YES
The TRS could be proven terminating. The proof took 41898 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (2053ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| | – Problem 14 was processed with processor SubtermCriterion (0ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor SubtermCriterion (0ms).
| – Problem 5 was processed with processor SubtermCriterion (0ms).
| | – Problem 15 was processed with processor SubtermCriterion (8ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| – Problem 7 was processed with processor SubtermCriterion (0ms).
| – Problem 8 was processed with processor SubtermCriterion (1ms).
| – Problem 9 was processed with processor SubtermCriterion (0ms).
| – Problem 10 was processed with processor SubtermCriterion (1ms).
| – Problem 11 was processed with processor SubtermCriterion (1ms).
| – Problem 12 was processed with processor SubtermCriterion (0ms).
| – Problem 13 was processed with processor ReductionPairSAT (3666ms).
| | – Problem 16 was processed with processor ReductionPairSAT (1873ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| proper#(cons(X1, X2)) | → | proper#(X1) | | top#(ok(X)) | → | top#(active(X)) |
| active#(sqr(s(X))) | → | s#(add(sqr(X), dbl(X))) | | add#(X1, mark(X2)) | → | add#(X1, X2) |
| active#(dbl(s(X))) | → | s#(dbl(X)) | | cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) |
| recip#(ok(X)) | → | recip#(X) | | active#(terms(N)) | → | terms#(s(N)) |
| active#(first(s(X), cons(Y, Z))) | → | cons#(Y, first(X, Z)) | | active#(cons(X1, X2)) | → | cons#(active(X1), X2) |
| terms#(mark(X)) | → | terms#(X) | | active#(add(X1, X2)) | → | add#(active(X1), X2) |
| active#(dbl(s(X))) | → | dbl#(X) | | first#(mark(X1), X2) | → | first#(X1, X2) |
| top#(mark(X)) | → | proper#(X) | | proper#(add(X1, X2)) | → | proper#(X2) |
| active#(first(X1, X2)) | → | active#(X2) | | top#(mark(X)) | → | top#(proper(X)) |
| active#(sqr(s(X))) | → | dbl#(X) | | proper#(cons(X1, X2)) | → | proper#(X2) |
| active#(dbl(s(X))) | → | s#(s(dbl(X))) | | proper#(first(X1, X2)) | → | first#(proper(X1), proper(X2)) |
| active#(sqr(s(X))) | → | sqr#(X) | | proper#(add(X1, X2)) | → | proper#(X1) |
| active#(terms(N)) | → | sqr#(N) | | add#(mark(X1), X2) | → | add#(X1, X2) |
| active#(sqr(X)) | → | sqr#(active(X)) | | active#(first(X1, X2)) | → | active#(X1) |
| proper#(first(X1, X2)) | → | proper#(X2) | | proper#(s(X)) | → | proper#(X) |
| active#(half(s(s(X)))) | → | half#(X) | | active#(add(s(X), Y)) | → | add#(X, Y) |
| active#(add(X1, X2)) | → | active#(X2) | | active#(recip(X)) | → | active#(X) |
| active#(sqr(s(X))) | → | add#(sqr(X), dbl(X)) | | active#(half(X)) | → | half#(active(X)) |
| proper#(terms(X)) | → | proper#(X) | | proper#(recip(X)) | → | proper#(X) |
| active#(sqr(X)) | → | active#(X) | | active#(cons(X1, X2)) | → | active#(X1) |
| active#(terms(X)) | → | active#(X) | | terms#(ok(X)) | → | terms#(X) |
| proper#(add(X1, X2)) | → | add#(proper(X1), proper(X2)) | | proper#(sqr(X)) | → | proper#(X) |
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | active#(add(X1, X2)) | → | add#(X1, active(X2)) |
| active#(first(X1, X2)) | → | first#(X1, active(X2)) | | proper#(half(X)) | → | half#(proper(X)) |
| proper#(recip(X)) | → | recip#(proper(X)) | | add#(ok(X1), ok(X2)) | → | add#(X1, X2) |
| top#(ok(X)) | → | active#(X) | | active#(first(s(X), cons(Y, Z))) | → | first#(X, Z) |
| active#(add(s(X), Y)) | → | s#(add(X, Y)) | | active#(dbl(X)) | → | active#(X) |
| half#(mark(X)) | → | half#(X) | | proper#(half(X)) | → | proper#(X) |
| first#(X1, mark(X2)) | → | first#(X1, X2) | | active#(add(X1, X2)) | → | active#(X1) |
| active#(recip(X)) | → | recip#(active(X)) | | proper#(sqr(X)) | → | sqr#(proper(X)) |
| proper#(dbl(X)) | → | proper#(X) | | half#(ok(X)) | → | half#(X) |
| active#(terms(N)) | → | cons#(recip(sqr(N)), terms(s(N))) | | dbl#(mark(X)) | → | dbl#(X) |
| active#(s(X)) | → | s#(active(X)) | | proper#(terms(X)) | → | terms#(proper(X)) |
| active#(terms(N)) | → | recip#(sqr(N)) | | active#(dbl(X)) | → | dbl#(active(X)) |
| s#(ok(X)) | → | s#(X) | | s#(mark(X)) | → | s#(X) |
| active#(half(s(s(X)))) | → | s#(half(X)) | | dbl#(ok(X)) | → | dbl#(X) |
| first#(ok(X1), ok(X2)) | → | first#(X1, X2) | | sqr#(mark(X)) | → | sqr#(X) |
| active#(half(X)) | → | active#(X) | | active#(first(X1, X2)) | → | first#(active(X1), X2) |
| proper#(cons(X1, X2)) | → | cons#(proper(X1), proper(X2)) | | proper#(dbl(X)) | → | dbl#(proper(X)) |
| active#(s(X)) | → | active#(X) | | active#(terms(N)) | → | s#(N) |
| active#(terms(X)) | → | terms#(active(X)) | | proper#(s(X)) | → | s#(proper(X)) |
| proper#(first(X1, X2)) | → | proper#(X1) | | sqr#(ok(X)) | → | sqr#(X) |
| recip#(mark(X)) | → | recip#(X) |
Rewrite Rules
| active(terms(N)) | → | mark(cons(recip(sqr(N)), terms(s(N)))) | | active(sqr(0)) | → | mark(0) |
| active(sqr(s(X))) | → | mark(s(add(sqr(X), dbl(X)))) | | active(dbl(0)) | → | mark(0) |
| active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(half(0)) | → | mark(0) |
| active(half(s(0))) | → | mark(0) | | active(half(s(s(X)))) | → | mark(s(half(X))) |
| active(half(dbl(X))) | → | mark(X) | | active(terms(X)) | → | terms(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(recip(X)) | → | recip(active(X)) |
| active(sqr(X)) | → | sqr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(add(X1, X2)) | → | add(active(X1), X2) | | active(add(X1, X2)) | → | add(X1, active(X2)) |
| active(dbl(X)) | → | dbl(active(X)) | | active(first(X1, X2)) | → | first(active(X1), X2) |
| active(first(X1, X2)) | → | first(X1, active(X2)) | | active(half(X)) | → | half(active(X)) |
| terms(mark(X)) | → | mark(terms(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| recip(mark(X)) | → | mark(recip(X)) | | sqr(mark(X)) | → | mark(sqr(X)) |
| s(mark(X)) | → | mark(s(X)) | | add(mark(X1), X2) | → | mark(add(X1, X2)) |
| add(X1, mark(X2)) | → | mark(add(X1, X2)) | | dbl(mark(X)) | → | mark(dbl(X)) |
| first(mark(X1), X2) | → | mark(first(X1, X2)) | | first(X1, mark(X2)) | → | mark(first(X1, X2)) |
| half(mark(X)) | → | mark(half(X)) | | proper(terms(X)) | → | terms(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(recip(X)) | → | recip(proper(X)) |
| proper(sqr(X)) | → | sqr(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(dbl(X)) | → | dbl(proper(X)) | | proper(first(X1, X2)) | → | first(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(half(X)) | → | half(proper(X)) |
| terms(ok(X)) | → | ok(terms(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| recip(ok(X)) | → | ok(recip(X)) | | sqr(ok(X)) | → | ok(sqr(X)) |
