TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60000 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (175ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (0ms).
| – Problem 4 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (28ms), PolynomialLinearRange4iUR (681ms), DependencyGraph (27ms), PolynomialLinearRange8NegiUR (4321ms), DependencyGraph (21ms), ReductionPairSAT (516ms), DependencyGraph (21ms), SizeChangePrinciple (timeout)].
The following open problems remain:
Open Dependency Pair Problem 4
Dependency Pairs
| activate#(n__fact(X)) | → | activate#(X) | | activate#(n__prod(X1, X2)) | → | activate#(X2) |
| fact#(X) | → | if#(zero(X), n__s(n__0), n__prod(X, n__fact(n__p(X)))) | | if#(true, X, Y) | → | activate#(X) |
| if#(false, X, Y) | → | activate#(Y) | | activate#(n__s(X)) | → | activate#(X) |
| activate#(n__fact(X)) | → | fact#(activate(X)) | | activate#(n__p(X)) | → | activate#(X) |
| activate#(n__prod(X1, X2)) | → | activate#(X1) |
Rewrite Rules
| fact(X) | → | if(zero(X), n__s(n__0), n__prod(X, n__fact(n__p(X)))) | | add(0, X) | → | X |
| add(s(X), Y) | → | s(add(X, Y)) | | prod(0, X) | → | 0 |
| prod(s(X), Y) | → | add(Y, prod(X, Y)) | | if(true, X, Y) | → | activate(X) |
| if(false, X, Y) | → | activate(Y) | | zero(0) | → | true |
| zero(s(X)) | → | false | | p(s(X)) | → | X |
| s(X) | → | n__s(X) | | 0 | → | n__0 |
| prod(X1, X2) | → | n__prod(X1, X2) | | fact(X) | → | n__fact(X) |
| p(X) | → | n__p(X) | | activate(n__s(X)) | → | s(activate(X)) |
| activate(n__0) | → | 0 | | activate(n__prod(X1, X2)) | → | prod(activate(X1), activate(X2)) |
| activate(n__fact(X)) | → | fact(activate(X)) | | activate(n__p(X)) | → | p(activate(X)) |
| activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: true, zero, add, n__s, activate, n__prod, n__0, fact, 0, s, if, p, n__p, false, n__fact, prod
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| activate#(n__fact(X)) | → | activate#(X) | | activate#(n__prod(X1, X2)) | → | prod#(activate(X1), activate(X2)) |
| activate#(n__prod(X1, X2)) | → | activate#(X2) | | fact#(X) | → | if#(zero(X), n__s(n__0), n__prod(X, n__fact(n__p(X)))) |
| activate#(n__s(X)) | → | activate#(X) | | activate#(n__fact(X)) | → | fact#(activate(X)) |
| add#(s(X), Y) | → | s#(add(X, Y)) | | activate#(n__p(X)) | → | p#(activate(X)) |
| add#(s(X), Y) | → | add#(X, Y) | | prod#(0, X) | → | 0# |
| prod#(s(X), Y) | → | prod#(X, Y) | | prod#(s(X), Y) | → | add#(Y, prod(X, Y)) |
| if#(false, X, Y) | → | activate#(Y) | | if#(true, X, Y) | → | activate#(X) |
| fact#(X) | → | zero#(X) | | activate#(n__0) | → | 0# |
| activate#(n__p(X)) | → | activate#(X) | | activate#(n__s(X)) | → | s#(activate(X)) |
| activate#(n__prod(X1, X2)) | → | activate#(X1) |
Rewrite Rules
| fact(X) | → | if(zero(X), n__s(n__0), n__prod(X, n__fact(n__p(X)))) | | add(0, X) | → | X |
| add(s(X), Y) | → | s(add(X, Y)) | | prod(0, X) | → | 0 |
| prod(s(X), Y) | → | add(Y, prod(X, Y)) | | if(true, X, Y) | → | activate(X) |
| if(false, X, Y) | → | activate(Y) | | zero(0) | → | true |
| zero(s(X)) | → | false | | p(s(X)) | → | X |
| s(X) | → | n__s(X) | | 0 | → | n__0 |
