TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60063 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (444ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (10ms), PolynomialLinearRange4iUR (1121ms), DependencyGraph (9ms), PolynomialLinearRange8NegiUR (3443ms), DependencyGraph (8ms), ReductionPairSAT (1506ms)].
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (2ms), PolynomialLinearRange4iUR (3334ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (10046ms), DependencyGraph (1ms), ReductionPairSAT (2852ms)].
| – Problem 5 was processed with processor SubtermCriterion (0ms).
| – Problem 6 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (5ms), PolynomialLinearRange4iUR (21ms), DependencyGraph (4ms), PolynomialLinearRange8NegiUR (0ms), DependencyGraph (4ms), ReductionPairSAT (37109ms)].
| – Problem 7 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), y) | → | plus#(x, 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))) |
| plus(0, y) | → | y | | plus(s(x), y) | → | s(plus(x, y)) |
| minus(minus(x, y), z) | → | minus(x, plus(y, z)) | | app(nil, k) | → | k |
| app(l, nil) | → | l | | app(cons(x, l), k) | → | cons(x, app(l, k)) |
| sum(cons(x, nil)) | → | cons(x, nil) | | sum(cons(x, cons(y, l))) | → | sum(cons(plus(x, y), l)) |
| sum(app(l, cons(x, cons(y, k)))) | → | sum(app(l, sum(cons(x, cons(y, k))))) | | 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: plus, app, minus, true, sum, zero, id, not, 0, s, if, false, gt, quot, nil, cons
Open Dependency Pair Problem 4
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))) |
| plus(0, y) | → | y | | plus(s(x), y) | → | s(plus(x, y)) |
| minus(minus(x, y), z) | → | minus(x, plus(y, z)) | | app(nil, k) | → | k |
| app(l, nil) | → | l | | app(cons(x, l), k) | → | cons(x, app(l, k)) |
| sum(cons(x, nil)) | → | cons(x, nil) | | sum(cons(x, cons(y, l))) | → | sum(cons(plus(x, y), l)) |
| sum(app(l, cons(x, cons(y, k)))) | → | sum(app(l, sum(cons(x, cons(y, k))))) | | 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: plus, app, minus, true, sum, zero, id, not, 0, s, if, false, gt, quot, nil, cons
Open Dependency Pair Problem 6
Dependency Pairs
| sum#(cons(x, cons(y, l))) | → | sum#(cons(plus(x, y), l)) | | sum#(app(l, cons(x, cons(y, k)))) | → | sum#(app(l, sum(cons(x, cons(y, k))))) |
| sum#(app(l, cons(x, cons(y, k)))) | → | sum#(cons(x, cons(y, k))) |
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))) |
| plus(0, y) | → | y | | plus(s(x), y) | → | s(plus(x, y)) |
| minus(minus(x, y), z) | → | minus(x, plus(y, z)) | | app(nil, k) | → | k |
| app(l, nil) | → | l | | app(cons(x, l), k) | → | cons(x, app(l, k)) |
| sum(cons(x, nil)) | → | cons(x, nil) | | sum(cons(x, cons(y, l))) | → | sum(cons(plus(x, y), l)) |
| sum(app(l, cons(x, cons(y, k)))) | → | sum(app(l, sum(cons(x, cons(y, k))))) | | 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: plus, app, minus, true, sum, zero, id, not, 0, s, if, false, gt, quot, nil, cons
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| plus#(s(x), s(y)) | → | gt#(x, y) | | 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)) | → | if#(not(gt(x, y)), id(x), id(y)) | | plus#(s(x), y) | → | plus#(x, y) |
| gt#(s(x), s(y)) | → | gt#(x, y) | | quot#(s(x), s(y)) | → | quot#(minus(x, y), s(y)) |
| not#(x) | → | if#(x, false, true) | | app#(cons(x, l), k) | → | app#(l, k) |
| 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) | | sum#(cons(x, cons(y, l))) | → | sum#(cons(plus(x, y), l)) |
| plus#(id(x), s(y)) | → | gt#(s(y), 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), s(y)) | → | id#(x) |
