TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60020 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (175ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (4ms), PolynomialLinearRange4iUR (1618ms), DependencyGraph (4ms), PolynomialLinearRange8NegiUR (3403ms), DependencyGraph (3ms), PolynomialLinearRange8NegiUR (3390ms), DependencyGraph (4ms), ReductionPairSAT (1374ms), DependencyGraph (4ms), SizeChangePrinciple (459ms), ForwardNarrowing (1ms), BackwardInstantiation (2ms), ForwardInstantiation (2ms), Propagation (1ms)].
| – Problem 3 was processed with processor PolynomialLinearRange8NegiUR (13321ms).
| – Problem 4 was processed with processor SubtermCriterion (0ms).
| – Problem 5 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| plus#(id(x), s(y)) | → | plus#(x, if(gt(s(y), y), y, s(y))) | | plus#(s(x), s(y)) | → | plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))) |
| plus#(s(x), x) | → | plus#(if(gt(x, x), id(x), id(x)), s(x)) |
Rewrite Rules
| minus(x, 0) | → | x | | minus(s(x), s(y)) | → | minus(x, y) |
| quot(0, s(y)) | → | 0 | | quot(s(x), s(y)) | → | s(quot(minus(x, y), s(y))) |
| minus(minus(x, y), z) | → | minus(x, plus(y, z)) | | plus(s(x), s(y)) | → | s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) |
| plus(s(x), x) | → | plus(if(gt(x, x), id(x), id(x)), s(x)) | | plus(zero, y) | → | y |
| plus(id(x), s(y)) | → | s(plus(x, if(gt(s(y), y), y, s(y)))) | | id(x) | → | x |
| if(true, x, y) | → | x | | if(false, x, y) | → | y |
| not(x) | → | if(x, false, true) | | gt(s(x), zero) | → | true |
| gt(zero, y) | → | false | | gt(s(x), s(y)) | → | gt(x, y) |
Original Signature
Termination of terms over the following signature is verified: not, id, plus, 0, minus, s, if, false, true, gt, zero, quot
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| plus#(s(x), x) | → | id#(x) | | plus#(id(x), s(y)) | → | plus#(x, if(gt(s(y), y), y, s(y))) |
| minus#(minus(x, y), z) | → | plus#(y, z) | | plus#(id(x), s(y)) | → | gt#(s(y), y) |
| plus#(s(x), s(y)) | → | gt#(x, y) | | plus#(s(x), s(y)) | → | if#(gt(x, y), x, y) |
| plus#(s(x), x) | → | plus#(if(gt(x, x), id(x), id(x)), s(x)) | | plus#(s(x), x) | → | gt#(x, x) |
| plus#(id(x), s(y)) | → | if#(gt(s(y), y), y, s(y)) | | minus#(minus(x, y), z) | → | minus#(x, plus(y, z)) |
| plus#(s(x), x) | → | if#(gt(x, x), id(x), id(x)) | | quot#(s(x), s(y)) | → | minus#(x, y) |
| plus#(s(x), s(y)) | → | id#(x) | | plus#(s(x), s(y)) | → | plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))) |
| plus#(s(x), s(y)) | → | if#(not(gt(x, y)), id(x), id(y)) | | gt#(s(x), s(y)) | → | gt#(x, y) |
| quot#(s(x), s(y)) | → | quot#(minus(x, y), s(y)) | | minus#(s(x), s(y)) | → | minus#(x, y) |
| not#(x) | → | if#(x, false, true) | | plus#(s(x), s(y)) | → | id#(y) |
| plus#(s(x), s(y)) | → | not#(gt(x, y)) |
Rewrite Rules
| minus(x, 0) | → | x | | minus(s(x), s(y)) | → | minus(x, y) |
| quot(0, s(y)) | → | 0 | | quot(s(x), s(y)) | → | s(quot(minus(x, y), s(y))) |
| minus(minus(x, y), z) | → | minus(x, plus(y, z)) | | plus(s(x), s(y)) | → | s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) |
| plus(s(x), x) | → | plus(if(gt(x, x), id(x), id(x)), s(x)) | | plus(zero, y) | → | y |
| plus(id(x), s(y)) | → | s(plus(x, if(gt(s(y), y), y, s(y)))) | | id(x) | → | x |
| if(true, x, y) | → | x | | if(false, x, y) | → | y |
| not(x) | → | if(x, false, true) | | gt(s(x), zero) | → | true |
| gt(zero, y) | → | false | | gt(s(x), s(y)) | → | gt(x, y) |
Original Signature
Termination of terms over the following signature is verified: id, not, plus, minus, 0, s, if, true, false, gt, zero, quot
Strategy
The following SCCs where found
| minus#(s(x), s(y)) → minus#(x, y) | minus#(minus(x, y), z) → minus#(x, plus(y, z)) |