| s(ok(X)) | → | ok(s(X)) | | add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) |
| dbl(ok(X)) | → | ok(dbl(X)) | | first(ok(X1), ok(X2)) | → | ok(first(X1, X2)) |
| half(ok(X)) | → | ok(half(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, half, mark, dbl, recip, add, 0, s, active, ok, proper, first, nil, cons, top
Strategy
The following SCCs where found
| cons#(mark(X1), X2) → cons#(X1, X2) | cons#(ok(X1), ok(X2)) → cons#(X1, X2) |
| proper#(first(X1, X2)) → proper#(X2) | proper#(s(X)) → proper#(X) |
| proper#(cons(X1, X2)) → proper#(X1) | proper#(cons(X1, X2)) → proper#(X2) |
| proper#(dbl(X)) → proper#(X) | proper#(sqr(X)) → proper#(X) |
| proper#(first(X1, X2)) → proper#(X1) | proper#(add(X1, X2)) → proper#(X1) |
| proper#(terms(X)) → proper#(X) | proper#(recip(X)) → proper#(X) |
| proper#(half(X)) → proper#(X) | proper#(add(X1, X2)) → proper#(X2) |
| half#(ok(X)) → half#(X) | half#(mark(X)) → half#(X) |
| dbl#(ok(X)) → dbl#(X) | dbl#(mark(X)) → dbl#(X) |
| active#(terms(X)) → active#(X) | active#(first(X1, X2)) → active#(X2) |
| active#(add(X1, X2)) → active#(X1) | active#(add(X1, X2)) → active#(X2) |
| active#(half(X)) → active#(X) | active#(recip(X)) → active#(X) |
| active#(s(X)) → active#(X) | active#(dbl(X)) → active#(X) |
| active#(first(X1, X2)) → active#(X1) | active#(sqr(X)) → active#(X) |
| active#(cons(X1, X2)) → active#(X1) |
| add#(X1, mark(X2)) → add#(X1, X2) | add#(mark(X1), X2) → add#(X1, X2) |
| add#(ok(X1), ok(X2)) → add#(X1, X2) |
| sqr#(mark(X)) → sqr#(X) | sqr#(ok(X)) → sqr#(X) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
| s#(mark(X)) → s#(X) | s#(ok(X)) → s#(X) |
| terms#(mark(X)) → terms#(X) | terms#(ok(X)) → terms#(X) |
| recip#(ok(X)) → recip#(X) | recip#(mark(X)) → recip#(X) |
| first#(ok(X1), ok(X2)) → first#(X1, X2) | first#(mark(X1), X2) → first#(X1, X2) |
| first#(X1, mark(X2)) → first#(X1, X2) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| first#(ok(X1), ok(X2)) | → | first#(X1, X2) | | first#(mark(X1), X2) | → | first#(X1, X2) |
| first#(X1, mark(X2)) | → | first#(X1, X2) |
Rewrite Rules
| active(terms(N)) | → | mark(cons(recip(sqr(N)), terms(s(N)))) | | active(sqr(0)) | → | mark(0) |
| active(sqr(s(X))) | → | mark(s(add(sqr(X), dbl(X)))) | | active(dbl(0)) | → | mark(0) |
| active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(half(0)) | → | mark(0) |
| active(half(s(0))) | → | mark(0) | | active(half(s(s(X)))) | → | mark(s(half(X))) |
| active(half(dbl(X))) | → | mark(X) | | active(terms(X)) | → | terms(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(recip(X)) | → | recip(active(X)) |
| active(sqr(X)) | → | sqr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(add(X1, X2)) | → | add(active(X1), X2) | | active(add(X1, X2)) | → | add(X1, active(X2)) |
| active(dbl(X)) | → | dbl(active(X)) | | active(first(X1, X2)) | → | first(active(X1), X2) |
| active(first(X1, X2)) | → | first(X1, active(X2)) | | active(half(X)) | → | half(active(X)) |
| terms(mark(X)) | → | mark(terms(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| recip(mark(X)) | → | mark(recip(X)) | | sqr(mark(X)) | → | mark(sqr(X)) |
| s(mark(X)) | → | mark(s(X)) | | add(mark(X1), X2) | → | mark(add(X1, X2)) |
| add(X1, mark(X2)) | → | mark(add(X1, X2)) | | dbl(mark(X)) | → | mark(dbl(X)) |
| first(mark(X1), X2) | → | mark(first(X1, X2)) | | first(X1, mark(X2)) | → | mark(first(X1, X2)) |
| half(mark(X)) | → | mark(half(X)) | | proper(terms(X)) | → | terms(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(recip(X)) | → | recip(proper(X)) |
| proper(sqr(X)) | → | sqr(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(dbl(X)) | → | dbl(proper(X)) | | proper(first(X1, X2)) | → | first(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(half(X)) | → | half(proper(X)) |
| terms(ok(X)) | → | ok(terms(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| recip(ok(X)) | → | ok(recip(X)) | | sqr(ok(X)) | → | ok(sqr(X)) |
| s(ok(X)) | → | ok(s(X)) | | add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) |
| dbl(ok(X)) | → | ok(dbl(X)) | | first(ok(X1), ok(X2)) | → | ok(first(X1, X2)) |
| half(ok(X)) | → | ok(half(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, half, mark, dbl, recip, add, 0, s, active, ok, proper, first, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| first#(ok(X1), ok(X2)) | → | first#(X1, X2) | | first#(mark(X1), X2) | → | first#(X1, X2) |
Problem 14: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| first#(X1, mark(X2)) | → | first#(X1, X2) |
Rewrite Rules
| active(terms(N)) | → | mark(cons(recip(sqr(N)), terms(s(N)))) | | active(sqr(0)) | → | mark(0) |
| active(sqr(s(X))) | → | mark(s(add(sqr(X), dbl(X)))) | | active(dbl(0)) | → | mark(0) |
| active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(half(0)) | → | mark(0) |
| active(half(s(0))) | → | mark(0) | | active(half(s(s(X)))) | → | mark(s(half(X))) |
| active(half(dbl(X))) | → | mark(X) | | active(terms(X)) | → | terms(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(recip(X)) | → | recip(active(X)) |
| active(sqr(X)) | → | sqr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(add(X1, X2)) | → | add(active(X1), X2) | | active(add(X1, X2)) | → | add(X1, active(X2)) |
| active(dbl(X)) | → | dbl(active(X)) | | active(first(X1, X2)) | → | first(active(X1), X2) |
| active(first(X1, X2)) | → | first(X1, active(X2)) | | active(half(X)) | → | half(active(X)) |
| terms(mark(X)) | → | mark(terms(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| recip(mark(X)) | → | mark(recip(X)) | | sqr(mark(X)) | → | mark(sqr(X)) |
| s(mark(X)) | → | mark(s(X)) | | add(mark(X1), X2) | → | mark(add(X1, X2)) |
| add(X1, mark(X2)) | → | mark(add(X1, X2)) | | dbl(mark(X)) | → | mark(dbl(X)) |
| first(mark(X1), X2) | → | mark(first(X1, X2)) | | first(X1, mark(X2)) | → | mark(first(X1, X2)) |
| half(mark(X)) | → | mark(half(X)) | | proper(terms(X)) | → | terms(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(recip(X)) | → | recip(proper(X)) |
| proper(sqr(X)) | → | sqr(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(dbl(X)) | → | dbl(proper(X)) | | proper(first(X1, X2)) | → | first(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(half(X)) | → | half(proper(X)) |
| terms(ok(X)) | → | ok(terms(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| recip(ok(X)) | → | ok(recip(X)) | | sqr(ok(X)) | → | ok(sqr(X)) |
| s(ok(X)) | → | ok(s(X)) | | add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) |
| dbl(ok(X)) | → | ok(dbl(X)) | | first(ok(X1), ok(X2)) | → | ok(first(X1, X2)) |
| half(ok(X)) | → | ok(half(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, half, mark, dbl, recip, add, 0, s, active, ok, proper, first, top, cons, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| first#(X1, mark(X2)) | → | first#(X1, X2) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(first(X1, X2)) | → | proper#(X2) | | proper#(s(X)) | → | proper#(X) |
| proper#(cons(X1, X2)) | → | proper#(X1) | | proper#(cons(X1, X2)) | → | proper#(X2) |
| proper#(dbl(X)) | → | proper#(X) | | proper#(sqr(X)) | → | proper#(X) |
| proper#(first(X1, X2)) | → | proper#(X1) | | proper#(terms(X)) | → | proper#(X) |
| proper#(add(X1, X2)) | → | proper#(X1) | | proper#(recip(X)) | → | proper#(X) |
| proper#(half(X)) | → | proper#(X) | | proper#(add(X1, X2)) | → | proper#(X2) |
Rewrite Rules
| active(terms(N)) | → | mark(cons(recip(sqr(N)), terms(s(N)))) | | active(sqr(0)) | → | mark(0) |
| active(sqr(s(X))) | → | mark(s(add(sqr(X), dbl(X)))) | | active(dbl(0)) | → | mark(0) |
| active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(half(0)) | → | mark(0) |
| active(half(s(0))) | → | mark(0) | | active(half(s(s(X)))) | → | mark(s(half(X))) |
| active(half(dbl(X))) | → | mark(X) | | active(terms(X)) | → | terms(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(recip(X)) | → | recip(active(X)) |
| active(sqr(X)) | → | sqr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(add(X1, X2)) | → | add(active(X1), X2) | | active(add(X1, X2)) | → | add(X1, active(X2)) |
| active(dbl(X)) | → | dbl(active(X)) | | active(first(X1, X2)) | → | first(active(X1), X2) |
| active(first(X1, X2)) | → | first(X1, active(X2)) | | active(half(X)) | → | half(active(X)) |
| terms(mark(X)) | → | mark(terms(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| recip(mark(X)) | → | mark(recip(X)) | | sqr(mark(X)) | → | mark(sqr(X)) |
| s(mark(X)) | → | mark(s(X)) | | add(mark(X1), X2) | → | mark(add(X1, X2)) |
| add(X1, mark(X2)) | → | mark(add(X1, X2)) | | dbl(mark(X)) | → | mark(dbl(X)) |
| first(mark(X1), X2) | → | mark(first(X1, X2)) | | first(X1, mark(X2)) | → | mark(first(X1, X2)) |
| half(mark(X)) | → | mark(half(X)) | | proper(terms(X)) | → | terms(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(recip(X)) | → | recip(proper(X)) |
| proper(sqr(X)) | → | sqr(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(dbl(X)) | → | dbl(proper(X)) | | proper(first(X1, X2)) | → | first(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(half(X)) | → | half(proper(X)) |
| terms(ok(X)) | → | ok(terms(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| recip(ok(X)) | → | ok(recip(X)) | | sqr(ok(X)) | → | ok(sqr(X)) |
| s(ok(X)) | → | ok(s(X)) | | add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) |
| dbl(ok(X)) | → | ok(dbl(X)) | | first(ok(X1), ok(X2)) | → | ok(first(X1, X2)) |
| half(ok(X)) | → | ok(half(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, half, mark, dbl, recip, add, 0, s, active, ok, proper, first, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(first(X1, X2)) | → | proper#(X2) | | proper#(s(X)) | → | proper#(X) |
| proper#(cons(X1, X2)) | → | proper#(X1) | | proper#(cons(X1, X2)) | → | proper#(X2) |
| proper#(sqr(X)) | → | proper#(X) | | proper#(dbl(X)) | → | proper#(X) |
| proper#(first(X1, X2)) | → | proper#(X1) | | proper#(add(X1, X2)) | → | proper#(X1) |
| proper#(terms(X)) | → | proper#(X) | | proper#(recip(X)) | → | proper#(X) |
| proper#(half(X)) | → | proper#(X) | | proper#(add(X1, X2)) | → | proper#(X2) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) |
Rewrite Rules
| active(terms(N)) | → | mark(cons(recip(sqr(N)), terms(s(N)))) | | active(sqr(0)) | → | mark(0) |
| active(sqr(s(X))) | → | mark(s(add(sqr(X), dbl(X)))) | | active(dbl(0)) | → | mark(0) |
| active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(half(0)) | → | mark(0) |
| active(half(s(0))) | → | mark(0) | | active(half(s(s(X)))) | → | mark(s(half(X))) |
| active(half(dbl(X))) | → | mark(X) | | active(terms(X)) | → | terms(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(recip(X)) | → | recip(active(X)) |
| active(sqr(X)) | → | sqr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(add(X1, X2)) | → | add(active(X1), X2) | | active(add(X1, X2)) | → | add(X1, active(X2)) |
| active(dbl(X)) | → | dbl(active(X)) | | active(first(X1, X2)) | → | first(active(X1), X2) |
| active(first(X1, X2)) | → | first(X1, active(X2)) | | active(half(X)) | → | half(active(X)) |
| terms(mark(X)) | → | mark(terms(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| recip(mark(X)) | → | mark(recip(X)) | | sqr(mark(X)) | → | mark(sqr(X)) |
| s(mark(X)) | → | mark(s(X)) | | add(mark(X1), X2) | → | mark(add(X1, X2)) |
| add(X1, mark(X2)) | → | mark(add(X1, X2)) | | dbl(mark(X)) | → | mark(dbl(X)) |
| first(mark(X1), X2) | → | mark(first(X1, X2)) | | first(X1, mark(X2)) | → | mark(first(X1, X2)) |
| half(mark(X)) | → | mark(half(X)) | | proper(terms(X)) | → | terms(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(recip(X)) | → | recip(proper(X)) |
| proper(sqr(X)) | → | sqr(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(dbl(X)) | → | dbl(proper(X)) | | proper(first(X1, X2)) | → | first(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(half(X)) | → | half(proper(X)) |
| terms(ok(X)) | → | ok(terms(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| recip(ok(X)) | → | ok(recip(X)) | | sqr(ok(X)) | → | ok(sqr(X)) |
| s(ok(X)) | → | ok(s(X)) | | add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) |
| dbl(ok(X)) | → | ok(dbl(X)) | | first(ok(X1), ok(X2)) | → | ok(first(X1, X2)) |
| half(ok(X)) | → | ok(half(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, half, mark, dbl, recip, add, 0, s, active, ok, proper, first, nil, 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 5: 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(terms(N)) | → | mark(cons(recip(sqr(N)), terms(s(N)))) | | active(sqr(0)) | → | mark(0) |
| active(sqr(s(X))) | → | mark(s(add(sqr(X), dbl(X)))) | | active(dbl(0)) | → | mark(0) |
| active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(half(0)) | → | mark(0) |
| active(half(s(0))) | → | mark(0) | | active(half(s(s(X)))) | → | mark(s(half(X))) |
| active(half(dbl(X))) | → | mark(X) | | active(terms(X)) | → | terms(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(recip(X)) | → | recip(active(X)) |
| active(sqr(X)) | → | sqr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(add(X1, X2)) | → | add(active(X1), X2) | | active(add(X1, X2)) | → | add(X1, active(X2)) |
| active(dbl(X)) | → | dbl(active(X)) | | active(first(X1, X2)) | → | first(active(X1), X2) |
| active(first(X1, X2)) | → | first(X1, active(X2)) | | active(half(X)) | → | half(active(X)) |
