YES
The TRS could be proven terminating. The proof took 19 ms.
The following DP Processors were used
Problem 1 was processed with processor SubtermCriterion (0ms).
| – Problem 2 was processed with processor DependencyGraph (1ms).
| | – Problem 3 was processed with processor SubtermCriterion (0ms).
Problem 1: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| u21#(ackout(X), Y) | → | ackin#(Y, X) | | ackin#(s(X), s(Y)) | → | ackin#(s(X), Y) |
| ackin#(s(X), s(Y)) | → | u21#(ackin(s(X), Y), X) |
Rewrite Rules
| ackin(s(X), s(Y)) | → | u21(ackin(s(X), Y), X) | | u21(ackout(X), Y) | → | u22(ackin(Y, X)) |
Original Signature
Termination of terms over the following signature is verified: u22, u21, s, ackin, ackout
Strategy
Projection
The following projection was used:
- π (u21#): 2
- π (ackin#): 1
Thus, the following dependency pairs are removed:
| ackin#(s(X), s(Y)) | → | u21#(ackin(s(X), Y), X) |
Problem 2: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| u21#(ackout(X), Y) | → | ackin#(Y, X) | | ackin#(s(X), s(Y)) | → | ackin#(s(X), Y) |
Rewrite Rules
| ackin(s(X), s(Y)) | → | u21(ackin(s(X), Y), X) | | u21(ackout(X), Y) | → | u22(ackin(Y, X)) |
Original Signature
Termination of terms over the following signature is verified: u22, u21, s, ackin, ackout
Strategy
The following SCCs where found
| ackin#(s(X), s(Y)) → ackin#(s(X), Y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| ackin#(s(X), s(Y)) | → | ackin#(s(X), Y) |
Rewrite Rules
| ackin(s(X), s(Y)) | → | u21(ackin(s(X), Y), X) | | u21(ackout(X), Y) | → | u22(ackin(Y, X)) |
Original Signature
Termination of terms over the following signature is verified: u22, u21, s, ackin, ackout
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| ackin#(s(X), s(Y)) | → | ackin#(s(X), Y) |