | sel#(X1, active(X2)) |
| active#(fib(N)) | → | sel#(N, fib1(s(0), s(0))) | | active#(s(X)) | → | s#(active(X)) |
| s#(ok(X)) | → | s#(X) | | active#(fib(N)) | → | s#(0) |
| proper#(sel(X1, X2)) | → | proper#(X1) | | s#(mark(X)) | → | s#(X) |
| proper#(cons(X1, X2)) | → | cons#(proper(X1), proper(X2)) | | active#(s(X)) | → | active#(X) |
| proper#(s(X)) | → | s#(proper(X)) | | proper#(fib(X)) | → | fib#(proper(X)) |
Rewrite Rules
| active(fib(N)) | → | mark(sel(N, fib1(s(0), s(0)))) | | active(fib1(X, Y)) | → | mark(cons(X, fib1(Y, add(X, Y)))) |
| active(add(0, X)) | → | mark(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(sel(0, cons(X, XS))) | → | mark(X) | | active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) |
| active(fib(X)) | → | fib(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(fib1(X1, X2)) | → | fib1(active(X1), X2) |
| active(fib1(X1, X2)) | → | fib1(X1, active(X2)) | | active(s(X)) | → | s(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fib(mark(X)) | → | mark(fib(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| fib1(mark(X1), X2) | → | mark(fib1(X1, X2)) | | fib1(X1, mark(X2)) | → | mark(fib1(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fib(X)) | → | fib(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(fib1(X1, X2)) | → | fib1(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) | | fib(ok(X)) | → | ok(fib(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | fib1(ok(X1), ok(X2)) | → | ok(fib1(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| add(ok(X1), ok(X2)) | → | ok(add(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: 0, s, active, mark, ok, proper, fib1, add, sel, fib, cons, top
Strategy
The following SCCs where found
| cons#(mark(X1), X2) → cons#(X1, X2) | cons#(ok(X1), ok(X2)) → cons#(X1, X2) |
| proper#(sel(X1, X2)) → proper#(X1) | proper#(s(X)) → proper#(X) |
| proper#(fib(X)) → proper#(X) | proper#(cons(X1, X2)) → proper#(X1) |
| proper#(fib1(X1, X2)) → proper#(X1) | proper#(cons(X1, X2)) → proper#(X2) |
| proper#(add(X1, X2)) → proper#(X1) | proper#(fib1(X1, X2)) → proper#(X2) |
| proper#(sel(X1, X2)) → proper#(X2) | proper#(add(X1, X2)) → proper#(X2) |
| active#(add(X1, X2)) → active#(X1) | active#(add(X1, X2)) → active#(X2) |
| active#(sel(X1, X2)) → active#(X2) | active#(sel(X1, X2)) → active#(X1) |
| active#(s(X)) → active#(X) | active#(fib(X)) → active#(X) |
| active#(fib1(X1, X2)) → active#(X2) | active#(fib1(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) |
| fib#(ok(X)) → fib#(X) | fib#(mark(X)) → fib#(X) |
| fib1#(ok(X1), ok(X2)) → fib1#(X1, X2) | fib1#(X1, mark(X2)) → fib1#(X1, X2) |
| fib1#(mark(X1), X2) → fib1#(X1, X2) |
| add#(X1, mark(X2)) → add#(X1, X2) | add#(mark(X1), X2) → add#(X1, X2) |
| add#(ok(X1), ok(X2)) → add#(X1, X2) |
| 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
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Rewrite Rules
| active(fib(N)) | → | mark(sel(N, fib1(s(0), s(0)))) | | active(fib1(X, Y)) | → | mark(cons(X, fib1(Y, add(X, Y)))) |
| active(add(0, X)) | → | mark(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(sel(0, cons(X, XS))) | → | mark(X) | | active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) |
| active(fib(X)) | → | fib(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(fib1(X1, X2)) | → | fib1(active(X1), X2) |
| active(fib1(X1, X2)) | → | fib1(X1, active(X2)) | | active(s(X)) | → | s(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fib(mark(X)) | → | mark(fib(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| fib1(mark(X1), X2) | → | mark(fib1(X1, X2)) | | fib1(X1, mark(X2)) | → | mark(fib1(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fib(X)) | → | fib(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(fib1(X1, X2)) | → | fib1(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) | | fib(ok(X)) | → | ok(fib(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | fib1(ok(X1), ok(X2)) | → | ok(fib1(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| add(ok(X1), ok(X2)) | → | ok(add(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: 0, s, active, mark, ok, proper, fib1, add, sel, fib, cons, top
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 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| fib1#(ok(X1), ok(X2)) | → | fib1#(X1, X2) | | fib1#(X1, mark(X2)) | → | fib1#(X1, X2) |
| fib1#(mark(X1), X2) | → | fib1#(X1, X2) |
Rewrite Rules
| active(fib(N)) | → | mark(sel(N, fib1(s(0), s(0)))) | | active(fib1(X, Y)) | → | mark(cons(X, fib1(Y, add(X, Y)))) |
| active(add(0, X)) | → | mark(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(sel(0, cons(X, XS))) | → | mark(X) | | active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) |
| active(fib(X)) | → | fib(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(fib1(X1, X2)) | → | fib1(active(X1), X2) |
| active(fib1(X1, X2)) | → | fib1(X1, active(X2)) | | active(s(X)) | → | s(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fib(mark(X)) | → | mark(fib(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| fib1(mark(X1), X2) | → | mark(fib1(X1, X2)) | | fib1(X1, mark(X2)) | → | mark(fib1(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fib(X)) | → | fib(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(fib1(X1, X2)) | → | fib1(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) | | fib(ok(X)) | → | ok(fib(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | fib1(ok(X1), ok(X2)) | → | ok(fib1(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| add(ok(X1), ok(X2)) | → | ok(add(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: 0, s, active, mark, ok, proper, fib1, add, sel, fib, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| fib1#(ok(X1), ok(X2)) | → | fib1#(X1, X2) | | fib1#(mark(X1), X2) | → | fib1#(X1, X2) |
Problem 11: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| fib1#(X1, mark(X2)) | → | fib1#(X1, X2) |
Rewrite Rules
| active(fib(N)) | → | mark(sel(N, fib1(s(0), s(0)))) | | active(fib1(X, Y)) | → | mark(cons(X, fib1(Y, add(X, Y)))) |
| active(add(0, X)) | → | mark(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(sel(0, cons(X, XS))) | → | mark(X) | | active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) |
| active(fib(X)) | → | fib(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(fib1(X1, X2)) | → | fib1(active(X1), X2) |
| active(fib1(X1, X2)) | → | fib1(X1, active(X2)) | | active(s(X)) | → | s(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fib(mark(X)) | → | mark(fib(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| fib1(mark(X1), X2) | → | mark(fib1(X1, X2)) | | fib1(X1, mark(X2)) | → | mark(fib1(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fib(X)) | → | fib(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(fib1(X1, X2)) | → | fib1(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) | | fib(ok(X)) | → | ok(fib(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | fib1(ok(X1), ok(X2)) | → | ok(fib1(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| add(ok(X1), ok(X2)) | → | ok(add(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: 0, s, active, ok, mark, proper, add, fib1, sel, fib, top, cons
Strategy
Function Precedence
mark < fib1 = add = fib1# = fib = 0 = s = active = ok = proper = sel = cons = top
Argument Filtering
mark: 1
fib1: all arguments are removed from fib1
add: all arguments are removed from add
fib1#: collapses to 2
fib: all arguments are removed from fib
0: all arguments are removed from 0
s: all arguments are removed from s
active: all arguments are removed from active
ok: collapses to 1
proper: all arguments are removed from proper
sel: 1 2
cons: 1 2
top: 1
Status
mark: multiset
fib1: multiset
add: multiset
fib: multiset
0: multiset
s: multiset
active: multiset
proper: multiset
sel: lexicographic with permutation 1 → 2 2 → 1
cons: lexicographic with permutation 1 → 2 2 → 1
top: lexicographic with permutation 1 → 1
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.
