TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60127 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (165ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – 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 (4ms).
| – Problem 7 was processed with processor SubtermCriterion (1ms).
| – Problem 8 was processed with processor SubtermCriterion (0ms).
| – Problem 9 was processed with processor PolynomialLinearRange4iUR (436ms).
| | – Problem 11 was processed with processor BackwardInstantiation (3ms).
| | | – Problem 12 remains open; application of the following processors failed [ForwardInstantiation (4ms), Propagation (1ms), ForwardNarrowing (1ms), BackwardInstantiation (1ms), ForwardInstantiation (2ms), Propagation (0ms)].
| – Problem 10 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 11
Dependency Pairs
| ify#(true, x, y) | → | if#(ge(x, y), x, y) | | div#(x, y) | → | ify#(ge(y, s(0)), x, y) |
| if#(true, x, y) | → | div#(minus(x, y), y) |
Rewrite Rules
| ge(0, 0) | → | true | | ge(s(x), 0) | → | ge(x, 0) |
| ge(0, s(0)) | → | false | | ge(0, s(s(x))) | → | ge(0, s(x)) |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(0, 0) | → | 0 |
| minus(0, s(x)) | → | minus(0, x) | | minus(s(x), 0) | → | s(minus(x, 0)) |
| minus(s(x), s(y)) | → | minus(x, y) | | plus(0, 0) | → | 0 |
| plus(0, s(x)) | → | s(plus(0, x)) | | plus(s(x), y) | → | s(plus(x, y)) |
| div(x, y) | → | ify(ge(y, s(0)), x, y) | | ify(false, x, y) | → | divByZeroError |
| ify(true, x, y) | → | if(ge(x, y), x, y) | | if(false, x, y) | → | 0 |
| if(true, x, y) | → | s(div(minus(x, y), y)) | | div(plus(x, y), z) | → | plus(div(x, z), div(y, z)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, minus, s, ify, if, div, divByZeroError, true, false, ge
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| ify#(true, x, y) | → | ge#(x, y) | | div#(plus(x, y), z) | → | div#(x, z) |
| ge#(s(x), 0) | → | ge#(x, 0) | | minus#(0, s(x)) | → | minus#(0, x) |
| minus#(s(x), 0) | → | minus#(x, 0) | | ify#(true, x, y) | → | if#(ge(x, y), x, y) |
| div#(plus(x, y), z) | → | div#(y, z) | | ge#(0, s(s(x))) | → | ge#(0, s(x)) |
| div#(plus(x, y), z) | → | plus#(div(x, z), div(y, z)) | | div#(x, y) | → | ify#(ge(y, s(0)), x, y) |
| plus#(s(x), y) | → | plus#(x, y) | | div#(x, y) | → | ge#(y, s(0)) |
| minus#(s(x), s(y)) | → | minus#(x, y) | | ge#(s(x), s(y)) | → | ge#(x, y) |
| if#(true, x, y) | → | minus#(x, y) | | if#(true, x, y) | → | div#(minus(x, y), y) |
| plus#(0, s(x)) | → | plus#(0, x) |
Rewrite Rules
| ge(0, 0) | → | true | | ge(s(x), 0) | → | ge(x, 0) |
| ge(0, s(0)) | → | false | | ge(0, s(s(x))) | → | ge(0, s(x)) |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(0, 0) | → | 0 |
| minus(0, s(x)) | → | minus(0, x) | | minus(s(x), 0) | → | s(minus(x, 0)) |
| minus(s(x), s(y)) | → | minus(x, y) | | plus(0, 0) | → | 0 |
| plus(0, s(x)) | → | s(plus(0, x)) | | plus(s(x), y) | → | s(plus(x, y)) |
| div(x, y) | → | ify(ge(y, s(0)), x, y) | | ify(false, x, y) | → | divByZeroError |
| ify(true, x, y) | → | if(ge(x, y), x, y) | | if(false, x, y) | → | 0 |
| if(true, x, y) | → | s(div(minus(x, y), y)) | | div(plus(x, y), z) | → | plus(div(x, z), div(y, z)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, minus, s, ify, if, div, divByZeroError, true, false, ge
Strategy
The following SCCs where found
| ge#(0, s(s(x))) → ge#(0, s(x)) |
| plus#(s(x), y) → plus#(x, y) |
| minus#(0, s(x)) → minus#(0, x) |
| minus#(s(x), s(y)) → minus#(x, y) |
| ge#(s(x), s(y)) → ge#(x, y) |
| plus#(0, s(x)) → plus#(0, x) |
| minus#(s(x), 0) → minus#(x, 0) |
| ify#(true, x, y) → if#(ge(x, y), x, y) | div#(plus(x, y), z) → div#(x, z) |