| sum#(app(l, cons(x, cons(y, k)))) | → | app#(l, sum(cons(x, cons(y, k)))) | | plus#(s(x), s(y)) | → | plus#(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y))) |
| sum#(cons(x, cons(y, l))) | → | plus#(x, y) | | sum#(app(l, cons(x, cons(y, k)))) | → | sum#(cons(x, cons(y, k))) |
| sum#(app(l, cons(x, cons(y, k)))) | → | sum#(app(l, sum(cons(x, cons(y, k))))) | | minus#(s(x), s(y)) | → | minus#(x, y) |
| 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))) |
| plus(0, y) | → | y | | plus(s(x), y) | → | s(plus(x, y)) |
| minus(minus(x, y), z) | → | minus(x, plus(y, z)) | | app(nil, k) | → | k |
| app(l, nil) | → | l | | app(cons(x, l), k) | → | cons(x, app(l, k)) |
| sum(cons(x, nil)) | → | cons(x, nil) | | sum(cons(x, cons(y, l))) | → | sum(cons(plus(x, y), l)) |
| sum(app(l, cons(x, cons(y, k)))) | → | sum(app(l, sum(cons(x, cons(y, k))))) | | 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: plus, app, minus, true, sum, zero, id, not, 0, s, if, false, gt, quot, cons, nil
Strategy
The following SCCs where found
| sum#(cons(x, cons(y, l))) → sum#(cons(plus(x, y), l)) | sum#(app(l, cons(x, cons(y, k)))) → sum#(cons(x, cons(y, k))) |
| sum#(app(l, cons(x, cons(y, k)))) → sum#(app(l, sum(cons(x, cons(y, k))))) |
| minus#(s(x), s(y)) → minus#(x, y) | minus#(minus(x, y), z) → minus#(x, plus(y, z)) |
| gt#(s(x), s(y)) → gt#(x, y) |
| quot#(s(x), s(y)) → quot#(minus(x, y), s(y)) |
| app#(cons(x, l), k) → app#(l, k) |
| 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), y) → plus#(x, y) | plus#(s(x), x) → plus#(if(gt(x, x), id(x), id(x)), s(x)) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| app#(cons(x, l), k) | → | app#(l, k) |
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))) |
| plus(0, y) | → | y | | plus(s(x), y) | → | s(plus(x, y)) |
| minus(minus(x, y), z) | → | minus(x, plus(y, z)) | | app(nil, k) | → | k |
| app(l, nil) | → | l | | app(cons(x, l), k) | → | cons(x, app(l, k)) |
| sum(cons(x, nil)) | → | cons(x, nil) | | sum(cons(x, cons(y, l))) | → | sum(cons(plus(x, y), l)) |
| sum(app(l, cons(x, cons(y, k)))) | → | sum(app(l, sum(cons(x, cons(y, k))))) | | 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: plus, app, minus, true, sum, zero, id, not, 0, s, if, false, gt, quot, cons, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| app#(cons(x, l), k) | → | app#(l, k) |
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))) |
| plus(0, y) | → | y | | plus(s(x), y) | → | s(plus(x, y)) |
| minus(minus(x, y), z) | → | minus(x, plus(y, z)) | | app(nil, k) | → | k |
| app(l, nil) | → | l | | app(cons(x, l), k) | → | cons(x, app(l, k)) |
| sum(cons(x, nil)) | → | cons(x, nil) | | sum(cons(x, cons(y, l))) | → | sum(cons(plus(x, y), l)) |
| sum(app(l, cons(x, cons(y, k)))) | → | sum(app(l, sum(cons(x, cons(y, k))))) | | 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: plus, app, minus, true, sum, zero, id, not, 0, s, if, false, gt, quot, cons, nil
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)) |
Problem 7: 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))) |
| plus(0, y) | → | y | | plus(s(x), y) | → | s(plus(x, y)) |
| minus(minus(x, y), z) | → | minus(x, plus(y, z)) | | app(nil, k) | → | k |
| app(l, nil) | → | l | | app(cons(x, l), k) | → | cons(x, app(l, k)) |
| sum(cons(x, nil)) | → | cons(x, nil) | | sum(cons(x, cons(y, l))) | → | sum(cons(plus(x, y), l)) |
| sum(app(l, cons(x, cons(y, k)))) | → | sum(app(l, sum(cons(x, cons(y, k))))) | | 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: plus, app, minus, true, sum, zero, id, not, 0, s, if, false, gt, quot, cons, nil
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| gt#(s(x), s(y)) | → | gt#(x, y) |