| terms(mark(X)) | → | mark(terms(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| recip(mark(X)) | → | mark(recip(X)) | | sqr(mark(X)) | → | mark(sqr(X)) |
| s(mark(X)) | → | mark(s(X)) | | add(mark(X1), X2) | → | mark(add(X1, X2)) |
| add(X1, mark(X2)) | → | mark(add(X1, X2)) | | dbl(mark(X)) | → | mark(dbl(X)) |
| first(mark(X1), X2) | → | mark(first(X1, X2)) | | first(X1, mark(X2)) | → | mark(first(X1, X2)) |
| half(mark(X)) | → | mark(half(X)) | | proper(terms(X)) | → | terms(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(recip(X)) | → | recip(proper(X)) |
| proper(sqr(X)) | → | sqr(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(dbl(X)) | → | dbl(proper(X)) | | proper(first(X1, X2)) | → | first(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(half(X)) | → | half(proper(X)) |
| terms(ok(X)) | → | ok(terms(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| recip(ok(X)) | → | ok(recip(X)) | | sqr(ok(X)) | → | ok(sqr(X)) |
| s(ok(X)) | → | ok(s(X)) | | add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) |
| dbl(ok(X)) | → | ok(dbl(X)) | | first(ok(X1), ok(X2)) | → | ok(first(X1, X2)) |
| half(ok(X)) | → | ok(half(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, half, mark, dbl, recip, add, 0, s, active, ok, proper, first, nil, 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 15: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| add#(X1, mark(X2)) | → | add#(X1, X2) |
Rewrite Rules
| active(terms(N)) | → | mark(cons(recip(sqr(N)), terms(s(N)))) | | active(sqr(0)) | → | mark(0) |
| active(sqr(s(X))) | → | mark(s(add(sqr(X), dbl(X)))) | | active(dbl(0)) | → | mark(0) |
| active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(half(0)) | → | mark(0) |
| active(half(s(0))) | → | mark(0) | | active(half(s(s(X)))) | → | mark(s(half(X))) |
| active(half(dbl(X))) | → | mark(X) | | active(terms(X)) | → | terms(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(recip(X)) | → | recip(active(X)) |
| active(sqr(X)) | → | sqr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(add(X1, X2)) | → | add(active(X1), X2) | | active(add(X1, X2)) | → | add(X1, active(X2)) |
| active(dbl(X)) | → | dbl(active(X)) | | active(first(X1, X2)) | → | first(active(X1), X2) |
| active(first(X1, X2)) | → | first(X1, active(X2)) | | active(half(X)) | → | half(active(X)) |
| terms(mark(X)) | → | mark(terms(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| recip(mark(X)) | → | mark(recip(X)) | | sqr(mark(X)) | → | mark(sqr(X)) |
| s(mark(X)) | → | mark(s(X)) | | add(mark(X1), X2) | → | mark(add(X1, X2)) |
| add(X1, mark(X2)) | → | mark(add(X1, X2)) | | dbl(mark(X)) | → | mark(dbl(X)) |
| first(mark(X1), X2) | → | mark(first(X1, X2)) | | first(X1, mark(X2)) | → | mark(first(X1, X2)) |
| half(mark(X)) | → | mark(half(X)) | | proper(terms(X)) | → | terms(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(recip(X)) | → | recip(proper(X)) |
| proper(sqr(X)) | → | sqr(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(dbl(X)) | → | dbl(proper(X)) | | proper(first(X1, X2)) | → | first(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(half(X)) | → | half(proper(X)) |
| terms(ok(X)) | → | ok(terms(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| recip(ok(X)) | → | ok(recip(X)) | | sqr(ok(X)) | → | ok(sqr(X)) |
| s(ok(X)) | → | ok(s(X)) | | add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) |
| dbl(ok(X)) | → | ok(dbl(X)) | | first(ok(X1), ok(X2)) | → | ok(first(X1, X2)) |
| half(ok(X)) | → | ok(half(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, half, mark, dbl, recip, add, 0, s, active, ok, proper, first, top, cons, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| add#(X1, mark(X2)) | → | add#(X1, X2) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| half#(ok(X)) | → | half#(X) | | half#(mark(X)) | → | half#(X) |
Rewrite Rules
| active(terms(N)) | → | mark(cons(recip(sqr(N)), terms(s(N)))) | | active(sqr(0)) | → | mark(0) |
| active(sqr(s(X))) | → | mark(s(add(sqr(X), dbl(X)))) | | active(dbl(0)) | → | mark(0) |
| active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(half(0)) | → | mark(0) |
| active(half(s(0))) | → | mark(0) | | active(half(s(s(X)))) | → | mark(s(half(X))) |
| active(half(dbl(X))) | → | mark(X) | | active(terms(X)) | → | terms(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(recip(X)) | → | recip(active(X)) |
| active(sqr(X)) | → | sqr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(add(X1, X2)) | → | add(active(X1), X2) | | active(add(X1, X2)) | → | add(X1, active(X2)) |
| active(dbl(X)) | → | dbl(active(X)) | | active(first(X1, X2)) | → | first(active(X1), X2) |
| active(first(X1, X2)) | → | first(X1, active(X2)) | | active(half(X)) | → | half(active(X)) |
| terms(mark(X)) | → | mark(terms(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| recip(mark(X)) | → | mark(recip(X)) | | sqr(mark(X)) | → | mark(sqr(X)) |
| s(mark(X)) | → | mark(s(X)) | | add(mark(X1), X2) | → | mark(add(X1, X2)) |
| add(X1, mark(X2)) | → | mark(add(X1, X2)) | | dbl(mark(X)) | → | mark(dbl(X)) |
| first(mark(X1), X2) | → | mark(first(X1, X2)) | | first(X1, mark(X2)) | → | mark(first(X1, X2)) |
| half(mark(X)) | → | mark(half(X)) | | proper(terms(X)) | → | terms(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(recip(X)) | → | recip(proper(X)) |
| proper(sqr(X)) | → | sqr(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(dbl(X)) | → | dbl(proper(X)) | | proper(first(X1, X2)) | → | first(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(half(X)) | → | half(proper(X)) |
| terms(ok(X)) | → | ok(terms(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| recip(ok(X)) | → | ok(recip(X)) | | sqr(ok(X)) | → | ok(sqr(X)) |
| s(ok(X)) | → | ok(s(X)) | | add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) |
| dbl(ok(X)) | → | ok(dbl(X)) | | first(ok(X1), ok(X2)) | → | ok(first(X1, X2)) |
| half(ok(X)) | → | ok(half(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, half, mark, dbl, recip, add, 0, s, active, ok, proper, first, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| half#(ok(X)) | → | half#(X) | | half#(mark(X)) | → | half#(X) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| terms#(mark(X)) | → | terms#(X) | | terms#(ok(X)) | → | terms#(X) |
Rewrite Rules
| active(terms(N)) | → | mark(cons(recip(sqr(N)), terms(s(N)))) | | active(sqr(0)) | → | mark(0) |
| active(sqr(s(X))) | → | mark(s(add(sqr(X), dbl(X)))) | | active(dbl(0)) | → | mark(0) |
| active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(half(0)) | → | mark(0) |
| active(half(s(0))) | → | mark(0) | | active(half(s(s(X)))) | → | mark(s(half(X))) |
