YES
The TRS could be proven terminating. The proof took 39 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (22ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| | – Problem 4 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| and#(x, or(y, z)) | → | and#(x, z) | | and#(x, and(y, y)) | → | and#(x, y) |
| or#(x, or(y, y)) | → | or#(x, y) | | and#(x, or(y, z)) | → | or#(and(x, y), and(x, z)) |
| and#(x, or(y, z)) | → | and#(x, y) | | or#(or(x, y), and(y, z)) | → | or#(x, y) |
Rewrite Rules
| and(x, or(y, z)) | → | or(and(x, y), and(x, z)) | | and(x, and(y, y)) | → | and(x, y) |
| or(or(x, y), and(y, z)) | → | or(x, y) | | or(x, and(x, y)) | → | x |
| or(true, y) | → | true | | or(x, false) | → | x |
| or(x, x) | → | x | | or(x, or(y, y)) | → | or(x, y) |
| and(x, true) | → | x | | and(false, y) | → | false |
| and(x, x) | → | x |
Original Signature
Termination of terms over the following signature is verified: or, true, false, and
Strategy
The following SCCs where found
| and#(x, or(y, z)) → and#(x, z) | and#(x, and(y, y)) → and#(x, y) |
| and#(x, or(y, z)) → and#(x, y) |
| or#(x, or(y, y)) → or#(x, y) | or#(or(x, y), and(y, z)) → or#(x, y) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| or#(x, or(y, y)) | → | or#(x, y) | | or#(or(x, y), and(y, z)) | → | or#(x, y) |
Rewrite Rules
| and(x, or(y, z)) | → | or(and(x, y), and(x, z)) | | and(x, and(y, y)) | → | and(x, y) |
| or(or(x, y), and(y, z)) | → | or(x, y) | | or(x, and(x, y)) | → | x |
| or(true, y) | → | true | | or(x, false) | → | x |
| or(x, x) | → | x | | or(x, or(y, y)) | → | or(x, y) |
| and(x, true) | → | x | | and(false, y) | → | false |
| and(x, x) | → | x |
Original Signature
Termination of terms over the following signature is verified: or, true, false, and
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| or#(or(x, y), and(y, z)) | → | or#(x, y) |
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| or#(x, or(y, y)) | → | or#(x, y) |
Rewrite Rules
| and(x, or(y, z)) | → | or(and(x, y), and(x, z)) | | and(x, and(y, y)) | → | and(x, y) |
| or(or(x, y), and(y, z)) | → | or(x, y) | | or(x, and(x, y)) | → | x |
| or(true, y) | → | true | | or(x, false) | → | x |
| or(x, x) | → | x | | or(x, or(y, y)) | → | or(x, y) |
| and(x, true) | → | x | | and(false, y) | → | false |
| and(x, x) | → | x |
Original Signature
Termination of terms over the following signature is verified: or, false, true, and
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| or#(x, or(y, y)) | → | or#(x, y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| and#(x, or(y, z)) | → | and#(x, z) | | and#(x, and(y, y)) | → | and#(x, y) |
| and#(x, or(y, z)) | → | and#(x, y) |
Rewrite Rules
| and(x, or(y, z)) | → | or(and(x, y), and(x, z)) | | and(x, and(y, y)) | → | and(x, y) |
| or(or(x, y), and(y, z)) | → | or(x, y) | | or(x, and(x, y)) | → | x |
| or(true, y) | → | true | | or(x, false) | → | x |
| or(x, x) | → | x | | or(x, or(y, y)) | → | or(x, y) |
| and(x, true) | → | x | | and(false, y) | → | false |
| and(x, x) | → | x |
Original Signature
Termination of terms over the following signature is verified: or, true, false, and
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| and#(x, or(y, z)) | → | and#(x, z) | | and#(x, and(y, y)) | → | and#(x, y) |
| and#(x, or(y, z)) | → | and#(x, y) |