YES
The TRS could be proven terminating. The proof took 2707 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (176ms).
| – Problem 2 was processed with processor PolynomialLinearRange4iUR (1182ms).
| | – Problem 3 was processed with processor DependencyGraph (13ms).
| | | – Problem 4 was processed with processor PolynomialLinearRange4iUR (422ms).
| | | | – Problem 5 was processed with processor PolynomialLinearRange4iUR (293ms).
| | | | | – Problem 6 was processed with processor PolynomialLinearRange4iUR (381ms).
| | | | | | – Problem 7 was processed with processor DependencyGraph (2ms).
| | | | | | | – Problem 8 was processed with processor PolynomialLinearRange4iUR (149ms).
| | | | | | | | – Problem 9 was processed with processor DependencyGraph (0ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(first(X1, X2)) | → | mark#(X2) | | mark#(if(X1, X2, X3)) | → | a__if#(mark(X1), X2, X3) |
| mark#(first(X1, X2)) | → | a__first#(mark(X1), mark(X2)) | | mark#(from(X)) | → | a__from#(X) |
| a__and#(true, X) | → | mark#(X) | | a__if#(true, X, Y) | → | mark#(X) |
| a__add#(0, X) | → | mark#(X) | | mark#(and(X1, X2)) | → | mark#(X1) |
| mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) | | mark#(add(X1, X2)) | → | a__add#(mark(X1), X2) |
| a__if#(false, X, Y) | → | mark#(Y) | | mark#(first(X1, X2)) | → | mark#(X1) |
| mark#(if(X1, X2, X3)) | → | mark#(X1) | | mark#(add(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| a__and(true, X) | → | mark(X) | | a__and(false, Y) | → | false |
| a__if(true, X, Y) | → | mark(X) | | a__if(false, X, Y) | → | mark(Y) |
| a__add(0, X) | → | mark(X) | | a__add(s(X), Y) | → | s(add(X, Y)) |
| a__first(0, X) | → | nil | | a__first(s(X), cons(Y, Z)) | → | cons(Y, first(X, Z)) |
| a__from(X) | → | cons(X, from(s(X))) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(if(X1, X2, X3)) | → | a__if(mark(X1), X2, X3) | | mark(add(X1, X2)) | → | a__add(mark(X1), X2) |
| mark(first(X1, X2)) | → | a__first(mark(X1), mark(X2)) | | mark(from(X)) | → | a__from(X) |
| mark(true) | → | true | | mark(false) | → | false |
| mark(0) | → | 0 | | mark(s(X)) | → | s(X) |
| mark(nil) | → | nil | | mark(cons(X1, X2)) | → | cons(X1, X2) |
| a__and(X1, X2) | → | and(X1, X2) | | a__if(X1, X2, X3) | → | if(X1, X2, X3) |
| a__add(X1, X2) | → | add(X1, X2) | | a__first(X1, X2) | → | first(X1, X2) |
| a__from(X) | → | from(X) |
Original Signature
Termination of terms over the following signature is verified: true, mark, from, a__add, add, and, a__and, 0, a__if, s, if, false, first, a__first, a__from, cons, nil
Strategy
The following SCCs where found
| mark#(and(X1, X2)) → mark#(X1) | mark#(first(X1, X2)) → mark#(X2) |
| mark#(if(X1, X2, X3)) → a__if#(mark(X1), X2, X3) | mark#(add(X1, X2)) → a__add#(mark(X1), X2) |
| mark#(and(X1, X2)) → a__and#(mark(X1), X2) | a__if#(false, X, Y) → mark#(Y) |
| mark#(first(X1, X2)) → mark#(X1) | mark#(if(X1, X2, X3)) → mark#(X1) |
| a__and#(true, X) → mark#(X) | a__if#(true, X, Y) → mark#(X) |
| mark#(add(X1, X2)) → mark#(X1) | a__add#(0, X) → mark#(X) |
Problem 2: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(and(X1, X2)) | → | mark#(X1) | | mark#(first(X1, X2)) | → | mark#(X2) |
| mark#(add(X1, X2)) | → | a__add#(mark(X1), X2) | | mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) |
