TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60010 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (3791ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| | – Problem 10 was processed with processor ReductionPairSAT (111ms).
| | | – Problem 14 was processed with processor ReductionPairSAT (41ms).
| – Problem 3 was processed with processor SubtermCriterion (0ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| | – Problem 11 was processed with processor ReductionPairSAT (429ms).
| | | – Problem 15 was processed with processor ReductionPairSAT (237ms).
| | | | – Problem 18 was processed with processor ReductionPairSAT (108ms).
| – Problem 5 was processed with processor SubtermCriterion (3ms).
| | – Problem 12 was processed with processor ReductionPairSAT (105ms).
| | | – Problem 16 was processed with processor ReductionPairSAT (26ms).
| – 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 ReductionPairSAT (8402ms).
| | – Problem 13 was processed with processor ReductionPairSAT (4475ms).
| | | – Problem 17 remains open; application of the following processors failed [DependencyGraph (411ms), ReductionPairSAT (16140ms), DependencyGraph (428ms), ReductionPairSAT (12629ms)].
The following open problems remain:
Open Dependency Pair Problem 17
Dependency Pairs
| mark#(if(X1, X2, X3)) | → | active#(if(mark(X1), X2, X3)) | | mark#(zero(X)) | → | mark#(X) |
| active#(if(false, X, Y)) | → | mark#(Y) | | active#(prod(0, X)) | → | mark#(0) |
| active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) | | mark#(prod(X1, X2)) | → | mark#(X2) |
| mark#(fact(X)) | → | mark#(X) | | active#(if(true, X, Y)) | → | mark#(X) |
| mark#(prod(X1, X2)) | → | mark#(X1) | | mark#(zero(X)) | → | active#(zero(mark(X))) |
| mark#(fact(X)) | → | active#(fact(mark(X))) | | mark#(add(X1, X2)) | → | mark#(X2) |
| mark#(prod(X1, X2)) | → | active#(prod(mark(X1), mark(X2))) | | mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) |
| mark#(s(X)) | → | mark#(X) | | mark#(add(X1, X2)) | → | mark#(X1) |
| active#(prod(s(X), Y)) | → | mark#(add(Y, prod(X, Y))) | | active#(fact(X)) | → | mark#(if(zero(X), s(0), prod(X, fact(p(X))))) |
| active#(add(0, X)) | → | mark#(X) | | mark#(p(X)) | → | active#(p(mark(X))) |
| mark#(p(X)) | → | mark#(X) | | active#(zero(s(X))) | → | mark#(false) |
| active#(p(s(X))) | → | mark#(X) | | mark#(if(X1, X2, X3)) | → | mark#(X1) |
| active#(zero(0)) | → | mark#(true) |
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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: fact, 0, s, if, p, false, true, active, mark, add, zero, prod
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(prod(s(X), Y)) | → | prod#(X, Y) | | mark#(zero(X)) | → | mark#(X) |
| fact#(active(X)) | → | fact#(X) | | mark#(fact(X)) | → | fact#(mark(X)) |
| active#(fact(X)) | → | if#(zero(X), s(0), prod(X, fact(p(X)))) | | active#(prod(0, X)) | → | mark#(0) |
| add#(X1, mark(X2)) | → | add#(X1, X2) | | active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) |
| active#(fact(X)) | → | s#(0) | | if#(X1, X2, active(X3)) | → | if#(X1, X2, X3) |
| mark#(fact(X)) | → | mark#(X) | | mark#(s(X)) | → | s#(mark(X)) |
| active#(if(true, X, Y)) | → | mark#(X) | | prod#(mark(X1), X2) | → | prod#(X1, X2) |
| mark#(add(X1, X2)) | → | mark#(X2) | | prod#(X1, active(X2)) | → | prod#(X1, X2) |
| mark#(prod(X1, X2)) | → | active#(prod(mark(X1), mark(X2))) | | mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) |
| mark#(s(X)) | → | mark#(X) | | mark#(add(X1, X2)) | → | mark#(X1) |
| active#(fact(X)) | → | p#(X) | | active#(fact(X)) | → | prod#(X, fact(p(X))) |
| mark#(true) | → | active#(true) | | active#(prod(s(X), Y)) | → | mark#(add(Y, prod(X, Y))) |
| prod#(X1, mark(X2)) | → | prod#(X1, X2) | | active#(fact(X)) | → | mark#(if(zero(X), s(0), prod(X, fact(p(X))))) |
| active#(add(0, X)) | → | mark#(X) | | add#(mark(X1), X2) | → | add#(X1, X2) |
| mark#(p(X)) | → | active#(p(mark(X))) | | active#(add(s(X), Y)) | → | add#(X, Y) |
| if#(X1, mark(X2), X3) | → | if#(X1, X2, X3) | | mark#(add(X1, X2)) | → | add#(mark(X1), mark(X2)) |
| mark#(p(X)) | → | mark#(X) | | active#(zero(s(X))) | → | mark#(false) |
| if#(X1, X2, mark(X3)) | → | if#(X1, X2, X3) | | add#(active(X1), X2) | → | add#(X1, X2) |
| if#(mark(X1), X2, X3) | → | if#(X1, X2, X3) | | mark#(if(X1, X2, X3)) | → | mark#(X1) |
| active#(fact(X)) | → | fact#(p(X)) | | mark#(false) | → | active#(false) |
| mark#(if(X1, X2, X3)) | → | active#(if(mark(X1), X2, X3)) | | active#(if(false, X, Y)) | → | mark#(Y) |
| mark#(zero(X)) | → | zero#(mark(X)) | | if#(X1, active(X2), X3) | → | if#(X1, X2, X3) |