| fib1#(X1, mark(X2)) → fib1#(X1, X2) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) |
Rewrite Rules
| active(fib(N)) | → | mark(sel(N, fib1(s(0), s(0)))) | | active(fib1(X, Y)) | → | mark(cons(X, fib1(Y, add(X, Y)))) |
| active(add(0, X)) | → | mark(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(sel(0, cons(X, XS))) | → | mark(X) | | active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) |
| active(fib(X)) | → | fib(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(fib1(X1, X2)) | → | fib1(active(X1), X2) |
| active(fib1(X1, X2)) | → | fib1(X1, active(X2)) | | active(s(X)) | → | s(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fib(mark(X)) | → | mark(fib(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| fib1(mark(X1), X2) | → | mark(fib1(X1, X2)) | | fib1(X1, mark(X2)) | → | mark(fib1(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fib(X)) | → | fib(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(fib1(X1, X2)) | → | fib1(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) | | fib(ok(X)) | → | ok(fib(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | fib1(ok(X1), ok(X2)) | → | ok(fib1(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| add(ok(X1), ok(X2)) | → | ok(add(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: 0, s, active, mark, ok, proper, fib1, add, sel, fib, cons, top
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 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(add(X1, X2)) | → | active#(X1) | | active#(add(X1, X2)) | → | active#(X2) |
| active#(sel(X1, X2)) | → | active#(X2) | | active#(sel(X1, X2)) | → | active#(X1) |
| active#(s(X)) | → | active#(X) | | active#(fib(X)) | → | active#(X) |
| active#(fib1(X1, X2)) | → | active#(X2) | | active#(fib1(X1, X2)) | → | active#(X1) |
| active#(cons(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(fib(N)) | → | mark(sel(N, fib1(s(0), s(0)))) | | active(fib1(X, Y)) | → | mark(cons(X, fib1(Y, add(X, Y)))) |
| active(add(0, X)) | → | mark(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(sel(0, cons(X, XS))) | → | mark(X) | | active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) |
| active(fib(X)) | → | fib(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(fib1(X1, X2)) | → | fib1(active(X1), X2) |
| active(fib1(X1, X2)) | → | fib1(X1, active(X2)) | | active(s(X)) | → | s(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fib(mark(X)) | → | mark(fib(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| fib1(mark(X1), X2) | → | mark(fib1(X1, X2)) | | fib1(X1, mark(X2)) | → | mark(fib1(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fib(X)) | → | fib(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(fib1(X1, X2)) | → | fib1(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) | | fib(ok(X)) | → | ok(fib(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | fib1(ok(X1), ok(X2)) | → | ok(fib1(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| add(ok(X1), ok(X2)) | → | ok(add(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: 0, s, active, mark, ok, proper, fib1, add, sel, fib, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(add(X1, X2)) | → | active#(X1) | | active#(add(X1, X2)) | → | active#(X2) |
| active#(sel(X1, X2)) | → | active#(X2) | | active#(sel(X1, X2)) | → | active#(X1) |
| active#(s(X)) | → | active#(X) | | active#(fib(X)) | → | active#(X) |
| active#(fib1(X1, X2)) | → | active#(X2) | | active#(fib1(X1, X2)) | → | active#(X1) |
| active#(cons(X1, X2)) | → | active#(X1) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(sel(X1, X2)) | → | proper#(X1) | | proper#(s(X)) | → | proper#(X) |
| proper#(fib(X)) | → | proper#(X) | | proper#(cons(X1, X2)) | → | proper#(X1) |
| proper#(fib1(X1, X2)) | → | proper#(X1) | | proper#(cons(X1, X2)) | → | proper#(X2) |
| proper#(add(X1, X2)) | → | proper#(X1) | | proper#(fib1(X1, X2)) | → | proper#(X2) |
| proper#(sel(X1, X2)) | → | proper#(X2) | | proper#(add(X1, X2)) | → | proper#(X2) |
Rewrite Rules
| active(fib(N)) | → | mark(sel(N, fib1(s(0), s(0)))) | | active(fib1(X, Y)) | → | mark(cons(X, fib1(Y, add(X, Y)))) |
| active(add(0, X)) | → | mark(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(sel(0, cons(X, XS))) | → | mark(X) | | active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) |
| active(fib(X)) | → | fib(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(fib1(X1, X2)) | → | fib1(active(X1), X2) |
