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 (172ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (6ms), PolynomialLinearRange4iUR (1205ms), DependencyGraph (7ms), PolynomialLinearRange8NegiUR (27629ms), DependencyGraph (4ms), ReductionPairSAT (795ms), DependencyGraph (5ms), SizeChangePrinciple (timeout)].
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| | – Problem 7 was processed with processor DependencyGraph (1ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| | – Problem 8 was processed with processor DependencyGraph (1ms).
| – Problem 5 was processed with processor SubtermCriterion (1ms).
| – Problem 6 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| if_qs#(false, x, n, y) | → | quicksort#(x) | | quicksort#(x) | → | if_qs#(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) |
| if_qs#(false, x, n, y) | → | quicksort#(y) |
Rewrite Rules
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | app(nil, y) | → | y |
| app(add(n, x), y) | → | add(n, app(x, y)) | | low(n, nil) | → | nil |
| low(n, add(m, x)) | → | if_low(le(m, n), n, add(m, x)) | | if_low(true, n, add(m, x)) | → | add(m, low(n, x)) |
| if_low(false, n, add(m, x)) | → | low(n, x) | | high(n, nil) | → | nil |
| high(n, add(m, x)) | → | if_high(le(m, n), n, add(m, x)) | | if_high(true, n, add(m, x)) | → | high(n, x) |
| if_high(false, n, add(m, x)) | → | add(m, high(n, x)) | | head(add(n, x)) | → | n |
| tail(add(n, x)) | → | x | | isempty(nil) | → | true |
| isempty(add(n, x)) | → | false | | quicksort(x) | → | if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) |
| if_qs(true, x, n, y) | → | nil | | if_qs(false, x, n, y) | → | app(quicksort(x), add(n, quicksort(y))) |
Original Signature
Termination of terms over the following signature is verified: if_qs, app, true, if_low, add, tail, 0, le, s, isempty, false, if_high, high, low, head, quicksort, nil
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| if_qs#(false, x, n, y) | → | quicksort#(x) | | high#(n, add(m, x)) | → | le#(m, n) |
| if_low#(true, n, add(m, x)) | → | low#(n, x) | | app#(add(n, x), y) | → | app#(x, y) |
| if_high#(true, n, add(m, x)) | → | high#(n, x) | | if_qs#(false, x, n, y) | → | app#(quicksort(x), add(n, quicksort(y))) |
| quicksort#(x) | → | low#(head(x), tail(x)) | | if_qs#(false, x, n, y) | → | quicksort#(y) |
| if_high#(false, n, add(m, x)) | → | high#(n, x) | | if_low#(false, n, add(m, x)) | → | low#(n, x) |
| quicksort#(x) | → | head#(x) | | low#(n, add(m, x)) | → | if_low#(le(m, n), n, add(m, x)) |
| high#(n, add(m, x)) | → | if_high#(le(m, n), n, add(m, x)) | | le#(s(x), s(y)) | → | le#(x, y) |
| quicksort#(x) | → | isempty#(x) | | quicksort#(x) | → | if_qs#(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) |
| quicksort#(x) | → | high#(head(x), tail(x)) | | low#(n, add(m, x)) | → | le#(m, n) |
| quicksort#(x) | → | tail#(x) |
Rewrite Rules
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | app(nil, y) | → | y |
| app(add(n, x), y) | → | add(n, app(x, y)) | | low(n, nil) | → | nil |
| low(n, add(m, x)) | → | if_low(le(m, n), n, add(m, x)) | | if_low(true, n, add(m, x)) | → | add(m, low(n, x)) |
| if_low(false, n, add(m, x)) | → | low(n, x) | | high(n, nil) | → | nil |
| high(n, add(m, x)) | → | if_high(le(m, n), n, add(m, x)) | | if_high(true, n, add(m, x)) | → | high(n, x) |
| if_high(false, n, add(m, x)) | → | add(m, high(n, x)) | | head(add(n, x)) | → | n |
| tail(add(n, x)) | → | x | | isempty(nil) | → | true |
| isempty(add(n, x)) | → | false | | quicksort(x) | → | if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) |
