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 (1755ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| – Problem 3 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (4ms), PolynomialLinearRange4iUR (3333ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (10000ms), DependencyGraph (5ms), ReductionPairSAT (4107ms), DependencyGraph (28ms), ReductionPairSAT (3944ms), DependencyGraph (5ms), SizeChangePrinciple (timeout)].
| – Problem 4 was processed with processor SubtermCriterion (3ms).
| | – Problem 10 was processed with processor ReductionPairSAT (34ms).
| – Problem 5 was processed with processor SubtermCriterion (1ms).
| | – Problem 11 was processed with processor ReductionPairSAT (64ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
| – Problem 7 was processed with processor SubtermCriterion (1ms).
| – Problem 8 was processed with processor SubtermCriterion (1ms).
| – Problem 9 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| top#(mark(X)) | → | top#(proper(X)) | | top#(ok(X)) | → | top#(active(X)) |
Rewrite Rules
| active(p(0)) | → | mark(0) | | active(p(s(X))) | → | mark(X) |
| active(leq(0, Y)) | → | mark(true) | | active(leq(s(X), 0)) | → | mark(false) |
| active(leq(s(X), s(Y))) | → | mark(leq(X, Y)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(diff(X, Y)) | → | mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) |
| active(p(X)) | → | p(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(leq(X1, X2)) | → | leq(active(X1), X2) | | active(leq(X1, X2)) | → | leq(X1, active(X2)) |
| active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) | | active(diff(X1, X2)) | → | diff(active(X1), X2) |
| active(diff(X1, X2)) | → | diff(X1, active(X2)) | | p(mark(X)) | → | mark(p(X)) |
| s(mark(X)) | → | mark(s(X)) | | leq(mark(X1), X2) | → | mark(leq(X1, X2)) |
| leq(X1, mark(X2)) | → | mark(leq(X1, X2)) | | if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) |
| diff(mark(X1), X2) | → | mark(diff(X1, X2)) | | diff(X1, mark(X2)) | → | mark(diff(X1, X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(0) | → | ok(0) |
| proper(s(X)) | → | s(proper(X)) | | proper(leq(X1, X2)) | → | leq(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | proper(diff(X1, X2)) | → | diff(proper(X1), proper(X2)) |
| p(ok(X)) | → | ok(p(X)) | | s(ok(X)) | → | ok(s(X)) |
| leq(ok(X1), ok(X2)) | → | ok(leq(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| diff(ok(X1), ok(X2)) | → | ok(diff(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: diff, leq, true, mark, 0, s, if, p, active, false, ok, proper, top
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(diff(X, Y)) | → | leq#(X, Y) | | top#(ok(X)) | → | top#(active(X)) |
| proper#(p(X)) | → | proper#(X) | | active#(if(X1, X2, X3)) | → | active#(X1) |
| active#(p(X)) | → | p#(active(X)) | | proper#(p(X)) | → | p#(proper(X)) |
| top#(mark(X)) | → | proper#(X) | | active#(diff(X, Y)) | → | p#(X) |
| top#(mark(X)) | → | top#(proper(X)) | | leq#(X1, mark(X2)) | → | leq#(X1, X2) |
| active#(leq(X1, X2)) | → | leq#(active(X1), X2) | | active#(p(X)) | → | active#(X) |
| proper#(s(X)) | → | proper#(X) | | active#(diff(X1, X2)) | → | diff#(X1, active(X2)) |
| if#(mark(X1), X2, X3) | → | if#(X1, X2, X3) | | proper#(leq(X1, X2)) | → | proper#(X2) |
| active#(diff(X, Y)) | → | if#(leq(X, Y), 0, s(diff(p(X), Y))) | | active#(diff(X, Y)) | → | s#(diff(p(X), Y)) |
| proper#(diff(X1, X2)) | → | proper#(X2) | | leq#(mark(X1), X2) | → | leq#(X1, X2) |
| active#(diff(X1, X2)) | → | active#(X2) | | p#(mark(X)) | → | p#(X) |
| top#(ok(X)) | → | active#(X) | | leq#(ok(X1), ok(X2)) | → | leq#(X1, X2) |
| proper#(leq(X1, X2)) | → | leq#(proper(X1), proper(X2)) | | active#(leq(s(X), s(Y))) | → | leq#(X, Y) |
| active#(diff(X1, X2)) | → | active#(X1) | | proper#(diff(X1, X2)) | → | diff#(proper(X1), proper(X2)) |
| active#(diff(X, Y)) | → | diff#(p(X), Y) | | if#(ok(X1), ok(X2), ok(X3)) | → | if#(X1, X2, X3) |
| active#(diff(X1, X2)) | → | diff#(active(X1), X2) | | active#(leq(X1, X2)) | → | leq#(X1, active(X2)) |
| active#(leq(X1, X2)) | → | active#(X1) | | proper#(leq(X1, X2)) | → | proper#(X1) |
| proper#(if(X1, X2, X3)) | → | proper#(X1) | | diff#(ok(X1), ok(X2)) | → | diff#(X1, X2) |
| proper#(if(X1, X2, X3)) | → | proper#(X2) | | diff#(mark(X1), X2) | → | diff#(X1, X2) |
| diff#(X1, mark(X2)) | → | diff#(X1, X2) | | active#(s(X)) | → | s#(active(X)) |
| active#(leq(X1, X2)) | → | active#(X2) | | 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) |
| proper#(diff(X1, X2)) | → | proper#(X1) | | active#(if(X1, X2, X3)) | → | if#(active(X1), X2, X3) |
| p#(ok(X)) | → | p#(X) | | proper#(if(X1, X2, X3)) | → | if#(proper(X1), proper(X2), proper(X3)) |
Rewrite Rules
| active(p(0)) | → | mark(0) | | active(p(s(X))) | → | mark(X) |
| active(leq(0, Y)) | → | mark(true) | | active(leq(s(X), 0)) | → | mark(false) |
| active(leq(s(X), s(Y))) | → | mark(leq(X, Y)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(diff(X, Y)) | → | mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) |
| active(p(X)) | → | p(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(leq(X1, X2)) | → | leq(active(X1), X2) | | active(leq(X1, X2)) | → | leq(X1, active(X2)) |
| active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) | | active(diff(X1, X2)) | → | diff(active(X1), X2) |
| active(diff(X1, X2)) | → | diff(X1, active(X2)) | | p(mark(X)) | → | mark(p(X)) |
| s(mark(X)) | → | mark(s(X)) | | leq(mark(X1), X2) | → | mark(leq(X1, X2)) |
| leq(X1, mark(X2)) | → | mark(leq(X1, X2)) | | if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) |
| diff(mark(X1), X2) | → | mark(diff(X1, X2)) | | diff(X1, mark(X2)) | → | mark(diff(X1, X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(0) | → | ok(0) |
| proper(s(X)) | → | s(proper(X)) | | proper(leq(X1, X2)) | → | leq(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | proper(diff(X1, X2)) | → | diff(proper(X1), proper(X2)) |
| p(ok(X)) | → | ok(p(X)) | | s(ok(X)) | → | ok(s(X)) |
| leq(ok(X1), ok(X2)) | → | ok(leq(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| diff(ok(X1), ok(X2)) | → | ok(diff(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: diff, leq, true, mark, 0, s, if, p, active, false, ok, proper, top
Strategy
The following SCCs where found
| p#(ok(X)) → p#(X) | p#(mark(X)) → p#(X) |
| leq#(ok(X1), ok(X2)) → leq#(X1, X2) | leq#(mark(X1), X2) → leq#(X1, X2) |
| leq#(X1, mark(X2)) → leq#(X1, X2) |
| diff#(ok(X1), ok(X2)) → diff#(X1, X2) | diff#(mark(X1), X2) → diff#(X1, X2) |
| diff#(X1, mark(X2)) → diff#(X1, X2) |
| if#(mark(X1), X2, X3) → if#(X1, X2, X3) | if#(ok(X1), ok(X2), ok(X3)) → if#(X1, X2, X3) |
| proper#(s(X)) → proper#(X) | proper#(leq(X1, X2)) → proper#(X1) |
| proper#(if(X1, X2, X3)) → proper#(X1) | proper#(if(X1, X2, X3)) → proper#(X2) |
| proper#(if(X1, X2, X3)) → proper#(X3) | proper#(diff(X1, X2)) → proper#(X1) |
| proper#(leq(X1, X2)) → proper#(X2) | proper#(p(X)) → proper#(X) |
| proper#(diff(X1, X2)) → proper#(X2) |
| s#(mark(X)) → s#(X) | s#(ok(X)) → s#(X) |
