TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60003 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (3624ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (8ms), PolynomialLinearRange4iUR (3334ms), DependencyGraph (6ms), PolynomialLinearRange8NegiUR (10001ms), DependencyGraph (5ms), ReductionPairSAT (5279ms), DependencyGraph (3ms), ReductionPairSAT (4886ms), DependencyGraph (25ms), SizeChangePrinciple (timeout)].
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| – Problem 5 was processed with processor SubtermCriterion (3ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| – Problem 7 was processed with processor SubtermCriterion (1ms).
| | – Problem 12 was processed with processor ReductionPairSAT (118ms).
| – Problem 8 was processed with processor SubtermCriterion (0ms).
| | – Problem 13 was processed with processor ReductionPairSAT (77ms).
| – Problem 9 was processed with processor SubtermCriterion (0ms).
| – Problem 10 was processed with processor SubtermCriterion (0ms).
| – Problem 11 was processed with processor SubtermCriterion (3ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(fact(X)) | → | mark(if(zero(X), s(0), prod(X, fact(p(X))))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(prod(0, X)) | → | mark(0) |
| active(prod(s(X), Y)) | → | mark(add(Y, prod(X, Y))) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(zero(0)) | → | mark(true) |
| active(zero(s(X))) | → | mark(false) | | active(p(s(X))) | → | mark(X) |
| active(fact(X)) | → | fact(active(X)) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| active(zero(X)) | → | zero(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(prod(X1, X2)) | → | prod(active(X1), X2) | | active(prod(X1, X2)) | → | prod(X1, active(X2)) |
| active(p(X)) | → | p(active(X)) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fact(mark(X)) | → | mark(fact(X)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | zero(mark(X)) | → | mark(zero(X)) |
| s(mark(X)) | → | mark(s(X)) | | prod(mark(X1), X2) | → | mark(prod(X1, X2)) |
| prod(X1, mark(X2)) | → | mark(prod(X1, X2)) | | p(mark(X)) | → | mark(p(X)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fact(X)) | → | fact(proper(X)) | | proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) |
| proper(zero(X)) | → | zero(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(prod(X1, X2)) | → | prod(proper(X1), proper(X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| fact(ok(X)) | → | ok(fact(X)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| zero(ok(X)) | → | ok(zero(X)) | | s(ok(X)) | → | ok(s(X)) |
| prod(ok(X1), ok(X2)) | → | ok(prod(X1, X2)) | | p(ok(X)) | → | ok(p(X)) |
| add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: true, mark, zero, add, 0, fact, s, if, p, active, false, ok, proper, top, prod
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(prod(s(X), Y)) | → | prod#(X, Y) | | top#(ok(X)) | → | top#(active(X)) |
| zero#(ok(X)) | → | zero#(X) | | active#(zero(X)) | → | zero#(active(X)) |
| active#(fact(X)) | → | if#(zero(X), s(0), prod(X, fact(p(X)))) | | add#(X1, mark(X2)) | → | add#(X1, X2) |
| active#(fact(X)) | → | s#(0) | | proper#(p(X)) | → | proper#(X) |
| active#(if(X1, X2, X3)) | → | active#(X1) | | active#(add(X1, X2)) | → | add#(active(X1), X2) |
| active#(p(X)) | → | p#(active(X)) | | prod#(mark(X1), X2) | → | prod#(X1, X2) |
| active#(prod(X1, X2)) | → | prod#(active(X1), X2) | | proper#(p(X)) | → | p#(proper(X)) |
| active#(prod(X1, X2)) | → | active#(X2) | | active#(prod(X1, X2)) | → | prod#(X1, active(X2)) |
| top#(mark(X)) | → | proper#(X) | | active#(fact(X)) | → | p#(X) |
| proper#(add(X1, X2)) | → | proper#(X2) | | active#(fact(X)) | → | prod#(X, fact(p(X))) |
| proper#(fact(X)) | → | fact#(proper(X)) | | active#(fact(X)) | → | fact#(active(X)) |
| prod#(X1, mark(X2)) | → | prod#(X1, X2) | | top#(mark(X)) | → | top#(proper(X)) |
| active#(p(X)) | → | active#(X) | | proper#(add(X1, X2)) | → | proper#(X1) |
| add#(mark(X1), X2) | → | add#(X1, X2) | | proper#(zero(X)) | → | proper#(X) |
| proper#(s(X)) | → | proper#(X) | | active#(add(s(X), Y)) | → | add#(X, Y) |
| active#(add(X1, X2)) | → | active#(X2) | | active#(zero(X)) | → | active#(X) |