| div#(plus(x, y), z) → div#(y, z) | div#(x, y) → ify#(ge(y, s(0)), x, y) |
| if#(true, x, y) → div#(minus(x, y), y) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| minus#(s(x), s(y)) | → | minus#(x, y) |
Rewrite Rules
| ge(0, 0) | → | true | | ge(s(x), 0) | → | ge(x, 0) |
| ge(0, s(0)) | → | false | | ge(0, s(s(x))) | → | ge(0, s(x)) |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(0, 0) | → | 0 |
| minus(0, s(x)) | → | minus(0, x) | | minus(s(x), 0) | → | s(minus(x, 0)) |
| minus(s(x), s(y)) | → | minus(x, y) | | plus(0, 0) | → | 0 |
| plus(0, s(x)) | → | s(plus(0, x)) | | plus(s(x), y) | → | s(plus(x, y)) |
| div(x, y) | → | ify(ge(y, s(0)), x, y) | | ify(false, x, y) | → | divByZeroError |
| ify(true, x, y) | → | if(ge(x, y), x, y) | | if(false, x, y) | → | 0 |
| if(true, x, y) | → | s(div(minus(x, y), y)) | | div(plus(x, y), z) | → | plus(div(x, z), div(y, z)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, minus, s, ify, if, div, divByZeroError, true, false, ge
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| minus#(s(x), s(y)) | → | minus#(x, y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| minus#(s(x), 0) | → | minus#(x, 0) |
Rewrite Rules
| ge(0, 0) | → | true | | ge(s(x), 0) | → | ge(x, 0) |
| ge(0, s(0)) | → | false | | ge(0, s(s(x))) | → | ge(0, s(x)) |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(0, 0) | → | 0 |
| minus(0, s(x)) | → | minus(0, x) | | minus(s(x), 0) | → | s(minus(x, 0)) |
| minus(s(x), s(y)) | → | minus(x, y) | | plus(0, 0) | → | 0 |
| plus(0, s(x)) | → | s(plus(0, x)) | | plus(s(x), y) | → | s(plus(x, y)) |
| div(x, y) | → | ify(ge(y, s(0)), x, y) | | ify(false, x, y) | → | divByZeroError |
| ify(true, x, y) | → | if(ge(x, y), x, y) | | if(false, x, y) | → | 0 |
| if(true, x, y) | → | s(div(minus(x, y), y)) | | div(plus(x, y), z) | → | plus(div(x, z), div(y, z)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, minus, s, ify, if, div, divByZeroError, true, false, ge
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| minus#(s(x), 0) | → | minus#(x, 0) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| plus#(s(x), y) | → | plus#(x, y) |
Rewrite Rules
| ge(0, 0) | → | true | | ge(s(x), 0) | → | ge(x, 0) |
| ge(0, s(0)) | → | false | | ge(0, s(s(x))) | → | ge(0, s(x)) |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(0, 0) | → | 0 |
| minus(0, s(x)) | → | minus(0, x) | | minus(s(x), 0) | → | s(minus(x, 0)) |
| minus(s(x), s(y)) | → | minus(x, y) | | plus(0, 0) | → | 0 |
| plus(0, s(x)) | → | s(plus(0, x)) | | plus(s(x), y) | → | s(plus(x, y)) |
| div(x, y) | → | ify(ge(y, s(0)), x, y) | | ify(false, x, y) | → | divByZeroError |
| ify(true, x, y) | → | if(ge(x, y), x, y) | | if(false, x, y) | → | 0 |
| if(true, x, y) | → | s(div(minus(x, y), y)) | | div(plus(x, y), z) | → | plus(div(x, z), div(y, z)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, minus, s, ify, if, div, divByZeroError, true, false, ge
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| plus#(s(x), y) | → | plus#(x, y) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| ge(0, 0) | → | true | | ge(s(x), 0) | → | ge(x, 0) |
| ge(0, s(0)) | → | false | | ge(0, s(s(x))) | → | ge(0, s(x)) |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(0, 0) | → | 0 |
| minus(0, s(x)) | → | minus(0, x) | | minus(s(x), 0) | → | s(minus(x, 0)) |
| minus(s(x), s(y)) | → | minus(x, y) | | plus(0, 0) | → | 0 |
| plus(0, s(x)) | → | s(plus(0, x)) | | plus(s(x), y) | → | s(plus(x, y)) |
| div(x, y) | → | ify(ge(y, s(0)), x, y) | | ify(false, x, y) | → | divByZeroError |
| ify(true, x, y) | → | if(ge(x, y), x, y) | | if(false, x, y) | → | 0 |
