YES
The TRS could be proven terminating. The proof took 22820 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (453ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| | – Problem 9 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 (1ms).
| | – Problem 10 was processed with processor SubtermCriterion (0ms).
| – Problem 6 was processed with processor SubtermCriterion (0ms).
| – Problem 7 was processed with processor ReductionPairSAT (1013ms).
| | – Problem 11 was processed with processor ReductionPairSAT (839ms).
| – Problem 8 was processed with processor SubtermCriterion (1ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(x(N, s(M))) | → | plus#(x(N, M), N) | | top#(ok(X)) | → | top#(active(X)) |
| proper#(x(X1, X2)) | → | proper#(X1) | | and#(ok(X1), ok(X2)) | → | and#(X1, X2) |
| active#(x(X1, X2)) | → | x#(active(X1), X2) | | proper#(and(X1, X2)) | → | proper#(X1) |
| proper#(and(X1, X2)) | → | and#(proper(X1), proper(X2)) | | top#(ok(X)) | → | active#(X) |
| x#(mark(X1), X2) | → | x#(X1, X2) | | active#(and(X1, X2)) | → | and#(active(X1), X2) |
| proper#(and(X1, X2)) | → | proper#(X2) | | plus#(X1, mark(X2)) | → | plus#(X1, X2) |
| proper#(plus(X1, X2)) | → | proper#(X1) | | x#(ok(X1), ok(X2)) | → | x#(X1, X2) |
| proper#(x(X1, X2)) | → | x#(proper(X1), proper(X2)) | | proper#(plus(X1, X2)) | → | proper#(X2) |
| x#(X1, mark(X2)) | → | x#(X1, X2) | | top#(mark(X)) | → | proper#(X) |
| proper#(plus(X1, X2)) | → | plus#(proper(X1), proper(X2)) | | active#(plus(N, s(M))) | → | plus#(N, M) |
| plus#(ok(X1), ok(X2)) | → | plus#(X1, X2) | | top#(mark(X)) | → | top#(proper(X)) |
| active#(x(N, s(M))) | → | x#(N, M) | | active#(x(X1, X2)) | → | active#(X1) |
| and#(mark(X1), X2) | → | and#(X1, X2) | | active#(s(X)) | → | s#(active(X)) |
| active#(x(X1, X2)) | → | x#(X1, active(X2)) | | proper#(x(X1, X2)) | → | proper#(X2) |
| s#(ok(X)) | → | s#(X) | | s#(mark(X)) | → | s#(X) |
| active#(plus(N, s(M))) | → | s#(plus(N, M)) | | active#(plus(X1, X2)) | → | plus#(X1, active(X2)) |
| proper#(s(X)) | → | proper#(X) | | active#(plus(X1, X2)) | → | active#(X1) |
| active#(plus(X1, X2)) | → | plus#(active(X1), X2) | | active#(s(X)) | → | active#(X) |
| proper#(s(X)) | → | s#(proper(X)) | | active#(x(X1, X2)) | → | active#(X2) |
| active#(plus(X1, X2)) | → | active#(X2) | | active#(and(X1, X2)) | → | active#(X1) |
| plus#(mark(X1), X2) | → | plus#(X1, X2) |
Rewrite Rules
| active(and(tt, X)) | → | mark(X) | | active(plus(N, 0)) | → | mark(N) |
| active(plus(N, s(M))) | → | mark(s(plus(N, M))) | | active(x(N, 0)) | → | mark(0) |
| active(x(N, s(M))) | → | mark(plus(x(N, M), N)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(s(X)) | → | s(active(X)) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | x(mark(X1), X2) | → | mark(x(X1, X2)) |
| x(X1, mark(X2)) | → | mark(x(X1, X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| x(ok(X1), ok(X2)) | → | ok(x(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, 0, s, tt, active, mark, ok, proper, and, x, 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) |
| x#(mark(X1), X2) → x#(X1, X2) | x#(ok(X1), ok(X2)) → x#(X1, X2) |
| x#(X1, mark(X2)) → x#(X1, X2) |
| active#(plus(X1, X2)) → active#(X1) | active#(s(X)) → active#(X) |
| active#(x(X1, X2)) → active#(X1) | active#(x(X1, X2)) → active#(X2) |
| active#(plus(X1, X2)) → active#(X2) | active#(and(X1, X2)) → active#(X1) |
| proper#(s(X)) → proper#(X) | proper#(and(X1, X2)) → proper#(X2) |
| proper#(plus(X1, X2)) → proper#(X1) | proper#(x(X1, X2)) → proper#(X1) |
| proper#(plus(X1, X2)) → proper#(X2) | proper#(and(X1, X2)) → proper#(X1) |
| proper#(x(X1, X2)) → proper#(X2) |
| 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
| 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(and(tt, X)) | → | mark(X) | | active(plus(N, 0)) | → | mark(N) |
| active(plus(N, s(M))) | → | mark(s(plus(N, M))) | | active(x(N, 0)) | → | mark(0) |
