TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60000 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (2210ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor SubtermCriterion (3ms).
| – Problem 5 was processed with processor SubtermCriterion (2ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| – Problem 7 was processed with processor SubtermCriterion (1ms).
| – Problem 8 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (6ms), PolynomialLinearRange4iUR (5001ms), DependencyGraph (6ms), PolynomialLinearRange8NegiUR (15000ms), DependencyGraph (16ms), ReductionPairSAT (6014ms), DependencyGraph (42ms), ReductionPairSAT (5910ms), DependencyGraph (4ms), SizeChangePrinciple (timeout)].
| – Problem 9 was processed with processor SubtermCriterion (1ms).
| – Problem 10 was processed with processor SubtermCriterion (1ms).
| | – Problem 11 was processed with processor ReductionPairSAT (93ms).
The following open problems remain:
Open Dependency Pair Problem 8
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(U11(tt, N)) | → | mark(N) | | active(U21(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(and(isNat(V1), isNat(V2))) | | active(isNat(s(V1))) | → | mark(isNat(V1)) |
| active(plus(N, 0)) | → | mark(U11(isNat(N), N)) | | active(plus(N, s(M))) | → | mark(U21(and(isNat(M), isNat(N)), M, N)) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U21(X1, X2, X3)) | → | U21(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)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U21(mark(X1), X2, X3) | → | mark(U21(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)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| 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, and, isNat, 0, s, tt, active, U11, ok, proper, U21, top
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(plus(N, s(M))) | → | and#(isNat(M), isNat(N)) | | top#(ok(X)) | → | top#(active(X)) |
| proper#(U11(X1, X2)) | → | proper#(X1) | | U21#(ok(X1), ok(X2), ok(X3)) | → | U21#(X1, X2, X3) |
| U11#(ok(X1), ok(X2)) | → | U11#(X1, X2) | | U11#(mark(X1), X2) | → | U11#(X1, X2) |
| proper#(U11(X1, X2)) | → | U11#(proper(X1), proper(X2)) | | proper#(and(X1, X2)) | → | and#(proper(X1), proper(X2)) |
| active#(U21(tt, M, N)) | → | s#(plus(N, M)) | | proper#(U11(X1, X2)) | → | proper#(X2) |
| proper#(and(X1, X2)) | → | proper#(X2) | | plus#(X1, mark(X2)) | → | plus#(X1, X2) |
| proper#(plus(X1, X2)) | → | proper#(X1) | | active#(plus(N, 0)) | → | isNat#(N) |
| active#(U11(X1, X2)) | → | active#(X1) | | top#(mark(X)) | → | proper#(X) |
| proper#(plus(X1, X2)) | → | plus#(proper(X1), proper(X2)) | | active#(isNat(plus(V1, V2))) | → | isNat#(V2) |
| top#(mark(X)) | → | top#(proper(X)) | | active#(isNat(plus(V1, V2))) | → | and#(isNat(V1), isNat(V2)) |
| U21#(mark(X1), X2, X3) | → | U21#(X1, X2, X3) | | active#(U21(X1, X2, X3)) | → | U21#(active(X1), X2, X3) |
| isNat#(ok(X)) | → | isNat#(X) | | and#(mark(X1), X2) | → | and#(X1, X2) |
| active#(U21(X1, X2, X3)) | → | active#(X1) | | proper#(s(X)) | → | proper#(X) |
| active#(plus(X1, X2)) | → | active#(X1) | | active#(plus(X1, X2)) | → | active#(X2) |
| plus#(mark(X1), X2) | → | plus#(X1, X2) | | active#(plus(N, s(M))) | → | isNat#(N) |
| proper#(U21(X1, X2, X3)) | → | proper#(X3) | | and#(ok(X1), ok(X2)) | → | and#(X1, X2) |
| active#(isNat(plus(V1, V2))) | → | isNat#(V1) | | proper#(and(X1, X2)) | → | proper#(X1) |
| top#(ok(X)) | → | active#(X) | | active#(and(X1, X2)) | → | and#(active(X1), X2) |
| proper#(U21(X1, X2, X3)) | → | proper#(X1) | | proper#(isNat(X)) | → | isNat#(proper(X)) |
| proper#(plus(X1, X2)) | → | proper#(X2) | | active#(isNat(s(V1))) | → | isNat#(V1) |
| active#(plus(N, s(M))) | → | U21#(and(isNat(M), isNat(N)), M, N) | | plus#(ok(X1), ok(X2)) | → | plus#(X1, X2) |
| proper#(isNat(X)) | → | proper#(X) | | active#(U11(X1, X2)) | → | U11#(active(X1), X2) |
| active#(s(X)) | → | s#(active(X)) | | proper#(U21(X1, X2, X3)) | → | U21#(proper(X1), proper(X2), proper(X3)) |
| s#(ok(X)) | → | s#(X) | | s#(mark(X)) | → | s#(X) |
| active#(plus(X1, X2)) | → | plus#(X1, active(X2)) | | active#(U21(tt, M, N)) | → | plus#(N, M) |
| active#(plus(N, s(M))) | → | isNat#(M) | | active#(s(X)) | → | active#(X) |
| active#(plus(X1, X2)) | → | plus#(active(X1), X2) | | proper#(s(X)) | → | s#(proper(X)) |
| proper#(U21(X1, X2, X3)) | → | proper#(X2) | | active#(and(X1, X2)) | → | active#(X1) |
| active#(plus(N, 0)) | → | U11#(isNat(N), N) |
Rewrite Rules
| active(U11(tt, N)) | → | mark(N) | | active(U21(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(and(isNat(V1), isNat(V2))) | | active(isNat(s(V1))) | → | mark(isNat(V1)) |
| active(plus(N, 0)) | → | mark(U11(isNat(N), N)) | | active(plus(N, s(M))) | → | mark(U21(and(isNat(M), isNat(N)), M, N)) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U21(X1, X2, X3)) | → | U21(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)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U21(mark(X1), X2, X3) | → | mark(U21(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)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| 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, and, isNat, 0, s, tt, active, U11, ok, proper, U21, top
Strategy
The following SCCs where found
| plus#(ok(X1), ok(X2)) → plus#(X1, X2) | plus#(X1, mark(X2)) → plus#(X1, X2) |
| plus#(mark(X1), X2) → plus#(X1, X2) |
| U21#(ok(X1), ok(X2), ok(X3)) → U21#(X1, X2, X3) | U21#(mark(X1), X2, X3) → U21#(X1, X2, X3) |
| proper#(s(X)) → proper#(X) | proper#(isNat(X)) → proper#(X) |
| proper#(U11(X1, X2)) → proper#(X2) | proper#(and(X1, X2)) → proper#(X2) |
| proper#(U11(X1, X2)) → proper#(X1) | proper#(U21(X1, X2, X3)) → proper#(X1) |
| proper#(U21(X1, X2, X3)) → proper#(X3) | proper#(U21(X1, X2, X3)) → proper#(X2) |
| proper#(plus(X1, X2)) → proper#(X1) | proper#(plus(X1, X2)) → proper#(X2) |
| proper#(and(X1, X2)) → proper#(X1) |
| active#(plus(X1, X2)) → active#(X1) | active#(s(X)) → active#(X) |
| active#(plus(X1, X2)) → active#(X2) | active#(and(X1, X2)) → active#(X1) |
| active#(U11(X1, X2)) → active#(X1) | active#(U21(X1, X2, X3)) → active#(X1) |
| U11#(ok(X1), ok(X2)) → U11#(X1, X2) | U11#(mark(X1), X2) → U11#(X1, X2) |
| isNat#(ok(X)) → isNat#(X) |
| and#(ok(X1), ok(X2)) → and#(X1, X2) | and#(mark(X1), X2) → and#(X1, X2) |
| s#(mark(X)) → s#(X) | s#(ok(X)) → s#(X) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| U21#(ok(X1), ok(X2), ok(X3)) | → | U21#(X1, X2, X3) | | U21#(mark(X1), X2, X3) | → | U21#(X1, X2, X3) |
Rewrite Rules
| active(U11(tt, N)) | → | mark(N) | | active(U21(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(and(isNat(V1), isNat(V2))) | | active(isNat(s(V1))) | → | mark(isNat(V1)) |
| active(plus(N, 0)) | → | mark(U11(isNat(N), N)) | | active(plus(N, s(M))) | → | mark(U21(and(isNat(M), isNat(N)), M, N)) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U21(X1, X2, X3)) | → | U21(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)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U21(mark(X1), X2, X3) | → | mark(U21(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)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| 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, and, isNat, 0, s, tt, active, U11, ok, proper, U21, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| U21#(ok(X1), ok(X2), ok(X3)) | → | U21#(X1, X2, X3) | | U21#(mark(X1), X2, X3) | → | U21#(X1, X2, X3) |
Problem 3: 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, N)) | → | mark(N) | | active(U21(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(and(isNat(V1), isNat(V2))) | | active(isNat(s(V1))) | → | mark(isNat(V1)) |
| active(plus(N, 0)) | → | mark(U11(isNat(N), N)) | | active(plus(N, s(M))) | → | mark(U21(and(isNat(M), isNat(N)), M, N)) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U21(X1, X2, X3)) | → | U21(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)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U21(mark(X1), X2, X3) | → | mark(U21(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)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| 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, and, isNat, 0, s, tt, active, U11, ok, 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 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| isNat#(ok(X)) | → | isNat#(X) |
Rewrite Rules
| active(U11(tt, N)) | → | mark(N) | | active(U21(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(and(isNat(V1), isNat(V2))) | | active(isNat(s(V1))) | → | mark(isNat(V1)) |
