TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60040 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (395ms).
| – Problem 2 was processed with processor ReductionPairSAT (3962ms).
| | – Problem 8 was processed with processor ReductionPairSAT (4171ms).
| | | – Problem 10 was processed with processor ReductionPairSAT (4255ms).
| | | | – Problem 11 was processed with processor ReductionPairSAT (5292ms).
| | | | | – Problem 12 was processed with processor ReductionPairSAT (3123ms).
| | | | | | – Problem 13 was processed with processor ReductionPairSAT (4242ms).
| | | | | | | – Problem 14 was processed with processor ReductionPairSAT (3688ms).
| | | | | | | | – Problem 15 was processed with processor ReductionPairSAT (3621ms).
| | | | | | | | | – Problem 16 was processed with processor ReductionPairSAT (4126ms).
| | | | | | | | | | – Problem 17 remains open; application of the following processors failed [DependencyGraph (2ms), ReductionPairSAT (timeout)].
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| | – Problem 7 was processed with processor ReductionPairSAT (63ms).
| | | – Problem 9 was processed with processor ReductionPairSAT (20ms).
| – Problem 4 was processed with processor SubtermCriterion (0ms).
| – Problem 5 was processed with processor SubtermCriterion (0ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 17
Dependency Pairs
| mark#(p(X)) | → | mark#(X) | | mark#(f(X)) | → | active#(f(mark(X))) |
| active#(f(s(0))) | → | mark#(f(p(s(0)))) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) | | cons#(mark(X1), X2) | → | cons#(X1, X2) |
| active#(f(s(0))) | → | f#(p(s(0))) | | f#(active(X)) | → | f#(X) |
| p#(mark(X)) | → | p#(X) | | active#(f(s(0))) | → | mark#(f(p(s(0)))) |
| mark#(s(X)) | → | s#(mark(X)) | | mark#(s(X)) | → | mark#(X) |
| cons#(X1, mark(X2)) | → | cons#(X1, X2) | | f#(mark(X)) | → | f#(X) |
| mark#(f(X)) | → | f#(mark(X)) | | mark#(cons(X1, X2)) | → | cons#(mark(X1), X2) |
| active#(f(0)) | → | s#(0) | | mark#(0) | → | active#(0) |
| mark#(s(X)) | → | active#(s(mark(X))) | | active#(f(s(0))) | → | p#(s(0)) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | cons#(active(X1), X2) | → | cons#(X1, X2) |
| mark#(f(X)) | → | mark#(X) | | mark#(p(X)) | → | active#(p(mark(X))) |
| mark#(f(X)) | → | active#(f(mark(X))) | | s#(mark(X)) | → | s#(X) |
| active#(f(s(0))) | → | s#(0) | | cons#(X1, active(X2)) | → | cons#(X1, X2) |
| active#(f(0)) | → | cons#(0, f(s(0))) | | mark#(p(X)) | → | p#(mark(X)) |
| mark#(p(X)) | → | mark#(X) | | active#(f(0)) | → | mark#(cons(0, f(s(0)))) |
| active#(p(s(X))) | → | mark#(X) | | active#(f(0)) | → | f#(s(0)) |
| s#(active(X)) | → | s#(X) | | p#(active(X)) | → | p#(X) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Strategy
The following SCCs where found
| p#(mark(X)) → p#(X) | p#(active(X)) → p#(X) |
| f#(active(X)) → f#(X) | f#(mark(X)) → f#(X) |
| mark#(0) → active#(0) | mark#(cons(X1, X2)) → active#(cons(mark(X1), X2)) |
| mark#(s(X)) → active#(s(mark(X))) | active#(f(0)) → mark#(cons(0, f(s(0)))) |
| mark#(p(X)) → mark#(X) | mark#(cons(X1, X2)) → mark#(X1) |
| active#(p(s(X))) → mark#(X) | mark#(s(X)) → mark#(X) |
| mark#(p(X)) → active#(p(mark(X))) | mark#(f(X)) → mark#(X) |
| mark#(f(X)) → active#(f(mark(X))) | active#(f(s(0))) → mark#(f(p(s(0)))) |
| cons#(X1, active(X2)) → cons#(X1, X2) | cons#(mark(X1), X2) → cons#(X1, X2) |
| cons#(X1, mark(X2)) → cons#(X1, X2) | cons#(active(X1), X2) → cons#(X1, X2) |
| s#(mark(X)) → s#(X) | s#(active(X)) → s#(X) |