| if_qs(true, x, n, y) | → | nil | | if_qs(false, x, n, y) | → | app(quicksort(x), add(n, quicksort(y))) |
Original Signature
Termination of terms over the following signature is verified: if_qs, app, true, if_low, add, tail, 0, le, s, isempty, false, if_high, high, head, low, quicksort, nil
Strategy
The following SCCs where found
| le#(s(x), s(y)) → le#(x, y) |
| if_qs#(false, x, n, y) → quicksort#(x) | quicksort#(x) → if_qs#(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) |
| if_qs#(false, x, n, y) → quicksort#(y) |
| app#(add(n, x), y) → app#(x, y) |
| low#(n, add(m, x)) → if_low#(le(m, n), n, add(m, x)) | if_low#(true, n, add(m, x)) → low#(n, x) |
| if_low#(false, n, add(m, x)) → low#(n, x) |
| high#(n, add(m, x)) → if_high#(le(m, n), n, add(m, x)) | if_high#(true, n, add(m, x)) → high#(n, x) |
| if_high#(false, n, add(m, x)) → high#(n, x) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| high#(n, add(m, x)) | → | if_high#(le(m, n), n, add(m, x)) | | if_high#(true, n, add(m, x)) | → | high#(n, x) |
| if_high#(false, n, add(m, x)) | → | high#(n, x) |
Rewrite Rules
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | app(nil, y) | → | y |
| app(add(n, x), y) | → | add(n, app(x, y)) | | low(n, nil) | → | nil |
| low(n, add(m, x)) | → | if_low(le(m, n), n, add(m, x)) | | if_low(true, n, add(m, x)) | → | add(m, low(n, x)) |
| if_low(false, n, add(m, x)) | → | low(n, x) | | high(n, nil) | → | nil |
| high(n, add(m, x)) | → | if_high(le(m, n), n, add(m, x)) | | if_high(true, n, add(m, x)) | → | high(n, x) |
| if_high(false, n, add(m, x)) | → | add(m, high(n, x)) | | head(add(n, x)) | → | n |
| tail(add(n, x)) | → | x | | isempty(nil) | → | true |
| isempty(add(n, x)) | → | false | | quicksort(x) | → | if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) |
| if_qs(true, x, n, y) | → | nil | | if_qs(false, x, n, y) | → | app(quicksort(x), add(n, quicksort(y))) |
Original Signature
Termination of terms over the following signature is verified: if_qs, app, true, if_low, add, tail, 0, le, s, isempty, false, if_high, high, head, low, quicksort, nil
Strategy
Projection
The following projection was used:
- π (high#): 2
- π (if_high#): 3
Thus, the following dependency pairs are removed:
| if_high#(true, n, add(m, x)) | → | high#(n, x) | | if_high#(false, n, add(m, x)) | → | high#(n, x) |
Problem 7: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| high#(n, add(m, x)) | → | if_high#(le(m, n), n, add(m, x)) |
Rewrite Rules
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | app(nil, y) | → | y |
| app(add(n, x), y) | → | add(n, app(x, y)) | | low(n, nil) | → | nil |
| low(n, add(m, x)) | → | if_low(le(m, n), n, add(m, x)) | | if_low(true, n, add(m, x)) | → | add(m, low(n, x)) |
| if_low(false, n, add(m, x)) | → | low(n, x) | | high(n, nil) | → | nil |
| high(n, add(m, x)) | → | if_high(le(m, n), n, add(m, x)) | | if_high(true, n, add(m, x)) | → | high(n, x) |
| if_high(false, n, add(m, x)) | → | add(m, high(n, x)) | | head(add(n, x)) | → | n |
| tail(add(n, x)) | → | x | | isempty(nil) | → | true |
| isempty(add(n, x)) | → | false | | quicksort(x) | → | if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) |
| if_qs(true, x, n, y) | → | nil | | if_qs(false, x, n, y) | → | app(quicksort(x), add(n, quicksort(y))) |
Original Signature
Termination of terms over the following signature is verified: if_qs, app, true, if_low, add, tail, 0, le, s, isempty, false, if_high, high, low, head, quicksort, nil
Strategy
There are no SCCs!
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| low#(n, add(m, x)) | → | if_low#(le(m, n), n, add(m, x)) | | if_low#(true, n, add(m, x)) | → | low#(n, x) |
| if_low#(false, n, add(m, x)) | → | low#(n, x) |
Rewrite Rules
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | app(nil, y) | → | y |
| app(add(n, x), y) | → | add(n, app(x, y)) | | low(n, nil) | → | nil |
| low(n, add(m, x)) | → | if_low(le(m, n), n, add(m, x)) | | if_low(true, n, add(m, x)) | → | add(m, low(n, x)) |
| if_low(false, n, add(m, x)) | → | low(n, x) | | high(n, nil) | → | nil |
| high(n, add(m, x)) | → | if_high(le(m, n), n, add(m, x)) | | if_high(true, n, add(m, x)) | → | high(n, x) |
| if_high(false, n, add(m, x)) | → | add(m, high(n, x)) | | head(add(n, x)) | → | n |
| tail(add(n, x)) | → | x | | isempty(nil) | → | true |
| isempty(add(n, x)) | → | false | | quicksort(x) | → | if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) |
| if_qs(true, x, n, y) | → | nil | | if_qs(false, x, n, y) | → | app(quicksort(x), add(n, quicksort(y))) |
Original Signature
Termination of terms over the following signature is verified: if_qs, app, true, if_low, add, tail, 0, le, s, isempty, false, if_high, high, head, low, quicksort, nil
Strategy
Projection
The following projection was used:
- π (if_low#): 3
- π (low#): 2
Thus, the following dependency pairs are removed:
| if_low#(true, n, add(m, x)) | → | low#(n, x) | | if_low#(false, n, add(m, x)) | → | low#(n, x) |
Problem 8: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| low#(n, add(m, x)) | → | if_low#(le(m, n), n, add(m, x)) |
Rewrite Rules
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | app(nil, y) | → | y |
| app(add(n, x), y) | → | add(n, app(x, y)) | | low(n, nil) | → | nil |
| low(n, add(m, x)) | → | if_low(le(m, n), n, add(m, x)) | | if_low(true, n, add(m, x)) | → | add(m, low(n, x)) |
| if_low(false, n, add(m, x)) | → | low(n, x) | | high(n, nil) | → | nil |
| high(n, add(m, x)) | → | if_high(le(m, n), n, add(m, x)) | | if_high(true, n, add(m, x)) | → | high(n, x) |
| if_high(false, n, add(m, x)) | → | add(m, high(n, x)) | | head(add(n, x)) | → | n |
| tail(add(n, x)) | → | x | | isempty(nil) | → | true |
| isempty(add(n, x)) | → | false | | quicksort(x) | → | if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) |
| if_qs(true, x, n, y) | → | nil | | if_qs(false, x, n, y) | → | app(quicksort(x), add(n, quicksort(y))) |
Original Signature
Termination of terms over the following signature is verified: if_qs, app, true, if_low, add, tail, 0, le, s, isempty, false, if_high, high, low, head, quicksort, nil
Strategy
There are no SCCs!
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| app#(add(n, x), y) | → | app#(x, y) |
Rewrite Rules
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | app(nil, y) | → | y |
| app(add(n, x), y) | → | add(n, app(x, y)) | | low(n, nil) | → | nil |
| low(n, add(m, x)) | → | if_low(le(m, n), n, add(m, x)) | | if_low(true, n, add(m, x)) | → | add(m, low(n, x)) |
| if_low(false, n, add(m, x)) | → | low(n, x) | | high(n, nil) | → | nil |
| high(n, add(m, x)) | → | if_high(le(m, n), n, add(m, x)) | | if_high(true, n, add(m, x)) | → | high(n, x) |
| if_high(false, n, add(m, x)) | → | add(m, high(n, x)) | | head(add(n, x)) | → | n |
| tail(add(n, x)) | → | x | | isempty(nil) | → | true |
| isempty(add(n, x)) | → | false | | quicksort(x) | → | if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) |
| if_qs(true, x, n, y) | → | nil | | if_qs(false, x, n, y) | → | app(quicksort(x), add(n, quicksort(y))) |
Original Signature
Termination of terms over the following signature is verified: if_qs, app, true, if_low, add, tail, 0, le, s, isempty, false, if_high, high, head, low, quicksort, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| app#(add(n, x), y) | → | app#(x, y) |
Problem 6: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| le#(s(x), s(y)) | → | le#(x, y) |
Rewrite Rules
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | app(nil, y) | → | y |
| app(add(n, x), y) | → | add(n, app(x, y)) | | low(n, nil) | → | nil |
| low(n, add(m, x)) | → | if_low(le(m, n), n, add(m, x)) | | if_low(true, n, add(m, x)) | → | add(m, low(n, x)) |
| if_low(false, n, add(m, x)) | → | low(n, x) | | high(n, nil) | → | nil |
| high(n, add(m, x)) | → | if_high(le(m, n), n, add(m, x)) | | if_high(true, n, add(m, x)) | → | high(n, x) |
| if_high(false, n, add(m, x)) | → | add(m, high(n, x)) | | head(add(n, x)) | → | n |
| tail(add(n, x)) | → | x | | isempty(nil) | → | true |
| isempty(add(n, x)) | → | false | | quicksort(x) | → | if_qs(isempty(x), low(head(x), tail(x)), head(x), high(head(x), tail(x))) |
| if_qs(true, x, n, y) | → | nil | | if_qs(false, x, n, y) | → | app(quicksort(x), add(n, quicksort(y))) |
Original Signature
Termination of terms over the following signature is verified: if_qs, app, true, if_low, add, tail, 0, le, s, isempty, false, if_high, high, head, low, quicksort, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| le#(s(x), s(y)) | → | le#(x, y) |