| mark#(prod(X1, X2)) | → | mark#(X2) | | p#(mark(X)) | → | p#(X) |
| mark#(prod(X1, X2)) | → | mark#(X1) | | mark#(zero(X)) | → | active#(zero(mark(X))) |
| active#(add(s(X), Y)) | → | s#(add(X, Y)) | | mark#(fact(X)) | → | active#(fact(mark(X))) |
| zero#(mark(X)) | → | zero#(X) | | add#(X1, active(X2)) | → | add#(X1, X2) |
| zero#(active(X)) | → | zero#(X) | | active#(prod(s(X), Y)) | → | add#(Y, prod(X, Y)) |
| mark#(prod(X1, X2)) | → | prod#(mark(X1), mark(X2)) | | mark#(if(X1, X2, X3)) | → | if#(mark(X1), X2, X3) |
| mark#(0) | → | active#(0) | | mark#(s(X)) | → | active#(s(mark(X))) |
| if#(active(X1), X2, X3) | → | if#(X1, X2, X3) | | s#(mark(X)) | → | s#(X) |
| mark#(p(X)) | → | p#(mark(X)) | | active#(p(s(X))) | → | mark#(X) |
| prod#(active(X1), X2) | → | prod#(X1, X2) | | s#(active(X)) | → | s#(X) |
| fact#(mark(X)) | → | fact#(X) | | active#(fact(X)) | → | zero#(X) |
| active#(zero(0)) | → | mark#(true) | | p#(active(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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, if, p, active, true, false, mark, zero, add, prod
Strategy
The following SCCs where found
| mark#(false) → active#(false) | mark#(zero(X)) → mark#(X) |
| mark#(if(X1, X2, X3)) → active#(if(mark(X1), X2, X3)) | active#(if(false, X, Y)) → mark#(Y) |
| active#(prod(0, X)) → mark#(0) | active#(add(s(X), Y)) → mark#(s(add(X, Y))) |
| mark#(prod(X1, X2)) → mark#(X2) | mark#(fact(X)) → mark#(X) |
| mark#(prod(X1, X2)) → mark#(X1) | active#(if(true, X, Y)) → mark#(X) |
| mark#(zero(X)) → active#(zero(mark(X))) | mark#(fact(X)) → active#(fact(mark(X))) |
| mark#(add(X1, X2)) → mark#(X2) | mark#(prod(X1, X2)) → active#(prod(mark(X1), mark(X2))) |
| mark#(s(X)) → mark#(X) | mark#(add(X1, X2)) → active#(add(mark(X1), mark(X2))) |
| mark#(add(X1, X2)) → mark#(X1) | mark#(0) → active#(0) |
| mark#(s(X)) → active#(s(mark(X))) | active#(prod(s(X), Y)) → mark#(add(Y, prod(X, Y))) |
| mark#(true) → active#(true) | active#(fact(X)) → mark#(if(zero(X), s(0), prod(X, fact(p(X))))) |
| active#(add(0, X)) → mark#(X) | mark#(p(X)) → active#(p(mark(X))) |
| mark#(p(X)) → mark#(X) | active#(zero(s(X))) → mark#(false) |
| active#(p(s(X))) → mark#(X) | mark#(if(X1, X2, X3)) → mark#(X1) |
| active#(zero(0)) → mark#(true) |
| p#(mark(X)) → p#(X) | p#(active(X)) → p#(X) |
| prod#(X1, mark(X2)) → prod#(X1, X2) | prod#(mark(X1), X2) → prod#(X1, X2) |
| prod#(active(X1), X2) → prod#(X1, X2) | prod#(X1, active(X2)) → prod#(X1, X2) |
| s#(mark(X)) → s#(X) | s#(active(X)) → s#(X) |
| if#(X1, mark(X2), X3) → if#(X1, X2, X3) | if#(X1, X2, mark(X3)) → if#(X1, X2, X3) |
| if#(mark(X1), X2, X3) → if#(X1, X2, X3) | if#(X1, active(X2), X3) → if#(X1, X2, X3) |
| if#(X1, X2, active(X3)) → if#(X1, X2, X3) | if#(active(X1), X2, X3) → if#(X1, X2, X3) |
| add#(active(X1), X2) → add#(X1, X2) | add#(X1, mark(X2)) → add#(X1, X2) |
| add#(X1, active(X2)) → add#(X1, X2) | add#(mark(X1), X2) → add#(X1, X2) |
| zero#(mark(X)) → zero#(X) | zero#(active(X)) → zero#(X) |
| fact#(active(X)) → fact#(X) | fact#(mark(X)) → fact#(X) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| add#(active(X1), X2) | → | add#(X1, X2) | | add#(X1, mark(X2)) | → | add#(X1, X2) |
| add#(X1, active(X2)) | → | add#(X1, X2) | | add#(mark(X1), 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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, if, p, active, true, false, mark, zero, add, prod
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| add#(active(X1), X2) | → | add#(X1, X2) | | add#(mark(X1), X2) | → | add#(X1, X2) |
Problem 10: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| add#(X1, mark(X2)) | → | add#(X1, X2) | | add#(X1, active(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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: fact, 0, s, if, p, false, true, active, mark, add, zero, prod
Strategy
Function Precedence
mark = active < true = zero = add = 0 = fact = s = if = p = false = add# = prod
Argument Filtering
true: all arguments are removed from true
mark: collapses to 1
zero: all arguments are removed from zero
add: all arguments are removed from add
0: all arguments are removed from 0
fact: collapses to 1
s: all arguments are removed from s
if: 2 3
p: 1
false: all arguments are removed from false
active: 1
add#: 1 2
prod: 1 2
Status
true: multiset
zero: multiset
add: multiset
0: multiset
s: multiset
if: lexicographic with permutation 2 → 2 3 → 1
p: lexicographic with permutation 1 → 1
false: multiset
active: multiset
add#: lexicographic with permutation 1 → 1 2 → 2
prod: lexicographic with permutation 1 → 2 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.