| if#(mark(X1), X2, X3) | → | if#(X1, X2, X3) | | fact#(ok(X)) | → | fact#(X) |
| active#(fact(X)) | → | fact#(p(X)) | | proper#(prod(X1, X2)) | → | proper#(X2) |
| proper#(zero(X)) | → | zero#(proper(X)) | | proper#(add(X1, X2)) | → | add#(proper(X1), proper(X2)) |
| proper#(prod(X1, X2)) | → | prod#(proper(X1), proper(X2)) | | active#(add(X1, X2)) | → | add#(X1, active(X2)) |
| add#(ok(X1), ok(X2)) | → | add#(X1, X2) | | p#(mark(X)) | → | p#(X) |
| top#(ok(X)) | → | active#(X) | | prod#(ok(X1), ok(X2)) | → | prod#(X1, X2) |
| active#(add(s(X), Y)) | → | s#(add(X, Y)) | | zero#(mark(X)) | → | zero#(X) |
| if#(ok(X1), ok(X2), ok(X3)) | → | if#(X1, X2, X3) | | active#(prod(X1, X2)) | → | active#(X1) |
| active#(prod(s(X), Y)) | → | add#(Y, prod(X, Y)) | | active#(fact(X)) | → | active#(X) |
| active#(add(X1, X2)) | → | active#(X1) | | proper#(if(X1, X2, X3)) | → | proper#(X1) |
| proper#(if(X1, X2, X3)) | → | proper#(X2) | | proper#(fact(X)) | → | proper#(X) |
| active#(s(X)) | → | s#(active(X)) | | s#(ok(X)) | → | s#(X) |
| s#(mark(X)) | → | s#(X) | | active#(s(X)) | → | active#(X) |
| proper#(s(X)) | → | s#(proper(X)) | | proper#(if(X1, X2, X3)) | → | proper#(X3) |
| active#(if(X1, X2, X3)) | → | if#(active(X1), X2, X3) | | p#(ok(X)) | → | p#(X) |
| fact#(mark(X)) | → | fact#(X) | | proper#(if(X1, X2, X3)) | → | if#(proper(X1), proper(X2), proper(X3)) |
| proper#(prod(X1, X2)) | → | proper#(X1) | | active#(fact(X)) | → | zero#(X) |
Rewrite Rules
| active(fact(X)) | → | mark(if(zero(X), s(0), prod(X, fact(p(X))))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(prod(0, X)) | → | mark(0) |
| active(prod(s(X), Y)) | → | mark(add(Y, prod(X, Y))) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(zero(0)) | → | mark(true) |
| active(zero(s(X))) | → | mark(false) | | active(p(s(X))) | → | mark(X) |
| active(fact(X)) | → | fact(active(X)) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| active(zero(X)) | → | zero(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(prod(X1, X2)) | → | prod(active(X1), X2) | | active(prod(X1, X2)) | → | prod(X1, active(X2)) |
| active(p(X)) | → | p(active(X)) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fact(mark(X)) | → | mark(fact(X)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | zero(mark(X)) | → | mark(zero(X)) |
| s(mark(X)) | → | mark(s(X)) | | prod(mark(X1), X2) | → | mark(prod(X1, X2)) |
| prod(X1, mark(X2)) | → | mark(prod(X1, X2)) | | p(mark(X)) | → | mark(p(X)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fact(X)) | → | fact(proper(X)) | | proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) |
| proper(zero(X)) | → | zero(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(prod(X1, X2)) | → | prod(proper(X1), proper(X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| fact(ok(X)) | → | ok(fact(X)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| zero(ok(X)) | → | ok(zero(X)) | | s(ok(X)) | → | ok(s(X)) |
| prod(ok(X1), ok(X2)) | → | ok(prod(X1, X2)) | | p(ok(X)) | → | ok(p(X)) |
| add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: true, mark, zero, add, 0, fact, s, if, p, active, false, ok, proper, prod, top
Strategy
The following SCCs where found
| p#(ok(X)) → p#(X) | p#(mark(X)) → p#(X) |
| zero#(mark(X)) → zero#(X) | zero#(ok(X)) → zero#(X) |
| prod#(X1, mark(X2)) → prod#(X1, X2) | prod#(mark(X1), X2) → prod#(X1, X2) |
| prod#(ok(X1), ok(X2)) → prod#(X1, X2) |
| add#(X1, mark(X2)) → add#(X1, X2) | add#(mark(X1), X2) → add#(X1, X2) |
| add#(ok(X1), ok(X2)) → add#(X1, X2) |
| if#(mark(X1), X2, X3) → if#(X1, X2, X3) | if#(ok(X1), ok(X2), ok(X3)) → if#(X1, X2, X3) |
| proper#(prod(X1, X2)) → proper#(X2) | proper#(s(X)) → proper#(X) |
| proper#(if(X1, X2, X3)) → proper#(X1) | proper#(if(X1, X2, X3)) → proper#(X2) |
| proper#(if(X1, X2, X3)) → proper#(X3) | proper#(add(X1, X2)) → proper#(X1) |
| proper#(fact(X)) → proper#(X) | proper#(zero(X)) → proper#(X) |
| proper#(p(X)) → proper#(X) | proper#(prod(X1, X2)) → proper#(X1) |
| proper#(add(X1, X2)) → proper#(X2) |
| active#(fact(X)) → active#(X) | active#(add(X1, X2)) → active#(X1) |
| active#(if(X1, X2, X3)) → active#(X1) | active#(add(X1, X2)) → active#(X2) |
