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 (189ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 remains open; application of the following processors failed [SubtermCriterion (2ms), DependencyGraph (6ms), PolynomialLinearRange4iUR (1458ms), DependencyGraph (10ms), PolynomialLinearRange8NegiUR (13487ms), DependencyGraph (5ms), ReductionPairSAT (19619ms), DependencyGraph (6ms), SizeChangePrinciple (timeout)].
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| – Problem 5 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| addLists#(xs, ys, zs) | → | if#(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys))) | | if#(false, false, false, xs, ys, xs2, ys2, zs, zs2) | → | addLists#(xs2, ys2, zs) |
| if#(false, false, true, xs, ys, xs2, ys2, zs, zs2) | → | addLists#(xs, ys, zs2) |
Rewrite Rules
| isEmpty(cons(x, xs)) | → | false | | isEmpty(nil) | → | true |
| isZero(0) | → | true | | isZero(s(x)) | → | false |
| head(cons(x, xs)) | → | x | | tail(cons(x, xs)) | → | xs |
| tail(nil) | → | nil | | append(nil, x) | → | cons(x, nil) |
| append(cons(y, ys), x) | → | cons(y, append(ys, x)) | | p(s(s(x))) | → | s(p(s(x))) |
| p(s(0)) | → | 0 | | p(0) | → | 0 |
| inc(s(x)) | → | s(inc(x)) | | inc(0) | → | s(0) |
| addLists(xs, ys, zs) | → | if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys))) | | if(true, true, b, xs, ys, xs2, ys2, zs, zs2) | → | zs |
| if(true, false, b, xs, ys, xs2, ys2, zs, zs2) | → | differentLengthError | | if(false, true, b, xs, ys, xs2, ys2, zs, zs2) | → | differentLengthError |
| if(false, false, false, xs, ys, xs2, ys2, zs, zs2) | → | addLists(xs2, ys2, zs) | | if(false, false, true, xs, ys, xs2, ys2, zs, zs2) | → | addLists(xs, ys, zs2) |
| addList(xs, ys) | → | addLists(xs, ys, nil) |
Original Signature
Termination of terms over the following signature is verified: append, true, tail, isEmpty, 0, s, inc, if, addLists, p, differentLengthError, false, head, addList, cons, nil, isZero
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| addLists#(xs, ys, zs) | → | isZero#(head(xs)) | | addLists#(xs, ys, zs) | → | if#(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys))) |
| addLists#(xs, ys, zs) | → | isEmpty#(ys) | | addLists#(xs, ys, zs) | → | p#(head(xs)) |
| addList#(xs, ys) | → | addLists#(xs, ys, nil) | | append#(cons(y, ys), x) | → | append#(ys, x) |
| addLists#(xs, ys, zs) | → | head#(xs) | | addLists#(xs, ys, zs) | → | tail#(xs) |
| addLists#(xs, ys, zs) | → | tail#(ys) | | addLists#(xs, ys, zs) | → | inc#(head(ys)) |
| addLists#(xs, ys, zs) | → | isEmpty#(xs) | | addLists#(xs, ys, zs) | → | append#(zs, head(ys)) |
| inc#(s(x)) | → | inc#(x) | | if#(false, false, false, xs, ys, xs2, ys2, zs, zs2) | → | addLists#(xs2, ys2, zs) |
| addLists#(xs, ys, zs) | → | head#(ys) | | p#(s(s(x))) | → | p#(s(x)) |
| if#(false, false, true, xs, ys, xs2, ys2, zs, zs2) | → | addLists#(xs, ys, zs2) |
Rewrite Rules
| isEmpty(cons(x, xs)) | → | false | | isEmpty(nil) | → | true |
| isZero(0) | → | true | | isZero(s(x)) | → | false |
| head(cons(x, xs)) | → | x | | tail(cons(x, xs)) | → | xs |
| tail(nil) | → | nil | | append(nil, x) | → | cons(x, nil) |
| append(cons(y, ys), x) | → | cons(y, append(ys, x)) | | p(s(s(x))) | → | s(p(s(x))) |
| p(s(0)) | → | 0 | | p(0) | → | 0 |
| inc(s(x)) | → | s(inc(x)) | | inc(0) | → | s(0) |
| addLists(xs, ys, zs) | → | if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys))) | | if(true, true, b, xs, ys, xs2, ys2, zs, zs2) | → | zs |
| if(true, false, b, xs, ys, xs2, ys2, zs, zs2) | → | differentLengthError | | if(false, true, b, xs, ys, xs2, ys2, zs, zs2) | → | differentLengthError |
| if(false, false, false, xs, ys, xs2, ys2, zs, zs2) | → | addLists(xs2, ys2, zs) | | if(false, false, true, xs, ys, xs2, ys2, zs, zs2) | → | addLists(xs, ys, zs2) |
| addList(xs, ys) | → | addLists(xs, ys, nil) |
Original Signature
Termination of terms over the following signature is verified: append, true, tail, isEmpty, 0, s, inc, if, addLists, p, differentLengthError, false, head, isZero, nil, cons, addList
Strategy
The following SCCs where found