Problem 2: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| mark#(0) | → | active#(0) | | mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) |
| mark#(s(X)) | → | active#(s(mark(X))) | | active#(f(0)) | → | mark#(cons(0, f(s(0)))) |
| mark#(p(X)) | → | mark#(X) | | active#(p(s(X))) | → | mark#(X) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(p(X)) | → | active#(p(mark(X))) | | mark#(f(X)) | → | mark#(X) |
| mark#(f(X)) | → | active#(f(mark(X))) | | active#(f(s(0))) | → | mark#(f(p(s(0)))) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Strategy
Function Precedence
active# < f = s = p = active = mark = mark# < 0 = cons
Argument Filtering
f: all arguments are removed from f
0: all arguments are removed from 0
s: all arguments are removed from s
p: all arguments are removed from p
active: all arguments are removed from active
mark: all arguments are removed from mark
active#: collapses to 1
mark#: all arguments are removed from mark#
cons: all arguments are removed from cons
Status
f: multiset
0: multiset
s: multiset
p: multiset
active: multiset
mark: multiset
mark#: multiset
cons: multiset
Usable Rules
| cons(active(X1), X2) → cons(X1, X2) | mark(s(X)) → active(s(mark(X))) |
| active(f(0)) → mark(cons(0, f(s(0)))) | active(f(s(0))) → mark(f(p(s(0)))) |
| cons(X1, mark(X2)) → cons(X1, X2) | mark(p(X)) → active(p(mark(X))) |
| active(p(s(X))) → mark(X) | p(mark(X)) → p(X) |
| cons(mark(X1), X2) → cons(X1, X2) | mark(cons(X1, X2)) → active(cons(mark(X1), X2)) |
| s(mark(X)) → s(X) | f(active(X)) → f(X) |
| p(active(X)) → p(X) | mark(f(X)) → active(f(mark(X))) |
| f(mark(X)) → f(X) | mark(0) → active(0) |
| s(active(X)) → s(X) | cons(X1, active(X2)) → cons(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.
| mark#(cons(X1, X2)) → active#(cons(mark(X1), X2)) |
Problem 8: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| mark#(0) | → | active#(0) | | mark#(s(X)) | → | active#(s(mark(X))) |
| mark#(p(X)) | → | mark#(X) | | active#(f(0)) | → | mark#(cons(0, f(s(0)))) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | active#(p(s(X))) | → | mark#(X) |
| mark#(s(X)) | → | mark#(X) | | mark#(f(X)) | → | mark#(X) |
| mark#(p(X)) | → | active#(p(mark(X))) | | active#(f(s(0))) | → | mark#(f(p(s(0)))) |
| mark#(f(X)) | → | active#(f(mark(X))) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Strategy
Function Precedence
active# < f = 0 = s = p = active = mark = mark# = cons
Argument Filtering
f: all arguments are removed from f
0: all arguments are removed from 0
s: all arguments are removed from s
p: all arguments are removed from p
active: all arguments are removed from active
mark: all arguments are removed from mark
active#: collapses to 1
mark#: all arguments are removed from mark#
cons: all arguments are removed from cons
Status
f: multiset
0: multiset
s: multiset
p: multiset
active: multiset
mark: multiset
mark#: multiset
cons: multiset
Usable Rules
| cons(active(X1), X2) → cons(X1, X2) | mark(s(X)) → active(s(mark(X))) |
| active(f(0)) → mark(cons(0, f(s(0)))) | active(f(s(0))) → mark(f(p(s(0)))) |
| cons(X1, mark(X2)) → cons(X1, X2) | mark(p(X)) → active(p(mark(X))) |
| active(p(s(X))) → mark(X) | p(mark(X)) → p(X) |
| cons(mark(X1), X2) → cons(X1, X2) | mark(cons(X1, X2)) → active(cons(mark(X1), X2)) |
| s(mark(X)) → s(X) | f(active(X)) → f(X) |
| p(active(X)) → p(X) | mark(f(X)) → active(f(mark(X))) |
| f(mark(X)) → f(X) | mark(0) → active(0) |
| s(active(X)) → s(X) | cons(X1, active(X2)) → cons(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.