| add#(X1, active(X2)) → add#(X1, X2) |
Problem 14: 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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, if, p, active, true, false, mark, zero, add, prod
Strategy
Function Precedence
true = mark = zero = add = 0 = fact = s = if = p = false = active = add# = prod
Argument Filtering
true: all arguments are removed from true
mark: 1
zero: all arguments are removed from zero
add: all arguments are removed from add
0: all arguments are removed from 0
fact: collapses to 1
s: all arguments are removed from s
if: collapses to 2
p: collapses to 1
false: all arguments are removed from false
active: collapses to 1
add#: 2
prod: collapses to 1
Status
true: multiset
mark: multiset
zero: multiset
add: multiset
0: multiset
s: multiset
false: multiset
add#: 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.
| add#(X1, mark(X2)) → add#(X1, X2) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| fact#(active(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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, if, p, active, true, false, mark, zero, add, prod
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| fact#(active(X)) | → | fact#(X) | | fact#(mark(X)) | → | fact#(X) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| if#(X1, mark(X2), X3) | → | if#(X1, X2, X3) | | if#(X1, X2, mark(X3)) | → | if#(X1, X2, X3) |
| if#(mark(X1), X2, X3) | → | if#(X1, X2, X3) | | if#(X1, active(X2), X3) | → | if#(X1, X2, X3) |
| if#(X1, X2, active(X3)) | → | if#(X1, X2, X3) | | if#(active(X1), X2, 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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, if, p, active, true, false, mark, zero, add, prod
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| if#(mark(X1), X2, X3) | → | if#(X1, X2, X3) | | if#(active(X1), X2, X3) | → | if#(X1, X2, X3) |
Problem 11: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| if#(X1, mark(X2), X3) | → | if#(X1, X2, X3) | | if#(X1, X2, mark(X3)) | → | if#(X1, X2, X3) |
| if#(X1, active(X2), X3) | → | if#(X1, X2, X3) | | if#(X1, X2, active(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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: fact, 0, s, if, p, false, true, active, mark, add, zero, prod
Strategy
Function Precedence
true = mark = zero = add = 0 = fact = s = if = p = if# = false = active = prod
Argument Filtering
true: all arguments are removed from true
mark: 1
zero: all arguments are removed from zero
add: 1 2
0: all arguments are removed from 0
fact: all arguments are removed from fact
s: all arguments are removed from s
if: 1 2 3
p: 1
if#: 3
false: all arguments are removed from false
active: 1
prod: all arguments are removed from prod
Status
true: multiset
mark: multiset
zero: multiset
add: lexicographic with permutation 1 → 2 2 → 1
0: multiset
fact: multiset
s: multiset
if: lexicographic with permutation 1 → 2 2 → 1 3 → 3
p: lexicographic with permutation 1 → 1
if#: lexicographic with permutation 3 → 1
false: multiset
active: multiset
prod: 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.
| if#(X1, X2, mark(X3)) → if#(X1, X2, X3) | if#(X1, X2, active(X3)) → if#(X1, X2, X3) |
Problem 15: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| if#(X1, mark(X2), X3) | → | if#(X1, X2, X3) | | if#(X1, active(X2), 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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, if, p, active, true, false, mark, zero, add, prod
Strategy
Function Precedence
mark = active < true = zero = add = 0 = fact = s = if = p = if# = false = prod
Argument Filtering
true: all arguments are removed from true
mark: collapses to 1
zero: collapses to 1
add: all arguments are removed from add
0: all arguments are removed from 0
fact: 1
s: all arguments are removed from s
if: 1 3
p: all arguments are removed from p
if#: 1 2 3
false: all arguments are removed from false
active: 1
prod: 1 2
Status
true: multiset
add: multiset
0: multiset
fact: lexicographic with permutation 1 → 1
s: multiset
if: lexicographic with permutation 1 → 2 3 → 1
p: multiset
if#: lexicographic with permutation 1 → 2 2 → 3 3 → 1
false: multiset
active: multiset
prod: 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.
| if#(X1, active(X2), X3) → if#(X1, X2, X3) |
Problem 18: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| if#(X1, mark(X2), 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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: fact, 0, s, if, p, false, true, active, mark, add, zero, prod
Strategy
Function Precedence
true = mark = zero = add = 0 = fact = s = if = p = if# = false = active = prod
Argument Filtering
true: all arguments are removed from true
mark: 1
zero: all arguments are removed from zero
add: all arguments are removed from add
0: all arguments are removed from 0
fact: all arguments are removed from fact
s: all arguments are removed from s
if: collapses to 2
p: all arguments are removed from p
if#: 2
false: all arguments are removed from false
active: all arguments are removed from active
prod: 1 2
Status
true: multiset
mark: multiset
zero: multiset
add: multiset
0: multiset
fact: multiset
s: multiset
p: multiset
if#: multiset
false: multiset
active: multiset
prod: 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.
| if#(X1, mark(X2), X3) → if#(X1, X2, X3) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| prod#(X1, mark(X2)) | → | prod#(X1, X2) | | prod#(mark(X1), X2) | → | prod#(X1, X2) |
| prod#(active(X1), X2) | → | prod#(X1, X2) | | prod#(X1, active(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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, if, p, active, true, false, mark, zero, add, prod
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| prod#(mark(X1), X2) | → | prod#(X1, X2) | | prod#(active(X1), X2) | → | prod#(X1, X2) |
Problem 12: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| prod#(X1, mark(X2)) | → | prod#(X1, X2) | | prod#(X1, active(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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: fact, 0, s, if, p, false, true, active, mark, add, zero, prod
Strategy
Function Precedence
active < mark < true = zero = add = prod# = 0 = fact = s = if = p = false = prod
Argument Filtering
true: all arguments are removed from true
mark: collapses to 1
zero: all arguments are removed from zero
add: collapses to 2
prod#: collapses to 2
0: all arguments are removed from 0
fact: all arguments are removed from fact
s: collapses to 1
if: all arguments are removed from if
p: 1
false: all arguments are removed from false
active: 1
prod: 2
Status
true: multiset
zero: multiset
0: multiset
fact: multiset
if: multiset
p: lexicographic with permutation 1 → 1
false: multiset
active: multiset
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, active(X2)) → prod#(X1, X2) |
Problem 16: 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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, if, p, active, true, false, mark, zero, add, prod
Strategy
Function Precedence
prod# < mark < true = zero = add = 0 = fact = s = if = p = false = active = prod
Argument Filtering
true: all arguments are removed from true
mark: 1
zero: all arguments are removed from zero
add: 1 2
prod#: collapses to 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: collapses to 1
false: all arguments are removed from false
active: 1
prod: all arguments are removed from prod
Status
true: multiset
mark: multiset
zero: multiset
add: lexicographic with permutation 1 → 1 2 → 2
0: multiset
fact: multiset
s: multiset
if: multiset
false: multiset
active: lexicographic with permutation 1 → 1
prod: 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.
| prod#(X1, mark(X2)) → prod#(X1, X2) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| p#(mark(X)) | → | p#(X) | | p#(active(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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, if, p, active, true, false, mark, zero, add, prod
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| p#(mark(X)) | → | p#(X) | | p#(active(X)) | → | p#(X) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| zero#(mark(X)) | → | zero#(X) | | zero#(active(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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, if, p, active, true, false, mark, zero, add, prod
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| zero#(mark(X)) | → | zero#(X) | | zero#(active(X)) | → | zero#(X) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| s#(mark(X)) | → | s#(X) | | s#(active(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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, if, p, active, true, false, mark, zero, add, prod