| prod(X1, X2) | → | n__prod(X1, X2) | | fact(X) | → | n__fact(X) |
| p(X) | → | n__p(X) | | activate(n__s(X)) | → | s(activate(X)) |
| activate(n__0) | → | 0 | | activate(n__prod(X1, X2)) | → | prod(activate(X1), activate(X2)) |
| activate(n__fact(X)) | → | fact(activate(X)) | | activate(n__p(X)) | → | p(activate(X)) |
| activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: true, zero, add, n__s, activate, n__prod, n__0, fact, 0, s, if, p, n__p, false, n__fact, prod
Strategy
The following SCCs where found
| add#(s(X), Y) → add#(X, Y) |
| prod#(s(X), Y) → prod#(X, Y) |
| activate#(n__fact(X)) → activate#(X) | activate#(n__prod(X1, X2)) → activate#(X2) |
| fact#(X) → if#(zero(X), n__s(n__0), n__prod(X, n__fact(n__p(X)))) | activate#(n__s(X)) → activate#(X) |
| if#(false, X, Y) → activate#(Y) | if#(true, X, Y) → activate#(X) |
| activate#(n__p(X)) → activate#(X) | activate#(n__fact(X)) → fact#(activate(X)) |
| activate#(n__prod(X1, X2)) → activate#(X1) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| prod#(s(X), Y) | → | prod#(X, Y) |
Rewrite Rules
| fact(X) | → | if(zero(X), n__s(n__0), n__prod(X, n__fact(n__p(X)))) | | add(0, X) | → | X |
| add(s(X), Y) | → | s(add(X, Y)) | | prod(0, X) | → | 0 |
| prod(s(X), Y) | → | add(Y, prod(X, Y)) | | if(true, X, Y) | → | activate(X) |
| if(false, X, Y) | → | activate(Y) | | zero(0) | → | true |
| zero(s(X)) | → | false | | p(s(X)) | → | X |
| s(X) | → | n__s(X) | | 0 | → | n__0 |
| prod(X1, X2) | → | n__prod(X1, X2) | | fact(X) | → | n__fact(X) |
| p(X) | → | n__p(X) | | activate(n__s(X)) | → | s(activate(X)) |
| activate(n__0) | → | 0 | | activate(n__prod(X1, X2)) | → | prod(activate(X1), activate(X2)) |
| activate(n__fact(X)) | → | fact(activate(X)) | | activate(n__p(X)) | → | p(activate(X)) |
| activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: true, zero, add, n__s, activate, n__prod, n__0, fact, 0, s, if, p, n__p, false, n__fact, prod
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| prod#(s(X), Y) | → | prod#(X, Y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| add#(s(X), Y) | → | add#(X, Y) |
Rewrite Rules
| fact(X) | → | if(zero(X), n__s(n__0), n__prod(X, n__fact(n__p(X)))) | | add(0, X) | → | X |
| add(s(X), Y) | → | s(add(X, Y)) | | prod(0, X) | → | 0 |
| prod(s(X), Y) | → | add(Y, prod(X, Y)) | | if(true, X, Y) | → | activate(X) |
| if(false, X, Y) | → | activate(Y) | | zero(0) | → | true |
| zero(s(X)) | → | false | | p(s(X)) | → | X |
| s(X) | → | n__s(X) | | 0 | → | n__0 |
| prod(X1, X2) | → | n__prod(X1, X2) | | fact(X) | → | n__fact(X) |
| p(X) | → | n__p(X) | | activate(n__s(X)) | → | s(activate(X)) |
| activate(n__0) | → | 0 | | activate(n__prod(X1, X2)) | → | prod(activate(X1), activate(X2)) |
| activate(n__fact(X)) | → | fact(activate(X)) | | activate(n__p(X)) | → | p(activate(X)) |
| activate(X) | → | X |
Original Signature
Termination of terms over the following signature is verified: true, zero, add, n__s, activate, n__prod, n__0, fact, 0, s, if, p, n__p, false, n__fact, prod
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| add#(s(X), Y) | → | add#(X, Y) |