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 (52ms).
| – Problem 2 was processed with processor BackwardInstantiation (42ms).
| | – Problem 7 remains open; application of the following processors failed [ForwardInstantiation (2ms), Propagation (0ms), ForwardNarrowing (0ms), BackwardInstantiation (1ms), ForwardInstantiation (0ms), Propagation (0ms)].
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor SubtermCriterion (0ms).
| – Problem 5 was processed with processor SubtermCriterion (1ms).
| – Problem 6 was processed with processor SubtermCriterion (0ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| iffact#(x, true) | → | fact#(-(x, s(0))) | | fact#(x) | → | iffact#(x, ge(x, s(s(0)))) |
Rewrite Rules
| +(x, 0) | → | x | | +(x, s(y)) | → | s(+(x, y)) |
| *(x, 0) | → | 0 | | *(x, s(y)) | → | +(*(x, y), x) |
| ge(x, 0) | → | true | | ge(0, s(y)) | → | false |
| ge(s(x), s(y)) | → | ge(x, y) | | -(x, 0) | → | x |
| -(s(x), s(y)) | → | -(x, y) | | fact(x) | → | iffact(x, ge(x, s(s(0)))) |
| iffact(x, true) | → | *(x, fact(-(x, s(0)))) | | iffact(x, false) | → | s(0) |
Original Signature
Termination of terms over the following signature is verified: fact, 0, s, iffact, *, false, true, +, ge, -
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| *#(x, s(y)) | → | *#(x, y) | | fact#(x) | → | ge#(x, s(s(0))) |
| iffact#(x, true) | → | fact#(-(x, s(0))) | | +#(x, s(y)) | → | +#(x, y) |
| *#(x, s(y)) | → | +#(*(x, y), x) | | iffact#(x, true) | → | -#(x, s(0)) |
| ge#(s(x), s(y)) | → | ge#(x, y) | | fact#(x) | → | iffact#(x, ge(x, s(s(0)))) |
| iffact#(x, true) | → | *#(x, fact(-(x, s(0)))) | | -#(s(x), s(y)) | → | -#(x, y) |
Rewrite Rules
| +(x, 0) | → | x | | +(x, s(y)) | → | s(+(x, y)) |
| *(x, 0) | → | 0 | | *(x, s(y)) | → | +(*(x, y), x) |
| ge(x, 0) | → | true | | ge(0, s(y)) | → | false |
| ge(s(x), s(y)) | → | ge(x, y) | | -(x, 0) | → | x |
| -(s(x), s(y)) | → | -(x, y) | | fact(x) | → | iffact(x, ge(x, s(s(0)))) |
| iffact(x, true) | → | *(x, fact(-(x, s(0)))) | | iffact(x, false) | → | s(0) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, iffact, *, +, true, false, ge, -
Strategy
The following SCCs where found
| iffact#(x, true) → fact#(-(x, s(0))) | fact#(x) → iffact#(x, ge(x, s(s(0)))) |
| ge#(s(x), s(y)) → ge#(x, y) |
| -#(s(x), s(y)) → -#(x, y) |
Problem 2: BackwardInstantiation
Dependency Pair Problem
Dependency Pairs
| iffact#(x, true) | → | fact#(-(x, s(0))) | | fact#(x) | → | iffact#(x, ge(x, s(s(0)))) |
Rewrite Rules
| +(x, 0) | → | x | | +(x, s(y)) | → | s(+(x, y)) |
| *(x, 0) | → | 0 | | *(x, s(y)) | → | +(*(x, y), x) |
| ge(x, 0) | → | true | | ge(0, s(y)) | → | false |
| ge(s(x), s(y)) | → | ge(x, y) | | -(x, 0) | → | x |
| -(s(x), s(y)) | → | -(x, y) | | fact(x) | → | iffact(x, ge(x, s(s(0)))) |
| iffact(x, true) | → | *(x, fact(-(x, s(0)))) | | iffact(x, false) | → | s(0) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, iffact, *, +, true, false, ge, -
Strategy
Instantiation
For all potential predecessors l → r of the rule fact
#(
x) → iffact
#(
x, ge(
x, s(s(0)))) on dependency pair chains it holds that:
- fact#(x) matches r,
- all variables of fact#(x) are embedded in constructor contexts, i.e., each subterm of fact#(x), containing a variable is rooted by a constructor symbol.
Thus, fact
#(
x) → iffact
#(
x, ge(
x, s(s(0)))) is replaced by instances determined through the above matching. These instances are:
| fact#(-(_x, s(0))) → iffact#(-(_x, s(0)), ge(-(_x, s(0)), s(s(0)))) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| ge#(s(x), s(y)) | → | ge#(x, y) |
Rewrite Rules
| +(x, 0) | → | x | | +(x, s(y)) | → | s(+(x, y)) |
| *(x, 0) | → | 0 | | *(x, s(y)) | → | +(*(x, y), x) |
| ge(x, 0) | → | true | | ge(0, s(y)) | → | false |
| ge(s(x), s(y)) | → | ge(x, y) | | -(x, 0) | → | x |
| -(s(x), s(y)) | → | -(x, y) | | fact(x) | → | iffact(x, ge(x, s(s(0)))) |
| iffact(x, true) | → | *(x, fact(-(x, s(0)))) | | iffact(x, false) | → | s(0) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, iffact, *, +, true, false, ge, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| ge#(s(x), s(y)) | → | ge#(x, y) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| +(x, 0) | → | x | | +(x, s(y)) | → | s(+(x, y)) |
| *(x, 0) | → | 0 | | *(x, s(y)) | → | +(*(x, y), x) |
| ge(x, 0) | → | true | | ge(0, s(y)) | → | false |
| ge(s(x), s(y)) | → | ge(x, y) | | -(x, 0) | → | x |
| -(s(x), s(y)) | → | -(x, y) | | fact(x) | → | iffact(x, ge(x, s(s(0)))) |
| iffact(x, true) | → | *(x, fact(-(x, s(0)))) | | iffact(x, false) | → | s(0) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, iffact, *, +, true, false, ge, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| -#(s(x), s(y)) | → | -#(x, y) |
Rewrite Rules
| +(x, 0) | → | x | | +(x, s(y)) | → | s(+(x, y)) |
| *(x, 0) | → | 0 | | *(x, s(y)) | → | +(*(x, y), x) |
| ge(x, 0) | → | true | | ge(0, s(y)) | → | false |
| ge(s(x), s(y)) | → | ge(x, y) | | -(x, 0) | → | x |
| -(s(x), s(y)) | → | -(x, y) | | fact(x) | → | iffact(x, ge(x, s(s(0)))) |
| iffact(x, true) | → | *(x, fact(-(x, s(0)))) | | iffact(x, false) | → | s(0) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, iffact, *, +, true, false, ge, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| -#(s(x), s(y)) | → | -#(x, y) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| +(x, 0) | → | x | | +(x, s(y)) | → | s(+(x, y)) |
| *(x, 0) | → | 0 | | *(x, s(y)) | → | +(*(x, y), x) |
| ge(x, 0) | → | true | | ge(0, s(y)) | → | false |
| ge(s(x), s(y)) | → | ge(x, y) | | -(x, 0) | → | x |
| -(s(x), s(y)) | → | -(x, y) | | fact(x) | → | iffact(x, ge(x, s(s(0)))) |
| iffact(x, true) | → | *(x, fact(-(x, s(0)))) | | iffact(x, false) | → | s(0) |
Original Signature
Termination of terms over the following signature is verified: 0, fact, s, iffact, *, +, true, false, ge, -
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed: