YES
The TRS could be proven terminating. The proof took 369 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (8ms).
| – Problem 2 was processed with processor SubtermCriterion (0ms).
| – Problem 3 was processed with processor PolynomialLinearRange4iUR (194ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| less_leaves#(cons(u, v), cons(w, z)) | → | less_leaves#(concat(u, v), concat(w, z)) | | less_leaves#(cons(u, v), cons(w, z)) | → | concat#(u, v) |
| less_leaves#(cons(u, v), cons(w, z)) | → | concat#(w, z) | | concat#(cons(u, v), y) | → | concat#(v, y) |
Rewrite Rules
| concat(leaf, y) | → | y | | concat(cons(u, v), y) | → | cons(u, concat(v, y)) |
| less_leaves(x, leaf) | → | false | | less_leaves(leaf, cons(w, z)) | → | true |
| less_leaves(cons(u, v), cons(w, z)) | → | less_leaves(concat(u, v), concat(w, z)) |
Original Signature
Termination of terms over the following signature is verified: false, true, leaf, concat, less_leaves, cons
Strategy
The following SCCs where found
| less_leaves#(cons(u, v), cons(w, z)) → less_leaves#(concat(u, v), concat(w, z)) |
| concat#(cons(u, v), y) → concat#(v, y) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| concat#(cons(u, v), y) | → | concat#(v, y) |
Rewrite Rules
| concat(leaf, y) | → | y | | concat(cons(u, v), y) | → | cons(u, concat(v, y)) |
| less_leaves(x, leaf) | → | false | | less_leaves(leaf, cons(w, z)) | → | true |
| less_leaves(cons(u, v), cons(w, z)) | → | less_leaves(concat(u, v), concat(w, z)) |
Original Signature
Termination of terms over the following signature is verified: false, true, leaf, concat, less_leaves, cons
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| concat#(cons(u, v), y) | → | concat#(v, y) |
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| less_leaves#(cons(u, v), cons(w, z)) | → | less_leaves#(concat(u, v), concat(w, z)) |
Rewrite Rules
| concat(leaf, y) | → | y | | concat(cons(u, v), y) | → | cons(u, concat(v, y)) |
| less_leaves(x, leaf) | → | false | | less_leaves(leaf, cons(w, z)) | → | true |
| less_leaves(cons(u, v), cons(w, z)) | → | less_leaves(concat(u, v), concat(w, z)) |
Original Signature
Termination of terms over the following signature is verified: false, true, leaf, concat, less_leaves, cons
Strategy
Polynomial Interpretation
- concat(x,y): y + x + 1
- cons(x,y): y + 2x + 2
- false: 0
- leaf: 0
- less_leaves(x,y): 0
- less_leaves#(x,y): x + 1
- true: 0
Improved Usable rules
| concat(leaf, y) | → | y | | concat(cons(u, v), y) | → | cons(u, concat(v, y)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| less_leaves#(cons(u, v), cons(w, z)) | → | less_leaves#(concat(u, v), concat(w, z)) |