TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60017 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (144ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
| – Problem 5 remains open; application of the following processors failed [SubtermCriterion (3ms), DependencyGraph (10ms), PolynomialLinearRange4iUR (554ms), DependencyGraph (9ms), PolynomialLinearRange8NegiUR (3040ms), DependencyGraph (9ms), ReductionPairSAT (56195ms)].
The following open problems remain:
Open Dependency Pair Problem 5
Dependency Pairs
| if2#(true, x, y, c, edge(u, v, i), j) | → | reach#(v, y, s(c), j, j) | | if1#(false, x, y, c, i, j) | → | if2#(le(c, size(j)), x, y, c, i, j) |
| reach#(x, y, c, i, j) | → | if1#(eq(x, y), x, y, c, i, j) | | if2#(true, x, y, c, edge(u, v, i), j) | → | if2#(true, x, y, c, i, j) |
Rewrite Rules
| eq(0, 0) | → | true | | eq(0, s(x)) | → | false |
| eq(s(x), 0) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| or(true, y) | → | true | | or(false, y) | → | y |
| and(true, y) | → | y | | and(false, y) | → | false |
| size(empty) | → | 0 | | size(edge(x, y, i)) | → | s(size(i)) |
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | reachable(x, y, i) | → | reach(x, y, 0, i, i) |
| reach(x, y, c, i, j) | → | if1(eq(x, y), x, y, c, i, j) | | if1(true, x, y, c, i, j) | → | true |
| if1(false, x, y, c, i, j) | → | if2(le(c, size(j)), x, y, c, i, j) | | if2(false, x, y, c, i, j) | → | false |
| if2(true, x, y, c, empty, j) | → | false | | if2(true, x, y, c, edge(u, v, i), j) | → | or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) |
Original Signature
Termination of terms over the following signature is verified: edge, or, true, if1, if2, and, size, reach, 0, s, le, reachable, empty, false, eq
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| reachable#(x, y, i) | → | reach#(x, y, 0, i, i) | | reach#(x, y, c, i, j) | → | eq#(x, y) |
| if2#(true, x, y, c, edge(u, v, i), j) | → | eq#(x, u) | | if1#(false, x, y, c, i, j) | → | if2#(le(c, size(j)), x, y, c, i, j) |
| if1#(false, x, y, c, i, j) | → | le#(c, size(j)) | | if2#(true, x, y, c, edge(u, v, i), j) | → | and#(eq(x, u), reach(v, y, s(c), j, j)) |
| if1#(false, x, y, c, i, j) | → | size#(j) | | if2#(true, x, y, c, edge(u, v, i), j) | → | or#(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) |
| le#(s(x), s(y)) | → | le#(x, y) | | if2#(true, x, y, c, edge(u, v, i), j) | → | reach#(v, y, s(c), j, j) |
| size#(edge(x, y, i)) | → | size#(i) | | eq#(s(x), s(y)) | → | eq#(x, y) |
| if2#(true, x, y, c, edge(u, v, i), j) | → | if2#(true, x, y, c, i, j) | | reach#(x, y, c, i, j) | → | if1#(eq(x, y), x, y, c, i, j) |
Rewrite Rules
| eq(0, 0) | → | true | | eq(0, s(x)) | → | false |
| eq(s(x), 0) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| or(true, y) | → | true | | or(false, y) | → | y |
| and(true, y) | → | y | | and(false, y) | → | false |
| size(empty) | → | 0 | | size(edge(x, y, i)) | → | s(size(i)) |
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | reachable(x, y, i) | → | reach(x, y, 0, i, i) |
| reach(x, y, c, i, j) | → | if1(eq(x, y), x, y, c, i, j) | | if1(true, x, y, c, i, j) | → | true |
| if1(false, x, y, c, i, j) | → | if2(le(c, size(j)), x, y, c, i, j) | | if2(false, x, y, c, i, j) | → | false |
| if2(true, x, y, c, empty, j) | → | false | | if2(true, x, y, c, edge(u, v, i), j) | → | or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) |
Original Signature
Termination of terms over the following signature is verified: edge, or, true, if1, if2, and, size, reach, 0, s, le, reachable, empty, false, eq
Strategy
The following SCCs where found
| le#(s(x), s(y)) → le#(x, y) |
| size#(edge(x, y, i)) → size#(i) |
