TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60003 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (70ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 remains open; application of the following processors failed [SubtermCriterion (0ms), DependencyGraph (4ms), PolynomialLinearRange4iUR (978ms), DependencyGraph (3ms), PolynomialLinearRange8NegiUR (20327ms), DependencyGraph (4ms), ReductionPairSAT (38553ms)].
The following open problems remain:
Open Dependency Pair Problem 4
Dependency Pairs
| cond2#(false, x, y, z) | → | cond1#(and(eq(x, y), gr(x, z)), x, y, z) | | cond1#(true, x, y, z) | → | cond2#(gr(y, z), x, y, z) |
| cond2#(true, x, y, z) | → | cond2#(gr(y, z), p(x), p(y), z) |
Rewrite Rules
| cond1(true, x, y, z) | → | cond2(gr(y, z), x, y, z) | | cond2(true, x, y, z) | → | cond2(gr(y, z), p(x), p(y), z) |
| cond2(false, x, y, z) | → | cond1(and(eq(x, y), gr(x, z)), x, y, z) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| p(0) | → | 0 | | p(s(x)) | → | x |
| eq(0, 0) | → | true | | eq(s(x), 0) | → | false |
| eq(0, s(x)) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| and(true, true) | → | true | | and(false, x) | → | false |
| and(x, false) | → | false |
Original Signature
Termination of terms over the following signature is verified: cond2, 0, s, p, false, true, gr, cond1, and, eq
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| cond2#(false, x, y, z) | → | and#(eq(x, y), gr(x, z)) | | cond2#(false, x, y, z) | → | gr#(x, z) |
| cond2#(true, x, y, z) | → | p#(x) | | cond2#(true, x, y, z) | → | p#(y) |
| cond2#(false, x, y, z) | → | cond1#(and(eq(x, y), gr(x, z)), x, y, z) | | cond2#(false, x, y, z) | → | eq#(x, y) |
| cond1#(true, x, y, z) | → | gr#(y, z) | | gr#(s(x), s(y)) | → | gr#(x, y) |
| cond2#(true, x, y, z) | → | gr#(y, z) | | cond1#(true, x, y, z) | → | cond2#(gr(y, z), x, y, z) |
| eq#(s(x), s(y)) | → | eq#(x, y) | | cond2#(true, x, y, z) | → | cond2#(gr(y, z), p(x), p(y), z) |
Rewrite Rules
| cond1(true, x, y, z) | → | cond2(gr(y, z), x, y, z) | | cond2(true, x, y, z) | → | cond2(gr(y, z), p(x), p(y), z) |
| cond2(false, x, y, z) | → | cond1(and(eq(x, y), gr(x, z)), x, y, z) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| p(0) | → | 0 | | p(s(x)) | → | x |
| eq(0, 0) | → | true | | eq(s(x), 0) | → | false |
| eq(0, s(x)) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| and(true, true) | → | true | | and(false, x) | → | false |
| and(x, false) | → | false |
Original Signature
Termination of terms over the following signature is verified: cond2, 0, s, p, true, false, gr, cond1, eq, and
Strategy
The following SCCs where found
| gr#(s(x), s(y)) → gr#(x, y) |
| eq#(s(x), s(y)) → eq#(x, y) |
| cond2#(false, x, y, z) → cond1#(and(eq(x, y), gr(x, z)), x, y, z) | cond1#(true, x, y, z) → cond2#(gr(y, z), x, y, z) |
| cond2#(true, x, y, z) → cond2#(gr(y, z), p(x), p(y), z) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| eq#(s(x), s(y)) | → | eq#(x, y) |
Rewrite Rules
| cond1(true, x, y, z) | → | cond2(gr(y, z), x, y, z) | | cond2(true, x, y, z) | → | cond2(gr(y, z), p(x), p(y), z) |
| cond2(false, x, y, z) | → | cond1(and(eq(x, y), gr(x, z)), x, y, z) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| p(0) | → | 0 | | p(s(x)) | → | x |
| eq(0, 0) | → | true | | eq(s(x), 0) | → | false |
| eq(0, s(x)) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| and(true, true) | → | true | | and(false, x) | → | false |
| and(x, false) | → | false |
Original Signature
Termination of terms over the following signature is verified: cond2, 0, s, p, true, false, gr, cond1, eq, and
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
| gr#(s(x), s(y)) | → | gr#(x, y) |
Rewrite Rules
| cond1(true, x, y, z) | → | cond2(gr(y, z), x, y, z) | | cond2(true, x, y, z) | → | cond2(gr(y, z), p(x), p(y), z) |
| cond2(false, x, y, z) | → | cond1(and(eq(x, y), gr(x, z)), x, y, z) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| p(0) | → | 0 | | p(s(x)) | → | x |
| eq(0, 0) | → | true | | eq(s(x), 0) | → | false |
| eq(0, s(x)) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| and(true, true) | → | true | | and(false, x) | → | false |
| and(x, false) | → | false |
Original Signature
Termination of terms over the following signature is verified: cond2, 0, s, p, true, false, gr, cond1, eq, and
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| gr#(s(x), s(y)) | → | gr#(x, y) |