| active#(s(X)) → active#(X) | active#(p(X)) → active#(X) |
| active#(prod(X1, X2)) → active#(X2) | active#(zero(X)) → active#(X) |
| active#(prod(X1, X2)) → active#(X1) |
| fact#(ok(X)) → fact#(X) | fact#(mark(X)) → fact#(X) |
| s#(mark(X)) → s#(X) | s#(ok(X)) → s#(X) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| if#(mark(X1), X2, X3) | → | if#(X1, X2, X3) | | if#(ok(X1), ok(X2), ok(X3)) | → | if#(X1, X2, X3) |
Rewrite Rules
| active(fact(X)) | → | mark(if(zero(X), s(0), prod(X, fact(p(X))))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(prod(0, X)) | → | mark(0) |
| active(prod(s(X), Y)) | → | mark(add(Y, prod(X, Y))) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(zero(0)) | → | mark(true) |
| active(zero(s(X))) | → | mark(false) | | active(p(s(X))) | → | mark(X) |
| active(fact(X)) | → | fact(active(X)) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| active(zero(X)) | → | zero(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(prod(X1, X2)) | → | prod(active(X1), X2) | | active(prod(X1, X2)) | → | prod(X1, active(X2)) |
| active(p(X)) | → | p(active(X)) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fact(mark(X)) | → | mark(fact(X)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | zero(mark(X)) | → | mark(zero(X)) |
| s(mark(X)) | → | mark(s(X)) | | prod(mark(X1), X2) | → | mark(prod(X1, X2)) |
| prod(X1, mark(X2)) | → | mark(prod(X1, X2)) | | p(mark(X)) | → | mark(p(X)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fact(X)) | → | fact(proper(X)) | | proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) |
| proper(zero(X)) | → | zero(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(prod(X1, X2)) | → | prod(proper(X1), proper(X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| fact(ok(X)) | → | ok(fact(X)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| zero(ok(X)) | → | ok(zero(X)) | | s(ok(X)) | → | ok(s(X)) |
| prod(ok(X1), ok(X2)) | → | ok(prod(X1, X2)) | | p(ok(X)) | → | ok(p(X)) |
| add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: true, mark, zero, add, 0, fact, s, if, p, active, false, ok, proper, prod, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| if#(mark(X1), X2, X3) | → | if#(X1, X2, X3) | | if#(ok(X1), ok(X2), ok(X3)) | → | if#(X1, X2, X3) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(fact(X)) | → | active#(X) | | active#(add(X1, X2)) | → | active#(X1) |
| active#(if(X1, X2, X3)) | → | active#(X1) | | active#(add(X1, X2)) | → | active#(X2) |
| active#(s(X)) | → | active#(X) | | active#(p(X)) | → | active#(X) |
| active#(prod(X1, X2)) | → | active#(X2) | | active#(zero(X)) | → | active#(X) |
| active#(prod(X1, X2)) | → | active#(X1) |
Rewrite Rules
| active(fact(X)) | → | mark(if(zero(X), s(0), prod(X, fact(p(X))))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(prod(0, X)) | → | mark(0) |
| active(prod(s(X), Y)) | → | mark(add(Y, prod(X, Y))) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(zero(0)) | → | mark(true) |
| active(zero(s(X))) | → | mark(false) | | active(p(s(X))) | → | mark(X) |
| active(fact(X)) | → | fact(active(X)) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| active(zero(X)) | → | zero(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(prod(X1, X2)) | → | prod(active(X1), X2) | | active(prod(X1, X2)) | → | prod(X1, active(X2)) |
| active(p(X)) | → | p(active(X)) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fact(mark(X)) | → | mark(fact(X)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | zero(mark(X)) | → | mark(zero(X)) |
| s(mark(X)) | → | mark(s(X)) | | prod(mark(X1), X2) | → | mark(prod(X1, X2)) |
| prod(X1, mark(X2)) | → | mark(prod(X1, X2)) | | p(mark(X)) | → | mark(p(X)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fact(X)) | → | fact(proper(X)) | | proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) |
| proper(zero(X)) | → | zero(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(prod(X1, X2)) | → | prod(proper(X1), proper(X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| fact(ok(X)) | → | ok(fact(X)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| zero(ok(X)) | → | ok(zero(X)) | | s(ok(X)) | → | ok(s(X)) |
| prod(ok(X1), ok(X2)) | → | ok(prod(X1, X2)) | | p(ok(X)) | → | ok(p(X)) |
