TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60027 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (81ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| | – Problem 6 was processed with processor DependencyGraph (1ms).
| – Problem 4 was processed with processor SubtermCriterion (0ms).
| – Problem 5 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (464ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (6537ms), DependencyGraph (0ms), ReductionPairSAT (531ms), DependencyGraph (2ms), SizeChangePrinciple (2705ms), ForwardNarrowing (1ms), BackwardInstantiation (1ms), ForwardInstantiation (1ms), Propagation (1ms)].
The following open problems remain:
Open Dependency Pair Problem 5
Dependency Pairs
| doublelist#(cons(x, xs)) | → | doublelist#(del(first(cons(x, xs)), cons(x, xs))) |
Rewrite Rules
| double(0) | → | 0 | | double(s(x)) | → | s(s(double(x))) |
| del(x, nil) | → | nil | | del(x, cons(y, xs)) | → | if(eq(x, y), x, y, xs) |
| if(true, x, y, xs) | → | xs | | if(false, x, y, xs) | → | cons(y, del(x, xs)) |
| eq(0, 0) | → | true | | eq(0, s(y)) | → | false |
| eq(s(x), 0) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| first(nil) | → | 0 | | first(cons(x, xs)) | → | x |
| doublelist(nil) | → | nil | | doublelist(cons(x, xs)) | → | cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) |
Original Signature
Termination of terms over the following signature is verified: 0, s, if, false, true, del, doublelist, double, first, cons, eq, nil
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| doublelist#(cons(x, xs)) | → | del#(first(cons(x, xs)), cons(x, xs)) | | del#(x, cons(y, xs)) | → | if#(eq(x, y), x, y, xs) |
| doublelist#(cons(x, xs)) | → | doublelist#(del(first(cons(x, xs)), cons(x, xs))) | | del#(x, cons(y, xs)) | → | eq#(x, y) |
| doublelist#(cons(x, xs)) | → | double#(x) | | double#(s(x)) | → | double#(x) |
| eq#(s(x), s(y)) | → | eq#(x, y) | | doublelist#(cons(x, xs)) | → | first#(cons(x, xs)) |
| if#(false, x, y, xs) | → | del#(x, xs) |
Rewrite Rules
| double(0) | → | 0 | | double(s(x)) | → | s(s(double(x))) |
| del(x, nil) | → | nil | | del(x, cons(y, xs)) | → | if(eq(x, y), x, y, xs) |
| if(true, x, y, xs) | → | xs | | if(false, x, y, xs) | → | cons(y, del(x, xs)) |
| eq(0, 0) | → | true | | eq(0, s(y)) | → | false |
| eq(s(x), 0) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| first(nil) | → | 0 | | first(cons(x, xs)) | → | x |
| doublelist(nil) | → | nil | | doublelist(cons(x, xs)) | → | cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) |
Original Signature
Termination of terms over the following signature is verified: 0, s, if, true, false, del, double, doublelist, first, nil, eq, cons
Strategy
The following SCCs where found
| doublelist#(cons(x, xs)) → doublelist#(del(first(cons(x, xs)), cons(x, xs))) |
| del#(x, cons(y, xs)) → if#(eq(x, y), x, y, xs) | if#(false, x, y, xs) → del#(x, xs) |
| double#(s(x)) → double#(x) |
| eq#(s(x), s(y)) → eq#(x, y) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| eq#(s(x), s(y)) | → | eq#(x, y) |
Rewrite Rules
| double(0) | → | 0 | | double(s(x)) | → | s(s(double(x))) |
| del(x, nil) | → | nil | | del(x, cons(y, xs)) | → | if(eq(x, y), x, y, xs) |
| if(true, x, y, xs) | → | xs | | if(false, x, y, xs) | → | cons(y, del(x, xs)) |
| eq(0, 0) | → | true | | eq(0, s(y)) | → | false |
| eq(s(x), 0) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| first(nil) | → | 0 | | first(cons(x, xs)) | → | x |
| doublelist(nil) | → | nil | | doublelist(cons(x, xs)) | → | cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) |
Original Signature
Termination of terms over the following signature is verified: 0, s, if, true, false, del, double, doublelist, first, nil, eq, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| eq#(s(x), s(y)) | → | eq#(x, y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| del#(x, cons(y, xs)) | → | if#(eq(x, y), x, y, xs) | | if#(false, x, y, xs) | → | del#(x, xs) |
Rewrite Rules
| double(0) | → | 0 | | double(s(x)) | → | s(s(double(x))) |
| del(x, nil) | → | nil | | del(x, cons(y, xs)) | → | if(eq(x, y), x, y, xs) |
| if(true, x, y, xs) | → | xs | | if(false, x, y, xs) | → | cons(y, del(x, xs)) |
| eq(0, 0) | → | true | | eq(0, s(y)) | → | false |
| eq(s(x), 0) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| first(nil) | → | 0 | | first(cons(x, xs)) | → | x |
| doublelist(nil) | → | nil | | doublelist(cons(x, xs)) | → | cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) |
Original Signature
Termination of terms over the following signature is verified: 0, s, if, true, false, del, double, doublelist, first, nil, eq, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| del#(x, cons(y, xs)) | → | if#(eq(x, y), x, y, xs) |
Problem 6: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| if#(false, x, y, xs) | → | del#(x, xs) |
Rewrite Rules
| double(0) | → | 0 | | double(s(x)) | → | s(s(double(x))) |
| del(x, nil) | → | nil | | del(x, cons(y, xs)) | → | if(eq(x, y), x, y, xs) |
| if(true, x, y, xs) | → | xs | | if(false, x, y, xs) | → | cons(y, del(x, xs)) |
| eq(0, 0) | → | true | | eq(0, s(y)) | → | false |
| eq(s(x), 0) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| first(nil) | → | 0 | | first(cons(x, xs)) | → | x |
| doublelist(nil) | → | nil | | doublelist(cons(x, xs)) | → | cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) |
Original Signature
Termination of terms over the following signature is verified: 0, s, if, false, true, del, doublelist, double, first, cons, eq, nil
Strategy
There are no SCCs!
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| double#(s(x)) | → | double#(x) |
Rewrite Rules
| double(0) | → | 0 | | double(s(x)) | → | s(s(double(x))) |
| del(x, nil) | → | nil | | del(x, cons(y, xs)) | → | if(eq(x, y), x, y, xs) |
| if(true, x, y, xs) | → | xs | | if(false, x, y, xs) | → | cons(y, del(x, xs)) |
| eq(0, 0) | → | true | | eq(0, s(y)) | → | false |
| eq(s(x), 0) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| first(nil) | → | 0 | | first(cons(x, xs)) | → | x |
| doublelist(nil) | → | nil | | doublelist(cons(x, xs)) | → | cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) |
Original Signature
Termination of terms over the following signature is verified: 0, s, if, true, false, del, double, doublelist, first, nil, eq, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| double#(s(x)) | → | double#(x) |