| active(plus(N, 0)) | → | mark(U11(isNat(N), N)) | | active(plus(N, s(M))) | → | mark(U21(and(isNat(M), isNat(N)), M, N)) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U21(X1, X2, X3)) | → | U21(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)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U21(mark(X1), X2, X3) | → | mark(U21(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)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| 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, and, isNat, 0, s, tt, active, U11, ok, proper, U21, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| isNat#(ok(X)) | → | isNat#(X) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(plus(X1, X2)) | → | active#(X1) | | active#(s(X)) | → | active#(X) |
| active#(plus(X1, X2)) | → | active#(X2) | | active#(and(X1, X2)) | → | active#(X1) |
| active#(U11(X1, X2)) | → | active#(X1) | | active#(U21(X1, X2, X3)) | → | active#(X1) |
Rewrite Rules
| active(U11(tt, N)) | → | mark(N) | | active(U21(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(and(isNat(V1), isNat(V2))) | | active(isNat(s(V1))) | → | mark(isNat(V1)) |
| active(plus(N, 0)) | → | mark(U11(isNat(N), N)) | | active(plus(N, s(M))) | → | mark(U21(and(isNat(M), isNat(N)), M, N)) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U21(X1, X2, X3)) | → | U21(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)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U21(mark(X1), X2, X3) | → | mark(U21(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)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| 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, and, isNat, 0, s, tt, active, U11, ok, 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#(plus(X1, X2)) | → | active#(X2) | | active#(and(X1, X2)) | → | active#(X1) |
| active#(U11(X1, X2)) | → | active#(X1) | | active#(U21(X1, X2, X3)) | → | active#(X1) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(s(X)) | → | proper#(X) | | proper#(isNat(X)) | → | proper#(X) |
| proper#(U11(X1, X2)) | → | proper#(X2) | | proper#(and(X1, X2)) | → | proper#(X2) |
| proper#(U11(X1, X2)) | → | proper#(X1) | | proper#(U21(X1, X2, X3)) | → | proper#(X1) |
| proper#(U21(X1, X2, X3)) | → | proper#(X3) | | proper#(U21(X1, X2, X3)) | → | proper#(X2) |
| proper#(plus(X1, X2)) | → | proper#(X1) | | proper#(plus(X1, X2)) | → | proper#(X2) |
| proper#(and(X1, X2)) | → | proper#(X1) |
Rewrite Rules
| active(U11(tt, N)) | → | mark(N) | | active(U21(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(and(isNat(V1), isNat(V2))) | | active(isNat(s(V1))) | → | mark(isNat(V1)) |
| active(plus(N, 0)) | → | mark(U11(isNat(N), N)) | | active(plus(N, s(M))) | → | mark(U21(and(isNat(M), isNat(N)), M, N)) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U21(X1, X2, X3)) | → | U21(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)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U21(mark(X1), X2, X3) | → | mark(U21(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)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| 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, and, isNat, 0, s, tt, active, U11, ok, proper, U21, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(s(X)) | → | proper#(X) | | proper#(isNat(X)) | → | proper#(X) |
| proper#(U11(X1, X2)) | → | proper#(X2) | | proper#(and(X1, X2)) | → | proper#(X2) |
| proper#(U11(X1, X2)) | → | proper#(X1) | | proper#(U21(X1, X2, X3)) | → | proper#(X1) |
| proper#(U21(X1, X2, X3)) | → | proper#(X3) | | proper#(U21(X1, X2, X3)) | → | proper#(X2) |
| proper#(plus(X1, X2)) | → | proper#(X1) | | proper#(plus(X1, X2)) | → | proper#(X2) |
| proper#(and(X1, X2)) | → | proper#(X1) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Rewrite Rules
| active(U11(tt, N)) | → | mark(N) | | active(U21(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(and(isNat(V1), isNat(V2))) | | active(isNat(s(V1))) | → | mark(isNat(V1)) |
| active(plus(N, 0)) | → | mark(U11(isNat(N), N)) | | active(plus(N, s(M))) | → | mark(U21(and(isNat(M), isNat(N)), M, N)) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U21(X1, X2, X3)) | → | U21(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)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U21(mark(X1), X2, X3) | → | mark(U21(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)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| 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, and, isNat, 0, s, tt, active, U11, ok, 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 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| and#(ok(X1), ok(X2)) | → | and#(X1, X2) | | and#(mark(X1), X2) | → | and#(X1, X2) |