| mark#(if(X1, X2, X3)) | → | a__if#(mark(X1), X2, X3) | | a__if#(false, X, Y) | → | mark#(Y) |
| mark#(first(X1, X2)) | → | mark#(X1) | | mark#(if(X1, X2, X3)) | → | mark#(X1) |
| a__and#(true, X) | → | mark#(X) | | a__if#(true, X, Y) | → | mark#(X) |
| mark#(add(X1, X2)) | → | mark#(X1) | | a__add#(0, X) | → | mark#(X) |
Rewrite Rules
| a__and(true, X) | → | mark(X) | | a__and(false, Y) | → | false |
| a__if(true, X, Y) | → | mark(X) | | a__if(false, X, Y) | → | mark(Y) |
| a__add(0, X) | → | mark(X) | | a__add(s(X), Y) | → | s(add(X, Y)) |
| a__first(0, X) | → | nil | | a__first(s(X), cons(Y, Z)) | → | cons(Y, first(X, Z)) |
| a__from(X) | → | cons(X, from(s(X))) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(if(X1, X2, X3)) | → | a__if(mark(X1), X2, X3) | | mark(add(X1, X2)) | → | a__add(mark(X1), X2) |
| mark(first(X1, X2)) | → | a__first(mark(X1), mark(X2)) | | mark(from(X)) | → | a__from(X) |
| mark(true) | → | true | | mark(false) | → | false |
| mark(0) | → | 0 | | mark(s(X)) | → | s(X) |
| mark(nil) | → | nil | | mark(cons(X1, X2)) | → | cons(X1, X2) |
| a__and(X1, X2) | → | and(X1, X2) | | a__if(X1, X2, X3) | → | if(X1, X2, X3) |
| a__add(X1, X2) | → | add(X1, X2) | | a__first(X1, X2) | → | first(X1, X2) |
| a__from(X) | → | from(X) |
Original Signature
Termination of terms over the following signature is verified: true, mark, from, a__add, add, and, a__and, 0, a__if, s, if, false, first, a__first, a__from, cons, nil
Strategy
Polynomial Interpretation
- 0: 0
- a__add(x,y): 2y + 2x
- a__add#(x,y): 2y + 1
- a__and(x,y): 2y + x
- a__and#(x,y): 2y + 1
- a__first(x,y): y + 2x
- a__from(x): 1
- a__if(x,y,z): 2z + 2y + x
- a__if#(x,y,z): 2z + 3y + x
- add(x,y): y + 2x
- and(x,y): y + x
- cons(x,y): 1
- false: 1
- first(x,y): y + 2x
- from(x): 1
- if(x,y,z): 2z + 2y + x
- mark(x): 2x
- mark#(x): 2x + 1
- nil: 0
- s(x): x + 1
- true: 1
Improved Usable rules
| a__first(X1, X2) | → | first(X1, X2) | | a__from(X) | → | cons(X, from(s(X))) |
| a__from(X) | → | from(X) | | a__and(true, X) | → | mark(X) |
| a__if(X1, X2, X3) | → | if(X1, X2, X3) | | a__first(0, X) | → | nil |
| a__add(X1, X2) | → | add(X1, X2) | | a__first(s(X), cons(Y, Z)) | → | cons(Y, first(X, Z)) |
| mark(nil) | → | nil | | a__add(0, X) | → | mark(X) |
| a__and(false, Y) | → | false | | mark(s(X)) | → | s(X) |
| mark(0) | → | 0 | | mark(cons(X1, X2)) | → | cons(X1, X2) |
| a__and(X1, X2) | → | and(X1, X2) | | a__if(false, X, Y) | → | mark(Y) |
| mark(false) | → | false | | mark(first(X1, X2)) | → | a__first(mark(X1), mark(X2)) |
| mark(if(X1, X2, X3)) | → | a__if(mark(X1), X2, X3) | | mark(true) | → | true |
| mark(from(X)) | → | a__from(X) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| a__add(s(X), Y) | → | s(add(X, Y)) | | a__if(true, X, Y) | → | mark(X) |
| mark(add(X1, X2)) | → | a__add(mark(X1), X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(if(X1, X2, X3)) | → | a__if#(mark(X1), X2, X3) |
Problem 3: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(and(X1, X2)) | → | mark#(X1) | | mark#(first(X1, X2)) | → | mark#(X2) |
| mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) | | mark#(add(X1, X2)) | → | a__add#(mark(X1), X2) |
| a__if#(false, X, Y) | → | mark#(Y) | | mark#(first(X1, X2)) | → | mark#(X1) |
| a__and#(true, X) | → | mark#(X) | | mark#(if(X1, X2, X3)) | → | mark#(X1) |