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 9: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| mark#(false) | → | active#(false) | | mark#(zero(X)) | → | mark#(X) |
| mark#(if(X1, X2, X3)) | → | active#(if(mark(X1), X2, X3)) | | active#(if(false, X, Y)) | → | mark#(Y) |
| active#(prod(0, X)) | → | mark#(0) | | active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) |
| mark#(prod(X1, X2)) | → | mark#(X2) | | mark#(fact(X)) | → | mark#(X) |
| mark#(prod(X1, X2)) | → | mark#(X1) | | active#(if(true, X, Y)) | → | mark#(X) |
| mark#(zero(X)) | → | active#(zero(mark(X))) | | mark#(fact(X)) | → | active#(fact(mark(X))) |
| mark#(add(X1, X2)) | → | mark#(X2) | | mark#(prod(X1, X2)) | → | active#(prod(mark(X1), mark(X2))) |
| mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) | | mark#(s(X)) | → | mark#(X) |
| mark#(add(X1, X2)) | → | mark#(X1) | | mark#(0) | → | active#(0) |
| mark#(s(X)) | → | active#(s(mark(X))) | | mark#(true) | → | active#(true) |
| active#(prod(s(X), Y)) | → | mark#(add(Y, prod(X, Y))) | | active#(fact(X)) | → | mark#(if(zero(X), s(0), prod(X, fact(p(X))))) |
| active#(add(0, X)) | → | mark#(X) | | mark#(p(X)) | → | active#(p(mark(X))) |
| mark#(p(X)) | → | mark#(X) | | active#(zero(s(X))) | → | mark#(false) |
| active#(p(s(X))) | → | mark#(X) | | mark#(if(X1, X2, X3)) | → | mark#(X1) |
| active#(zero(0)) | → | mark#(true) |
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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, if, p, active, true, false, mark, zero, add, prod
Strategy
Function Precedence
active# < mark = add = zero = mark# = fact = if = p = active = prod < true = 0 = s = false
Argument Filtering
true: all arguments are removed from true
mark: all arguments are removed from mark
add: all arguments are removed from add
zero: all arguments are removed from zero
mark#: all arguments are removed from mark#
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
active#: collapses to 1
prod: all arguments are removed from prod
Status
true: multiset
mark: multiset
add: multiset
zero: multiset
mark#: multiset
0: multiset
fact: multiset
s: multiset
if: multiset
p: multiset
false: multiset
active: multiset
prod: multiset
Usable Rules
| active(prod(0, X)) → mark(0) | active(prod(s(X), Y)) → mark(add(Y, prod(X, Y))) |
| zero(mark(X)) → zero(X) | prod(active(X1), X2) → prod(X1, X2) |
| mark(add(X1, X2)) → active(add(mark(X1), mark(X2))) | mark(s(X)) → active(s(mark(X))) |
| mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3)) | if(active(X1), X2, X3) → if(X1, X2, X3) |
| add(X1, mark(X2)) → add(X1, X2) | mark(fact(X)) → active(fact(mark(X))) |
| mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2))) | prod(X1, active(X2)) → prod(X1, X2) |
| mark(true) → active(true) | fact(active(X)) → fact(X) |
| mark(p(X)) → active(p(mark(X))) | active(p(s(X))) → mark(X) |
| if(X1, X2, active(X3)) → if(X1, X2, X3) | add(X1, active(X2)) → add(X1, X2) |
| if(X1, X2, mark(X3)) → if(X1, X2, X3) | prod(X1, mark(X2)) → prod(X1, X2) |
| add(mark(X1), X2) → add(X1, X2) | mark(0) → active(0) |
| s(active(X)) → s(X) | fact(mark(X)) → fact(X) |
| active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X))))) | if(X1, active(X2), X3) → if(X1, X2, X3) |
| if(mark(X1), X2, X3) → if(X1, X2, X3) | add(active(X1), X2) → add(X1, X2) |
| prod(mark(X1), X2) → prod(X1, X2) | active(add(s(X), Y)) → mark(s(add(X, Y))) |
| active(if(false, X, Y)) → mark(Y) | zero(active(X)) → zero(X) |
| if(X1, mark(X2), X3) → if(X1, X2, X3) | p(mark(X)) → p(X) |
| mark(false) → active(false) | active(zero(s(X))) → mark(false) |
| active(add(0, X)) → mark(X) | mark(zero(X)) → active(zero(mark(X))) |
| s(mark(X)) → s(X) | active(if(true, X, Y)) → mark(X) |
| p(active(X)) → p(X) | active(zero(0)) → mark(true) |
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#(0) → active#(0) | mark#(false) → active#(false) |
| mark#(s(X)) → active#(s(mark(X))) |
Problem 13: ReductionPairSAT
Dependency Pair Problem
Dependency Pairs
| mark#(if(X1, X2, X3)) | → | active#(if(mark(X1), X2, X3)) | | mark#(zero(X)) | → | mark#(X) |
| active#(if(false, X, Y)) | → | mark#(Y) | | active#(prod(0, X)) | → | mark#(0) |
| active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) | | mark#(prod(X1, X2)) | → | mark#(X2) |
| mark#(fact(X)) | → | mark#(X) | | active#(if(true, X, Y)) | → | mark#(X) |
| mark#(prod(X1, X2)) | → | mark#(X1) | | mark#(zero(X)) | → | active#(zero(mark(X))) |
| mark#(fact(X)) | → | active#(fact(mark(X))) | | mark#(add(X1, X2)) | → | mark#(X2) |
| mark#(prod(X1, X2)) | → | active#(prod(mark(X1), mark(X2))) | | mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) |
| mark#(s(X)) | → | mark#(X) | | mark#(add(X1, X2)) | → | mark#(X1) |
| mark#(true) | → | active#(true) | | active#(prod(s(X), Y)) | → | mark#(add(Y, prod(X, Y))) |
| active#(fact(X)) | → | mark#(if(zero(X), s(0), prod(X, fact(p(X))))) | | active#(add(0, X)) | → | mark#(X) |
| mark#(p(X)) | → | active#(p(mark(X))) | | mark#(p(X)) | → | mark#(X) |
| active#(zero(s(X))) | → | mark#(false) | | active#(p(s(X))) | → | mark#(X) |
| mark#(if(X1, X2, X3)) | → | mark#(X1) | | active#(zero(0)) | → | mark#(true) |
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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: fact, 0, s, if, p, false, true, active, mark, add, zero, prod
Strategy
Function Precedence
0 = s = false < mark = active < true = add = zero = mark# = fact = if = p = active# = prod
Argument Filtering
true: all arguments are removed from true
mark: all arguments are removed from mark
add: all arguments are removed from add
zero: all arguments are removed from zero
mark#: all arguments are removed from mark#
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
active#: collapses to 1
prod: all arguments are removed from prod
Status
true: multiset
mark: multiset
add: multiset
zero: multiset
mark#: multiset
0: multiset
fact: multiset
s: multiset
if: multiset
p: multiset
false: multiset
active: multiset
prod: multiset
Usable Rules
| active(prod(0, X)) → mark(0) | active(prod(s(X), Y)) → mark(add(Y, prod(X, Y))) |
| zero(mark(X)) → zero(X) | prod(active(X1), X2) → prod(X1, X2) |
| mark(add(X1, X2)) → active(add(mark(X1), mark(X2))) | mark(s(X)) → active(s(mark(X))) |
| mark(if(X1, X2, X3)) → active(if(mark(X1), X2, X3)) | if(active(X1), X2, X3) → if(X1, X2, X3) |
| add(X1, mark(X2)) → add(X1, X2) | mark(fact(X)) → active(fact(mark(X))) |
| mark(prod(X1, X2)) → active(prod(mark(X1), mark(X2))) | prod(X1, active(X2)) → prod(X1, X2) |
| mark(true) → active(true) | fact(active(X)) → fact(X) |
| mark(p(X)) → active(p(mark(X))) | active(p(s(X))) → mark(X) |
| if(X1, X2, active(X3)) → if(X1, X2, X3) | add(X1, active(X2)) → add(X1, X2) |
| if(X1, X2, mark(X3)) → if(X1, X2, X3) | prod(X1, mark(X2)) → prod(X1, X2) |
| add(mark(X1), X2) → add(X1, X2) | mark(0) → active(0) |
| s(active(X)) → s(X) | fact(mark(X)) → fact(X) |
| active(fact(X)) → mark(if(zero(X), s(0), prod(X, fact(p(X))))) | if(X1, active(X2), X3) → if(X1, X2, X3) |
| if(mark(X1), X2, X3) → if(X1, X2, X3) | add(active(X1), X2) → add(X1, X2) |
| prod(mark(X1), X2) → prod(X1, X2) | active(add(s(X), Y)) → mark(s(add(X, Y))) |
| active(if(false, X, Y)) → mark(Y) | zero(active(X)) → zero(X) |
| if(X1, mark(X2), X3) → if(X1, X2, X3) | p(mark(X)) → p(X) |
| mark(false) → active(false) | active(zero(s(X))) → mark(false) |
| active(add(0, X)) → mark(X) | mark(zero(X)) → active(zero(mark(X))) |
| s(mark(X)) → s(X) | active(if(true, X, Y)) → mark(X) |
| p(active(X)) → p(X) | active(zero(0)) → mark(true) |
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#(true) → active#(true) |
Problem 17: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| mark#(zero(X)) | → | mark#(X) | | mark#(if(X1, X2, X3)) | → | active#(if(mark(X1), X2, X3)) |
| active#(if(false, X, Y)) | → | mark#(Y) | | active#(prod(0, X)) | → | mark#(0) |
| active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) | | mark#(prod(X1, X2)) | → | mark#(X2) |
| mark#(fact(X)) | → | mark#(X) | | mark#(prod(X1, X2)) | → | mark#(X1) |
| active#(if(true, X, Y)) | → | mark#(X) | | mark#(zero(X)) | → | active#(zero(mark(X))) |
| mark#(fact(X)) | → | active#(fact(mark(X))) | | mark#(add(X1, X2)) | → | mark#(X2) |
| mark#(prod(X1, X2)) | → | active#(prod(mark(X1), mark(X2))) | | mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) |
| mark#(s(X)) | → | mark#(X) | | mark#(add(X1, X2)) | → | mark#(X1) |
| active#(prod(s(X), Y)) | → | mark#(add(Y, prod(X, Y))) | | active#(fact(X)) | → | mark#(if(zero(X), s(0), prod(X, fact(p(X))))) |
| active#(add(0, X)) | → | mark#(X) | | mark#(p(X)) | → | active#(p(mark(X))) |
| mark#(p(X)) | → | mark#(X) | | active#(zero(s(X))) | → | mark#(false) |
