YES
The TRS could be proven terminating. The proof took 659 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (41ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor PolynomialLinearRange4iUR (308ms).
| | – Problem 5 was processed with processor PolynomialLinearRange4iUR (153ms).
| | | – Problem 6 was processed with processor DependencyGraph (15ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| if#(true, s(X), s(Y)) | → | minus#(X, Y) | | minus#(X, s(Y)) | → | pred#(minus(X, Y)) |
| gcd#(s(X), s(Y)) | → | le#(Y, X) | | if#(true, s(X), s(Y)) | → | gcd#(minus(X, Y), s(Y)) |
| if#(false, s(X), s(Y)) | → | gcd#(minus(Y, X), s(X)) | | gcd#(s(X), s(Y)) | → | if#(le(Y, X), s(X), s(Y)) |
| if#(false, s(X), s(Y)) | → | minus#(Y, X) | | minus#(X, s(Y)) | → | minus#(X, Y) |
| le#(s(X), s(Y)) | → | le#(X, Y) |
Rewrite Rules
| minus(X, s(Y)) | → | pred(minus(X, Y)) | | minus(X, 0) | → | X |
| pred(s(X)) | → | X | | le(s(X), s(Y)) | → | le(X, Y) |
| le(s(X), 0) | → | false | | le(0, Y) | → | true |
| gcd(0, Y) | → | 0 | | gcd(s(X), 0) | → | s(X) |
| gcd(s(X), s(Y)) | → | if(le(Y, X), s(X), s(Y)) | | if(true, s(X), s(Y)) | → | gcd(minus(X, Y), s(Y)) |
| if(false, s(X), s(Y)) | → | gcd(minus(Y, X), s(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, 0, s, le, if, false, true, pred, gcd
Strategy
The following SCCs where found
| if#(true, s(X), s(Y)) → gcd#(minus(X, Y), s(Y)) | if#(false, s(X), s(Y)) → gcd#(minus(Y, X), s(X)) |
| gcd#(s(X), s(Y)) → if#(le(Y, X), s(X), s(Y)) |
| minus#(X, s(Y)) → minus#(X, Y) |
| le#(s(X), s(Y)) → le#(X, Y) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| le#(s(X), s(Y)) | → | le#(X, Y) |
Rewrite Rules
| minus(X, s(Y)) | → | pred(minus(X, Y)) | | minus(X, 0) | → | X |
| pred(s(X)) | → | X | | le(s(X), s(Y)) | → | le(X, Y) |
| le(s(X), 0) | → | false | | le(0, Y) | → | true |
| gcd(0, Y) | → | 0 | | gcd(s(X), 0) | → | s(X) |
| gcd(s(X), s(Y)) | → | if(le(Y, X), s(X), s(Y)) | | if(true, s(X), s(Y)) | → | gcd(minus(X, Y), s(Y)) |
| if(false, s(X), s(Y)) | → | gcd(minus(Y, X), s(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, 0, s, le, if, false, true, pred, gcd
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| le#(s(X), s(Y)) | → | le#(X, Y) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| minus#(X, s(Y)) | → | minus#(X, Y) |
Rewrite Rules
| minus(X, s(Y)) | → | pred(minus(X, Y)) | | minus(X, 0) | → | X |
| pred(s(X)) | → | X | | le(s(X), s(Y)) | → | le(X, Y) |
| le(s(X), 0) | → | false | | le(0, Y) | → | true |
| gcd(0, Y) | → | 0 | | gcd(s(X), 0) | → | s(X) |
| gcd(s(X), s(Y)) | → | if(le(Y, X), s(X), s(Y)) | | if(true, s(X), s(Y)) | → | gcd(minus(X, Y), s(Y)) |
| if(false, s(X), s(Y)) | → | gcd(minus(Y, X), s(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, 0, s, le, if, false, true, pred, gcd
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| minus#(X, s(Y)) | → | minus#(X, Y) |
