TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60001 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (148ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (16ms), PolynomialLinearRange4iUR (3545ms), DependencyGraph (14ms), PolynomialLinearRange8NegiUR (timeout), DependencyGraph (12ms), ReductionPairSAT (22942ms), DependencyGraph (14ms), SizeChangePrinciple (209ms), ForwardNarrowing (2ms), BackwardInstantiation (3ms), ForwardInstantiation (3ms), Propagation (2ms)].
| – Problem 3 was processed with processor SubtermCriterion (1ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| cond3#(true, x, y) | → | cond3#(gr(x, 0), p(x), y) | | cond2#(false, x, y) | → | cond4#(gr(y, 0), x, y) |
| cond4#(false, x, y) | → | cond1#(and(gr(x, 0), gr(y, 0)), x, y) | | cond2#(true, x, y) | → | cond3#(gr(x, 0), x, y) |
| cond1#(true, x, y) | → | cond2#(gr(x, y), x, y) | | cond3#(false, x, y) | → | cond1#(and(gr(x, 0), gr(y, 0)), x, y) |
| cond4#(true, x, y) | → | cond4#(gr(y, 0), x, p(y)) |
Rewrite Rules
| cond1(true, x, y) | → | cond2(gr(x, y), x, y) | | cond2(true, x, y) | → | cond3(gr(x, 0), x, y) |
| cond2(false, x, y) | → | cond4(gr(y, 0), x, y) | | cond3(true, x, y) | → | cond3(gr(x, 0), p(x), y) |
| cond3(false, x, y) | → | cond1(and(gr(x, 0), gr(y, 0)), x, y) | | cond4(true, x, y) | → | cond4(gr(y, 0), x, p(y)) |
| cond4(false, x, y) | → | cond1(and(gr(x, 0), gr(y, 0)), x, y) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| and(true, true) | → | true | | and(false, x) | → | false |
| and(x, false) | → | false | | p(0) | → | 0 |
| p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: cond2, cond3, cond4, 0, s, p, false, true, gr, cond1, and
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| cond1#(true, x, y) | → | gr#(x, y) | | cond2#(false, x, y) | → | cond4#(gr(y, 0), x, y) |
| cond4#(true, x, y) | → | gr#(y, 0) | | cond3#(true, x, y) | → | gr#(x, 0) |
| cond3#(true, x, y) | → | p#(x) | | cond4#(false, x, y) | → | gr#(x, 0) |
| cond3#(false, x, y) | → | gr#(y, 0) | | cond2#(true, x, y) | → | gr#(x, 0) |
| cond4#(false, x, y) | → | and#(gr(x, 0), gr(y, 0)) | | cond3#(true, x, y) | → | cond3#(gr(x, 0), p(x), y) |
| cond4#(false, x, y) | → | cond1#(and(gr(x, 0), gr(y, 0)), x, y) | | cond3#(false, x, y) | → | gr#(x, 0) |
| cond2#(true, x, y) | → | cond3#(gr(x, 0), x, y) | | cond2#(false, x, y) | → | gr#(y, 0) |
| cond1#(true, x, y) | → | cond2#(gr(x, y), x, y) | | gr#(s(x), s(y)) | → | gr#(x, y) |
| cond3#(false, x, y) | → | cond1#(and(gr(x, 0), gr(y, 0)), x, y) | | cond4#(true, x, y) | → | cond4#(gr(y, 0), x, p(y)) |
| cond4#(true, x, y) | → | p#(y) | | cond3#(false, x, y) | → | and#(gr(x, 0), gr(y, 0)) |
| cond4#(false, x, y) | → | gr#(y, 0) |
Rewrite Rules
| cond1(true, x, y) | → | cond2(gr(x, y), x, y) | | cond2(true, x, y) | → | cond3(gr(x, 0), x, y) |
| cond2(false, x, y) | → | cond4(gr(y, 0), x, y) | | cond3(true, x, y) | → | cond3(gr(x, 0), p(x), y) |
| cond3(false, x, y) | → | cond1(and(gr(x, 0), gr(y, 0)), x, y) | | cond4(true, x, y) | → | cond4(gr(y, 0), x, p(y)) |
| cond4(false, x, y) | → | cond1(and(gr(x, 0), gr(y, 0)), x, y) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| and(true, true) | → | true | | and(false, x) | → | false |
| and(x, false) | → | false | | p(0) | → | 0 |
| p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: cond2, cond3, cond4, 0, s, p, true, false, gr, cond1, and
Strategy
The following SCCs where found
| cond3#(true, x, y) → cond3#(gr(x, 0), p(x), y) | cond2#(false, x, y) → cond4#(gr(y, 0), x, y) |
| cond4#(false, x, y) → cond1#(and(gr(x, 0), gr(y, 0)), x, y) | cond2#(true, x, y) → cond3#(gr(x, 0), x, y) |
| cond1#(true, x, y) → cond2#(gr(x, y), x, y) | cond3#(false, x, y) → cond1#(and(gr(x, 0), gr(y, 0)), x, y) |
| cond4#(true, x, y) → cond4#(gr(y, 0), x, p(y)) |
| gr#(s(x), s(y)) → gr#(x, y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| gr#(s(x), s(y)) | → | gr#(x, y) |
Rewrite Rules
| cond1(true, x, y) | → | cond2(gr(x, y), x, y) | | cond2(true, x, y) | → | cond3(gr(x, 0), x, y) |
| cond2(false, x, y) | → | cond4(gr(y, 0), x, y) | | cond3(true, x, y) | → | cond3(gr(x, 0), p(x), y) |
| cond3(false, x, y) | → | cond1(and(gr(x, 0), gr(y, 0)), x, y) | | cond4(true, x, y) | → | cond4(gr(y, 0), x, p(y)) |
| cond4(false, x, y) | → | cond1(and(gr(x, 0), gr(y, 0)), x, y) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| and(true, true) | → | true | | and(false, x) | → | false |
| and(x, false) | → | false | | p(0) | → | 0 |
| p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: cond2, cond3, cond4, 0, s, p, true, false, gr, cond1, and
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| gr#(s(x), s(y)) | → | gr#(x, y) |