| active(half(dbl(X))) | → | mark(X) | | active(terms(X)) | → | terms(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(recip(X)) | → | recip(active(X)) |
| active(sqr(X)) | → | sqr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(add(X1, X2)) | → | add(active(X1), X2) | | active(add(X1, X2)) | → | add(X1, active(X2)) |
| active(dbl(X)) | → | dbl(active(X)) | | active(first(X1, X2)) | → | first(active(X1), X2) |
| active(first(X1, X2)) | → | first(X1, active(X2)) | | active(half(X)) | → | half(active(X)) |
| terms(mark(X)) | → | mark(terms(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| recip(mark(X)) | → | mark(recip(X)) | | sqr(mark(X)) | → | mark(sqr(X)) |
| s(mark(X)) | → | mark(s(X)) | | add(mark(X1), X2) | → | mark(add(X1, X2)) |
| add(X1, mark(X2)) | → | mark(add(X1, X2)) | | dbl(mark(X)) | → | mark(dbl(X)) |
| first(mark(X1), X2) | → | mark(first(X1, X2)) | | first(X1, mark(X2)) | → | mark(first(X1, X2)) |
| half(mark(X)) | → | mark(half(X)) | | proper(terms(X)) | → | terms(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(recip(X)) | → | recip(proper(X)) |
| proper(sqr(X)) | → | sqr(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(dbl(X)) | → | dbl(proper(X)) | | proper(first(X1, X2)) | → | first(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(half(X)) | → | half(proper(X)) |
| terms(ok(X)) | → | ok(terms(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| recip(ok(X)) | → | ok(recip(X)) | | sqr(ok(X)) | → | ok(sqr(X)) |
| s(ok(X)) | → | ok(s(X)) | | add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) |
| dbl(ok(X)) | → | ok(dbl(X)) | | first(ok(X1), ok(X2)) | → | ok(first(X1, X2)) |
| half(ok(X)) | → | ok(half(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, half, mark, dbl, recip, add, 0, s, active, ok, proper, first, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| terms#(mark(X)) | → | terms#(X) | | terms#(ok(X)) | → | terms#(X) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| recip#(ok(X)) | → | recip#(X) | | recip#(mark(X)) | → | recip#(X) |
Rewrite Rules
| active(terms(N)) | → | mark(cons(recip(sqr(N)), terms(s(N)))) | | active(sqr(0)) | → | mark(0) |
| active(sqr(s(X))) | → | mark(s(add(sqr(X), dbl(X)))) | | active(dbl(0)) | → | mark(0) |
| active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(half(0)) | → | mark(0) |
| active(half(s(0))) | → | mark(0) | | active(half(s(s(X)))) | → | mark(s(half(X))) |
| active(half(dbl(X))) | → | mark(X) | | active(terms(X)) | → | terms(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(recip(X)) | → | recip(active(X)) |
| active(sqr(X)) | → | sqr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(add(X1, X2)) | → | add(active(X1), X2) | | active(add(X1, X2)) | → | add(X1, active(X2)) |
| active(dbl(X)) | → | dbl(active(X)) | | active(first(X1, X2)) | → | first(active(X1), X2) |
| active(first(X1, X2)) | → | first(X1, active(X2)) | | active(half(X)) | → | half(active(X)) |
| terms(mark(X)) | → | mark(terms(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| recip(mark(X)) | → | mark(recip(X)) | | sqr(mark(X)) | → | mark(sqr(X)) |
| s(mark(X)) | → | mark(s(X)) | | add(mark(X1), X2) | → | mark(add(X1, X2)) |
| add(X1, mark(X2)) | → | mark(add(X1, X2)) | | dbl(mark(X)) | → | mark(dbl(X)) |
| first(mark(X1), X2) | → | mark(first(X1, X2)) | | first(X1, mark(X2)) | → | mark(first(X1, X2)) |
| half(mark(X)) | → | mark(half(X)) | | proper(terms(X)) | → | terms(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(recip(X)) | → | recip(proper(X)) |
| proper(sqr(X)) | → | sqr(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(dbl(X)) | → | dbl(proper(X)) | | proper(first(X1, X2)) | → | first(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(half(X)) | → | half(proper(X)) |
| terms(ok(X)) | → | ok(terms(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| recip(ok(X)) | → | ok(recip(X)) | | sqr(ok(X)) | → | ok(sqr(X)) |
| s(ok(X)) | → | ok(s(X)) | | add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) |
| dbl(ok(X)) | → | ok(dbl(X)) | | first(ok(X1), ok(X2)) | → | ok(first(X1, X2)) |
| half(ok(X)) | → | ok(half(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, half, mark, dbl, recip, add, 0, s, active, ok, proper, first, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| recip#(ok(X)) | → | recip#(X) | | recip#(mark(X)) | → | recip#(X) |
Problem 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Rewrite Rules
| active(terms(N)) | → | mark(cons(recip(sqr(N)), terms(s(N)))) | | active(sqr(0)) | → | mark(0) |
| active(sqr(s(X))) | → | mark(s(add(sqr(X), dbl(X)))) | | active(dbl(0)) | → | mark(0) |
| active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(half(0)) | → | mark(0) |
| active(half(s(0))) | → | mark(0) | | active(half(s(s(X)))) | → | mark(s(half(X))) |
| active(half(dbl(X))) | → | mark(X) | | active(terms(X)) | → | terms(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(recip(X)) | → | recip(active(X)) |
| active(sqr(X)) | → | sqr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(add(X1, X2)) | → | add(active(X1), X2) | | active(add(X1, X2)) | → | add(X1, active(X2)) |
| active(dbl(X)) | → | dbl(active(X)) | | active(first(X1, X2)) | → | first(active(X1), X2) |
| active(first(X1, X2)) | → | first(X1, active(X2)) | | active(half(X)) | → | half(active(X)) |
| terms(mark(X)) | → | mark(terms(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| recip(mark(X)) | → | mark(recip(X)) | | sqr(mark(X)) | → | mark(sqr(X)) |
| s(mark(X)) | → | mark(s(X)) | | add(mark(X1), X2) | → | mark(add(X1, X2)) |
| add(X1, mark(X2)) | → | mark(add(X1, X2)) | | dbl(mark(X)) | → | mark(dbl(X)) |
| first(mark(X1), X2) | → | mark(first(X1, X2)) | | first(X1, mark(X2)) | → | mark(first(X1, X2)) |
| half(mark(X)) | → | mark(half(X)) | | proper(terms(X)) | → | terms(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(recip(X)) | → | recip(proper(X)) |
| proper(sqr(X)) | → | sqr(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(dbl(X)) | → | dbl(proper(X)) | | proper(first(X1, X2)) | → | first(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(half(X)) | → | half(proper(X)) |
| terms(ok(X)) | → | ok(terms(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| recip(ok(X)) | → | ok(recip(X)) | | sqr(ok(X)) | → | ok(sqr(X)) |
| s(ok(X)) | → | ok(s(X)) | | add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) |
| dbl(ok(X)) | → | ok(dbl(X)) | | first(ok(X1), ok(X2)) | → | ok(first(X1, X2)) |
| half(ok(X)) | → | ok(half(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, half, mark, dbl, recip, add, 0, s, active, ok, proper, first, nil, 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 10: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| sqr#(mark(X)) | → | sqr#(X) | | sqr#(ok(X)) | → | sqr#(X) |