| active#(p(s(X))) | → | mark#(X) | | mark#(if(X1, X2, X3)) | → | mark#(X1) |
| active#(zero(0)) | → | mark#(true) |
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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, if, p, active, true, false, mark, zero, add, prod
Strategy
The right-hand side of the rule mark
#(if(
X1,
X2,
X3)) → active
#(if(mark(
X1),
X2,
X3)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
| Relevant Terms | Irrelevant Terms |
|---|
| active#(if(active(s(mark(_x31))), X2, X3)) | |
| active#(if(active(0), X2, X3)) | |
| active#(if(mark(X1), _x22, _x23)) | |
| active#(if(active(zero(mark(_x31))), X2, X3)) | |
| active#(if(active(p(mark(_x31))), X2, X3)) | |
| active#(if(active(add(mark(_x31), mark(_x32))), X2, X3)) | |
| active#(if(active(if(mark(_x31), _x32, _x33)), X2, X3)) | |
| active#(if(active(prod(mark(_x31), mark(_x32))), X2, X3)) | |
| active#(if(_x21, _x22, _x23)) | |
| active#(if(active(true), X2, X3)) | |
| active#(if(active(false), X2, X3)) | |
| active#(if(active(fact(mark(_x31))), X2, X3)) | |
Thus, the rule mark
#(if(
X1,
X2,
X3)) → active
#(if(mark(
X1),
X2,
X3)) is replaced by the following rules:
| mark#(if(p(_x31), X2, X3)) → active#(if(active(p(mark(_x31))), X2, X3)) | mark#(if(0, X2, X3)) → active#(if(active(0), X2, X3)) |
| mark#(if(if(_x31, _x32, _x33), X2, X3)) → active#(if(active(if(mark(_x31), _x32, _x33)), X2, X3)) | mark#(if(zero(_x31), X2, X3)) → active#(if(active(zero(mark(_x31))), X2, X3)) |
| mark#(if(fact(_x31), X2, X3)) → active#(if(active(fact(mark(_x31))), X2, X3)) | mark#(if(X1, _x22, mark(_x23))) → active#(if(mark(X1), _x22, _x23)) |
| mark#(if(s(_x31), X2, X3)) → active#(if(active(s(mark(_x31))), X2, X3)) | mark#(if(prod(_x31, _x32), X2, X3)) → active#(if(active(prod(mark(_x31), mark(_x32))), X2, X3)) |
| mark#(if(X1, mark(_x22), _x23)) → active#(if(mark(X1), _x22, _x23)) | mark#(if(true, X2, X3)) → active#(if(active(true), X2, X3)) |
| mark#(if(add(_x31, _x32), X2, X3)) → active#(if(active(add(mark(_x31), mark(_x32))), X2, X3)) | mark#(if(_x21, _x22, _x23)) → active#(if(_x21, _x22, _x23)) |
| mark#(if(X1, active(_x22), _x23)) → active#(if(mark(X1), _x22, _x23)) | mark#(if(false, X2, X3)) → active#(if(active(false), X2, X3)) |
| mark#(if(X1, _x22, active(_x23))) → active#(if(mark(X1), _x22, _x23)) |
Problem 19: ForwardNarrowing
Dependency Pair Problem
Dependency Pairs
| mark#(if(p(_x31), X2, X3)) | → | active#(if(active(p(mark(_x31))), X2, X3)) | | mark#(zero(X)) | → | mark#(X) |
| active#(if(false, X, Y)) | → | mark#(Y) | | mark#(if(0, X2, X3)) | → | active#(if(active(0), X2, X3)) |
| mark#(if(if(_x31, _x32, _x33), X2, X3)) | → | active#(if(active(if(mark(_x31), _x32, _x33)), X2, X3)) | | active#(prod(0, X)) | → | mark#(0) |
| active#(add(s(X), Y)) | → | mark#(s(add(X, Y))) | | mark#(prod(X1, X2)) | → | mark#(X2) |
| mark#(fact(X)) | → | mark#(X) | | mark#(if(X1, _x22, mark(_x23))) | → | active#(if(mark(X1), _x22, _x23)) |
| active#(if(true, X, Y)) | → | mark#(X) | | mark#(prod(X1, X2)) | → | mark#(X1) |
| mark#(zero(X)) | → | active#(zero(mark(X))) | | mark#(if(s(_x31), X2, X3)) | → | active#(if(active(s(mark(_x31))), X2, X3)) |
| mark#(fact(X)) | → | active#(fact(mark(X))) | | mark#(if(X1, mark(_x22), _x23)) | → | active#(if(mark(X1), _x22, _x23)) |
| mark#(add(X1, X2)) | → | mark#(X2) | | mark#(prod(X1, X2)) | → | active#(prod(mark(X1), mark(X2))) |
| mark#(add(X1, X2)) | → | active#(add(mark(X1), mark(X2))) | | mark#(s(X)) | → | mark#(X) |
| mark#(add(X1, X2)) | → | mark#(X1) | | mark#(if(X1, active(_x22), _x23)) | → | active#(if(mark(X1), _x22, _x23)) |
| active#(prod(s(X), Y)) | → | mark#(add(Y, prod(X, Y))) | | active#(fact(X)) | → | mark#(if(zero(X), s(0), prod(X, fact(p(X))))) |
| active#(add(0, X)) | → | mark#(X) | | mark#(if(zero(_x31), X2, X3)) | → | active#(if(active(zero(mark(_x31))), X2, X3)) |
| mark#(p(X)) | → | active#(p(mark(X))) | | mark#(if(fact(_x31), X2, X3)) | → | active#(if(active(fact(mark(_x31))), X2, X3)) |
| mark#(p(X)) | → | mark#(X) | | mark#(if(prod(_x31, _x32), X2, X3)) | → | active#(if(active(prod(mark(_x31), mark(_x32))), X2, X3)) |
| active#(zero(s(X))) | → | mark#(false) | | active#(p(s(X))) | → | mark#(X) |
| mark#(if(true, X2, X3)) | → | active#(if(active(true), X2, X3)) | | mark#(if(X1, X2, X3)) | → | mark#(X1) |
| mark#(if(add(_x31, _x32), X2, X3)) | → | active#(if(active(add(mark(_x31), mark(_x32))), X2, X3)) | | mark#(if(_x21, _x22, _x23)) | → | active#(if(_x21, _x22, _x23)) |