Problem 10: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| mark#(s(X)) | → | active#(s(mark(X))) | | active#(f(0)) | → | mark#(cons(0, f(s(0)))) |
| mark#(p(X)) | → | mark#(X) | | active#(p(s(X))) | → | mark#(X) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(p(X)) | → | active#(p(mark(X))) | | mark#(f(X)) | → | mark#(X) |
| mark#(f(X)) | → | active#(f(mark(X))) | | active#(f(s(0))) | → | mark#(f(p(s(0)))) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Strategy
Function Precedence
p < f < active = active# < 0 = s = mark = mark# = cons
Argument Filtering
f: collapses to 1
0: all arguments are removed from 0
s: 1
p: collapses to 1
active: collapses to 1
mark: collapses to 1
active#: collapses to 1
mark#: collapses to 1
cons: collapses to 1
Status
0: multiset
s: lexicographic with permutation 1 → 1
Usable Rules
| cons(active(X1), X2) → cons(X1, X2) | mark(s(X)) → active(s(mark(X))) |
| active(f(0)) → mark(cons(0, f(s(0)))) | active(f(s(0))) → mark(f(p(s(0)))) |
| cons(X1, mark(X2)) → cons(X1, X2) | mark(p(X)) → active(p(mark(X))) |
| active(p(s(X))) → mark(X) | p(mark(X)) → p(X) |
| cons(mark(X1), X2) → cons(X1, X2) | mark(cons(X1, X2)) → active(cons(mark(X1), X2)) |
| s(mark(X)) → s(X) | f(active(X)) → f(X) |
| p(active(X)) → p(X) | mark(f(X)) → active(f(mark(X))) |
| f(mark(X)) → f(X) | mark(0) → active(0) |
| s(active(X)) → s(X) | cons(X1, active(X2)) → cons(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.
| active#(p(s(X))) → mark#(X) |
Problem 11: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(p(X)) | → | mark#(X) |
| active#(f(0)) | → | mark#(cons(0, f(s(0)))) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(f(X)) | → | mark#(X) |
| mark#(p(X)) | → | active#(p(mark(X))) | | active#(f(s(0))) | → | mark#(f(p(s(0)))) |
| mark#(f(X)) | → | active#(f(mark(X))) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Strategy
Function Precedence
active# < f = s = mark# < cons < 0 = p = active = mark
Argument Filtering
f: all arguments are removed from f
0: all arguments are removed from 0
s: all arguments are removed from s
p: all arguments are removed from p
active: all arguments are removed from active
mark: all arguments are removed from mark
active#: collapses to 1
mark#: all arguments are removed from mark#
cons: all arguments are removed from cons
Status
f: multiset
0: multiset
s: multiset
p: multiset
active: multiset
mark: multiset
mark#: multiset
cons: multiset
Usable Rules
| cons(active(X1), X2) → cons(X1, X2) | mark(s(X)) → active(s(mark(X))) |
| active(f(0)) → mark(cons(0, f(s(0)))) | active(f(s(0))) → mark(f(p(s(0)))) |
| cons(X1, mark(X2)) → cons(X1, X2) | mark(p(X)) → active(p(mark(X))) |
| active(p(s(X))) → mark(X) | p(mark(X)) → p(X) |
| cons(mark(X1), X2) → cons(X1, X2) | mark(cons(X1, X2)) → active(cons(mark(X1), X2)) |
| s(mark(X)) → s(X) | f(active(X)) → f(X) |
| p(active(X)) → p(X) | mark(f(X)) → active(f(mark(X))) |
| f(mark(X)) → f(X) | mark(0) → active(0) |
| s(active(X)) → s(X) | cons(X1, active(X2)) → cons(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.