| plus#(id(x), s(y)) → plus#(x, if(gt(s(y), y), y, s(y))) | plus#(s(x), s(y)) → plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))) |
| plus#(s(x), x) → plus#(if(gt(x, x), id(x), id(x)), s(x)) |
| gt#(s(x), s(y)) → gt#(x, y) |
| quot#(s(x), s(y)) → quot#(minus(x, y), s(y)) |
Problem 3: PolynomialLinearRange8NegiUR
Dependency Pair Problem
Dependency Pairs
| quot#(s(x), s(y)) | → | quot#(minus(x, y), s(y)) |
Rewrite Rules
| minus(x, 0) | → | x | | minus(s(x), s(y)) | → | minus(x, y) |
| quot(0, s(y)) | → | 0 | | quot(s(x), s(y)) | → | s(quot(minus(x, y), s(y))) |
| minus(minus(x, y), z) | → | minus(x, plus(y, z)) | | plus(s(x), s(y)) | → | s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) |
| plus(s(x), x) | → | plus(if(gt(x, x), id(x), id(x)), s(x)) | | plus(zero, y) | → | y |
| plus(id(x), s(y)) | → | s(plus(x, if(gt(s(y), y), y, s(y)))) | | id(x) | → | x |
| if(true, x, y) | → | x | | if(false, x, y) | → | y |
| not(x) | → | if(x, false, true) | | gt(s(x), zero) | → | true |
| gt(zero, y) | → | false | | gt(s(x), s(y)) | → | gt(x, y) |
Original Signature
Termination of terms over the following signature is verified: id, not, plus, minus, 0, s, if, true, false, gt, zero, quot
Strategy
Polynomial Interpretation
- 0: 1
- false: 0
- gt(x,y): 3y - 2
- id(x): 7x - 2
- if(x,y,z): 5y + 3x + 1
- minus(x,y): x
- not(x): -2
- plus(x,y): 2
- quot(x,y): -2
- quot#(x,y): x + 2
- s(x): x + 1
- true: 1
- zero: 2
Improved Usable rules
| minus(s(x), s(y)) | → | minus(x, y) | | minus(minus(x, y), z) | → | minus(x, plus(y, z)) |
| minus(x, 0) | → | x |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| quot#(s(x), s(y)) | → | quot#(minus(x, y), s(y)) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| gt#(s(x), s(y)) | → | gt#(x, y) |
Rewrite Rules
| minus(x, 0) | → | x | | minus(s(x), s(y)) | → | minus(x, y) |
| quot(0, s(y)) | → | 0 | | quot(s(x), s(y)) | → | s(quot(minus(x, y), s(y))) |
| minus(minus(x, y), z) | → | minus(x, plus(y, z)) | | plus(s(x), s(y)) | → | s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) |
| plus(s(x), x) | → | plus(if(gt(x, x), id(x), id(x)), s(x)) | | plus(zero, y) | → | y |
| plus(id(x), s(y)) | → | s(plus(x, if(gt(s(y), y), y, s(y)))) | | id(x) | → | x |
| if(true, x, y) | → | x | | if(false, x, y) | → | y |
| not(x) | → | if(x, false, true) | | gt(s(x), zero) | → | true |
| gt(zero, y) | → | false | | gt(s(x), s(y)) | → | gt(x, y) |
Original Signature
Termination of terms over the following signature is verified: id, not, plus, minus, 0, s, if, true, false, gt, zero, quot
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| gt#(s(x), s(y)) | → | gt#(x, y) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| minus#(s(x), s(y)) | → | minus#(x, y) | | minus#(minus(x, y), z) | → | minus#(x, plus(y, z)) |
Rewrite Rules
| minus(x, 0) | → | x | | minus(s(x), s(y)) | → | minus(x, y) |
| quot(0, s(y)) | → | 0 | | quot(s(x), s(y)) | → | s(quot(minus(x, y), s(y))) |
| minus(minus(x, y), z) | → | minus(x, plus(y, z)) | | plus(s(x), s(y)) | → | s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))))) |
| plus(s(x), x) | → | plus(if(gt(x, x), id(x), id(x)), s(x)) | | plus(zero, y) | → | y |
| plus(id(x), s(y)) | → | s(plus(x, if(gt(s(y), y), y, s(y)))) | | id(x) | → | x |
| if(true, x, y) | → | x | | if(false, x, y) | → | y |
| not(x) | → | if(x, false, true) | | gt(s(x), zero) | → | true |
| gt(zero, y) | → | false | | gt(s(x), s(y)) | → | gt(x, y) |
Original Signature
Termination of terms over the following signature is verified: id, not, plus, minus, 0, s, if, true, false, gt, zero, quot
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| minus#(s(x), s(y)) | → | minus#(x, y) | | minus#(minus(x, y), z) | → | minus#(x, plus(y, z)) |