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 (121ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 remains open; application of the following processors failed [SubtermCriterion (0ms), DependencyGraph (6ms), PolynomialLinearRange4iUR (954ms), DependencyGraph (4ms), PolynomialLinearRange8NegiUR (5369ms), DependencyGraph (3ms), ReductionPairSAT (1636ms), DependencyGraph (4ms), SizeChangePrinciple (398ms), ForwardNarrowing (1ms), BackwardInstantiation (1ms), ForwardInstantiation (4ms), Propagation (1ms)].
| – 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 3
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) |
| double(0) | → | 0 | | double(s(x)) | → | s(s(double(x))) |
| 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, double
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))) |
| 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)) |
| plus#(s(x), s(y)) | → | id#(x) | | plus#(s(x), x) | → | if#(gt(x, x), id(x), id(x)) |
| plus#(s(x), s(y)) | → | plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))) | | gt#(s(x), s(y)) | → | gt#(x, y) |
| plus#(s(x), s(y)) | → | if#(not(gt(x, y)), id(x), id(y)) | | minus#(s(x), s(y)) | → | minus#(x, y) |
| double#(s(x)) | → | double#(x) | | 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) |
| double(0) | → | 0 | | double(s(x)) | → | s(s(double(x))) |
| 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, double, zero
Strategy
The following SCCs where found
| 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) |
| minus#(s(x), s(y)) → minus#(x, y) |
| double#(s(x)) → double#(x) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| double#(s(x)) | → | double#(x) |
Rewrite Rules
| minus(x, 0) | → | x | | minus(s(x), s(y)) | → | minus(x, y) |
| double(0) | → | 0 | | double(s(x)) | → | s(s(double(x))) |
| 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, double, zero
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| double#(s(x)) | → | double#(x) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| minus#(s(x), s(y)) | → | minus#(x, y) |
Rewrite Rules
| minus(x, 0) | → | x | | minus(s(x), s(y)) | → | minus(x, y) |
| double(0) | → | 0 | | double(s(x)) | → | s(s(double(x))) |
| 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, double, zero
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| minus#(s(x), s(y)) | → | minus#(x, y) |
Problem 5: 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) |
| double(0) | → | 0 | | double(s(x)) | → | s(s(double(x))) |
| 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, double, zero
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| gt#(s(x), s(y)) | → | gt#(x, y) |