| active(fib1(X1, X2)) | → | fib1(X1, active(X2)) | | active(s(X)) | → | s(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fib(mark(X)) | → | mark(fib(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| fib1(mark(X1), X2) | → | mark(fib1(X1, X2)) | | fib1(X1, mark(X2)) | → | mark(fib1(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fib(X)) | → | fib(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(fib1(X1, X2)) | → | fib1(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) | | fib(ok(X)) | → | ok(fib(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | fib1(ok(X1), ok(X2)) | → | ok(fib1(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| add(ok(X1), ok(X2)) | → | ok(add(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: 0, s, active, mark, ok, proper, fib1, add, sel, fib, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(sel(X1, X2)) | → | proper#(X1) | | proper#(s(X)) | → | proper#(X) |
| proper#(fib(X)) | → | proper#(X) | | proper#(cons(X1, X2)) | → | proper#(X1) |
| proper#(fib1(X1, X2)) | → | proper#(X1) | | proper#(cons(X1, X2)) | → | proper#(X2) |
| proper#(add(X1, X2)) | → | proper#(X1) | | proper#(fib1(X1, X2)) | → | proper#(X2) |
| proper#(sel(X1, X2)) | → | proper#(X2) | | proper#(add(X1, X2)) | → | proper#(X2) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| add#(X1, mark(X2)) | → | add#(X1, X2) | | add#(mark(X1), X2) | → | add#(X1, X2) |
| add#(ok(X1), ok(X2)) | → | add#(X1, X2) |
Rewrite Rules
| active(fib(N)) | → | mark(sel(N, fib1(s(0), s(0)))) | | active(fib1(X, Y)) | → | mark(cons(X, fib1(Y, add(X, Y)))) |
| active(add(0, X)) | → | mark(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(sel(0, cons(X, XS))) | → | mark(X) | | active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) |
| active(fib(X)) | → | fib(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(fib1(X1, X2)) | → | fib1(active(X1), X2) |
| active(fib1(X1, X2)) | → | fib1(X1, active(X2)) | | active(s(X)) | → | s(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fib(mark(X)) | → | mark(fib(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| fib1(mark(X1), X2) | → | mark(fib1(X1, X2)) | | fib1(X1, mark(X2)) | → | mark(fib1(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fib(X)) | → | fib(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(fib1(X1, X2)) | → | fib1(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) | | fib(ok(X)) | → | ok(fib(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | fib1(ok(X1), ok(X2)) | → | ok(fib1(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| add(ok(X1), ok(X2)) | → | ok(add(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: 0, s, active, mark, ok, proper, fib1, add, sel, fib, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| add#(mark(X1), X2) | → | add#(X1, X2) | | add#(ok(X1), ok(X2)) | → | add#(X1, X2) |
Problem 12: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| add#(X1, mark(X2)) | → | add#(X1, X2) |
Rewrite Rules
| active(fib(N)) | → | mark(sel(N, fib1(s(0), s(0)))) | | active(fib1(X, Y)) | → | mark(cons(X, fib1(Y, add(X, Y)))) |
| active(add(0, X)) | → | mark(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(sel(0, cons(X, XS))) | → | mark(X) | | active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) |
| active(fib(X)) | → | fib(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(fib1(X1, X2)) | → | fib1(active(X1), X2) |
| active(fib1(X1, X2)) | → | fib1(X1, active(X2)) | | active(s(X)) | → | s(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fib(mark(X)) | → | mark(fib(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| fib1(mark(X1), X2) | → | mark(fib1(X1, X2)) | | fib1(X1, mark(X2)) | → | mark(fib1(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fib(X)) | → | fib(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(fib1(X1, X2)) | → | fib1(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) | | fib(ok(X)) | → | ok(fib(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | fib1(ok(X1), ok(X2)) | → | ok(fib1(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| add(ok(X1), ok(X2)) | → | ok(add(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: 0, s, active, ok, mark, proper, add, fib1, sel, fib, top, cons
Strategy
Function Precedence
mark < add# < fib1 = add = fib = 0 = s = active = ok = proper = sel = cons = top
Argument Filtering
mark: 1
fib1: 2
add: all arguments are removed from add
fib: all arguments are removed from fib
0: all arguments are removed from 0
s: all arguments are removed from s
active: all arguments are removed from active
ok: all arguments are removed from ok
proper: all arguments are removed from proper
add#: collapses to 2
sel: 1 2
cons: 1 2
top: all arguments are removed from top
Status
mark: multiset
fib1: lexicographic with permutation 2 → 1
add: multiset
fib: multiset
0: multiset
s: multiset
active: multiset
ok: multiset
proper: multiset
sel: lexicographic with permutation 1 → 1 2 → 2
cons: lexicographic with permutation 1 → 2 2 → 1
top: 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.