| top#(mark(X)) → top#(proper(X)) | top#(ok(X)) → top#(active(X)) |
| active#(if(X1, X2, X3)) → active#(X1) | active#(leq(X1, X2)) → active#(X1) |
| active#(diff(X1, X2)) → active#(X2) | active#(s(X)) → active#(X) |
| active#(p(X)) → active#(X) | active#(diff(X1, X2)) → active#(X1) |
| active#(leq(X1, X2)) → active#(X2) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| p#(ok(X)) | → | p#(X) | | p#(mark(X)) | → | p#(X) |
Rewrite Rules
| active(p(0)) | → | mark(0) | | active(p(s(X))) | → | mark(X) |
| active(leq(0, Y)) | → | mark(true) | | active(leq(s(X), 0)) | → | mark(false) |
| active(leq(s(X), s(Y))) | → | mark(leq(X, Y)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(diff(X, Y)) | → | mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) |
| active(p(X)) | → | p(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(leq(X1, X2)) | → | leq(active(X1), X2) | | active(leq(X1, X2)) | → | leq(X1, active(X2)) |
| active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) | | active(diff(X1, X2)) | → | diff(active(X1), X2) |
| active(diff(X1, X2)) | → | diff(X1, active(X2)) | | p(mark(X)) | → | mark(p(X)) |
| s(mark(X)) | → | mark(s(X)) | | leq(mark(X1), X2) | → | mark(leq(X1, X2)) |
| leq(X1, mark(X2)) | → | mark(leq(X1, X2)) | | if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) |
| diff(mark(X1), X2) | → | mark(diff(X1, X2)) | | diff(X1, mark(X2)) | → | mark(diff(X1, X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(0) | → | ok(0) |
| proper(s(X)) | → | s(proper(X)) | | proper(leq(X1, X2)) | → | leq(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | proper(diff(X1, X2)) | → | diff(proper(X1), proper(X2)) |
| p(ok(X)) | → | ok(p(X)) | | s(ok(X)) | → | ok(s(X)) |
| leq(ok(X1), ok(X2)) | → | ok(leq(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| diff(ok(X1), ok(X2)) | → | ok(diff(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: diff, leq, true, mark, 0, s, if, p, active, false, ok, proper, 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 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| diff#(ok(X1), ok(X2)) | → | diff#(X1, X2) | | diff#(mark(X1), X2) | → | diff#(X1, X2) |
| diff#(X1, mark(X2)) | → | diff#(X1, X2) |
Rewrite Rules
| active(p(0)) | → | mark(0) | | active(p(s(X))) | → | mark(X) |
| active(leq(0, Y)) | → | mark(true) | | active(leq(s(X), 0)) | → | mark(false) |
| active(leq(s(X), s(Y))) | → | mark(leq(X, Y)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(diff(X, Y)) | → | mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) |
| active(p(X)) | → | p(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(leq(X1, X2)) | → | leq(active(X1), X2) | | active(leq(X1, X2)) | → | leq(X1, active(X2)) |
| active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) | | active(diff(X1, X2)) | → | diff(active(X1), X2) |
| active(diff(X1, X2)) | → | diff(X1, active(X2)) | | p(mark(X)) | → | mark(p(X)) |
| s(mark(X)) | → | mark(s(X)) | | leq(mark(X1), X2) | → | mark(leq(X1, X2)) |
| leq(X1, mark(X2)) | → | mark(leq(X1, X2)) | | if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) |
| diff(mark(X1), X2) | → | mark(diff(X1, X2)) | | diff(X1, mark(X2)) | → | mark(diff(X1, X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(0) | → | ok(0) |
| proper(s(X)) | → | s(proper(X)) | | proper(leq(X1, X2)) | → | leq(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | proper(diff(X1, X2)) | → | diff(proper(X1), proper(X2)) |
| p(ok(X)) | → | ok(p(X)) | | s(ok(X)) | → | ok(s(X)) |
