TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60002 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (5315ms).
| – Problem 2 was processed with processor SubtermCriterion (4ms).
| – Problem 3 was processed with processor SubtermCriterion (2ms).
| – Problem 4 was processed with processor SubtermCriterion (2ms).
| | – Problem 14 was processed with processor ReductionPairSAT (61ms).
| – Problem 5 was processed with processor SubtermCriterion (1ms).
| – Problem 6 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (5ms), PolynomialLinearRange4iUR (5001ms), DependencyGraph (28ms), PolynomialLinearRange8NegiUR (15000ms), DependencyGraph (6ms), ReductionPairSAT (17528ms), DependencyGraph (5ms), ReductionPairSAT (timeout)].
| – Problem 7 was processed with processor SubtermCriterion (3ms).
| – Problem 8 was processed with processor SubtermCriterion (1ms).
| – Problem 9 was processed with processor SubtermCriterion (1ms).
| – Problem 10 was processed with processor SubtermCriterion (2ms).
| – Problem 11 was processed with processor SubtermCriterion (1ms).
| – Problem 12 was processed with processor SubtermCriterion (3ms).
| – Problem 13 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 6
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(U42(isNat(N), M, N)) | | active(U42(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(plus(N, 0)) | → | mark(U31(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U41(isNat(M), M, N)) | | active(U11(X1, X2)) | → | U11(active(X1), X2) |
| active(U12(X)) | → | U12(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) |
| active(U42(X1, X2, X3)) | → | U42(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U12(mark(X)) | → | mark(U12(X)) |
| U21(mark(X)) | → | mark(U21(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) | | U42(mark(X1), X2, X3) | → | mark(U42(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(U12(X)) | → | U12(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U21(X)) | → | U21(proper(X)) |
| proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2, X3)) | → | U42(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| U42(ok(X1), ok(X2), ok(X3)) | → | ok(U42(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(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: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, ok, U12, proper, U31, U21, top
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(U41(tt, M, N)) | → | isNat#(N) | | top#(ok(X)) | → | top#(active(X)) |
| proper#(U11(X1, X2)) | → | proper#(X1) | | proper#(U31(X1, X2)) | → | proper#(X1) |
| active#(U12(X)) | → | active#(X) | | U11#(ok(X1), ok(X2)) | → | U11#(X1, X2) |
| active#(U42(X1, X2, X3)) | → | U42#(active(X1), X2, X3) | | active#(U21(X)) | → | U21#(active(X)) |
| active#(plus(N, 0)) | → | U31#(isNat(N), N) | | active#(U42(X1, X2, X3)) | → | active#(X1) |
| U11#(mark(X1), X2) | → | U11#(X1, X2) | | proper#(U11(X1, X2)) | → | U11#(proper(X1), proper(X2)) |
| proper#(U41(X1, X2, X3)) | → | proper#(X2) | | U12#(mark(X)) | → | U12#(X) |
| proper#(U11(X1, X2)) | → | proper#(X2) | | plus#(X1, mark(X2)) | → | plus#(X1, X2) |
| active#(plus(N, s(M))) | → | U41#(isNat(M), M, N) | | active#(U41(tt, M, N)) | → | U42#(isNat(N), M, N) |
| proper#(plus(X1, X2)) | → | proper#(X1) | | active#(plus(N, 0)) | → | isNat#(N) |
| active#(U11(X1, X2)) | → | active#(X1) | | top#(mark(X)) | → | proper#(X) |
| proper#(U41(X1, X2, X3)) | → | U41#(proper(X1), proper(X2), proper(X3)) | | proper#(plus(X1, X2)) | → | plus#(proper(X1), proper(X2)) |
| U21#(ok(X)) | → | U21#(X) | | top#(mark(X)) | → | top#(proper(X)) |
| U42#(mark(X1), X2, X3) | → | U42#(X1, X2, X3) | | proper#(U12(X)) | → | U12#(proper(X)) |
| isNat#(ok(X)) | → | isNat#(X) | | active#(U41(X1, X2, X3)) | → | active#(X1) |
| active#(U11(tt, V2)) | → | U12#(isNat(V2)) | | proper#(s(X)) | → | proper#(X) |
| active#(plus(X1, X2)) | → | active#(X1) | | proper#(U21(X)) | → | U21#(proper(X)) |
| proper#(U42(X1, X2, X3)) | → | proper#(X3) | | proper#(U42(X1, X2, X3)) | → | U42#(proper(X1), proper(X2), proper(X3)) |
| active#(plus(X1, X2)) | → | active#(X2) | | plus#(mark(X1), X2) | → | plus#(X1, X2) |