Problem 4: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| if#(true, s(X), s(Y)) | → | gcd#(minus(X, Y), s(Y)) | | if#(false, s(X), s(Y)) | → | gcd#(minus(Y, X), s(X)) |
| gcd#(s(X), s(Y)) | → | if#(le(Y, X), s(X), s(Y)) |
Rewrite Rules
| minus(X, s(Y)) | → | pred(minus(X, Y)) | | minus(X, 0) | → | X |
| pred(s(X)) | → | X | | le(s(X), s(Y)) | → | le(X, Y) |
| le(s(X), 0) | → | false | | le(0, Y) | → | true |
| gcd(0, Y) | → | 0 | | gcd(s(X), 0) | → | s(X) |
| gcd(s(X), s(Y)) | → | if(le(Y, X), s(X), s(Y)) | | if(true, s(X), s(Y)) | → | gcd(minus(X, Y), s(Y)) |
| if(false, s(X), s(Y)) | → | gcd(minus(Y, X), s(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, 0, s, le, if, false, true, pred, gcd
Strategy
Polynomial Interpretation
- 0: 2
- false: 1
- gcd(x,y): 0
- gcd#(x,y): y + 2x
- if(x,y,z): 0
- if#(x,y,z): z + y + 2x
- le(x,y): y
- minus(x,y): x
- pred(x): x
- s(x): 2x
- true: 0
Improved Usable rules
| pred(s(X)) | → | X | | minus(X, 0) | → | X |
| le(s(X), s(Y)) | → | le(X, Y) | | minus(X, s(Y)) | → | pred(minus(X, Y)) |
| le(0, Y) | → | true | | le(s(X), 0) | → | false |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| if#(false, s(X), s(Y)) | → | gcd#(minus(Y, X), s(X)) |
Problem 5: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| if#(true, s(X), s(Y)) | → | gcd#(minus(X, Y), s(Y)) | | gcd#(s(X), s(Y)) | → | if#(le(Y, X), s(X), s(Y)) |
Rewrite Rules
| minus(X, s(Y)) | → | pred(minus(X, Y)) | | minus(X, 0) | → | X |
| pred(s(X)) | → | X | | le(s(X), s(Y)) | → | le(X, Y) |
| le(s(X), 0) | → | false | | le(0, Y) | → | true |
| gcd(0, Y) | → | 0 | | gcd(s(X), 0) | → | s(X) |
| gcd(s(X), s(Y)) | → | if(le(Y, X), s(X), s(Y)) | | if(true, s(X), s(Y)) | → | gcd(minus(X, Y), s(Y)) |
| if(false, s(X), s(Y)) | → | gcd(minus(Y, X), s(X)) |
Original Signature
Termination of terms over the following signature is verified: 0, minus, le, s, if, true, false, pred, gcd
Strategy
Polynomial Interpretation
- 0: 3
- false: 2
- gcd(x,y): 0
- gcd#(x,y): x
- if(x,y,z): 0
- if#(x,y,z): y
- le(x,y): 0
- minus(x,y): x + 1
- pred(x): x
- s(x): x + 3
- true: 0
Improved Usable rules
| pred(s(X)) | → | X | | minus(X, 0) | → | X |
| minus(X, s(Y)) | → | pred(minus(X, Y)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| if#(true, s(X), s(Y)) | → | gcd#(minus(X, Y), s(Y)) |
Problem 6: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| gcd#(s(X), s(Y)) | → | if#(le(Y, X), s(X), s(Y)) |
Rewrite Rules
| minus(X, s(Y)) | → | pred(minus(X, Y)) | | minus(X, 0) | → | X |
| pred(s(X)) | → | X | | le(s(X), s(Y)) | → | le(X, Y) |
| le(s(X), 0) | → | false | | le(0, Y) | → | true |
| gcd(0, Y) | → | 0 | | gcd(s(X), 0) | → | s(X) |
| gcd(s(X), s(Y)) | → | if(le(Y, X), s(X), s(Y)) | | if(true, s(X), s(Y)) | → | gcd(minus(X, Y), s(Y)) |
| if(false, s(X), s(Y)) | → | gcd(minus(Y, X), s(X)) |
Original Signature
Termination of terms over the following signature is verified: minus, 0, s, le, if, false, true, pred, gcd
Strategy
There are no SCCs!