Rewrite Rules
| active(terms(N)) | → | mark(cons(recip(sqr(N)), terms(s(N)))) | | active(sqr(0)) | → | mark(0) |
| active(sqr(s(X))) | → | mark(s(add(sqr(X), dbl(X)))) | | active(dbl(0)) | → | mark(0) |
| active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(half(0)) | → | mark(0) |
| active(half(s(0))) | → | mark(0) | | active(half(s(s(X)))) | → | mark(s(half(X))) |
| active(half(dbl(X))) | → | mark(X) | | active(terms(X)) | → | terms(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(recip(X)) | → | recip(active(X)) |
| active(sqr(X)) | → | sqr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(add(X1, X2)) | → | add(active(X1), X2) | | active(add(X1, X2)) | → | add(X1, active(X2)) |
| active(dbl(X)) | → | dbl(active(X)) | | active(first(X1, X2)) | → | first(active(X1), X2) |
| active(first(X1, X2)) | → | first(X1, active(X2)) | | active(half(X)) | → | half(active(X)) |
| terms(mark(X)) | → | mark(terms(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| recip(mark(X)) | → | mark(recip(X)) | | sqr(mark(X)) | → | mark(sqr(X)) |
| s(mark(X)) | → | mark(s(X)) | | add(mark(X1), X2) | → | mark(add(X1, X2)) |
| add(X1, mark(X2)) | → | mark(add(X1, X2)) | | dbl(mark(X)) | → | mark(dbl(X)) |
| first(mark(X1), X2) | → | mark(first(X1, X2)) | | first(X1, mark(X2)) | → | mark(first(X1, X2)) |
| half(mark(X)) | → | mark(half(X)) | | proper(terms(X)) | → | terms(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(recip(X)) | → | recip(proper(X)) |
| proper(sqr(X)) | → | sqr(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(dbl(X)) | → | dbl(proper(X)) | | proper(first(X1, X2)) | → | first(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(half(X)) | → | half(proper(X)) |
| terms(ok(X)) | → | ok(terms(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| recip(ok(X)) | → | ok(recip(X)) | | sqr(ok(X)) | → | ok(sqr(X)) |
| s(ok(X)) | → | ok(s(X)) | | add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) |
| dbl(ok(X)) | → | ok(dbl(X)) | | first(ok(X1), ok(X2)) | → | ok(first(X1, X2)) |
| half(ok(X)) | → | ok(half(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, half, mark, dbl, recip, add, 0, s, active, ok, proper, first, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| sqr#(mark(X)) | → | sqr#(X) | | sqr#(ok(X)) | → | sqr#(X) |
Problem 11: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(terms(X)) | → | active#(X) | | active#(first(X1, X2)) | → | active#(X2) |
| active#(add(X1, X2)) | → | active#(X1) | | active#(add(X1, X2)) | → | active#(X2) |
| active#(half(X)) | → | active#(X) | | active#(recip(X)) | → | active#(X) |
| active#(s(X)) | → | active#(X) | | active#(dbl(X)) | → | active#(X) |
| active#(first(X1, X2)) | → | active#(X1) | | active#(sqr(X)) | → | active#(X) |
| active#(cons(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(terms(N)) | → | mark(cons(recip(sqr(N)), terms(s(N)))) | | active(sqr(0)) | → | mark(0) |
| active(sqr(s(X))) | → | mark(s(add(sqr(X), dbl(X)))) | | active(dbl(0)) | → | mark(0) |
| active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(half(0)) | → | mark(0) |
| active(half(s(0))) | → | mark(0) | | active(half(s(s(X)))) | → | mark(s(half(X))) |
| active(half(dbl(X))) | → | mark(X) | | active(terms(X)) | → | terms(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(recip(X)) | → | recip(active(X)) |
| active(sqr(X)) | → | sqr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(add(X1, X2)) | → | add(active(X1), X2) | | active(add(X1, X2)) | → | add(X1, active(X2)) |
| active(dbl(X)) | → | dbl(active(X)) | | active(first(X1, X2)) | → | first(active(X1), X2) |
| active(first(X1, X2)) | → | first(X1, active(X2)) | | active(half(X)) | → | half(active(X)) |
| terms(mark(X)) | → | mark(terms(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| recip(mark(X)) | → | mark(recip(X)) | | sqr(mark(X)) | → | mark(sqr(X)) |
| s(mark(X)) | → | mark(s(X)) | | add(mark(X1), X2) | → | mark(add(X1, X2)) |
| add(X1, mark(X2)) | → | mark(add(X1, X2)) | | dbl(mark(X)) | → | mark(dbl(X)) |
| first(mark(X1), X2) | → | mark(first(X1, X2)) | | first(X1, mark(X2)) | → | mark(first(X1, X2)) |
| half(mark(X)) | → | mark(half(X)) | | proper(terms(X)) | → | terms(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(recip(X)) | → | recip(proper(X)) |
| proper(sqr(X)) | → | sqr(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(dbl(X)) | → | dbl(proper(X)) | | proper(first(X1, X2)) | → | first(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(half(X)) | → | half(proper(X)) |
| terms(ok(X)) | → | ok(terms(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| recip(ok(X)) | → | ok(recip(X)) | | sqr(ok(X)) | → | ok(sqr(X)) |
| s(ok(X)) | → | ok(s(X)) | | add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) |
| dbl(ok(X)) | → | ok(dbl(X)) | | first(ok(X1), ok(X2)) | → | ok(first(X1, X2)) |
| half(ok(X)) | → | ok(half(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, half, mark, dbl, recip, add, 0, s, active, ok, proper, first, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(first(X1, X2)) | → | active#(X2) | | active#(terms(X)) | → | active#(X) |
| active#(add(X1, X2)) | → | active#(X1) | | active#(add(X1, X2)) | → | active#(X2) |
| active#(half(X)) | → | active#(X) | | active#(recip(X)) | → | active#(X) |
| active#(s(X)) | → | active#(X) | | active#(dbl(X)) | → | active#(X) |
| active#(first(X1, X2)) | → | active#(X1) | | active#(sqr(X)) | → | active#(X) |
| active#(cons(X1, X2)) | → | active#(X1) |
Problem 12: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| dbl#(ok(X)) | → | dbl#(X) | | dbl#(mark(X)) | → | dbl#(X) |
Rewrite Rules
| active(terms(N)) | → | mark(cons(recip(sqr(N)), terms(s(N)))) | | active(sqr(0)) | → | mark(0) |
| active(sqr(s(X))) | → | mark(s(add(sqr(X), dbl(X)))) | | active(dbl(0)) | → | mark(0) |
| active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(half(0)) | → | mark(0) |
| active(half(s(0))) | → | mark(0) | | active(half(s(s(X)))) | → | mark(s(half(X))) |
| active(half(dbl(X))) | → | mark(X) | | active(terms(X)) | → | terms(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(recip(X)) | → | recip(active(X)) |
| active(sqr(X)) | → | sqr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(add(X1, X2)) | → | add(active(X1), X2) | | active(add(X1, X2)) | → | add(X1, active(X2)) |
| active(dbl(X)) | → | dbl(active(X)) | | active(first(X1, X2)) | → | first(active(X1), X2) |
| active(first(X1, X2)) | → | first(X1, active(X2)) | | active(half(X)) | → | half(active(X)) |
| terms(mark(X)) | → | mark(terms(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| recip(mark(X)) | → | mark(recip(X)) | | sqr(mark(X)) | → | mark(sqr(X)) |
| s(mark(X)) | → | mark(s(X)) | | add(mark(X1), X2) | → | mark(add(X1, X2)) |
| add(X1, mark(X2)) | → | mark(add(X1, X2)) | | dbl(mark(X)) | → | mark(dbl(X)) |
| first(mark(X1), X2) | → | mark(first(X1, X2)) | | first(X1, mark(X2)) | → | mark(first(X1, X2)) |
| half(mark(X)) | → | mark(half(X)) | | proper(terms(X)) | → | terms(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(recip(X)) | → | recip(proper(X)) |
| proper(sqr(X)) | → | sqr(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(dbl(X)) | → | dbl(proper(X)) | | proper(first(X1, X2)) | → | first(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(half(X)) | → | half(proper(X)) |
| terms(ok(X)) | → | ok(terms(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| recip(ok(X)) | → | ok(recip(X)) | | sqr(ok(X)) | → | ok(sqr(X)) |
| s(ok(X)) | → | ok(s(X)) | | add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) |
| dbl(ok(X)) | → | ok(dbl(X)) | | first(ok(X1), ok(X2)) | → | ok(first(X1, X2)) |
| half(ok(X)) | → | ok(half(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, half, mark, dbl, recip, add, 0, s, active, ok, proper, first, nil, cons, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| dbl#(ok(X)) | → | dbl#(X) | | dbl#(mark(X)) | → | dbl#(X) |
Problem 13: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(terms(N)) | → | mark(cons(recip(sqr(N)), terms(s(N)))) | | active(sqr(0)) | → | mark(0) |
| active(sqr(s(X))) | → | mark(s(add(sqr(X), dbl(X)))) | | active(dbl(0)) | → | mark(0) |
| active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(half(0)) | → | mark(0) |
| active(half(s(0))) | → | mark(0) | | active(half(s(s(X)))) | → | mark(s(half(X))) |
| active(half(dbl(X))) | → | mark(X) | | active(terms(X)) | → | terms(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(recip(X)) | → | recip(active(X)) |
| active(sqr(X)) | → | sqr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(add(X1, X2)) | → | add(active(X1), X2) | | active(add(X1, X2)) | → | add(X1, active(X2)) |
| active(dbl(X)) | → | dbl(active(X)) | | active(first(X1, X2)) | → | first(active(X1), X2) |
| active(first(X1, X2)) | → | first(X1, active(X2)) | | active(half(X)) | → | half(active(X)) |
| terms(mark(X)) | → | mark(terms(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| recip(mark(X)) | → | mark(recip(X)) | | sqr(mark(X)) | → | mark(sqr(X)) |
| s(mark(X)) | → | mark(s(X)) | | add(mark(X1), X2) | → | mark(add(X1, X2)) |
| add(X1, mark(X2)) | → | mark(add(X1, X2)) | | dbl(mark(X)) | → | mark(dbl(X)) |
| first(mark(X1), X2) | → | mark(first(X1, X2)) | | first(X1, mark(X2)) | → | mark(first(X1, X2)) |
| half(mark(X)) | → | mark(half(X)) | | proper(terms(X)) | → | terms(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(recip(X)) | → | recip(proper(X)) |
| proper(sqr(X)) | → | sqr(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(dbl(X)) | → | dbl(proper(X)) | | proper(first(X1, X2)) | → | first(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(half(X)) | → | half(proper(X)) |
| terms(ok(X)) | → | ok(terms(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| recip(ok(X)) | → | ok(recip(X)) | | sqr(ok(X)) | → | ok(sqr(X)) |
| s(ok(X)) | → | ok(s(X)) | | add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) |
| dbl(ok(X)) | → | ok(dbl(X)) | | first(ok(X1), ok(X2)) | → | ok(first(X1, X2)) |
| half(ok(X)) | → | ok(half(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, half, mark, dbl, recip, add, 0, s, active, ok, proper, first, nil, cons, top
Strategy
Function Precedence
recip < terms < 0 < nil < sqr < active < dbl < half = first < add < proper = top# < mark = s = ok = top = cons
Argument Filtering
terms: 1
sqr: 1
half: 1
mark: 1
dbl: 1
recip: collapses to 1
add: 1 2
0: all arguments are removed from 0
s: 1
active: collapses to 1
ok: collapses to 1
proper: collapses to 1
first: 1 2
top: collapses to 1
cons: collapses to 1
nil: all arguments are removed from nil
top#: collapses to 1
Status
terms: lexicographic with permutation 1 → 1
sqr: lexicographic with permutation 1 → 1
half: lexicographic with permutation 1 → 1
mark: lexicographic with permutation 1 → 1
dbl: lexicographic with permutation 1 → 1
add: lexicographic with permutation 1 → 1 2 → 2
0: multiset
s: lexicographic with permutation 1 → 1
first: lexicographic with permutation 1 → 2 2 → 1
nil: multiset
Usable Rules
| active(half(X)) → half(active(X)) | proper(sqr(X)) → sqr(proper(X)) |
| dbl(mark(X)) → mark(dbl(X)) | active(s(X)) → s(active(X)) |
| proper(dbl(X)) → dbl(proper(X)) | cons(mark(X1), X2) → mark(cons(X1, X2)) |
| active(terms(X)) → terms(active(X)) | active(recip(X)) → recip(active(X)) |
| proper(recip(X)) → recip(proper(X)) | add(mark(X1), X2) → mark(add(X1, X2)) |
| terms(ok(X)) → ok(terms(X)) | active(dbl(0)) → mark(0) |
| sqr(ok(X)) → ok(sqr(X)) | add(X1, mark(X2)) → mark(add(X1, X2)) |
| first(ok(X1), ok(X2)) → ok(first(X1, X2)) | proper(terms(X)) → terms(proper(X)) |
| sqr(mark(X)) → mark(sqr(X)) | active(half(dbl(X))) → mark(X) |
| first(mark(X1), X2) → mark(first(X1, X2)) | active(sqr(X)) → sqr(active(X)) |
| active(add(s(X), Y)) → mark(s(add(X, Y))) | cons(ok(X1), ok(X2)) → ok(cons(X1, X2)) |
| recip(ok(X)) → ok(recip(X)) | active(half(s(0))) → mark(0) |
| active(cons(X1, X2)) → cons(active(X1), X2) | proper(cons(X1, X2)) → cons(proper(X1), proper(X2)) |
| proper(nil) → ok(nil) | proper(0) → ok(0) |
| half(ok(X)) → ok(half(X)) | active(add(X1, X2)) → add(X1, active(X2)) |
| active(sqr(0)) → mark(0) | active(dbl(X)) → dbl(active(X)) |
| active(first(0, X)) → mark(nil) | first(X1, mark(X2)) → mark(first(X1, X2)) |
| proper(add(X1, X2)) → add(proper(X1), proper(X2)) | active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z))) |
| s(mark(X)) → mark(s(X)) | active(first(X1, X2)) → first(active(X1), X2) |
| active(add(X1, X2)) → add(active(X1), X2) | proper(s(X)) → s(proper(X)) |
| active(dbl(s(X))) → mark(s(s(dbl(X)))) | active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N)))) |
| active(half(s(s(X)))) → mark(s(half(X))) | dbl(ok(X)) → ok(dbl(X)) |
| proper(first(X1, X2)) → first(proper(X1), proper(X2)) | terms(mark(X)) → mark(terms(X)) |
| s(ok(X)) → ok(s(X)) | proper(half(X)) → half(proper(X)) |
| active(half(0)) → mark(0) | recip(mark(X)) → mark(recip(X)) |
| add(ok(X1), ok(X2)) → ok(add(X1, X2)) | half(mark(X)) → mark(half(X)) |
| active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X)))) | active(add(0, X)) → mark(X) |
| active(first(X1, X2)) → first(X1, active(X2)) |
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.
| top#(mark(X)) → top#(proper(X)) |
Problem 16: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(terms(N)) | → | mark(cons(recip(sqr(N)), terms(s(N)))) | | active(sqr(0)) | → | mark(0) |
| active(sqr(s(X))) | → | mark(s(add(sqr(X), dbl(X)))) | | active(dbl(0)) | → | mark(0) |
| active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(first(0, X)) | → | mark(nil) |
| active(first(s(X), cons(Y, Z))) | → | mark(cons(Y, first(X, Z))) | | active(half(0)) | → | mark(0) |
| active(half(s(0))) | → | mark(0) | | active(half(s(s(X)))) | → | mark(s(half(X))) |
| active(half(dbl(X))) | → | mark(X) | | active(terms(X)) | → | terms(active(X)) |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | | active(recip(X)) | → | recip(active(X)) |
| active(sqr(X)) | → | sqr(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(add(X1, X2)) | → | add(active(X1), X2) | | active(add(X1, X2)) | → | add(X1, active(X2)) |
| active(dbl(X)) | → | dbl(active(X)) | | active(first(X1, X2)) | → | first(active(X1), X2) |
| active(first(X1, X2)) | → | first(X1, active(X2)) | | active(half(X)) | → | half(active(X)) |
| terms(mark(X)) | → | mark(terms(X)) | | cons(mark(X1), X2) | → | mark(cons(X1, X2)) |
| recip(mark(X)) | → | mark(recip(X)) | | sqr(mark(X)) | → | mark(sqr(X)) |
| s(mark(X)) | → | mark(s(X)) | | add(mark(X1), X2) | → | mark(add(X1, X2)) |
| add(X1, mark(X2)) | → | mark(add(X1, X2)) | | dbl(mark(X)) | → | mark(dbl(X)) |
| first(mark(X1), X2) | → | mark(first(X1, X2)) | | first(X1, mark(X2)) | → | mark(first(X1, X2)) |
| half(mark(X)) | → | mark(half(X)) | | proper(terms(X)) | → | terms(proper(X)) |
| proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | | proper(recip(X)) | → | recip(proper(X)) |
| proper(sqr(X)) | → | sqr(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(dbl(X)) | → | dbl(proper(X)) | | proper(first(X1, X2)) | → | first(proper(X1), proper(X2)) |
| proper(nil) | → | ok(nil) | | proper(half(X)) | → | half(proper(X)) |
| terms(ok(X)) | → | ok(terms(X)) | | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) |
| recip(ok(X)) | → | ok(recip(X)) | | sqr(ok(X)) | → | ok(sqr(X)) |
| s(ok(X)) | → | ok(s(X)) | | add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) |
| dbl(ok(X)) | → | ok(dbl(X)) | | first(ok(X1), ok(X2)) | → | ok(first(X1, X2)) |
| half(ok(X)) | → | ok(half(X)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: terms, sqr, half, mark, dbl, recip, add, 0, s, active, ok, proper, first, top, cons, nil
Strategy
Function Precedence
half < recip < s < cons < first < mark < 0 < terms = sqr = dbl = add = active = ok = proper = top = nil = top#
Argument Filtering
terms: 1
sqr: 1
half: collapses to 1
mark: collapses to 1
dbl: 1
recip: collapses to 1
add: collapses to 2
0: all arguments are removed from 0
s: collapses to 1
active: collapses to 1
ok: 1
proper: all arguments are removed from proper
first: 1
top: 1
cons: collapses to 2
nil: all arguments are removed from nil
top#: 1
Status
terms: lexicographic with permutation 1 → 1
sqr: lexicographic with permutation 1 → 1
dbl: lexicographic with permutation 1 → 1
0: multiset
ok: lexicographic with permutation 1 → 1
proper: multiset
first: lexicographic with permutation 1 → 1
top: lexicographic with permutation 1 → 1
nil: multiset
top#: multiset
Usable Rules
| active(sqr(0)) → mark(0) | active(dbl(X)) → dbl(active(X)) |
| active(half(X)) → half(active(X)) | dbl(mark(X)) → mark(dbl(X)) |
| active(s(X)) → s(active(X)) | cons(mark(X1), X2) → mark(cons(X1, X2)) |
| active(terms(X)) → terms(active(X)) | first(X1, mark(X2)) → mark(first(X1, X2)) |
| active(first(0, X)) → mark(nil) | active(recip(X)) → recip(active(X)) |
| active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z))) | add(mark(X1), X2) → mark(add(X1, X2)) |
| terms(ok(X)) → ok(terms(X)) | s(mark(X)) → mark(s(X)) |
| active(first(X1, X2)) → first(active(X1), X2) | active(add(X1, X2)) → add(active(X1), X2) |
| active(dbl(0)) → mark(0) | active(dbl(s(X))) → mark(s(s(dbl(X)))) |
| sqr(ok(X)) → ok(sqr(X)) | active(terms(N)) → mark(cons(recip(sqr(N)), terms(s(N)))) |
| add(X1, mark(X2)) → mark(add(X1, X2)) | first(ok(X1), ok(X2)) → ok(first(X1, X2)) |
| active(half(s(s(X)))) → mark(s(half(X))) | dbl(ok(X)) → ok(dbl(X)) |
| sqr(mark(X)) → mark(sqr(X)) | terms(mark(X)) → mark(terms(X)) |
| active(half(dbl(X))) → mark(X) | first(mark(X1), X2) → mark(first(X1, X2)) |
| s(ok(X)) → ok(s(X)) | active(sqr(X)) → sqr(active(X)) |
| active(add(s(X), Y)) → mark(s(add(X, Y))) | cons(ok(X1), ok(X2)) → ok(cons(X1, X2)) |
| recip(ok(X)) → ok(recip(X)) | active(half(0)) → mark(0) |
| recip(mark(X)) → mark(recip(X)) | half(mark(X)) → mark(half(X)) |
| add(ok(X1), ok(X2)) → ok(add(X1, X2)) | active(half(s(0))) → mark(0) |
| active(sqr(s(X))) → mark(s(add(sqr(X), dbl(X)))) | active(cons(X1, X2)) → cons(active(X1), X2) |
| active(add(0, X)) → mark(X) | half(ok(X)) → ok(half(X)) |
| active(add(X1, X2)) → add(X1, active(X2)) | active(first(X1, X2)) → first(X1, active(X2)) |
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.
| top#(ok(X)) → top#(active(X)) |