| active#(zero(0)) | → | mark#(true) | | mark#(if(false, X2, X3)) | → | active#(if(active(false), X2, X3)) |
| mark#(if(X1, _x22, active(_x23))) | → | active#(if(mark(X1), _x22, _x23)) |
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) |
| mark(fact(X)) | → | active(fact(mark(X))) | | mark(if(X1, X2, X3)) | → | active(if(mark(X1), X2, X3)) |
| mark(zero(X)) | → | active(zero(mark(X))) | | mark(s(X)) | → | active(s(mark(X))) |
| mark(0) | → | active(0) | | mark(prod(X1, X2)) | → | active(prod(mark(X1), mark(X2))) |
| mark(p(X)) | → | active(p(mark(X))) | | mark(add(X1, X2)) | → | active(add(mark(X1), mark(X2))) |
| mark(true) | → | active(true) | | mark(false) | → | active(false) |
| fact(mark(X)) | → | fact(X) | | fact(active(X)) | → | fact(X) |
| if(mark(X1), X2, X3) | → | if(X1, X2, X3) | | if(X1, mark(X2), X3) | → | if(X1, X2, X3) |
| if(X1, X2, mark(X3)) | → | if(X1, X2, X3) | | if(active(X1), X2, X3) | → | if(X1, X2, X3) |
| if(X1, active(X2), X3) | → | if(X1, X2, X3) | | if(X1, X2, active(X3)) | → | if(X1, X2, X3) |
| zero(mark(X)) | → | zero(X) | | zero(active(X)) | → | zero(X) |
| s(mark(X)) | → | s(X) | | s(active(X)) | → | s(X) |
| prod(mark(X1), X2) | → | prod(X1, X2) | | prod(X1, mark(X2)) | → | prod(X1, X2) |
| prod(active(X1), X2) | → | prod(X1, X2) | | prod(X1, active(X2)) | → | prod(X1, X2) |
| p(mark(X)) | → | p(X) | | p(active(X)) | → | p(X) |
| add(mark(X1), X2) | → | add(X1, X2) | | add(X1, mark(X2)) | → | add(X1, X2) |
| add(active(X1), X2) | → | add(X1, X2) | | add(X1, active(X2)) | → | add(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: fact, 0, s, if, p, false, true, active, mark, add, zero, prod
Strategy
The right-hand side of the rule mark
#(if(p(
_x31),
X2,
X3)) → active
#(if(active(p(mark(
_x31))),
X2,
X3)) is narrowed to the following relevant and irrelevant terms (a narrowing is irrelevant if by dropping it the correctness (and completeness) of the processor is not influenced).
| Relevant Terms | Irrelevant Terms |
|---|
| active#(if(active(p(active(p(mark(_x61))))), X2, X3)) | |
| active#(if(active(p(active(0))), X2, X3)) | |
| active#(if(active(p(mark(_x31))), _x22, _x23)) | |
| active#(if(active(p(active(s(mark(_x61))))), X2, X3)) | |
| active#(if(active(p(active(fact(mark(_x61))))), X2, X3)) | |
| active#(if(active(p(_x51)), X2, X3)) | |
| active#(if(active(p(active(false))), X2, X3)) | |
| active#(if(active(p(active(prod(mark(_x61), mark(_x62))))), X2, X3)) | |
| active#(if(active(p(active(zero(mark(_x61))))), X2, X3)) | |
| active#(if(active(p(active(add(mark(_x61), mark(_x62))))), X2, X3)) | |
| active#(if(active(p(active(true))), X2, X3)) | |
| active#(if(active(p(active(if(mark(_x61), _x62, _x63)))), X2, X3)) | |
| active#(if(p(mark(_x31)), _x22, _x23)) | |
Thus, the rule mark
#(if(p(
_x31),
X2,
X3)) → active
#(if(active(p(mark(
_x31))),
X2,
X3)) is replaced by the following rules:
| mark#(if(p(false), X2, X3)) → active#(if(active(p(active(false))), X2, X3)) | mark#(if(p(s(_x61)), X2, X3)) → active#(if(active(p(active(s(mark(_x61))))), X2, X3)) |
| mark#(if(p(if(_x61, _x62, _x63)), X2, X3)) → active#(if(active(p(active(if(mark(_x61), _x62, _x63)))), X2, X3)) | mark#(if(p(_x31), active(_x22), _x23)) → active#(if(active(p(mark(_x31))), _x22, _x23)) |
| mark#(if(p(add(_x61, _x62)), X2, X3)) → active#(if(active(p(active(add(mark(_x61), mark(_x62))))), X2, X3)) | mark#(if(p(_x31), _x22, mark(_x23))) → active#(if(active(p(mark(_x31))), _x22, _x23)) |
| mark#(if(p(_x31), _x22, active(_x23))) → active#(if(active(p(mark(_x31))), _x22, _x23)) | mark#(if(p(zero(_x61)), X2, X3)) → active#(if(active(p(active(zero(mark(_x61))))), X2, X3)) |
| mark#(if(p(_x31), mark(_x22), _x23)) → active#(if(active(p(mark(_x31))), _x22, _x23)) | mark#(if(p(prod(_x61, _x62)), X2, X3)) → active#(if(active(p(active(prod(mark(_x61), mark(_x62))))), X2, X3)) |
| mark#(if(p(_x51), X2, X3)) → active#(if(active(p(_x51)), X2, X3)) | mark#(if(p(true), X2, X3)) → active#(if(active(p(active(true))), X2, X3)) |
| mark#(if(p(0), X2, X3)) → active#(if(active(p(active(0))), X2, X3)) | mark#(if(p(_x31), _x22, _x23)) → active#(if(p(mark(_x31)), _x22, _x23)) |
| mark#(if(p(p(_x61)), X2, X3)) → active#(if(active(p(active(p(mark(_x61))))), X2, X3)) | mark#(if(p(fact(_x61)), X2, X3)) → active#(if(active(p(active(fact(mark(_x61))))), X2, X3)) |