| a__if#(true, X, Y) | → | mark#(X) | | a__add#(0, X) | → | mark#(X) |
| mark#(add(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| a__and(true, X) | → | mark(X) | | a__and(false, Y) | → | false |
| a__if(true, X, Y) | → | mark(X) | | a__if(false, X, Y) | → | mark(Y) |
| a__add(0, X) | → | mark(X) | | a__add(s(X), Y) | → | s(add(X, Y)) |
| a__first(0, X) | → | nil | | a__first(s(X), cons(Y, Z)) | → | cons(Y, first(X, Z)) |
| a__from(X) | → | cons(X, from(s(X))) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(if(X1, X2, X3)) | → | a__if(mark(X1), X2, X3) | | mark(add(X1, X2)) | → | a__add(mark(X1), X2) |
| mark(first(X1, X2)) | → | a__first(mark(X1), mark(X2)) | | mark(from(X)) | → | a__from(X) |
| mark(true) | → | true | | mark(false) | → | false |
| mark(0) | → | 0 | | mark(s(X)) | → | s(X) |
| mark(nil) | → | nil | | mark(cons(X1, X2)) | → | cons(X1, X2) |
| a__and(X1, X2) | → | and(X1, X2) | | a__if(X1, X2, X3) | → | if(X1, X2, X3) |
| a__add(X1, X2) | → | add(X1, X2) | | a__first(X1, X2) | → | first(X1, X2) |
| a__from(X) | → | from(X) |
Original Signature
Termination of terms over the following signature is verified: true, mark, from, a__add, add, and, a__and, 0, a__if, s, if, false, a__first, first, a__from, nil, cons
Strategy
The following SCCs where found
| mark#(and(X1, X2)) → mark#(X1) | mark#(first(X1, X2)) → mark#(X2) |
| mark#(add(X1, X2)) → a__add#(mark(X1), X2) | mark#(and(X1, X2)) → a__and#(mark(X1), X2) |
| mark#(first(X1, X2)) → mark#(X1) | mark#(if(X1, X2, X3)) → mark#(X1) |
| a__and#(true, X) → mark#(X) | mark#(add(X1, X2)) → mark#(X1) |
| a__add#(0, X) → mark#(X) |
Problem 4: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(and(X1, X2)) | → | mark#(X1) | | mark#(first(X1, X2)) | → | mark#(X2) |
| mark#(add(X1, X2)) | → | a__add#(mark(X1), X2) | | mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) |
| mark#(first(X1, X2)) | → | mark#(X1) | | mark#(if(X1, X2, X3)) | → | mark#(X1) |
| a__and#(true, X) | → | mark#(X) | | mark#(add(X1, X2)) | → | mark#(X1) |
| a__add#(0, X) | → | mark#(X) |
Rewrite Rules
| a__and(true, X) | → | mark(X) | | a__and(false, Y) | → | false |
| a__if(true, X, Y) | → | mark(X) | | a__if(false, X, Y) | → | mark(Y) |
| a__add(0, X) | → | mark(X) | | a__add(s(X), Y) | → | s(add(X, Y)) |
| a__first(0, X) | → | nil | | a__first(s(X), cons(Y, Z)) | → | cons(Y, first(X, Z)) |
| a__from(X) | → | cons(X, from(s(X))) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(if(X1, X2, X3)) | → | a__if(mark(X1), X2, X3) | | mark(add(X1, X2)) | → | a__add(mark(X1), X2) |
| mark(first(X1, X2)) | → | a__first(mark(X1), mark(X2)) | | mark(from(X)) | → | a__from(X) |
| mark(true) | → | true | | mark(false) | → | false |
| mark(0) | → | 0 | | mark(s(X)) | → | s(X) |
| mark(nil) | → | nil | | mark(cons(X1, X2)) | → | cons(X1, X2) |
| a__and(X1, X2) | → | and(X1, X2) | | a__if(X1, X2, X3) | → | if(X1, X2, X3) |
| a__add(X1, X2) | → | add(X1, X2) | | a__first(X1, X2) | → | first(X1, X2) |
| a__from(X) | → | from(X) |
Original Signature
Termination of terms over the following signature is verified: true, mark, from, a__add, add, and, a__and, 0, a__if, s, if, false, a__first, first, a__from, nil, cons
Strategy
Polynomial Interpretation
- 0: 0
- a__add(x,y): 3y + 3x
- a__add#(x,y): y
- a__and(x,y): 3x
- a__and#(x,y): 2y
- a__first(x,y): 3y + x
- a__from(x): 3x + 1
- a__if(x,y,z): z + 3y + 3x
- add(x,y): y + x
- and(x,y): 2y + x
- cons(x,y): 3x
- false: 0
- first(x,y): y + 2x + 2
- from(x): 0
- if(x,y,z): 2z + 2x
- mark(x): 0
- mark#(x): x
- nil: 3
- s(x): 0
- true: 0
Improved Usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(first(X1, X2)) | → | mark#(X2) | | mark#(first(X1, X2)) | → | mark#(X1) |
Problem 5: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(and(X1, X2)) | → | mark#(X1) | | mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) |
| mark#(add(X1, X2)) | → | a__add#(mark(X1), X2) | | a__and#(true, X) | → | mark#(X) |
| mark#(if(X1, X2, X3)) | → | mark#(X1) | | a__add#(0, X) | → | mark#(X) |
| mark#(add(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| a__and(true, X) | → | mark(X) | | a__and(false, Y) | → | false |
| a__if(true, X, Y) | → | mark(X) | | a__if(false, X, Y) | → | mark(Y) |
| a__add(0, X) | → | mark(X) | | a__add(s(X), Y) | → | s(add(X, Y)) |
| a__first(0, X) | → | nil | | a__first(s(X), cons(Y, Z)) | → | cons(Y, first(X, Z)) |
| a__from(X) | → | cons(X, from(s(X))) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(if(X1, X2, X3)) | → | a__if(mark(X1), X2, X3) | | mark(add(X1, X2)) | → | a__add(mark(X1), X2) |
| mark(first(X1, X2)) | → | a__first(mark(X1), mark(X2)) | | mark(from(X)) | → | a__from(X) |
| mark(true) | → | true | | mark(false) | → | false |
| mark(0) | → | 0 | | mark(s(X)) | → | s(X) |
| mark(nil) | → | nil | | mark(cons(X1, X2)) | → | cons(X1, X2) |
| a__and(X1, X2) | → | and(X1, X2) | | a__if(X1, X2, X3) | → | if(X1, X2, X3) |
| a__add(X1, X2) | → | add(X1, X2) | | a__first(X1, X2) | → | first(X1, X2) |
| a__from(X) | → | from(X) |
Original Signature
Termination of terms over the following signature is verified: true, mark, from, a__add, add, and, a__and, 0, a__if, s, if, false, first, a__first, a__from, cons, nil
Strategy
Polynomial Interpretation
- 0: 0
- a__add(x,y): 2y + 2
- a__add#(x,y): 2y
- a__and(x,y): 1
- a__and#(x,y): y
- a__first(x,y): 2y
- a__from(x): 3x + 3
- a__if(x,y,z): 3z + y + 3
- add(x,y): 2y + x
- and(x,y): 2y + 2x
- cons(x,y): 1
- false: 2
- first(x,y): 2y + 2x + 3
- from(x): 0
- if(x,y,z): 2y + x + 2
- mark(x): 0
- mark#(x): x
- nil: 2
- s(x): 3
- true: 0
Improved Usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(if(X1, X2, X3)) | → | mark#(X1) |
Problem 6: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(and(X1, X2)) | → | mark#(X1) | | mark#(add(X1, X2)) | → | a__add#(mark(X1), X2) |
| mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) | | a__and#(true, X) | → | mark#(X) |
| mark#(add(X1, X2)) | → | mark#(X1) | | a__add#(0, X) | → | mark#(X) |
Rewrite Rules
| a__and(true, X) | → | mark(X) | | a__and(false, Y) | → | false |
| a__if(true, X, Y) | → | mark(X) | | a__if(false, X, Y) | → | mark(Y) |
| a__add(0, X) | → | mark(X) | | a__add(s(X), Y) | → | s(add(X, Y)) |
| a__first(0, X) | → | nil | | a__first(s(X), cons(Y, Z)) | → | cons(Y, first(X, Z)) |
| a__from(X) | → | cons(X, from(s(X))) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(if(X1, X2, X3)) | → | a__if(mark(X1), X2, X3) | | mark(add(X1, X2)) | → | a__add(mark(X1), X2) |
| mark(first(X1, X2)) | → | a__first(mark(X1), mark(X2)) | | mark(from(X)) | → | a__from(X) |
| mark(true) | → | true | | mark(false) | → | false |
