YES
The TRS could be proven terminating. The proof took 28 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (13ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| | – Problem 4 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (0ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| odd#(s(x)) | → | odd#(x) | | +#(x, s(y)) | → | +#(x, y) |
| odd#(s(x)) | → | not#(odd(x)) | | +#(s(x), y) | → | +#(x, y) |
Rewrite Rules
| not(true) | → | false | | not(false) | → | true |
| odd(0) | → | false | | odd(s(x)) | → | not(odd(x)) |
| +(x, 0) | → | x | | +(x, s(y)) | → | s(+(x, y)) |
| +(s(x), y) | → | s(+(x, y)) |
Original Signature
Termination of terms over the following signature is verified: not, 0, s, false, true, +, odd
Strategy
The following SCCs where found
| +#(x, s(y)) → +#(x, y) | +#(s(x), y) → +#(x, y) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| +#(x, s(y)) | → | +#(x, y) | | +#(s(x), y) | → | +#(x, y) |
Rewrite Rules
| not(true) | → | false | | not(false) | → | true |
| odd(0) | → | false | | odd(s(x)) | → | not(odd(x)) |
| +(x, 0) | → | x | | +(x, s(y)) | → | s(+(x, y)) |
| +(s(x), y) | → | s(+(x, y)) |
Original Signature
Termination of terms over the following signature is verified: not, 0, s, false, true, +, odd
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| not(true) | → | false | | not(false) | → | true |
| odd(0) | → | false | | odd(s(x)) | → | not(odd(x)) |
| +(x, 0) | → | x | | +(x, s(y)) | → | s(+(x, y)) |
| +(s(x), y) | → | s(+(x, y)) |
Original Signature
Termination of terms over the following signature is verified: not, 0, s, +, true, false, odd
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| not(true) | → | false | | not(false) | → | true |
| odd(0) | → | false | | odd(s(x)) | → | not(odd(x)) |
| +(x, 0) | → | x | | +(x, s(y)) | → | s(+(x, y)) |
| +(s(x), y) | → | s(+(x, y)) |
Original Signature
Termination of terms over the following signature is verified: not, 0, s, false, true, +, odd
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed: