TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60045 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (37ms).
| – Problem 2 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (2ms), PolynomialLinearRange4iUR (193ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (538ms), DependencyGraph (3ms), ReductionPairSAT (timeout)].
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor SubtermCriterion (0ms).
The following open problems remain:
Open Dependency Pair Problem 2
Dependency Pairs
| if#(true, b, c, x, y) | → | fibiter#(b, s(c), y, plus(x, y)) | | fibiter#(b, c, x, y) | → | if#(lt(c, b), b, c, x, y) |
Rewrite Rules
| plus(0, y) | → | y | | plus(s(x), y) | → | s(plus(x, y)) |
| lt(0, s(y)) | → | true | | lt(x, 0) | → | false |
| lt(s(x), s(y)) | → | lt(x, y) | | fib(x) | → | fibiter(x, 0, 0, s(0)) |
| fibiter(b, c, x, y) | → | if(lt(c, b), b, c, x, y) | | if(false, b, c, x, y) | → | x |
| if(true, b, c, x, y) | → | fibiter(b, s(c), y, plus(x, y)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, if, false, true, lt, fibiter, fib
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| fibiter#(b, c, x, y) | → | lt#(c, b) | | if#(true, b, c, x, y) | → | plus#(x, y) |
| plus#(s(x), y) | → | plus#(x, y) | | if#(true, b, c, x, y) | → | fibiter#(b, s(c), y, plus(x, y)) |
| fib#(x) | → | fibiter#(x, 0, 0, s(0)) | | fibiter#(b, c, x, y) | → | if#(lt(c, b), b, c, x, y) |
| lt#(s(x), s(y)) | → | lt#(x, y) |
Rewrite Rules
| plus(0, y) | → | y | | plus(s(x), y) | → | s(plus(x, y)) |
| lt(0, s(y)) | → | true | | lt(x, 0) | → | false |
| lt(s(x), s(y)) | → | lt(x, y) | | fib(x) | → | fibiter(x, 0, 0, s(0)) |
| fibiter(b, c, x, y) | → | if(lt(c, b), b, c, x, y) | | if(false, b, c, x, y) | → | x |
| if(true, b, c, x, y) | → | fibiter(b, s(c), y, plus(x, y)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, if, true, false, lt, fibiter, fib
Strategy
The following SCCs where found
| if#(true, b, c, x, y) → fibiter#(b, s(c), y, plus(x, y)) | fibiter#(b, c, x, y) → if#(lt(c, b), b, c, x, y) |
| plus#(s(x), y) → plus#(x, y) |
| lt#(s(x), s(y)) → lt#(x, y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| lt#(s(x), s(y)) | → | lt#(x, y) |
Rewrite Rules
| plus(0, y) | → | y | | plus(s(x), y) | → | s(plus(x, y)) |
| lt(0, s(y)) | → | true | | lt(x, 0) | → | false |
| lt(s(x), s(y)) | → | lt(x, y) | | fib(x) | → | fibiter(x, 0, 0, s(0)) |
| fibiter(b, c, x, y) | → | if(lt(c, b), b, c, x, y) | | if(false, b, c, x, y) | → | x |
| if(true, b, c, x, y) | → | fibiter(b, s(c), y, plus(x, y)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, if, true, false, lt, fibiter, fib
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| lt#(s(x), s(y)) | → | lt#(x, y) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| plus#(s(x), y) | → | plus#(x, y) |
Rewrite Rules
| plus(0, y) | → | y | | plus(s(x), y) | → | s(plus(x, y)) |
| lt(0, s(y)) | → | true | | lt(x, 0) | → | false |
| lt(s(x), s(y)) | → | lt(x, y) | | fib(x) | → | fibiter(x, 0, 0, s(0)) |
| fibiter(b, c, x, y) | → | if(lt(c, b), b, c, x, y) | | if(false, b, c, x, y) | → | x |
| if(true, b, c, x, y) | → | fibiter(b, s(c), y, plus(x, y)) |
Original Signature
Termination of terms over the following signature is verified: plus, 0, s, if, true, false, lt, fibiter, fib
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| plus#(s(x), y) | → | plus#(x, y) |