Rewrite Rules
| active(U11(tt, N)) | → | mark(N) | | active(U21(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(and(isNat(V1), isNat(V2))) | | active(isNat(s(V1))) | → | mark(isNat(V1)) |
| active(plus(N, 0)) | → | mark(U11(isNat(N), N)) | | active(plus(N, s(M))) | → | mark(U21(and(isNat(M), isNat(N)), M, N)) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U21(X1, X2, X3)) | → | U21(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)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U21(mark(X1), X2, X3) | → | mark(U21(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)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| 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, and, isNat, 0, s, tt, active, U11, ok, proper, U21, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| and#(ok(X1), ok(X2)) | → | and#(X1, X2) | | and#(mark(X1), X2) | → | and#(X1, X2) |
Problem 10: 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, N)) | → | mark(N) | | active(U21(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(and(isNat(V1), isNat(V2))) | | active(isNat(s(V1))) | → | mark(isNat(V1)) |
| active(plus(N, 0)) | → | mark(U11(isNat(N), N)) | | active(plus(N, s(M))) | → | mark(U21(and(isNat(M), isNat(N)), M, N)) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U21(X1, X2, X3)) | → | U21(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)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U21(mark(X1), X2, X3) | → | mark(U21(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)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| 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, and, isNat, 0, s, tt, active, U11, ok, 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 11: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| plus#(X1, mark(X2)) | → | plus#(X1, X2) |
Rewrite Rules
| active(U11(tt, N)) | → | mark(N) | | active(U21(tt, M, N)) | → | mark(s(plus(N, M))) |
| active(and(tt, X)) | → | mark(X) | | active(isNat(0)) | → | mark(tt) |
| active(isNat(plus(V1, V2))) | → | mark(and(isNat(V1), isNat(V2))) | | active(isNat(s(V1))) | → | mark(isNat(V1)) |
| active(plus(N, 0)) | → | mark(U11(isNat(N), N)) | | active(plus(N, s(M))) | → | mark(U21(and(isNat(M), isNat(N)), M, N)) |
| active(U11(X1, X2)) | → | U11(active(X1), X2) | | active(U21(X1, X2, X3)) | → | U21(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)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| U11(mark(X1), X2) | → | mark(U11(X1, X2)) | | U21(mark(X1), X2, X3) | → | mark(U21(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)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | | proper(tt) | → | ok(tt) |
| proper(U21(X1, X2, X3)) | → | U21(proper(X1), proper(X2), proper(X3)) | | proper(s(X)) | → | s(proper(X)) |
| proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(isNat(X)) | → | isNat(proper(X)) | | proper(0) | → | ok(0) |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | | U21(ok(X1), ok(X2), ok(X3)) | → | ok(U21(X1, X2, X3)) |
| s(ok(X)) | → | ok(s(X)) | | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | | isNat(ok(X)) | → | ok(isNat(X)) |
| 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, and, isNat, 0, s, tt, active, U11, ok, proper, U21, top
Strategy
Function Precedence
mark < plus = and = isNat = 0 = s = tt = U11 = plus# = active = ok = proper = U21 = top
Argument Filtering
plus: all arguments are removed from plus
mark: 1
and: 1 2
isNat: all arguments are removed from isNat
0: all arguments are removed from 0
s: all arguments are removed from s
tt: all arguments are removed from tt
U11: all arguments are removed from U11
plus#: collapses to 2
active: all arguments are removed from active
ok: all arguments are removed from ok
proper: all arguments are removed from proper
U21: 2 3
top: all arguments are removed from top
Status
plus: multiset
mark: multiset
and: lexicographic with permutation 1 → 2 2 → 1
isNat: multiset
0: multiset
s: multiset
tt: multiset
U11: multiset
active: multiset
ok: multiset
proper: multiset
U21: lexicographic with permutation 2 → 1 3 → 2
top: multiset
Usable Rules
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
Eliminated dependency pairs
The following dependency pairs (at least) can be eliminated according to the given precedence.
| plus#(X1, mark(X2)) → plus#(X1, X2) |