| mark#(p(X)) → active#(p(mark(X))) |
Problem 12: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| mark#(s(X)) | → | active#(s(mark(X))) | | active#(f(0)) | → | mark#(cons(0, f(s(0)))) |
| mark#(p(X)) | → | mark#(X) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(s(X)) | → | mark#(X) | | mark#(f(X)) | → | mark#(X) |
| mark#(f(X)) | → | active#(f(mark(X))) | | active#(f(s(0))) | → | mark#(f(p(s(0)))) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Strategy
Function Precedence
active = mark < f = mark# < 0 = s = p = active# = cons
Argument Filtering
f: all arguments are removed from f
0: all arguments are removed from 0
s: all arguments are removed from s
p: all arguments are removed from p
active: all arguments are removed from active
mark: all arguments are removed from mark
active#: collapses to 1
mark#: all arguments are removed from mark#
cons: all arguments are removed from cons
Status
f: multiset
0: multiset
s: multiset
p: multiset
active: multiset
mark: multiset
mark#: multiset
cons: multiset
Usable Rules
| cons(active(X1), X2) → cons(X1, X2) | mark(s(X)) → active(s(mark(X))) |
| active(f(0)) → mark(cons(0, f(s(0)))) | active(f(s(0))) → mark(f(p(s(0)))) |
| cons(X1, mark(X2)) → cons(X1, X2) | mark(p(X)) → active(p(mark(X))) |
| active(p(s(X))) → mark(X) | p(mark(X)) → p(X) |
| cons(mark(X1), X2) → cons(X1, X2) | mark(cons(X1, X2)) → active(cons(mark(X1), X2)) |
| s(mark(X)) → s(X) | f(active(X)) → f(X) |
| p(active(X)) → p(X) | mark(f(X)) → active(f(mark(X))) |
| f(mark(X)) → f(X) | mark(0) → active(0) |
| s(active(X)) → s(X) | cons(X1, active(X2)) → cons(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.
| mark#(s(X)) → active#(s(mark(X))) |
Problem 13: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| mark#(p(X)) | → | mark#(X) | | active#(f(0)) | → | mark#(cons(0, f(s(0)))) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | mark#(s(X)) | → | mark#(X) |
| mark#(f(X)) | → | mark#(X) | | active#(f(s(0))) | → | mark#(f(p(s(0)))) |
| mark#(f(X)) | → | active#(f(mark(X))) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Strategy
Function Precedence
p = active < active# < f < mark < 0 = s = mark# = cons
Argument Filtering
f: collapses to 1
0: all arguments are removed from 0
s: 1
p: collapses to 1
active: collapses to 1
mark: collapses to 1
active#: collapses to 1
mark#: collapses to 1
cons: collapses to 1
Status
0: multiset
s: lexicographic with permutation 1 → 1
Usable Rules
| cons(active(X1), X2) → cons(X1, X2) | mark(s(X)) → active(s(mark(X))) |
| active(f(0)) → mark(cons(0, f(s(0)))) | active(f(s(0))) → mark(f(p(s(0)))) |
| cons(X1, mark(X2)) → cons(X1, X2) | mark(p(X)) → active(p(mark(X))) |
| active(p(s(X))) → mark(X) | p(mark(X)) → p(X) |
| cons(mark(X1), X2) → cons(X1, X2) | mark(cons(X1, X2)) → active(cons(mark(X1), X2)) |
| s(mark(X)) → s(X) | f(active(X)) → f(X) |
| p(active(X)) → p(X) | mark(f(X)) → active(f(mark(X))) |
| f(mark(X)) → f(X) | mark(0) → active(0) |
| s(active(X)) → s(X) | cons(X1, active(X2)) → cons(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.
Problem 14: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| active#(f(0)) | → | mark#(cons(0, f(s(0)))) | | mark#(p(X)) | → | mark#(X) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | mark#(f(X)) | → | mark#(X) |
| mark#(f(X)) | → | active#(f(mark(X))) | | active#(f(s(0))) | → | mark#(f(p(s(0)))) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Strategy
Function Precedence
active < 0 < active# < p < s < mark < f = mark# = cons
Argument Filtering
f: 1
0: all arguments are removed from 0
s: collapses to 1
p: collapses to 1
active: collapses to 1
mark: collapses to 1
active#: collapses to 1
mark#: collapses to 1
cons: collapses to 1
Status
f: lexicographic with permutation 1 → 1
0: multiset
Usable Rules
| cons(active(X1), X2) → cons(X1, X2) | mark(s(X)) → active(s(mark(X))) |
| active(f(0)) → mark(cons(0, f(s(0)))) | active(f(s(0))) → mark(f(p(s(0)))) |
| cons(X1, mark(X2)) → cons(X1, X2) | mark(p(X)) → active(p(mark(X))) |
| active(p(s(X))) → mark(X) | p(mark(X)) → p(X) |
| cons(mark(X1), X2) → cons(X1, X2) | mark(cons(X1, X2)) → active(cons(mark(X1), X2)) |
| s(mark(X)) → s(X) | f(active(X)) → f(X) |
| p(active(X)) → p(X) | mark(f(X)) → active(f(mark(X))) |
| f(mark(X)) → f(X) | mark(0) → active(0) |
| s(active(X)) → s(X) | cons(X1, active(X2)) → cons(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.
Problem 15: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| mark#(p(X)) | → | mark#(X) | | active#(f(0)) | → | mark#(cons(0, f(s(0)))) |
| mark#(cons(X1, X2)) | → | mark#(X1) | | active#(f(s(0))) | → | mark#(f(p(s(0)))) |
| mark#(f(X)) | → | active#(f(mark(X))) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Strategy
Function Precedence
s < p < active < f = 0 = mark = active# = mark# = cons
Argument Filtering
f: all arguments are removed from f
0: all arguments are removed from 0
s: collapses to 1
p: collapses to 1
active: collapses to 1
mark: collapses to 1
active#: collapses to 1
mark#: collapses to 1
cons: collapses to 1
Status
f: multiset
0: multiset
Usable Rules
| cons(active(X1), X2) → cons(X1, X2) | mark(s(X)) → active(s(mark(X))) |
| active(f(0)) → mark(cons(0, f(s(0)))) | active(f(s(0))) → mark(f(p(s(0)))) |
| cons(X1, mark(X2)) → cons(X1, X2) | mark(p(X)) → active(p(mark(X))) |
| active(p(s(X))) → mark(X) | p(mark(X)) → p(X) |
| cons(mark(X1), X2) → cons(X1, X2) | mark(cons(X1, X2)) → active(cons(mark(X1), X2)) |
| s(mark(X)) → s(X) | f(active(X)) → f(X) |
| p(active(X)) → p(X) | mark(f(X)) → active(f(mark(X))) |
| f(mark(X)) → f(X) | mark(0) → active(0) |
| s(active(X)) → s(X) | cons(X1, active(X2)) → cons(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.
| active#(f(0)) → mark#(cons(0, f(s(0)))) |
Problem 16: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| mark#(p(X)) | → | mark#(X) | | mark#(cons(X1, X2)) | → | mark#(X1) |
| mark#(f(X)) | → | active#(f(mark(X))) | | active#(f(s(0))) | → | mark#(f(p(s(0)))) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Strategy
Function Precedence
active < mark < s < active# < mark# < f = 0 = p = cons
Argument Filtering
f: 1
0: all arguments are removed from 0
s: collapses to 1
p: collapses to 1
active: collapses to 1
mark: collapses to 1
active#: collapses to 1
mark#: collapses to 1
cons: 1
Status
f: lexicographic with permutation 1 → 1
0: multiset
cons: lexicographic with permutation 1 → 1
Usable Rules
| cons(active(X1), X2) → cons(X1, X2) | mark(s(X)) → active(s(mark(X))) |
| active(f(0)) → mark(cons(0, f(s(0)))) | active(f(s(0))) → mark(f(p(s(0)))) |
| cons(X1, mark(X2)) → cons(X1, X2) | mark(p(X)) → active(p(mark(X))) |
| active(p(s(X))) → mark(X) | p(mark(X)) → p(X) |
| cons(mark(X1), X2) → cons(X1, X2) | mark(cons(X1, X2)) → active(cons(mark(X1), X2)) |
| s(mark(X)) → s(X) | f(active(X)) → f(X) |
| mark(f(X)) → active(f(mark(X))) | f(mark(X)) → f(X) |
| p(active(X)) → p(X) | mark(0) → active(0) |
| s(active(X)) → s(X) | cons(X1, active(X2)) → cons(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.
| mark#(cons(X1, X2)) → mark#(X1) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| cons#(X1, active(X2)) | → | cons#(X1, X2) | | cons#(mark(X1), X2) | → | cons#(X1, X2) |
| cons#(X1, mark(X2)) | → | cons#(X1, X2) | | cons#(active(X1), X2) | → | cons#(X1, X2) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| cons#(mark(X1), X2) | → | cons#(X1, X2) | | cons#(active(X1), X2) | → | cons#(X1, X2) |
Problem 7: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| cons#(X1, active(X2)) | → | cons#(X1, X2) | | cons#(X1, mark(X2)) | → | cons#(X1, X2) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Strategy
Function Precedence
mark < active < f = cons# = 0 = s = p = cons
Argument Filtering
f: all arguments are removed from f
cons#: collapses to 2
0: all arguments are removed from 0
s: all arguments are removed from s
p: all arguments are removed from p
active: collapses to 1
mark: 1
cons: 1 2
Status
f: multiset
0: multiset
s: multiset
p: multiset
mark: multiset
cons: lexicographic with permutation 1 → 1 2 → 2
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.
| cons#(X1, mark(X2)) → cons#(X1, X2) |
Problem 9: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| cons#(X1, active(X2)) | → | cons#(X1, X2) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Strategy
Function Precedence
active < f = cons# = 0 = s = p = mark = cons
Argument Filtering
f: all arguments are removed from f
cons#: collapses to 2
0: all arguments are removed from 0
s: all arguments are removed from s
p: all arguments are removed from p
active: 1
mark: all arguments are removed from mark
cons: 2
Status
f: multiset
0: multiset
s: multiset
p: multiset
active: multiset
mark: multiset
cons: 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.
| cons#(X1, active(X2)) → cons#(X1, X2) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(active(X)) | → | s#(X) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| s#(mark(X)) | → | s#(X) | | s#(active(X)) | → | s#(X) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| f#(active(X)) | → | f#(X) | | f#(mark(X)) | → | f#(X) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| f#(active(X)) | → | f#(X) | | f#(mark(X)) | → | f#(X) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| p#(mark(X)) | → | p#(X) | | p#(active(X)) | → | p#(X) |
Rewrite Rules
| active(f(0)) | → | mark(cons(0, f(s(0)))) | | active(f(s(0))) | → | mark(f(p(s(0)))) |
| active(p(s(X))) | → | mark(X) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(0) | → | active(0) | | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) |
| mark(s(X)) | → | active(s(mark(X))) | | mark(p(X)) | → | active(p(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| cons(mark(X1), X2) | → | cons(X1, X2) | | cons(X1, mark(X2)) | → | cons(X1, X2) |
| cons(active(X1), X2) | → | cons(X1, X2) | | cons(X1, active(X2)) | → | cons(X1, X2) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, p, active, mark, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| p#(mark(X)) | → | p#(X) | | p#(active(X)) | → | p#(X) |