| eq#(s(x), s(y)) → eq#(x, y) |
| if2#(true, x, y, c, edge(u, v, i), j) → reach#(v, y, s(c), j, j) | if1#(false, x, y, c, i, j) → if2#(le(c, size(j)), x, y, c, i, j) |
| if2#(true, x, y, c, edge(u, v, i), j) → if2#(true, x, y, c, i, j) | reach#(x, y, c, i, j) → if1#(eq(x, y), x, y, c, i, j) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| size#(edge(x, y, i)) | → | size#(i) |
Rewrite Rules
| eq(0, 0) | → | true | | eq(0, s(x)) | → | false |
| eq(s(x), 0) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| or(true, y) | → | true | | or(false, y) | → | y |
| and(true, y) | → | y | | and(false, y) | → | false |
| size(empty) | → | 0 | | size(edge(x, y, i)) | → | s(size(i)) |
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | reachable(x, y, i) | → | reach(x, y, 0, i, i) |
| reach(x, y, c, i, j) | → | if1(eq(x, y), x, y, c, i, j) | | if1(true, x, y, c, i, j) | → | true |
| if1(false, x, y, c, i, j) | → | if2(le(c, size(j)), x, y, c, i, j) | | if2(false, x, y, c, i, j) | → | false |
| if2(true, x, y, c, empty, j) | → | false | | if2(true, x, y, c, edge(u, v, i), j) | → | or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) |
Original Signature
Termination of terms over the following signature is verified: edge, or, true, if1, if2, and, size, reach, 0, s, le, reachable, empty, false, eq
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| size#(edge(x, y, i)) | → | size#(i) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| le#(s(x), s(y)) | → | le#(x, y) |
Rewrite Rules
| eq(0, 0) | → | true | | eq(0, s(x)) | → | false |
| eq(s(x), 0) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| or(true, y) | → | true | | or(false, y) | → | y |
| and(true, y) | → | y | | and(false, y) | → | false |
| size(empty) | → | 0 | | size(edge(x, y, i)) | → | s(size(i)) |
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | reachable(x, y, i) | → | reach(x, y, 0, i, i) |
| reach(x, y, c, i, j) | → | if1(eq(x, y), x, y, c, i, j) | | if1(true, x, y, c, i, j) | → | true |
| if1(false, x, y, c, i, j) | → | if2(le(c, size(j)), x, y, c, i, j) | | if2(false, x, y, c, i, j) | → | false |
| if2(true, x, y, c, empty, j) | → | false | | if2(true, x, y, c, edge(u, v, i), j) | → | or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) |
Original Signature
Termination of terms over the following signature is verified: edge, or, true, if1, if2, and, size, reach, 0, s, le, reachable, empty, false, eq
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| le#(s(x), s(y)) | → | le#(x, y) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| eq#(s(x), s(y)) | → | eq#(x, y) |
Rewrite Rules
| eq(0, 0) | → | true | | eq(0, s(x)) | → | false |
| eq(s(x), 0) | → | false | | eq(s(x), s(y)) | → | eq(x, y) |
| or(true, y) | → | true | | or(false, y) | → | y |
| and(true, y) | → | y | | and(false, y) | → | false |
| size(empty) | → | 0 | | size(edge(x, y, i)) | → | s(size(i)) |
| le(0, y) | → | true | | le(s(x), 0) | → | false |
| le(s(x), s(y)) | → | le(x, y) | | reachable(x, y, i) | → | reach(x, y, 0, i, i) |
| reach(x, y, c, i, j) | → | if1(eq(x, y), x, y, c, i, j) | | if1(true, x, y, c, i, j) | → | true |
| if1(false, x, y, c, i, j) | → | if2(le(c, size(j)), x, y, c, i, j) | | if2(false, x, y, c, i, j) | → | false |
| if2(true, x, y, c, empty, j) | → | false | | if2(true, x, y, c, edge(u, v, i), j) | → | or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j))) |
Original Signature
Termination of terms over the following signature is verified: edge, or, true, if1, if2, and, size, reach, 0, s, le, reachable, empty, false, eq
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| eq#(s(x), s(y)) | → | eq#(x, y) |