YES
The TRS could be proven terminating. The proof took 1663 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (19ms).
| – Problem 2 was processed with processor SubtermCriterion (3ms).
| – Problem 3 was processed with processor PolynomialLinearRange4iUR (691ms).
| | – Problem 4 was processed with processor PolynomialLinearRange4iUR (542ms).
| | | – Problem 5 was processed with processor PolynomialLinearRange4iUR (339ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| g#(x, c(y)) | → | g#(s(x), y) | | g#(x, c(y)) | → | g#(x, if(f(x), c(g(s(x), y)), c(y))) |
| g#(x, c(y)) | → | if#(f(x), c(g(s(x), y)), c(y)) | | g#(x, c(y)) | → | g#(x, y) |
| g#(x, c(y)) | → | f#(x) | | f#(s(x)) | → | f#(x) |
Rewrite Rules
| f(0) | → | true | | f(1) | → | false |
| f(s(x)) | → | f(x) | | if(true, s(x), s(y)) | → | s(x) |
| if(false, s(x), s(y)) | → | s(y) | | g(x, c(y)) | → | c(g(x, y)) |
| g(x, c(y)) | → | g(x, if(f(x), c(g(s(x), y)), c(y))) |
Original Signature
Termination of terms over the following signature is verified: f, g, 1, 0, s, c, if, true, false
Strategy
The following SCCs where found
| g#(x, c(y)) → g#(s(x), y) | g#(x, c(y)) → g#(x, if(f(x), c(g(s(x), y)), c(y))) |
| g#(x, c(y)) → g#(x, y) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| f(0) | → | true | | f(1) | → | false |
| f(s(x)) | → | f(x) | | if(true, s(x), s(y)) | → | s(x) |
| if(false, s(x), s(y)) | → | s(y) | | g(x, c(y)) | → | c(g(x, y)) |
| g(x, c(y)) | → | g(x, if(f(x), c(g(s(x), y)), c(y))) |
Original Signature
Termination of terms over the following signature is verified: f, g, 1, 0, s, c, if, true, false
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| g#(x, c(y)) | → | g#(s(x), y) | | g#(x, c(y)) | → | g#(x, if(f(x), c(g(s(x), y)), c(y))) |
| g#(x, c(y)) | → | g#(x, y) |
Rewrite Rules
| f(0) | → | true | | f(1) | → | false |
| f(s(x)) | → | f(x) | | if(true, s(x), s(y)) | → | s(x) |
| if(false, s(x), s(y)) | → | s(y) | | g(x, c(y)) | → | c(g(x, y)) |
| g(x, c(y)) | → | g(x, if(f(x), c(g(s(x), y)), c(y))) |
Original Signature
Termination of terms over the following signature is verified: f, g, 1, 0, s, c, if, true, false
Strategy
Polynomial Interpretation
- 0: 2
- 1: 3
- c(x): x + 2
- f(x): 0
- false: 3
- g(x,y): 0
- g#(x,y): y + 2x
- if(x,y,z): 2
- s(x): 1
- true: 3
Improved Usable rules
| if(false, s(x), s(y)) | → | s(y) | | if(true, s(x), s(y)) | → | s(x) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
Problem 4: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| g#(x, c(y)) | → | g#(s(x), y) | | g#(x, c(y)) | → | g#(x, if(f(x), c(g(s(x), y)), c(y))) |
Rewrite Rules
| f(0) | → | true | | f(1) | → | false |
| f(s(x)) | → | f(x) | | if(true, s(x), s(y)) | → | s(x) |
| if(false, s(x), s(y)) | → | s(y) | | g(x, c(y)) | → | c(g(x, y)) |
| g(x, c(y)) | → | g(x, if(f(x), c(g(s(x), y)), c(y))) |
Original Signature
Termination of terms over the following signature is verified: f, g, 1, 0, s, c, if, false, true
Strategy
Polynomial Interpretation
- 0: 3
- 1: 2
- c(x): 2x + 1
- f(x): 1
- false: 3
- g(x,y): y
- g#(x,y): 2y + x
- if(x,y,z): y
- s(x): 1
- true: 3
Improved Usable rules
| if(false, s(x), s(y)) | → | s(y) | | if(true, s(x), s(y)) | → | s(x) |
| g(x, c(y)) | → | c(g(x, y)) | | g(x, c(y)) | → | g(x, if(f(x), c(g(s(x), y)), c(y))) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| g#(x, c(y)) | → | g#(s(x), y) |
Problem 5: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| g#(x, c(y)) | → | g#(x, if(f(x), c(g(s(x), y)), c(y))) |
Rewrite Rules
| f(0) | → | true | | f(1) | → | false |
| f(s(x)) | → | f(x) | | if(true, s(x), s(y)) | → | s(x) |
| if(false, s(x), s(y)) | → | s(y) | | g(x, c(y)) | → | c(g(x, y)) |
| g(x, c(y)) | → | g(x, if(f(x), c(g(s(x), y)), c(y))) |
Original Signature
Termination of terms over the following signature is verified: f, g, 1, 0, s, c, if, true, false
Strategy
Polynomial Interpretation
- 0: 1
- 1: 3
- c(x): 1
- f(x): 0
- false: 2
- g(x,y): 2x + 1
- g#(x,y): 2y + 1
- if(x,y,z): 0
- s(x): 0
- true: 3
Improved Usable rules
| if(false, s(x), s(y)) | → | s(y) | | if(true, s(x), s(y)) | → | s(x) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| g#(x, c(y)) | → | g#(x, if(f(x), c(g(s(x), y)), c(y))) |