| if(true, x, y) | → | s(div(minus(x, y), y)) | | div(plus(x, y), z) | → | plus(div(x, z), div(y, z)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, minus, s, ify, if, div, divByZeroError, true, false, ge
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| plus#(0, s(x)) | → | plus#(0, x) |
Rewrite Rules
| ge(0, 0) | → | true | | ge(s(x), 0) | → | ge(x, 0) |
| ge(0, s(0)) | → | false | | ge(0, s(s(x))) | → | ge(0, s(x)) |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(0, 0) | → | 0 |
| minus(0, s(x)) | → | minus(0, x) | | minus(s(x), 0) | → | s(minus(x, 0)) |
| minus(s(x), s(y)) | → | minus(x, y) | | plus(0, 0) | → | 0 |
| plus(0, s(x)) | → | s(plus(0, x)) | | plus(s(x), y) | → | s(plus(x, y)) |
| div(x, y) | → | ify(ge(y, s(0)), x, y) | | ify(false, x, y) | → | divByZeroError |
| ify(true, x, y) | → | if(ge(x, y), x, y) | | if(false, x, y) | → | 0 |
| if(true, x, y) | → | s(div(minus(x, y), y)) | | div(plus(x, y), z) | → | plus(div(x, z), div(y, z)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, minus, s, ify, if, div, divByZeroError, true, false, ge
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| plus#(0, s(x)) | → | plus#(0, x) |
Problem 7: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| minus#(0, s(x)) | → | minus#(0, x) |
Rewrite Rules
| ge(0, 0) | → | true | | ge(s(x), 0) | → | ge(x, 0) |
| ge(0, s(0)) | → | false | | ge(0, s(s(x))) | → | ge(0, s(x)) |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(0, 0) | → | 0 |
| minus(0, s(x)) | → | minus(0, x) | | minus(s(x), 0) | → | s(minus(x, 0)) |
| minus(s(x), s(y)) | → | minus(x, y) | | plus(0, 0) | → | 0 |
| plus(0, s(x)) | → | s(plus(0, x)) | | plus(s(x), y) | → | s(plus(x, y)) |
| div(x, y) | → | ify(ge(y, s(0)), x, y) | | ify(false, x, y) | → | divByZeroError |
| ify(true, x, y) | → | if(ge(x, y), x, y) | | if(false, x, y) | → | 0 |
| if(true, x, y) | → | s(div(minus(x, y), y)) | | div(plus(x, y), z) | → | plus(div(x, z), div(y, z)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, minus, s, ify, if, div, divByZeroError, true, false, ge
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| minus#(0, s(x)) | → | minus#(0, x) |
Problem 8: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| ge#(s(x), s(y)) | → | ge#(x, y) |
Rewrite Rules
| ge(0, 0) | → | true | | ge(s(x), 0) | → | ge(x, 0) |
| ge(0, s(0)) | → | false | | ge(0, s(s(x))) | → | ge(0, s(x)) |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(0, 0) | → | 0 |
| minus(0, s(x)) | → | minus(0, x) | | minus(s(x), 0) | → | s(minus(x, 0)) |
| minus(s(x), s(y)) | → | minus(x, y) | | plus(0, 0) | → | 0 |
| plus(0, s(x)) | → | s(plus(0, x)) | | plus(s(x), y) | → | s(plus(x, y)) |
| div(x, y) | → | ify(ge(y, s(0)), x, y) | | ify(false, x, y) | → | divByZeroError |
| ify(true, x, y) | → | if(ge(x, y), x, y) | | if(false, x, y) | → | 0 |
| if(true, x, y) | → | s(div(minus(x, y), y)) | | div(plus(x, y), z) | → | plus(div(x, z), div(y, z)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, minus, s, ify, if, div, divByZeroError, 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 9: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| ify#(true, x, y) | → | if#(ge(x, y), x, y) | | div#(plus(x, y), z) | → | div#(x, z) |
| div#(plus(x, y), z) | → | div#(y, z) | | div#(x, y) | → | ify#(ge(y, s(0)), x, y) |
| if#(true, x, y) | → | div#(minus(x, y), y) |
Rewrite Rules
| ge(0, 0) | → | true | | ge(s(x), 0) | → | ge(x, 0) |
| ge(0, s(0)) | → | false | | ge(0, s(s(x))) | → | ge(0, s(x)) |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(0, 0) | → | 0 |
| minus(0, s(x)) | → | minus(0, x) | | minus(s(x), 0) | → | s(minus(x, 0)) |
| minus(s(x), s(y)) | → | minus(x, y) | | plus(0, 0) | → | 0 |