| proper#(U41(X1, X2, X3)) | → | proper#(X3) | | proper#(U31(X1, X2)) | → | proper#(X2) |
| active#(U41(X1, X2, X3)) | → | U41#(active(X1), X2, X3) | | active#(U42(tt, M, N)) | → | s#(plus(N, M)) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | top#(ok(X)) | → | active#(X) |
| active#(U21(X)) | → | active#(X) | | active#(U31(X1, X2)) | → | U31#(active(X1), X2) |
| active#(U12(X)) | → | U12#(active(X)) | | proper#(U41(X1, X2, X3)) | → | proper#(X1) |
| U41#(mark(X1), X2, X3) | → | U41#(X1, X2, X3) | | active#(U31(X1, X2)) | → | active#(X1) |
| proper#(isNat(X)) | → | isNat#(proper(X)) | | proper#(plus(X1, X2)) | → | proper#(X2) |
| proper#(U42(X1, X2, X3)) | → | proper#(X2) | | proper#(U21(X)) | → | proper#(X) |
| active#(isNat(s(V1))) | → | isNat#(V1) | | U41#(ok(X1), ok(X2), ok(X3)) | → | U41#(X1, X2, X3) |
| plus#(ok(X1), ok(X2)) | → | plus#(X1, X2) | | proper#(isNat(X)) | → | proper#(X) |
| proper#(U12(X)) | → | proper#(X) | | active#(U11(tt, V2)) | → | isNat#(V2) |
| U21#(mark(X)) | → | U21#(X) | | active#(isNat(s(V1))) | → | U21#(isNat(V1)) |
| active#(U11(X1, X2)) | → | U11#(active(X1), X2) | | active#(s(X)) | → | s#(active(X)) |
| proper#(U42(X1, X2, X3)) | → | proper#(X1) | | U31#(mark(X1), X2) | → | U31#(X1, X2) |
| s#(ok(X)) | → | s#(X) | | active#(U42(tt, M, N)) | → | plus#(N, M) |
| s#(mark(X)) | → | s#(X) | | active#(plus(X1, X2)) | → | plus#(X1, active(X2)) |
| active#(isNat(plus(V1, V2))) | → | U11#(isNat(V1), V2) | | U12#(ok(X)) | → | U12#(X) |
| active#(plus(X1, X2)) | → | plus#(active(X1), X2) | | active#(s(X)) | → | active#(X) |
| active#(plus(N, s(M))) | → | isNat#(M) | | proper#(s(X)) | → | s#(proper(X)) |
| proper#(U31(X1, X2)) | → | U31#(proper(X1), proper(X2)) | | U31#(ok(X1), ok(X2)) | → | U31#(X1, X2) |
| U42#(ok(X1), ok(X2), ok(X3)) | → | U42#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(U42(isNat(N), M, N)) | | active(U42(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(plus(N, 0)) | → | mark(U31(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U41(isNat(M), M, N)) | | active(U11(X1, X2)) | → | U11(active(X1), X2) |
| active(U12(X)) | → | U12(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) |
| active(U42(X1, X2, X3)) | → | U42(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U12(mark(X)) | → | mark(U12(X)) |
| U21(mark(X)) | → | mark(U21(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) | | U42(mark(X1), X2, X3) | → | mark(U42(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(U12(X)) | → | U12(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U21(X)) | → | U21(proper(X)) |
| proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2, X3)) | → | U42(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| U42(ok(X1), ok(X2), ok(X3)) | → | ok(U42(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(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: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, ok, U12, U31, proper, U21, top
Strategy
The following SCCs where found
| active#(plus(X1, X2)) → active#(X1) | active#(s(X)) → active#(X) |
| active#(U12(X)) → active#(X) | active#(U31(X1, X2)) → active#(X1) |
| active#(plus(X1, X2)) → active#(X2) | active#(U42(X1, X2, X3)) → active#(X1) |
| active#(U11(X1, X2)) → active#(X1) | active#(U21(X)) → active#(X) |
| active#(U41(X1, X2, X3)) → active#(X1) |
| U42#(mark(X1), X2, X3) → U42#(X1, X2, X3) | U42#(ok(X1), ok(X2), ok(X3)) → U42#(X1, X2, X3) |
| plus#(ok(X1), ok(X2)) → plus#(X1, X2) | plus#(X1, mark(X2)) → plus#(X1, X2) |
| plus#(mark(X1), X2) → plus#(X1, X2) |
| U12#(ok(X)) → U12#(X) | U12#(mark(X)) → U12#(X) |
| proper#(isNat(X)) → proper#(X) | proper#(U11(X1, X2)) → proper#(X1) |
| proper#(U12(X)) → proper#(X) | proper#(U31(X1, X2)) → proper#(X1) |
| proper#(U41(X1, X2, X3)) → proper#(X3) | proper#(U31(X1, X2)) → proper#(X2) |
| proper#(U42(X1, X2, X3)) → proper#(X1) | proper#(U41(X1, X2, X3)) → proper#(X2) |
| proper#(s(X)) → proper#(X) | proper#(U11(X1, X2)) → proper#(X2) |
| proper#(U42(X1, X2, X3)) → proper#(X3) | proper#(U41(X1, X2, X3)) → proper#(X1) |
| proper#(plus(X1, X2)) → proper#(X1) | proper#(plus(X1, X2)) → proper#(X2) |
| proper#(U42(X1, X2, X3)) → proper#(X2) | proper#(U21(X)) → proper#(X) |
| U11#(ok(X1), ok(X2)) → U11#(X1, X2) | U11#(mark(X1), X2) → U11#(X1, X2) |
| isNat#(ok(X)) → isNat#(X) |
| U21#(ok(X)) → U21#(X) | U21#(mark(X)) → U21#(X) |
| s#(mark(X)) → s#(X) | s#(ok(X)) → s#(X) |
| U31#(mark(X1), X2) → U31#(X1, X2) | U31#(ok(X1), ok(X2)) → U31#(X1, X2) |
| U41#(mark(X1), X2, X3) → U41#(X1, X2, X3) | U41#(ok(X1), ok(X2), ok(X3)) → U41#(X1, X2, X3) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U21#(ok(X)) | → | U21#(X) | | U21#(mark(X)) | → | U21#(X) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(U42(isNat(N), M, N)) | | active(U42(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(plus(N, 0)) | → | mark(U31(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U41(isNat(M), M, N)) | | active(U11(X1, X2)) | → | U11(active(X1), X2) |
| active(U12(X)) | → | U12(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) |
| active(U42(X1, X2, X3)) | → | U42(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U12(mark(X)) | → | mark(U12(X)) |
| U21(mark(X)) | → | mark(U21(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) | | U42(mark(X1), X2, X3) | → | mark(U42(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(U12(X)) | → | U12(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U21(X)) | → | U21(proper(X)) |
| proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2, X3)) | → | U42(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| U42(ok(X1), ok(X2), ok(X3)) | → | ok(U42(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(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: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, ok, U12, U31, proper, U21, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U21#(ok(X)) | → | U21#(X) | | U21#(mark(X)) | → | U21#(X) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(isNat(X)) | → | proper#(X) | | proper#(U12(X)) | → | proper#(X) |
| proper#(U11(X1, X2)) | → | proper#(X1) | | proper#(U41(X1, X2, X3)) | → | proper#(X3) |
| proper#(U31(X1, X2)) | → | proper#(X1) | | proper#(U31(X1, X2)) | → | proper#(X2) |
| proper#(U42(X1, X2, X3)) | → | proper#(X1) | | proper#(U41(X1, X2, X3)) | → | proper#(X2) |
| proper#(s(X)) | → | proper#(X) | | proper#(U11(X1, X2)) | → | proper#(X2) |
| proper#(U42(X1, X2, X3)) | → | proper#(X3) | | proper#(U41(X1, X2, X3)) | → | proper#(X1) |
| proper#(plus(X1, X2)) | → | proper#(X1) | | proper#(plus(X1, X2)) | → | proper#(X2) |
| proper#(U42(X1, X2, X3)) | → | proper#(X2) | | proper#(U21(X)) | → | proper#(X) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(U42(isNat(N), M, N)) | | active(U42(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(plus(N, 0)) | → | mark(U31(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U41(isNat(M), M, N)) | | active(U11(X1, X2)) | → | U11(active(X1), X2) |
| active(U12(X)) | → | U12(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) |
| active(U42(X1, X2, X3)) | → | U42(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U12(mark(X)) | → | mark(U12(X)) |
| U21(mark(X)) | → | mark(U21(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) | | U42(mark(X1), X2, X3) | → | mark(U42(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(U12(X)) | → | U12(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U21(X)) | → | U21(proper(X)) |
| proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2, X3)) | → | U42(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| U42(ok(X1), ok(X2), ok(X3)) | → | ok(U42(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(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: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, ok, U12, U31, proper, U21, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(isNat(X)) | → | proper#(X) | | proper#(U12(X)) | → | proper#(X) |
| proper#(U11(X1, X2)) | → | proper#(X1) | | proper#(U31(X1, X2)) | → | proper#(X1) |
| proper#(U41(X1, X2, X3)) | → | proper#(X3) | | proper#(U31(X1, X2)) | → | proper#(X2) |
| proper#(U42(X1, X2, X3)) | → | proper#(X1) | | proper#(U41(X1, X2, X3)) | → | proper#(X2) |
| proper#(s(X)) | → | proper#(X) | | proper#(U11(X1, X2)) | → | proper#(X2) |
| proper#(U42(X1, X2, X3)) | → | proper#(X3) | | proper#(U41(X1, X2, X3)) | → | proper#(X1) |
| proper#(plus(X1, X2)) | → | proper#(X1) | | proper#(plus(X1, X2)) | → | proper#(X2) |
| proper#(U42(X1, X2, X3)) | → | proper#(X2) | | proper#(U21(X)) | → | proper#(X) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| plus#(ok(X1), ok(X2)) | → | plus#(X1, X2) | | plus#(X1, mark(X2)) | → | plus#(X1, X2) |
| plus#(mark(X1), X2) | → | plus#(X1, X2) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(U42(isNat(N), M, N)) | | active(U42(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(plus(N, 0)) | → | mark(U31(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U41(isNat(M), M, N)) | | active(U11(X1, X2)) | → | U11(active(X1), X2) |
| active(U12(X)) | → | U12(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) |
| active(U42(X1, X2, X3)) | → | U42(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U12(mark(X)) | → | mark(U12(X)) |
| U21(mark(X)) | → | mark(U21(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) | | U42(mark(X1), X2, X3) | → | mark(U42(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(U12(X)) | → | U12(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U21(X)) | → | U21(proper(X)) |
| proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2, X3)) | → | U42(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| U42(ok(X1), ok(X2), ok(X3)) | → | ok(U42(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(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: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, ok, U12, U31, proper, U21, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| plus#(ok(X1), ok(X2)) | → | plus#(X1, X2) | | plus#(mark(X1), X2) | → | plus#(X1, X2) |
Problem 14: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| plus#(X1, mark(X2)) | → | plus#(X1, X2) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(U42(isNat(N), M, N)) | | active(U42(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(plus(N, 0)) | → | mark(U31(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U41(isNat(M), M, N)) | | active(U11(X1, X2)) | → | U11(active(X1), X2) |
| active(U12(X)) | → | U12(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) |
| active(U42(X1, X2, X3)) | → | U42(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U12(mark(X)) | → | mark(U12(X)) |
| U21(mark(X)) | → | mark(U21(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) | | U42(mark(X1), X2, X3) | → | mark(U42(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(U12(X)) | → | U12(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U21(X)) | → | U21(proper(X)) |
| proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2, X3)) | → | U42(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| U42(ok(X1), ok(X2), ok(X3)) | → | ok(U42(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(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: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, ok, U12, proper, U31, U21, top
Strategy
Function Precedence
mark < plus = isNat = 0 = s = U42 = tt = U41 = active = U11 = plus# = ok = U12 = U31 = proper = U21 = top
Argument Filtering
plus: all arguments are removed from plus
mark: 1
isNat: all arguments are removed from isNat
0: all arguments are removed from 0
s: collapses to 1
U42: all arguments are removed from U42
tt: all arguments are removed from tt
U41: all arguments are removed from U41
active: all arguments are removed from active
U11: 1 2
plus#: collapses to 2
ok: all arguments are removed from ok
U12: all arguments are removed from U12
U31: all arguments are removed from U31
proper: all arguments are removed from proper
U21: all arguments are removed from U21
top: 1
Status
plus: multiset
mark: lexicographic with permutation 1 → 1
isNat: multiset
0: multiset
U42: multiset
tt: multiset
U41: multiset
active: multiset
U11: lexicographic with permutation 1 → 2 2 → 1
ok: multiset
U12: multiset
U31: multiset
proper: multiset
U21: multiset
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.
| plus#(X1, mark(X2)) → plus#(X1, X2) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U41#(mark(X1), X2, X3) | → | U41#(X1, X2, X3) | | U41#(ok(X1), ok(X2), ok(X3)) | → | U41#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(U42(isNat(N), M, N)) | | active(U42(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(plus(N, 0)) | → | mark(U31(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U41(isNat(M), M, N)) | | active(U11(X1, X2)) | → | U11(active(X1), X2) |
| active(U12(X)) | → | U12(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) |
| active(U42(X1, X2, X3)) | → | U42(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U12(mark(X)) | → | mark(U12(X)) |
| U21(mark(X)) | → | mark(U21(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) | | U42(mark(X1), X2, X3) | → | mark(U42(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(U12(X)) | → | U12(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U21(X)) | → | U21(proper(X)) |
| proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2, X3)) | → | U42(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| U42(ok(X1), ok(X2), ok(X3)) | → | ok(U42(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(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: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, ok, U12, U31, proper, U21, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U41#(mark(X1), X2, X3) | → | U41#(X1, X2, X3) | | U41#(ok(X1), ok(X2), ok(X3)) | → | U41#(X1, X2, X3) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(U42(isNat(N), M, N)) | | active(U42(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(plus(N, 0)) | → | mark(U31(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U41(isNat(M), M, N)) | | active(U11(X1, X2)) | → | U11(active(X1), X2) |
| active(U12(X)) | → | U12(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) |
| active(U42(X1, X2, X3)) | → | U42(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U12(mark(X)) | → | mark(U12(X)) |
| U21(mark(X)) | → | mark(U21(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) | | U42(mark(X1), X2, X3) | → | mark(U42(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(U12(X)) | → | U12(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U21(X)) | → | U21(proper(X)) |
| proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2, X3)) | → | U42(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| U42(ok(X1), ok(X2), ok(X3)) | → | ok(U42(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(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: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, ok, U12, U31, proper, U21, 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 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U12#(ok(X)) | → | U12#(X) | | U12#(mark(X)) | → | U12#(X) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(U42(isNat(N), M, N)) | | active(U42(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(plus(N, 0)) | → | mark(U31(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U41(isNat(M), M, N)) | | active(U11(X1, X2)) | → | U11(active(X1), X2) |
| active(U12(X)) | → | U12(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) |
| active(U42(X1, X2, X3)) | → | U42(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U12(mark(X)) | → | mark(U12(X)) |
| U21(mark(X)) | → | mark(U21(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) | | U42(mark(X1), X2, X3) | → | mark(U42(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(U12(X)) | → | U12(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U21(X)) | → | U21(proper(X)) |
| proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2, X3)) | → | U42(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| U42(ok(X1), ok(X2), ok(X3)) | → | ok(U42(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(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: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, ok, U12, U31, proper, U21, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U12#(ok(X)) | → | U12#(X) | | U12#(mark(X)) | → | U12#(X) |
Problem 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| isNat#(ok(X)) | → | isNat#(X) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(U42(isNat(N), M, N)) | | active(U42(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(plus(N, 0)) | → | mark(U31(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U41(isNat(M), M, N)) | | active(U11(X1, X2)) | → | U11(active(X1), X2) |