| leq(ok(X1), ok(X2)) | → | ok(leq(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| diff(ok(X1), ok(X2)) | → | ok(diff(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: diff, leq, true, mark, 0, s, if, p, active, false, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| diff#(ok(X1), ok(X2)) | → | diff#(X1, X2) | | diff#(mark(X1), X2) | → | diff#(X1, X2) |
Problem 10: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| diff#(X1, mark(X2)) | → | diff#(X1, X2) |
Rewrite Rules
| active(p(0)) | → | mark(0) | | active(p(s(X))) | → | mark(X) |
| active(leq(0, Y)) | → | mark(true) | | active(leq(s(X), 0)) | → | mark(false) |
| active(leq(s(X), s(Y))) | → | mark(leq(X, Y)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(diff(X, Y)) | → | mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) |
| active(p(X)) | → | p(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(leq(X1, X2)) | → | leq(active(X1), X2) | | active(leq(X1, X2)) | → | leq(X1, active(X2)) |
| active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) | | active(diff(X1, X2)) | → | diff(active(X1), X2) |
| active(diff(X1, X2)) | → | diff(X1, active(X2)) | | p(mark(X)) | → | mark(p(X)) |
| s(mark(X)) | → | mark(s(X)) | | leq(mark(X1), X2) | → | mark(leq(X1, X2)) |
| leq(X1, mark(X2)) | → | mark(leq(X1, X2)) | | if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) |
| diff(mark(X1), X2) | → | mark(diff(X1, X2)) | | diff(X1, mark(X2)) | → | mark(diff(X1, X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(0) | → | ok(0) |
| proper(s(X)) | → | s(proper(X)) | | proper(leq(X1, X2)) | → | leq(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | proper(diff(X1, X2)) | → | diff(proper(X1), proper(X2)) |
| p(ok(X)) | → | ok(p(X)) | | s(ok(X)) | → | ok(s(X)) |
| leq(ok(X1), ok(X2)) | → | ok(leq(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| diff(ok(X1), ok(X2)) | → | ok(diff(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: diff, leq, true, mark, 0, s, if, p, active, false, ok, proper, top
Strategy
Function Precedence
diff# = diff = leq = true = mark = 0 = s = if = p = false = active = ok = proper = top
Argument Filtering
diff#: 2
diff: all arguments are removed from diff
leq: 2
true: all arguments are removed from true
mark: 1
0: all arguments are removed from 0
s: collapses to 1
if: all arguments are removed from if
p: collapses to 1
false: all arguments are removed from false
active: all arguments are removed from active
ok: all arguments are removed from ok
proper: collapses to 1
top: all arguments are removed from top
Status
diff#: multiset
diff: multiset
leq: lexicographic with permutation 2 → 1
true: multiset
mark: multiset
0: multiset
if: multiset
false: multiset
active: multiset
ok: multiset
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.
| diff#(X1, mark(X2)) → diff#(X1, X2) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| leq#(ok(X1), ok(X2)) | → | leq#(X1, X2) | | leq#(mark(X1), X2) | → | leq#(X1, X2) |
| leq#(X1, mark(X2)) | → | leq#(X1, X2) |
Rewrite Rules
| active(p(0)) | → | mark(0) | | active(p(s(X))) | → | mark(X) |
| active(leq(0, Y)) | → | mark(true) | | active(leq(s(X), 0)) | → | mark(false) |
| active(leq(s(X), s(Y))) | → | mark(leq(X, Y)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(diff(X, Y)) | → | mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) |
| active(p(X)) | → | p(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(leq(X1, X2)) | → | leq(active(X1), X2) | | active(leq(X1, X2)) | → | leq(X1, active(X2)) |
| active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) | | active(diff(X1, X2)) | → | diff(active(X1), X2) |
| active(diff(X1, X2)) | → | diff(X1, active(X2)) | | p(mark(X)) | → | mark(p(X)) |
| s(mark(X)) | → | mark(s(X)) | | leq(mark(X1), X2) | → | mark(leq(X1, X2)) |
| leq(X1, mark(X2)) | → | mark(leq(X1, X2)) | | if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) |
| diff(mark(X1), X2) | → | mark(diff(X1, X2)) | | diff(X1, mark(X2)) | → | mark(diff(X1, X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(0) | → | ok(0) |
| proper(s(X)) | → | s(proper(X)) | | proper(leq(X1, X2)) | → | leq(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | proper(diff(X1, X2)) | → | diff(proper(X1), proper(X2)) |
| p(ok(X)) | → | ok(p(X)) | | s(ok(X)) | → | ok(s(X)) |
| leq(ok(X1), ok(X2)) | → | ok(leq(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| diff(ok(X1), ok(X2)) | → | ok(diff(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: diff, leq, true, mark, 0, s, if, p, active, false, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| leq#(ok(X1), ok(X2)) | → | leq#(X1, X2) | | leq#(mark(X1), X2) | → | leq#(X1, X2) |
Problem 11: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| leq#(X1, mark(X2)) | → | leq#(X1, X2) |
Rewrite Rules
| active(p(0)) | → | mark(0) | | active(p(s(X))) | → | mark(X) |
| active(leq(0, Y)) | → | mark(true) | | active(leq(s(X), 0)) | → | mark(false) |
| active(leq(s(X), s(Y))) | → | mark(leq(X, Y)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(diff(X, Y)) | → | mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) |
| active(p(X)) | → | p(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(leq(X1, X2)) | → | leq(active(X1), X2) | | active(leq(X1, X2)) | → | leq(X1, active(X2)) |
| active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) | | active(diff(X1, X2)) | → | diff(active(X1), X2) |
| active(diff(X1, X2)) | → | diff(X1, active(X2)) | | p(mark(X)) | → | mark(p(X)) |
| s(mark(X)) | → | mark(s(X)) | | leq(mark(X1), X2) | → | mark(leq(X1, X2)) |
| leq(X1, mark(X2)) | → | mark(leq(X1, X2)) | | if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) |
| diff(mark(X1), X2) | → | mark(diff(X1, X2)) | | diff(X1, mark(X2)) | → | mark(diff(X1, X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(0) | → | ok(0) |
| proper(s(X)) | → | s(proper(X)) | | proper(leq(X1, X2)) | → | leq(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | proper(diff(X1, X2)) | → | diff(proper(X1), proper(X2)) |
| p(ok(X)) | → | ok(p(X)) | | s(ok(X)) | → | ok(s(X)) |
| leq(ok(X1), ok(X2)) | → | ok(leq(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| diff(ok(X1), ok(X2)) | → | ok(diff(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: diff, leq, true, mark, 0, s, if, p, active, false, ok, proper, top
Strategy
Function Precedence
mark < leq# = diff = leq = true = 0 = s = if = p = false = active = ok = proper = top
Argument Filtering
leq#: collapses to 2
diff: collapses to 1
leq: collapses to 2
true: all arguments are removed from true
mark: 1
0: all arguments are removed from 0
s: all arguments are removed from s
if: all arguments are removed from if
p: collapses to 1
false: all arguments are removed from false
active: all arguments are removed from active
ok: all arguments are removed from ok
proper: collapses to 1
top: all arguments are removed from top
Status
true: multiset
mark: multiset
0: multiset
s: multiset
if: multiset
false: multiset
active: multiset
ok: multiset
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.
| leq#(X1, mark(X2)) → leq#(X1, X2) |
Problem 6: 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(p(0)) | → | mark(0) | | active(p(s(X))) | → | mark(X) |
| active(leq(0, Y)) | → | mark(true) | | active(leq(s(X), 0)) | → | mark(false) |
| active(leq(s(X), s(Y))) | → | mark(leq(X, Y)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(diff(X, Y)) | → | mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) |
| active(p(X)) | → | p(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(leq(X1, X2)) | → | leq(active(X1), X2) | | active(leq(X1, X2)) | → | leq(X1, active(X2)) |
| active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) | | active(diff(X1, X2)) | → | diff(active(X1), X2) |
| active(diff(X1, X2)) | → | diff(X1, active(X2)) | | p(mark(X)) | → | mark(p(X)) |
| s(mark(X)) | → | mark(s(X)) | | leq(mark(X1), X2) | → | mark(leq(X1, X2)) |
| leq(X1, mark(X2)) | → | mark(leq(X1, X2)) | | if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) |
| diff(mark(X1), X2) | → | mark(diff(X1, X2)) | | diff(X1, mark(X2)) | → | mark(diff(X1, X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(0) | → | ok(0) |
| proper(s(X)) | → | s(proper(X)) | | proper(leq(X1, X2)) | → | leq(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | proper(diff(X1, X2)) | → | diff(proper(X1), proper(X2)) |
| p(ok(X)) | → | ok(p(X)) | | s(ok(X)) | → | ok(s(X)) |
| leq(ok(X1), ok(X2)) | → | ok(leq(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| diff(ok(X1), ok(X2)) | → | ok(diff(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: diff, leq, true, mark, 0, s, if, p, active, false, ok, proper, 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 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| active#(if(X1, X2, X3)) | → | active#(X1) | | active#(leq(X1, X2)) | → | active#(X1) |
| active#(diff(X1, X2)) | → | active#(X2) | | active#(s(X)) | → | active#(X) |
| active#(p(X)) | → | active#(X) | | active#(diff(X1, X2)) | → | active#(X1) |
| active#(leq(X1, X2)) | → | active#(X2) |
Rewrite Rules
| active(p(0)) | → | mark(0) | | active(p(s(X))) | → | mark(X) |
| active(leq(0, Y)) | → | mark(true) | | active(leq(s(X), 0)) | → | mark(false) |
| active(leq(s(X), s(Y))) | → | mark(leq(X, Y)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(diff(X, Y)) | → | mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) |
| active(p(X)) | → | p(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(leq(X1, X2)) | → | leq(active(X1), X2) | | active(leq(X1, X2)) | → | leq(X1, active(X2)) |
| active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) | | active(diff(X1, X2)) | → | diff(active(X1), X2) |
| active(diff(X1, X2)) | → | diff(X1, active(X2)) | | p(mark(X)) | → | mark(p(X)) |
| s(mark(X)) | → | mark(s(X)) | | leq(mark(X1), X2) | → | mark(leq(X1, X2)) |
| leq(X1, mark(X2)) | → | mark(leq(X1, X2)) | | if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) |
| diff(mark(X1), X2) | → | mark(diff(X1, X2)) | | diff(X1, mark(X2)) | → | mark(diff(X1, X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(0) | → | ok(0) |
| proper(s(X)) | → | s(proper(X)) | | proper(leq(X1, X2)) | → | leq(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | proper(diff(X1, X2)) | → | diff(proper(X1), proper(X2)) |
| p(ok(X)) | → | ok(p(X)) | | s(ok(X)) | → | ok(s(X)) |
| leq(ok(X1), ok(X2)) | → | ok(leq(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| diff(ok(X1), ok(X2)) | → | ok(diff(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: diff, leq, true, mark, 0, s, if, p, active, false, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| active#(if(X1, X2, X3)) | → | active#(X1) | | active#(leq(X1, X2)) | → | active#(X1) |
| active#(diff(X1, X2)) | → | active#(X2) | | active#(s(X)) | → | active#(X) |
| active#(p(X)) | → | active#(X) | | active#(diff(X1, X2)) | → | active#(X1) |
| active#(leq(X1, X2)) | → | active#(X2) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(ok(X)) | → | s#(X) |
Rewrite Rules
| active(p(0)) | → | mark(0) | | active(p(s(X))) | → | mark(X) |