| plus(0, s(x)) | → | s(plus(0, x)) | | plus(s(x), y) | → | s(plus(x, y)) |
| div(x, y) | → | ify(ge(y, s(0)), x, y) | | ify(false, x, y) | → | divByZeroError |
| ify(true, x, y) | → | if(ge(x, y), x, y) | | if(false, x, y) | → | 0 |
| if(true, x, y) | → | s(div(minus(x, y), y)) | | div(plus(x, y), z) | → | plus(div(x, z), div(y, z)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, minus, s, ify, if, div, divByZeroError, true, false, ge
Strategy
Polynomial Interpretation
- 0: 0
- div(x,y): 0
- div#(x,y): x + 2
- divByZeroError: 0
- false: 2
- ge(x,y): 2
- if(x,y,z): 0
- if#(x,y,z): x
- ify(x,y,z): 0
- ify#(x,y,z): x
- minus(x,y): 0
- plus(x,y): y + x + 1
- s(x): 0
- true: 2
Improved Usable rules
| minus(s(x), 0) | → | s(minus(x, 0)) | | minus(s(x), s(y)) | → | minus(x, y) |
| minus(0, s(x)) | → | minus(0, x) | | minus(0, 0) | → | 0 |
| ge(0, s(0)) | → | false | | ge(0, 0) | → | true |
| ge(0, s(s(x))) | → | ge(0, s(x)) | | ge(s(x), 0) | → | ge(x, 0) |
| ge(s(x), s(y)) | → | ge(x, y) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| div#(plus(x, y), z) | → | div#(x, z) | | div#(plus(x, y), z) | → | div#(y, z) |
Problem 11: BackwardInstantiation
Dependency Pair Problem
Dependency Pairs
| ify#(true, x, y) | → | if#(ge(x, y), x, y) | | div#(x, y) | → | ify#(ge(y, s(0)), x, y) |
| if#(true, x, y) | → | div#(minus(x, y), y) |
Rewrite Rules
| ge(0, 0) | → | true | | ge(s(x), 0) | → | ge(x, 0) |
| ge(0, s(0)) | → | false | | ge(0, s(s(x))) | → | ge(0, s(x)) |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(0, 0) | → | 0 |
| minus(0, s(x)) | → | minus(0, x) | | minus(s(x), 0) | → | s(minus(x, 0)) |
| minus(s(x), s(y)) | → | minus(x, y) | | plus(0, 0) | → | 0 |
| plus(0, s(x)) | → | s(plus(0, x)) | | plus(s(x), y) | → | s(plus(x, y)) |
| div(x, y) | → | ify(ge(y, s(0)), x, y) | | ify(false, x, y) | → | divByZeroError |
| ify(true, x, y) | → | if(ge(x, y), x, y) | | if(false, x, y) | → | 0 |
| if(true, x, y) | → | s(div(minus(x, y), y)) | | div(plus(x, y), z) | → | plus(div(x, z), div(y, z)) |
Original Signature
Termination of terms over the following signature is verified: plus, minus, 0, s, if, ify, divByZeroError, div, false, true, ge
Strategy
Instantiation
For all potential predecessors l → r of the rule div
#(
x,
y) → ify
#(ge(
y, s(0)),
x,
y) on dependency pair chains it holds that:
- div#(x, y) matches r,
- all variables of div#(x, y) are embedded in constructor contexts, i.e., each subterm of div#(x, y), containing a variable is rooted by a constructor symbol.
Thus, div
#(
x,
y) → ify
#(ge(
y, s(0)),
x,
y) is replaced by instances determined through the above matching. These instances are:
| div#(minus(_x, _y), _y) → ify#(ge(_y, s(0)), minus(_x, _y), _y) |
Problem 10: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| ge#(0, s(s(x))) | → | ge#(0, s(x)) |
Rewrite Rules
| ge(0, 0) | → | true | | ge(s(x), 0) | → | ge(x, 0) |
| ge(0, s(0)) | → | false | | ge(0, s(s(x))) | → | ge(0, s(x)) |
| ge(s(x), s(y)) | → | ge(x, y) | | minus(0, 0) | → | 0 |
| minus(0, s(x)) | → | minus(0, x) | | minus(s(x), 0) | → | s(minus(x, 0)) |
| minus(s(x), s(y)) | → | minus(x, y) | | plus(0, 0) | → | 0 |
| plus(0, s(x)) | → | s(plus(0, x)) | | plus(s(x), y) | → | s(plus(x, y)) |
| div(x, y) | → | ify(ge(y, s(0)), x, y) | | ify(false, x, y) | → | divByZeroError |
| ify(true, x, y) | → | if(ge(x, y), x, y) | | if(false, x, y) | → | 0 |
| if(true, x, y) | → | s(div(minus(x, y), y)) | | div(plus(x, y), z) | → | plus(div(x, z), div(y, z)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, minus, s, ify, if, div, divByZeroError, true, false, ge
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| ge#(0, s(s(x))) | → | ge#(0, s(x)) |