| mark(0) | → | 0 | | mark(s(X)) | → | s(X) |
| mark(nil) | → | nil | | mark(cons(X1, X2)) | → | cons(X1, X2) |
| a__and(X1, X2) | → | and(X1, X2) | | a__if(X1, X2, X3) | → | if(X1, X2, X3) |
| a__add(X1, X2) | → | add(X1, X2) | | a__first(X1, X2) | → | first(X1, X2) |
| a__from(X) | → | from(X) |
Original Signature
Termination of terms over the following signature is verified: true, mark, from, a__add, add, and, a__and, 0, a__if, s, if, false, a__first, first, a__from, nil, cons
Strategy
Polynomial Interpretation
- 0: 1
- a__add(x,y): 2y + 2x
- a__add#(x,y): 2y
- a__and(x,y): 2y + 2x
- a__and#(x,y): 2y + 2x
- a__first(x,y): 0
- a__from(x): 2x + 1
- a__if(x,y,z): 2z + 2y
- add(x,y): y + 2x
- and(x,y): 2y + 2x
- cons(x,y): 0
- false: 0
- first(x,y): 0
- from(x): 2x + 1
- if(x,y,z): z + 2y
- mark(x): 2x
- mark#(x): 2x
- nil: 0
- s(x): 0
- true: 1
Improved Usable rules
| a__first(X1, X2) | → | first(X1, X2) | | a__from(X) | → | cons(X, from(s(X))) |
| a__from(X) | → | from(X) | | a__and(true, X) | → | mark(X) |
| a__if(X1, X2, X3) | → | if(X1, X2, X3) | | a__first(0, X) | → | nil |
| a__add(X1, X2) | → | add(X1, X2) | | a__first(s(X), cons(Y, Z)) | → | cons(Y, first(X, Z)) |
| mark(nil) | → | nil | | a__add(0, X) | → | mark(X) |
| a__and(false, Y) | → | false | | mark(s(X)) | → | s(X) |
| mark(0) | → | 0 | | mark(cons(X1, X2)) | → | cons(X1, X2) |
| a__and(X1, X2) | → | and(X1, X2) | | a__if(false, X, Y) | → | mark(Y) |
| mark(false) | → | false | | mark(first(X1, X2)) | → | a__first(mark(X1), mark(X2)) |
| mark(if(X1, X2, X3)) | → | a__if(mark(X1), X2, X3) | | mark(true) | → | true |
| mark(from(X)) | → | a__from(X) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| a__add(s(X), Y) | → | s(add(X, Y)) | | a__if(true, X, Y) | → | mark(X) |
| mark(add(X1, X2)) | → | a__add(mark(X1), X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| a__and#(true, X) | → | mark#(X) |
Problem 7: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(and(X1, X2)) | → | mark#(X1) | | mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) |
| mark#(add(X1, X2)) | → | a__add#(mark(X1), X2) | | a__add#(0, X) | → | mark#(X) |
| mark#(add(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| a__and(true, X) | → | mark(X) | | a__and(false, Y) | → | false |
| a__if(true, X, Y) | → | mark(X) | | a__if(false, X, Y) | → | mark(Y) |
| a__add(0, X) | → | mark(X) | | a__add(s(X), Y) | → | s(add(X, Y)) |
| a__first(0, X) | → | nil | | a__first(s(X), cons(Y, Z)) | → | cons(Y, first(X, Z)) |
| a__from(X) | → | cons(X, from(s(X))) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(if(X1, X2, X3)) | → | a__if(mark(X1), X2, X3) | | mark(add(X1, X2)) | → | a__add(mark(X1), X2) |
| mark(first(X1, X2)) | → | a__first(mark(X1), mark(X2)) | | mark(from(X)) | → | a__from(X) |
| mark(true) | → | true | | mark(false) | → | false |
| mark(0) | → | 0 | | mark(s(X)) | → | s(X) |
| mark(nil) | → | nil | | mark(cons(X1, X2)) | → | cons(X1, X2) |
| a__and(X1, X2) | → | and(X1, X2) | | a__if(X1, X2, X3) | → | if(X1, X2, X3) |
| a__add(X1, X2) | → | add(X1, X2) | | a__first(X1, X2) | → | first(X1, X2) |
| a__from(X) | → | from(X) |
Original Signature
Termination of terms over the following signature is verified: true, mark, from, a__add, add, and, a__and, 0, a__if, s, if, false, first, a__first, a__from, cons, nil