| active(leq(0, Y)) | → | mark(true) | | active(leq(s(X), 0)) | → | mark(false) |
| active(leq(s(X), s(Y))) | → | mark(leq(X, Y)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(diff(X, Y)) | → | mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) |
| active(p(X)) | → | p(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(leq(X1, X2)) | → | leq(active(X1), X2) | | active(leq(X1, X2)) | → | leq(X1, active(X2)) |
| active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) | | active(diff(X1, X2)) | → | diff(active(X1), X2) |
| active(diff(X1, X2)) | → | diff(X1, active(X2)) | | p(mark(X)) | → | mark(p(X)) |
| s(mark(X)) | → | mark(s(X)) | | leq(mark(X1), X2) | → | mark(leq(X1, X2)) |
| leq(X1, mark(X2)) | → | mark(leq(X1, X2)) | | if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) |
| diff(mark(X1), X2) | → | mark(diff(X1, X2)) | | diff(X1, mark(X2)) | → | mark(diff(X1, X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(0) | → | ok(0) |
| proper(s(X)) | → | s(proper(X)) | | proper(leq(X1, X2)) | → | leq(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | proper(diff(X1, X2)) | → | diff(proper(X1), proper(X2)) |
| p(ok(X)) | → | ok(p(X)) | | s(ok(X)) | → | ok(s(X)) |
| leq(ok(X1), ok(X2)) | → | ok(leq(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| diff(ok(X1), ok(X2)) | → | ok(diff(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: diff, leq, true, mark, 0, s, if, p, active, false, ok, proper, 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
| proper#(s(X)) | → | proper#(X) | | proper#(leq(X1, X2)) | → | proper#(X1) |
| proper#(if(X1, X2, X3)) | → | proper#(X1) | | proper#(if(X1, X2, X3)) | → | proper#(X2) |
| proper#(if(X1, X2, X3)) | → | proper#(X3) | | proper#(diff(X1, X2)) | → | proper#(X1) |
| proper#(leq(X1, X2)) | → | proper#(X2) | | proper#(p(X)) | → | proper#(X) |
| proper#(diff(X1, X2)) | → | proper#(X2) |
Rewrite Rules
| active(p(0)) | → | mark(0) | | active(p(s(X))) | → | mark(X) |
| active(leq(0, Y)) | → | mark(true) | | active(leq(s(X), 0)) | → | mark(false) |
| active(leq(s(X), s(Y))) | → | mark(leq(X, Y)) | | active(if(true, X, Y)) | → | mark(X) |
| active(if(false, X, Y)) | → | mark(Y) | | active(diff(X, Y)) | → | mark(if(leq(X, Y), 0, s(diff(p(X), Y)))) |
| active(p(X)) | → | p(active(X)) | | active(s(X)) | → | s(active(X)) |
| active(leq(X1, X2)) | → | leq(active(X1), X2) | | active(leq(X1, X2)) | → | leq(X1, active(X2)) |
| active(if(X1, X2, X3)) | → | if(active(X1), X2, X3) | | active(diff(X1, X2)) | → | diff(active(X1), X2) |
| active(diff(X1, X2)) | → | diff(X1, active(X2)) | | p(mark(X)) | → | mark(p(X)) |
| s(mark(X)) | → | mark(s(X)) | | leq(mark(X1), X2) | → | mark(leq(X1, X2)) |
| leq(X1, mark(X2)) | → | mark(leq(X1, X2)) | | if(mark(X1), X2, X3) | → | mark(if(X1, X2, X3)) |
| diff(mark(X1), X2) | → | mark(diff(X1, X2)) | | diff(X1, mark(X2)) | → | mark(diff(X1, X2)) |
| proper(p(X)) | → | p(proper(X)) | | proper(0) | → | ok(0) |
| proper(s(X)) | → | s(proper(X)) | | proper(leq(X1, X2)) | → | leq(proper(X1), proper(X2)) |
| proper(true) | → | ok(true) | | proper(false) | → | ok(false) |
| proper(if(X1, X2, X3)) | → | if(proper(X1), proper(X2), proper(X3)) | | proper(diff(X1, X2)) | → | diff(proper(X1), proper(X2)) |
| p(ok(X)) | → | ok(p(X)) | | s(ok(X)) | → | ok(s(X)) |
| leq(ok(X1), ok(X2)) | → | ok(leq(X1, X2)) | | if(ok(X1), ok(X2), ok(X3)) | → | ok(if(X1, X2, X3)) |
| diff(ok(X1), ok(X2)) | → | ok(diff(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: diff, leq, true, mark, 0, s, if, p, active, false, ok, proper, top
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| proper#(s(X)) | → | proper#(X) | | proper#(leq(X1, X2)) | → | proper#(X1) |
| proper#(if(X1, X2, X3)) | → | proper#(X1) | | proper#(if(X1, X2, X3)) | → | proper#(X2) |
| proper#(if(X1, X2, X3)) | → | proper#(X3) | | proper#(diff(X1, X2)) | → | proper#(X1) |
| proper#(leq(X1, X2)) | → | proper#(X2) | | proper#(p(X)) | → | proper#(X) |
| proper#(diff(X1, X2)) | → | proper#(X2) |