| add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: true, mark, zero, add, 0, fact, s, if, p, active, false, ok, proper, prod, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(fact(X)) | → | active#(X) | | active#(add(X1, X2)) | → | active#(X1) |
| active#(if(X1, X2, X3)) | → | active#(X1) | | active#(add(X1, X2)) | → | active#(X2) |
| active#(p(X)) | → | active#(X) | | active#(s(X)) | → | active#(X) |
| active#(zero(X)) | → | active#(X) | | active#(prod(X1, X2)) | → | active#(X2) |
| active#(prod(X1, X2)) | → | active#(X1) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| zero#(mark(X)) | → | zero#(X) | | zero#(ok(X)) | → | zero#(X) |
Rewrite Rules
| active(fact(X)) | → | mark(if(zero(X), s(0), prod(X, fact(p(X))))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(prod(0, X)) | → | mark(0) |
| active(prod(s(X), Y)) | → | mark(add(Y, prod(X, Y))) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(zero(0)) | → | mark(true) |
| active(zero(s(X))) | → | mark(false) | | active(p(s(X))) | → | mark(X) |
| active(fact(X)) | → | fact(active(X)) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| active(zero(X)) | → | zero(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(prod(X1, X2)) | → | prod(active(X1), X2) | | active(prod(X1, X2)) | → | prod(X1, active(X2)) |
| active(p(X)) | → | p(active(X)) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fact(mark(X)) | → | mark(fact(X)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | zero(mark(X)) | → | mark(zero(X)) |
| s(mark(X)) | → | mark(s(X)) | | prod(mark(X1), X2) | → | mark(prod(X1, X2)) |
| prod(X1, mark(X2)) | → | mark(prod(X1, X2)) | | p(mark(X)) | → | mark(p(X)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fact(X)) | → | fact(proper(X)) | | proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) |
| proper(zero(X)) | → | zero(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(prod(X1, X2)) | → | prod(proper(X1), proper(X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| fact(ok(X)) | → | ok(fact(X)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| zero(ok(X)) | → | ok(zero(X)) | | s(ok(X)) | → | ok(s(X)) |
| prod(ok(X1), ok(X2)) | → | ok(prod(X1, X2)) | | p(ok(X)) | → | ok(p(X)) |
| add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: true, mark, zero, add, 0, fact, s, if, p, active, false, ok, proper, prod, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| zero#(ok(X)) | → | zero#(X) | | zero#(mark(X)) | → | zero#(X) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| proper#(prod(X1, X2)) | → | proper#(X2) | | proper#(s(X)) | → | proper#(X) |
| proper#(if(X1, X2, X3)) | → | proper#(X1) | | proper#(if(X1, X2, X3)) | → | proper#(X2) |
| proper#(if(X1, X2, X3)) | → | proper#(X3) | | proper#(add(X1, X2)) | → | proper#(X1) |
| proper#(fact(X)) | → | proper#(X) | | proper#(zero(X)) | → | proper#(X) |
| proper#(p(X)) | → | proper#(X) | | proper#(prod(X1, X2)) | → | proper#(X1) |
| proper#(add(X1, X2)) | → | proper#(X2) |
Rewrite Rules
| active(fact(X)) | → | mark(if(zero(X), s(0), prod(X, fact(p(X))))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(prod(0, X)) | → | mark(0) |
| active(prod(s(X), Y)) | → | mark(add(Y, prod(X, Y))) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(zero(0)) | → | mark(true) |
| active(zero(s(X))) | → | mark(false) | | active(p(s(X))) | → | mark(X) |
| active(fact(X)) | → | fact(active(X)) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| active(zero(X)) | → | zero(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(prod(X1, X2)) | → | prod(active(X1), X2) | | active(prod(X1, X2)) | → | prod(X1, active(X2)) |
| active(p(X)) | → | p(active(X)) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fact(mark(X)) | → | mark(fact(X)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | zero(mark(X)) | → | mark(zero(X)) |
| s(mark(X)) | → | mark(s(X)) | | prod(mark(X1), X2) | → | mark(prod(X1, X2)) |
| prod(X1, mark(X2)) | → | mark(prod(X1, X2)) | | p(mark(X)) | → | mark(p(X)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fact(X)) | → | fact(proper(X)) | | proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) |