| add#(X1, mark(X2)) → add#(X1, X2) |
Problem 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| fib#(ok(X)) | → | fib#(X) | | fib#(mark(X)) | → | fib#(X) |
Rewrite Rules
| active(fib(N)) | → | mark(sel(N, fib1(s(0), s(0)))) | | active(fib1(X, Y)) | → | mark(cons(X, fib1(Y, add(X, Y)))) |
| active(add(0, X)) | → | mark(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(sel(0, cons(X, XS))) | → | mark(X) | | active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) |
| active(fib(X)) | → | fib(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(fib1(X1, X2)) | → | fib1(active(X1), X2) |
| active(fib1(X1, X2)) | → | fib1(X1, active(X2)) | | active(s(X)) | → | s(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fib(mark(X)) | → | mark(fib(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| fib1(mark(X1), X2) | → | mark(fib1(X1, X2)) | | fib1(X1, mark(X2)) | → | mark(fib1(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fib(X)) | → | fib(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(fib1(X1, X2)) | → | fib1(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) | | fib(ok(X)) | → | ok(fib(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | fib1(ok(X1), ok(X2)) | → | ok(fib1(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| add(ok(X1), ok(X2)) | → | ok(add(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: 0, s, active, mark, ok, proper, fib1, add, sel, fib, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| fib#(ok(X)) | → | fib#(X) | | fib#(mark(X)) | → | fib#(X) |
Problem 10: 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(fib(N)) | → | mark(sel(N, fib1(s(0), s(0)))) | | active(fib1(X, Y)) | → | mark(cons(X, fib1(Y, add(X, Y)))) |
| active(add(0, X)) | → | mark(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(sel(0, cons(X, XS))) | → | mark(X) | | active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) |
| active(fib(X)) | → | fib(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(fib1(X1, X2)) | → | fib1(active(X1), X2) |
| active(fib1(X1, X2)) | → | fib1(X1, active(X2)) | | active(s(X)) | → | s(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fib(mark(X)) | → | mark(fib(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| fib1(mark(X1), X2) | → | mark(fib1(X1, X2)) | | fib1(X1, mark(X2)) | → | mark(fib1(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fib(X)) | → | fib(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(fib1(X1, X2)) | → | fib1(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) | | fib(ok(X)) | → | ok(fib(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | fib1(ok(X1), ok(X2)) | → | ok(fib1(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| add(ok(X1), ok(X2)) | → | ok(add(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: 0, s, active, mark, ok, proper, fib1, add, sel, fib, cons, top
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 13: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| sel#(X1, mark(X2)) | → | sel#(X1, X2) |
Rewrite Rules
| active(fib(N)) | → | mark(sel(N, fib1(s(0), s(0)))) | | active(fib1(X, Y)) | → | mark(cons(X, fib1(Y, add(X, Y)))) |
| active(add(0, X)) | → | mark(X) | | active(add(s(X), Y)) | → | mark(s(add(X, Y))) |
| active(sel(0, cons(X, XS))) | → | mark(X) | | active(sel(s(N), cons(X, XS))) | → | mark(sel(N, XS)) |
| active(fib(X)) | → | fib(active(X)) | | active(sel(X1, X2)) | → | sel(active(X1), X2) |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | | active(fib1(X1, X2)) | → | fib1(active(X1), X2) |
| active(fib1(X1, X2)) | → | fib1(X1, active(X2)) | | active(s(X)) | → | s(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fib(mark(X)) | → | mark(fib(X)) |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) |
| fib1(mark(X1), X2) | → | mark(fib1(X1, X2)) | | fib1(X1, mark(X2)) | → | mark(fib1(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fib(X)) | → | fib(proper(X)) | | proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) |
| proper(fib1(X1, X2)) | → | fib1(proper(X1), proper(X2)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) |
| proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) | | fib(ok(X)) | → | ok(fib(X)) |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | | fib1(ok(X1), ok(X2)) | → | ok(fib1(X1, X2)) |
| s(ok(X)) | → | ok(s(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| add(ok(X1), ok(X2)) | → | ok(add(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: 0, s, active, ok, mark, proper, add, fib1, sel, fib, top, cons
Strategy
Function Precedence
sel# < mark < fib1 = add = fib = 0 = s = active = ok = proper = sel = cons = top
Argument Filtering
mark: 1
sel#: collapses to 2
fib1: all arguments are removed from fib1
add: collapses to 1
fib: all arguments are removed from fib
0: all arguments are removed from 0
s: all arguments are removed from s
active: all arguments are removed from active
ok: collapses to 1
proper: all arguments are removed from proper
sel: all arguments are removed from sel
cons: all arguments are removed from cons
top: collapses to 1
Status
mark: multiset
fib1: multiset
fib: multiset
0: multiset
s: multiset
active: multiset
proper: multiset
sel: multiset
cons: 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) |