| active(x(N, s(M))) | → | mark(plus(x(N, M), N)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(s(X)) | → | s(active(X)) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | x(mark(X1), X2) | → | mark(x(X1, X2)) |
| x(X1, mark(X2)) | → | mark(x(X1, X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| x(ok(X1), ok(X2)) | → | ok(x(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, 0, s, tt, active, mark, ok, proper, and, x, 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 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| plus#(X1, mark(X2)) | → | plus#(X1, X2) |
Rewrite Rules
| active(and(tt, X)) | → | mark(X) | | active(plus(N, 0)) | → | mark(N) |
| active(plus(N, s(M))) | → | mark(s(plus(N, M))) | | active(x(N, 0)) | → | mark(0) |
| active(x(N, s(M))) | → | mark(plus(x(N, M), N)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(s(X)) | → | s(active(X)) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | x(mark(X1), X2) | → | mark(x(X1, X2)) |
| x(X1, mark(X2)) | → | mark(x(X1, X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| x(ok(X1), ok(X2)) | → | ok(x(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, 0, s, tt, active, ok, mark, proper, top, x, and
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| plus#(X1, mark(X2)) | → | plus#(X1, X2) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Rewrite Rules
| active(and(tt, X)) | → | mark(X) | | active(plus(N, 0)) | → | mark(N) |
| active(plus(N, s(M))) | → | mark(s(plus(N, M))) | | active(x(N, 0)) | → | mark(0) |
| active(x(N, s(M))) | → | mark(plus(x(N, M), N)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(s(X)) | → | s(active(X)) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | x(mark(X1), X2) | → | mark(x(X1, X2)) |
| x(X1, mark(X2)) | → | mark(x(X1, X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| x(ok(X1), ok(X2)) | → | ok(x(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, 0, s, tt, active, mark, ok, proper, and, x, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(plus(X1, X2)) | → | active#(X1) | | active#(s(X)) | → | active#(X) |
| active#(x(X1, X2)) | → | active#(X1) | | active#(x(X1, X2)) | → | active#(X2) |
| active#(plus(X1, X2)) | → | active#(X2) | | active#(and(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(and(tt, X)) | → | mark(X) | | active(plus(N, 0)) | → | mark(N) |
| active(plus(N, s(M))) | → | mark(s(plus(N, M))) | | active(x(N, 0)) | → | mark(0) |
| active(x(N, s(M))) | → | mark(plus(x(N, M), N)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(s(X)) | → | s(active(X)) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | x(mark(X1), X2) | → | mark(x(X1, X2)) |
| x(X1, mark(X2)) | → | mark(x(X1, X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| x(ok(X1), ok(X2)) | → | ok(x(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, 0, s, tt, active, mark, ok, proper, and, x, 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#(x(X1, X2)) | → | active#(X1) | | active#(x(X1, X2)) | → | active#(X2) |
| active#(plus(X1, X2)) | → | active#(X2) | | active#(and(X1, X2)) | → | active#(X1) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| x#(mark(X1), X2) | → | x#(X1, X2) | | x#(ok(X1), ok(X2)) | → | x#(X1, X2) |
| x#(X1, mark(X2)) | → | x#(X1, X2) |
Rewrite Rules
| active(and(tt, X)) | → | mark(X) | | active(plus(N, 0)) | → | mark(N) |
| active(plus(N, s(M))) | → | mark(s(plus(N, M))) | | active(x(N, 0)) | → | mark(0) |
| active(x(N, s(M))) | → | mark(plus(x(N, M), N)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(s(X)) | → | s(active(X)) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | x(mark(X1), X2) | → | mark(x(X1, X2)) |
| x(X1, mark(X2)) | → | mark(x(X1, X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| x(ok(X1), ok(X2)) | → | ok(x(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, 0, s, tt, active, mark, ok, proper, and, x, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| x#(mark(X1), X2) | → | x#(X1, X2) | | x#(ok(X1), ok(X2)) | → | x#(X1, X2) |
Problem 10: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| x#(X1, mark(X2)) | → | x#(X1, X2) |
Rewrite Rules
| active(and(tt, X)) | → | mark(X) | | active(plus(N, 0)) | → | mark(N) |
| active(plus(N, s(M))) | → | mark(s(plus(N, M))) | | active(x(N, 0)) | → | mark(0) |
| active(x(N, s(M))) | → | mark(plus(x(N, M), N)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(s(X)) | → | s(active(X)) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | x(mark(X1), X2) | → | mark(x(X1, X2)) |
| x(X1, mark(X2)) | → | mark(x(X1, X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| x(ok(X1), ok(X2)) | → | ok(x(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, 0, s, tt, active, ok, mark, proper, top, x, and
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| x#(X1, mark(X2)) | → | x#(X1, X2) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| and#(ok(X1), ok(X2)) | → | and#(X1, X2) | | and#(mark(X1), X2) | → | and#(X1, X2) |
Rewrite Rules
| active(and(tt, X)) | → | mark(X) | | active(plus(N, 0)) | → | mark(N) |
| active(plus(N, s(M))) | → | mark(s(plus(N, M))) | | active(x(N, 0)) | → | mark(0) |
| active(x(N, s(M))) | → | mark(plus(x(N, M), N)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(s(X)) | → | s(active(X)) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | x(mark(X1), X2) | → | mark(x(X1, X2)) |
| x(X1, mark(X2)) | → | mark(x(X1, X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| x(ok(X1), ok(X2)) | → | ok(x(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, 0, s, tt, active, mark, ok, proper, and, x, 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 7: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(and(tt, X)) | → | mark(X) | | active(plus(N, 0)) | → | mark(N) |
| active(plus(N, s(M))) | → | mark(s(plus(N, M))) | | active(x(N, 0)) | → | mark(0) |
| active(x(N, s(M))) | → | mark(plus(x(N, M), N)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(s(X)) | → | s(active(X)) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | x(mark(X1), X2) | → | mark(x(X1, X2)) |
| x(X1, mark(X2)) | → | mark(x(X1, X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| x(ok(X1), ok(X2)) | → | ok(x(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, 0, s, tt, active, mark, ok, proper, and, x, top
Strategy
Function Precedence
tt < ok = and < x < plus = active < 0 = s = mark = proper = top = top#
Argument Filtering
plus: 1 2
0: all arguments are removed from 0
s: 1
tt: all arguments are removed from tt
active: collapses to 1
ok: collapses to 1
mark: 1
proper: collapses to 1
top: all arguments are removed from top
x: 1 2
and: 1 2
top#: collapses to 1
Status
plus: lexicographic with permutation 1 → 2 2 → 1
0: multiset
s: lexicographic with permutation 1 → 1
tt: multiset
mark: lexicographic with permutation 1 → 1
top: multiset
x: lexicographic with permutation 1 → 2 2 → 1
and: lexicographic with permutation 1 → 2 2 → 1
Usable Rules
| active(plus(X1, X2)) → plus(X1, active(X2)) | plus(X1, mark(X2)) → mark(plus(X1, X2)) |
| active(plus(X1, X2)) → plus(active(X1), X2) | active(s(X)) → s(active(X)) |
| active(x(N, 0)) → mark(0) | and(ok(X1), ok(X2)) → ok(and(X1, X2)) |
| active(x(N, s(M))) → mark(plus(x(N, M), N)) | plus(mark(X1), X2) → mark(plus(X1, X2)) |
| active(x(X1, X2)) → x(X1, active(X2)) | proper(and(X1, X2)) → and(proper(X1), proper(X2)) |
| plus(ok(X1), ok(X2)) → ok(plus(X1, X2)) | s(mark(X)) → mark(s(X)) |
| proper(s(X)) → s(proper(X)) | active(and(X1, X2)) → and(active(X1), X2) |
| proper(x(X1, X2)) → x(proper(X1), proper(X2)) | proper(plus(X1, X2)) → plus(proper(X1), proper(X2)) |
| s(ok(X)) → ok(s(X)) | proper(tt) → ok(tt) |
| active(x(X1, X2)) → x(active(X1), X2) | active(plus(N, s(M))) → mark(s(plus(N, M))) |
| x(mark(X1), X2) → mark(x(X1, X2)) | active(and(tt, X)) → mark(X) |
| x(X1, mark(X2)) → mark(x(X1, X2)) | active(plus(N, 0)) → mark(N) |
| and(mark(X1), X2) → mark(and(X1, X2)) | proper(0) → ok(0) |