| proper(zero(X)) | → | zero(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(prod(X1, X2)) | → | prod(proper(X1), proper(X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| fact(ok(X)) | → | ok(fact(X)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| zero(ok(X)) | → | ok(zero(X)) | | s(ok(X)) | → | ok(s(X)) |
| prod(ok(X1), ok(X2)) | → | ok(prod(X1, X2)) | | p(ok(X)) | → | ok(p(X)) |
| add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: true, mark, zero, add, 0, fact, s, if, p, active, false, ok, proper, prod, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(s(X)) | → | proper#(X) | | proper#(prod(X1, X2)) | → | proper#(X2) |
| proper#(if(X1, X2, X3)) | → | proper#(X1) | | proper#(if(X1, X2, X3)) | → | proper#(X2) |
| proper#(if(X1, X2, X3)) | → | proper#(X3) | | proper#(add(X1, X2)) | → | proper#(X1) |
| proper#(fact(X)) | → | proper#(X) | | proper#(zero(X)) | → | proper#(X) |
| proper#(p(X)) | → | proper#(X) | | proper#(prod(X1, X2)) | → | proper#(X1) |
| proper#(add(X1, X2)) | → | proper#(X2) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| add#(X1, mark(X2)) | → | add#(X1, X2) | | add#(mark(X1), X2) | → | add#(X1, X2) |
| add#(ok(X1), ok(X2)) | → | add#(X1, X2) |
Rewrite Rules
| active(fact(X)) | → | mark(if(zero(X), s(0), prod(X, fact(p(X))))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(prod(0, X)) | → | mark(0) |
| active(prod(s(X), Y)) | → | mark(add(Y, prod(X, Y))) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(zero(0)) | → | mark(true) |
| active(zero(s(X))) | → | mark(false) | | active(p(s(X))) | → | mark(X) |
| active(fact(X)) | → | fact(active(X)) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| active(zero(X)) | → | zero(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(prod(X1, X2)) | → | prod(active(X1), X2) | | active(prod(X1, X2)) | → | prod(X1, active(X2)) |
| active(p(X)) | → | p(active(X)) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fact(mark(X)) | → | mark(fact(X)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | zero(mark(X)) | → | mark(zero(X)) |
| s(mark(X)) | → | mark(s(X)) | | prod(mark(X1), X2) | → | mark(prod(X1, X2)) |
| prod(X1, mark(X2)) | → | mark(prod(X1, X2)) | | p(mark(X)) | → | mark(p(X)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fact(X)) | → | fact(proper(X)) | | proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) |
| proper(zero(X)) | → | zero(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(prod(X1, X2)) | → | prod(proper(X1), proper(X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| fact(ok(X)) | → | ok(fact(X)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| zero(ok(X)) | → | ok(zero(X)) | | s(ok(X)) | → | ok(s(X)) |
| prod(ok(X1), ok(X2)) | → | ok(prod(X1, X2)) | | p(ok(X)) | → | ok(p(X)) |
| add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: true, mark, zero, add, 0, fact, s, if, p, active, false, ok, proper, prod, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| add#(mark(X1), X2) | → | add#(X1, X2) | | add#(ok(X1), ok(X2)) | → | add#(X1, X2) |
Problem 12: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| add#(X1, mark(X2)) | → | add#(X1, X2) |
Rewrite Rules
| active(fact(X)) | → | mark(if(zero(X), s(0), prod(X, fact(p(X))))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(prod(0, X)) | → | mark(0) |
| active(prod(s(X), Y)) | → | mark(add(Y, prod(X, Y))) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(zero(0)) | → | mark(true) |
| active(zero(s(X))) | → | mark(false) | | active(p(s(X))) | → | mark(X) |
| active(fact(X)) | → | fact(active(X)) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| active(zero(X)) | → | zero(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(prod(X1, X2)) | → | prod(active(X1), X2) | | active(prod(X1, X2)) | → | prod(X1, active(X2)) |
| active(p(X)) | → | p(active(X)) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fact(mark(X)) | → | mark(fact(X)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | zero(mark(X)) | → | mark(zero(X)) |
| s(mark(X)) | → | mark(s(X)) | | prod(mark(X1), X2) | → | mark(prod(X1, X2)) |
| prod(X1, mark(X2)) | → | mark(prod(X1, X2)) | | p(mark(X)) | → | mark(p(X)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fact(X)) | → | fact(proper(X)) | | proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) |
| proper(zero(X)) | → | zero(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(prod(X1, X2)) | → | prod(proper(X1), proper(X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| fact(ok(X)) | → | ok(fact(X)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| zero(ok(X)) | → | ok(zero(X)) | | s(ok(X)) | → | ok(s(X)) |
| prod(ok(X1), ok(X2)) | → | ok(prod(X1, X2)) | | p(ok(X)) | → | ok(p(X)) |
| add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: true, mark, zero, add, 0, fact, s, if, p, active, false, ok, proper, top, prod
Strategy
Function Precedence
true = mark = add = zero = 0 = fact = s = if = p = false = active = ok = proper = add# = top = prod
Argument Filtering
true: all arguments are removed from true
mark: 1
add: all arguments are removed from add
zero: all arguments are removed from zero
0: all arguments are removed from 0
fact: all arguments are removed from fact
s: all arguments are removed from s
if: all arguments are removed from if
p: all arguments are removed from p
false: all arguments are removed from false
active: all arguments are removed from active
ok: all arguments are removed from ok
proper: all arguments are removed from proper
add#: collapses to 2
top: all arguments are removed from top
prod: 1
Status
true: multiset
mark: multiset
add: multiset
zero: multiset
0: multiset
fact: multiset
s: multiset
if: multiset
p: multiset
false: multiset
active: multiset
ok: multiset
proper: multiset
top: multiset
prod: 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.
| add#(X1, mark(X2)) → add#(X1, X2) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| prod#(X1, mark(X2)) | → | prod#(X1, X2) | | prod#(mark(X1), X2) | → | prod#(X1, X2) |
| prod#(ok(X1), ok(X2)) | → | prod#(X1, X2) |
Rewrite Rules
| active(fact(X)) | → | mark(if(zero(X), s(0), prod(X, fact(p(X))))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(prod(0, X)) | → | mark(0) |
| active(prod(s(X), Y)) | → | mark(add(Y, prod(X, Y))) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(zero(0)) | → | mark(true) |
| active(zero(s(X))) | → | mark(false) | | active(p(s(X))) | → | mark(X) |
| active(fact(X)) | → | fact(active(X)) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| active(zero(X)) | → | zero(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(prod(X1, X2)) | → | prod(active(X1), X2) | | active(prod(X1, X2)) | → | prod(X1, active(X2)) |
| active(p(X)) | → | p(active(X)) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fact(mark(X)) | → | mark(fact(X)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | zero(mark(X)) | → | mark(zero(X)) |
| s(mark(X)) | → | mark(s(X)) | | prod(mark(X1), X2) | → | mark(prod(X1, X2)) |
| prod(X1, mark(X2)) | → | mark(prod(X1, X2)) | | p(mark(X)) | → | mark(p(X)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fact(X)) | → | fact(proper(X)) | | proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) |
| proper(zero(X)) | → | zero(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(prod(X1, X2)) | → | prod(proper(X1), proper(X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| fact(ok(X)) | → | ok(fact(X)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| zero(ok(X)) | → | ok(zero(X)) | | s(ok(X)) | → | ok(s(X)) |
| prod(ok(X1), ok(X2)) | → | ok(prod(X1, X2)) | | p(ok(X)) | → | ok(p(X)) |
| add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: true, mark, zero, add, 0, fact, s, if, p, active, false, ok, proper, prod, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| prod#(mark(X1), X2) | → | prod#(X1, X2) | | prod#(ok(X1), ok(X2)) | → | prod#(X1, X2) |
Problem 13: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| prod#(X1, mark(X2)) | → | prod#(X1, X2) |
Rewrite Rules
| active(fact(X)) | → | mark(if(zero(X), s(0), prod(X, fact(p(X))))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(prod(0, X)) | → | mark(0) |
| active(prod(s(X), Y)) | → | mark(add(Y, prod(X, Y))) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(zero(0)) | → | mark(true) |
| active(zero(s(X))) | → | mark(false) | | active(p(s(X))) | → | mark(X) |