Strategy
The following SCCs where found
| mark#(and(X1, X2)) → mark#(X1) | mark#(add(X1, X2)) → a__add#(mark(X1), X2) |
| a__add#(0, X) → mark#(X) | mark#(add(X1, X2)) → mark#(X1) |
Problem 8: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(and(X1, X2)) | → | mark#(X1) | | mark#(add(X1, X2)) | → | a__add#(mark(X1), X2) |
| a__add#(0, X) | → | mark#(X) | | mark#(add(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| a__and(true, X) | → | mark(X) | | a__and(false, Y) | → | false |
| a__if(true, X, Y) | → | mark(X) | | a__if(false, X, Y) | → | mark(Y) |
| a__add(0, X) | → | mark(X) | | a__add(s(X), Y) | → | s(add(X, Y)) |
| a__first(0, X) | → | nil | | a__first(s(X), cons(Y, Z)) | → | cons(Y, first(X, Z)) |
| a__from(X) | → | cons(X, from(s(X))) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(if(X1, X2, X3)) | → | a__if(mark(X1), X2, X3) | | mark(add(X1, X2)) | → | a__add(mark(X1), X2) |
| mark(first(X1, X2)) | → | a__first(mark(X1), mark(X2)) | | mark(from(X)) | → | a__from(X) |
| mark(true) | → | true | | mark(false) | → | false |
| mark(0) | → | 0 | | mark(s(X)) | → | s(X) |
| mark(nil) | → | nil | | mark(cons(X1, X2)) | → | cons(X1, X2) |
| a__and(X1, X2) | → | and(X1, X2) | | a__if(X1, X2, X3) | → | if(X1, X2, X3) |
| a__add(X1, X2) | → | add(X1, X2) | | a__first(X1, X2) | → | first(X1, X2) |
| a__from(X) | → | from(X) |
Original Signature
Termination of terms over the following signature is verified: true, mark, from, a__add, add, and, a__and, 0, a__if, s, if, false, first, a__first, a__from, cons, nil
Strategy
Polynomial Interpretation
- 0: 1
- a__add(x,y): 0
- a__add#(x,y): y
- a__and(x,y): x + 1
- a__first(x,y): y
- a__from(x): 2x + 3
- a__if(x,y,z): 3z + y
- add(x,y): 2y + x + 2
- and(x,y): x + 2
- cons(x,y): y + x + 2
- false: 1
- first(x,y): 0
- from(x): x
- if(x,y,z): x + 2
- mark(x): x + 1
- mark#(x): x
- nil: 1
- s(x): 0
- true: 0
Improved Usable rules
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(and(X1, X2)) | → | mark#(X1) | | mark#(add(X1, X2)) | → | a__add#(mark(X1), X2) |
| mark#(add(X1, X2)) | → | mark#(X1) |
Problem 9: DependencyGraph
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| a__and(true, X) | → | mark(X) | | a__and(false, Y) | → | false |
| a__if(true, X, Y) | → | mark(X) | | a__if(false, X, Y) | → | mark(Y) |
| a__add(0, X) | → | mark(X) | | a__add(s(X), Y) | → | s(add(X, Y)) |
| a__first(0, X) | → | nil | | a__first(s(X), cons(Y, Z)) | → | cons(Y, first(X, Z)) |
| a__from(X) | → | cons(X, from(s(X))) | | mark(and(X1, X2)) | → | a__and(mark(X1), X2) |
| mark(if(X1, X2, X3)) | → | a__if(mark(X1), X2, X3) | | mark(add(X1, X2)) | → | a__add(mark(X1), X2) |
| mark(first(X1, X2)) | → | a__first(mark(X1), mark(X2)) | | mark(from(X)) | → | a__from(X) |
| mark(true) | → | true | | mark(false) | → | false |
| mark(0) | → | 0 | | mark(s(X)) | → | s(X) |
| mark(nil) | → | nil | | mark(cons(X1, X2)) | → | cons(X1, X2) |
| a__and(X1, X2) | → | and(X1, X2) | | a__if(X1, X2, X3) | → | if(X1, X2, X3) |
| a__add(X1, X2) | → | add(X1, X2) | | a__first(X1, X2) | → | first(X1, X2) |
| a__from(X) | → | from(X) |
Original Signature
Termination of terms over the following signature is verified: true, mark, from, a__add, add, and, a__and, 0, a__if, s, if, false, a__first, first, a__from, nil, cons
Strategy
There are no SCCs!