| x(ok(X1), ok(X2)) → ok(x(X1, 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 11: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(and(tt, X)) | → | mark(X) | | active(plus(N, 0)) | → | mark(N) |
| active(plus(N, s(M))) | → | mark(s(plus(N, M))) | | active(x(N, 0)) | → | mark(0) |
| active(x(N, s(M))) | → | mark(plus(x(N, M), N)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(s(X)) | → | s(active(X)) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | x(mark(X1), X2) | → | mark(x(X1, X2)) |
| x(X1, mark(X2)) | → | mark(x(X1, X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| x(ok(X1), ok(X2)) | → | ok(x(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, 0, s, tt, active, ok, mark, proper, top, x, and
Strategy
Function Precedence
x = top# < plus = 0 = s = tt = and < active = mark < ok = proper = top
Argument Filtering
plus: collapses to 1
0: all arguments are removed from 0
s: collapses to 1
tt: all arguments are removed from tt
active: collapses to 1
ok: 1
mark: collapses to 1
proper: collapses to 1
top: all arguments are removed from top
x: collapses to 2
and: collapses to 2
top#: collapses to 1
Status
0: multiset
tt: multiset
ok: lexicographic with permutation 1 → 1
top: multiset
Usable Rules
| active(plus(X1, X2)) → plus(X1, active(X2)) | plus(X1, mark(X2)) → mark(plus(X1, X2)) |
| active(plus(X1, X2)) → plus(active(X1), X2) | active(x(N, 0)) → mark(0) |
| active(s(X)) → s(active(X)) | and(ok(X1), ok(X2)) → ok(and(X1, X2)) |
| s(ok(X)) → ok(s(X)) | active(x(N, s(M))) → mark(plus(x(N, M), N)) |
| plus(mark(X1), X2) → mark(plus(X1, X2)) | active(x(X1, X2)) → x(active(X1), X2) |
| active(plus(N, s(M))) → mark(s(plus(N, M))) | x(mark(X1), X2) → mark(x(X1, X2)) |
| active(x(X1, X2)) → x(X1, active(X2)) | active(and(tt, X)) → mark(X) |
| s(mark(X)) → mark(s(X)) | x(X1, mark(X2)) → mark(x(X1, X2)) |
| plus(ok(X1), ok(X2)) → ok(plus(X1, X2)) | active(plus(N, 0)) → mark(N) |
| and(mark(X1), X2) → mark(and(X1, X2)) | x(ok(X1), ok(X2)) → ok(x(X1, X2)) |
| active(and(X1, X2)) → and(active(X1), 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)) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(s(X)) | → | proper#(X) | | proper#(and(X1, X2)) | → | proper#(X2) |
| proper#(plus(X1, X2)) | → | proper#(X1) | | proper#(x(X1, X2)) | → | proper#(X1) |
| proper#(plus(X1, X2)) | → | proper#(X2) | | proper#(and(X1, X2)) | → | proper#(X1) |
| proper#(x(X1, X2)) | → | proper#(X2) |
Rewrite Rules
| active(and(tt, X)) | → | mark(X) | | active(plus(N, 0)) | → | mark(N) |
| active(plus(N, s(M))) | → | mark(s(plus(N, M))) | | active(x(N, 0)) | → | mark(0) |
| active(x(N, s(M))) | → | mark(plus(x(N, M), N)) | | active(and(X1, X2)) | → | and(active(X1), X2) |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | | active(plus(X1, X2)) | → | plus(X1, active(X2)) |
| active(s(X)) | → | s(active(X)) | | active(x(X1, X2)) | → | x(active(X1), X2) |
| active(x(X1, X2)) | → | x(X1, active(X2)) | | and(mark(X1), X2) | → | mark(and(X1, X2)) |
| plus(mark(X1), X2) | → | mark(plus(X1, X2)) | | plus(X1, mark(X2)) | → | mark(plus(X1, X2)) |
| s(mark(X)) | → | mark(s(X)) | | x(mark(X1), X2) | → | mark(x(X1, X2)) |
| x(X1, mark(X2)) | → | mark(x(X1, X2)) | | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) |
| proper(tt) | → | ok(tt) | | proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) |
| proper(0) | → | ok(0) | | proper(s(X)) | → | s(proper(X)) |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) |
| plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | | s(ok(X)) | → | ok(s(X)) |
| x(ok(X1), ok(X2)) | → | ok(x(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, 0, s, tt, active, mark, ok, proper, and, x, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(s(X)) | → | proper#(X) | | proper#(and(X1, X2)) | → | proper#(X2) |
| proper#(plus(X1, X2)) | → | proper#(X1) | | proper#(x(X1, X2)) | → | proper#(X1) |
| proper#(plus(X1, X2)) | → | proper#(X2) | | proper#(and(X1, X2)) | → | proper#(X1) |
| proper#(x(X1, X2)) | → | proper#(X2) |