| active(fact(X)) | → | fact(active(X)) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| active(zero(X)) | → | zero(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(prod(X1, X2)) | → | prod(active(X1), X2) | | active(prod(X1, X2)) | → | prod(X1, active(X2)) |
| active(p(X)) | → | p(active(X)) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fact(mark(X)) | → | mark(fact(X)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | zero(mark(X)) | → | mark(zero(X)) |
| s(mark(X)) | → | mark(s(X)) | | prod(mark(X1), X2) | → | mark(prod(X1, X2)) |
| prod(X1, mark(X2)) | → | mark(prod(X1, X2)) | | p(mark(X)) | → | mark(p(X)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fact(X)) | → | fact(proper(X)) | | proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) |
| proper(zero(X)) | → | zero(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(prod(X1, X2)) | → | prod(proper(X1), proper(X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| fact(ok(X)) | → | ok(fact(X)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| zero(ok(X)) | → | ok(zero(X)) | | s(ok(X)) | → | ok(s(X)) |
| prod(ok(X1), ok(X2)) | → | ok(prod(X1, X2)) | | p(ok(X)) | → | ok(p(X)) |
| add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: true, mark, zero, add, 0, fact, s, if, p, active, false, ok, proper, top, prod
Strategy
Function Precedence
true = mark = add = zero = prod# = 0 = fact = s = if = p = false = active = ok = proper = top = prod
Argument Filtering
true: all arguments are removed from true
mark: 1
add: all arguments are removed from add
zero: all arguments are removed from zero
prod#: 2
0: all arguments are removed from 0
fact: all arguments are removed from fact
s: all arguments are removed from s
if: all arguments are removed from if
p: 1
false: all arguments are removed from false
active: 1
ok: all arguments are removed from ok
proper: collapses to 1
top: 1
prod: 2
Status
true: multiset
mark: multiset
add: multiset
zero: multiset
prod#: lexicographic with permutation 2 → 1
0: multiset
fact: multiset
s: multiset
if: multiset
p: lexicographic with permutation 1 → 1
false: multiset
active: lexicographic with permutation 1 → 1
ok: multiset
top: lexicographic with permutation 1 → 1
prod: lexicographic with permutation 2 → 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.
| prod#(X1, mark(X2)) → prod#(X1, X2) |
Problem 9: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| fact#(ok(X)) | → | fact#(X) | | fact#(mark(X)) | → | fact#(X) |
Rewrite Rules
| active(fact(X)) | → | mark(if(zero(X), s(0), prod(X, fact(p(X))))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(prod(0, X)) | → | mark(0) |
| active(prod(s(X), Y)) | → | mark(add(Y, prod(X, Y))) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(zero(0)) | → | mark(true) |
| active(zero(s(X))) | → | mark(false) | | active(p(s(X))) | → | mark(X) |
| active(fact(X)) | → | fact(active(X)) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| active(zero(X)) | → | zero(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(prod(X1, X2)) | → | prod(active(X1), X2) | | active(prod(X1, X2)) | → | prod(X1, active(X2)) |
| active(p(X)) | → | p(active(X)) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fact(mark(X)) | → | mark(fact(X)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | zero(mark(X)) | → | mark(zero(X)) |
| s(mark(X)) | → | mark(s(X)) | | prod(mark(X1), X2) | → | mark(prod(X1, X2)) |
| prod(X1, mark(X2)) | → | mark(prod(X1, X2)) | | p(mark(X)) | → | mark(p(X)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fact(X)) | → | fact(proper(X)) | | proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) |
| proper(zero(X)) | → | zero(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(prod(X1, X2)) | → | prod(proper(X1), proper(X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| fact(ok(X)) | → | ok(fact(X)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| zero(ok(X)) | → | ok(zero(X)) | | s(ok(X)) | → | ok(s(X)) |
| prod(ok(X1), ok(X2)) | → | ok(prod(X1, X2)) | | p(ok(X)) | → | ok(p(X)) |
| add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: true, mark, zero, add, 0, fact, s, if, p, active, false, ok, proper, prod, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| fact#(ok(X)) | → | fact#(X) | | fact#(mark(X)) | → | fact#(X) |
Problem 10: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| p#(ok(X)) | → | p#(X) | | p#(mark(X)) | → | p#(X) |
Rewrite Rules
| active(fact(X)) | → | mark(if(zero(X), s(0), prod(X, fact(p(X))))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(prod(0, X)) | → | mark(0) |
| active(prod(s(X), Y)) | → | mark(add(Y, prod(X, Y))) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(zero(0)) | → | mark(true) |
| active(zero(s(X))) | → | mark(false) | | active(p(s(X))) | → | mark(X) |
| active(fact(X)) | → | fact(active(X)) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| active(zero(X)) | → | zero(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(prod(X1, X2)) | → | prod(active(X1), X2) | | active(prod(X1, X2)) | → | prod(X1, active(X2)) |
| active(p(X)) | → | p(active(X)) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fact(mark(X)) | → | mark(fact(X)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | zero(mark(X)) | → | mark(zero(X)) |
| s(mark(X)) | → | mark(s(X)) | | prod(mark(X1), X2) | → | mark(prod(X1, X2)) |
| prod(X1, mark(X2)) | → | mark(prod(X1, X2)) | | p(mark(X)) | → | mark(p(X)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fact(X)) | → | fact(proper(X)) | | proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) |
| proper(zero(X)) | → | zero(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(prod(X1, X2)) | → | prod(proper(X1), proper(X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| fact(ok(X)) | → | ok(fact(X)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| zero(ok(X)) | → | ok(zero(X)) | | s(ok(X)) | → | ok(s(X)) |
| prod(ok(X1), ok(X2)) | → | ok(prod(X1, X2)) | | p(ok(X)) | → | ok(p(X)) |
| add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: true, mark, zero, add, 0, fact, s, if, p, active, false, ok, proper, prod, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| p#(ok(X)) | → | p#(X) | | p#(mark(X)) | → | p#(X) |
Problem 11: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Rewrite Rules
| active(fact(X)) | → | mark(if(zero(X), s(0), prod(X, fact(p(X))))) | | active(add(0, X)) | → | mark(X) |
| active(add(s(X), Y)) | → | mark(s(add(X, Y))) | | active(prod(0, X)) | → | mark(0) |
| active(prod(s(X), Y)) | → | mark(add(Y, prod(X, Y))) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(zero(0)) | → | mark(true) |
| active(zero(s(X))) | → | mark(false) | | active(p(s(X))) | → | mark(X) |
| active(fact(X)) | → | fact(active(X)) | | active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) |
| active(zero(X)) | → | zero(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(prod(X1, X2)) | → | prod(active(X1), X2) | | active(prod(X1, X2)) | → | prod(X1, active(X2)) |
| active(p(X)) | → | p(active(X)) | | active(add(X1, X2)) | → | add(active(X1), X2) |
| active(add(X1, X2)) | → | add(X1, active(X2)) | | fact(mark(X)) | → | mark(fact(X)) |
| if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) | | zero(mark(X)) | → | mark(zero(X)) |
| s(mark(X)) | → | mark(s(X)) | | prod(mark(X1), X2) | → | mark(prod(X1, X2)) |
| prod(X1, mark(X2)) | → | mark(prod(X1, X2)) | | p(mark(X)) | → | mark(p(X)) |
| add(mark(X1), X2) | → | mark(add(X1, X2)) | | add(X1, mark(X2)) | → | mark(add(X1, X2)) |
| proper(fact(X)) | → | fact(proper(X)) | | proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) |
| proper(zero(X)) | → | zero(proper(X)) | | proper(s(X)) | → | s(proper(X)) |
| proper(0) | → | ok(0) | | proper(prod(X1, X2)) | → | prod(proper(X1), proper(X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(add(X1, X2)) | → | add(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| fact(ok(X)) | → | ok(fact(X)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| zero(ok(X)) | → | ok(zero(X)) | | s(ok(X)) | → | ok(s(X)) |
| prod(ok(X1), ok(X2)) | → | ok(prod(X1, X2)) | | p(ok(X)) | → | ok(p(X)) |
| add(ok(X1), ok(X2)) | → | ok(add(X1, X2)) | | top(mark(X)) | → | top(proper(X)) |
| top(ok(X)) | → | top(active(X)) |
Original Signature
Termination of terms over the following signature is verified: true, mark, zero, add, 0, fact, s, if, p, active, false, ok, proper, prod, 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) |