| active(U12(X)) | → | U12(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) |
| active(U42(X1, X2, X3)) | → | U42(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U12(mark(X)) | → | mark(U12(X)) |
| U21(mark(X)) | → | mark(U21(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) | | U42(mark(X1), X2, X3) | → | mark(U42(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(U12(X)) | → | U12(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U21(X)) | → | U21(proper(X)) |
| proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2, X3)) | → | U42(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| U42(ok(X1), ok(X2), ok(X3)) | → | ok(U42(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(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: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, ok, U12, U31, proper, U21, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNat#(ok(X)) | → | isNat#(X) |
Problem 10: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U31#(mark(X1), X2) | → | U31#(X1, X2) | | U31#(ok(X1), ok(X2)) | → | U31#(X1, X2) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(U42(isNat(N), M, N)) | | active(U42(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(plus(N, 0)) | → | mark(U31(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U41(isNat(M), M, N)) | | active(U11(X1, X2)) | → | U11(active(X1), X2) |
| active(U12(X)) | → | U12(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) |
| active(U42(X1, X2, X3)) | → | U42(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U12(mark(X)) | → | mark(U12(X)) |
| U21(mark(X)) | → | mark(U21(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) | | U42(mark(X1), X2, X3) | → | mark(U42(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(U12(X)) | → | U12(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U21(X)) | → | U21(proper(X)) |
| proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2, X3)) | → | U42(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| U42(ok(X1), ok(X2), ok(X3)) | → | ok(U42(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(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: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, ok, U12, U31, proper, U21, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U31#(mark(X1), X2) | → | U31#(X1, X2) | | U31#(ok(X1), ok(X2)) | → | U31#(X1, X2) |
Problem 11: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U11#(ok(X1), ok(X2)) | → | U11#(X1, X2) | | U11#(mark(X1), X2) | → | U11#(X1, X2) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(U42(isNat(N), M, N)) | | active(U42(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(plus(N, 0)) | → | mark(U31(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U41(isNat(M), M, N)) | | active(U11(X1, X2)) | → | U11(active(X1), X2) |
| active(U12(X)) | → | U12(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) |
| active(U42(X1, X2, X3)) | → | U42(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U12(mark(X)) | → | mark(U12(X)) |
| U21(mark(X)) | → | mark(U21(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) | | U42(mark(X1), X2, X3) | → | mark(U42(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(U12(X)) | → | U12(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U21(X)) | → | U21(proper(X)) |
| proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2, X3)) | → | U42(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| U42(ok(X1), ok(X2), ok(X3)) | → | ok(U42(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(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: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, ok, U12, U31, proper, U21, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U11#(ok(X1), ok(X2)) | → | U11#(X1, X2) | | U11#(mark(X1), X2) | → | U11#(X1, X2) |
Problem 12: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(plus(X1, X2)) | → | active#(X1) | | active#(s(X)) | → | active#(X) |
| active#(U12(X)) | → | active#(X) | | active#(U31(X1, X2)) | → | active#(X1) |
| active#(plus(X1, X2)) | → | active#(X2) | | active#(U42(X1, X2, X3)) | → | active#(X1) |
| active#(U11(X1, X2)) | → | active#(X1) | | active#(U21(X)) | → | active#(X) |
| active#(U41(X1, X2, X3)) | → | active#(X1) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(U42(isNat(N), M, N)) | | active(U42(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(plus(N, 0)) | → | mark(U31(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U41(isNat(M), M, N)) | | active(U11(X1, X2)) | → | U11(active(X1), X2) |
| active(U12(X)) | → | U12(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) |
| active(U42(X1, X2, X3)) | → | U42(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U12(mark(X)) | → | mark(U12(X)) |
| U21(mark(X)) | → | mark(U21(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) | | U42(mark(X1), X2, X3) | → | mark(U42(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(U12(X)) | → | U12(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U21(X)) | → | U21(proper(X)) |
| proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2, X3)) | → | U42(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| U42(ok(X1), ok(X2), ok(X3)) | → | ok(U42(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(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: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, ok, U12, U31, proper, U21, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(plus(X1, X2)) | → | active#(X1) | | active#(s(X)) | → | active#(X) |
| active#(U12(X)) | → | active#(X) | | active#(U31(X1, X2)) | → | active#(X1) |
| active#(plus(X1, X2)) | → | active#(X2) | | active#(U42(X1, X2, X3)) | → | active#(X1) |
| active#(U11(X1, X2)) | → | active#(X1) | | active#(U21(X)) | → | active#(X) |
| active#(U41(X1, X2, X3)) | → | active#(X1) |
Problem 13: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U42#(mark(X1), X2, X3) | → | U42#(X1, X2, X3) | | U42#(ok(X1), ok(X2), ok(X3)) | → | U42#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, V2)) | → | mark(U12(isNat(V2))) | | active(U12(tt)) | → | mark(tt) |
| active(U21(tt)) | → | mark(tt) | | active(U31(tt, N)) | → | mark(N) |
| active(U41(tt, M, N)) | → | mark(U42(isNat(N), M, N)) | | active(U42(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(isNat(0)) | → | mark(tt) | | active(isNat(plus(V1, V2))) | → | mark(U11(isNat(V1), V2)) |
| active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | | active(plus(N, 0)) | → | mark(U31(isNat(N), N)) |
| active(plus(N, s(M))) | → | mark(U41(isNat(M), M, N)) | | active(U11(X1, X2)) | → | U11(active(X1), X2) |
| active(U12(X)) | → | U12(active(X)) | | active(U21(X)) | → | U21(active(X)) |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | | active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) |
| active(U42(X1, X2, X3)) | → | U42(active(X1), X2, X3) | | active(s(X)) | → | s(active(X)) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U12(mark(X)) | → | mark(U12(X)) |
| U21(mark(X)) | → | mark(U21(X)) | | U31(mark(X1), X2) | → | mark(U31(X1, X2)) |
| U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) | | U42(mark(X1), X2, X3) | → | mark(U42(X1, X2, X3)) |
| s(mark(X)) | → | mark(s(X)) | | plus(mark(X1), X2) | → | mark(plus(X1, X2)) |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | | proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(U12(X)) | → | U12(proper(X)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(U21(X)) | → | U21(proper(X)) |
| proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) |
| proper(U42(X1, X2, X3)) | → | U42(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U12(ok(X)) | → | ok(U12(X)) |
| isNat(ok(X)) | → | ok(isNat(X)) | | U21(ok(X)) | → | ok(U21(X)) |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) |
| U42(ok(X1), ok(X2), ok(X3)) | → | ok(U42(X1, X2, X3)) | | s(ok(X)) | → | ok(s(X)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(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: plus, mark, isNat, 0, U42, s, tt, U41, active, U11, ok, U12, U31, proper, U21, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U42#(mark(X1), X2, X3) | → | U42#(X1, X2, X3) | | U42#(ok(X1), ok(X2), ok(X3)) | → | U42#(X1, X2, X3) |