| append#(cons(y, ys), x) → append#(ys, x) |
| addLists#(xs, ys, zs) → if#(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys))) | if#(false, false, false, xs, ys, xs2, ys2, zs, zs2) → addLists#(xs2, ys2, zs) |
| if#(false, false, true, xs, ys, xs2, ys2, zs, zs2) → addLists#(xs, ys, zs2) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| isEmpty(cons(x, xs)) | → | false | | isEmpty(nil) | → | true |
| isZero(0) | → | true | | isZero(s(x)) | → | false |
| head(cons(x, xs)) | → | x | | tail(cons(x, xs)) | → | xs |
| tail(nil) | → | nil | | append(nil, x) | → | cons(x, nil) |
| append(cons(y, ys), x) | → | cons(y, append(ys, x)) | | p(s(s(x))) | → | s(p(s(x))) |
| p(s(0)) | → | 0 | | p(0) | → | 0 |
| inc(s(x)) | → | s(inc(x)) | | inc(0) | → | s(0) |
| addLists(xs, ys, zs) | → | if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys))) | | if(true, true, b, xs, ys, xs2, ys2, zs, zs2) | → | zs |
| if(true, false, b, xs, ys, xs2, ys2, zs, zs2) | → | differentLengthError | | if(false, true, b, xs, ys, xs2, ys2, zs, zs2) | → | differentLengthError |
| if(false, false, false, xs, ys, xs2, ys2, zs, zs2) | → | addLists(xs2, ys2, zs) | | if(false, false, true, xs, ys, xs2, ys2, zs, zs2) | → | addLists(xs, ys, zs2) |
| addList(xs, ys) | → | addLists(xs, ys, nil) |
Original Signature
Termination of terms over the following signature is verified: append, true, tail, isEmpty, 0, s, inc, if, addLists, p, differentLengthError, false, head, isZero, nil, cons, addList
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| append#(cons(y, ys), x) | → | append#(ys, x) |
Rewrite Rules
| isEmpty(cons(x, xs)) | → | false | | isEmpty(nil) | → | true |
| isZero(0) | → | true | | isZero(s(x)) | → | false |
| head(cons(x, xs)) | → | x | | tail(cons(x, xs)) | → | xs |
| tail(nil) | → | nil | | append(nil, x) | → | cons(x, nil) |
| append(cons(y, ys), x) | → | cons(y, append(ys, x)) | | p(s(s(x))) | → | s(p(s(x))) |
| p(s(0)) | → | 0 | | p(0) | → | 0 |
| inc(s(x)) | → | s(inc(x)) | | inc(0) | → | s(0) |
| addLists(xs, ys, zs) | → | if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys))) | | if(true, true, b, xs, ys, xs2, ys2, zs, zs2) | → | zs |
| if(true, false, b, xs, ys, xs2, ys2, zs, zs2) | → | differentLengthError | | if(false, true, b, xs, ys, xs2, ys2, zs, zs2) | → | differentLengthError |
| if(false, false, false, xs, ys, xs2, ys2, zs, zs2) | → | addLists(xs2, ys2, zs) | | if(false, false, true, xs, ys, xs2, ys2, zs, zs2) | → | addLists(xs, ys, zs2) |
| addList(xs, ys) | → | addLists(xs, ys, nil) |
Original Signature
Termination of terms over the following signature is verified: append, true, tail, isEmpty, 0, s, inc, if, addLists, p, differentLengthError, false, head, isZero, nil, cons, addList
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| append#(cons(y, ys), x) | → | append#(ys, x) |
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| isEmpty(cons(x, xs)) | → | false | | isEmpty(nil) | → | true |
| isZero(0) | → | true | | isZero(s(x)) | → | false |
| head(cons(x, xs)) | → | x | | tail(cons(x, xs)) | → | xs |
| tail(nil) | → | nil | | append(nil, x) | → | cons(x, nil) |
| append(cons(y, ys), x) | → | cons(y, append(ys, x)) | | p(s(s(x))) | → | s(p(s(x))) |
| p(s(0)) | → | 0 | | p(0) | → | 0 |
| inc(s(x)) | → | s(inc(x)) | | inc(0) | → | s(0) |
| addLists(xs, ys, zs) | → | if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys))) | | if(true, true, b, xs, ys, xs2, ys2, zs, zs2) | → | zs |
| if(true, false, b, xs, ys, xs2, ys2, zs, zs2) | → | differentLengthError | | if(false, true, b, xs, ys, xs2, ys2, zs, zs2) | → | differentLengthError |
| if(false, false, false, xs, ys, xs2, ys2, zs, zs2) | → | addLists(xs2, ys2, zs) | | if(false, false, true, xs, ys, xs2, ys2, zs, zs2) | → | addLists(xs, ys, zs2) |
| addList(xs, ys) | → | addLists(xs, ys, nil) |
Original Signature
Termination of terms over the following signature is verified: append, true, tail, isEmpty, 0, s, inc, if, addLists, p, differentLengthError, false, head, isZero, nil, cons, addList
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed: