YES
The TRS could be proven terminating. The proof took 238 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (63ms).
| – Problem 2 was processed with processor SubtermCriterion (11ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| or#(x, y) | → | xor#(x, y) | | impl#(x, y) | → | xor#(and(x, y), xor(x, T)) |
| and#(xor(x, y), z) | → | xor#(and(x, z), and(y, z)) | | or#(x, y) | → | and#(x, y) |
| neg#(x) | → | xor#(x, T) | | and#(xor(x, y), z) | → | and#(y, z) |
| equiv#(x, y) | → | xor#(x, xor(y, T)) | | impl#(x, y) | → | xor#(x, T) |
| and#(xor(x, y), z) | → | and#(x, z) | | impl#(x, y) | → | and#(x, y) |
| or#(x, y) | → | xor#(and(x, y), xor(x, y)) | | equiv#(x, y) | → | xor#(y, T) |
Rewrite Rules
| xor(x, F) | → | x | | xor(x, neg(x)) | → | F |
| and(x, T) | → | x | | and(x, F) | → | F |
| and(x, x) | → | x | | and(xor(x, y), z) | → | xor(and(x, z), and(y, z)) |
| xor(x, x) | → | F | | impl(x, y) | → | xor(and(x, y), xor(x, T)) |
| or(x, y) | → | xor(and(x, y), xor(x, y)) | | equiv(x, y) | → | xor(x, xor(y, T)) |
| neg(x) | → | xor(x, T) |
Original Signature
Termination of terms over the following signature is verified: neg, T, xor, F, impl, or, equiv, and
Strategy
The following SCCs where found
| and#(xor(x, y), z) → and#(y, z) | and#(xor(x, y), z) → and#(x, z) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| and#(xor(x, y), z) | → | and#(y, z) | | and#(xor(x, y), z) | → | and#(x, z) |
Rewrite Rules
| xor(x, F) | → | x | | xor(x, neg(x)) | → | F |
| and(x, T) | → | x | | and(x, F) | → | F |
| and(x, x) | → | x | | and(xor(x, y), z) | → | xor(and(x, z), and(y, z)) |
| xor(x, x) | → | F | | impl(x, y) | → | xor(and(x, y), xor(x, T)) |
| or(x, y) | → | xor(and(x, y), xor(x, y)) | | equiv(x, y) | → | xor(x, xor(y, T)) |
| neg(x) | → | xor(x, T) |
Original Signature
Termination of terms over the following signature is verified: neg, T, xor, F, impl, or, equiv, and
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| and#(xor(x, y), z) | → | and#(y, z) | | and#(xor(x, y), z) | → | and#(x, z) |